Progress in Optics, Volume 48

Preface The present volume of Progress in Optics contains five review articles on various topics of current interest to...

Author: WOLF E.

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Preface The present volume of Progress in Optics contains five review articles on various topics of current interest to optical scientists and optical engineers. The first article by B.J. Eggleton, P. Domachuk, C. Grillet, E.C. Magi, H.C. Nguyen, M.J. Steel and P. Steinvurzel discusses recent developments in the field of microstructured optical fibers. It provides an account of some of their interesting properties such as new types of nonlinearities and dispersion profiles and discusses how such fibers may be used to produce novel devices by postfabrication engineering techniques. The second article by F.Kh. Abdullaev and J. Garnier concerns solitons in random media. It discusses soliton dynamics in optical fibers with randomly varying dispersion, amplifications and birefringence, as well as the propagation of dispersion-managed solitons in fiber links with random dispersion. The influence of fluctuations on transverse variables is also discussed, for spatial solitons in planar wavefields and 2D optical bullets. The article which follows, by N. Bokor and N. Davidson, presents a review of research on diffractive optical elements that are fabricated on curved surfaces. For the design of such elements, the substrate shape as well as the grating function may be used as free parameters, leading to a more flexible design than is possible for flat diffractive optical elements. Curved diffractive optical elements have become of special interest in recent years because of the development of sophisticated fabrication techniques. They have a wide range of applications, for example in imaging, for concentration of diffuse light on targets of arbitrary shape and in connection with focusing by systems of high numerical apertures. The fourth article, by P. Hariharan, reviews researches on the geometrical phase. The origin of this subject is related to the fact that the wave function of a quantum system can exhibit a phase shift when parameter of the system undergoes a cyclic change. Two manifestations of the phase shift, known generally as the geometric phase shift, are commonly observed in optics. One is due to a cyclic change in the direction of propagation of a light beam – the spin-redirection phase. The other is due to a cyclic change in the state of polarization of a beam, called sometimes the Pancharatnam phase. The geometric phase persists at the single photon level, but the results obtained with fourth-order interference, involving v

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entangled photon pairs, are very different from what one might expect from classical theory. The understanding of such effects has opened up new possibilities in interference microscopy, and in stellar interferometry, for example. Some of these developments are discussed in this article. The last article written by A. Uchida, F. Rogister, J. García-Ojalvo and R. Roy reviews the progress in the field of synchronization and communication with chaotic laser systems. In the first part of the article the origins of chaotic dynamics in lasers is explained and are illustrated with accounts of experimental observations and with numerical computations on suitable models. It is then shown how synchronization of chaotic laser systems leads to the possibility of communicating information in both digital and in analog form. The article also presents an account of the latest advances in the field which employ semiconductor and fiber lasers, as well as gas and solid state laser systems. The possibility is also discussed of developing secure communication systems, using chaos and nonlinear dynamics. It is with deep sadness that I conclude this Preface by paying tribute to the memory of Richard M. Sillitto who passed away a few months ago. He was a member of the Editorial Advisory Board of Progress in Optics for ten years and was a very dear friend for more than fifty years. He will be greatly missed. Emil Wolf Department of Physics and Astronomy and The Institute of Optics, University of Rochester, Rochester, NY 14627, USA August 2005

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1. Laboratory post-engineering of microstructured optical fibers, B.J. Eggleton, P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel (Sydney, Australia) and M.J. Steel (Chippendale, Australia)

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§ 1. Introduction . . . . . . . . . . . . . . . . . . § 2. Microstructured optical fibers: a brief review 2.1. Photonic crystal fibers . . . . . . . . . 2.2. Other MOFs . . . . . . . . . . . . . . § 3. Transverse MOFs . . . . . . . . . . . . . . . 3.1. Modeling . . . . . . . . . . . . . . . . 3.2. Experimental results . . . . . . . . . . § 4. Tapered MOFs . . . . . . . . . . . . . . . . . 4.1. Tapering principles . . . . . . . . . . . 4.2. Low air-fill fraction PCF . . . . . . . . 4.3. High air-fill fraction PCF . . . . . . . 4.4. Silica nanowires . . . . . . . . . . . . § 5. Microfluidic MOFs . . . . . . . . . . . . . . 5.1. Static tuning . . . . . . . . . . . . . . 5.2. Dynamic tuning . . . . . . . . . . . . § 6. Microfluidic interferometer . . . . . . . . . . § 7. Conclusion . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . .

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Chapter 2. Optical solitons in random media, Fatkhulla Abdullaev (Tashkent, Uzbekistan) and Josselin Garnier (Toulouse, France) . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . § 2. The main equations . . . . . . . . . . . . . . . . . . . . 2.1. Derivation of the nonlinear Schrödinger equation 2.2. Derivation of χ (2) system . . . . . . . . . . . . . § 3. Solitons in random single-mode fibers . . . . . . . . . . 3.1. Different approaches . . . . . . . . . . . . . . . . 3.2. Single soliton driven by random perturbations . . 3.3. Interaction of solitons in random medium . . . . 3.4. Beyond the white-noise model . . . . . . . . . . 3.5. Femtosecond solitons in random fibers . . . . . . vii

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§ 4. Dispersion-managed solitons under random perturbations . . . . . . . . . . . . . . 4.1. Periodic and random dispersion modulations . . . . . . . . . . . . . . . . . . 4.2. Different approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Random dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Pinning schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5. Randomly birefringent fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Derivation of the perturbed Manakov system . . . . . . . . . . . . . . . . . . 5.2. Radiative damping of solitons . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Dispersion-managed solitons and PMD effects . . . . . . . . . . . . . . . . . § 6. Solitons in random quadratic media . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1. Mean field method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Nonlinear damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. Linear damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 7. Spatial solitons in random waveguides . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Spatial solitons in planar waveguides with random parameters . . . . . . . . 7.2. Spatial solitons under random dispersive perturbations . . . . . . . . . . . . 7.3. Pulse propagation in nonlinear waveguides with fluctuating refraction index § 8. Two-dimensional solitons in random media . . . . . . . . . . . . . . . . . . . . . . § 9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: The inverse scattering transform for the nonlinear Schrödinger equation . . A.1. The unperturbed nonlinear Schrödinger equation . . . . . . . . . . . . . . . A.2. Perturbation theory for the NLSE . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: The inverse scattering transform for the Manakov system . . . . . . . . . . B.1. The unperturbed Manakov system . . . . . . . . . . . . . . . . . . . . . . . B.2. Perturbation theory for the Manakov system . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 3. Curved diffractive optical elements: Design and applications, Nándor Bokor (Budapest, Hungary) and Nir Davidson (Rehovot, Israel) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. Spherical/cylindrical CDOEs for imaging and Fourier transform . . . . . . . . . . . . . 2.1. Spherical CDOE for imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Spherical CDOE for Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . § 3. Spherical/cylindrical CDOEs for concentration of diffuse light on flat targets . . . . . . § 4. Uniform collimation and ideal concentration for arbitrary source and target shapes . . . 4.1. Uniform 1D collimation and concentration of diffuse light at the thermodynamic limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Extension to finite distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 5. CDOEs for controlling the geometrical apodization factor . . . . . . . . . . . . . . . . . § 6. Curved gratings for spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 7. CDOEs for optical coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . § 8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 4. The geometric phase, P. Hariharan (Sydney, Australia) . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . § 2. The geometric phase in optics . . . . . . . . . . . . . . . . . 2.1. The spin-redirection phase . . . . . . . . . . . . . . . . 2.2. The Pancharatnam phase . . . . . . . . . . . . . . . . . 2.3. The Pancharatnam phase in optical rotation . . . . . . 2.4. The Pancharatnam phase with subwavelength gratings 2.5. Features of the Pancharatnam phase . . . . . . . . . . 2.6. Combined effects . . . . . . . . . . . . . . . . . . . . . § 3. The geometric phase with single photons . . . . . . . . . . . 3.1. Observations at the single-photon level . . . . . . . . . 3.2. Observations with single-photon states . . . . . . . . . § 4. The geometric phase with photon pairs . . . . . . . . . . . . § 5. The Pancharatnam phase as a geometric phase . . . . . . . . § 6. The Pancharatnam phase with white light . . . . . . . . . . . § 7. Achromatic phase shifters . . . . . . . . . . . . . . . . . . . § 8. Switchable achromatic phase shifters . . . . . . . . . . . . . § 9. Polarization interferometers . . . . . . . . . . . . . . . . . . 9.1. Two-wavelength interferometry . . . . . . . . . . . . . 9.2. Switchable achromatic phase shifters . . . . . . . . . . § 10. White-light phase-shifting interferometry . . . . . . . . . . . 10.1. Coherence-probe microscopy . . . . . . . . . . . . . . 10.2. Spectrally resolved interferometry . . . . . . . . . . . § 11. Stellar interferometry . . . . . . . . . . . . . . . . . . . . . . § 12. Nulling interferometry . . . . . . . . . . . . . . . . . . . . . 12.1. Nulling using the spin-redirection phase . . . . . . . . 12.2. Nulling using the Pancharatnam phase . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 5. Synchronization and communication with chaotic laser systems, Atsushi Uchida (Tokyo, Japan), Fabien Rogister (Mons, Belgium), Jordi García-Ojalvo (Terrassa, Spain) and Rajarshi Roy (College Park, MD, USA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . 1.1. Laser fluctuations and dynamics . . . . . . . . 1.2. Chaotic dynamics . . . . . . . . . . . . . . . 1.3. Laser chaos: the Haken–Lorenz equations . . 1.4. Characterization of chaos . . . . . . . . . . . 1.5. Synchronization of chaos . . . . . . . . . . . 1.6. Private communications . . . . . . . . . . . . 1.7. Privacy in chaotic communications . . . . . . 1.8. Communication with chaotic lasers . . . . . . § 2. Synchronization of chaotic lasers . . . . . . . . . . 2.1. Why should chaotic systems synchronize? . . 2.2. Coupling schemes and synchronization types 2.3. Synchronization in feedback systems . . . . .

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2.4. One-way versus mutual coupling . . . . . . . . 2.5. Examples of chaotic laser synchronization . . . 2.6. Phase and generalized synchronization . . . . . § 3. Communication with chaotic lasers . . . . . . . . . . 3.1. Introduction of chaotic communications . . . . 3.2. Encoding and decoding techniques . . . . . . . 3.3. Examples of chaotic communication systems . 3.4. Decoding quality in chaos modulation . . . . . 3.5. Communications with electronic circuits . . . . 3.6. Examples of communication with chaotic lasers 3.7. Spatiotemporal communication . . . . . . . . . 3.8. Polarization encoding . . . . . . . . . . . . . . 3.9. Multiplexing . . . . . . . . . . . . . . . . . . . § 4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Motivations and achievements . . . . . . . . . . 4.2. Short-term goals . . . . . . . . . . . . . . . . . 4.3. Longer-term perspective and open questions . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Author index for Volume 48 . . . . Subject index for Volume 48 . . . . Contents of previous volumes . . . Cumulative index – Volumes 1–48

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E. Wolf, Progress in Optics 48 © 2005 Elsevier B.V. All rights reserved

Chapter 1

Laboratory post-engineering of microstructured optical fibers by

B.J. Eggleton, P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel ARC Centre of Excellence for Ultrahigh-bandwidth Devices for Optical Systems, School of Physics, University of Sydney, NSW 2006, Australia e-mail: [email protected] url: http://www.physics.usyd.edu.au/cudos

M.J. Steel RSoft Design Group, 65 O’Connor St, Chippendale, NSW 2008, Australia and University of Sydney, NSW 2006, Australia

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)48001-4 1

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 2. Microstructured optical fibers: a brief review . . . . . . . . . . . . .

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§ 3. Transverse MOFs . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 4. Tapered MOFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 5. Microfluidic MOFs . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 6. Microfluidic interferometer . . . . . . . . . . . . . . . . . . . . . . .

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§ 7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2

§ 1. Introduction After maturation in long-haul telecommunications (Nagel, Macchesney and Walker [1982], Croft, Ritter and Bhagavatula [1985] and Ramaswami [1993]), fiber-optics technology is enjoying a renaissance in the form of microstructured optical fibers (MOFs). These fibers, unlike conventional single-mode fibers, have air inclusions running along their length. These air inclusions modify the transmission properties of the fiber, providing a high degree of control over the propagation of light in the fiber and leading to numerous applications (Knight, Birks, Russell and Atkin [1996], Broeng, Mogilevstev, Barkou and Bjarklev [1999], Eggleton, Kerbage, Westbrook, Windeler and Hale [2001], Monro, Belardi, Furusawa, Baggett, Broderick and Richardson [2001], Larsen, Bjarklev, Hermann and Broeng [2003], Windeler, Wagener and DiGiovanni [1999] and Steel, White, deSterke, McPhedran and Botten [2001]). Whilst MOFs have many interesting properties in their own right, such as exotic nonlinearities and dispersion profiles, a variety of novel devices may be realized by post-fabrication engineering these fibers. Through modification of the MOF itself, via tapering (Birks and Li [1992]), the introduction of fluids into the microstructure of the fiber (microfluidics Nguyen and Wereley [2002]) or using the MOFs in novel geometries (Nguyen, Domachuk, Eggleton, Steel, Straub, Gu and Sumetsky [2004]), devices from photonic crystal switches (Domachuk, Nguyen and Eggleton [2004]) to ultracompact interferometers (Grillet, Domachuk, Ta’eed, Mägi, Bolger, Eggleton, Rodd and Cooper-White [2004]) may be fabricated using these fibers. In combination, these technologies allow for almost limitless scope in device design. In this chapter we demonstrate the postengineering of MOFs and also demonstrate devices fabricated using these postengineering techniques. This chapter is structured as follows. We begin in Section 2 by providing a brief history of MOFs and a review of various types of MOFs, particularly PCFs, their guiding mechanisms and uses. In Section 3 we present a novel transverse interrogation technique for MOFs, with experimental results from transversely probed PCFs, in comparison with numerical simulations of the geometry. In Section 4 we discuss tapering as applied to MOFs and PCFs, both in the transversely 3

4

Laboratory post-engineering of microstructured optical fibers

[1, § 2

probed and traditional longitudinal regimes and, specifically, the creation of silica nanowires (Tong, Gattass, Ashcon, He, Lou, Shen, Maxwell and Mazur [2003]). In Section 5 we examine microfluidic tuning of MOFs and PCFs, in a variety of device geometries. In Section 6 we present the microfluidic interferometer, which uses the post-engineering methods outlined earlier to create a novel MOF-based device of enhanced functionality. We conclude with Section 7.

§ 2. Microstructured optical fibers: a brief review As noted above, optical fibers have become the standard medium for long-haul telecommunications. However, while the single-mode fiber was being developed in the 1970s (Nagel, Macchesney and Walker [1982]), several other designs were being considered. These fibers relied on a microstructure to aid in light confinement, earning the acronym Microstructured Optical Fibers (MOFs). Perhaps the two most notable were the silica–air fiber (an example is shown in fig. 1) (Kaiser and Astle [1974]), where a core of silica is suspended in air by thin silica filaments, and the Bragg fiber (Yeh, Yariv and Marom [1978]), where the light is confined by a concentric series of refractive-index modulations. Single-mode fiber came to dominance due to its relative ease of fabrication (through the advent of MCVD (Croft, Ritter and Bhagavatula [1985]) and robustness, something which these other designs lacked at the time. However, as fiber fabrication technology has advanced, a renewed interest in MOFs has emerged.

Fig. 1. An example of an early microstructured optical fiber.

1, § 2]

Microstructured optical fibers: a brief review

5

2.1. Photonic crystal fibers A specific type of MOF that has enjoyed much study of late for its interesting properties is the photonic crystal fiber (PCF) (Russell [2003]). These fibers have a matrix of periodically spaced air inclusions running along their length. Guidance may be achieved either by total internal reflection (Knight, Birks, Russell and Atkin [1996]), coherent Bragg scattering from the matrix of air inclusions (Cregan, Mangan, Knight, Birks, Russell, Roberts and Allan [1999]) or through ARROW guidance (Litchinitser, Abeeluck, Headley and Eggleton [2002]). These fibers may be fabricated through a number of methods, an example of which is the stack and draw technique: a stack of hollow silica capillaries are bundled and heated until the silica becomes plastic. It is then drawn, reducing the dimensions of the stack by up to 50,000 times. Figure 2 shows a schematic of this technique, pioneered at the University of Bath (Knight, Birks, Russell and Atkin [1996]). As PCF fabrication has become more sophisticated, more complex structures have been demonstrated. Figure 3 shows a PCF with a hollow core that was fabricated in 1999. Guidance in this hollow core is achieved through coherent reflection (Knight, Birks, Russell and Atkin [1996]). Such fibers have a very low nonlinearity, allowing for propagation of intense pulses, an issue in single-mode fiber communications (Gobel, Nimmerjahn and Helmchen [2004]). In comparison to single-mode fiber, however, PCFs, in general, have high transmission loss. Indeed, one of the challenges of PCF design is to minimize this loss. More recently, PCFs with a high air-fill fraction photonic crystal microstructure was demonstrated, exhibiting significantly reduced loss over previous designs (Benabid, Knight, Antonopoulos and Russell [2002] and Ouzounov, Ahmad, Müller, Venkataraman, Gallagher, Thomas, Silcox, Koch and Gaeta [2003]), also

Fig. 2. A schematic representation of the stack and draw technique. At left, a stack of hollow glass capillaries is made. At right, the stack is heated and drawn down to the required dimensions.

6


[1, § 2

Fig. 3. Examples of more recent MOFs. Clockwise, from top left: The first hollow-core PCF using the photonic band gap effect to guide a mode in air. The Corning group low-loss photonic crystal fiber. The hollow, air guiding, OmniGuide fiber; details of the layered microstructure shown inset. A polarization-maintaining PANDA fiber.

shown in fig. 3. The transmission properties of PCFs are significantly enhanced over single-mode fiber as well. Adjusting the geometry of the air inclusions at the design stage allows control over the dispersion characteristics of these fibers. Further, the PCF can be designed so that it retains single-mode propagation over the band of its functionality (Birks, Knight and Russell [1997]).

2.2. Other MOFs MOFs are not limited to PCFs, though PCFs represent a large subset of total MOFs. Other designs include the hollow-core Bragg fiber (Temelkuran, Hart, Benoit, Joannopoulos and Fink [2002]). Similar to the Bragg fiber mentioned

1, § 3]

Transverse MOFs

7

above, the hollow-core variant guides a mode in an air core though resonant reflection from concentric refractive-index modulations in the cladding. These fibers are used for high power delivery in surgical or machining applications (Konorov, Mitrokhin, Fedotov, Sidorov-Biryukov, Beloglazov, Skibina, Shcherbakov, Wintner, Scalora and Zheltikov [2003]), where the power carried in the field would damage a solid-core fiber. Figure 3 shows an example of the hollow-core Bragg fiber. MOF design may also use air inclusions to modify light propagation in a solid core. One such fiber is polarization-maintaining fiber, which uses point asymmetry in its microstructure to favor propagation in one polarization, thus maintaining polarization (Noda, Okamoto and Sasaki [1986]). An example is shown in fig. 3. Another MOF is the hollow-core optical fiber. This fiber has a hollow core surrounded by a ring of doped silica where the light is guided (Choi, Oh, Shin and Ryu [2001]). This design of fiber has seen many applications, from sensing (Saggese, Harrington and Sigel [1991]) to mode conversion (Choi and Oh [2003]). From the examples above, the great utility of MOFs by themselves is evident. Many designs exist to cater to a variety of application and geometries. It is because of this variety that MOFs are ideal candidates for laboratory post-processing, allowing their already broad scope for application to be further extended into an almost limitless array of geometries and devices.

§ 3. Transverse MOFs One of the simplest, yet most versatile, ways to post process MOFs is to use them in novel geometries. Traditionally, optical fibers are used for transmission, confining light in the transverse direction for propagation in the longitudinal direction. However, it is possible to probe the optical fiber transversely (Nguyen, Domachuk, Eggleton, Steel, Straub, Gu and Sumetsky [2004], Domachuk, Nguyen and Eggleton [2004]) and have the incident light interacting with the microstructure in the core of the fiber. Whilst oblique side probing has been used for characterization of fiber microstructure (Knight, Birks, Russell and Rarity [1998]), it is possible to realize a quasi-two-dimensional photonic crystal by simply transversely probing a PCF. This methodology has several advantages over traditional planar fabrication in the context of microphotonic devices. Firstly, surface roughness is an issue in planar devices and is the major source of loss in these geometries (Ladoucer and Love [1996]). In the transverse MOF, however, the inside surfaces of the fiber are drawn to nearly atomic smoothness by the very high surface tension of plastic

8


[1, § 3

Fig. 4. Schematic representation of the creation of transverse fiber devices through the fiber drawing and cleaving process.

silica during the draw process. A second advantage lies in the high longitudinal regularity of drawn fibers. Because of this, many identical units may be made from a single length of fiber using standard fiber cleaving techniques. Figure 4 illustrates this. Third, one may take advantage of the longitudinal dimension. This allows light to be guided perpendicular to the semiplanar region. Also, this third dimension may be used to introduce fluids into the optical beam path. This form of fluid tuning is discussed at greater length below. Transverse probing may also be used as a diagnostic tool for determining the longitudinal regularity in a fiber during the draw process or during tapering. This is achieved by observing the strength of the resonant reflection and the wavelength at which it occurs. This allows the dimension and the relative structure of the photonic crystal to be inferred. We demonstrate the transverse MOF by realizing a quasi-two-dimensional photonic crystal though the transverse probing of a PCF. Figure 5 illustrates the device geometry used in this experiment. The object of study is the center of the PCF, consisting of a photonic crystal made of an array of air holes embedded in

1, § 3]

Transverse MOFs

9

Fig. 5. Photograph of the transverse PCF experiment. The PCF is placed between two SMFs, visible in the photograph. One SMF delivers light from a broadband source and the other collects this light for analysis, after passing through the photonic crystal material. Shown inset is an SEM of the fiber microstructure, with the symmetry points of the photonic crystal indicated.

a silica cladding. The fiber itself was fabricated by Crystal Fiber A/S. It has a hexagonally packed array of air holes that are 0.8 µm in diameter with a period of 1.4 µm, with the main symmetry directions in reciprocal space indicated in fig. 5. The central hole of the crystal lattice has been removed, creating a “defect” in the PBG material which, as shown by simulations, has no discernible effect in the transverse geometry. The regularity of the fiber in the usual propagation direction of the fiber ensures that the microstructured core may be considered essentially a two-dimensional photonic crystal. Note also the irregularity in the hole diameter farther away from the central guiding region; the effect of this irregularity will be discussed later. The crystal fiber is placed between two standard single-mode fibers (SMF) that are used to guide light into, and take light out of, the PCF. The light guided into the structure was from a white light source and the outgoing light was analyzed by an optical spectrum analyzer (OSA). The PCF was characterized only in the Γ –M direction due to the large-diameter air holes at the vertex of the hexagon, scattering from which detracts from the observation of resonant effects. Two polarizations are also defined, where TM has its electric field parallel to the length of the fiber and TE is perpendicular.

10


[1, § 3

3.1. Modeling Two numerical methods were used to investigate the properties of the transverse PCF: the plane-wave expansion method and the Finite Difference Time Domain (FDTD) technique. These simulations use a two-dimensional approximation of the experimental geometry. Both of these tools are used throughout this review to characterize transverse PCFs in various applications. 3.1.1. Plane-wave expansion The plane-wave method calculates dispersion relations of regular, infinite photonic crystals (Joannopoulos, Meade and Winn [1995]). The transverse PCF is neither of these; however, the band gaps of a regular, infinite approximation to the PCF microstructure provide a guide as to the location of the experimental band gaps. We use a simulation geometry that consists of a hexagonal lattice of circular air inclusions in silica. The dimensions of the idealized crystal are taken as the average over the inner, most regular rings of air inclusions of the PCF. Figure 6 shows the resulting dispersion relation, given in terms of polarizations defined earlier. In the Γ –M direction, a number of partial band gaps appear in both polarizations.

Fig. 6. The dispersion relation for the PCF, calculated by the plane-wave method. A number of partial band gaps appear for both polarizations.

1, § 3]

Transverse MOFs

11

3.1.2. FDTD From the plane-wave expansion calculation, the approximate wavelength and width of the expected resonant features are known. To more precisely simulate the experiment, the FDTD method is used (Tavlove [1995]). As the FDTD method solves the full Maxwell equations over space and time grids, no approximations are made using this method, other than the planar approximation of the experimental geometry. We digitize a scanning electron micrograph (SEM) of the PCF in fig. 5 and use this as the refractive-index profile for the simulation (Nguyen, Domachuk, Eggleton, Steel, Straub, Gu and Sumetsky [2004]). The PCF sits evenly between the two SMFs separated by the width of the fiber.

3.2. Experimental results Figure 7 shows spectra obtained using the transverse method described above, for both polarizations and along the Γ –M direction, in comparison with the FDTD

Fig. 7. The experimental (grey) and FDTD calculated (black) spectra for the transverse PCF. Also indicated are the positions of the band gaps predicted by the plane-wave method. The FDTD calculation provides a rough prediction of position and shape of the major resonant features.

12


[1, § 4

spectral results. Superimposed on each plot is the position and width of the resonant feature calculated using the plane-wave method. In both polarizations, the position and width of the gap at 1.6 µm is much more accurately represented using the FDTD method than with the plane-wave expansion method. This is due to the gradation of hole diameter in the PCF causing a “chirping” effect of the photonic crystal, thus broadening the band gap from that predicted using the idealized representation. The insertion loss of this system is relatively high, at 5 dB. This may be attributed to several factors: the divergence of the input beam, reflection and refraction off the various component interfaces and the diffraction of the beam by the photonic crystal itself. The transverse probing of fibers provides a convenient way to access the microstructure of MOFs. Using this method it is possible to create semiplanar photonic crystals by simply probing a PCF from the side. Indeed, a whole range of semiplanar structures could be accessed in this way by choosing the correct MOF.

§ 4. Tapered MOFs Tapering of optical fibers (Leon-Saval, Birks, Wadsworth, Russell and Mason [2004], Lizé, Mägi, Ta’eed, Bolger, Steinvurzel and Eggleton [2004] and Mägi, Steinvurzel and Eggleton [2003]) is a mature technology, developed during the 1980s, that has enjoyed a recent resurgence in the post processing of PCFs (LeonSaval, Birks, Wadsworth, Russell and Mason [2004]) and the fabrication of silica nanowires (Lizé, Mägi, Ta’eed, Bolger, Steinvurzel and Eggleton [2004]). We apply tapering to MOFs in order create a number of unique photonic devices. By tapering PCFs, we are able shift transmission band gaps to different wavelengths and by tapering MOFs we create silica nanowires.

4.1. Tapering principles Figure 8 is a schematic of the tapering apparatus, which is based on a standard flame-brushing technique (Mägi, Steinvurzel and Eggleton [2003]). The fiber is clamped in place and pulled on either side by motorized stages while a butane flame brushes back and forth along the fiber. The entire process is computercontrolled and the pull rate, total elongation, and flame brushing profile can all be varied to realize different sorts of tapers. When the fiber is heated during the tapering process, the viscosity of the silica decreases and the glass is able to flow. The dynamics of the material flow depend

1, § 4]

Tapered MOFs

13

Fig. 8. A schematic of the fiber tapering process. Tension is applied to both ends of the fiber to be tapered while a flame is moved along the fiber (“brushed”) to create an effective hot zone, resulting in the representative taper shape shown at the bottom of the figure.

on the local heat distribution, the surface tension of the silica, the gas pressure within the air voids, and the elongation rate of the taper draw. For a taper that preserves the relative dimension of the microstructure, most of the material flow is along the fiber axis, and this is primarily governed by the elongation rate and the flame temperature. When the elongation rate and gas pressure within the voids are too low for a given temperature, however, surface tension dominates and material flow in the transverse plane will increase, reducing the photonic crystal dimensions. As a consequence of this hole collapse, the diameter of the waist becomes smaller as compared with a taper produced with single-mode fiber for the same tapering parameters.

4.2. Low air-fill fraction PCF The fiber that is described in Section 3 (air-fill fraction 0.45) is tapered down to an outer diameter of 38.7 µm (a factor of 0.31) and probed transversely. Figure 9 shows a schematic of the experimental setup. The taper is placed between two pieces of SMF and probed at different positions along the taper length. Since only a short length of fiber is necessary for carrying out the transverse probing experiments, the taper consists of a 3 mm length of PCF (before elongation) fusion spliced between two pieces of SMF. The flame is only applied to the PCF region during the tapering process. There are two advantages to making the tapers in this manner. By keeping the volume of the air voids to a minimum and sealing the fiber on either end via fusion splicing, we were able to maintain sufficient gas pressure within the voids to produce tapers which helped preserve their aspect ratio during tapering.

14


[1, § 4

Fig. 9. Schematic of the transverse probed taper experiment. As for usual transverse MOF probing, the taper is placed between two single-mode fibers acting as source and detector. To probe different taper dimensions, the taper is moved longitudinally between the SMFs, out of the plane of the diagram.

Fig. 10. A series of spectra obtained through probing the transverse taper at different longitudinal positions. As the local taper outer diameter, and the internal microstructure dimensions, become smaller the resonant feature moves to smaller wavelengths.

Figure 10 shows a series of spectra corresponding to the resonant responses of the taper probed at different positions along its length, again in the Γ –M di-

1, § 4]

Tapered MOFs

15

rection. As the taper diameter (and the photonic crystal dimension) increases, the observed resonant feature moves to higher wavelengths. The regularity of this motion implies the microstructure of the taper is well preserved throughout its length. The rate of taper in the longitudinal direction is sufficiently low that, when considered transversely, the dimensions of the microstructure are essentially unvarying within the scale of the probe beam. Through tapering a PCF, we have created a device which has continuously varying photonic crystal dimensions. To observe a band gap at a given wavelength, we merely probe the taper at an appropriate longitudinal position.

4.3. High air-fill fraction PCF In Section 4.1, a PCF with a relatively low air-fill fraction was tapered, demonstrating good reproduction of the microstructure throughout the taper length. In this fiber, the size of the air inclusions is approximately the same as the silica regions between them. This makes the fiber relatively robust against hole collapse. In the present subsection, we apply the tapering methods outlined earlier to a high air-fill fraction PCF (0.93), which has very fine silica structure. Figure 11(left) shows a cleaved end-face of a PCF with a high air-fill fraction photonic crystal core. It consists of ten rings of approximately circular air inclusions of diameter 3.15 µm arranged in a two-dimensional hexagonal crystal lattice of period 3.50 µm. This results in an air-fill fraction of 0.93. In transmission, this high air-fill fraction is responsible for low loss propagation. We taper this fiber using the method above. These values for width and period are taken as averages over the inner few rings, as the inclusions become smaller and more irregular further away from the core. The high symmetry points of the reciprocal lattice are

Fig. 11. Comparative SEMs of the tapered high air-fill fraction photonic crystal fiber. At left is the untapered fiber, with the symmetry points of the photonic crystal microstructure shown. At right is the tapered PCF, with an outer diameter of 27.1 µm, showing good reproduction of the microstructure.

16


[1, § 4

superimposed. Again, there are large, irregular holes at the K symmetry points. As such we only consider propagation in the Γ –M direction, as scattering from these inclusions detract from the observation of resonant effects in the Γ –K direction. There is also a germanium-doped ring surrounding the air core, whose refractive index is raised by ∼1% above that of the surrounding material. We find through simulation that this doped ring has a negligible effect in the transverse orientation, thus we ignore the refractive-index change brought about by the doping. Figure 11(right) shows a comparison of SEMs of the untapered and tapered fibers. The taper has an outer diameter of 27.1 µm, which is a six times reduction in dimension. The microstructure is well reproduced even at this large level of reduction. The profile of the taper was analyzed using optical microscopy. Figure 12 shows the profile of the high-fill taper. Again, the local variation over the size of the probe beam is sufficiently small to allow the taper to be considered locally flat. Figures 13, 14 and 15 show a series of spectra taken from different outer diameters of the tapered, high air-fill, PCF. Also shown is the predicted position of the band gaps from the plane-wave expansion method. As in Section 4.1, the resonant

Fig. 12. The profile of the high air-fill taper outer diameter. Notice that the local rate of change of the outer diameter is small enough to consider the taper flat over the width of the probe beam.

1, § 4]

Tapered MOFs

17

Fig. 13. The fundamental gap observed in the transverse high air-fill PCF, for both polarizations, showing an attenuation of approximately 7 dB. The shading indicates the gap position predicted by the plane-wave calculation. The outer diameter of the taper is given in the lower left-hand corner of the plot.

features are seen moving to longer wavelengths as the photonic crystal dimension increases. As predicted by the plane-wave calculation, a number of higher-order band gaps are observed. These band gaps only appear along the Γ –M direction rather than over the entire reciprocal lattice and are referred to as partial band gaps. Again, the regularity of gap motion with dimension indicates uniformity of the microstructure throughout the taper. We have demonstrated the application of the tapering process to two photonic crystal fibers, both of which display good reproduction of the microstructure throughout the tapered region, regardless of air-fill fraction. Using tapering, any MOF can have its dimensions scaled to create a device with the appropriate optical response, which can be tuned again by simply probing different longitudinal positions along the taper.

18


[1, § 4

Fig. 14. The second-order band gap for both polarizations, the plane-wave prediction shown shaded. Notice the TM partial gap is much deeper and broader than the TE one; this behavior is supported by the plane-wave calculations.

4.4. Silica nanowires Tapering of fibers was initially performed to modify the transmission of light in the traditional longitudinal direction of propagation. Recently this concept has been extended by the creation of silica nanowires, fiber tapers with diameters typically around the micron or submicron range (Tong, Gattass, Ashcon, He, Lou, Shen, Maxwell and Mazur [2003]). Typically, these devices are highly resistant to bend loss and demonstrate nonlinearity at relatively low powers. We demonstrate nanowires using MOFs which display unique transmission properties. Figure 16 shows the MOF that was tapered in the creation of these nanowires. This “grapefruit” fiber, so called for its citrus-shaped air inclusions, has a germanium-doped core, the same as regular single-mode fiber, surrounded by six large air inclusions. Similarly to other fibers, the grapefruit fiber has an outer diameter of 125 µm. This MOF is used to overcome traditional issues involved with

1, § 4]

Tapered MOFs

19

Fig. 15. The third-order partial band gap region, in both polarizations. Notice a distinctly deeper feature in the TM. Indeed, the plane-wave calculation does not predict a feature in the TE. There is, however, a very small attenuation in the TE that is dependent on taper position. We attribute this to a small lack of polarizer discrimination. There is also an offset in the position of the predicted partial gap compared to that of the experiment. We attribute this to higher-order gaps having higher dependence on spatial perturbation of the photonic crystal.

fine taper fabrication. Usually, when these nanowires are made from single-mode fiber, the guided field is quite extended, and scattering loss becomes an issue when contaminants attach to the surface of the wire. The grapefruit fiber overcomes this by confining the guided mode using the air inclusions around the core, thus isolating it from the environment. The taper is performed as described earlier, expect that the fiber is illuminated using a broadband source throughout the tapering process, and the output is monitored on an OSA. The modal properties of these structures were found to be strongly dependent upon the ratio of taper diameter to wavelength. Figure 17 shows the various propagation regimes in these nanowires, simulated using the beam propagation method (Yevick and Hermansson [1990]). At top, the diameter of the nanowire is significantly larger than the wavelength and the mode is

20


[1, § 4

Fig. 16. A series of SEMs showing the progressive tapering of the grapefruit fiber. Top left is a taper with 5 µm outer diameter, showing no deformation of the microstructure. As the taper diameter goes past 2.7 µm the inclusions begin to deform, until, at 1.5 µm, hole collapse begins.

strongly confined in the silica core, bound by the air cladding. In comparison to the SMF nanowire mode, the MOF mode is strongly isolated from the environment. However, as the taper diameter becomes a similar in scale to the wavelength guided, the mode leaks out of the microstructure and becomes highly evanescent. Whilst this implies the mode is very sensitive to external perturbation, this fact may be used to create robust, fiber-based environmental sensors. Because of the strong field confinement in the silica core of MOF nanowires, enhanced nonlinearity is expected. We calculate the nonlinear coefficient of this structure and find it peaks at a value 2.73 for the ratio of taper diameter to wavelength. This gives, for typical values of wavelength 800 nm and taper diameter 2.1 µm, a π nonlinear phase shift after 7 mm of propagation of a 1 kW pulse. This nonlinearity is about 5 times greater than commercial “highly nonlinear” fiber, and about 100 times larger than that of bulk silica. The fabrication of nanowires using MOFs adds several degrees of utility to the MOF itself. The resulting reduction in modal size means nonlinear effects can been seen at lower powers than conventionally used, allowing easier utilization of

1, § 4]

Tapered MOFs

21

Fig. 17. The various mode profiles of the tapered grapefruit fiber (left-hand column) in comparison with a similarly dimensioned tapered SMF (middle column). The right-hand column represents the mode of a silica nanowire the same dimension as the core of the grapefruit fiber. In the top three panels, the taper is significantly larger than the wavelength of the propagating light, making the mode tightly confined in all three cases. As the taper size becomes smaller, the mode begins to be more evanescent in nature, until, for dimensions significantly smaller than the light wavelength, the mode is highly evanescent.

nonlinear effects, such as supercontinuum generation. Also, the ability to control the model behavior from tightly confined to evanescent has applications to environmental sensing in a structure typically more robust than other optical sensors.

22


[1, § 5

§ 5. Microfluidic MOFs Whilst tapers provide a wealth of post-process customization for MOFs though alteration of dimension, fluid tuning via microfluidics provides a similar level of versatility though the alteration of refractive index. Also, according to the fluid chemistry, other optical properties may be tailored at will, such as the inclusion of gain media (Large, Ponrathnam, Argyros, Pujari and Cox [2004]). The fluid may also be moved dynamically in the sample, allowing for further temporal tunability. Microfluidics is a field recently emerged from biochemistry, where it arose from the need to perform very small volume assays on biological molecules, especially DNA (Cao, Tegenfeldt, Austin and Chou [2002]). From these origins, microfluidics now, much more generally, encompasses the study of the nature of fluid behavior in femtoliter volumes or less (Karniadakis [2002]). Microfluidics has a number of attractive aspects in its application to optics. Fluids are an inherently mobile phase and may be inserted or withdrawn from an optically active region by the application of a pressure gradient. Because of this mobility, a microfluidic tuned photonic device may incorporate a degree of temporal tuning. The optical properties of the fluid are highly dependent upon their chemistry, allowing a broad range of refractive indices, gain media or liquid metals to be inserted into the photonic device. Also, the refractive-index contrast between the body of the fluid and the surrounding air is typically much higher than that available through other tuning mechanisms. We classify microfluidic tuning of MOFs into two regimes: static and dynamic. Static fluid tuning involves the infiltration of the microstructure with a stationary fluid to modify the device’s optical response. Dynamic fluid tuning involves actuating the fluid inside the device, imbuing the device with a temporal response. Using these methods we tune the response of PCFs, in a variety of geometries, and create a PCF-based microfluidic switch.

5.1. Static tuning To demonstrate “static tuning”, that is, modifying the optical response of a structure by infiltrating it with a body of fluid that is stationary, we again utilize the transverse PCF. Figure 18 shows a series of photographs of the fluid delivery method used in this experiment. We use an all-fiber delivery system, as this minimizes contamination of the outside of the PCF with the fluid. The first frame of the figure shows two fibers; on the left is the unfilled PCF, with its microstructure clearly visible. On the right is an SMF with a droplet of fluid adhering to the

1, § 5]

Microfluidic MOFs

23

Fig. 18. A series of photographs illustrating the all-fiber fluid delivery method. On the left is the PCF to be filled. At right is an SMF with a drop of fluid on the end. As the fibers are brought together capillarity draws the fluid along the PCF. In this case, the refractive index of the fluid is matched to silica, so the PCF microstructure appears transparent when filled.

end of the fiber. In the second frame, the drop is brought into contact with the face of the PCF. The fluid is attracted to the silica of the fiber and is drawn into the microstructure by capillary force. As the fluid is drawn along the fiber, the microstructure becomes indistinguishable from the surrounding silica, shown in the third frame of fig. 18. The PCF is optically probed in the fluid filled section, 200 µm away from the cleaved end face. The PCF microstructure is now infiltrated by a series of fluids whose refractive indices vary from n = 1.45 to 1.8. The transverse PCF is now operating in the high-index inclusion regime, which has an inherently different dispersion relation to the photonic crystal with low index inclusions. The comparative dispersion relation between these two regimes is shown in fig. 19. On the left is the dispersion relation for air inclusions, on the right it is shown for inclusions of n = 1.75. Some differences between the two are immediately apparent. Firstly,

24


[1, § 5

Fig. 19. Comparative dispersion relations calculated for air inclusions and inclusions with refractive index 1.75, for both polarizations in the Γ –M direction. Notice that in the high-index case, the partial band gaps move to lower frequencies, and a third-order partial gap appears in the TM polarization.

there is some offset between the fundamental and second order partial gaps in the Γ –M direction between the two regimes, the high-index partial gaps occurring at higher wavelengths than the low-index ones. Also, a third-order partial gap is seen to appear in the high-index regime for the TM polarization. Figure 20 shows a series of spectra of the transverse PCF as the microstructure is filled with fluids of different indices. At n = 1.45, the fiber is essentially transparent, as the fluid is index-matched to the surrounding silica. As such, no resonant features are seen. However, as the refractive index increases, a resonant feature begins to grow in depth and shift toward longer wavelengths. This behavior is predicted by the plane-wave calculation: no gap is predicted for n = 1.45, but one appears and grows wider as the inclusion index increases. In the planewave expansion, the crystal is assumed to be infinite. When applying the predictions of this calculation to a finite photonic crystal, the gap depth in a finite crystal is related to the gap width in an infinite crystal. Similarly to the untuned case, the plane-wave predicted gap has a slight offset compared to the observed gap. The wavelength of the resonant feature increasing with refractive index can be thought of in analogous terms to a one-dimensional Bragg grating. The Bragg grating resonance wavelength is given by ¯ λB = 2nd,

(1)

where n¯ is the average refractive index and d is the period of the grating. So as the refractive-index contrast of the grating increases, the resonant wavelength also

1, § 5]

Microfluidic MOFs

25

Fig. 20. The position of the second-order transverse band gap, in both polarizations, as the refractive index of the inclusions is increased. Indicated are the positions of the gaps predicted by the plane-wave calculation. Notice the gap becomes broader and deeper, and moves to longer wavelengths, as the refractive index is increased.

increases. Similarly, the resonant wavelength of the photonic crystal increases as the inclusion index increases. Figure 21 also shows a series of spectra, this time for a PCF whose period is 1.1 µm but whose inclusion diameter is the same as the fiber treated above. This change of periodicity allows the third partial gap, predicted above, to be probed within the limits of the experimental setup. Similarly to fig. 20, the third-order partial gap can be seen becoming deeper and moving to higher wavelengths as the inclusion refractive index increases. However, the third-order partial gap only becomes visible at much higher index contrasts than the second-order partial gap. This behavior is also predicted by the plane-wave simulation.

26


[1, § 5

Fig. 21. The appearance of the third-order partial gap. Similarly to the second-order case, the gap appears at higher refractive-index contrast and moves to longer wavelengths as the index of the fluid increases.

5.2. Dynamic tuning Fluids are a naturally mobile phase and it is possible to take advantage of this mobility to create dynamically tunable microfluidic photonic devices. We once again utilize the transverse PCF geometry for this demonstration. Figure 22 shows the experimental setup. A fluid plug of 500 µm (refractive index n = 1.50) is introduced into the PBG structure (Nguyen, Domachuk, Eggleton, Steel, Straub, Gu and Sumetsky [2004]). Figure 23 shows a series of OSA traces comparing the optical properties of the transverse PCF in various states of tuning. The solid curve (top) gives the transmission of probe fibers butt-coupled, with no PCF inserted. The dotted curve (mid) gives the transmission of PBG fiber in the “on” state, with the fluid plugs in the optical interaction region. This provides an insertion loss of around 7 dB. The slight rise in the transmission intensity at 1100 nm is due to the SMF reaching the single-mode cut-off. The dot–dashed

1, § 5]

Microfluidic MOFs

27

Fig. 22. A schematic representation of the dynamic tuning experiment. The fluid is moved by an applied temperature gradient produced by a thin film resistive heater, driven by a square voltage.

Fig. 23. The states of the dynamically tuned photonic device. When the fluid is in the optical path, the device is relatively transparent across a broad band (dotted curve). When the fluid is taken out of the optical path, the bandgap of the fiber manifests itself as high resonant loss at wavelengths between 1400 and 1600 nm.

28


[1, § 5

curve (bottom) is the transmission for the fiber with the fluid moved out of the optical interaction region, the “off” state, with the previously mentioned resonant features present. Figure 24(top) shows oscilloscope traces of the transmission of the fluidic device being driven by a step voltage of 7.0 V (which provides a temperature range of 50–80◦ C) with a period of 4.5 seconds. We use a tunable laser to probe λ ≈ 1.47 µm, at the bottom of the observed partial bandgap. The output of

Fig. 24. The temporal response of the dynamically tuned device. The top panel shows three periods of the device, illustrating the regularity of the response. The bottom panel shows detail from one period. This device has a relatively slow response time of 1.5 seconds, due to the high thermal mass of silica fibers.

1, § 6]

Microfluidic interferometer

29

the probe fiber is registered on an oscilloscope that is triggered by the rise of the square wave driving the heater. The top graph shows three periods of the device, illustrating the device’s regularity of transmission over its operation. The “ears” on each side of the step are caused by unevenness in the fluid distribution, leading to the fluid mass being more transparent at the air–fluid interface than at the fluid’s resting position. The bottom of Figure 24 shows detail of a single period. The initial step voltage is at the origin, giving a response time of the device of 1.5 seconds with a rise (and fall) time of 0.33 seconds. This response time is due to the finite thermal rise time of the capillary heater and the fact that the fluid plugs have several hundred µm’s to travel before intersecting the beam. While this is slow for an optical switch, it is conceivable that much faster switching times can be achieved using smaller lengths of fluid, smaller total device volumes and alternative fluid motivation methods.

§ 6. Microfluidic interferometer In this chapter we have presented two tuning regimes for extending the utility of MOFs. We have altered the dimensions of the MOF using tapering and we have altered its refractive-index distribution using fluid infiltration. We bring these two methodologies together to demonstrate the compact, microfluidic interferometer. This device can be considered as a precursor to the next generation highly integrated microfluidic systems based around tunable interferometric configurations (Wolfe, Conroy, Garstecki, Mayers, Fischbach, Paul, Prentiss and Whitesides [2004]), with potential applications in biosensing, safe chemical detection systems as well as communication systems. The specific embodiment we present uses a hollow-core square silica capillary, which is used to contain a fluid plug, and is optically probed transversely. Light propagates across the fluid–air interface (meniscus) and the phase difference is introduced through the different optical path lengths on either side of the meniscus. This phase difference leads to a Mach–Zehnder interference-like response, with resonance position and depth obviously related to the meniscus position and shape. To optimize the geometry we post-process the capillary by manipulating the meniscus shape, via surface chemistry, and tapering. We simulate the geometry using the beam propagation technique and find good agreement between simulation and experiment. The experimental setup is shown in fig. 25. In the center is the silica capillary used to contain the fluid. The capillary is square in cross-section; this shape avoids scattering via diffraction of the beam by a curved core. Silica capillaries are ideal environments for microfluidics as their surfaces are naturally atomically smooth.

30


[1, § 6

Fig. 25. A schematic of the experimental geometry used to demonstrate the single-beam interferometer. A square capillary is placed between two SMFs. The fluid inside the capillary is then moved so that the meniscus bisects the optical beam, creating the resonant optical effect.

The square capillary has an outer width of 381 µm and a core width of 52 µm. Light from a broadband source (range 0.8–1.7 µm) is brought to the capillary, and collected from the other side, using standard single-mode fiber (SMF). It is then analyzed on an Optical Spectrum Analyzer (OSA). The orientation of the capillary is viewed through a CCD camera. The fluid used is silica matching oil (n = 1.45). We create a pressure differential by coupling the square capillary to a syringe pump, moving the fluid interface into the beam. The central position of the interface is inferred from the maximum reflection at the resonant wavelengths. Initially, the experiment was performed with the square capillary untreated in any way. The loss across the capillary was very high, and resonant effects across the capillary were hard to see. From simulations we attribute this very high loss to two factors. First, due to the curvature of the meniscus, there is a high degree of scattering from the interface. Second, loss arises from the large width of the capillary: by the time the beam has traversed the capillary, only a small fraction of light is captured by the output SMF, resulting in 6 dB attenuation. These two loss mechanisms can be addressed by surface conditioning and capillary tapering, respectively. The first loss mechanism to address is the curvature of the meniscus. The origin of this curvature is the hydrophilic nature of the glass surface. It is composed of highly polar ions that are very attractive to water molecules. We change the surface chemistry of the glass using a process called silanization, which attaches molecules of dodecyltrichlorosilane to the glass, creating a neutral surface (McGovern, Kallury and Thompson [1994]). This results in a flat interface between the water and the air. Figure 26 illustrates the contrast between the shape of

1, § 6]

Microfluidic interferometer

31

the treated and untreated menisci. The other loss mechanism of this device is due to the size of the square capillary. We alter this by tapering: previous work has demonstrated the utilization of tapering capillaries to modify their optical properties (Tavlove [1995]). We taper the square capillary to an outer width of 80 µm (core width of 12 µm). This width is sufficiently small so as to significantly increase the out-of-resonance transmission. Figure 27 shows the transmission spectra of the optimized device, shown in comparison to the simulated spectra. Good

Fig. 26. Illustration of the meniscus modulation through surface chemistry. At top is the untreated meniscus, with a contact angle of ∼40◦ . At bottom is the channel with the surface treatment, displaying a nearly flat meniscus.

Fig. 27. The spectrum of the final, optimized device, with an out-of-band insertion loss of ∼4 dB, with an extinction ratio of 20 dB. The beam propagation simulation matches well the experimental results.

32


[1, § 7

agreement is seen between the predicted spectra and that observed. Notice the out-of-resonance transmission has increased to −4 dB, due to the optimization we have performed while maintaining the strong dip resonance to less than −25 dB.

§ 7. Conclusion In this review, we have extended the already considerable utility of MOFs by probing them in novel geometries, modifying their dimensions using tapering and tuning their refractive indices using microfluidics. We exemplify this postengineering in the microfluidic interferometer, a device that utilizes tapered transverse MOFs tuned by microfluidics to achieve higher functionality.

References Benabid, F., Knight, J.C., Antonopoulos, G., Russell, P.St.J., 2002, Stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber, Science 298, 399. Birks, T.A., Knight, J.C., Russell, P.S., 1997, Endlessly single-mode photonic crystal fiber, Opt. Lett. 22, 961. Birks, T.A., Li, Y.W., 1992, The shape of fiber tapers, J. Lightwave Technol. 10, 432. Broeng, J., Mogilevstev, D., Barkou, S.E., Bjarklev, A., 1999, Photonic crystal fibers: A new class of optical waveguides, Opt. Fiber Technol. 5, 305–330. Cao, H., Tegenfeldt, J.O., Austin, R.H., Chou, S.Y., 2002, Gradient nanostructures for interfacing microfluidics and nanofluidics, Appl. Phys. Lett. 81, 3058. Choi, S., Oh, K., 2003, A new LP02 mode dispersion compensation scheme based on mode converter using hollow optical fiber, Opt. Commun. 222, 213. Choi, S., Oh, K., Shin, W., Ryu, U.C., 2001, Low loss mode converter based on adiabatically tapered hollow optical fiber, Electron. Lett. 37, 823. Cregan, R.F., Mangan, B.J., Knight, J.C., Birks, T.A., Russell, P.St.J., Roberts, P.J., Allan, D.C., 1999, Single-mode photonic band gap guidance of light in air, Science 285, 1537. Croft, T.D., Ritter, J.E., Bhagavatula, V.A., 1985, Low-loss dispersion-shifted single-mode fiber manufactured by the OVD process, J. Lightwave Technol. 3, 931. Domachuk, P., Nguyen, H.C., Eggleton, B.J., 2004, Transverse probed microfluidic switchable photonic crystal fiber devices, IEEE Photonics Technol. Lett. 16, 1900. Eggleton, B.J., Kerbage, C., Westbrook, P., Windeler, R., Hale, A., 2001, Microstructured optical fiber devices, Opt. Exp. 9, 698. Gobel, W., Nimmerjahn, A., Helmchen, F., 2004, Distortion-free delivery of nanojoule femtosecond pulses from a Ti:sapphire laser through a hollow-core photonic crystal fiber, Opt. Lett. 29, 1285. Grillet, C., Domachuk, P., Ta’eed, V., Mägi, E., Bolger, J.A., Eggleton, B.J., Rodd, L.E., CooperWhite, J., 2004, Compact tunable microfluidic interferometer, Opt. Exp. 12, 5440. Joannopoulos, J.D., Meade, R.D., Winn, J.N., 1995, Photonic Crystals: Molding the Flow of Light, Princeton University Press, Princeton, NY. Kaiser, P., Astle, H.W., 1974, Low-loss single-material fibers made from pure fused silica, Bell Syst. Tech. J. 53, 1021. Karniadakis, G.E., 2002, Micro Flows, Springer-Verlag, New York.

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Knight, J.C., Birks, T.A., Russell, P.St.J., Atkin, D.M., 1996, All-silica single mode optical fiber with photonic crystal cladding, Opt. Lett. 21, 1547–1549. Knight, J.C., Birks, T.A., Russell, P.St.J., Rarity, J.G., 1998, Bragg scattering from an obliquely illuminated photonic crystal fiber, App. Opt. 37, 449–452. Konorov, S.O., Mitrokhin, V.P., Fedotov, A.B., Sidorov-Biryukov, D.A., Beloglazov, V.I., Skibina, N.B., Shcherbakov, A.V., Wintner, E., Scalora, M., Zheltikov, A.M., 2003, Laser ablation of dental tissues with picosecond pulses of 1.06 µm radiation transmitted through a hollow-core photonic-crystal fiber, App. Opt. 43, 2251. Ladoucer, F., Love, J.D., 1996, Silica-Based Buried Channel Waveguides and Devices, Chapman and Hall, London. Large, M.C.J., Ponrathnam, S., Argyros, A., Pujari, N.S., Cox, F., 2004, Solution doping of microstructured polymer optical fibres, Opt. Exp. 12, 1966. Larsen, T.T., Bjarklev, A., Hermann, D.S., Broeng, J., 2003, Optical devices based on liquid crystal photonic bandgap fibres, Opt. Exp. 11, 2589. Leon-Saval, S.G., Birks, T.A., Wadsworth, W.J., Russell, P.S.J., Mason, M.W., 2004, Supercontinuum generation in submicron fibre waveguides, Opt. Exp. 12, 2864. Litchinitser, N.M., Abeeluck, A.K., Headley, C., Eggleton, B.J., 2002, Antiresonant reflecting photonic crystal optical waveguides, Opt. Lett. 27, 1592. Lizé, Y.K., Mägi, E.C., Ta’eed, V.G., Bolger, J.A., Steinvurzel, P., Eggleton, B.J., 2004, Microstructured optical fiber photonic wires with subwavelength core diameter, Opt. Exp. 12, 3209. Mägi, E.C., Steinvurzel, P., Eggleton, B.J., 2003, Tapered photonic crystal fibers, Opt. Exp. 12, 776. McGovern, M.E., Kallury, K.M.R., Thompson, M., 1994, Role of solvent on the silanization of glass with octadecyltrichlorosilane, Langmuir 10, 3607. Monro, T.M., Belardi, W., Furusawa, K., Baggett, J.C., Broderick, N.G.R., Richardson, D.J., 2001, Sensing with microstructured optical fibres, Meas. Sci. Technol. 12, 854. Nagel, S.R., Macchesney, J.B., Walker, K.L., 1982, An overview of the modified chemical vapordeposition (MCVD) process and performance, IEEE J. Quantum Electron. 18, 459. Nguyen, H.C., Domachuk, P., Eggleton, B.J., Steel, M.J., Straub, M., Gu, M., Sumetsky, M., 2004, A new slant on photonic crystal fibers, Opt. Exp. 12, 1528. Nguyen, N., Wereley, S., 2002, Microfluidics, Artech House, Boston, MA. Noda, J., Okamoto, K., Sasaki, Y., 1986, Polarization maintaining fibers and their applications, J. Lightwave Technol. 4, 1071. Ouzounov, D.G., Ahmad, F.R., Müller, D., Venkataraman, N., Gallagher, M.T., Thomas, M.G., Silcox, J., Koch, K.W., Gaeta, A.L., 2003, Generation of megawatt optical solitons in hollow-core photonic band-gap fibers, Science 301, 1702. Ramaswami, R., 1993, Multiwavelength lightwave networks for computer-communication, IEEE Comm. Mag. 31, 78. Russell, P.St.J., 2003, Photonic crystal fibers, Science 299, 358. Saggese, S.J., Harrington, J.A., Sigel, G.H., 1991, Attenuation of incoherent infrared radiation in hollow sapphire and silica wave-guides, Opt. Lett. 16, 27. Steel, M.J., White, T.P., deSterke, C.M., McPhedran, R.C., Botten, L.C., 2001, Symmetry and degeneracy in microstructured optical fibers, Opt. Lett. 26, 488. Tavlove, A., 1995, Computational Electrodynamics: The Finite-Difference Time Domain Method, Artech House, Norwood, MA. Temelkuran, B., Hart, S.D., Benoit, G., Joannopoulos, J.D., Fink, Y., 2002, Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission, Nature 420, 650. Tong, L., Gattass, R., Ashcon, J.B., He, S., Lou, J., Shen, M., Maxwell, I., Mazur, E., 2003, Subwavelength-diameter silica wires for low-loss optical wave guiding, Nature 426, 816. Windeler, R.S., Wagener, J.L., DiGiovanni, D.J., 1999, Silica-air microstructured fibers: Properties and applications, Optical Fiber Communications conference, San Diego.

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Chapter 2

Optical solitons in random media by

Fatkhulla Abdullaev Physical-Technical Institute, Uzbek Academy of Sciences, G. Mavlyanov str., 2-b, 700084, Tashkent-84, Uzbekistan and Dipartimento di Fisica “E.R. Caianiello”, Universitá di Salerno, I-84081 Baronissi (SA), Italy e-mail: [email protected]

Josselin Garnier Laboratoire de Statistique et Probabilités, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse cedex 4, France

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)48002-6 35

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

§ 2. The main equations . . . . . . . . . . . . . . . . . . . . . . . . . .

41

§ 3. Solitons in random single-mode fibers . . . . . . . . . . . . . . . .

46

§ 4. Dispersion-managed solitons under random perturbations . . . . .

62

§ 5. Randomly birefringent fibers . . . . . . . . . . . . . . . . . . . . .

73

§ 6. Solitons in random quadratic media . . . . . . . . . . . . . . . . .

79

§ 7. Spatial solitons in random waveguides . . . . . . . . . . . . . . . .

83

§ 8. Two-dimensional solitons in random media . . . . . . . . . . . . .

92

§ 9. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

94

Appendix A: The inverse scattering transform for the nonlinear Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

Appendix B: The inverse scattering transform for the Manakov system . .

99

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

103

36

§ 1. Introduction Solitons are now considered to be among the most important objects in nonlinear physics. They can be found in many areas of physics, including hydrodynamics, condensed matter, nonlinear optics, and atomic Bose–Einstein condensates. Optical solitons in fibers and spatial solitons (self-supported optical beams) in nonlinear media have attracted special attention. It is widely believed that optical solitons will be key elements for the next generation of high-speed optical communication systems and ultrafast optical devices. Spatial solitons are considered as the base elements for all-optical logical and switching devices. The existence of solitons is connected with the balance between the fiber dispersion, or linear diffraction, and the medium nonlinearity. Optical solitons in fibers have been predicted by Hasegawa and Tappert [1973] and observed experimentally by Mollenauer, Stolen and Gordon [1980]. Optical fibers are media with cubic nonlinearity. Mathematically the optical pulse propagation in a slowlyvarying-amplitude approximation is described by the nonlinear Schrödinger equation (NLSE) for the electric field. This equation is integrable and can be solved by the use of the Inverse Scattering Transform (IST). Fortunately, many properties of real optical fiber systems, such as dissipation, amplification, Raman processes, self-steepening and higher-order dispersion, can be described as weak perturbations of the NLSE. It is therefore possible to develop a perturbation theory for explaining many phenomena connected with optical solitons in fibers and related devices, like soliton couplers, soliton lasers, etc. Finally, the propagation of short pulses in birefringent fibers has been investigated. In these media cross-phase modulation processes play important roles. Mathematically the pulse propagation can be described by the Manakov system and its nearly integrable and nonintegrable modifications, which supports vector optical solitons. Perturbations of the Manakov system can be dealt with by similar approaches as for the scalar NLSE. The efficiency of using solitons in optical communication systems is limited by several factors. The most important ones are: amplifier noise, inhomogeneities of the fiber material and the fiber transverse profile along the propagation, and random birefringence. The nonlinear interaction processes in trains of solitons also cause information loss in single-mode fibers. As was shown by Gordon and Haus 37

38

Optical solitons in random media

[2, § 1

[1986] and by Elgin [1985], the distributed noise leads to the soliton Gordon– Haus (GH) jitter and to fundamental limits on the bit error rate in soliton communication systems. Some estimates of the effects of random variations of fibers on soliton propagation were also obtained at that time, mainly by using an adiabatic approximation of the perturbation theory for solitons (see the book by Abdullaev, Darmanyan and Khabibullaev [1993]). More precise results were derived later by Baker and Elgin [1998], Falkovich, Kolokolov, Lebedev and Turitsyn [2001], Biswas [2001]. Finally, optical soliton propagation in fibers with random birefringence was investigated. It was shown that the scale of variations of birefringence is much smaller than the characteristic scales of the problem like dispersion and nonlinear lengths. As shown by Wai, Menyuk and Chen [1991] and Evangelides, Mollenauer, Gordon and Bergano [1992], averaging over the fast variations can be performed, and the averaged dynamics are described by the Manakov system. Using collective-variable methods, Ueda and Kath [1992] derived the Fokker–Planck equation to find the probability densities for the polarization state of a propagating vector soliton. Higher-order corrections to the averaged Manakov system lead to random wave generation by the soliton that limits the error free bit rate in optical communication systems as shown by Lakoba and Kaup [1997], Chen and Haus [2000a] and Chung, Lebedev and Vergelis [2004]. Investigation of this Polarization Mode Dispersion (PMD) process requires the development of a perturbation theory for the Manakov system that admits the calculation of radiative effects. The analyses of the perturbed Manakov system performed by Midrio, Wabnitz and Franco [1996] and Shechnovich and Doktorov [1997] dealt with the adiabatic approximation, the radiation by Horikis and Elgin [2003]. In Appendix B we present our results for the calculation of the radiation effects in the perturbed Manakov system, based on the Gel’fand–Levitan–Marchenko representation and the Hamiltonian formulation. Many stochastic problems are connected with the long-scale propagation of optical solitons: generation of waves by picosecond and femtosecond solitons in random fibers, interaction of solitons, and existence of dissipative solitons. Some of these problems have been solved recently, by Abdullaev, Caputo and Flytzanis [1994], Abdullaev and Baizakov [2000], Chertkov, Chung, Dyachenko, Gabitov, Kolokolov and Lebedev [2003], Abdullaev, Navotny and Baizakov [2004] and Chung, Lebedev and Vergelis [2004], while others remain open. The discovery of dispersion-managed (DM) solitons by Smith, Knox, Doran, Blow and Bennion [1996] and Gabitov and Turitsyn [1996] has opened new possibilities for the soliton optical communication. Periodic and strong modulations of the fiber dispersion are used to generate a DM soliton. The DM soliton is breathing during propagation and has enhanced energy in comparison with the standard

2, § 1]

Introduction

39

optical soliton. This type of pulse is more robust against the action of the GH jitter. But since dispersion is modulated, new types of stochastic problems appear. It is important to understand the stability of DM solitons against random variations of the chromatic and waveguide dispersions existing in fibers. As measurements show, the random variations of the dispersion can be of the order of its average value, and so they play an important role in short-pulse degradation. Mathematically this problem is more complicated since we have to deal with an NLSE with strong periodic and random variations in dispersion coefficient. Analysis of this problem can be undertaken effectively by a variational method (used by Abdullaev and Baizakov [2000], Malomed and Berntson [2001], Garnier [2002] and Schafer, Moore and Jones [2002]) or a frequency-domain approach (used by Chertkov, Gabitov, Lushnikov, Moeser and Toroczkai [2002]). The evolution of DM solitons under PMD has been examined theoretically by Karlsson [1998], Chen and Haus [2000b] and Abdullaev, Umarov, Wahiddin and Navotny [2000] and experimentally by Xie, Sunnerud, Karlsson and Andrekson [2001]. Other areas of optical solitons are connected with self-supported optical beams (spatial solitons) and spatiotemporal pulses (optical bullets). In media supporting such structures the nonlinearity can be much larger than in optical fibers and so the effective scales are much shorter. Optical solitons become important as they are used as elements of all-optical logical devices and switches. Only cubic nonlinearity has been discussed in the preceding paragraphs, but quadratic nonlinear media are also important for theoretical and practical purposes. The formation of quadratic solitons in such media is possible due to the balance between the linear diffraction (dispersion) and nonlinear terms involving phase mismatch. The soliton has a more complicated form here than in the fiber case, and it involves in general the interaction between three waves. An important partial case considered in this review is the interaction of a fundamental wave (FW) and a second-harmonic (SH) wave. The system of equations describing this process is not integrable in general. One-dimensional (1D) solitons have been predicted by Karamzin and Sukhorukov [1974] and were observed experimentally in a planar nonlinear waveguide by Baek, Schiek, Stegeman, Bauman and Sohler [1997] (see also the reviews by Buryak, Di Trapani, Skryabin and Trillo [2002] and Etrich, Lederer, Malomed, Peschel and Peschel [2000]). The stability of such solitons in random media has not yet been investigated. In contrast to solitons in cubic nonlinear media, quadratic solitons are very sensitive to the phase-matching condition. The randomness of the medium should lead to the distortion of phase matching and the degradation of the quadratic soliton. As the equations are nonintegrable, exact analytical methods are absent. Numerical simulations for the random mismatch and quadratic nonlinearity performed by

40


[2, § 1

Torner and Stegeman [1997], Clausen, Bang, Kivshar and Christiansen [1997] and Abdullaev, Darmanyan, Kobyakov, Schmidt and Lederer [1999] show the pure dissipative nature of the soliton decay. The problems mentioned above involve pulse dynamics driven by perturbations which are random in the evolutional variable. In the case of spatial solitons it is important to consider the propagation in a medium which exhibits random variations in the spatial transverse variable. It corresponds to the random linear, V1 (x), and/or nonlinear, V2 (x)|u|2 , potentials in the NLSE. This problem is also important for many areas of physics. It was investigated by many authors (Gredeskul and Kivshar [1992], Bronski [1998], Knapp [1995]), and has been solved recently using the IST approach for nonlinear and dispersive perturbations Garnier [1998, 2001]. The evolution of optical bullets in random media remains virtually uninvestigated. Some results were obtained recently by Gaididei and Christiansen [1998], Fibich and Papanicolaou [1999], Abdullaev, Bronski and Galimzyanov [2003] and Yannacopoulos, Frantzeskakis, Polymilis and Hizanidis [2002]. The dynamics of discrete optical solitons in disordered 1D arrays of planar waveguides and 2D arrays of fibers have recently attracted attention – see the experiments performed by Pertsch, Peschel, Kobelke, Schuster, Bartelt, Nolte, Tunnermann and Lederer [2004]. However, a theory for wave phenomena in such structures is absent. Only some numerical simulations for 1D nonlinear disordered arrays have been performed by Kopidakis and Aubry [2000]. In this review we intend to give a description of the present status of theory and experiment about propagation and interaction of optical solitons in random media. We start with a short derivation of the main equations describing the evolution of optical solitons in the presence of a cubic or a quadratic nonlinearity (Section 2). The first part of the review is devoted to the dynamics of solitonic pulses in fibers with random parameters. The propagation of standard optical solitons (Section 3) as well as dispersion-managed solitons (Section 4) under random fluctuations of dispersion and nonlinearity (amplification) is examined. The inverse scattering transform, the variational approach, and averaging of the NLSE in the frequency domain are applied for the investigation of solitonic processes in fibers with these types of inhomogeneities. We also address other related issues: the interaction of solitons in single-mode fibers with fluctuating parameters and the propagation of femtosecond pulses using a randomly perturbed modified NLSE. In Section 5 we study the influence of random birefringence on the propagation of optical solitons. PMD is a limiting factor for long-distance optical fiber communication systems. It can be overcome by solitonic propagation. The propagation can be analyzed by averaging the original system of coupled NLSEs over

2, § 2]

The main equations

41

the rapid variations of the birefringence. The resulting system is the randomly perturbed Manakov system. For vector optical solitons, radiative effects, internal motion, etc., we describe their propagation using the IST approach. At the end of Section 5 we study the influence of PMD on dispersion-managed solitons. In Section 6 we address the propagation of optical solitons in quadratic random media. Analytical and numerical results are described for two types of fluctuations along the propagation direction: random mismatch and fluctuating nonlinearity. Section 7 is devoted to the propagation of spatial solitons (beams) in media with spatially varying parameters. We first address the propagation of a soliton in a long slab with an index of refraction that is random in the transverse direction. The problem is treated analytically and numerically. The role of random dispersive perturbations in the propagation of solitons is studied through the example of an array of planar waveguides with random tunnel coupling. The results of recent experiments in 2D disordered fiber arrays are discussed. The case where the index of refraction varies randomly in the transverse and longitudinal directions is considered using the moments method. In the final section the evolution of 2D solitons under fluctuations of the medium parameters is investigated. The analysis is based on the modulation theory for the Townes soliton. The Appendices A and B present the IST approach for the scalar NLSE and the vector NLSE. Throughout this review we present different approaches (inverse scattering transform, moments approach, variational approach, averaging of partial differential equations). We discuss their respective domains of applicability from a theoretical point of view.

§ 2. The main equations 2.1. Derivation of the nonlinear Schrödinger equation In this section we reproduce standard arguments that can be found for instance in the article by Haus and Wong [1996]. The transverse profile of the index of refraction of an optical fiber can be designed so that the fiber supports only one electromagnetic mode, with two possible polarizations. Field patterns of greater transverse variation are not guided but are lost to radiation. A single-mode fiber propagates modes of two polarizations. Polarization-maintaining fibers are sufficiently birefringent that a mode launched in one linear polarization along one of the axes of birefringence remains polarized along this axis. However, polarizationmaintaining fibers are expensive and have higher losses than regular fibers, and hence are not used in long-distance fiber communications. Regular fibers still possess birefringence of the order of 10−7 . This means that within 107 wavelengths,

42


[2, § 2

i.e. about 10 m, the polarization changes uncontrollably. The depolarization length is much shorter than the dispersion length, and the length within which the optical Kerr effect produces self-phase modulation. On these larger distance scales, the mode can be treated in a first approximation as a mode of a single average polarization. A pulse propagating along the fiber experiences dispersion and the optical Kerr effect. Dispersion is called normal, or positive, if the group velocity decreases with increasing frequency; it is anomalous, or negative, if the change is in the opposite direction. Dispersion acting alone is a linear effect that does not change the spectrum of the pulse. Different frequencies travel with different group velocities and thus a pulse spreads in time. The optical Kerr effect is an intensity-induced change in refractive index. The Kerr coefficient is defined to be positive if the refractive index increases with increasing intensity. This index change leads to a time-dependent phase shift. Since the time derivative of phase is related to the frequency, the optical Kerr effect usually produces a change of the pulse spectrum. The combination of positive dispersion and a positive Kerr coefficient leads to temporal and spectral broadening of the pulse. A fiber with negative dispersion and a positive Kerr coefficient can propagate a pulse with no distortion. This may be surprising, at first, since dispersion affects the pulse in the time domain, while the Kerr effect does so in the frequency domain. However, a small time-dependent phase shift added to a Fourier-transform-limited pulse does not change the spectrum to first order. If this phase shift is canceled by dispersion in the same fiber, the pulse does not change its shape or its spectrum as it propagates. This propagation of an optical soliton is governed by the nonlinear Schrödinger equation (NLSE), which we will now derive. Consider the propagation of an electromagnetic mode of one polarization along a single-mode optical fiber. The fact that the polarization varies along the fiber due to the natural birefringence of the fiber will be taken up in Section 5. In the frequency domain, the amplitude a(z, ˆ ω) of the mode obeys the differential equation aˆ z (z, ω) = iβ(ω)a(z, ˆ ω), where β(ω) is the propagation constant. The spectrum of a(z, ˆ ω) is assumed to be confined to a frequency region around ω0 . A Taylor expansion of β(ω0 ) in frequency around the carrier frequency ω0 gives β(ω) β(ω0 ) + β ω + β ω2 /2, where ω = ω − ω0 . If we introduce the new envelope variable a(z, ˆ ω) = exp(iβ(ω0 )z − iω0 t)v(z, ˆ ω), we obtain the equation for v: ˆ vˆz = i(β ω + β ω2 /2)v. ˆ In the time domain this equation reads 1 vz = −β vt − iβ vtt . 2

2, § 2]

The main equations

43

Transformation of the independent variables z and t into a frame represented by z = z, t = t −β z comoving with the group velocity 1/β removes the first-order time derivative. To keep the notation simple, we shall drop the primes on z and t, getting as a result the simple equation 1 vz = − iβ vtt . (2.1) 2 This is the linear Schrödinger equation for a free particle in one dimension (with z and t interchanged). If the fiber is nonlinear, the propagation constant acquires an intensitydependent contribution, and β(ω) has to be supplemented by the Kerr contribution, which is a change of the refractive index proportional to the optical intensity. The process is known as four-wave mixing, since Fourier components at ω1 and ω2 mix with a Fourier component at ω3 to produce a phase shift at ω = ω3 − ω2 + ω1 . If a distribution of frequencies is involved, a convolution has to be carried out in the frequency domain. The nonlinear index n2 produces a phase shift proportional to n2 dω1 dω2 v(z, ω1 )v ∗ (z, ω2 )v(z, ω + ω2 − ω1 ). When transformed back into the time domain, this integral becomes 2 n2 ∂t v(z, t) v(z, t). Thus, the Kerr effect is expressed much more simply in the time domain, as long as the Kerr coefficient is frequency independent. The change of amplitude v of the pulse propagating through a differential length of fiber z can be derived from standard perturbation theory: v = i

ω0 |v(z, t)|2 n2 v(z, t)z. c Aeff

Here n2 is the Kerr coefficient (n2 3 × 10−20 m2 /W for silica fiber), Aeff is the effective mode area, obtained by averaging the phase shift using the mode profile e(x, y) over the fiber, that is, by integrating over the transverse dimensions x and y: |e(x, y)|4 dx dy 1 . = Aeff |e(x, y)|2 dx dy We have assumed that |v(z, t)|2 is normalized to the power and that the mode pattern e(x, y) has been so normalized that |e(x, y)|2 dx dy is dimensionless.

44


[2, § 2

When eq. (2.1) is supplemented by the Kerr effect, we obtain the NLSE 1 vz = − iβ vtt + iκ|v|2 v, (2.2) 2 where κ = ω0 n2 /(cAeff ). The nonlinearity in this equation can compensate for the dispersion: a pulse need not disperse since it can dig its own potential well, which provides confinement. This happens when β < 0. Indeed, a solitary-wave solution of eq. (2.2) is κ|A0 |2 + β ω12 t − β ω1 z exp i z , vS (z, t) = A0 sech τ 2 where 1/τ 2 = −κ|A0 |2 /β . This constraint shows that dispersion has to be negative indeed. We next introduce a reference time t0 (of the order of the pulse width) and we define the dimensionless time T = t/t0 and distance Z = z/z0 , where z0 = t02 /|β | is the dispersion distance. The dimensionless field √ u(T , Z) = κz0 u(T t0 , Zz0 ) satisfies the normalized NLSE 1 iuZ + uT T + |u|2 u = 0. (2.3) 2 The NLSE is integrable (see Appendix A) and supports soliton solutions with a sech envelope. 2.2. Derivation of χ (2) system The induced polarization of the medium P can be expanded as powers of the electric field amplitude P = ε0 χ (1) · E + χ (2) : (E, E) + · · · , (2.4) where ε0 is the vacuum permittivity and χ (i) , i = 1, 2, . . . , is the ith-order susceptibility. The second-order susceptibility is responsible for the second-harmonic generation. Let us derive the equations for the fundamental waves (FW) and second-harmonic (SH) fields. The lightwave with the frequency ω can feel the nonlinear response of the χ 2 terms through the interaction of ω and 2ω components. In a film waveguide geometry we can look for fields of the form E(X, Y, Z; t) = ei E¯ i + c.c., i

E¯ 1 (X, Y, Z; t) = A1 (X, Z)fA (Y )ei(k1 Z−ω1 t) , E¯ 2 (X, Y, Z; t) = A2 (X, Z)fB (Y )ei(k2 Z−ω2 t) ,

(2.5)

2, § 2]

The main equations

45

where c.c. stands for complex conjugate, the ei are unit polarization vectors, k1,2 = ωn1,2 /c are propagation constants, n1,2 are effective mode indices, fA (Y ), fB (Y ) are the propagating modes in the transverse Y -direction, and A1,2 are the slowly varying envelopes for FW and SH fields, respectively. It is assumed that the frequencies of the interacting waves are exactly matched (2ω1 = ω2 ) and the wavevectors are almost matched (2k1 (ω1 ) − k2 (ω2 ) = k, k ki ). Substituting this expansion into the Maxwell equation 1 ∂2 (2.6) (χ ∗ E) − ∇(∇ · E) = 0, c2 ∂t 2 and averaging over the transverse profile we obtain the system of equations for the FW (A1 ) and SH (A2 ) fields: E −

1 A1XX + Γ (X, Z)A∗1 A2 e−ikZ = 0, (2.7) 2k1 1 A2XX + Γ (X, Z)A21 eikZ = 0. iA2Z + (2.8) 2k2 The effective nonlinear coefficient Γ (X, Z) has the form (fA (Y ))2 fB (Y ) dY 2

ωd Γ (X, Z) = , (2.9) eff ∞ ∞ ε0 c3 n21 n2 2 2 −∞ |fA (Y )| dY −∞ |fB (Y )| dY iA1Z +

where deff is an effective component of the nonlinear susceptibility tensor χ (2) and fA,B (Y ) are dimensionless mode profiles. Equations (2.7)–(2.8) can be recast into a convenient dimensionless form with the following transformation: P0 A1 = √ E1 , 2

A2 = P0 E2 eikZ ,

Z = LD z,

X = X0 x,

where P0 = (Γ Ld )−1 , LD = ωnω X02 /c is the diffraction length, and X0 is the initial beam width. Then eqs. (2.7)–(2.8) take the form 1 iE1z + E1xx + E1∗ E2 = 0, 2 1 1 iE2z + E2xx + E12 − qE2 = 0, 4 2 where q = kLD is the dimensionless mismatch. The total energy ∞ E= |E1 |2 + 2|E2 |2 dx −∞

(2.10)

(2.11)

and the Hamiltonian ∞ 1 1 1 ∗2 2 2 2 ∗ 2 |E1x | + |E2x | − E1 E2 + E1 E2 + q|E2 | dx H= 4 2 −∞ 2 (2.12)

46


[2, § 3

are two conserved quantities. The equations of motion can be written in the form iE1z =

δH , δE1∗

iE2z =

δH . δE2∗

(2.13)

Although the system is not integrable, it has different types of solitonic solutions. A bright soliton solution known for phase mismatch q = −3 has the form E1 (z, x) = 3 sech2 (x)e2iz ,

E2 (z, x) = 3 sech2 (x)e4iz .

(2.14)

This solution will be used in Section 6 in the numerical simulations of the stochastic version of the χ (2) system. This analytical solution holds only for negative mismatch q < 0, whereas the whole family of one-parametric solitons can be found by numerical means. For details we refer to the reviews by Buryak, Di Trapani, Skryabin and Trillo [2002] and Etrich, Lederer, Malomed, Peschel and Peschel [2000]. We can obtain moving solutions E1 = 3 sech2 (x − vz)e2iz+ivx−i E2 = 3 sech (x − vz)e 2

v2 z 2

,

4iz+2ivx−iv 2 z

(2.15)

.

This transformation to moving solutions is valid only for this system, describing spatial χ (2) solitons. When the mismatch is positive and q 1, we obtain from the second equation in the system (2.10) that E2 ≈ E12 /(2q). Substituting this result into the first equation we get the NLSE for E1 : 1 1 |E1 |2 E1 = 0. iE1z + E1xx + 2 2q Thus the solutions in this limit are E1 = 2 2q ν sech(2νx)eiφs , E2 = 4ν 2 sech2 (2νx)e2iφs .

(2.16)

(2.17)

It is seen that the amplitude of the SH soliton is smaller than that of the FW soliton.

§ 3. Solitons in random single-mode fibers Let us consider an optical pulse propagating in a single-mode optical fiber with randomly varying nonlinear and dispersion parameters. The dimensionless envelope of the electric field obeys the modified NLSE iuz +

d(z) utt + c(z)|u|2 u = 0, 2

(3.1)

2, § 3]

Solitons in random single-mode fibers

47

where d(z) = 1 + m(z) and c(z) = 1 + n(z) model the fluctuations of the dispersion and nonlinearity, and m and n are assumed to be zero-mean random processes. We address the propagation of a soliton or a train of solitons governed by eq. (3.1).

3.1. Different approaches At the present time a universal approach to describe pulse propagation in a random medium is absent. For randomly perturbed nonintegrable systems we can use the variational approach. For systems close to integrable a perturbation technique based on the IST can be applied. Expansion of the solution over a basis of eigenfunctions of the unperturbed system is also possible. We review these techniques in this section. 3.1.1. The variational approach Let us consider the propagation of a single soliton in a random optical medium. The Lagrangian density for eq. (3.1) is d(z) c(z) 4 i ∗ u uz − uu∗z − |ut |2 + |u| . (3.2) 2 2 2 The key to what follows is the choice of the trial function for u to substitute into the Lagrangian (Anderson [1983] and Malomed [2002]). As pointed out in different papers, the chirped soliton trial function turns out to give good predictions:

t exp ib(z)t 2 , u(z, t) = A(z) sech (3.3) a(z) L=

where A(z), a(z) and b(z) describe the complex amplitude, the width and the chirp of the soliton, respectively. When substituting this ansatz into the Lagrangian density, the average is ∞ π2 dt L dt = − |A|2 a 3 bz − 2|A|2 a Arg(A) z 6 −∞

2 2 π2 2 3 2 |A| + |A| a b + c(z)|A|4 a. − d(z) 3a 3 3 L ∞ By equating to zero the variations of 0 dz −∞ dt L we obtain the equations for the soliton parameters A, a and b: b(z) =

az , 2ad(z)

(3.4)

48


a|A|2 z = 0,

N 2 = a|A|2 ,

[2, § 3

(3.5)

4d 2 (z) 4N 2 d(z)c(z) az dz (z) (3.6) , − + d(z) π2 a 3 π2 a 2 d(z) 5N 2 c(z) arg(A) z = − 2 + (3.7) . 6a 3a As can be seen from eq. (3.6), the evolution of the soliton width under random perturbations is described by the motion equation of a unit-mass particle in a nonstationary anharmonic potential. We shall see that this analogy opens ways to investigate the soliton dynamics. It is well known that the validity of the variational approach is limited. Its main drawback is the same as its main advantage: inserting a trial function into the Lagrangian dramatically reduces the number of degrees of freedom of the system. This simplifies the problem, but it may lead to wrong predictions as the trajectories of the system are forced to follow the constraints imposed by the choice of the trial function. In the case of the random NLSE the variational approach can be improved by taking a trial function which consists of a soliton-like pulse with variable parameters plus a linear dispersive term taking into account the generation of dispersive radiation upon pulse evolution (Kath and Smith [1995]). In any case the predictions of the variational approach should be confirmed by full numerical simulations of the random NLSE. azz =

3.1.2. Inverse scattering transform We now study the propagation of a single soliton in a random medium by applying a perturbation theory of the inverse scattering transform (IST, see Appendix A). The propagating soliton emits radiation due to scattering with the random medium so the total field can be decomposed as the sum of a localized soliton part uS (associated with the discrete eigenvalue λS = µ + iν) and delocalized radiation (associated with the continuous spectrum). uS has the form uS (z, t) = 2ν sech 2ν t − tS (z) exp iφS (z) , where the amplitude 2ν is slowly varying. The total energy E is preserved by the perturbed equation (3.1) and it can be written as the sum of the soliton energy and the radiative energy: ∞ 2 1 ∞ |u|2 dt = 4ν − ln 1 − b(z, λ) dλ, E := (3.8) π −∞ −∞ where λ is the spectral parameter and b is the Jost coefficient introduced in IST theory (see Appendix A). The space evolution of the Jost coefficient is

2, § 3]


49

bz = −2iλ2 b for the unperturbed NLSE, which shows that the radiative energy, and consequently the soliton energy, are preserved. In the case of the random NLSE (3.1) the evolution of the Jost coefficient can be found from the perturbation theory based on the IST (see eq. (A.10)). Qualitatively, the energy of the radiation increases, which involves a decay of the soliton energy. Quantitatively, in the first approximation the right-hand side of eq. (A.10) can be computed by substituting the corresponding expressions for the soliton part: 2πν[(λ − µ)2 + ν 2 ] bz = −2iλ2 b − exp iφS (z) − 2iλtS (z) cosh[ 12 π(λ − µ)/ν] × m(z) − n(z) . From eq. (3.8) the mean emitted spectral power is P (λ) = 2 Re b∗ bz /π, and we find the evolution equation 1 ∞ P (λ) dλ. νz = − (3.9) 4 −∞ In the white-noise case, (m − n)(z)(m − n)(z ) = Dδ(z − z ), the mean emitted spectral power is P (λ) =

Dπ[(λ − µ)2 + ν 2 ]2 cosh2 [ 12 π(λ − µ)/ν]

.

(3.10)

Substituting into eq. (3.9) and integrating with respect to λ gives a power law for the soliton decay. This decay will be analyzed in detail in the forthcoming sections. 3.1.3. Kaup perturbation technique Motivated by the weakness of the disorder we can write the solution as u = uS + uL , where uS is the soliton (or multi-soliton) part and uL is the radiative part. We can expand the radiative part uL over the eigenfunctions of the operator describing the evolution of a linear perturbation about the single-soliton profile of the unperturbed NLSE. This is possible because the complete system of eigenfunctions was found by Kaup [1990]. In the presence of random perturbations the coefficients of the decomposition of uL are slowly varying, and using the orthogonality properties of the eigenfunctions we can write the evolution equations of these coefficients by a projection technique. Chertkov, Chung, Dyachenko, Gabitov, Kolokolov and Lebedev [2003] applied this method to the study of the impact of random dispersion, and moreover extended the technique for addressing the evolution of a multi-soliton solution.

50


[2, § 3

3.2. Single soliton driven by random perturbations 3.2.1. Random dispersion Random fluctuations of dispersion may occur as shown by Kodama, Maruta and Hasegawa [1994] and Wabnitz, Kodama and Aceves [1995]. A general theory was presented by Lin and Agrawal [2002] to describe the effects of dispersion fluctuations on optical pulses propagating inside single-mode fibers modeled as a linear dispersive medium. Random dispersion has been shown by Abdullaev, Darmanyan, Kobyakov and Lederer [1996] to involve dramatic effects on the modulational instability of stationary waves because of a stochastic parametric resonance phenomenon. In this subsection we shall analyze the stability of the soliton with respect to random fluctuations of the dispersion. Approximate scale characteristics of the dispersion noise present in real fibers can be extracted from experimental results (Mollenauer, Mamyshev and Neubelt [1996], Nakajima, Ohashi and Tateda [1997], Mollenauer and Gripp [1998]). These results show that the typical distance of noticeable change in the dispersion value is shorter than 100 m. (The resolution of the experimental methods varies from 100 m to 1 km, while one expects that the typical scale of the variations is actually 10–100 m, which is the size of the production facility.) For constant-dispersion fibers, the amplifier spacing is of the order of 50 km, and for dispersion-managed fibers, the period of a typical dispersion map is also of the order of 50 km. These scales are much longer than that of the dispersion variation. Therefore, according to the approximation-diffusion theory (Papanicolaou and Kohler [1974]), the natural m at the larger scales can be treated as a homogeneous Gaussian random process with zero mean and delta-correlated correlation function m(z)m(z ) = Dδ(z − z ), where D = m(z)m(z + h) dh. Measurements show that fluctuations of the dispersion coefficient in dispersion-shifted fibers are of the order of its averaged value δβ2 ∼ 0.5 ps2 /km. Therefore, for a pulse duration of ∼7 ps with the nonlinear length znl = 1/κP0 ≈ 250 km, the noise intensity in dimensionless variables is estimated as D ≈ ld d 2 ≈ 10−2√ –10−3 . The scale at which the interacting solitons feel the noise effect is ∼1/ D ≈ 10znl ≈ 2500 km. A soliton propagating through a fiber emits radiation due to disorder and, consequently, loses energy. However, in the case of weak disorder (weak disorder is actually required for successful fiber performance) the destruction of the soliton is slow, thus permitting an adiabatic description of this problem. The adiabaticity implies separation of dynamical degrees of freedom into slow and fast modes. Slow modes describe the evolution of the soliton itself while fast modes correspond to the radiation. The soliton keeps its shape so that, at each instant, the

2, § 3]


51

soliton is close to a stationary solution of the noiseless NLSE, with the soliton parameters (position, width, phase, and phase velocity) evolving slowly. Waves shed by a soliton are moving away from it. One finds that at any z, however large, the radiation in the immediate vicinity of the soliton is much less intense than the soliton itself, i.e., the soliton is always distinguishable from the radiation. Since the soliton energy is converted into radiation, its amplitude 2ν decays with z. The degradation law is deterministic in spite of the original stochastic setting. This is due to the fact that the variation of ν is determined by an integral over z, which is a self-averaged quantity at large z. The soliton degradation law is −1/4 ν(z) = ν0 1 + 128ν04 Dz/15 (3.11) . This formula was derived using the perturbed IST approach by Abdullaev, Caputo and Flytzanis [1994], the variational approach by Abdullaev, Bronski and Papanicolaou [2000], and the Kaup perturbation technique by Chertkov, Chung, Dyachenko, Gabitov, Kolokolov and Lebedev [2003]. It was confirmed by numerical simulations carried out by Chertkov, Gabitov, Lushnikov, Moeser and Toroczkai [2002] and Abdullaev, Navotny and Baizakov [2004]. Equation (3.11) shows that the soliton starts to degrade essentially at z ∼ 1/(Dν04 ) (see fig. 1). 3.2.2. Random nonlinearity Random fluctuations of the nonlinear coefficient can have several origins. The nonlinear index of refraction of the fiber may be varying, but this is not generally the main factor. Imperfections of the geometry of the fiber core may cause variations of the transverse profile of the mode of the fiber, which in turn induce fluctuations of the nonlinear coefficient of the NLSE because the effective core area plays a role in this coefficient (Bauer and Melnikov [1995]). Finally, losses in the fiber are usually compensated by a series of amplifiers. As a result, a z-varying damping term appears in the right-hand side of the NLSE, which is equivalent after a change of variable to a z-varying nonlinear coefficient (Gordon [1992]). The purpose of this section is the study of the propagation of optical solitons in fibers with a randomly modulated nonlinear parameter. In the framework of the variational approach this problem can be reduced to the dynamics of a unit-mass particle in a randomly perturbed central potential. In this case the governing equation for the soliton width coincides with the Kepler problem in a randomly perturbed potential. Using this analogy we obtain a system of equations in action-angle variables which describes the behavior of a chirped pulse in a randomly inhomogeneous fiber. Abdullaev, Abdumalikov and Baizakov [1997], Abdullaev, Bronski and Papanicolaou [2000] studied the behavior of soliton-like pulses in long-distance optical lines on the basis of asymptotic

52


[2, § 3

Fig. 1. Dependence of the soliton amplitude η = 2ν, measured in units of its initial value, on the dimensionless coordinate along the fiber z, for disorder strength D = 0.18. Dashed and solid curves represent theory, eq. (3.11), and numerics for a representative realization of the disorder, respectively. (From Chertkov, Chung, Dyachenko, Gabitov, Kolokolov and Lebedev [2003].)

properties of the soliton width evolution. They derived analytic estimations for the decay length of solitons propagating in randomly inhomogeneous fibers in the case where n is assumed to be a zero-mean Gaussian random function with correlation function n(z)n(z ) = Dδ(z − z ). The formulas thus obtained were confirmed first by numerical simulations, and then also by the perturbed IST. Indeed, there exists a point at which the results obtained by the variational approach and the IST coincide. In the effective particle model this point corresponds to the minimum of the potential well, i.e. the single-soliton initial state. Then the estimation of the decay length can be deduced by the IST and it is found that the soliton amplitude decay has exactly the same rate (3.11) as that one triggered by random dispersion. 3.2.3. A random Kepler problem The variational approach shows that the soliton-width dynamics under random perturbations can be represented as particle dynamics in a random anharmonic

2, § 3]


53

potential. This is useful for studying the oscillations of the soliton width, but also the soliton disintegration distance that can be expressed in terms of the mean exit time of the equivalent particle from the Kepler potential (Abdullaev, Bronski and Papanicolaou [2000]). The resulting equation for the soliton width a is the randomly perturbed Kepler problem azz = −U (a) − γ (z)V (a),

(3.12)

where U (a) is the Kepler potential: 2 4N 2 (3.13) . − π2 a 2 π2 a γ (z) is assumed to be a Gaussian white noise with noise level D, and V is the perturbation potential, which is equal to a 2 for a fluctuating quadratic potential and to 4N 2 /(π2 a) for a fluctuating nonlinearity. The Hamiltonian for the unperturbed Kepler problem is H0 = 12 (az )2 +U (a). The Hamiltonian for the perturbed Kepler problem (3.12) is U (a) =

1 (az )2 + U (a) + γ (z)V . (3.14) 2 In order to continue the analysis of the random Kepler problem, we must first transform the unperturbed Kepler problem into action-angle variables. We summarize here the relevant facts that we need; details are given in the texts by Landau and Lifshitz [1974]. The minimum of the potential U (a) occurs at ac = 1/N 2 , and is equal to U0 = −2N 4 /π2 . The frequency ω0 of small oscillations about the minimum is given by ω0 = 2N 4 /π. This is the frequency of the width oscillations of an unperturbed soliton as it propagates down a homogeneous fiber. For large oscillations the qualitative behavior of the orbits of the Kepler problem is determined by the energy H ≡ H0 + γ (z)V ≡

E0 = 2a02 b02 +

2 4N 2 − 2 , 2 π a0 π2 a0

where b0 is the initial chirp and a0 is the initial width. When E0 < 0 (i.e. 1 + π2 a04 b02 < 2N 2 a0 , corresponding to a sufficiently weak initial chirp) the orbits are closed, corresponding to oscillatory motion. When E0 > 0, corresponding to sufficiently large initial chirp, the orbits are unbounded, and the asymptotic motion is qualitatively like the motion of a free particle. In this regime the soliton does not persist, but instead spreads out and is lost (a → ∞). The interesting

54


[2, § 3

questions in soliton propagation arise when the unperturbed motion is oscillatory, and so we consider the regime E0 < 0. We now change over to action-angle variables. For the Kepler problem, where the phase space is two dimensional, there is one action variable J and the conjugate angle variable Θ. The action variable is given by √ 1 2 2N 2 2 J = − . az da = √ 2π π π2 −E Solving for the total energy E in terms of the action J gives the unperturbed Hamiltonian H0 = E = −

8N 4 , + 2)2

(3.15)

π2 (πJ

which is a function of J only. The change of variables to action-angle variables is canonical, so that Hamilton’s equations retain their form ∂H0 (J ) dJ =− = 0, dz ∂Θ

dΘ ∂H0 (J ) = = ω(J ), dz ∂J

where ω(J ) =

dH0 16N 4 π2 = = (−E)3/2 . √ 2 dJ π(πJ + 2)3 2N

(3.16)

The following implicit representation for the orbits is useful in the perturbation calculations. The position a and “time” z can be expressed parametrically in terms of a variable ξ (called the Kepler parameter) by a = b(1 − e0 cos ξ ),

Θ = ω(J )z = ξ − e0 sin ξ,

(3.17)

where the eccentricity e0 > 0 and the semi-major axis b are given in terms of the constant E as e02 = 1 −

π2 |E| 4 =1− , 2N 4 (πJ + 2)2

b=

(πJ + 2)2 2N 2 = . π2 |E| 4N 2

This parametric representation determines a implicitly as a function of z. Note that as Θ varies over (0, 2π), ξ also varies over (0, 2π). From eq. (3.17) it follows that the soliton width oscillates between the minimum value amin = b(1 − e0 ) and the maximum value amax = b(1 + e0 ). The value of the action J varies from 0 for oscillations near the bottom of the potential well to ∞ for oscillations near the separatrix E = 0. The frequency ω(J ) varies from 2N 4 /π for oscillations near the bottom of the potential well to 0 for oscillations near the separatrix.

2, § 3]


55

It is also useful to relate the initial action J0 to the initial chirp b0 and initial width a0 : πJ0 + 2 =

2N 2 2N 2 a0

−

1 a02

.

(3.18)

− π2 b02 a02

Let us address the case where the coefficients of the NLSE vary randomly. In this case the soliton parameters evolve according to a set of random ODEs, and we can derive a Fokker–Planck equation for the evolution of the probability distribution for the soliton parameters. The perturbations of the fluctuating quadratic potential and of the fluctuating nonlinear term can be analyzed with the white-noise model directly, using the Stratonovich interpretation. In the case of fluctuating dispersion the treatment is slightly more subtle (for details, see Abdullaev, Bronski and Papanicolaou [2000]). In action-angle variables, the perturbed Hamiltonian has the form H = H0 (J ) + γ (z)V (J, Θ),

(3.19)

where V (J, Θ) is obtained from eq. (3.14). In general V , while simple in the original variables, is quite complicated in the action-angle variables. However for weak disorder we do not need the explicit form of V , but only averages over Θ. These averages are simple to calculate using the implicit representation given in eq. (3.17). In the presence of weak disorder the action-angle variables for the unperturbed Kepler problem provide a convenient framework for the analysis of the perturbed problem because the change in the action is proportional to the small parameter. Since the change to action-angle variables is canonical the Hamiltonian structure is preserved, and the perturbed equations become dJ ∂V = −γ (z) , dz ∂Θ (3.20) dΘ ∂V = ω(J ) + γ (z) . dz ∂J The Fokker–Planck equation for the evolution of the probability distribution function P (z, J ) of J (z), after averaging over Θ, is given by 2 ∂P 1 2π ∂ ∂P A(J ) , A(J ) = VΘ (J, Θ) dΘ, = (3.21) ∂z ∂J ∂J 2π 0 with P (0, J ) = δ(J − J0 ). We note that this averaging calculation breaks down very near the separatrix, since ω(J ) vanishes there, and the perturbations can no longer be considered small.

56


[2, § 3

The mean exit time (distance) to reach ∞ starting from J0 is ∞ J dJ τ= . A(J ) J0 For the fluctuating nonlinearity, the perturbation of the Hamiltonian is V = (4N 2 )/[π2 a(J, θ )] and, starting from a single-state soliton state J = 0, the mean disintegration distance is proportional to a04 /D ∼ 1/(Dν04 ). The same result holds true for the fluctuating dispersion, and different quantitative estimates of the soliton disintegration distance have been provided by Abdullaev, Bronski and Papanicolaou [2000].

3.3. Interaction of solitons in random medium 3.3.1. Interaction induced by radiation shedding Using the Kaup perturbation technique Chertkov, Chung, Dyachenko, Gabitov, Kolokolov and Lebedev [2003] studied the interaction of solitons when successive solitons are far from each other so that the interaction through the tails can be considered as negligible. Radiation emitted by a soliton acts as multiplicative noise on another soliton. The interaction is extremely long-range, due to the onedimensional nature of the system and also because of the reflectionless feature of the radiation. At any z all solitons separated from a given one by |t| z act on this soliton with a force that is zero on average. Fluctuations of the force result in a Gaussian jitter of the soliton position. It was found that in the two-soliton case, i.e. for a pattern consisting of only two solitons, so that no other solitons are present anywhere in the |t| z vicinity of the pair, the fluctuation in their relative position δy is a zero-mean Gaussian random variable with variance 2 δy 1.5ν08 1 + cos(2φ) D 2 z3 , (3.22) where φ is the intersoliton phase mismatch (see fig. 2). In the general multisoliton case, intersoliton interaction caused by radiation leads to an essential shift of the solitons at the distance z ∼ N −1/3 D −2/3 , where N is the number of solitons in the channel. The interaction causes the soliton to jitter randomly. The soliton displacement δy is a zero-mean Gaussian random variable with variance

δy 2 ∼ D 2 z3 N. If N does not grow with z (e.g., there is only a finite number of solitons propagating in the channel) the z-dependence of the jitter is the same as that given by the Elgin–Gordon–Haus jitter (Elgin [1985], Gordon and Haus [1986], Falkovich, Kolokolov, Lebedev and Turitsyn [2001]) developed under the action of random additive noise (amplifier noise in the system that is short-

2, § 3]


57

Fig. 2. Dependence of the mean square value of the intersoliton separation δy 2 , measured in units of the soliton width squared, on the dimensionless position along the fiber z, for disorder strength D = 0.00125. Three different sets of curves for three different values of the intersoliton phase mismatch, φ = 0, 14 π, 12 π, are presented. Dashed curves represent the analytical result, eq. (3.22). Solid curves represent results of numerics. Each curve is the result of averaging over 15 different realizations of disorder. (From Chertkov, Chung, Dyachenko, Gabitov, Kolokolov and Lebedev [2003].)

correlated in both t and z). However, if the flow of information is continuous, i.e., if the front of radiation shed by the given soliton sweeps more and more solitons with increasing z, N ∼ z, the efficiency of the interaction grows with z in a faster way, δy ∼ z2 , thus overwhelming the Elgin–Gordon–Haus jitter in long-haul transmission. 3.3.2. Interaction induced by tail overlap Using the perturbation theory for soliton interaction, Karpman and Solov’ev [1981] studied the interaction of solitons induced by overlap of their tails. The soliton separation should be sufficiently large for the interaction to be weak so that it can be treated by expansion methods, but it should not be so large that the interaction with the induced radiation dominates the interaction through the tails. The analysis performed by Abdullaev, Hensen, Bischoff, Sorensen and Smeltink [1998] for a two-soliton configuration with a random nonlinearity exhibits complicated phenomena. When the solitons have the same phase and amplitude 2ν, the same result as in the unperturbed NLSE is found, that is, solitons expe-

58


[2, § 3

rience small oscillations with period ∼ exp(−2νtS ), where tS is the initial soliton separation. If the solitons do not have the same amplitude initially, the mean square of the amplitude difference ν grows diffusively during propaga√ tion, ν − ν0 ∼ ν0 exp(−(ν1 + ν2 )tS ) Dz. This asymmetry in the two soliton shapes in media with random fluctuations has also been observed numerically.

3.4. Beyond the white-noise model 3.4.1. Pinning schemes Periodic pinning of disorder was suggested recently by Chertkov, Gabitov, Lushnikov, Moeser and Toroczkai [2002] as a way to compensate for the random dispersion. This method comes in two modifications, of distributed and point pinning. Distributed pinning applies to new fiber lines (not yet laid in the ground). The method requires controlling the (fluctuating part of the) integral dispersion of a fiber piece prior to its connection to the line. First, the profile of the integral of the fluctuating part of the dispersion coefficient should be established, and then the suggestion is to cut this fiber at the zero of this fluctuating part closest to the end of the fiber piece. The other type of pinning, point pinning, was suggested for implementation in already installed fiber-optics lines. At the points of access to the fiber optics line (e.g., at amplifier stations placed periodically along the fiber) it is suggested to measure the integral of the fluctuating part of the dispersion, and then to compensate it by inserting a small piece of fiber with a very well controlled integral dispersion. If the pinning period, l, is short (i.e., if it is the shortest scale in the problem), the noise term m can be replaced by m ˜ described by m(z) ˜ m(z ˜ ) = −(Dl 2 /12)δ (z − z ). (This m ˜ actually corresponds to the case of “distributed pinning”, while in the case of “point pinning” the replacement should be m ˜ → 2m.) ˜ Recalculation of all the major results for the pinned noise is straightforward. First of all, one gets the decay rate of a single soliton: −1/8 ν(z) = ν0 1 + 214 ν08 Dl 2 z/315 .

(3.23)

This expression, when contrasted with eq. (3.11), shows an essential reduction in the soliton decay since the critical propagation distance becomes z ∼ 1/[(ν04 D)(ν02 l)2 ] 1/(ν04 D). Second, in the multi-soliton case, the radiation√ mediated jitter can be estimated by δy ∼ N z, which becomes less important asymptotically than the Elgin–Gordon–Haus jitter.

2, § 3]


59

3.4.2. Colored noise It may happen that the correlation length of the fluctuations of the medium parameters is not the smallest length scale present in the problem. A more careful analysis taking into account the noise spectrum is then necessary. Let us consider a random dispersion or nonlinearity with autocorrelation function

D |z − z | . m(z)m(z ) = exp − 2lc lc The mean emitted spectral power is then (Abdullaev, Hensen, Bischoff, Sorensen and Smeltink [1998]) P (λ) =

Dπ[(λ − µ)2 + ν 2 ]2 {1 + 16lc2 [(λ − µ)2 + ν 2 ]2 } cosh2 ( π2 λ−µ ν )

.

(3.24)

When 4ν 2 lc 1 we recover the white-noise model. When 4ν 2 lc 1 we get, by integrating eq. (3.9), that the decay rate of the soliton amplitude is Dz ν(z) = ν0 exp − 2 , 16lc which shows that increasing the correlation length leads to less damping. The difference between radiative emission in media with white-noise fluctuations and media with colored-noise fluctuations can be explained as follows. In a periodically perturbed fiber there exists a resonance between the characteristic frequencies of the soliton and the modulation frequency of the medium, leading to the emission of waves generated by the soliton (Hasegawa and Kodama [1991]). The associated resonance condition is 2nπ/L = 4ν 2 , n ∈ N, where L is the modulation period. When the soliton amplitude decays, the soliton parameters detune from resonance, leading to stabilization of the soliton. In the case of a white-noise perturbation, all frequencies are present in the spectrum. Therefore the soliton cannot detune from resonance and it is continuously damped. In the case of a colored-noise perturbation, the spectrum has a cut-off frequency corresponding to the correlation length lc . If the soliton phase velocity 2ν 2 is smaller than lc , the soliton is detuned from resonance. 3.4.3. Dissipation and filters The influence of dissipation and amplification on soliton propagation under random perturbations can be analyzed. Malomed [1996] considered the effects of weak amplification and filtering. The unchirped autosoliton under noise has been studied by Hasegawa and Kodama [1995]. The governing equation in presence of

60


weak amplification, filtering, and noise is 1 1 2 iuz + d(z)utt + c(z)|u| u = iΓ0 u + utt . 2 3

[2, § 3

(3.25)

When d is constant there exists a chirped autosoliton solution. The evolution of the width of the chirped soliton is described by the random Kepler problem with a damping term proportional to az , which in turn implies adding to the right-hand side of eq. (3.20) a linear damping term equal to −Γ J , Γ 4.3Γ0 (Abdullaev, Bronski and Papanicolaou [2000]). The result is the following expression for the mean exit time (distance): J ∞ J 1 Γz dz dy dJ. exp τ= J0 A(J ) 0 y A(z) This expression shows that the exit time is increasing with Γ , which means that the interplay of noise and dissipation can lead to stabilization of the soliton. This is also a confirmation of the stabilizing role of filters. The existence of an autosoliton in random media can be shown by the IST approach. In a fiber with fluctuating dispersion the soliton radiative decay can be compensated by the linear/nonlinear amplifiers. The action of the distributed gain can be described by the terms iδl u + iδnl |u|2 u in the right-hand side of eq. (3.1). A distinctive property of the solitons created is that they recover the stable waveform when deformed. Numerical simulations show that linear amplification gives rise to more pronounced oscillations around the fixed point during the transient period compared to nonlinear amplification. At longer propagation distances the oscillations decay and a stable dissipative soliton is formed. The amplitude of the autosoliton at the fixed point νst can be found from the balance equation for the soliton energy, ∞ dES 64 =− (3.26) P (λ) dλ + 8δl ν + δnl ν 3 . dz 3 −∞ Taking the right-hand side equal to zero we get 5δnl 3δl D 2 ν1,2st = . 1± 1+ 2 4D 5δnl

(3.27)

Two types of fixed points exist. The first type ν1st is defined by the competition between the dissipation (induced by the randomness of the fiber dispersion) and the linear/nonlinear amplifications. For δl = 0, δnl > 0 (nonlinear ampli2 = (5δ )/(2D), and similarly for δ fication dominating) we find ν1st nl nl = 0, 2 = √(15δ )/(8D). The second δl > 0 (linear amplification dominating) ν1st l

2, § 3]


61

type of fixed points is defined by the competition between the linear dissipation δl < 0 and nonlinear amplification δnl > 0. For small D we can expand 2 ≈ (3|δ |)/(8δ ) + D(9δ 2 )/(160δ 3 ), which shows that the fluctuations inν2st l nl l nl crease the autosoliton amplitude in this case. Simulations of the stochastic NLSE with nonconservative terms have been performed by Abdullaev, Navotny and Baizakov [2004], and they confirm the theoretical predictions. However, it should be pointed out that for the case of linear amplification, stable pulse propagation is limited by the growth of the zero mode under amplification. When the initial pulse amplitude is close to this solution the instability growth becomes noticeable at distances z ∼ 1/δnl (see, for instance, Akhmediev and Ankiewicz [2000]).

3.5. Femtosecond solitons in random fibers The preceding analysis based on the randomly perturbed NLSE shows that the influence of random dispersion is very strong for very short pulses. For solitons with durations < 100 fs it is necessary to consider a modified NLSE, taking into account the finite response time of the Kerr nonlinearity, the Raman effect, and third- and higher-order dispersion. Below we consider a simplified model that takes into account only the dispersion, nonlinearity, and self-steepening. The governing equation is 1 iuz + utt + |u|2 u + iα |u|2 u t = ε(z, t)R(u), (3.28) 2 where ε(z, t) is a random process. When ε = 0 this equation is integrable for any α by means of the IST and the spectral information can be processed to study the soliton dynamics. It is important to notice that the additional term in α can be large, which allows us to work beyond perturbation theory. A nontrivial example of application is the soliton jitter driven by distributed amplifier noise (Doktorov and Kuten [2001]). In distinction from the Elgin–Gordon–Haus (EGH) effect which is based on the standard NLSE for picosecond pulses, the jitter can be suppressed by a proper choice of the pulse and fiber parameters. The distributed amplifier noise is represented by the additive white-noise source in the right-hand side of eq. (3.28), εR(u) = s(z, t), with autocorrelation function

s ∗ (z, t)s(z , t ) = Dδ(z − z )δ(t − t ). The soliton solution of eq. (3.28) with s = 0 is u(z, t) =

i ke−x + k ∗ ex iψ e , w (kex + k ∗ e−x )2

(3.29)

62


[2, § 4

where x = q(z) − t/w, ψ = vt + φ(z), q(z) = (t0 + vz)/w, and φ(z) = φ0 − 12 (v 2 − 1/w 2 )z. Here t0 and φ0 determine the initial position and phase of the soliton, and v and w are the soliton velocity and width: 1 2(k 2 + k ∗ 2 ) iα (3.30) , k = ξ − iη. − , w=− 2 α α 2(k − k ∗ 2 ) The integrals of motion behave quite differently from the standard NLSE case. For example, the soliton power is 1 4γ , γ = Arg k ∗ , 0 < γ < π. E = |u|2 dt = (3.31) α 2 We recover the standard IST for the NLS equation by taking the limit α → 0, k ≈ 12 − 12 αλnlse , λnlse = µnlse + iνnlse , so v → 2µnlse , w → 1/(2νnlse ). Using the perturbation theory for the IST of the modified NLSE, Doktorov and Kuten [2001] found the expression of the soliton jitter: v=

Dz3 F (α, v, γ ). q(z)2 = (3.32) 9 Other contributions exist, but those are proportional to z so they can be neglected for large propagation distances. The EGH theory gives the same formula for the jitter with F = 1. If v < 1/α, then γ can be chosen so that F < 1. So by the proper choice of parameters the EGH jitter can be essentially reduced. Many problems for ultrashort pulses on the base of eq. (3.28) are still unsolved. One of them is the radiation of a femtosecond soliton under fluctuations of dispersion. We should note that the interesting effects in this model are observed for nonsmall values of α. But in this case the validity of the model is questionable, since it is necessary to take into account higher-order dispersion of nonlinearity as well.

§ 4. Dispersion-managed solitons under random perturbations The modified NLSE has the form 1 iuz + d(z)utt + c(z)|u|2 u = 0, (4.1) 2 where d and c possess periodic modulations dp and cp with period L (the socalled dispersion map) as well as random fluctuations modeled by zero-mean random processes dr and cr : d(z) = dp (z) + dr (z),

c(z) = cp (z) + cr (z).

The amplitudes of the periodic modulations can be large, while we consider smallamplitude random fluctuations.

2, § 4]

Dispersion-managed solitons under random perturbations

63

4.1. Periodic and random dispersion modulations In dimensional units, the optical pulse propagation in a system with varying dispersion is governed by the NLSE (2.2): 1 ivZ − β2 (Z)vT T + κ|v|2 v = 0, 2 where Z is the propagation distance (in km), T is the time in the frame moving with the group velocity (in ps), P = |v|2 is the optical power (in W), and β2 is the group-velocity dispersion (GVD) coefficient (in ps2 /km). The coefficient β2 is related to the usual dispersion parameter D by β2 = −λ20 D/(2πc), where c = 0.3 mm/ps is the speed of light, λ0 is the carrier wavelength (in µm), and D is measured in ps/(nm km). The pulse energy Epulse = |v|2 (Z, T ) dT is independent of Z. The carrier wavelength is λ0 = 1.55 µm for telecommunication applications. The nonlinear coefficient κ is given by κ = 2πn2 /(λ0 Aeff ) (in W−1 m−1 ), where n2 is the nonlinear refractive index (n2 ≈ 3 × 10−2 nm2 /W in glass) and Aeff is the effective fiber area (in µm2 ). Typically Aeff ≈ 60 µm2 (Agrawal [1995]). We then introduce the typical nonlinear length Z0 := 1/(κP0 ), where P0 is the typical pulse power. In most practical applications P0 equals a few milliwatts. Say P0 = 2 × 10−3 W, so that Z0 = 250 km. We normalize the coordinate along √ the fiber z = Z/Z0 and the envelope of the electric field u = E/ P0 . We also normalize the time t = T /T0 , where T0 is the typical pulse width, say T0 = 5 ps, so that the propagation is governed by the dimensionless NLSE (4.1) with c = 1 and d(z) =

λ20 D(Z0 z)Z0 −β2 (Z0 z)Z0 = . T02 2πcT02

Let us first describe the periodic dispersion management. We assume that the fiber consists of a periodic concatenation of segments of alternately normal and anomalous fibers, so that the physical dispersion management is of the type   D+ if Z mod Lmap ∈ [0, L+ /2), D(Z) = D− if Z mod Lmap ∈ [L+ /2, Lmap − L+ /2),  D+ if Z mod Lmap ∈ [Lmap − L+ /2, Lmap ). The map period is Lmap = L− + L+ (see fig. 3). The multi-scale analysis that will be used in the following sections is based on the separation of the scales Lmap Z0 . In dimensionless units the GVD coefficient dp is of the form

64


[2, § 4

Fig. 3. Two-step dispersion map.

dp (z) = d˜p (z) + d0 where   d+ if z mod lmap ∈ [0, l+ /2), d˜p (z) = d− if z mod lmap ∈ [l+ /2, lmap − l+ /2),  d+ if z mod lmap ∈ [lmap − l+ /2, lmap ), lmap = Lmap /Z0 , l± = L± /Z0 , d± = λ20 (D± − Dm )Z0 /(2πcT02 ), and d0 = λ20 Dm Z0 /(2πcT02 ) is the residual dispersion. Typical values are |D± | = 2–20 ps/(nm km), L± = 20–200 km, and D0 = 0–0.1 ps/(nm km). In dimensionless units, |d± | = 20–200, l± = 0.1–1, and d0 = 0–1. The so-called dispersion management (DM) strength is DL := d+ l+ = −d− l− . In this section we address the influence of small random dispersive fluctuations on the pulse propagation in DM systems. Considering the same type of fluctuations as in Section 3.2.1, we have the white-noise model dr (z)dr (z ) = Dδ(z − z ) with D ≈ 10−2 –10−3 . 4.2. Different approaches A reduction of the NLSE to simpler equations is important for two reasons. First, the generation of reliable statistics quantifying the effect of random coefficients on pulse evolution requires a large number of simulations. This is particularly true if the most relevant occurrences are exactly those that are least likely to occur in a straightforward Monte Carlo approach, such as bit errors. It is therefore of obvious benefit to reduce the computational time for each simulation to produce meaningful results in a minimum of time. The second argument in favor of finding a reduction of the NLSE is that it is often difficult to obtain deep insight from a partial differential equation. A reduction to a finite-dimensional system is generally more conducive to obtaining an analytical description of the pulse behavior. The

2, § 4]


65

resulting ordinary differential equations can be linearized about the dispersionmanaged soliton for approximations valid over short distances, or they can be averaged over the dispersion map period to capture the behavior over longer distance scales (Turitsyn, Aceves, Jones and Zharnitsky [1998]). In this subsection we present a review of different methods used to reduce the NLSE. The first two approaches consist in reducing the partial differential equation to a finite-dimensional problem by deriving a set of ordinary differential equations for the pulse parameters. The last two approaches derive effective partial differential equations after averaging of the NLSE over one map. 4.2.1. The variational approach Employing the variational approach, the underlying NLSE can be reduced to a system of ordinary differential equations for the DM soliton parameters. The trial localized waveform is taken as

b(z) t 2 t exp i , u(z, t) = A(z)Q a(z) a(z) 2

(4.2)

where A, a and b/(2a) are the complex amplitude, width, and chirp, respectively, and Q is the localized function specifying the pulse profile. Substituting the ansatz (4.2) into the average Lagrangian (4.1) and equating its variations to zero, we arrive at the following variational equations: az = d(z)b,

bz =

C1 d(z) 2C2 Ec(z) − , a3 a2

(4.3)

where the conserved quantity is the pulse energy E = |u|2 dt = C0 |A|2 (z)a(z), and the constants Cj depend only on the shape function Q: C0 =

|Q| ds, 2

|Qs |2 ds C1 = 2 2 , s |Q| ds

|Q|4 ds C2 = . 4( s 2 |Q|2 ds)( |Q|2 ds)

For the Gaussian approximation of the pulse function Q(s) = exp(−s 2 ) we have √ √ C0 = π/2, C1 = 4, and C2 = 1/ π (see Turitsyn, Gabitov, Laedke, Mezentsev, Musher, Shapiro, Schäfer and Spatschek [1998]).

66


[2, § 4

4.2.2. Moment equations Following Turitsyn, Aceves, Jones and Zharnitsky [1998] we introduce the rms pulse width and the rms chirp of the pulse as: 2 2 1/2 t |u| dt Mrms i t (uu∗t − u∗ ut ) dt . Trms = , = Trms 4 |u|2 dt t 2 |u|2 dt It is easy to check that the evolutions of Trms and Rrms are (Trms )z = 2d(z)Mrms , where

|ut |2 dt Ωrms (z) = 2 , |u| dt

(Trms Mrms )z =

d(z) c(z) Ωrms (z) − Prms , 2 4

4 |u| dt Prms (z) = 2 . |u| dt

We thus get exact but nonclosed equations. To close this system, we can: (1) assume that the pulse has the self-similar structure (4.2). It is then possible to derive a closed-form system of ordinary differential equations for Trms and Mrms , or alternatively a and b since both pairs are related to each other through the identities a(z)/a(0) = Trms (z)/Trms (0) and b(z)/a(z) = Mrms (z)/Trms (z). The system so derived is the same as that obtained with the variational approach. (2) Write the differential equations satisfied by Ωrms and Prms and make use of higher-order momentum equations. It is not possible to close the equations, but Turitsyn [1998] has succeeded in finding a five-dimensional system with five momenta (including Trms , Mrms , Ωrms and Prms ) that can be closed by using only one assumption (a parabolic approximation of the phase near the location of the pulse peak power). After integration, it turns out that this system is identical to the system (4.3). This theoretical analysis and subsequent numerical simulations show that the system (4.3) describes the DM soliton dynamics with good accuracy. 4.2.3. Path-averaged theory in the time domain This method aims at estimating the deviations of a true DM soliton from the selfsimilar structure assumed in the variational or moment approaches. The large variation of the dispersion coefficient within one map does not allow a direct averaging of eq. (4.1). The main idea is to first use a transformation that accounts for the fast pulse dynamics and then to apply an averaging procedure to the transformed equation. Let us make the following transformation:

bp (z) t 2 N t u(z, t) = (4.4) exp i . Q z, ap (z) ap (z) 2 ap (z)

2, § 4]


67

The rapid oscillations of pulse width and chirp are accounted for by the functions ap and bp satisfying (ap )z = dp (z)bp ,

(bp )z =

dp (z) 2N 2 cp (z) − , ap3 ap2

(4.5)

where N is a constant to be determined by the requirement that ap and bp be periodic functions. Note that we only take the periodic components of the dispersion and nonlinear coefficients in the system (4.5). Substituting eqs. (4.4) and (4.5) into eq. (4.1) gives the modified NLSE cp N 2 2 dp 2 |Q| + x 2 Q Q − x Q + xx 2 ap (z) 2ap

2

2 bp 2 ibp ibp N 1 = dr Q− xQx − cr |Q|2 Q . x Q − 2 Qxx − 2 2ap ap ap 2ap (4.6) We can expand the function Q(z, x) over the system of Gauss–Hermite functions Q(z, x) = n qn (z)fn (x). Multiplying eq. (4.6) by fn and integrating with respect to x, we obtain a system of coupled nonlinear ordinary differential equations for the coefficients qn of the form iQz +

(qn )z +

dp cp (z)Fn (q) + (z)Gn (q) = dr (z)Hn (q, z) + cr (z)In (q, z), ap ap2 (4.7)

where dp /ap2 (z), cp /ap (z), Hn (q, z) and In (q, z) are z-periodic functions. In the absence of random perturbations it is possible to average this system over a map and to exhibit a stationary solution which is a pure DM soliton (Turitsyn, Schäfer, Spatschek and Mezentsev [1999]). In the presence of perturbations, we can apply an adiabatic approach and write the slow evolutions of the coefficients qn by averaging the right-hand side of eq. (4.7) over a map. Note that, in practice, the system (4.7) is simplified by neglecting modes n n0 for a given n0 . If only the highest mode of this Gauss–Hermite expansion is considered, the resulting ordinary differential equations are identical to those found previously by the variational approach or the moment approach (Schafer, Mezentsev, Spatschek and Turitsyn [2001]). 4.2.4. Path-averaged theory in the frequency domain If we neglect the nonlinear term in eq. (4.1) we get that, in the spectral domain, the pulse phase experiences rapid oscillations during a map period. Nonlinear effects can be accounted for as a modification of these quasi-linear dynamics, and

68


[2, § 4

the parameters of the quasi-linear solution play the role of adiabatic invariants of the system. For simplicity we consider a version of eq. (4.1) where the nonlinear coefficient is constant, c(z) ≡ c0 . We expand the periodic component of the dispersion coefficient as dp (z) = d0 + d˜p (z), where d˜p has zero mean. We introduce the accumulated dispersion function associated with the zero-mean periodic component and the (statistically) zero-mean random component dr : z D(z) = d˜p (ζ ) + dr (ζ ) dζ. 0

Chertkov, Gabitov, Lushnikov, Moeser and Toroczkai [2002] applied a Fourierlike transform taking into account the zero-mean dispersion fluctuations 1 u(z, t) = u(z, ˆ ω) exp −iωt − iω2 D(z) dω. 2π The resulting equation can be averaged by splitting uˆ into a small-amplitude rapidly varying component and a large-amplitude slowly varying component vˆ which satisfies an integro-differential equation: ivˆz − ω2 d0 vˆ + c0 Lvˆ = 0, Lvˆ = δ(ω1 + ω2 − ω − ω3 ) exp −iD(z)

(4.8)

ˆ 2 )v(ω ˆ 3 )∗ dω1 dω2 dω3 , × v(ω ˆ 1 )v(ω where = ω12 + ω22 − ω2 − ω32 . In the absence of random fluctuations this pathaveraged equation was first obtained by Gabitov and Turitsyn [1996] and Gabitov, Shapiro and Turitsyn [1996].

4.3. Random dispersion 4.3.1. Pulse broadening induced by random dispersion Initially, particular configurations were addressed that led to simplified problems. Abdullaev and Baizakov [2000] assumed weak DM strength so that the averaged dynamics approach developed by Kutz, Holmes, Evangelidis and Gordon [1998] is valid. In this approach a real map with fiber segments of alternating anomalous and normal dispersion is replaced by a uniform fiber with path-averaged dispersion dav . The problem can be reduced to a randomly perturbed NLSE, and using the same approach as in Section 3.2.3 it can be considered as a particular case of the random Kepler problem in the context of optical solitons (Abdullaev and Caputo [1998]). An explicit analytical expression is derived for the expected distance that a soliton propagates along a fiber with randomly varying dispersion

2, § 4]


69

Fig. 4. Disintegration of a soliton propagating in a DM line with randomly varying dispersion magnitudes of spans. The dispersion map corresponds to the concatenation of two segments of different fibers with dispersion D± and lengths L± . The map parameters are D+ = 2.446 ps/(nm km), D− = 2.258 ps/(nm km), and L+ = L− = 50 km. The initial pulse width is 10 ps. In dimensionless units the parameters are d+ = 26, d− = −24, l+ = l− = 0.14 and the initial pulse is u(0, t) = 2 = 0.0852 . 2.45 exp(−0.427t 2 ). The magnitudes of the spans are randomly varying with δd 2 /d±

before it disintegrates: Ldis = a04 /D, where a0 is the initial pulse width. Numerical simulations also show that magnitude fluctuations of dispersion of spans have more dramatic effects on DM solitons than do length fluctuations of spans (see fig. 4). This observation has been confirmed by Xie, Mollenauer and Mamysheva [2003]. Abdullaev and Navotny [2002] developed a mean field theory that is valid for very weak nonlinearity, as the averaging |u|2 u = | u|2 u is performed, so that the interplay between random dispersion and nonlinearity is not perfectly captured. In a white-noise model for the dispersion fluctuations the effect of the dispersion fluctuations can be described within the framework of a modified nonlinear Schrödinger equation with a frequency-dependent damping term (∼iutttt ). The presence of randomly modulated dispersion leads to the damping of optical pulses. A second generation of work consists in reducing the problem by using one of the approaches developed in Section 4.2, and then to carry out numerical simulations of the reduced problems. Malomed and Berntson [2001] considered a model of a long optical communication line consisting of alternating segments with anomalous and normal dispersion, whose lengths are picked randomly from a certain interval. As the first stage of the analysis, small changes in parameters of a quasi-Gaussian pulse passing a double-segment cell are calculated by means

70


[2, § 4

of the variational approach. The evolution of the pulse passing many cells is approximated by smoothed ordinary differential equations with random coefficients, which are solved numerically. Simulations reveal slow long-scale dynamics of the pulse, frequently in the form of long-period oscillations of its width. It is found that the soliton is most stable in the case of zero path-average dispersion (PAD), less stable in the case of anomalous PAD, and least stable in the case of normal PAD. The soliton’s stability also strongly depends on its energy, a low-energy soliton being much more robust than its high-energy counterpart. This result has been confirmed by numerical simulations performed by Poutrina and Agrawal [2003]. Chertkov, Gabitov, Lushnikov, Moeser and Toroczkai [2002] used the approach developed in Section 4.2.4 in the white-noise case dr (z)dr (z ) = Dδ(z − z ). They show that the solution is not a self-averaging quantity so that a deterministic path-averaged approach is not applicable. They also predict that the length of the pulse destruction is Ldis ∼ a04 /D, and they perform full numerical simulations of the path-averaged equation (4.8) to confirm the theoretical predictions. These results based on numerical simulations have been confirmed by further work addressing the problem from a full theoretical point of view (Garnier [2002], Schafer, Moore and Jones [2002]). For short distances, a quasi-linear approach in the form of a simple perturbation expansion is sufficient to capture the broadening for each realization of the random dispersion. For intermediate distances, over which the noise and nonlinearity interact, the partial differential equation can be reduced to a relatively simple system of nonlinear stochastic ordinary differential equations. Finally, over long distances, the slow evolution of the pulse width can be obtained by applying an appropriate multiple-scales averaging procedure to yield a new, scaled noise process effecting pulse broadening. The main features exhibited by this analysis are the following: A low-energy soliton is more robust than a high-energy soliton with equivalent characteristics; the soliton robustness is also enhanced by strong dispersion management. More quantitatively, the impact of the random dispersion is proportional to the strength of disorder D and inversely proportional to the fourth power of the pulse width a0 . The mean pulse broadening grows like Dz/a04 , while the standard deviation of the pulse broadening grows like (Dz/a04 )1/2 . Therefore the noise in dispersion presents a potential source of serious limitations for the next generation of highspeed communication. 4.3.2. Separation-of-scales technique We present in more detail the results obtained by the application of the variational or moment approach and the technique of separation of scales which is valid for

2, § 4]


71

z strong DM. We write D˜ p (z) = 0 d˜p (s) ds (which is a periodic function with period lmap ) and introduce the periodic orbits

A(a0 , b0 , z) = a02 + 2a0 b0 D˜ p (z) + b02 + 4a0−2 D˜ p (z)2 , B(a0 , b0 , z) =

a0 b0 + (b02 + 4a0−2 )D˜ p (z) a02 + 2a0 b0 D˜ p (z) + (b02 + 4a0−2 )D˜ p (z)2

.

If c = 0, d0 = 0, and dr ≡ 0, then a(z) = A(z) and b(z) = B(z) are the solutions of system (4.3) starting from (a0 , b0 ). If the variations of the coefficients a and b over a period lmap characterized by the nonlinearity c, the residual dispersion d0 and the random dispersion dr are small, then (1) the short-scale dynamics of the parameters (a, b) are driven by d˜p and follow the periodic orbits (A, B); (2) the long-scale dynamics of the parameters (a, b) are driven by an effective system where the fast periodic oscillations have been averaged. The derivation of this effective system has been performed by Garnier [2002]: ¯ ¯ a(z) = A a(z), ¯ b(z), z , b(z) = B a(z), ¯ b(z), z , (4.9) ¯ obeys the effective system where (a, ¯ b) da¯ ¯ + d0 + dr (z) b, ¯ = f (a, ¯ b) dz db¯ ¯ + 4 d0 + dr (z) a¯ −3 , = g(a, ¯ b) dz and the explicit expressions of the functions f and g are f (a, b) =

g(a, b) =

(4.10)

2CE a 3 b 4CE a 3 ln ψ(a, b) + DL (4 + a 2 b2 )3/2 DL (4 + a 2 b2 ) a + bDL a − bDL , −

×

(2a − bDL )2 a 2 + 4DL2 (2a + bDL )2 a 2 + 4DL2 8CE 4CE ln ψ(a, b) + 2 2 3/2 DL (4 + a b ) DL (4 + a 2 b2 ) ba 3 − 4DL ba 3 + 4DL ×

, −

(2a + bDL )2 a 2 + 4DL2 (2a − bDL )2 a 2 + 4DL2

√ 4 + a 2 b2 (2a + bDL )2 a 2 + 4DL2 + DL (4 + a 2 b2 )

ψ(a, b) = . √ 2a 3 b + 4 + a 2 b2 (2a − bDL )2 a 2 + 4DL2 − DL (4 + a 2 b2 ) 2a 3 b +

72


[2, § 4

√ Here CE = E/ π and DL = d+ l+ = −d− l− is the DM strength. The pair ¯ is a diffusion Markov process whose probability density function satisfies (a, ¯ b) a Fokker–Planck equation. These equations can only be integrated numerically due to the complexity of the functions f and g, but expansions for small or large DM strengths can be performed. Also, a linear stability analysis a = as + a1 can be applied around the stable point (a = as , b = 0) that corresponds to a DM soliton solution of eq. (4.10) with dr = 0. All moments have been computed by Garnier [2002], and we report the expression for the second moment of the pulse broadening: √ 2 sin(2 γ z) D∂b f , z− a1 = 8 6 √ 2 γ as |∂a g| where γ = −∂b f ∂a g > 0 and the partial derivatives are computed at the stable point (a = as , b = 0) (see Garnier [2002] for closed-form formulas). Comparisons of these theoretical predictions with full numerical simulations of the random NLSE are shown in fig. 5.

Fig. 5. Pulse broadening in dimensionless units. The normalized first, second and fourth moments of the pulse-width increments are plotted, DTj = (Trms (z) − Trms0 )j 1/j , j = 1, 2, 4. The solid lines represent theoretical values, the dotted lines represent the numerical values averaged over 104 realizations. Here d+ = −d− = 100, d0 = 1, l+ = l− = 0.1, D = 0.033, Trms0 = 1.68.

2, § 5]

Randomly birefringent fibers

73

4.4. Pinning schemes The pinning method for a DM system consists of periodic compensation of accumulated fiber dispersion by insertion of an additional piece of fiber with a wellcontrolled length and dispersion value. All components required for implementation of this method, including measurements of accumulated dispersion of a fiber span, are standard and well established in optical fiber communications. The pinning method can be implemented both for the upgrading of existing links and for the production of new optical cables. The pinning method is found to prevent pulse deterioration and is capable of improving the performance of high-speed optical fiber links (Chertkov, Gabitov, Lushnikov, Moeser and Toroczkai [2002]). The effect is even more dramatic than in the case of the standard soliton. Indeed, for a pinning period below some critical value, one observes a tendency toward statistically steady behavior: the average pulse width does not decay and the probability density function of the pulse width and amplitude does not change shape with z. This critical period is increasing with the noise level D. These theoretical findings are verified by direct numerical simulation, where no emission of radiation by the DM soliton can be observed. An independent set of numerical simulations performed by Xie, Mollenauer and Mamysheva [2003] has recently confirmed these results. Thus, the pinning method is effective in optical fibers with and without dispersion management.

§ 5. Randomly birefringent fibers Experiments have shown that polarization mode dispersion (PMD) is one of the main limitations on fiber transmission links (Gisin, Pellaux and Von der Weid [1991]). PMD has its origin in the birefringence, i.e. the fact that the electric field is a vector field and the index of refraction of the medium depends on the polarization state (i.e. the unit vector pointing in the direction of the electric vector field). For a fixed position in the fiber, there are two orthogonal polarization eigenstates corresponding to the maximum and the minimum of the index of refraction. These two polarization states are parametrized by an angle with respect to a fixed pair of axes that is called the birefringence angle (Born and Wolf [1980]). The difference between the maximum and the minimum of the index of refraction is the birefringence strength. If the birefringence angle and strength were constant along the fiber, then a pulse polarized along one of the eigenstates would travel at constant velocity. However the birefringence angle is randomly varying which involves coupling between the two polarized modes. The modes travel

74


[2, § 5

with different velocities, which involves pulse spreading. Random birefringence results from variations of the fiber parameters such as the core radius or geometry. There exist various physical reasons for the fluctuations of the fiber parameters. They may be induced by mechanical distortions on fibers in practical use, such as point-like pressures or twists (Rashleigh [1983]). They may also result from variations of ambient temperature or other environmental parameters (Biondini, Kath and Menyuk [2002]). In linear media PMD causes a separation between the two components of the pulse into the two principal polarizations. In nonlinear media a soliton may counteract this separation through the effective potential well produced by the local Kerr index. As a consequence, it is expected that solitons could withstand PMD (Menyuk [1988], Mollenauer, Smith, Gordon and Menyuk [1989]).

5.1. Derivation of the perturbed Manakov system The evolution of polarized fields in randomly birefringent fibers is governed by coupled nonlinear Schrödinger equations with random PMD between two modes (polarizations) (Menyuk [1989]): 1 iAz + K0 A + iK1 At − β Att + κN = 0, (5.1) 2 where A is the column vector (Ax , Ay )T that denotes the envelopes of the electric field in the two eigenmodes. The z-dependent 2 × 2 matrices K0 and K1 describe random fiber birefringence. The dispersion coefficient β is the second derivative of the propagation constant with respect to frequency, which is negative (respectively positive) for anomalous (respectively normal) dispersion. In this section we assume that β < 0. Finally κ is the Kerr coefficient. The N term stands for the nonlinear terms: (|Ax |2 + α|Ay |2 )Ax + α2 A2y A∗x , N= (5.2) (|Ay |2 + α|Ax |2 )Ay + α2 A2x A∗y where the cross-phase modulation is α = 23 for linearly birefringent fiber. As shown by Menyuk and Wai [1994] and Wai and Menyuk [1996], one can eliminate the fast random birefringence variations that appear in eq. (5.1) by means of a change of variables, leading to the new vector equation 1 8 iA1z − β A1tt + κN1 = iRl A1t + Rnl (A1 ), (5.3) 2 9 where A1 ≡ M −1 A, A1 = (A1x , A1y )T represents the slow evolution of the field envelopes in the reference frame of the local polarization eigenmodes, and the

2, § 5]


75

matrix M obeys the equation iMz + K0 M = 0. The nonlinear term N1 is (|A1x |2 + |A1y |2 )A1x . N1 = (|A1x |2 + |A1y |2 )A1y

(5.4)

The left-hand side of eq. (5.3) is known as the Manakov system, which is an integrable system supporting soliton solutions (see Appendix B). The first (second) term of the right-hand side is the linear (nonlinear) PMD. They are corrections to the Manakov system involving high-order PMD and they have zero mean. In the absence of losses M is unitary and Rl is a combination of three Pauli matrices, Rl (z) = m1 (z)Σ1 + m2 (z)Σ2 + m3 (z)Σ3 , where Σ1 =

0 1 1 0

,

Σ2 =

0 −i i 0

(5.5)

,

Σ3 =

1 0 0 −1

,

and mj are real-valued random processes. Rl is associated with the linear coupling between the modes, as well as an accumulation of phase mismatch. Rnl is the nonlinear PMD investigated by Wai and Menyuk [1996]; as shown by Lakoba and Kaup [1997] it is usually smaller than the linear PMD. 5.1.1. The white-noise model Different PMD models have been developed by Menyuk and Wai [1994], Wai and Menyuk [1996] and Lakoba and Kaup [1997]. These models become equivalent as soon as the correlation length of the random fluctuations of birefringence parameters is much smaller than the other characteristic lengths of the problem. The processes mj can then be described by independent white random processes with zero mean and autocorrelation functions mi (z)mj (z ) = σ 2 δij δ(z − z ). 5.1.2. The retarded-plate model For numerical simulations, we may think of the commonly used retarded-plate model Marcuse, Menyuk and Wai [1997]. The birefringence strength β and its derivative β are constant; the birefringence angle is constant over elementary intervals with length z; at junctions between the fiber pieces with length z, random axial rotations are incorporated as well as additions of random phase differences between the two field components, so that the Stokes vector obeys a random walk over the Poincaré sphere. If z is small enough we can model this configuration by the white-noise model with σ 2 = β 2 z/12. In this model the so-called polarization-mode dispersion parameter is given by

76


[2, § 5

√ √ Dp = 8/(3π)β z (Poole and Wagner [1986]). The parameter σ 2 is simply related to Dp through σ 2 = πDp2 /32. Chen and Haus [2000a] studied the system (5.3) with a white-noise model for PMD excitation mj in order to obtain an analytical expression for the jitter due to the interaction with the continuum component. Analogies with the noise-driven harmonic oscillator were observed and confirmed by numerical simulations of eq. (5.3) with white-noise perturbations. We shall focus our attention in the next section on the radiative soliton decay. 5.1.3. Dimensionless form We introduce a reference time t0 (of the order of the pulse width) and we define the dimensionless time T = t/t0 and distance Z = z/z0 , where z0 = t02 /|β | is the dispersion distance. The dimensionless field 8κz0 A1 (T t0 , Zz0 ) U(T , Z) = 9 satisfies the normalized equation 1 iUZ + UT T + |U|2 U = iεR(Z)UT , 2 √ where ε = σ/ |β |,

(5.6)

R(Z) = η1 (Z)Σ1 + η2 (Z)Σ2 + η3 (Z)Σ3 , and the normalized processes ηj (Z) = z0 /σ 2 mj (z0 Z) are white noises with autocorrelation function mj (Z)mj (Z ) = δij δ(Z − Z ). For consistency, note that the usual GVD parameter is D = 2πc|β |/λ2 , with λ the carrier wavelength of the√pulse (1.55 µm for standard optical √ fiber applications). Thus ε = √ π cDp /(4λ √D ). If Dp is expressed in ps/ km and D is in ps/(nm km) then ε = 0.28Dp / D for λ = 1.55 µm. The GVD parameter D is usually in the range of 1–20 ps/(nm km). Typical values of the PMD parameter Dp have been mea√ sured in the range of 0.1–1 ps/ km (de Lignie, Nagel and van Deventer [1994]). Dispersion-shifted fibers (which are of particular interest for soliton transmission) have been found to have particular high values (Galtarossa, Gianello, Someda and Schiano [1996]). The correlation length z of PMD varies between 0.1 and 1 km, which shows that the white-noise model is valid.

5.2. Radiative damping of solitons A perturbation theory of the IST for the Manakov system is presented in Appendix B. This theory is valid for a large class of perturbations as soon as the

2, § 5]


77

total energy is preserved. It is formally equivalent to the method presented in Section 3.1.2 for the perturbed NLSE. It can applied to the analysis of soliton propagation driven by random PMD. It is found that the mean emitted spectral power by PMD is P (λ) =

2ε 2 π(ν 2 + (µ − λ)2 ) cosh2 [( 12 π(µ − λ)/ν)]

.

The conservation of the total energy allows us to find the soliton decay −1/2 ν(Z) = ν0 1 + 16ε 2 ν02 Z/3 ,

(5.7)

z−1/2 .

Besides, the mowhich shows that the amplitude of the soliton decays as mentum of the soliton is not affected in leading order. This formula can also be obtained by means of the perturbation theory of the IST for the scalar NLSE (Lakoba and Kaup [1997]). In the dimensional variables, we can consider an optical soliton with rms pulse width T0 . The time jitter is the prevailing phenomenon if the propagation distance is short, z T02 /Dp2 . For long propagation distances, z T02 /Dp2 , the radiative soliton decay is of order 1. As a consequence, the inverse of the rms pulse width (proportional to the soliton energy) for long propagation distances is given by T 2 (z) = T02 + π3 Dp2 z/288 = T02 + π2 σ 2 z/9.

(5.8)

This formula confirms the results obtained by a direct expansion technique by Matsumoto, Akagi and Hasegawa [1997]. If we compare this expression with the one describing the growth of the width of a linear pulse (Karlsson and Brentel [1999]),

σ 2 zT02 2 T = T02 + 3σ 2 z − T02 4 + σ 2 z/T02 − 2 T02 + 2σ 2 z, we get that the soliton is more stable with respect to PMD fluctuations. Besides, the variance of the soliton width is also smaller than the variance of the linear pulse width (see fig. 6).

5.3. Dispersion-managed solitons and PMD effects DM solitons propagate along segments of fiber that are alternately positively and negatively dispersive. The scale of the dispersion management is usually much shorter than the soliton period. The scale of PMD is much shorter than that. PMDinduced broadening and outage probability have been investigated by means of intensive numerical simulations. The qualitative picture is that the DM soliton system can achieve better performance than both the conventional soliton system and

78


[2, § 5

Fig. 6. Normalized pulse width versus distance for a soliton and a linear pulse (i.e. a sech √ pulse in absence of dispersion and nonlinearity). Here D = 0.25 ps/(nm km) and Dp = 0.2062 ps/ km. The distance is normalized by the dispersion distance. (From Matsumoto, Akagi and Hasegawa [1997].)

the linear system with respect to PMD (Xie, Sunnerud, Karlsson and Andrekson [2001], Kylemark, Sunnerud, Karlsson and Andrekson [2003]). The quantitative picture is complicated because, apart from the PMD itself and the pulse energy and width, the performance of the conventional system depends on the average dispersion while the performance of the DM system also depends on the dispersion map strength (Sunnerud, Li, Xie and Andrekson [2001]). A complete analytical picture of the DM soliton dynamics under PMD is still lacking. Partial theoretical results are available. The DM soliton was treated as an average Manakov soliton by Chen and Haus [2000b]. Abdullaev, Umarov, Wahiddin and Navotny [2000] applied a variational approach to describe the dynamics of vector solitons in dispersion-managed optical fiber with periodically and randomly inhomogeneous birefringence. With the help of a Lagrangian formalism an approximate system of equations was derived for the soliton parameters. The predictions for the evolution of the vector soliton parameters from variational equations were compared with the results of numerical modeling of the governing coupled nonlinear Schrödinger equations, and the variational approach was shown to give reliable results. Inhomogeneous birefringence affects primarily the relative distance and frequency of solitons, whereas chirp and intensity are only slightly affected. Randomly inhomogeneous birefringence leads to diffusion growth of the mean square of the relative distance, and may split vector solitons into their constitutional components. The dependence of the mean decay length Ld of the vector soliton on the strength of random birefringence and on the energy of the initial pulse was obtained numerically. There exists a threshold value of the soliton energy below which the dependence of Ld on the energy is weak and above which Ld decreases quickly.

2, § 6]

Solitons in random quadratic media

79

§ 6. Solitons in random quadratic media In the preceding sections we considered the propagation of solitons in Kerr nonlinear media with random parameters. From both the theoretical and the experimental point of view it is important to study the dynamics of optical solitons in quadratic media with fluctuating parameters. Spatial solitons in quadratic (or χ (2) ) media were studied for the first time two decades ago by Karamzin and Sukhorukov [1974]. Later they became the object of intensive investigations (Kanashov and Rubenchik [1984], Torner [1995]). The first observations of χ (2) solitons in bulk media (Torruellas, Wang, Hagan, Van Stryland, Stegeman, Torner and Menyuk [1995]) and in film waveguides (Schiek, Baek and Stegeman [1996]) triggered further experimental efforts. Exciting effects such as various scenarios of soliton interaction and switching operations have been observed by Constantini, De Angelis, Barthelemy, Bourliaguet and Kermene [1998], Fuerst, Lawrence, Torruellas and Stegeman [1997] and Baek, Schiek, Stegeman, Bauman and Sohler [1997]. It should be mentioned, however, that these experiments have been performed in samples where the wavevector mismatch as well as the effective nonlinear coefficient can vary due to fluctuations of the impurity density of the material, the temperature, the effective mode area, etc. Although spatial χ (2) solitons have been proven to be stable both in bulk medium and in waveguides, they may be unstable under some perturbations, especially phase-sensitive ones. This feature distinguishes χ (2) solitons from their Kerr-like counterparts as the latter are not sensitive to random phase variations. Thus the dynamics of χ (2) solitons under random, in particular phase-sensitive, perturbations need to be investigated. In general, random variations of the linear dielectric constant lead to fluctuations of the wavevector mismatch. In addition, if the orientation of the crystallographic axes in a bulk medium or the effective mode area in planar waveguides vary with propagation distance, then the effective quadratic nonlinear coefficient is randomly modulated. Stochastic effects become even more pronounced when quasi-phase-matched (QPM) geometries are used. In QPM geometries a periodic modulation of the quadratic nonlinearity is used to compensate for the large mismatch. Randomly varying domain lengths contribute to deviations of the phase mismatch and/or the effective quadratic nonlinearity from their mean values (Torner and Stegeman [1997]). Thus, it is important to understand the effect of such fluctuations on nonlinear waves and, in particular, on quadratic solitons. The dynamics of χ (2) solitons in media with fluctuating parameters are very complicated and have been studied mainly numerically. Numerical simulations

80


[2, § 6

of soliton propagation in quadratic media with a fluctuating phase mismatch were carried out by Torner and Stegeman [1997], Clausen, Bang, Kivshar and Christiansen [1997] and Abdullaev, Darmanyan, Kobyakov, Schmidt and Lederer [1999]. It was observed that such fluctuations lead to a soliton decay which has a dissipative nature. Qualitatively, the effect of fluctuating parameters on soliton propagation can be understood intuitively. Indeed, an important prerequisite for soliton formation in χ (2) media is a balanced phase between the fundamental wave (FW) and the second harmonic (SH). The medium fluctuations distort the phase balance and cause a net flow of energy between harmonics. As a result, the soliton emits radiation and decays upon propagation. These qualitative arguments, however, have to be supported by theoretical estimations.

6.1. Mean field method In what follows we deal with one-dimensional spatial solitons, thus assuming a film waveguide geometry. Beam propagation in quadratic nonlinear waveguides with variable parameters is described by two coupled equations for the FW (E1 ) and SH (E2 ) fields, and in dimensionless variables has the form 1 iE1z + E1xx + d(x, z)E1∗ E2 + β1 (x, z)E1 = 0, (6.1) 2 1 1 iE2z + E2xx + d(x, z)E12 + β2 (x, z) − q E2 = 0, (6.2) 4 2 where q and β1,2 are the deterministic and stochastic parts of the mismatch, respectively, and d = 1 + ε(x, z) denotes the randomly varying nonlinearity. We assume that ε(x, z) and βj (x, z), j = 1, 2, are independent white-noise random processes. The correlation function for βj (z) is assumed to be βj (z)βj (z ) = σj2 δ(z − z ), where σj2 = lj,c δn2j /n2j and lj,c (j = 1, 2) are the correlation lengths of the linear refraction indices. The correlation function for ε(z) is 2 /d 2 , with l the correassumed to be ε(z)ε(z ) = σ 2 δ(x), where σ 2 = lc δdeff c eff lation length of fluctuations of the quadratic nonlinearity. The χ (2) system is not integrable even without fluctuating parameters. Therefore some special, approximate methods, such as the mean field method (MFM) has to be applied. Although the applicability of MFM is only justified for linear waves propagating in random media (Klyatzkin [1980]), recent studies have shown that this method can be successfully extended toward nonlinear problems, e.g., the dynamics of excitations in disordered condensed matter (Zaiman [1980]) or of nonlinear waves in random weakly dispersive media (Abdullaev, Abdumalikov and Baizakov [1997]). As the MFM results can be incorrect in circum-

2, § 6]

Solitons in random quadratic media

81

Fig. 7. Typical evolution of the initial profile E1 (0, x) = E2 (0, x) = 3 sech2 (x) in a quadratic medium with fluctuating nonlinearity. Intensities of the FW and SH components are shown. Here σ = 0.042, σ1,2 = 0.

stances, they have to be double-checked by numerical means (see fig. 7). Typically, MFM provides correct results with regard to damping and its accuracy compares to that of the Born approximation as shown by Konotop and Vasquez [1994] and Abdullaev, Darmanyan and Khabibullaev [1993]. Averaging the system (6.1)–(6.2) over all realizations of the stochastic process and using MFM we obtain a system of equations for the mean fields E1 and E2 : 1 1 i E1 z + E1 xx + E1 ∗ E2 + iσ12 E1 2 2 2 2 1 2 1

E1 − E2 E1 = 0, + iσ 2 2 1 1 1 i E2 z − q E2 + E2 xx + E1 2 + iσ22 E2 4 2 2 2 1 2 + iσ E1 E2 = 0. (6.3) 2 This shows that, in the MFM approximation, the influence of fluctuations is described by the effective linear and nonlinear losses acting on propagating soli-

82


[2, § 6

tons. Using these results it is possible to develop an analytical model describing the soliton evolution in quadratic media with fluctuating parameters based on the MFM equations (6.3) (Torner and Stegeman [1997]). Numerical observations performed by Abdullaev, Darmanyan, Kobyakov, Schmidt and Lederer [1999] indicate that the mean amplitudes decrease significantly but the mean widths of solitons change only slightly. Motivated by these observations we introduce the following ansatz to be substituted into eqs. (6.3):

E1 = a(z)As (z, x),

E2 = b(z)Bs (z, x),

(6.4)

where As (z, x) and Bs (z, x) are the soliton solutions of the unperturbed system. The complex functions a(z) = a0 (z)eiφ(z) and b(z) = b0 (z)eiψ(z) describe the changes of the soliton’s amplitudes and phases caused by the perturbation.

6.2. Nonlinear damping Substituting eqs. (6.4) into eqs. (6.3) and averaging over the transverse variable we get the following equations for the complex amplitudes: iaξ − a + a ∗ b + iγ |a|2 − 2|b|2 a = 0, (6.5) 1 1 ibξ − b + a 2 + 2iγ |a|2 b = 0, 2 2 where ξ = 4A0 z/5, γ = 3A0 σ 2 /14. The corresponding initial conditions (at z = ξ = 0) for the system (6.5) read a0 = b0 = 1, ϕ = ψ = 0. An approximate solution to the system (6.5) for γ 1 can be written in the form (Abdullaev, Darmanyan, Kobyakov, Schmidt and Lederer [1999]) 8 3 θ = 2φ − ψ ≈ − γ 1 − cos ξ , 3 2 3 8 1 b0 ≈ 2 + e−2γ ξ − γ sin ξ , (6.6) 3 9 2 3 1/2 a0 ≈ 4b02 − 3b0 + 8γ b0 sin ξ . 2 Comparison of the solution (6.6) with numerical results shows good agreement.

6.3. Linear damping Additionally we briefly discuss the behavior of χ (2) solitons for fluctuations of the phase mismatch. As shown in the preceding section in the framework of MFM this

2, § 7]

Spatial solitons in random waveguides

83

influence is described by the linear effective damping acting on both components. Applying the analytical approach described above, we obtain a system of ODEs for the functions a and b: 1 1 iaξ − a + a ∗ b = −iγ1 a, (6.7) ibξ − b + a 2 = −iγ2 b, 2 2 2 /(8A ). The approximate solutions to this syswhere ξ = 4A0 z/5, γ1,2 = 5σ1,2 0 tem are 2 3 θ ≈ (2γ1 − 5γ2 ) 1 − cos ξ , 9 2 2 2 3 b0 ≈ exp − (γ1 + 2γ2 ) + (2γ1 − 5γ2 ) sin ξ , (6.8) 9 27 2 2 2 1/2 2 a0 ≈ 4b0 − 3b0 − b0 (2γ1 − 5γ2 ) sin ξ . 3 3

Note that θ = 0 is not a solution to the system (6.7), therefore for 2γ1 = 5γ2 one should take higher-order terms into account. The details of the theory of quadratic soliton propagation in lossy media can be found in the work of (Darmanyan, Kobyakov and Lederer [1999]). It should be noted that for mean fields the total energy decreases, while the original system (6.1)–(6.2) conserves the total energy. This discrepancy can be explained by the fact that the conserved total energy is the sum of the solitonic (mean field) and random linear radiation parts. The decreasing soliton energy in MFM reflects a decreasing soliton content of the field. Numerical simulations for the fluctuating phase mismatch (Clausen, Bang, Kivshar and Christiansen [1997]) and nonlinearity fluctuations (Abdullaev [1999]) confirm this picture qualitatively. However, in the case of fluctuating mismatch, MFM and numerics agree only qualitatively in the sense that MFM overestimates damping. The reason for this is that the stochastically varying mismatch enters z into eqs. (6.1)–(6.2) via the Brownian motion W (z) = 0 (2β1 (z ) − β2 (z )) dz in the exponent. Thus the effective nonlinearity is d = exp(−i(W (z) + q)). A characteristic feature of the Brownian motion is that the deviations grow with distance. For the randomly modulated nonlinearity considered above this deviation is constant. Thus, MFM predictions for fluctuations of the phase mismatch are less accurate here compared to the case of randomly varying nonlinearity.

§ 7. Spatial solitons in random waveguides In this section we consider the propagation of stable self-focused beams (called spatial solitons) in media with random parameters. We mainly deal with planar

84


[2, § 7

beam propagation in nonlinear waveguides with random parameters, since the governing equation is equivalent to the perturbed NLSE. As is well known, a spatial soliton is unstable in free space with respect to perturbations in the transverse direction (see the discussion in the book by Hasegawa and Kodama [1995] and in the review by Kuznetzov, Rubenchik and Zakharov [1986]). The case of propagation in planar waveguides is free from these difficulties. The beam is constrained in the transverse (say y-) direction by the profile of the linear refraction index, while the nonlinear refraction index defines the field distribution in the other transverse (say x-) direction.

7.1. Spatial solitons in planar waveguides with random parameters The electromagnetic field in a planar waveguide can be represented as E(x, z, t) = F (x, z)A(y)ei(k0 z−ω0 t) ,

(7.1)

where A is the transverse profile of the propagating mode. In thin films this mode is described by the linear equation Ayy + k 2 f 2 (y) − β02 A = 0, (7.2) where f 2 (y) describes the profile of the refraction index in the y-direction, and β0 is the propagation constant that corresponds to the ground state A. Here it is assumed that the waveguide supports only one propagating mode. Substituting this expression into the equation for the electric field, ω2 n0 ± n2 |E|2 E = 0, (7.3) 2 c and averaging the wave equation over the transverse distribution we get the nonlinear Schrödinger equation for F (x, z) in the form E +

2ik0 F˜z + F˜xx + 2k02 σ S|F˜ |2 F˜ = 0, (7.4) where F˜ = F exp(iβ02 z/(2k0 )), S = A(y)4 dy/ A2 dy, and σ = +1 (−1) corresponds to a focusing (defocusing) Kerr nonlinearity. Here the second derivative corresponds to spatial diffraction. Introducing the dimensionless variables z = z/zd , x = x/x0 , where zd = k0 x02 is the diffraction length and x0 is the characteristic size of the beam, and √ introducing the normalized amplitude u(z, x) = k0 zd n2 S/n0 F˜ , we obtain 1 iuz + uxx + σ |u|2 u = V1 (x)u + V2 (x)|u|2 u. 2

(7.5)

2, § 7]


85

The unperturbed equation (7.5) (with zero right-hand side) supports bright soliton solutions in the focusing case, σ = +1. The single spatial soliton solution describing the self-focused beam in the waveguide is us (z, x) = 2ν sech 2ν(x − 2µz) exp 2iµ(x − 2µz) + 2i ν 2 + µ2 z . (7.6) The velocity 2µ is related to the angle θ in the (z, x)-plane of the beam propagation in the waveguide by the identity 2µ = tan(θ ). We now address the effects of random fluctuations of the waveguide parameters on the soliton dynamics. We add to the right-hand side of eq. (7.5) the functions V1 (x) and V2 (x) which describe transverse random inhomogeneities in the linear, n0 , and nonlinear, n2 , refractive indices. It is assumed that Vi (x) = 0 and

Vi (x)Vi (y) = Di δ(x − y), i = 1, 2. In general the spatial random fluctuations could be dependent on both variables x and z. The influence of such fluctuations on pulse propagation will be analyzed in Section 7.3. In the present subsection we consider in detail the simplified model where the random processes Vi depend only on the transverse variable x, since it is a fundamental model for many branches of physics. In contrast to the soliton propagation problem in optical fibers, where fluctuations were dependent on the evolutional variable (the time in standard NLSE), fluctuations in the transverse direction x correspond to the spatial variable in the standard NLSE and so the soliton dynamics are very different. For the case of wave propagation in disordered 1D linear media the solution of the problem for the white-noise disorder model is well known: Anderson localization occurs with a localization length of the order of Lloc ∼ 8k 2 /D1 for a pulse with carrier wavenumber k. Application of the IST approach to the randomly perturbed NLSE (see Appendix A) gives equations for the evolution of soliton amplitude and velocity: dµ dν = F (ν, µ), = G(ν, µ), (7.7) dz dz with the initial conditions ν(z = 0) = ν0 , µ(z = 0) = µ0 corresponding to the input soliton. These equations take into account the emission of radiative waves. The expressions for F , G are complicated and are not shown here. The analysis performed by Garnier [1998] shows that two regimes exist: (1) For small-amplitude solitons ν0 µ0 , the soliton velocity is almost constant and the soliton amplitude decays exponentially: ν(L) ≈ ν0 e−L/L1 ,

L1 =

32µ20 D1

86


[2, § 7

in the case of random linear refraction index V1 , and algebraically ν(L) ≈

ν0 , (1 + L/L2 )1/4

L2 =

3µ20 4D2 ν04

in the case of random nonlinear refraction index V2 . (2) If the soliton amplitude is large enough, ν0 µ0 , then the soliton amplitude is almost constant and the soliton velocity decays to zero. The decay rate of the velocity is almost logarithmic. It can be noted that in the limit ν0 /µ0 → 0 the incoming soliton can be approximated by a linear wavepacket ∞ 2 u0 (z, x) ≈ dk f (k)eikx−ik z/2 , −∞ 1 k − 2µ0 1 , with f (k) = cosh−1 π 2 4 ν0 whose spectrum is narrow around the carrier wavenumber k˜ = 2µ0 . The result for the small-amplitude soliton decay is in agreement with the linear approximation, since the length L1 corresponding to a perturbation of the linear potential can be written in terms of the carrier wavenumber k˜ as L1 = 8k˜ 2 /D1 . It is equal to the localization length of a narrow-band pulse with carrier wavenumber k˜ scattered by a random slab. Garnier [1998] also compared the predictions of the effective system (7.7) with full numerical simulations of the random NLSE. As shown by figs. 8 and 9, the agreement is very good. The existence of two regimes in the soliton propagation in disordered media can be understood by comparing the different length scales existing in this problem. The characteristic length scales are the localization length Lloc and the nonlinear length Lnl . If the pulse energy is small (weak nonlinear regime), then the nonlinear length is larger then the localization length, Lnl > Lloc , so that the exponential decay of the amplitude dominates. If the pulse energy is large, then Lnl Lloc , the disorder intensity is not enough to produce localization, and the pulse decay is much weaker than the exponential decay. If Lnl ≈ Lloc then for short propagation distances the soliton amplitude decay is moderate and for large propagation distances exponential decay takes place.

7.2. Spatial solitons under random dispersive perturbations Spatial random dispersive perturbations arise naturally when one considers the electromagnetic beam propagation in planar waveguide arrays with randomly

2, § 7]


87

(a)

(b) Fig. 8. (a) Envelope of the initial soliton (solid line) whose initial coefficients are ν0 = 0.5, µ0 = 0.4. The dashed line plots a realization of the random potential V1 . (b) Soliton envelopes for its center crossing different depth lines l for one of the realizations of the random potential. The coordinate x is normalized around the depth line l.

varying tunnel-coupling term. The evolution of the amplitude of the complex electromagnetic field is described by the following equation: iun,z + Vn,n+1 un+1 + Vn,n−1 un−1 + |un |2 un = 0,

(7.8)

88


[2, § 7

(a)

(b) Fig. 9. Evolution of the soliton whose initial coefficients are ν0 = 0.5, µ0 = 0.4. The mass N = 4ν and the velocity v = 2µ are plotted in (a) and (b), respectively. The thick solid lines represent the theoretical predictions. The thin dashed and dotted lines plot the simulated masses and velocities for 7 different realizations of the random potential.

where Vn,n±1 are the linear tunnel coupling coefficients. Here, two types of disorder are possible. If we have disorder in the z-direction (i.e. Vn,n±1 = Vn,n±1 (z)) then we deal with the discrete analog of dispersion management in optical fibers – the so-

2, § 7]


89

called diffraction management for spatial solitons. This was studied recently by Ablowitz and Musslimani [2001]. If we have disorder in the x-direction (x = nh), then we deal with the evolution of a spatial soliton under spatial diffractive perturbations. If the envelope occupies a sufficiently large number of waveguides (typically 6), then we can take a continuum limit and obtain the equation iuz + uxx + 2|u|2 u = αV (x)uxx + βVx ux + γ Vxx u,

(7.9)

with α = β = 1, γ = 2 (see Mischall, Schmidt-Hattenberger and Lederer [1994]). V (x) is now assumed to be a colored noise with correlation function V (x)V (y) = R(x − y; lc ), (7.10) where lc is the correlation length. We can assume a noise with Gaussian autocorrelation function R(x) = σ 2 exp(−x 2 / lc2 ), whose power-spectral density is √ ˆ R(k) = π(σ 2 lc exp(−k 2 lc2 /4)). For long propagation distances z ≈ 1/(σ 2 lc ) the IST analysis performed by Abdullaev and Garnier [1999] again predicts two regimes: (1) For small soliton amplitude, ν0 µ0 , the soliton velocity is almost constant and the decay law for the soliton amplitude is exponential, i.e. 2 ˆ 2 2 ν(L) ≈ ν0 exp(−L/L1 ), L−1 (7.11) 1 = µ0 R(4µ0 ) (α + 4γ ) + 4β . Two peaks exist in the spectrum of scattered radiation: a main peak at kr,1 = ˆ −2µ0 and a secondary peak at k2,r = 2µ0 with energy ∼ ν 3 R(0). If the carrier wavenumber of the soliton lies in the tail of the spectrum of perturbation then the contribution of the second peak dominates and the decay rate of the soliton amplitude is ν0 8 ˆ 2 2 (7.12) , L−1 2 = 27 R(0)ν0 α . 1/2 (1 + L/L2 ) The appearance of this law is connected with the random perturbation of the dispersion term V (x)uxx . When ν becomes small enough the decay rate is again exponential. (2) For large soliton amplitude, ν0 µ0 , the soliton amplitude remains almost constant during the propagation, but the velocity decays according to the law πν0 µ(L) ≈ (7.13) . 2 ln(L) If the spectrum of the noise decays faster than the Gaussian, then the velocity decays as ν(L) ≈

ν 2 lc µ(L) ≈ √ 0 . 2 ln(L)

(7.14)

90


[2, § 7

Recently, experimental beam propagation in a disordered 2D tunnel-coupled fiber array was performed by Pertsch, Peschel, Kobelke, Schuster, Bartelt, Nolte, Tunnermann and Lederer [2004]. The light dynamics in the hexagonal lattice are described by the coupled-mode equations (3) (1) um+1,n + V (1) um−1,n i∂z + βmn + χeff |umn |2 umn + Vm,n (2)

(2) (3) + Vm,n um,n+1 + Vm,n−1 um,n−1 + Vm,n um+1,n+1

+ V (3) um−1,n−1 um−1,n−1 = 0, where χ (3)eff = 5.5 km−1 W−1 . The coupling coefficient fluctuates around the −1 mean value V0 = 46 m with (δV )2 = 26.7 m−1 . The stochastic variation of the core diameter acrossthe fiber array leads to random variations of the propagation coefficients with (δβ)2 = 140 m−1 ≈ 3V0 . Although these fluctuations are 5 orders of magnitude smaller than the mean value, they have a dramatic influence on the field evolution. Indeed, the standard deviation is of the order of the width of the band of the homogeneous array 9V0 . The mean value β0 = 11.4 × 106 m−1 can be removed by a simple transformation of the field in the equations, but the fluctuations cannot. It was observed that the linear modes become localized for sufficiently strong disorder – the Anderson localization phenomenon, predicted for random arrays by Abdullaev and Abdullaev [1980]. When the input power increases the linearly localized states evolve in two different ways. Some linear modes first expand and later contract when the power is increased, while other modes contract immediately. The behavior actually depends on the wavenumber of the localized state. If it is in the upper part of the linear band, then the increasing power shifts the wavenumber out of the linear spectrum. If the initial wave number is at the lower edge of the linear band, then the nonlinearity shifts the wavenumber inside the band. The influence of nonlinearity on the delocalized state leads initially to a slight expansion of the beam with the rapid collapsing to a single guide and as a result to the formation of a discrete optical soliton. The explanation of this experiment requires a theoretical investigation of the dynamics of discrete solitons in disordered lattices. Some results in this direction have been obtained by Garnier [2001] and by Kopidakis and Aubry [2000].

7.3. Pulse propagation in nonlinear waveguides with fluctuating refraction index In this section we consider the influence of fluctuations of the medium parameters on the dynamics of intense optical pulses (optical bullets) in nonlinear

2, § 7]


91

waveguides. We address the case of the spatiotemporal evolution of intense optical pulses in nonlinear waveguides with a randomly varying linear refraction index (Gaididei and Christiansen [1998]). The pulse propagation is described by the 2D nonlinear Schrödinger equation iuz + u + |u|2 u + η(x, z)u = 0,

(7.15)

where = ∂x2 + ∂t2 and η(x, z) models the fluctuations of the linear refraction index. It is well known that in the absence of fluctuations the solutions of the 2D NLSE (7.15) become singular if the initial energy excesses some critical value, i.e., if E > Ec = 11.7, where E = dx dt ρ, ρ = |u|2 . The unperturbed Hamiltonian H is

∞ ∞ 1 4 2 2 H= (7.16) dx dt |ut | + |ux | − |u| . 2 −∞ −∞ It is interesting to investigate the influence of noise in η(x, z) on the collapse of intense optical pulses. In the case where the random process is δ-correlated with respect to z it is possible to obtain exact results by applying the virial theorem. It is assumed that η is a zero-mean Gaussian random process with autocorrelation function η(x, z), η(x , z ) = C(x − x )δ(z − z ). The analysis is based on the equation satisfied by the second virial integral ∞ 1 ∞ dx dt x 2 + t 2 ρ, I(z) = (7.17) E −∞ −∞ which is the pulse width. Together with H it obeys the differential equation ∞ ∞ E d2 I (7.18a) = 2H + dx dt ρηx , 4 dz2 −∞ −∞ ∞ 1 ∞ dH = (7.18b) dx dt ηρz . dz 2 −∞ −∞ Applying the Furutsu–Novikov formula to perform averaging over fluctuations (Klyatzkin [1980]) z δR , dz η(z)η(z ) η(z)R(η) = (7.19) δη(z ) 0 and using the principle of causality, we get the system of equations for the averaged values I(z) and H(z): E

d2 I = 8 H(z), dz2

d H = κE, dz

(7.20)

92


[2, § 8

where κ = −C (0) > 0. By integrating eq. (7.20) we get that the influence of fluctuations on the pulse width is described by 4

I(z) = I02 + 4εz2 + κz3 , 3

(7.21)

where I(0) = I0 , Iz (0) = 0, and H(0) = εE. If κ = 0 (no fluctuations) and ε < 0, then I(z) = I0 (1 − z2 /zc2 ), which shows that collapse occurs before the √ √ critical propagation distance zc = I0 /(2 |ε| ). Fluctuations lead to an increase of this distance. In particular, if κ > κcr with 4|ε|3/2 , κcr = √ 3I0

(7.22)

then I(z) 0 for all z > 0, and the collapse on average is arrested. As a result the pulse will spread instead of collapsing.

§ 8. Two-dimensional solitons in random media In this section we consider the influence of random fluctuations of the transverse profile of the waveguide on the dynamics of 2D solitons. The governing equation is the randomly perturbed 2D NLSE iuz + ⊥ u + |u|2 u + ε1 g(z) x 2 + y 2 u − ε2 |u|4 u = 0, (8.1) where ⊥ = ∂x2 + ∂y2 is the transverse Laplacian, g(z) = g0 (1 + η(z)) is the amplitude of the quadratic potential imposed by the waveguide, and η(z) is a random process that models the fluctuations of the waveguide. The quintic nonlinear term can appear for example by expanding a saturable nonlinearity. A modulation theory has been proposed by Fibich and Papanicolaou [1999] for the analysis of the 2D soliton dynamics. The unperturbed 2D NLSE (ε1 = ε2 = 0) has waveguide solutions of the form u(r, z) = eiz RT (r), where the function RT (r) is the solution of the boundary-value problem RT + r −1 RT − RT + RT3 = 0,

RT (0) = 0, RT (∞) = 0.

(8.2)

The solution of this equation with the lowest energy (power) is called Townes soliton. It plays an important role in self-focusing theory and it has exactly the critical power for self-focusing: ∞ ET ≡ (8.3) RT2 (r)r dr = Ec ≡ 1.862, 0

2, § 9]

Conclusion

93

while its Hamiltonian is equal to 0: ∞ 1 HT = (RT )2 − RT4 (r) r dr = 0. 2 0

(8.4)

The solution of the perturbed 2D NLSE is searched for in the form of a modulated Townes soliton, r 1 az r 2 RT eiS , S = σ + , σz = a −2 . u(r, z) ≈ (8.5) a(z) a(z) 4a If the initial power is just above the critical value, then it is possible to derive an evolution equation for the function a(z) starting from the approximation (8.5). The equation of modulation theory for the width a(z) is a 3 azz = −β0 −

1 f1 (z), 2M0

where β0 = β(0) − f1 (0)/(2M0 ), β(0) = (E − Ec )/M0 , and M0 ≡ 14 RT2 ≈ 0.55. The auxiliary function is given by

1 f1 (z) = 2a(z) Re dx dy F (u)e−iS RT + ρ∇RT (ρ) , 2π

(8.6) ∞ 0

r 3 dr ×

(8.7)

where F (u) = −ε1 g(z)(x 2 + y 2 )u + ε2 |u|4 u. In the lowest-order approximation, the equation for the width takes the form azz + 4ε1 g(z)a + U (a) = 0,

(8.8)

where the potential U is ε2 Ec β0 (8.9) − 2. 4 M0 a 2a Similar equations appear in the study of 2D and 3D Bose–Einstein condensates (BECs) with time-varying trap potential (Abdullaev, Baizakov and Konotop [2001], Abdullaev, Bronski and Galimzyanov [2003], Garnier, Abdullaev and Baizakov [2004]). Random modulations impose a distortion of the BEC due to the growth of focusing–defocusing oscillations. This destabilizing mechanism is similar to the random Kepler problem studied above. U (a) =

§ 9. Conclusion In this review we have considered the propagation and interaction of optical solitons in random media. The dynamics of optical solitons in optical fibers with randomly varying dispersion, amplification and birefringence have been studied.

94


[2, § 9

The propagation of dispersion-managed solitons in fiber links with random dispersion maps has been investigated. The influence of the fluctuations on the transverse variables has been considered for spatial solitons in waveguides and 2D optical bullets. Some problems have been not addressed in this chapter. First, we have not considered the evolution of dark solitons in random fibers and bulk media. While some numerical results for dark solitons in the random dispersion-managed regime have been obtained by Hong and De-Xiu [2003], this class of problems remains largely uninvestigated. Second, we have not considered the dynamics of partially coherent solitons, which have been intensively investigated in the last decade, in particular in photorefractive crystals (see the review by Chen, Segev and Christodoulides [2003]). However this problem involves the evolution of initial random conditions in nonlinear media and that is beyond the scope of this review. It should be noted that we have mainly modeled the soliton propagation in random media by stochastic perturbations of integrable nonlinear wave equations such as the NLSE and the Manakov system. Fortunately many physical problems are described by these equations. The case of stochastic perturbations of nonintegrable wave equations, such as the system for FW and SH waves, random dispersion-managed solitons, and multidimensional solitons is still under investigation. It should be of great interest to extend the approaches described in this review to the propagation of dissipative optical solitons in random media. Early work based on the adiabatic approximation shows that for such solitons the noise effects are essentially suppressed (see the book by Hasegawa and Kodama [1995] and recent work by Abdullaev, Navotny and Baizakov [2004]). It should also be interesting to investigate the properties of discrete optical solitons in disordered arrays of planar waveguides and 2D fiber arrays. Such investigations are important for studying the interplay between nonlinearity, disorder, and discreteness. The theory of discrete optical solitons for deterministic systems of fiber arrays has been developed by Aceves, Luther, De Angelis, Rubenchik and Turitsyn [1995]. We should also note the recent experiments performed by Pertsch, Peschel, Kobelke, Schuster, Bartelt, Nolte, Tunnermann and Lederer [2004], studying the properties of discrete solitons in disordered 2D fiber arrays.

Acknowledgements The authors are grateful for valuable collaboration and/or discussions to A.A. Abdumalikov, B.B. Baizakov, J.C. Bronski, J.G. Caputo, S.A. Darmanyan, I. Gabitov,

2, A]

Appendix A

95

A. Kobyakov, V.V. Konotop, F. Lederer, B.A. Malomed, G.C. Papanicolaou, M.P. Soerensen, S.K. Turitsyn, B.A. Umarov and S. Wabnitz.

Appendix A: The inverse scattering transform for the nonlinear Schrödinger equation A.1: The unperturbed nonlinear Schrödinger equation For more details about the following statements and their proofs we refer to Ablowitz and Segur [1981] and Manakov, Novikov, Pitaevskii and Zakharov [1984]. Let us consider the unperturbed NLSE: 1 iuz + utt + |u|2 u = 0. (A.1) 2 The inverse scattering transform consists in linearizing this nonlinear equation. It is based on the fact that u(z, ·) can be characterized by a set of spectral data of the operator L(u(z, ·)) in which u plays the role of a potential: ∂ 1 0 0 u∗ . + Q(u), with P = and Q(u) = L(u) = iP −u 0 0 −1 ∂t A.1.1: Essential spectrum of the operator L(u) Let us consider the spectral problem associated with the operator L(u): 1 0 L(u)ψ = λψ, ψ = ψ1 e1 + ψ2 e2 , e1 = , e2 = . 0 1

(A.2)

We introduce the so-called Jost functions f and g, defined as the eigenfunctions of L(u) associated with the real eigenvalue λ satisfying the following boundary conditions: t→+∞

f (t, λ) −→ e2 eiλt ,

t→−∞

g(t, λ) −→ e1 e−iλt .

If we denote by ψ¯ the vector (ψ2∗ , −ψ1∗ ) associated with a vector ψ solution of ¯ If the eigenvalue is real, ψ and ψ¯ eq. (A.2), then ψ¯ is a solution of Lψ¯ = λ∗ ψ. are linearly independent and form a base of the space of the solutions of eq. (A.2). It can then be proved that the Jost functions are related by g(t, λ) = a(λ)f¯(t, λ) + b(λ)f (t, λ), f (t, λ) = −a(λ)g(t, ¯ λ) + b∗ (λ)g(t, λ).

96


[2, A

Multiplying the first identity by the vector f¯∗ , we get an explicit representation of the coefficient a as the Wronskian of f and g: a(λ) = g1 (t, λ)f2 (t, λ) − g2 (t, λ)f1 (t, λ). Substituting the second identity into the first one we also get the conservation relation |a|2 + |b|2 = 1. A.1.2: Point spectrum of the operator L(u) It is possible to define an analytic continuation of a(λ) over the upper complex half-plane. A noticeable feature then appears. If λr is a zero of a(λ), then f and g are linearly dependent, so there exists a coefficient ρr such that g(t, λr ) = ρr f (t, λr ). The corresponding eigenfunction is bounded and decays exponentially as t → +∞ (because |f | ∼ e− Im λr t ) and as t → −∞ (because |g| ∼ e+ Im λr t ). Thus λr is an element of the point spectrum of L(u). It can then be proved that the set (a(λ), b(λ), λr , ρr , a (λr )) characterizes the Jost functions f and g as well as the solution u. The inverse transform is essentially based on the resolution of the linear integro-differential Gel’fand–Levitan–Marchenko (GLM) equation, whose entries are constituted by the set (a, b, λr , ρr , a (λr )): ∞ ∞ ∗ K1 (s, t) = Φ (s + t) − K1 (s, t ) Φ ∗ (t + t )Φ(t + t ) dt dt , s

s

(A.3)

where Φ(t) = −

iρr 1 iλr t e + a (λr ) 2π r

+∞ b(λ) −∞ a(λ)

eiλt dλ.

(A.4)

We then obtain u by the formula u(t) = −2iK1∗ (t, t). The study of the inverse problem associated with the operator L(u) has not yet been completed. In particular, the precise characterization of the spectral data which lead to well-defined potentials u has not yet been accomplished. However, in the case where the initial condition u0 is rapidly decaying so that it satisfies t → |t|n |u0 |(t) ∈ L1 for any n, the inverse scattering can be achieved rigorously (Ablowitz and Segur [1981]). A.1.3: Evolution of the scattering data The evolution equations of the scattering data are simple and uncoupled for the unperturbed NLSE: ∂z a(z, λ) = 0,

∂z b(z, λ) = −2iλ2 b(z, λ),

∂z ρr (z) = −2iλ2r ρr .

2, A]

Appendix A

97

To sum up, the scattering transform involves the following operations: u(z0 , t)

direct scatt.

(a, b, λr , ρr , a (λr ))(z0 ) uncoupled evolution equations.

NLSE

u(z, t)

inverse scatt.

(a, b, λr , ρr , a (λr ))(z)

A.1.4: Soliton The soliton solution of the NLSE is: uS (z, t) = 2ν

exp(i(2µ(t − 2µz) + 2(ν 2 + µ2 )z)) . cosh(2ν(t − 2µz))

(A.5)

The scattering coefficients are a(λ) = (λ − λS )/(λ − λ∗S ), where λS = µ + iν is the unique zero of a in the upper complex half-plane, while b = 0. A.1.5: Conserved quantities There exists an infinite number of quantities which are preserved by the unperturbed NLSE (Manakov, Novikov, Pitaevskii and Zakharov [1984]). They can be represented as functionals of the solution u or in terms of the scattering data. We present only two of them: The energy of the wave E = |u|2 dt can be written as ∗ E= (A.6) 2i λr − λr + n(λ) dλ, r

where n(λ) = −π−1 ln |a(λ)|2 . The Hamiltonian H = also be expressed as 4 ∗3 3 H= i λr − λr + 2 λ2 n(λ) dλ. 3 r

1 2

(|ut |2 − |u|4 ) dt can

(A.7)

A.2: Perturbation theory for the NLSE We consider a perturbed NLSE that in dimensionless units reads 1 iuz + utt + |u|2 u = εR(u). (A.8) 2 In the framework described in Section 3 u represents the complex envelope of the electromagnetic field in a cubic nonlinear medium. The initial condition corresponds to a single-soliton state. If ε is small we can assume that the propagating wave consists of one soliton plus radiation. The random perturbation R(u) induces variations of the spectral data. The evolution of the continuous component

98


[2, A

corresponding to radiation can be found from the evolution equations for the Jost coefficients (Karpman [1979]): ∂a(λ, z) = −ε a(λ, t)γ¯ (λ, z) + b(λ, z)γ (λ, z) , (A.9) ∂z ∂b(λ, z) = −2iλ2 b(λ, z) + ε a(λ, z)γ ∗ (λ, z) + b(λ, z)γ¯ (λ, z) , (A.10) ∂z where γ (λ, z) = dt R(u)∗ f22 + R(u)f12 and γ¯ (λ, z) = dt R(u)f1 f2∗ − R(u)∗ f1∗ f2 . The evolutions of the soliton parameters can then be derived from the conservation of some integrals of motion, such as the energy. We can obtain the expression for the total wave in the following way. The function Φ which appears in eq. (A.4) is equal to the sum ΦS + ΦL with −iρS iλS t e , ΦS (z, t) = a (λS )

1 ΦL (z, t) = 2π

+∞ b(z, λ) −∞ a(z, λ)

eiλt dλ.

ΦL plays the role of a perturbation of Φ. The kernel K associated with the total wave u(z, ·) can be represented as the sum KS + KL , where ν exp((ν + iµ)(s − t)) KS (s, t) = cosh(2ν(s − tS )) where λS = µ + iν, 1 ρS 1 , ln tS = 2ν 2ν a (λS )

−i exp i(2µ(tS − s) − φS ) − exp(2ν(tS − s))

φS = − arg

ρS a (λS )

,

+ 2µtS .

(A.11)

Neglecting higher-order terms, KL1 is a solution of KL1 (s, t) +

∞

KL1 (s, t )

s

∞ s

ΦS∗ (t + t )ΦS (t + t ) dt dt = Ψ (s, t),

Ψ (s, t) = ΦL∗ (s + t) ∞ KS1 (s, t ) − ×

s ∞ s

ΦS∗ (t + t )ΦL (t + t ) + ΦL∗ (t + t )ΦS (t + t ) dt dt .

Following the IST method, we obtain the transmitted wave by the formula u(z, t) = −2iK1∗ (t, t), which establishes that the total wave is given by the sum u(z, t) = uS (z, t) + uL (z, t), where uS is a soliton of energy 4ν and ve-

2, B]

Appendix B

99

locity 4µ: uS (z, t) = 2ν

exp i(2µ(t − tS ) + φS ) , cosh(2ν(t − tS ))

(A.12)

which shows that tS and φS are, respectively, the position and phase of the soliton. uL admits the following expression: (λ − µ + iν tanh(2ν(t − tS )))2 2iλt 1 ∞ b (λ) e dλ uL (z, t) = iπ −∞ a (λ − µ + iν)2 ν 2 exp 2i(2µ(t − tS ) + φS ) iπ cosh2 (2ν(t − tS )) ∞ ∗ b 1 (λ) e−2iλt dλ. × ∗ (λ − µ − iν)2 −∞ a −

(A.13)

Appendix B: The inverse scattering transform for the Manakov system B.1: The unperturbed Manakov system We consider the Manakov system that reads in dimensionless units as: 1 iuz + utt + |u|2 + |v|2 u = 0, 2 1 ivz + vtt + |u|2 + |v|2 v = 0. 2 This model can be derived from the Hamiltonian 2 1 2 H= |ut | + |vt |2 − |u|2 + |v|2 dt. 2

(B.1) (B.2)

(B.3)

B.1.1: Direct transform: the scattering problem The scattering problem associated with the Manakov system is the following eigenvalue problem (Manakov [1974]):   −iλ iu∗ iv ∗ ∂t |f = L|f , L :=  iu (B.4) iλ 0 . iv 0 iλ If λ ∈ R then we can construct two sets of special solutions of eq. (B.4), denoted by |φi (t, λ) and |ψi (t, λ) (i = 1, 2, 3) with the asymptotic behavior |φi j = δij exp(−iIj λt),

t → −∞,

|ψi j = δij exp(−iIj λt),

t → ∞,

100


[2, B

with I1 = 1 and I2 = I3 = −1. The kets |φi and |ψi are known as the Jost functions. They both represent a complete set of solutions to the problem (B.4) for the eigenvalue λ ∈ R. Hence we may write the elements of one basis in terms of the other: 3 φi (t, λ) = αij (λ)ψj (t, λ) . j =1

From the equivalence αij (λ) = ψj |φi we derive the conservation relations for i = 1, 2, 3: 3l=1 |αil |2 = 1, while for i = j , 3l=1 αil∗ αj l = 0. The functions |φ1 , |ψ2 and |ψ3 can be analytically continued in the upper complex half-plane Im(λ) > 0, whereas this holds true in the lower half-plane for the functions |φ2 , |ψ3 , and |ψ1 . The analyticity of the Jost functions implies the analyticity of α11 (λ) = ψ1 |φ1 in Im(λ) > 0. Let us denote by λk , k = 1, . . . , N , the zeros of the function α11 . For such an eigenvalue λk we have: φ1 (t, λk ) = ρk2 ψ2 (t, λk ) + ρk3 ψ3 (t, λk ) . Since |φ1 (t, λk ) decays exponentially to 0 as t → −∞, and |ψl (t, λk ), l = 2, 3, both decay exponentially to 0 as t → ∞, we get that the zeros of the function α11 correspond to the discrete eigenvalues of the problem (B.4). The main point of the IST method is that the solutions u and v that play the roles of potentials in the eigenvalue problem are completely characterized by the set of scattering data: # " α12 α13 (λ), (λ), λ ∈ R , S = (λk , γ2k , γ3k ), k = 1, . . . , N, α11 α11 (λ ), l = 2, 3. As in the case of the scalar NLSE, the evoluwhere γlk = ρlk /α11 k tion equations of the scattering data are uncoupled:

∂z α11 (λ) = 0,

λ ∈ R,

∂z α1l (λ) = −2iλ α1l (λ), 2

∂z γlk =

(B.5) λ ∈ R, l = 2, 3,

−2iλ2k γlk .

(B.6) (B.7)

B.1.2: The inverse transform: the GLM equation The inverse transform proposed by Manakov [1974] is based on concepts used to solve standard Riemann–Hilbert problems. Since global results are often difficult to obtain from such equations, we prefer to go to a Gel’fand–Levitan–Marchenko representation. The inverse problem then reads as the resolution of a system of linear integro-differential equations, whose entries are constituted by the set S of

2, B]

Appendix B

101

scattering data: K(1) (s, t) =

3

Fj (s + t)ej +

j =2

K

(j )

(s, t) =

Fl (t) = −

Fj∗ (s

+ t)e1 −

iγr eiλr t

r

K(j ) (s, t )Fj (t + t) dt ,

(B.8)

s ∞ s

1 + 2π

∞

K(1) (s, t )Fj∗ (t + t) dt ,

j = 2, 3, (B.9)

+∞

α1l (λ) iλt e dλ, −∞ α11 (λ)

(B.10)

where ej = (δij )i=1,2,3 . We can get the Jost functions |ψj from the solutions of the kernels K(j ) of the GLM equations: ∞ −Ij iλt + K(j ) (t, s)e−iIj λs ds, j = 1, 2, 3, |ψj (t, λ) = ej e t

with I1 = 1 and I2 = I3 = −1. We then obtain (u, v) by the formula u(t) = −2iK1 (t, t)∗ , (2)

v(t) = −2iK1 (t, t)∗ . (3)

(B.11)

Accordingly it is sufficient to solve the following two equations: K1 (s, t) = Fj∗ (s + t) − (j )

Gj l (s, t, t ) =

∞ s

3

∞

l=2

s

K1 (s, t )Glj (s, t, t ) dt , (l)

Fj (t + t )Fl∗ (t + t) dt .

j = 2, 3, (B.12)

B.1.3: Soliton The soliton solution of the Manakov system is: uS cos(θ ) exp i(2µ(t − 2µz) + 2(ν 2 + µ2 )z) = 2ν . sin(θ ) vS cosh(2ν(t − 2µz))

(B.13)

The scattering coefficients are α11 (λ) = (λ − λS )/(λ − λ∗S ), where λS = µ + iν is the unique zero of α11 in the upper complex half-plane, while α12 = α13 = 0. B.1.4: Conserved quantities Conserved quantities can be worked out as in any integrable system. The total energy and Hamiltonian (defined by eq. (B.3)), and more generally all conserved quantities, can be expressed in terms of scattering data. Let us define α12 2 α13 2 1 (λ) , n(λ) := log 1 − (B.14) (λ) − α π α11 11

102


[2, B

for λ ∈ R. The energy and Hamiltonian can be decomposed into the sums of continuous parts and discrete parts: E=

2 |u| + |v|2 dt =

H=2

R

λ2 n(λ) dλ + 8

R

n(λ) dλ + 4

J

νj ,

(B.15)

j =1

J νj3 νj µ2j − . 3

(B.16)

j =1

B.2: Perturbation theory for the Manakov system We consider a perturbed Manakov system that reads in dimensionless units as: 1 iuz + utt + |u|2 + |v|2 u = εRu (u, v), (B.17) 2 1 ivz + vtt + |u|2 + |v|2 v = εRv (u, v). (B.18) 2 In the framework described in Section 5.1 u and v represent the complex envelopes of the two orthogonal polarizations of a transverse electromagnetic field in a cubic nonlinear medium. The evolution equations of the Jost coefficients are (n = 1, 2, 3): dα1n ∗ ∗ ∗ ∗ =i uz φ11 ψn2 dt. (B.19) + u∗z φ12 ψn1 + vz φ11 ψn3 + vz∗ φ13 ψn1 dz Assume that the total wave consists of one soliton λS = µ + iν plus radiation. The functions F2 and F3 which appear in eq. (B.12) are equal to the sum FSj + FLj , with 1 +∞ α1j (λ) iλt e dλ. FSj (t) = −iγSj eiλS t , FLj (t) = 2π −∞ α11 (λ) Assume also that the radiation component is small enough so that FLj (t) is small. (2) As a consequence FLj plays the role of a perturbation of FSj . The kernels K1 and (3) (2) K1 associated with the total wave u, v can be represented as the sums K1 = (2) (2) (3) (3) (3) KS + KL and K1 = KS + KL , where exp(−ν(s + t − 2tS )) , 1 + exp(−4ν(s − tS )) exp(−ν(s + t − 2tS )) (3) KS (s, t) = −2iν sin(θ )ei(−µ(s+t−2tS )−φS3 ) , 1 + exp(−4ν(s − tS )) (2)

KS (s, t) = −2iν cos(θ )ei(−µ(s+t−2tS )−φS2 )

2]

References

103

where θ = arctant(|γS3 /γS2 |) is the soliton angle, and |γS2 |2 + |γS3 |2 1 , φSj = − arg(γSj ) + 2µTs ln tS = 4ν 4ν 2 are the soliton center and phases, respectively. Neglecting higher-order terms, (2) (3) KL and KL are solutions of (j ) KL (s, t)

=

(j ) ψL (s, t) −

3

∞

l=2

s

KL (s, t )GSS (s, t, t ) dt , (l)

(lj )

j = 2, 3,

where ∗ ψL (s, t) = FLj (s + t) − (j )

3

∞

l=2

s

(lj ) (lj ) (l) KS (s, t ) GLS + GSL (s, t, t ) dt ,

j = 2, 3, and GXY (s, t, t ) = (j l)

∞ s

FXj (t + t )FY∗l (t + t) dt .

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Chapter 3

Curved diffractive optical elements: Design and applications by

Nándor Bokor Department of Physics, Budapest University of Technology and Economics, Budafoki u. 8, 1111 Budapest, Hungary e-mail: [email protected]

Nir Davidson Department of Physics of Complex Systems, Weizmann Institute of Science, 76100 Rehovot, Israel

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)48003-8 107

Contents


109

§ 2. Spherical/cylindrical CDOEs for imaging and Fourier transform . .

112

§ 3. Spherical/cylindrical CDOEs for concentration of diffuse light on flat targets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121

§ 4. Uniform collimation and ideal concentration for arbitrary source and target shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125

§ 5. CDOEs for controlling the geometrical apodization factor . . . . . .

136

§ 6. Curved gratings for spectroscopy . . . . . . . . . . . . . . . . . . .

141

§ 7. CDOEs for optical coordinate transformations . . . . . . . . . . . .

143

§ 8. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

147

108

§ 1. Introduction Diffractive optical elements (DOEs) are used in many areas in optics such as imaging, Fourier transform, multifocusing lenses, spectroscopy, interferometry, optical signal processing and optical transformations. They have the advantage of light weight, small thickness, and the flexibility to encode complicated aspheric phase functions on a single element. The phase function, i.e. the shape of the grating grooves, can be used as an optimization parameter for specific tasks. In many cases, for simplicity, DOEs are fabricated on flat substrates, but when manufactured on curved surfaces they offer many additional important advantages. Curved spectroscopic gratings were invented and first produced by Rowland already in the 19th century (Rowland [1883]), thus they represent the first manmade curved diffractive optical elements (CDOEs). Rowland gratings combine the diffractive properties of a spectroscopic grating with the focusing properties of a spherical mirror in a single element, thereby minimizing throughput losses. They are widely used in spectroscopy, especially in the far-ultraviolet regime. Welford [1965] gave a detailed aberration analysis of spectroscopic gratings ruled on spherical substrates, and Schmahl and Rudolph [1976] reviewed spectroscopic gratings recorded on spherical and toroidal substrates by holography. DOEs serving for visual display applications, and fabricated on curved surfaces, were first considered much later, by McCauley, Simpson and Murbach [1973]. In their article, however, the substrate shape was not used as a free parameter, but was rather motivated and dictated by the specific application area (windshields, swimming goggles, etc.). Fairchild and Fienup [1982] proposed a numerical raytracing technique for analyzing and designing DOEs holographically recorded with aspheric wavefronts on both flat and curved substrate shapes. Hybrid refractive–diffractive lenses, in which a diffractive pattern is manufactured onto a curved lens surface, have been considered for combined CD–DVD readers, or thermal imagers operating in wide temperature ranges (Wood [1992], Zhang and Wang [2004]). Curved diffraction gratings can even be found in nature: curved surface relief gratings are partially responsible for the metallic colors of certain species of spiders (Parker and Hegedus [2003]). 109

110

Curved diffractive optical elements: Design and applications

[3, § 1

In this chapter we show that when a DOE is fabricated on a curved substrate, the shape of the substrate can provide additional degrees of freedom in the design. Figure 1 shows the schematic diagram of a CDOE, working in transmission. Both the substrate shape z(x, y) and the phase function φ[x, y, z(x, y)] are free parameters which can be controlled independently to optimize the CDOE, resulting in much better performance than is possible with DOEs fabricated on flat substrates. In a simplified ray-optics picture, the phase φ[x, y, z(x, y)] determines the directions into which the incoming rays are diffracted by the grating, whereas the shape z(x, y) determines the spatial density of rays propagating in the directions imposed by φ[x, y, z(x, y)], thereby serving as an effective local apodization factor. In recent years, improved fabrication methods, such as sophisticated ruling machines and interferometric recording with the help of computer-generated holography or computer-originated holography, have provided great flexibility in creating the phase function φ[x, y, z(x, y)]. Additionally, it is now feasible to produce and replicate arbitrary substrate shapes z(x, y) with high precision. These two practical developments, along with the appearance of new application areas and design principles, account for the recent increase of interest in CDOEs. For many applications either the CDOE shape or the phase function or both can have very simple analytical forms. For example, an aplanatic imaging CDOE satisfying the sine condition, and hence eliminating coma, can be fabricated on a spherical (in 3D) or cylindrical (in 2D) substrate, with the interference of two

Fig. 1. Schematic diagram of a CDOE, working in transmission.

3, § 1]

Introduction

111

simple spherical or cylindrical waves, as was first pointed out by Murty [1960] and Welford [1973]. First, in Section 2, we will review the properties of such spherical/cylindrical CDOEs, and their applications for aberration-free imaging and Fourier transformation. In Section 3 we will show that aplanatic spherical/cylindrical CDOEs satisfying the sine condition are also capable of concentrating quasi-monochromatic diffuse light onto a flat target at the theoretical limit imposed by brightness conservation (or equivalently, the second law of thermodynamics, see Bassett, Welford and Winston [1989]). By simply reversing the direction of the light rays, it can be shown that these spherical/cylindrical CDOEs can also uniformly collimate the light emitted from a flat Lambertian source. In Section 4 we will extend the sine condition, which leads to coma-free imaging and ideal diffuse light concentration and collimation, to arbitrary target and source shapes. This generalization leads to CDOE shapes other than spherical/cylindrical operating at the thermodynamic limit. The smallest possible spot sizes, essential for applications such as lithography, data storage, microscopy, optical tweezers and optical traps, require high numerical aperture focusing optics. An important factor for such systems is the so-called geometrical apodization factor, which determines the relative intensity of light rays coming towards the focus from different directions. In Section 5 we will show that the substrate shape of CDOEs can control the geometrical apodization factor and hence improve the performance of high numerical aperture focusing systems. In Section 6 we will discuss recent developments in the very old field of curved spectroscopic gratings where either aspheric substrate shape or aspheric phase function are shown to suppress various types of aberration. As mentioned above, earlier works on spherical spectroscopic gratings, both ruled and holographically recorded, were reviewed by Welford [1965] and Schmahl and Rudolph [1976], respectively, and are hence only briefly discussed here. In Section 7 we will discuss the application of CDOEs for optical transformations and illustrate the principle with optical systems capable of coordinate transformation. Here the CDOE has an appropriately designed substrate shape, and the recording waves are simple plane waves. Finally, we will conclude in Section 8. Throughout the chapter, except in Section 5, we resort only to geometrical optics, since the sizes of CDOEs and of all other elements in the systems considered are much larger than diffraction-limited sizes. Moreover, except in Section 6, we limit the discussion to quasi-monochromatic light to suppress the large chromatic aberration of DOEs. We do not put emphasis on fabrication methods. We briefly mention that holography is a very flexible tool for recording the phase function

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[3, § 2

on the CDOE, provided that a substrate of the given shape can be coated with the recording material. For surfaces with zero curvature, like a cylinder, this can be done, e.g., by glueing a photopolymer film to the substrate (Bokor and Davidson [2001b]), whereas for surfaces with nonzero curvature, like a sphere, dip-coating processes can be used. We note that for most applications the performance of the CDOE depends only weakly on the exact shape of the substrate, and hence the fabricated shape does not have to be accurate within optical precision, as long as it is identical during recording and readout. CDOEs with straight or nearly straight grating grooves can be formed using mechanical ruling engines. Finally, a lithographic method for fabricating CDOEs having axial symmetry was reported by Xie, Lu, Li, Zhao and Weng [2002].

§ 2. Spherical/cylindrical CDOEs for imaging and Fourier transform Consider an imaging system I with an object distance a and an image distance b, as shown in fig. 2 (all distances are taken to be positive when they are to the left of I and negative if they are to the right). For a thin imaging lens I is the principal plane, and for a thin CDOE it is the surface of the CDOE. Abbe’s sine condition (Born and Wolf [1993]) states that first-order aberrations (i.e. aberrations that grow linearly with the object point’s distance from the optical axis) are eliminated if sin α b (2.1) = const = − = −M, sin β a where M is the magnification and α and β are arbitrary angles of the input and the output rays, respectively, that connect the on-axis points A and B. A system that satisfies the Abbe sine condition (2.1) is called aplanatic and is free of

Fig. 2. An imaging system with a principal surface I; a and b are the object and image distances, respectively.

3, § 2]

Spherical/cylindrical CDOEs for imaging and Fourier transform

113

first-order coma aberration. Welford [1973] showed that the surface that satisfies condition (2.1) is a sphere with radius ab (2.2) . a+b The center of the sphere is at a distance of c = R from the origin (the common point of I and the optical axis). Hence, if a CDOE is located on such an Abbe surface, it is free of first-order aberrations for off-axis points at an object distance a and an image distance b. Let us consider some specific cases: 1. Transmission geometry (ab < 0) with unit magnification (M = −1). Here the Abbe surface is a plane that is situated symmetrically between A and B (in this case α = β; hence sin α/ sin β = tan α/ tan β = 1). 2. Reflection geometry (ab > 0) with unit magnification (M = 1). In this case α = −β. Moreover, the input and output rays for on-axis points coincide completely. Hence the Abbe surface can be any arbitrary surface. 3. Reflection geometry with arbitrary (not unit) magnification. Here it follows from eq. (2.2) that the radius of curvature of the Abbe sphere is half the radius of the equivalent spherical mirror, i.e. the one with the same object and image distances. 4. Plane-wave illumination, where, if a = ∞, then R = c = b for both transmission and reflection: Here the Abbe sine condition can simply be stated as h = R sin β, where h is the distance of the incoming ray from the optical axis. R=

2.1. Spherical CDOE for imaging Bokor and Davidson [2001b] investigated numerically and experimentally an aplanatic CDOE recorded as a reflection hologram. The performance of the CDOE was compared with that of a flat DOE, a reflection hologram recorded on a flat substrate, but having the same object and image distances. The object distance in both cases was infinity, hence the image distance was equal to the focal length of the CDOEs. Numerical raytracing results showed severe coma for the flat DOE, and a significantly larger spot size than for the spherical CDOE, as seen in fig. 3. These spot diagrams were obtained for NA = 0.7, a focal length of 35 mm, and an angular separation of 2◦ between adjacent spots. The method was experimentally tested in a configuration for 1D imaging (cylindrical CDOE). The recording setup is shown in fig. 4. To form the CDOE, the interference of a plane wave and a counterpropagating, diverging spherical wave was recorded on a thin (∼20 µm) Dupont photopolymer film that was coated on

114


(a)

[3, § 2

(b)

Fig. 3. Spot diagrams obtained from numerical raytracing for (a) flat DOE, and (b) a CDOE satisfying the sine condition (Bokor and Davidson [2001b]).

Fig. 4. Recording setup of a CDOE satisfying the sine condition.

the outside of a 1 mm thick glass surface with a half-cylinder shape with radius R = 35 mm. The spherical wave was produced by a 60× microscope objective whose focal point coincided with the center of the cylinder, yielding a beam with a large NA (>0.86). To ensure essentially 1D operation of the imaging CDOE – despite the spherical recording wave – the height of the CDOE was limited to 3 mm, as compared to the width of 60 mm. 1D configuration can also be obtained if a cylindrical lens, instead of the microscope objective, is used in the recording setup. For comparison, a flat DOE was also recorded with the same parameters. The focal length of both DOEs was F = 35 mm. The 488 nm line of an argon laser was used for both recording and reconstruction. The experimental focal images are shown in fig. 5. As expected, when the DOEs are reconstructed with an on-axis plane wave, no appreciable aberrations are present. However, when the reconstructing plane waves have an off-axis angle of θ = 5◦ , the flat DOE has se-

3, § 2]


115

Fig. 5. Experimental results of Bokor and Davidson [2001b], showing a large coma for the flat DOE when it is illuminated with an off-axis plane wave; for the aplanatic CDOE the coma is eliminated.

vere coma, whereas the CDOE still has no appreciable aberration. This outcome is all the more impressive because the two DOEs had the same size of 60 mm, yielding NA = 30/35 = 0.86 for the CDOE and NA = 30/(302 + 352 )1/2 = 0.65 for the flat DOE. Thus the CDOE, even when its NA is much higher, performs significantly better than the flat DOE. Welford [1975] investigated the transmission CDOE setup shown in fig. 6. Here the holographic layer is on the concave surface of a spherical meniscus substrate. Each reconstructing ray (shown with solid arrow in fig. 6) must pass the substrate before reaching the hologram, and the meniscus will in general introduce both spherical aberration and coma. The rays used for the recording are represented by broken lines in fig. 6. If the CDOE is reconstructed with a point source located at (xC , yC , zC ), the Gaussian image point will be located at (xI , yI , zI ). For paraxial rays (x, y z), 1 1 1 1 , = ±µ − (2.3) zI zC zO zR xI xC xO xR (2.4) , = ±µ − zI zC zO zR yI yC yO yR (2.5) , = ±µ − zI zC zO zR where (xO , yO , zO ) and (xR , yR , zR ) are the object and reference point locations, µ is the ratio between the reconstructing and the recording wavelengths, and the ± sign refers to the ±1st diffraction orders. The location of the paraxial image point is independent of R, the curvature of the CDOE. Both spherical aberration and coma can be eliminated in the following way (Welford [1975]): the radius of the inner surface of the meniscus and the distance of the reconstructing object point are considered given. The location of the reconstructed image point can be chosen such that eq. (2.2) is satisfied, where now a = zC and b = zI , hence coma

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[3, § 2

Fig. 6. Aplanatic transmission CDOE recorded on the inner surface of a meniscus substrate (Welford [1975]).

is canceled. The locations of the recording point sources must satisfy the paraxial imaging condition (2.3) which gives a relationship between zO and zR , but does not determine their values. They can be chosen such that the spherical aberration introduced by the meniscus is compensated, while relation (2.3) is maintained. The application of aplanatic transmission CDOEs was proposed for optical disk pick-ups by Tatsuno [1997]. The setup considered by Tatsuno [1997] is similar to the one shown in fig. 6, except that the CDOE is recorded on the outer surface, and the outer and inner surfaces of the substrate have a common center of curvature which coincides with the focal point of the CDOE. This geometry ensures that the finite thickness of the substrate does not introduce appreciable aberrations in the system. All the third-order aberration terms – spherical aberration (S), coma (C), astigmatism (A), field curvature (F ) and distortion (D) – were calculated by Jagoszewski [1985] and Jagoszewski and Klakoćar-Ciepacz [1986] for spherical CDOEs recorded and reconstructed with spherical waves. The aberrations have the following form: 2 1 1 1 1 1 1 1 1 , (2.6) − ± µ − S = 3 − 3 ±µ 3 − 3 + 2 R zC2 zI2 zO zR2 zC zI zO zR xC xI xR xO Cx = 3 − 3 ± µ 3 − 3 zC z zO z I R 1 xC xI xO xR + (2.7) , − ± µ − 2 R zC2 zI2 zO zR2 Ax =

xC2 zC3

−

xI2 zI3

±µ

xO2 3 zO

−

xR2 zR3

,

(2.8)

3, § 2]


Axy =

F =

xC yC zC3

−

xC2 + yC2

xI yI

±µ

zI3

xO yO 3 zO

xI2 + yI2

−

xR yR

zR3

xO2 + yO2

, xR2 + yR2

− ±µ − 3 zC3 zI3 zO zR3 2 xO2 + yO2 xI2 + yI2 xR2 + yR2 1 xC + yC2 , + − ±µ − 2 R zC2 zI2 zO zR2

Dx =

(xC2 + yC2 )xC zC3

±µ

−

(2.9)

(xI2 + yI2 )xI

(xO2 + yO2 )xO 3 zO

117

zI3 −

(xR2 + yR2 )xR zR3

.

(2.10)

As seen, S, C and F depend on R, while A and D are independent of it. Note that not only is coma canceled when the aplanatic condition (2.1) is satisfied, but the spherical aberration is also zero, provided that zC = zR (or zC = zO ) and µ = 1. For example, for a CDOE recorded with the setup shown in fig. 4, zero spherical aberration can be obtained when the CDOE is reconstructed either with a plane wave or with a point source that is situated in the front focal plane. A similar analytic calculation for the aberration terms of spherical CDOEs was performed by Peng and Frankena [1986]. Verboven and Lagasse [1986] provided a programmable formula for the numerical calculation of aberration coefficients up to arbitrary order, and for CDOEs with arbitrary shape. Third-order aberration calculations were also done for spherical CDOEs by Nowak and Zajac [1988] and Jagoszewski and Talatinian [1991b], but they also considered the effect of shifting the entrance pupil to a distance t from the CDOE, as shown in fig. 7. The radius of curvature of the CDOE and the position of the entrance pupil can be used as additional free parameters for numerical optimization, e.g., to achieve the smallest possible spot size. Several DOEs were compared numerically (flat DOE and CDOE, with and without a shifted entrance pupil). It was shown by Nowak and Zajac [1988] that the position of the entrance pupil has a stronger effect on the image quality than does the radius of curvature. The best results were obtained with a CDOE and a shifted entrance pupil. Bokor and Davidson [2001b] also showed that in terms of minimum spot size the optimal CDOE radius is not equal to the aplanatic radius given by eq. (2.2). However, for high NA (NA > 0.8) this discrepancy between optimal and aplanatic radii is negligible. At NA = 0.86 and θ = 5◦ , for example, the radius giving the minimum spot size is less than 2% different from the aplanatic radius.

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[3, § 2

Fig. 7. Imaging CDOE with an entrance pupil at a distance t from the CDOE.

As discussed above, spherical aberration can be eliminated by choosing zC = zR or zC = zO . Baskar and Singh [1992] investigated a CDOE which is recorded with a spherical object wave and a plane reference wave, and is reconstructed by a spherical wave for which zC = zR and zC = zO . The radius R for which spherical aberration is canceled was calculated from eq. (2.6). Note that for such a spherical aberration corrected CDOE the object field angle must be small, because coma – which grows linearly with object field angle – is not eliminated. In all of these investigations, the CDOE recording waves were considered to be perfect spherical or plane waves. Fisher [1989] proposed to use CDOEs for head-up displays in airplanes. Here the motivation to use CDOEs instead of flat DOEs is the need for increased field of view and large binocular overlap in a limited space. A spherical reflection CDOE was considered whose grating function, refractive index modulation and thickness were free parameters for iterative numerical optimization. The results of these numerical calculations showed that for best performance, one of the recording wavefronts has to be astigmatic, and the other has to have a complicated aspheric shape. A special application of CDOEs was reported for planar optics by Belostotsky and Leonov [1993]. Because of the plane geometry, astigmatism does not exist, and the only low-order aberrations that should be corrected in an imaging system are spherical aberration, coma and field curvature. Spherical aberration and coma can be simultaneously eliminated by a circular CDOE which satisfies the Abbe sine condition and for which zC = zR , as discussed above. Belostotsky

3, § 2]


119

and Leonov [1993] proposed to use an aplanatic planar Fresnel lens to act as the CDOE. Since an aplanatic Fresnel lens and an aplanatic refractive lens have field curvatures of opposite sign, they can be combined to form a hybrid system that eliminates field curvature too, as shown in fig. 8. Belostotsky and Leonov [1993] also gave a generalization of the Abbe sine condition (2.1) for the extraordinary waveguide mode in an anisotropic planar medium, for the special case of a uniaxial crystal with its crystal axis lying in the plane of propagation and perpendicular to the optical axis of the system:

2 2 2 sin α 1 + (no /ne − 1) sin β = const = −M, sin β 1 + (n2 /n2 − 1) sin2 α o e

(2.11)

where no and ne are the ordinary and extraordinary indices of refraction, respectively. A curve satisfying condition (2.11) is a section of an ellipse, instead of a circle as it was in the isotropic case. The semi-axes of the ellipse are ab/(a + b) and (ne /no )ab/(a + b), along the optical axis and along the crystal axis, respectively (see fig. 2 and cf. eq. (2.2)).

Fig. 8. The planar hybrid system of Belostotsky and Leonov [1993] that eliminates spherical aberration, coma and field curvature.

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[3, § 2

2.2. Spherical CDOE for Fourier transform An ideal Fourier-transform lens converts an incoming plane wave of arbitrary slant angle into a diffraction-limited spot at the back focal plane, also called the frequency plane. Each incoming plane wave represents a component in the angular spectrum of an input transparency which is placed in the front focal plane, and hence the back focal plane shows the 2D spectrum of the input transparency. The location of each diffraction-limited spot in the frequency plane should therefore be directly proportional to the spatial frequency component of the input transparency which is responsible for the given tilted plane wave. The spatial frequency of a grating is proportional to the sine of the tilt angle α of the diffracted plane wave, according to the grating relation 1 (2.12) , Λ where 1/Λ is the spatial frequency of the grating and λ is the illuminating wavelength. It follows that the location of spots at the frequency plane should be proportional to sin α. As was pointed out by Jagoszewski and Talatinian [1991a] and shown in fig. 9, this condition is exactly fulfilled for an aplanatic CDOE for principal rays. Therefore, aplanatic CDOEs, which are also free of aberrations that grow linearly with the angle, appear to be naturally suited for Fourier-transform applications. However, for simple aplanatic CDOEs that are recorded with perfect spherical waves, and whose phase function is therefore called “spherical”, the residual aberrations such as astigmatism and field curvature still impair the performance. In order to achieve minimum spot sizes over a given input angular range, the phase function of the CDOE needs to be optimized. sin α = λ

Fig. 9. Aplanatic CDOE used for Fourier transform (Jagoszewski and Talatinian [1991a]).

3, § 3]

Spherical/cylindrical CDOEs for concentration of diffuse light on flat targets

121

For the optimization of Fourier-transform CDOEs Talatinian [1991, 1992] proposed and demonstrated the so-called analytic raytracing technique developed first for flat Fourier-transform DOEs by Hasman and Friesem [1989]. This optimization technique yields a DOE with low overall aberrations, provided that there is an entrance pupil in front of the DOE. In this case each incoming plane wave illuminates only part of the DOE (a “local hologram”), as can be seen in fig. 9. Talatinian [1991, 1992] compared numerically three types of CDOEs: (1) having spherical phase function, (2) having quadratic phase function, and (3) having optimized phase function obtained from analytic ray tracing. In each case several CDOEs with different radii of curvature were considered. In terms of spot size, CDOEs with quadratic phase function perform almost as well as the optimized CDOEs. Additionally, the radius of curvature satisfying the aplanatic condition (2.2) does not yield the smallest spot sizes for a given input angular range, hence not only the grating function has to be optimized, but the radius of the CDOE too. Naturally, a CDOE having a spherical phase function can very easily be fabricated, since only spherical wave fronts are required for the holographic recording. On the other hand, CDOEs having optimized phase functions require complicated aspherical wavefronts for the recording. Such wavefronts can be produced, e.g., with computer-generated holograms.

§ 3. Spherical/cylindrical CDOEs for concentration of diffuse light on flat targets Concentration of diffuse light has many applications, including solar energy concentration, light collection for optical instruments and efficient light coupling into fiber optics (Winston and Welford [1989], Davidson and Bokor [2003]). In this section we show that aplanatic spherical/cylindrical CDOEs that eliminate firstorder coma are inherently capable of concentrating quasi-monochromatic diffuse light onto a flat target at the theoretical limit imposed by the second law of thermodynamics (or, equivalently, the conservation of brightness). This thermodynamic limit on the amount of concentration is 1/ sin αx (in beam width) in 1D and 1/(sin αx sin αy ) (in beam area) in 2D, where αx and αy are the half-divergence angles of the diffuse input beam in the transverse directions (Davidson and Bokor [2003]). Note that in the common terminology of the literature (Bassett, Welford and Winston [1989]) “1D concentration”, i.e. concentration along one spatial direction, is achieved by “2D concentrator shapes” z(x), whereas “2D concentration” is achieved by “3D concentrator shapes” z(x, y).

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[3, § 3

For 1D coma-free imaging the concentration ratio is (Bokor, Shechter, Friesem and Davidson [2001]) CRideal =

Xin NA = , Xout sin αx

(3.1)

where Xin and Xout are the input and output lateral beam sizes, respectively. Note that as NA approaches 1, eq. (3.1) approaches the 1D thermodynamic limit CRmax = 1/ sin αx . The normalized concentration ratio is defined as CR/CRmax = (Xin /Xout ) sin αx , so that at the thermodynamic limit it is equal to 1. In general, high-NA imaging concentrators, in particular those that use simple parabolic mirrors, suffer from large aberrations which reduce the concentration ratio far below the theoretical limit (for example, by a factor of 2 for 1D and a factor of 4 for 2D concentration by optimal parabolic mirrors). Aplanatic CDOEs cancel those aberrations that depend linearly on the incoming divergence angle, and hence are good candidates to achieve ideal concentration for quasi-monochromatic sources and small incoming diffusive angles. We note that nonimaging concentrators, that were also shown to achieve diffuse-light concentration close to the thermodynamic limit (Winston and Welford [1989]), are limited in use for small incoming diffuse angles, since they are extremely elongated in this case. Bokor, Shechter, Friesem and Davidson [2001] investigated an aplanatic CDOE theoretically and experimentally for diffuse-light concentration. The normalized concentration ratio was calculated analytically, using a paraxial expansion of the diffraction relation for small diffusive angles, and confirmed by numerical ray tracing. The results are shown for 1D concentration in fig. 10, together with results from two other concentrators: the flat DOE and the parabolic mirror (PM). The normalized concentration ratios are NA, NA · (1 − NA2 )1/2 and NA · (1 − NA2 ), for the CDOE, the PM and the flat DOE, respectively. As expected, for small diffusive angles the concentration capabilities of the CDOE are in accordance with eq. (3.1), and hence approach the thermodynamic limit at high NA. The maximum achievable concentration ratio for the PM and the flat DOE is 50% (at NA = 0.71) and 38% (at NA = 0.58) of the thermodynamic limit, respectively. For 2D concentration, the best normalized concentration ratio for the PM and the flat DOE is even lower, namely 25% and 14%, respectively, of the thermodynamic limit, whereas for the CDOE concentration at the thermodynamic limit is still achieved. The failure of the parabolic concentrator at high NA can be understood by noting that at each point an incident light cone with half-angle αx is reflected as an

3, § 3]

Spherical/cylindrical CDOEs for concentration of diffuse light on flat targets

123

Fig. 10. Calculated normalized concentration ratios as a function of numerical aperture (NA) for 1D concentration on a flat target with a CDOE, a parabolic mirror (PM) and a flat DOE (Bokor, Shechter, Friesem and Davidson [2001]).

identical cone towards the focus. However, as the cone intersects the optical axis at an angle θ (where NA = sin θmax ), the focal spot includes an inclination factor 1/ cos θ , which diverges for large NA, and the spot size reaches an optimum at NA = 0.71. For the CDOE the diffracted light cones for high θ are actually narrower than the incident cones by exactly cos θ . This result is directly obtained from the diffraction relations of the grating for small αx and is an excellent approximation even for large αx . Hence the effect of the inclination factor 1/ cos θ is exactly canceled, and an identical, diffraction-limited spot size is obtained even as the NA approaches 1 (and hence a higher concentration ratio). For the flat DOE, the reflected light cone from each point is 1/ cos θ times wider than the incident cone, resulting in an even worse concentration. Bokor, Shechter, Friesem and Davidson [2001] demonstrated diffuse light concentration experimentally with a CDOE using a cylindrical aplanatic CDOE, recorded as a reflection hologram on Dupont photopolymer film. The CDOE was recorded with a spherical wave and a plane wave and had NA = 0.86. An experimental normalized concentration ratio of 0.79 was obtained (which is 92% of the theoretical limit at this NA), for an input beam with a diffusive half-angle of αx = 1◦ . The aplanatic CDOE has thus the advantage of ideal concentration of diffuse light at the thermodynamic limit. Additionally, it is very simple to fabricate since a cylindrical/spherical substrate, and only readily available cylindrical/spherical and plane waves are needed for the recording. However, in order to reach high diffraction efficiency, Bragg CDOEs, or blazed surface-relief CDOEs should be used for concentration. In this case the angular selectivity of the CDOE puts an

124


[3, § 3

upper limit on the usable input diffusive angle αx . The angle αx is also limited by higher-order aberrations, which the aplanatic CDOE does not eliminate. Furthermore, by simply reversing the direction of the light rays, the spherical/cylindrical CDOEs can also uniformly collimate the light emitted from a flat Lambertian source. This result is obtained as a special case from the uniform collimator design principle, presented in Section 4 for general Lambertian sources. In a somewhat related approach, Kritchman, Friesem and Yekutieli [1979a, 1979b] suggested a curved refractive Fresnel lens for 1D concentration of sunlight. The system is shown schematically in fig. 11. The basic design principle is that two collimated light beams from the edges of the acceptance field should ideally be aimed by the curved Fresnel lens toward two stigmatic focus points, respectively: a/2 = f tan αx , where a is the size of the output aperture and f is the height of the center of the lens above the output plane. If this requirement is satisfied then every light ray included in the acceptance field will reach the output aperture. This principle is analogous to the so-called edge-ray principle (Winston and Welford [1989]) widely used for the design of nonimaging concentrators. From the design principle, after symmetry considerations and application of Snell’s law, the lens shape z(x) and the primary angle of the prismatic grooves Φ(x) can be found numerically for given αx and n, where n is the index of refraction. If n is small then the obtainable concentration ratio is much worse than the theoretical limit (e.g., by a factor of 2.6 for n = 1.1 and by a factor of 1.33 for n = 1.5, when αx → 0◦ ). In the limiting case of αx → 0◦ and n = ∞, the curved Fresnel

Fig. 11. The curved Fresnel lens of Kritchman, Friesem and Yekutieli [1979a] for 1D concentration of sunlight.

3, § 4]

Uniform collimation and ideal concentration for arbitrary source and target shapes 125

lens has no grooves and can be treated as a thin aplanatic CDOE whose shape is cylindrical, in accordance with relation (2.2).

§ 4. Uniform collimation and ideal concentration for arbitrary source and target shapes 4.1. Uniform 1D collimation and concentration of diffuse light at the thermodynamic limit Uniform plane wave illumination is of great importance in many areas of optics. A point source can be simply collimated with a parabolic mirror (PM) that reflects all rays originating from the focal point into the same direction without any aberrations. However, even for isotropic point sources, the PM yields a collimated beam of nonuniform intensity. The crux of the problem here is that the shape of the mirror required for collimation is uniquely determined by Snell’s reflection law to be a parabola, whereas uniform intensity of the reflected light requires a different shape. Following Bokor and Davidson [2002], in this section we first present the design principle of a CDOE reflector that can independently fulfill the two requirements for quasi-monochromatic light: its shape ensures uniform illumination for arbitrarily shaped diffuse sources, and its phase function ensures optimal collimation. Extension of the method for any specific nonuniform illumination is straightforward. The geometric arrangements for recording and readout of a curved holographic collimator operating in 1D are shown in fig. 12. Figure 12(a) shows a recording arrangement of the interference of a plane wave and a counter-propagating cylindrical wave on a holographic film with shape z(x). Figure 12(b) shows the readout arrangement with the same diverging cylindrical wave. As seen, the first diffraction order yields a perfectly collimated plane wave, regardless of the CDOEs shape. Note that this readout geometry also automatically fulfills the Bragg condition for the entire CDOE, hence permitting high diffraction efficiencies. The shape z(x) can now be designed to yield uniform intensity of the collimated beam for a given illumination source. Consider a small source located at x = 0, z = 0, and emitting radiation in the xz plane with angular (far-field) intensity distribution S(β), where β is the angle between the direction of the radiation and the z axis. For an isotropic point source S(β) = const, for a planar diffuse Lambertian source S(β) ∝ cos β, and for general elongated Lambertian sources S(β) ∝ ∆(β), where ∆(β) is the projected diameter of the source as seen from direction β. For a cylindrical Lambertian

126


(a)

[3, § 4

(b)

Fig. 12. (a) Recording and (b) readout geometry of a curved holographic collimator of arbitrary shape z(x).

source S(β) = const, the same as for an isotropic point source. If I (x) is the local intensity of the output beam at vertical coordinate x, then from energy conservation we get S(β) dβ = I (x) dx.

(4.1)

Thus to ensure uniform intensity along x, i.e. I (x) = const, light rays emitted from the source toward direction β must intersect the CDOE at x(β), obeying the relation dx = F · S(β), dβ

(4.2)

where the constant F is the paraxial focal length of the collimator, chosen to be much larger than the source size. Uniform diffraction efficiency was assumed in eq. (4.2). x(β) is obtained from eq. (4.2) by integration, and z(β) is simply given as (see fig. 12) z(β) =

x(β) . tan β

(4.3)

Hence the shape z(x) of the uniform collimator CDOE is obtained. As an example, consider the simplest case, of an isotropic “1D” light source (e.g., an elongated line source or a diffuse cylindrical source), where S(β) = const. The shape of the uniform collimator CDOE for this case, derived from eqs. (4.2) and (4.3), is z(x) =

x . tan(x/F )

(4.4)

3, § 4]


Equation (4.4) is defined for the entire angular distribution −180◦ β < 180◦ , where −πF < x < πF and −∞ < z F , and therefore uniformly collimates the entire source energy. Part of z(x) of eq. (4.4) is shown in fig. 13(b). As shown, the rays originating from the point source at equal angular density also have uniform spatial density along x, after collimation. On the other hand, such rays are collimated by a PM (whose shape is z(x) = F − x 2 /(4F )) with nonuniform spatial density along x, as seen in fig. 13(a). For the PM the intensity of the collimated beam satisfies I (x) ∝ 4/[(x/F )2 + 4)], having a maximum at β = 0◦ (x = 0) and dropping monotonically to zero for β = ±180◦ (x = ±∞), whereas for the CDOE I (x) = const for all β. Next, we consider the finite angular spread of the collimated beam resulting from a diffuse source with finite size. Here the goal is to design a collimator with a minimum angular spread. Alternatively, when the direction of the rays is simply reversed, this goal is equivalent to designing a concentrator that will concentrate a uniform incoming diffuse beam on a small target of the given shape. Although in many applications light is concentrated on flat targets (e.g., detectors, solar cells), for other applications, such as cylindrical water pipes and cylindrical laser rods, the target shape is curved. For curved targets the thermodynamic limit on concentration (discussed in Section 3) must be applied to the circumference length of the target. For example, a diffuse beam may be concentrated on a cylindrical target with diameter π times smaller than that of a flat target. It can be analytically proved that a CDOE that fulfills eqs. (4.2) and (4.3) to collimate light uniformly for a given light source shape is inherently also an ideal

(a)

(b)

Fig. 13. (a) The PM that gives nonuniform collimation, (b) the CDOE shape designed by Bokor and Davidson [2002] that yields uniform collimation.

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[3, § 4

diffuse light concentrator [collimator] for the same shape as a target [source], operating at the thermodynamic limit. For example, the CDOE of fig. 13(b) is an ideal collimator for small cylindrical sources and an ideal diffuse light concentrator onto cylindrical targets. In the concentrator application the ray directions of fig. 13(b) are simply reversed, and the target is placed in the focus so that its center coincides with the focal point. Figure 14(a) illustrates why a PM fails as a diffuse-light concentrator for 1D concentration on a uniform cylindrical target (see also our discussion of the PM for flat targets in Section 3). At each point on the PM the incident light cone with half angle α is reflected as an identical cone toward the focus. As |β| approaches 180◦ , the distance from the reflection point to the focal point approaches infinity, and hence the spread of the reflected light cone at the focus also approaches infinity, resulting in a dramatic decrease in concentration performance. On the other hand, for the uniform collimator CDOE, as |β| approaches 180◦ , the diffracted light cones are actually much narrower than the incident light cones (α α), as illustrated in fig. 14(b), and compensate for the increased distance. Expansion of the diffraction relations of the grating for small α and simple algebra yield for the CDOE that the extreme incident rays at angles ±α from the z axis are directed exactly tangent to the target, and all intermediate rays hit the target for all β. Just like for flat targets, discussed in Section 3, these considerations are similar to the edge-ray principle (Winston and Welford [1989]) used for the design of nonimaging concentrators. Figure 15 presents the calculated normalized concentration ratio onto a cylindrical target, as a function

(a)

(b)

Fig. 14. Diffuse light concentration onto a small cylindrical target by (a) the PM, and (b) the uniform collimator CDOE; the CDOE reaches the thermodynamic limit of light concentration (Bokor and Davidson [2002]).

3, § 4]


Fig. 15. Calculated normalized concentration ratios onto a small cylindrical target as a function of |βmax | for the uniform collimator CDOE and the PM (Bokor and Davidson [2002]).

of |βmax | for both the uniform-collimator CDOE and the PM. For the CDOE the normalized concentration ratio is |βmax |/π, which yields concentration at the thermodynamic limit for |βmax | = 180◦ . For the PM the normalized concentration ratio is sin |βmax |/π, resulting in an optimal concentration at |βmax | = 90◦ , the maximum concentration ratio being π times below the thermodynamic limit. Bokor and Davidson [2002] verified the design procedure experimentally by holographically recording a reflection CDOE of the shape given by eq. (4.4) (fig. 13(b)). The focal length of the CDOE was F = 20 mm. Several exposures were used to record a hologram for the range −90◦ β 90◦ (corresponding to NA = 1 and a lateral size of 2xmax = 58 mm). Since a spherical (instead of a cylindrical) wave was used for the recording, a 5 mm slit in the y direction ensured that the setup worked essentially as a 1D concentrator. To test the performance of the uniform collimator CDOE as an ideal concentrator, the entire aperture of the CDOE was illuminated with a monochromatic plane wave that was inclined at variable angle α with respect to the z axis. A cylindrical black target with radius r = 1.5 mm at the focal point of the CDOE blocked all the concentrated light until α reached the value of ±r/F = ±4.3◦ . The unblocked light power P was measured by integrating the imaged intensity on a white diffuse screen placed behind the target. The measured P (α) graph, shown in fig. 16 with diamonds, features a discontinuous sharp transition at α = r/F , as expected for an ideal concentrator. The theoretical P (α) curve for the CDOE concentrator (fig. 16, solid line) has a somewhat sharper, but also finite slope, which is due to the finite size of the target, corresponding to a finite input diffuse angle. Finally, the theoretical P (α) concentrator curve for the PM concentrator (fig. 16, dashed line) has a much gen-

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[3, § 4

Fig. 16. Experimental data by Bokor and Davidson [2002], characterizing the performance of the uniform collimator CDOE as a diffuse light concentrator onto a small cylindrical target: the measured light intensity not blocked by the cylindrical target as a function of the illumination angle (diamonds), compared with calculations for the uniform collimator CDOE (solid curve) and the PM (dashed curve).

tler slope, indicating concentration well below the thermodynamic limit (Bokor and Davidson [2002]). The design procedure for uniform collimator CDOE described by eqs. (4.2) and (4.3) can be applied for general Lambertian source shapes. For example, the uniform collimator [concentrator] CDOE for a one-sided flat source [target], common in many practical cases, is found to have the half-cylinder shape that satisfies the Abbe sine condition (see Sections 2 and 3). Equations (4.2) and (4.3) also yield simple analytic solutions for the CDOE shapes for Lambertian sources having oval, square, triangular, and many other cross-sections (Bokor and Davidson [2002]). We stress that the actual shape of the fabricated CDOE does not have to follow the analytical shape obtained from eqs. (4.2) and (4.3) within an optical precision. The reason for this is that the quality of concentration depends only weakly on the exact shape of the CDOE (as long as it is identical during recording and reconstruction), making this technique insensitive to small fabrication errors.

4.2. Extension to finite distances The design principle described above was implemented and generalized by Davidson and Bokor [2004a] for finite distances, both for reflection and transmission geometries in 1D. The problem to be solved can be stated as follows: the light of an elongated Lambertian source of given size and cross-sectional geometry is to be uniformly concentrated onto an elongated target of given cross-sectional shape

3, § 4]


and the smallest cross-sectional size that is permitted by brightness conservation. This general problem has great practical significance, e.g., for side pumping of solid-state laser rods, where high geometrical efficiency and uniform pump density profile are important. First consider a reflection CDOE. Figure 17(a) shows schematically the design geometry for a reflection CDOE for 1D diffuse-light concentration. A small source is centered at point A and a small target is located at a distance d, centered at point B. The source emits radiation with angular (farfield) intensity distribution S1 (α), where α is the angle between the direction of the radiation and the line connecting A and B. As discussed above, for general 2D Lambertian sources S1 (α) ∝ ∆1 (α), where ∆1 (α) is the projected diameter of the source as seen from direction α. Similarly, S2 (β) is the local intensity of the output beam in direction β, measured at the target, as shown in fig. 17(a). From energy conservation we get S1 (α) dα = S2 (β) dβ.

(4.5)

To ensure uniform illumination on a Lambertian target, S2 (β) must be proportional to ∆2 (β), where ∆2 (β) is the projected diameter of the target as seen from direction β. Hence for Lambertian source and target, eq. (4.5) takes the form ∆1 (α) dα = ∆2 (β) dβ.

(4.6)

The CDOE shape r(α) for specific ∆1 (α) and ∆2 (β) geometries can be obtained by solving differential eq. (4.6) for β, and then substituting it into the relation r(α) =

d sin β(α) , sin[α − β(α)]

(4.7)

as seen in fig. 17(a). The meaning of d as the scaling parameter is similar to that of F in eq. (4.2). This design principle together with eqs. (4.5) and (4.6) is also valid for the geometry of a transmission CDOE concentrator, presented in

(a)

(b)

Fig. 17. Design geometry of a CDOE working (a) in reflection, (b) in transmission, for finite distance concentration of diffuse light.

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[3, § 4

fig. 17(b). However, since here α is measured from the opposite direction, eq. (4.7) is modified to d sin β(α) r(α) = (4.8) . sin[α + β(α)] Similarly as for the uniform collimator CDOE described in Section 4.1, again by expanding the diffraction relations of the holographic grating for small diffuse angles, i.e. when the source and target sizes are much smaller than d, it can be shown analytically that for the CDOE designs of eqs. (4.6)–(4.8) the rays emitted tangentially by the source are directed exactly tangent to the target, and all intermediate rays emitted by the source hit the target for all α. Moreover, the phase-space area is conserved, i.e. (phase-space area)target = (phase-space area)source . Therefore, if there is no loss in power, such a device concentrates diffuse light on the theoretical limit of brightness conservation, i.e. on the smallest possible target size of the given geometry. The experimental realization of such a holographic element is extremely simple, since the curved hologram can be recorded using two simple cylindrical laser beams, originating from locations A and B (see figs. 17(a) and 17(b)). As the first example of a CDOE designed from eqs. (4.6) and (4.7), consider a cylindrical source emitting rays with uniform angular distribution in the entire angular range −180◦ α < 180◦ , and a target that receives rays with uniform angular distribution in the limited angular range (−180◦ /M) β < (180◦ /M), where M is the linear magnification of the CDOE. (The role of source and target can, of course, be interchanged, and then M becomes demagnification.) Here both ∆1 (α) and ∆2 (β) are constants, and eq. (4.6) yields dα = M dβ. The solution is β = α/M, and from eq. (4.7) we get r(α) = d sin(α/M)/sin[(1 − 1/M)α]. The CDOE shape corresponding to M = ∞ is the uniform collimator/concentrator discussed above and shown in fig. 13(b). The CDOE shape corresponding to M = 2 is a simple cylinder (r(α) = d = const), where A is at the center and B is on the circumference. As M → 1, the CDOE cross-sectional shape approaches the so-called cochleoid curve, for which r(α) ∝ (sin α)/α. The M < ∞ cases of the CDOE shapes can be applied to direct the rays coming from a cylindrical source onto a cylindrical target, where the diameter of the target is M times larger than the diameter of the source. Figure 18(b) shows how, for example, two CDOEs with M = 2 can be used in a double-lamp configuration in the pumping of solid-state lasers. The two cylindrical pump lamps are located in the centers of the two cylindrical CDOEs, and the axis of the cylindrical laser rod coincides with the straight line where the two CDOEs touch. To avoid shadowing effects, neither of the two CDOEs is completely closed, as shown in fig. 18(b). This leads to a small geometrical loss in

3, § 4]


(a)

(b) Fig. 18. Double-lamp pumping configuration using (a) two elliptical mirrors, (b) two reflective CDOEs (Davidson and Bokor [2004a]).

power, since some of the rays emitted by the sources do not reach the CDOE. However, if the source and the target sizes are made much smaller than the DOE cross-sectional size, this geometrical loss becomes negligible. Moreover, since the magnification of both CDOEs is M = 2, the combined phase-space area of the two sources is expected to be equal to the phase-space area of the target. To summarize, the double-CDOE configuration of fig. 18(b) conserves brightness and provides a uniform pump density profile. For comparison, fig. 18(a) shows a double elliptical pumping configuration, which is widely used for solid state lasers. In order to provide the same magnification M = 2 (and hence phase-space area conservation) as in the CDOE case, an eccentricity of 1/3 was chosen for both ellipses. As seen in fig. 18(a), combining the two ellipses causes relatively large portions of both mirrors to be missing, and hence a large percentage of the rays

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[3, § 4

emitted by the sources does not arrive at the proper elliptical mirror surface and is essentially lost. The geometrical loss in this case is ∼30%, which corresponds to an equal amount of loss in brightness. An additional drawback of the double elliptical cavity, and elliptical mirrors in general, is its highly nonuniform pump density profile, as illustrated by fig. 19 (Davidson and Bokor [2004a]). Figures 19(a) and 19(b) show phase-space spot diagrams of rays hitting the cylindrical target, for the double elliptical reflector and for the double CDOE, respectively, obtained from numerical ray tracing for the paraxial case (i.e. when the source and target sizes are much smaller than the size of the cavity). The horizontal axis l represents the location of the rays along the circumference of the target, and the vertical axis represents the sine of the angle εl between the rays and the surface normal. The density of dots in any region on the phase-space diagrams gives directly the local brightness of light in that region. The rays are emitted isotropically from the sources, half from the left source (solid circles in fig. 19) and half from the right source (open circles). Figure 19(a) is found to contain only ∼70% of the illuminating dots, in agreement with the geometrical loss calculated above. As also seen, the phase space of the double elliptical reflector is highly nonuniform and contains empty regions. On the other hand, the double-CDOE configuration yields uniform phase-space density, and light concentration at the thermodynamic limit of brightness conservation. Since uniform pump density profile, besides high pumping efficiency, is often desirable in a laser rod, especially when fundamental transverse modes are

(a)

(b)

Fig. 19. Phase-space spot diagrams obtained from numerical raytracing by Davidson and Bokor [2004a], for (a) the double elliptical mirror cavity of fig. 18(a), and (b) the double CDOE cavity of fig. 18(b).

3, § 4]


to be excited, CDOEs may give a practical alternative to elliptical reflector tubes as pump cavities. For the special case of a flat Lambertian source and a flat Lambertian target, eq. (4.6) takes the form cos α dα = M cos β dβ, which gives back exactly the Abbe sine condition sin α/ sin β = M, and the CDOE shape is a cylinder, in agreement with the aplanatic CDOE discussed in Sections 2 and 3. The phasespace area conservation can be proved for this case by noting that although the target size is M times larger than the source size, the sine of the angular range of rays hitting the target is exactly M times smaller than the sine of the angular range of rays emitted by the source. For M = 1 the radius of the cylinder becomes infinity (corresponding to a flat DOE, placed halfway between A and B), and for M → ∞ the circle is centered at point A. Equation (4.6) can hence be thought of as a generalization of the Abbe condition for arbitrary Lambertian target and source shapes. For sources and targets having an axial symmetry the technique can be extended to 3D CDOE shapes and 2D concentration/collimation in a straightforward way (Davidson and Bokor [2004a]). Because of the solid angles involved, the energy conservation equation (4.5) is modified to S1 (α) sin α dα = S2 (β) sin β dβ,

(4.9)

where S1 (α) and S2 (β) are defined in the same way as for the 1D concentration case. Interestingly, for a flat Lambertian source and a flat Lambertian target, the 3D-shape design (4.9), which takes the form cos α sin α dα = M cos √ β sin β dβ, leads to Abbe’s sine condition again (here its form is sin α/ sin β = M ), just as eq. (4.6) did for the 2D CDOE shapes. For this geometry, the 3D CDOE shape is spherical, and can be thought of as a simple rotated version of the 2D shape. Another simple analytic shape is given by a spherical source and a flat target. In this case, eq. (4.9) can be written as sin α dα = C cos β sin β dβ, and the solution is β = α/2. This is also a sphere, with the spherical source at its center and the flat target at its surface. For infinitesimal source sizes the uniform collimator design eq. (4.9) gives correct results, however, it can be shown analytically that the “edge ray principle” discussed above for elongated Lambertian sources and targets does not apply automatically for 3D CDOE shapes. The reason for this is the effect of skew rays, i.e. rays that are not in the same plane as the symmetry axis of the system (Winston and Welford [1989]). The diffuse ray pencils hitting the CDOE in the skew direction are unaffected by the grating and are directed towards the target as identical diffuse ray pencils, similarly to simple reflections. For the case of a flat source and a flat target, where Abbe’s sine condition is satisfied, uniform concentration

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[3, § 5

at the thermodynamic limit is achieved for the 3D CDOE shapes too, whereas in other cases, such as spherical sources and spherical targets, skew rays impair the performance of the CDOE and result in concentration below the thermodynamic limit (Davidson and Bokor [2004a]).

§ 5. CDOEs for controlling the geometrical apodization factor in vectorial diffraction problems In this section we demonstrate how a CDOE with an appropriate shape can control the principal surface, and hence the geometrical apodization factor, of a high NA focusing device, and thereby improve its performance (Davidson and Bokor [2004b], Bokor and Davidson [2004]). In our discussion both the focusing system and the illumination intensity are assumed to have axial symmetry. By use of vectorial diffraction theory for a focusing element, the electric field vector near the focus at point P can be obtained from the generalized Debye integral (Richards and Wolf [1959]), expressed as dsx dsy i

A2 (θ )

a (θ, φ) exp ik(sx x + sy y + sz z) . (5.1) E(P) =− λ sz Ω Here E is the electric field vector, k = 2π/λ with λ the wavelength of illumination, (sx , sy , sz ) is the propagation unit vector toward the focus, θ is the focusing angle (the angle between the optical axis and the propagation unit vector) so that NA = sin(θmax ), φ is the azimuth angle, and a (θ, φ) is a unit vector representing the polarization direction of the given electric field component. The amplitude factor A2 (θ ) can be expressed as A2 (θ ) = A1 (θ ) · Aap (θ ), where A1 (θ ) is the amplitude distribution of the collimated input beam at the entrance pupil, and Aap (θ ) is the so-called apodization factor, obtained from energy conservation and geometric considerations. The integration is done over a solid angle Ω that covers the entrance pupil of the system. The Debye approximation is valid for high-NA systems where the focal length is much larger than λ. The physical meaning of the geometrical apodization factor Aap (θ ) can be understood with the help of fig. 20. The focusing system can be characterized with the geometrical shape of its principal surface. The principal surface consists of the points of intersection between the incoming ray directions and the directions of the corresponding rays that are deflected toward the focus. For example, for a thin lens (or a flat DOE) the principal surface is flat, for a PM it is a paraboloid, and for an aplanatic system (which, as discussed before, eliminates first order transverse aberrations) it is a sphere centered on the focal point. The shape of the principal

3, § 5]

CDOEs for controlling the geometrical apodization factor

137

Fig. 20. Schematic diagram of a high-NA focusing system with principal surface ρ(θ); the principal surface can be controlled by CDOE shape.

surface is given by the function g(θ ) for which ρ = f · g(θ ),

(5.2)

where ρ is the distance of the input ray from the optical axis, and f is the focal length. For example, for an aplanatic system g(θ ) = sin θ , for a flat DOE g(θ ) = tan θ , for a PM g(θ ) = 2 tan(θ/2), and for a Herschel-type system (which eliminates first-order longitudinal aberration, Born and Wolf [1993]) g(θ ) = 2 sin(θ/2). As can be understood from fig. 20, the input amplitude distribution A1 (θ ), which depends on the density of rays in the collimated input beam, is different from the output amplitude distribution A2 (θ ), which depends on the angular density of the rays directed toward the focus. The geometrical intensity law yields the following relationship between A2 (θ ) and A1 (θ ): g(θ )g (θ ) 1/2 , A2 (θ ) = A1 (θ ) (5.3) sin θ where g (θ ) = dg/dθ . As seen, A2 (θ ) is the product of two terms, the first of which depends only on the incident illumination and is independent of the focusing system, while the second depends only on the geometrical properties of the focusing system. Hence g(θ )g (θ ) 1/2 . Aap (θ ) = (5.4) sin θ For an aplanatic system Aap (θ ) = cos1/2 θ , for a flat DOE Aap (θ ) = 1/ cos3/2 θ , for a PM Aap (θ ) = 2/(1 + cos θ ), and for a Herschel-type system Aap (θ ) = 1.

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[3, § 5

The apodization factor Aap (θ ) can be used as a free parameter when a system is designed to optimize various properties of the focused field, such as the focus area, the focal depth, and the height of the sidelobes. Lens systems can be designed to have a pre-described set of principal surfaces, and hence, apodization factors. A simple alternative way of controlling Aap (θ ) is made possible by the fact that for a thin curved diffractive focusing lens the principal surface is simply the shape of the lens substrate (Davidson and Bokor [2004b]), as seen in fig. 20. Hence a high-NA CDOE having an appropriate substrate shape can directly determine the apodization factor Aap (θ ) through eq. (5.4), where g(θ ) now describes the shape of the CDOE. As explained before, the fabrication of such a CDOE is extremely simple, since a plane wave and a spherical wave (originating from the focal point) can be used to record it holographically. However, since the grating period at the edge of a high-NA CDOE approaches λ (where incoming rays are deflected by a large angle toward the focus), special care must be taken to include in the calculation of eq. (5.1) the space-variant diffraction efficiencies and phase delays imposed by each local grating (Turunen, Kuittinen and Wyrowski [2000]). These can be calculated from vectorial rigorous coupled-wave analysis (Moharam and Gaylord [1982]), but are not considered here. The situation is further complicated by the fact that CDOEs can also drastically change the polarization state of the incoming beam. An important exception to this is when the CDOE receives purely TE or purely TM radiation, i.e. when the electric field vector is everywhere parallel with or everywhere perpendicular to the grating lines. In these cases, the polarization state of the input beam is unchanged upon diffraction. High-NA focusing CDOEs have axial symmetry and concentric grating grooves. For such CDOEs the purely TM case corresponds to radial polarization, and the purely TE case to azimuthal polarization. These two polarization states are relevant in many high-NA applications. For example, as Dorn, Quabis and Leuchs [2003] demonstrated both theoretically and experimentally for high-NA aplanatic lenses, a radially polarized input beam yields a sharper spot at the focus than a linearly polarized input beam, especially for the longitudinal electric field component. As an example for the application of CDOEs to control Aap (θ ), Bokor and Davidson [2004] considered the 4π focusing system shown schematically in fig. 21. The basic idea of the 4π geometry (Hell and Stelzer [1992]) is to enhance the axial (z-)resolution in 3D scanning confocal microscopy by coherently illuminating the sample with two counter-propagating waves. However, the improved axial resolution is usually accompanied by high-intensity sidelobes in the axial direction, especially when both focusing objectives are aplanatic lenses and the illuminating beams have linear polarization. Bokor and Davidson [2004] investigated

3, § 5]

CDOEs for controlling the geometrical apodization factor

139

Fig. 21. Schematic diagram of a 4π focusing system with principal surfaces ρ(θ), illuminated with two counterpropagating radially polarized doughnut beams with a relative π phase shift (Bokor and Davidson [2004]).

the applicability of radially polarized beams and properly designed principal surface shapes to achieve a sharp, nearly spherical focal spot with uniformly low sidelobe intensities in the longitudinal and transverse directions. In fig. 21 the instantaneous electric field vectors are represented by dotted arrows, and the layer of the specimen under investigation is represented by a dashed line. To have a strong longitudinal electric field component in the focus, the two counter-propagating radially polarized beams need to have opposite phases, as shown in fig. 21. Thus the polarization distribution of the focused field strongly resembles that of a dipole radiation pattern, propagating outward from the focal point. The principal surfaces of the two focusing objectives are characterized by their ρ(θ ) shape, according to relation (5.2). Several focusing systems, each having different apodization factors, were investigated. The focal spots having the best spherical symmetry and uniformly low sidelobes along the longitudinal and transverse directions were obtained for a focusing system that had two Herschel-type principal surfaces. The calculated logarithmic contour map of the electric field intensity near the focus of such a system is shown in fig. 22. Here R and Z are the radial and axial directions at the focus. Other geometries such as the Lagrange system (which can be thought of as the rotated version of the 2D uniform collimator of eq. (4.4), and for which Aap (θ ) = (θ/ sin θ )1/2 ) and the parabolic surfaces also performed significantly better than the widely used aplanatic objectives. Figure 23 presents the calculated radial and axial intensity profiles at the focus for the Herschel-type system, along with the corresponding intensity profiles for an aplanatic system, showing

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[3, § 5

Fig. 22. Calculated intensity contour map near the focus in the (R, Z) plane for a Herschel-type 4π focusing system and radial illumination polarization, yielding a nearly spherical central spot and low radial and axial sidelobes (Bokor and Davidson [2004]).

(a)

(b)

(c)

(d)

Fig. 23. Calculated normalized intensity profiles near the focus, along (a,c) the R direction, and (b,d) the Z direction; (a) and (b) are for a Herschel-type 4π focusing system, and (c) and (d) for an aplanatic 4π focusing system (Bokor and Davidson [2004]).

3, § 6]

Curved gratings for spectroscopy

141

the poorer spherical symmetry and higher axial sidelobes for the aplanatic system. These results demonstrate the importance of controlling the geometrical apodization factor for specific purposes.

§ 6. Curved gratings for spectroscopy As already stated in Section 1, curved spectroscopic gratings had been reviewed by Welford [1965] and Schmahl and Rudolph [1976], so we mention this application only briefly here. We focus on recent developments, where either the CDOE shape or the grating-line shapes or both are optimized to eliminate or suppress certain aberrations. The basic setup for a curved grating spectrograph is shown in fig. 24. The CDOE is a curved reflection grating and the dotted line represents the well-known Rowland circle which has a diameter equal to the radius of the CDOE. Figure 24 shows the spectrograph application, as opposed to the monochromator, where A is the entrance slit, and the spectrum is observed at the exit aperture B. The Rowland geometry has several important advantages. First, if the object A is placed on the Rowland circle then sharp spectroscopic lines, free of meridional coma, will be formed at the image B, also on the Rowland circle. Hence, the Rowland geometry is, in this sense, aplanatic. Second, the spectrograph consists of a single optical element, without any need for additional lenses or mirrors. The latter property is especially useful in far-ultraviolet applications, where high reflection and transmission losses are a key issue. Although the simple Rowland geometry eliminates

Fig. 24. The basic setup for a Rowland-type curved grating spectrograph.

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[3, § 6

meridional coma at all wavelengths, several other aberrations, such as astigmatism, sagittal coma and spherical aberration are still present. These aberrations in general lead to deteriorated spectrum quality. In several applications astigmatism is especially important to correct, because astigmatism-corrected spectrographs have an enhanced intensity, and hence an improved signal-to-noise ratio at the exit aperture. Note that in contrast with Sections 2 and 3, where both the substrate shape and the phase function of the spherical CDOE were symmetric about the optical axis, here the grating lines are straight or almost straight. Harada and Kita [1980] demonstrated a mechanical ruling engine which is capable of numerically controlling the spacing and curvature of the grating lines, in order to reduce aberrations associated with straight, equidistant grooves. Holography, which is capable of recording high-density gratings, is also a flexible tool to control the shape of the grating lines, and was proposed by several authors for the optimization of spectroscopic gratings. McKinney and Palmer [1987] considered a CDOE recorded with two point sources, and then used the location of the two recording sources, along with the location of the entrance slit and the detector (not on the Rowland circle), as the parameters for numerical optimization, for cases where the Rowland circle must be abandoned, e.g. when the detector is flat. Noda, Harada and Koike [1989] proposed a recording setup in which at least one of the two recording waves is aspherical, to provide free parameters for the optimization. To simplify fabrication, the aspheric wavefront is generated by having a spherical wave reflected from an off-axis spherical mirror of appropriate location and tilt angle. Grange [1992, 1993] demonstrated how a simple recording setup, involving only spherical waves (but one of which needs usually to be converging) can simultaneously reduce sagittal coma and astigmatism. Duban, Lemaître and Malina [1998] proposed a method where one of the recording wavefronts for the CDOE is aspherical, generated by reflecting a spherical wave from a deformable plane mirror. This technique gives many free parameters that can be used for optimization, however, the calculated mirror deformations are extremely small and must be reproduced with optical precision. Duban [1999] suggested a method in which the aspherical recording wavefront is generated by an auxiliary spherical holographic grating. This setup involves only spherical surfaces and spherical waves, and still gives enough controllable parameters for correcting aberrations up to the 4th order. So far, only spherical CDOE substrates were considered, and the grating lines were chosen to deviate from straight grooves. On the other hand, astigmatism, which is one of the dominant aberrations in the original Rowland configuration, can simply be eliminated if the grating shape is ellipsoidal or toroidal, as already noted by Welford [1965]. Although both the ellipsoidal and the toroidal shapes

3, § 7]

CDOEs for optical coordinate transformations

143

are capable of eliminating astigmatism, their behavior is different in terms of higher-order aberrations, e.g., the ellipsoidal substrate gives a better correction of spherical aberration, as noted by Grange and Laget [1991]. Cash [1984] considered an ellipsoidal CDOE with straight, mechanically ruled grating grooves. Besides eliminating astigmatism, sagittal coma can also be canceled by a slight 3rd-order modification of the ellipsoidal shape. Such modified ellipsoidal substrates are, however, extremely difficult to fabricate. A variation of the same method was proposed by Content, Trout, Davila and Wilson [1991], where a 4th-order deformation was added to the ellipsoid, leading to much better performance, but at the expense of more problematic fabrication issues due to small deviations from the ellipsoidal shape. Grange and Laget [1991] combined a regular ellipsoidal substrate, to eliminate astigmatism, with the method of Noda, Harada and Koike [1989], mentioned above, to optimize the grating lines holographically and eliminate sagittal coma. Content and Namioka [1993] considered the case when the Rowland geometry cannot be used (e.g., in case of multiple gratings), and suggested that an asymmetrically deformed ellipsoidal substrate with straight grating grooves be used to minimize aberrations, which in this case must include compensating for meridional coma. They also demonstrated the usefulness of a setup in which one recording spherical wave is convergent and the other is divergent. This is equivalent to the setup suggested by Grange [1992, 1993] and mentioned above. Namioka, Koike and Content [1994] gave analytical formulas to express ray-traced spot diagrams for CDOEs having a 4th-order deformed ellipsoidal substrate and several types of ruled grating lines. The formulas are derived from 3rd-order approximations and can be used in grating optimization tasks. Astigmatism can be corrected with a CDOE on a toroidal substrate too. For example, such a setup was proposed and demonstrated experimentally both for holographically recorded metallic gratings (Huber, Jannitti, Lemaître and Tondello [1981]) and mechanically ruled gratings (Huber, Timothy, Morgan, Lemaître, Tondello, Jannitti and Scarin [1988]). The grating was created by first fabricating a grating on a spherical substrate and then slightly deforming it by applying elastic forces at four points on its circumference. This flexible design makes it possible to use the same fabricated grating for different wavelength ranges by varying the elastic deformation.

§ 7. CDOEs for optical coordinate transformations Davidson, Friesem, Hasman and Shariv [1991] proposed and demonstrated an approach for obtaining optical coordinate transformations which is based on

144


[3, § 7

CDOEs. Unlike flat DOEs which must have complicated aspheric phase functions to achieve optical transformations, here the curvature provides an additional degree of freedom that makes it possible for CDOEs to have simple phase functions that can be simply recorded directly by optical means. The design method for such CDOEs is the following: consider a coordinate transformation for a 1D function f (x), f (x) → f u(x) , (7.1) where u(x) is the new coordinate. Such a transformation is illustrated graphically in fig. 25, where the input function f (x) is drawn along the x axis and the output function f [u(x)] along the z axis (for simplicity fig. 25 shows a binary function). From each point xi on the x axis a vertical line is drawn, and from its intersection with the curve z = u(x) a horizontal line is drawn that intersects the z axis at the point zi = u(xi ). The transformation of fig. 25 may be realized directly by optical means: a transparency with transmittance function f (x) at the input plane is illuminated by a coherent plane wave. If it is assumed that this plane wave is not diffracted by the input pattern (geometrical shadow approximation), the wavefront after the transparency can still be described by parallel rays, where the coordinate of each ray is x and the intensity of the ray is f (x). The geometrical shadow approximation is valid when the distance between the input and the output planes is small compared −2 , where f with λ−1 fmax max is the maximum spatial frequency of the input. These parallel rays are diffracted by a CDOE along a curve z = u(x) by exactly 90◦ and hence arrive at the output plane at the desired location z = u(x). This is a much larger diffraction angle than most computer-generated flat DOEs are capable of. Thus for a given input size the distance between the input and output planes will be much smaller for the CDOE than for the flat DOE. Besides making the total optical system more compact, this shorter distance considerably improves the optical performance of the coordinate transformation. Specifically, the spacebandwidth product capabilities of the transformation are inversely proportional to the square root of this distance (Davidson, Friesem and Hasman [1992]). The CDOE may be generated optically by simply recording the interference pattern of two perpendicular plane waves on a curved holographic film, as shown in fig. 26. The CDOE approach can be generalized to 2D. Quasi-1D coordinate transformations of the form [x, y] → [u(x, y), y] can be achieved with the same configuration as for the 1D case, except that now the CDOE is located on a surface z = u(x, y). However, a general 2D coordinate transformation of the form [x, y] → [u(x, y), v(x, y)] requires two cascaded CDOEs. The first generates the transformation [x, y] → [u(x, y), y] as discussed above, while the

3, § 7]

CDOEs for optical coordinate transformations

145

Fig. 25. Graphical illustration of a 1D coordinate transformation.

Fig. 26. Recording setup for a CDOE performing 1D optical transformation (Davidson, Friesem, Hasman and Shariv [1991]).

second generates the transformation [u(x, y), y] → [u(x, y), w(u, y)], where w(u, y) = v(x, y). This CDOE approach can also be applied for tasks that represent coordinate transformations only in a broader sense. One of the examples investigated experimentally by Davidson, Friesem, Hasman and Shariv [1991] was to transform a Gaussian beam into one that is uniform in one of the transverse dimensions. Such a conversion may be obtained by the following coordinate transformation: √ 2x ,y , [x, y] → erf r0

(7.2)

146


(a)

[3, § 8

(b)

Fig. 27. Experimental cross-sections for the Gaussian-to-uniform beam converter of Davidson, Friesem, Hasman and Shariv [1991]: (a) input, (b) output.

where erf is the error function (the integral of the Gaussian function). Experimental intensity cross-sections for the input and output of the Gaussian-to-uniform beam converter are shown in figs. 27(a) and 27(b), respectively, qualitatively indicating the validity of the concept. Davidson, Friesem, Hasman and Shariv [1991] have also used CDOEs recorded in a similar method, but having a different shape, for logarithmic coordinate transformation which is used for scale-invariant optical correlation. We note that a related approach of a beam converter for diffuse white light was reported by Bokor and Davidson [2001a], where, instead of using diffraction, the beam shaping is achieved by a single reflection from an appropriately shaped mirror consisting of many displaced parallel planar reflecting facets.

§ 8. Summary We have discussed the main application areas of diffractive optical elements fabricated on a curved substrate. The versatility of the CDOE comes from the fact that the substrate shape and the grating function can be controlled independently. CDOEs can be used to achieve aberration-free imaging, uniform collimation and ideal concentration of diffuse light at the thermodynamic limit, for general source and target shapes. In high-NA focusing CDOEs can be used to control the effective apodization factor of the focused light rays, and thereby yield uniform and tight focal spots with very low sidelobe intensities. For many applications – like the ideal concentration of diffuse light discussed in Sections 3 and 4, the vectorial diffraction problems mentioned in Section 5, and the optical coordinate transformation described in Section 7 – it is sufficient to optimize the shape of the CDOE, and the grating function can be recorded either

3]

References

147

holographically with simple spherical or plane waves, or by direct laser lithography. In other cases the grating function must have a more complicated form; here additional optics or computer-generated holograms may be required. Since CDOEs are diffractive elements, they usually have large chromatic aberrations which limit their use to quasi-monochromatic light, except for the spectrograph applications discussed in Section 6. The history of CDOEs began in the mid-19th century with Rowland’s curved spectroscopic grating. Thanks to their versatility, and to recent advances in applications, in new design methods, and in holographic recording materials that can be placed on arbitrary shapes and enable high diffraction efficiency, CDOEs should remain a lively area of research in the future too.

References Baskar, S., Singh, K., 1992, J. Mod. Opt. 39, 1533. Bassett, I.M., Welford, W.T., Winston, R., 1989, Nonimaging optics for flux concentration, in: Wolf, E. (Ed.), Progress in Optics, vol. 27, North-Holland, Amsterdam, pp. 161–226, ch. 3. Belostotsky, A.L., Leonov, A.S., 1993, J. Lightwave Technol. 11, 1314. Bokor, N., Davidson, N., 2001a, Appl. Opt. 40, 2132. Bokor, N., Davidson, N., 2001b, Appl. Opt. 40, 5825. Bokor, N., Davidson, N., 2002, J. Opt. Soc. Am. A 19, 2479. Bokor, N., Davidson, N., 2004, Opt. Lett. 29, 1968. Bokor, N., Shechter, R., Friesem, A.A., Davidson, N., 2001, Opt. Commun. 191, 141. Born, M., Wolf, E., 1993, Principles of Optics, Pergamon Press, Oxford, U.K. Cash, Jr., C.C., 1984, Appl. Opt. 23, 4518. Content, D., Namioka, T., 1993, Appl. Opt. 32, 4881. Content, D., Trout, C., Davila, P., Wilson, M., 1991, Appl. Opt. 30, 801. Davidson, N., Bokor, N., 2003, in: Wolf, E. (Ed.), Progress in Optics, vol. 45, North-Holland, Amsterdam, pp. 1–51, ch. 1. Davidson, N., Bokor, N., 2004a, J. Opt. Soc. Am. A 21, 656. Davidson, N., Bokor, N., 2004b, Opt. Lett. 29, 1318. Davidson, N., Friesem, A.A., Hasman, E., Shariv, I., 1991, Opt. Lett. 16, 1430. Davidson, N., Friesem, A.A., Hasman, E., 1992, Appl. Opt. 31, 1067. Dorn, R., Quabis, S., Leuchs, G., 2003, Phys. Rev. Lett. 91, 233901. Duban, M., 1999, Appl. Opt. 38, 3443. Duban, M., Lemaître, G.R., Malina, R.F., 1998, Appl. Opt. 37, 3438. Fairchild, R.C., Fienup, J.R., 1982, Opt. Eng. 21, 133. Fisher, R.L., 1989, Opt. Eng. 28, 616. Grange, R., 1992, Appl. Opt. 31, 3744. Grange, R., 1993, Appl. Opt. 32, 4875. Grange, R., Laget, M., 1991, Appl. Opt. 30, 3598. Harada, T., Kita, T., 1980, Appl. Opt. 19, 3987. Hasman, E., Friesem, A.A., 1989, J. Opt. Soc. Am. A 6, 62. Hell, S., Stelzer, E.H.K., 1992, J. Opt. Soc. Am. A 9, 2159. Huber, M.C.E., Jannitti, E., Lemaître, G., Tondello, G., 1981, Appl. Opt. 20, 2139. Huber, M.C.E., Timothy, J.G., Morgan, J.S., Lemaître, G., Tondello, G., Jannitti, E., Scarin, P., 1988, Appl. Opt. 27, 3503.

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Jagoszewski, E., 1985, Optik 69, 85. Jagoszewski, E., Klakoćar-Ciepacz, M., 1986, Optik 72, 165. Jagoszewski, E., Talatinian, A., 1991a, Optik 88, 20. Jagoszewski, E., Talatinian, A., 1991b, Optik 88, 155. Kritchman, E.M., Friesem, A.A., Yekutieli, G., 1979a, Appl. Opt. 18, 2688. Kritchman, E.M., Friesem, A.A., Yekutieli, G., 1979b, Solar Energy 22, 119. McCauley, D.G., Simpson, C.E., Murbach, W.J., 1973, Appl. Opt. 12, 232. McKinney, W.R., Palmer, C., 1987, Appl. Opt. 26, 3108. Moharam, M.G., Gaylord, T.K., 1982, J. Opt. Soc. Am. A 72, 1385. Murty, M.V.R., 1960, J. Opt. Soc. Am. 50, 923. Namioka, T., Koike, M., Content, D., 1994, Appl. Opt. 33, 7261. Noda, H., Harada, Y., Koike, M., 1989, Appl. Opt. 28, 4375. Nowak, J., Zajac, M., 1988, Opt. Appl. 18, 51. Parker, A.R., Hegedus, Z., 2003, J. Opt. A – Pure Appl. Opt. 5, S111. Peng, K.O., Frankena, H.J., 1986, Appl. Opt. 25, 1319. Richards, B., Wolf, E., 1959, Proc. R. Soc. London Ser. A 253, 358. Rowland, H.A., 1883, Amer. J. Sci. (3) 26, 87. Schmahl, G., Rudolph, D., 1976, Holographic diffraction gratings, in: Wolf, E. (Ed.), Progress in Optics, vol. 14, North-Holland, Amsterdam, pp. 197–244, ch. 5. Talatinian, A., 1991, Opt. Appl. 21, 19. Talatinian, A., 1992, Optik 89, 151. Tatsuno, K., 1997, Opt. Rev. 4, 203. Turunen, J., Kuittinen, M., Wyrowski, F., 2000, Diffractive optics: electromagnetic approach, in: Wolf, E. (Ed.), Progress in Optics, vol. 40, North-Holland, Amsterdam, pp. 343–388, ch. 5. Verboven, P.E., Lagasse, P.E., 1986, Appl. Opt. 25, 4150. Welford, W.T., 1965, Aberration theory of gratings and grating mountings, in: Wolf, E. (Ed.), Progress in Optics, vol. 4, North-Holland, Amsterdam, pp. 241–280, ch. 6. Welford, W.T., 1973, Opt. Commun. 9, 268. Welford, W.T., 1975, Opt. Commun. 15, 46. Winston, R., Welford, W.T., 1989, High Collection Nonimaging Optics, Academic Press, New York. Wood, A.P., 1992, Appl. Opt. 31, 2253. Xie, Y., Lu, T., Li, F., Zhao, J., Weng, Z., 2002, Opt. Exp. 10, 1043. Zhang, Y., Wang, Z., 2004, Optik 115, 169.


Chapter 4

The geometric phase by

P. Hariharan School of Physics, University of Sydney, Sydney, NSW 2006, Australia e-mail: [email protected]

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)48004-X 149

Contents

Page § 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

§ 2. The geometric phase in optics . . . . . . . . . . . . . . . . . . . . .

151

§ 3. The geometric phase with single photons . . . . . . . . . . . . . .

168

§ 4. The geometric phase with photon pairs . . . . . . . . . . . . . . . .

171

§ 5. The Pancharatnam phase as a geometric phase . . . . . . . . . . .

173

§ 6. The Pancharatnam phase with white light . . . . . . . . . . . . . .

175

§ 7. Achromatic phase shifters . . . . . . . . . . . . . . . . . . . . . . .

177

§ 8. Switchable achromatic phase shifters . . . . . . . . . . . . . . . . .

181

§ 9. Polarization interferometers . . . . . . . . . . . . . . . . . . . . . .

182

§ 10. White-light phase-shifting interferometry . . . . . . . . . . . . . .

184

§ 11. Stellar interferometry . . . . . . . . . . . . . . . . . . . . . . . . .

192

§ 12. Nulling interferometry . . . . . . . . . . . . . . . . . . . . . . . . .

194

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction In classical physics, rotation of an object by an integral number of complete revolutions about any axis returns the object to its original state. In other words, the operator for rotation through 2mπ radians, where m is an integer, is equivalent to the identity operation, and its effect cannot be observed. However, a general extension of the adiabatic theorem of quantum mechanics by Berry [1984] showed that the wave function of a quantum system could exhibit a phase shift when the parameters of the system undergo a cyclic change. An example is the precession of a neutron in a magnetic field (Klein and Werner [1983]). A rotation of the spin axis of the neutron through 360◦ is not, as expected, an identity operation, but results in a phase shift of 180◦ . This phase shift is a physical property of the system which can be observed by interference, if the cycled system is combined with another similar system that was separated from it at an earlier time and has not undergone any change. It has also been shown that this phase shift, now known as the geometric phase, has a mathematical interpretation in terms of anholonomy resulting from parallel transport (Simon [1983]). Subsequently, Berry’s results were generalized by Aharonov and Anandan [1987], by giving up the assumption of adiabaticity, and by Samuel and Bhandari [1988], who showed that the evolution of the quantum system need not be either unitary or cyclic and may be interrupted by quantum measurements. Effects due to the geometric phase are seen in many fields of physics and have been documented in several reviews (Shapere and Wilczek [1989], Markovski and Vinitsky [1989], Berry [1990], Anandan [1992], Anandan, Christian and Wanelik [1997]).

§ 2. The geometric phase in optics A number of manifestations of the geometric phase occur in optics. One results from a cycle of changes in the direction of propagation of a beam of light (the spin-redirection phase: Chiao and Wu [1986], Tomita and Chiao [1986]), another results from a cycle of changes in the state of polarization of a beam of light (the 151

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[4, § 2

Pancharatnam phase: Pancharatnam [1956]), and a third results from a cycle of changes in squeezed states of light (Chiao and Jordan [1988]). Other manifestations of a geometric phase are the phase appearing when light is reflected at optical multilayers (Vigoureux and Labeke [1998]) or transmitted through a smoothly inhomogeneous isotropic medium (Bliokh and Bliokh [2004]) and the phase arising when the transverse mode of a beam of light is transformed, following a closed path in the space of modes (Galvez, Crawford, Sztul, Pysher, Haglin and Williams [2003]). Of all these, the first two are the most commonly observed. The nature of the geometric phase in optics, as to whether it is a classical or quantum effect, has also been discussed in some detail (see Agarwal and Simon [1990], Tiwari [1992]).

2.1. The spin-redirection phase One manifestation of the geometric phase in optics is the phase change acquired due to a circuit on the sphere of directions in momentum space. This phase change is known as the spin-redirection phase and is equal to the solid angle subtended by the circuit at the center of the sphere. The most frequently observed effect of the spin-redirection phase is the rotation of the plane of polarization of a linearly polarized light beam propagating along a nonplanar path (Chiao and Wu [1986]). An experimental study of the optical activity arising from the spin-redirection phase was carried out by Tomita and Chiao [1986]. As shown in fig. 1, a 180-cm-long single-mode fiber was inserted loosely in a 175-cm-long Teflon sleeve and wound helically, care being taken not to introduce any torsional stress and to keep the propagation directions of the input and output identical, in order to form a closed path in momentum space. The pitch angle θ of the helix, i.e. the angle between the axis of the fiber and the axis of the helix, was varied by attaching the Teflon sleeve to the outer perimeter of a spring, which could be stretched. In this way, the pitch length p could be varied, while keeping the fiber length s constant. The solid angle in momentum space spanned by the closed path in the fiber, in this case a circle, is Ω(C) = 2π(1 − cos θ ),

(2.1)

where cos θ = p/s. The corresponding values of the geometric phase are γ (C) = −σ Ω(C),

(2.2)

4, § 2]

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153

Fig. 1. Observation of the rotation of the plane of polarization of a linearly polarized light beam propagating in an optical fiber coiled into a helix (Tomita and Chiao [1986]).

where σ = ±1 is the helicity quantum number of the photon, resulting in a rotation of the plane of polarization of a linearly polarized beam by an angle Θ = 2π(1 − cos θ ).

(2.3)

In another series of experiments, the fiber was wound around a cylinder with a fixed radius to form a nonuniform helix. It was found that the measured rotation of the plane of polarization depended only on the solid angle in momentum space subtended by the path traced out by the fiber and not on the actual path. This result confirmed the topological nature of the geometric phase. A similar effect can also be observed when a linearly polarized beam is reflected at a series of mirrors along a nonplanar path (Kitano, Yabuzaki and Ogawa [1987], Berry [1987b], Rozuvan and Tikhonov [1995]). More recently, an exact expression for the geometric phase for photons propagating inside a noncoplanarly curved optical fiber has been presented by Shen and Ma [2003]. Another effect of the geometric phase is the introduction of a phase difference between two beams with opposite senses of circular polarization traversing, as shown in fig. 2, a nonplanar Mach–Zehnder interferometer (Chiao, Antaramian, Ganga, Jiao, Wilkinson and Nathel [1988], Jiao, Wilkinson, Chiao and Nathel [1989]). In this arrangement, the unpolarized beam from a He–Ne laser was divided at the polarization-preserving beam splitter B1 into two beams which traversed the paths α and β, respectively, before they were recombined at a second polarizationpreserving beam splitter B2. Two circular analyzers (right- and left-handed), mounted side by side, were used to select the interference patterns formed by photons of opposite circular polarizations incident on B1. If we use the unit sphere shown in fig. 3 to represent all possible directions of the spin vector of the photon, the history of a photon traversing the path α is summarized by the closed circuit ABCDA, which subtends a solid angle Ω at the

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Fig. 2. Top view of the nonplanar Mach–Zehnder interferometer. The beams in the upper half of the diagram are at a greater height than the beams in the lower half of the diagram (Chiao, Antaramian, Ganga, Jiao, Wilkinson and Nathel [1988]).

Fig. 3. Sphere of spin directions of the photon. The heavily shaded area corresponds to the solid angle Ω (Chiao, Antaramian, Ganga, Jiao, Wilkinson and Nathel [1988]).

4, § 2]


155

center of the sphere given by the relation Ω=

π − θ. 2

(2.4)

The solid angle Ω can be varied by adjusting the mirrors M1, M6, etc. so as to change the angle θ . If, then, we follow the same photon along the other path β, which it would traverse if reflected at B1, we find that the same spherical triangle is traversed in the opposite sense. If the two arms of the interferometer are exactly symmetrical, the dynamic phases cancel, and the phase difference introduced between the two beams due to the spin-redirection phase is 2Ω. The effects of the dynamic phase can be eliminated, without equalizing the optical paths in the two arms, by reversing the sense of circular polarization selected by the circular analyzers. This operation does not affect the dynamic phase, but reverses the sign of the spin-redirection phase, producing a phase shift of 4Ω, which is entirely topological. A direct demonstration of this phase difference is also possible using an interferometer consisting of a single-mode fiber ring, with part of it forming a helix. The fringe shift observed when the pitch of the helix is changed is a measure of the spin-redirection phase (Frins and Dultz [1997]). We note that a linearly polarized light beam can be regarded as a superposition of left- and right-handed circularly polarized states. When the two circularly polarized states propagate along a coiled fiber, they acquire opposite phase shifts, so that the beam emerging from the fiber, while still linearly polarized, has its plane of polarization rotated. The angle of optical rotation is equal to the solid angle subtended by the path in momentum space. The geometric phase of coiled optics has applications in polarization rotators. One application, for which polarization rotators based on discrete reflections at mirrors are particularly suitable, is in situations where transmissive devices cannot be used. An example is a practical coiled-optics rotator for CO2 laser beams using metallic mirrors (Galvez and Koch [1997]). Smith and Koch [1996] have applied the concept of the geometric phase to show that a minimum of four reflections is required to rotate the linear polarization of a laser beam by an angle 0 < φ < π and satisfy the requirement that the output beam is collinear with the input beam. Subsequently, a detailed study of two types of variable pure polarization rotators, based on the concept of the geometric phase, was presented by Galvez, Cheyne, Stewart, Holmes and Sztul [1999]. They consist of two “pseudo rotators” (Galvez and Holmes [1999]) in series, which produce a geometric phase that varies with their relative orientation.

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These systems preserve all states of polarization and rotate the plane of polarization for linear polarization and the ellipse axes for elliptical polarization; one of them is also achromatic. An interesting fundamental issue is the behavior of an incoherent imagebearing beam, when it is redirected by a sequence of mirrors in a nonplanar configuration. Segev, Solomon and Yariv [1992] have shown that the image is rotated by an angle which is equal in magnitude to the geometric (spin-redirection) phase but with the opposite sign. This is a special case where the phase property of an incoherent ensemble of particles, of different energies (frequencies), momenta (directions of propagation), and spins (polarizations), is conserved.

2.2. The Pancharatnam phase Berry’s [1984] paper led to a reappraisal of earlier work by Pancharatnam [1956, 1975] on the interference of polarized light, that could now be seen as an example of the geometric phase (Ramaseshan and Nityananda [1986]). Pancharatnam’s work involved effects arising from changes in the state of polarization of light traveling in a fixed direction. His results are therefore complementary to those described above, which involve light, with a fixed state of polarization, whose direction of propagation is changing in three-dimensional space. Pancharatnam was studying the interference patterns produced in plates of an anisotropic crystal and was concerned with the problem of defining the phase difference between two beams in different states of polarization. He did this by considering the intensity obtained by coherent superposition of the two beams. As the phase of one beam is varied linearly, this intensity varies sinusoidally. He, therefore, defined the two beams as being “in phase” when the resultant intensity was a maximum. Since the two beams could also be regarded as representing different stages in the polarization history of a single beam, this approach made it possible to define how a beam changed its phase when its polarization state was altered. It also led to the observation that a beam could be taken from one polarization state to a second polarization state, then to a third polarization state and, finally, back to its original state, and end up with its phase shifted, the magnitude of this phase shift depending on the geometry of the cycle as represented on the Poincaré sphere. As shown in fig. 4, states of polarization are represented as points on the Poincaré sphere, the poles corresponding to left- and right-handed circular polarizations, while points on the equator represent linear polarizations (with the orientation of the plane of polarization rotating by 180◦ in a 360◦ circuit, since the

4, § 2]


157

Fig. 4. Poincaré sphere representation of polarization states. The poles represent circular polarizations, and points on the equator represent linear polarization. The phase change associated with the circuit ABCA is half the solid angle subtended by the geodesic triangle ABC at the center of the sphere.

polarization then returns to its original direction). All other points on the sphere represent elliptic polarizations (Jerrard [1954], Ramachandran and Ramaseshan [1961], Born and Wolf [1999]). The polarization associated with any point on the Poincaré sphere indicated by the unit vector r is given by the complex eigenvector of r · σ , where r is a unit vector (with polar angles θ and φ), whose components are the Stokes parameters of the polarization, and σ is the vector of Pauli spin matrices. Pancharatnam’s studies led to the important result that if a beam of light is returned to its original state of polarization via two intermediate states of polarization, as shown in fig. 4, its phase does not return to its original value but changes by −Ω/2, where Ω is the area (solid angle) spanned on the Poincaré sphere by the geodetic triangle whose vertices are the three polarizations. This phase change is now known as the Pancharatnam phase. The correspondence between the Pancharatnam phase and the geometric phase was formally established by Berry [1987a]. Subsequently, a simple proof of Pan-

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charatnam’s result has been formulated by Aravind [1992], while De Vito and Levrero [1994] have presented a theoretical analysis discussing the relationship of the Pancharatnam phase to Berry’s phase. The procedure for observing the Pancharatnam phase with an interferometer, as well as in a general noncyclic SU(2) evolution, has also been discussed by Wagh and Rakecha [1995a, 1995b]. Soon after Berry’s paper appeared, experiments were carried out by Bhandari and Samuel [1988] and by Chyba, Wang, Mandel and Simon [1988] to demonstrate the Pancharatnam phase.1 As shown in fig. 5, Bhandari and Samuel [1988] used a modified Hewlett– Packard interferometer to demonstrate the Pancharatnam phase. In this instrument (Dukes and Gordon [1970]), a He–Ne laser is forced to oscillate simultaneously on two frequencies, f1 and f2 , separated by a constant difference of about 2 MHz, by the application of an axial magnetic field. Normally, the output signal from the receiver is at the beat frequency, (f2 − f1 ). Any change in the phase of one beam

Fig. 5. Two-frequency interferometer used to demonstrate the Pancharatnam phase (Bhandari and Samuel [1988]).

1 An observation of what can now be recognized as a manifestation of the Pancharatnam phase was made in 1961 (see Hariharan and Singh [1961], Hariharan [1993a]).

4, § 2]


159

results in an instantaneous shift in the frequency of that beam, which is integrated by the counter to yield the phase change. One of the beams in this interferometer, initially linearly polarized along the x direction and represented by point A on the Poincaré sphere in fig. 6, passed through a quarter-wave plate QWP1 whose principal axis was oriented at an angle of 45◦ to the x direction. This took the state of polarization of the beam to point P, which represents right-hand circular polarization. The beam then passed through a second quarter-wave plate QWP2, with its principal axis oriented at an angle α/2 to the principal axis of QWP1, which took the polarization state of the beam to point B, which represents light linearly polarized at an angle α/2 to the x direction. Finally, the polarization state of the beam was brought back to A by passage through a linear polarizer P with its principal axis set along the x direction. The result of these operations was a change in the phase of the beam equal to half the solid angle subtended at the center of the sphere by the trajectory APBA, that is to say, α/2.

Fig. 6. Poincaré sphere representation of the trajectory of the polarization state of the test beam in the two-frequency interferometer used to demonstrate the Pancharatnam phase (Bhandari and Samuel [1988]).

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Fig. 7. Modified Michelson interferometer used to demonstrate the Pancharatnam phase (Chyba, Wang, Mandel and Simon [1988]).

Chyba, Wang, Mandel and Simon [1988] used a modified Michelson interferometer (see fig. 7) illuminated with linearly polarized light from an intensitystabilized, single-mode He–Ne laser. One of the beams traversed a quarter-wave plate QP1, whose principal axis was fixed at 45◦ to the incident polarization. This converted the incident linearly polarized beam into a right-circularly polarized beam that fell on a second quarter-wave plate QP2, whose principal axis made an angle β with the original linear polarization. The linearly polarized beam emerging from QP2 was reflected by the mirror M2 back along its original path. As shown in fig. 8, the linearly polarized beam incident on QP1 can be represented on the Poincaré sphere by point A. After passing through QP1, the light becomes right-circularly polarized and is represented by point B. After passing through QP2, it is again linearly polarized and is represented by point C. When the light is reflected from mirror M2 and retraces its path through QP2 and QP1, this point moves to D and then back to A. As a result, the beam suffers a phase

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Fig. 8. Poincaré sphere representation of the trajectory of the polarization state of the test beam in the Michelson interferometer used to demonstrate the Pancharatnam phase (Chyba, Wang, Mandel and Simon [1988]).

change 2β, equal to half the solid angle subtended by the closed circuit ABCDA at the center of the sphere. A drawback of the experiments described above, as well as other similar experiments (Simon, Kimble and Sudarshan [1988], Bhandari [1988], Bhandari and Dasgupta [1990]), is the possibility of a change in the length of one of the optical paths resulting in a change in the phase (a dynamic phase). These problems were avoided in an experiment using a Sagnac interferometer in which the two beams traversed the same path in opposite senses, and the phase difference between the beams could be varied only by operating on the Pancharatnam phase (Hariharan and Roy [1992]). As shown in fig. 9, light from a He–Ne laser, linearly polarized at 45◦ to the plane of the figure by a polarizer P1 , was divided at a polarizing beam splitter into two orthogonally polarized beams that traversed the same, closed, triangular path in opposite directions. A second polarizer P2 , with its principal axis at 45◦ to the plane of the figure, brought the two beams leaving the interferometer into a condition to interfere at a photo detector.

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Fig. 9. Common-path interferometer in which the phase difference between the beams can be varied only by operating on the Pancharatnam phase (Hariharan and Roy [1992]).

The phase difference between the beams was varied by a system consisting of a half-wave plate (HWP) located between two quarter-wave plates, QWP1 and QWP2 (a QHQ phase shifter: Martinelli and Vavassori [1990]). The two quarterwave plates had their principal axes fixed at an angle of 45◦ to the plane of the figure, while the half-wave plate could be rotated by known amounts. The operation of this interferometer can be followed by means of the Poincaré sphere. We consider, in the first instance, the p-polarized beam transmitted by the polarizing beam splitter. As shown in fig. 10, the first quarter-wave plate QWP1 converts this linearly polarized state, which is represented by point A1 on the equator, to the left-circularly polarized state represented by S, the south pole of the sphere. If, then, the half-wave plate HWP is set with its principal axis at an angle θ to the principal axis of QWP1 , it moves this left-circularly polarized state, through an arc that cuts the equator at point A2 , to the right-circularly polarized state represented by N, the north pole of the sphere. Finally, the second quarter-wave plate QWP2 brings this right-circularly polarized state back to the original linearly polarized state represented by A1 . Since the input state has been taken around the closed circuit A1 SA2 NA1 , which subtends a solid angle 4θ at the center of the Poincaré sphere, this beam acquires a phase shift equal to 2θ .

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Fig. 10. Poincaré sphere representation of the trajectories of the polarization states of the two beams in a common-path interferometer using a QHQ phase shifter (Hariharan and Roy [1992]).

We next consider the s-polarized beam reflected by the polarizing beam splitter. In this case, the input state represented by point B1 traverses the circuit B1 SB2 NB1 . Since this circuit is identical to the circuit A1 SA2 NA1 , but is traversed in the opposite sense, this beam acquires a phase shift of −2θ . The additional phase difference φ introduced between the s- and p-polarized beams is, therefore, 4θ .

2.3. The Pancharatnam phase in optical rotation As mentioned earlier, the simplest example of a geometric phase is a spin-1/2 particle, such as a neutron, in a magnetic field (Klein and Werner [1983]). An optical analog is the rotation of the plane of polarization of a light wave produced by an optically active medium (Berry [1987a]), or by the Faraday effect (Zak [1991]). This aspect was not addressed in the experiments described above, but was subsequently studied by Hariharan and Roy [1993], who described an interferometric arrangement for measurements of the geometric (Pancharatnam) phase arising from optical rotation.

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In their experiment, a linearly polarized beam was divided at a beam splitter into two beams that traversed the same triangular path in opposite directions before they were recombined at the same beam splitter and emerged from the interferometer. Two pairs of identical wedges cut from crystal quartz, with the optical axis at right angles to the outer face of the wedges, were inserted in the optical path. One of the wedges in each pair was cut from left-handed (L) quartz and the other from right-handed (R) quartz. As a result, the two beams emerged with their planes of polarization rotated by equal amounts in opposite directions. The phase difference between the two beams, arising from the rotation of their planes of polarization, is of the Aharonov–Anandan type (Aharonov and Anandan [1987]), and could be observed directly by introducing, in the beams leaving the interferometer, a quarter-wave plate, with its principal axis at 45◦ , followed by a linear polarizer.

2.4. The Pancharatnam phase with subwavelength gratings Recent experiments with space-variant subwavelength metal-stripe gratings have revealed interesting effects due to a geometric phase related to the Pancharatnam phase (Bomzon, Kleiner and Hasman [2001a, 2001b]. When the period of the grating is smaller than the incident wavelength, such gratings behave like layers of a uniaxial crystal with the principal axes perpendicular and parallel to the grating grooves. Accordingly, by controlling the local period and orientation of the grating, any desired output polarization can be obtained. Higher efficiency can be obtained by using space-variant subwavelength dielectric gratings (Bomzon, Biener, Kleiner and Hasman [2002a]). One application of such gratings has been the conversion of a circularly polarized beam, at a wavelength of 10.6 µm, into a radially or azimuthally polarized beam (Bomzon, Biener, Kleiner and Hasman [2002a]). Space-variant subwavelength gratings have also been utilized in novel optical phase elements in which a space-variant phase-front modification, originating from the geometric (Pancharatnam) phase, is produced by continuously controlling the local orientation and period of a subwavelength grating (Bomzon, Biener, Kleiner and Hasman [2002b]). We can regard a subwavelength grating in which the period is the same everywhere, and only the local orientation of the grooves varies, as equivalent to a wave plate with constant retardation and a spatially varying fast axis. If such a spacevariant subwavelength grating is illuminated with circularly polarized light, the beams incident at different points on the grating traverse different paths on the

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Poincaré sphere and can be represented, when they emerge, by the points |E0 (0) and |E0 (θ ) (see fig. 11). The resultant phase difference φp between the beams, which can be obtained from Pancharatnam’s definition for the phase difference between two beams with different polarizations (Pancharatnam [1956]), is then equal to half the area Ω of the geodesic triangle on the Poincaré sphere defined by the pole |R and the points |E0 (0) and |E0 (θ ). The resulting wave consists of two components: the zero order and the diffracted order. The zero order has the same polarization as the incident wave and does not undergo any phase modification. On the other hand, for a retardation of π, the diffracted order has a polarization orthogonal to that of the incident wave and acquires a phase at each point equal to twice the local orientation θ (x, y) of the grating. If we define the diffractive geometric phase (DGP) as the phase of the diffracted orders when the incident beam is circularly polarized, it follows that for incident |R and |L polarizations, the DGP is equal to −2θ (x, y) and 2θ (x, y), respectively. By choosing the local orientation of the grating appropriately, any desired DGP can be realized, enabling the production of optical elements such as blazed gratings and lenses (Hasman, Kleiner, Biener and Niv [2003]). However, since such an optical element operates in different ways on the two circular polarizations, a converging lens for |R polarization would act as a diverging lens for |L polarization.

Fig. 11. Generation of a space-variant phase front by a space-variant subwavelength grating (Bomzon, Biener, Kleiner and Hasman [2002b]).

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We note that, in contrast to the situation with diffractive and refractive elements, the phase is not introduced through optical path differences, but results from the geometric phase that accompanies space-variant polarization manipulation. Another application of computer-generated space-variant subwavelength gratings is in forming linearly polarized axially symmetric beams (LPASBs). LPASBs are characterized by their polarization orientation, ψ(ω) = mω + ψ0 , where m is the polarization order, ω is the azimuthal angle, and ψ0 is the initial polarization orientation for ω = 0. The most widely used LPASBs are radial (m = 1, ψ0 = 0) and azimuthal (m = 1, ψ0 = π/2) beams, which are used in particle acceleration, atom trapping, optical tweezers, materials processing and tight focusing. Design and fabrication procedures for computer-generated space-variant subwavelength dielectric gratings that can be used to transform circularly polarized beams into LPASBs with any multiple of a half-integer polarization order have been presented by Niv, Biener, Kleiner and Hasman [2003]. Propagation-invariant vectorial Bessel beams can be generated by inserting an axicon in such an LPASB (Niv, Biener, Kleiner and Hasman [2004]). The intensity profile of the vectorial Bessel beam is determined by m, the polarization order of the original LPASB. If the resultant beam is transmitted through a polarizer, a unique propagation-invariant scalar beam is obtained. Such beams exhibit a propeller-shaped intensity pattern that can be rotated by a simple rotation of the polarizer.

2.5. Features of the Pancharatnam phase The geometric phase exhibits several interesting features. The first is that, since it is a topological phase, it is intrinsically achromatic. This fact is apparent in the case of the spin-redirection phase. In the case of the Pancharatnam phase, the phase shift only depends on the solid angle subtended by the closed path on the Poincaré sphere. The achromatism of the Pancharatnam phase has been experimentally demonstrated and has found several applications (see Sections 6–12). Another interesting feature of the Pancharatnam phase is that it is unbounded. This fact was pointed out at an early stage by Bhandari [1988]. In addition, while the Pancharatnam phase is additive, it does not affect the optical path difference (Schmitzer, Klein and Dultz [1998]). The unbounded nature of the Pancharatnam phase has been used in devices for phase control (Martinelli and Vavassori [1990]), where it eliminates the need for reset operations because such devices have no ‘end stops’, as well as for active stabilization of an interferometer (Wehner, Ulm and Wegener [1997]). Another application, described in Section 7, has been in frequency shifters for heterodyne interferometry.

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The question then arises whether phase shifts greater than 2π produced by the Pancharatnam phase have a physical significance. This question has been discussed by Love [1996], who points out that Pancharatnam phase shifts are physically significant, in that one can set up an experiment to observe arbitrarily large fringe displacements; however, they do involve a cyclic sequence of operations, and a beam of light which has been geometrically phase-shifted by 2nπ is equivalent to one which has been shifted by 2(n + 1)π. It should also be noted that Pancharatnam’s theorem holds only for an area on the Poincaré sphere enclosed by geodesic arcs. In this respect, it differs from the Berry phase which does not restrict the nature of the contour. Accordingly, while the Pancharatnam phase includes the case of an adiabatic development of the polarization due to a stack of continuously varying analyzers, it does not hold for closed paths made up of segments of small circles due to (say) a stack of elliptically birefringent plates (Berry and Klein [1996], Schmitzer, Klein and Dultz [1998]). In this case, an additional phase gain occurs, which can compensate (in some cases, totally) for the Pancharatnam phase. The nonlinear behavior of geometric phases related to SU(2) transformations described by arbitrary circular loops on the Poincaré sphere has been discussed by Lifu, Qu and Yingli [1999]. A nonlinear behavior of the Pancharatnam phase has also been observed with certain interferometer arrangements (Bhandari [1991b], Schmitzer, Klein and Dultz [1993]). This nonlinear behavior can result in a rapid change from a bright fringe to a dark fringe for a relatively small rotation of a polarizing element and can be used in an optical switch (Tewari, Ashoka and Ramana [1995], Li, Gong, Gao and Chen [1999]). In addition, with an appropriately chosen sequence of birefringent plates placed in the path of a light beam, certain cycles of rotation of the wave plates that enclose a singularity in the parameter space result in a measurable phase shift of the wave equal to ±2π, whereas other cycles nearby in parameter space, that do not enclose the singularity, result in a zero phase shift (Bhandari [1992a, 1992b]). These phase shifts can be interpreted as examples of the “nodal singularities” discussed by Dirac (Bhandari [1993a, 1994]). An application of the nonlinear behavior of the Pancharatnam phase has been in a tunable, polarization-based interferometric filter (Frins and Dultz [1998]). The optical system consists essentially of a two-beam interferometer with birefringent plates in the two arms and linear polarizers at the input and output. In principle, the system has three free parameters – the thickness of the birefringent plates, the optical path difference, and the orientation of the analyzer placed at the interferometer output. With a suitable choice of parameters, the chromatic dispersion of the filter can be compensated over a given spectral range. The transfer function of

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the filter can then be varied by rotating the analyzer to tune the filter and obtain bandpass or band-stop behavior.

2.6. Combined effects The combined effects of the spin-redirection phase and the Pancharatnam phase have been studied by Jiao, Wilkinson, Chiao and Nathel [1989], using a nonplanar Mach–Zehnder interferometer. They were able to show that the two phases are additive. They were also able to determine, experimentally, the signs of these phases and show that they conform to the predictions of theory (Berry [1987a]). A more detailed analysis of these experiments has been presented by Bhandari [1989, 1991a], who developed a framework for analyzing the evolution of the state of a light beam traveling along an arbitrary space curve and undergoing arbitrary polarization changes. Subsequently, Tavrov, Kawabata, Miyamoto, Takeda and Andreev [1999] suggested a method for evaluating the geometric (spinredirection) phase for a nonplanar ray traversing a set of separated optical elements, which could be applied even if the ray trajectory traced a contour on the sphere of directions which was not closed. This was followed by the formulation of a generalized algorithm for the unified analysis and simultaneous evaluation of the geometric spin-redirection phase and the Pancharatnam phase in a complex interferometric system (Tavrov, Miyamoto, Kawabata, Takeda and Andreev [2000]), which makes it possible to trace the polarization-state changes, and the geometric phase shifts, caused by beam propagation along an arbitrary optical path that involves both reflection and refraction at surfaces which exhibit a Fresnel phase shift and birefringence. The geometric (spin-redirection) phase has also been examined in the context of the Pancharatnam phase by Siebert, Frins and Dultz [1998], using a modified Michelson interferometer which allowed measurements of the spin-redirection phase resulting from the propagation of light through a helically wound singlemode optical fiber. The operation of this interferometer could be interpreted in terms of Pancharatnam’s theorem.

§ 3. The geometric phase with single photons 3.1. Observations at the single-photon level A simple experiment to demonstrate the geometric (Pancharatnam) phase at the single-photon level has been described by Hariharan, Roy, Robinson and O’Byrne

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[1993] using a Sagnac interferometer, similar to that shown in fig. 9, with a QHQ phase shifter. In this experiment, the beam from the laser was attenuated by a set of neutral density filters, and measurements of the photon count rate at the output from the interferometer were made at an input power level < 1 pW, corresponding to a photon flux < 3.2 × 106 photons/s. Since the length of the optical path in the interferometer was about 1 m, it follows that if P (1) and P (n > 1) are, respectively, the probabilities for the presence of one photon and more than one photon in the interferometer, we have, at this flux level, P (n > 1)/P (1) = 0.005. The probability that more than one photon was present, at the same time, in the interferometer was, therefore, negligible. The counting rate was recorded for a number of settings of the rotation angle θ of the half-wave plate. The net count rate, after subtracting the dark count, exhibited a sinusoidal variation corresponding to a phase difference φ = 4θ . The visibility of the interference fringes was better than 0.97. In another experiment (Hariharan, Brown and Sanders [1993]), the two orthogonally polarized modes of a Zeeman laser were combined at a polarizing beam splitter, with its principal plane at 45◦ to the planes of polarization of the beams, to produce two outputs at the difference frequency. Measurements made at power levels ranging from 1.0 µW down to 4.8 pW confirmed that the phases of the two beat signals differed by 180◦ even at the lowest power level. At this power level, the probability of more than one photon being present simultaneously in the attenuated beam was only 0.5% of that for the presence of a single photon. The phase difference of 180◦ between the beats at the two outputs can be regarded as the geometric phase arising from the difference of 180◦ in the rotations of the planes of polarization of the two input fields required to generate the two outputs. The persistence of this phase difference down to the single-photon level confirms that the geometric phase operates on superposition states of two fields at the single-photon level in the same manner as it does on superposition states of two classical fields.

3.2. Observations with single-photon states An experiment to demonstrate the geometric (Pancharatnam) phase with singlephoton states has been performed by Kwiat and Chiao [1991], using a light source which produced pairs of photons (λ ≈ 702.2 nm) by parametric down-conversion. As shown in fig. 12, the idler beam from the nonlinear crystal was transmitted through the narrow-band filter F1 to the detector D1, while the signal beam

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Fig. 12. Optical system used to demonstrate the geometric (Pancharatnam) phase for single-photon states (Kwiat and Chiao [1991]).

entered a Michelson interferometer. The beam leaving the interferometer was incident on a second beam splitter B2, from which it was transmitted to the detector D2 through the filter F2, or reflected to the detector D3 through the filter F3. One arm of the interferometer contained a fixed quarter-wave plate Q1 with its principal axis at 45◦ , as well as a quarter-wave plate Q2 which could be rotated. A rotation of Q2 through an angle θ introduced an additional phase shift φ = 2θ in this arm. The count rates for coincidences between D1 and D2, and between D1 and D3, as well as for triple coincidences between D1, D2 and D3, were recorded for a series of angular settings of Q2. When data were recorded using filters with a bandwidth of 10 nm at F2 and F3, and an optical path difference of 220 µm, which was greater than the coherence length corresponding to this bandwidth (about 50 µm), the visibility of the fringes seen by the detectors D2 and D3, operating individually, was essentially zero. However, when a filter with a bandwidth of 0.86 nm was placed in front of D1, the count rate for coincidences between D1 and D3 varied sinusoidally with the angular setting θ of Q2, with a visibility of 0.60 ± 0.05. With a broad-band filter at F1, these coincidence fringes disappeared.

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Measurements of the anticorrelation parameter N123 N1 (3.1) , N12 N13 with N123 the rate of triple coincidences between D1, D2 and D3, N1 the rate of single counts by D1 alone, N12 the rate of coincidences between D1 and D2, and N13 the rate of coincidences between D1 and D3, confirmed that the observations essentially involved photons in n = 1 Fock states. A=

§ 4. The geometric phase with photon pairs In the experiments described above, there is little difference in the effects predicted by a classical treatment or a quantum-mechanical treatment. However, the effects produced by the geometric (Pancharatnam) phase in fourth-order interference have been shown to be quite different by Brendel, Dultz and Martienssen [1995], using the experimental arrangement shown in fig. 13. In this experiment, photon pairs produced by down-conversion of blue light (λ = 458 nm), from an Ar+ laser, in a beta barium borate (BBO) crystal were used as the input to a Michelson interferometer. One arm of the interferometer contained two quarter-wave plates, one of which was fixed at an azimuth of 45◦ , while the other could be rotated. Rotation of the second quarter-wave plate through an angle θ introduced phase shifts of ±2θ , respectively, for the two orthogonal polarizations. Experiments were carried out using two BBO crystals cut, respectively, for type-I and type-II phase matching, so that the photons of each pair could be prepared either in the same state of polarization (type-I phase matching), or in orthogonal states of polarization (type-II phase matching). The photon pairs emerging from the interferometer were incident on a second beam splitter BS2 which directed them to two photo detectors D1 and D2 . With type-I phase matching, BS2 was a normal beam splitter, while, with type-II phase matching, BS2 was a polarizing beam splitter. Measurements with this system showed that the effects observed depended on the initial states of polarization of the two photons in a pair and the optical path difference. With a near-zero optical path difference, second-order interference fringes were observed. Under these conditions, the effects of the dynamic phase and the geometric (Pancharatnam) phase were equivalent. However, with a large optical path difference and coincidence detection, no changes in the coincidence rate were observed, as the second quarter-wave plate was rotated, in the case of type-II phase matching. On the other hand, with type-I phase matching,

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Fig. 13. Experimental arrangement used to demonstrate the effects of the geometric (Pancharatnam) phase in fourth-order interference (Brendel, Dultz and Martienssen [1995]).

interference fringes with a visibility of 0.75 were obtained, with a period half that expected with a classical light field. These results imply that pairs of photons with parallel polarizations acquire twice the geometric phase of single photons and behave like single particles with spin 2. On the other hand, pairs of photons with orthogonal polarizations acquire geometric phases with opposite signs and behave like single particles with spin 0. A study has also been made by Siebert, Schmitzer and Dultz [2002], using a modified Michelson interferometer, of the spin-redirection phase of indistinguishable photon pairs resulting from their propagation through a helically wound single-mode optical fiber whose pitch angle could be varied continuously. They were able to confirm that the interference effects due to photon pairs had a doubled periodicity with respect to the solid angle spanned in momentum space by the closed path in the fiber, compared to the interference effects produced by single photons.

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More recently, a theoretical analysis of the geometric phase for photon pairs produced by parametric down-conversion has been made by Ben-Aryeh [2003], who has derived explicit expressions for the Pancharatnam, dynamic and geometric phases for the transformations produced by linear phase converters. The Pancharatnam phase for generalized squeezed states has also been discussed by Seshadri, Lakshmibala and Balakrishnan [1997] who have extended this study to multiphoton squeezed coherent states, and by Mendaš [1997], who has discussed the special case of a two-photon interaction process, such as parametric amplification.

§ 5. The Pancharatnam phase as a geometric phase A study of the possible optical configurations for an interferometer in which the two beams traverse the same path in opposite senses, and the phase difference between the beams can be varied only by operating on the Pancharatnam phase, has been presented by Hariharan [1993b]. However, in these interferometers, as well as in all the demonstrations of the Pancharatnam phase described so far, a polarization state is transported around a circuit on the Poincaré sphere by means of a series of retarders (which effect unitary transformations on the input polarization state) and analyzers (which project the input polarization state on to another polarization state). While the beam leaving an analyzer is in phase with the beam incident on it, the beam leaving a retarder is, in general, not in phase with the incident beam. Accordingly, all these experiments could be criticized on the ground that the observed phase may contain contributions due to changes in the dynamic phase (De Vito and Levrero [1994]). There is, therefore, a need for an experiment that demonstrates the Pancharatnam phase arising purely from projections. The theory for such an experiment was discussed by Ramaseshan and Nityananda [1986], and by Samuel and Bhandari [1988]. Subsequently, Berry and Klein [1996] performed an experiment to demonstrate the geometric (Pancharatnam) phase, without unitary elements, using a twisted stack of linear polarizers. However, this arrangement has the limitation that it can only cycle a polarization state along the equator of the Poincaré sphere. Consequently, the only nontrivial value of the Pancharatnam phase that can be obtained with this arrangement is π. The use of beam splitters and mirrors was avoided in experiments using a system based on the Hewlett–Packard interferometer (Bhandari [1993b]). In this arrangement, the two beams emerging from the laser, with frequencies f1 and f2 and linearly polarized along the x and y directions, respectively, traveled along

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the same path through a quarter-wave plate (QWP) and through a linear polarizer, after which they were incident on a photodetector. If the QWP was set with its principal axis at 45◦ , the two beams emerging from it had opposite circular polarizations, and a rotation of the linear polarizer through an angle α resulted in a change in the phase difference between the beams of 2α. The need to use retarders was avoided by Hariharan, Ramachandran, Suresh and Samuel [1997] by using a circular analyzer – a cholesteric liquid-crystal film enclosed between two treated glass plates separated by a mylar spacer. The direction of alignment of the rod-like cholesteric liquid-crystal molecules twists uniformly about an axis oriented perpendicular to the surface of the glass plates, resulting in a helical arrangement of the molecules. At wavelengths of the order of the pitch of the helix, one circular polarization (in this case the left-circular polarization) is strongly Bragg reflected, while the other is freely transmitted (Chandrasekhar [1992]). The experimental arrangement consisted of a Sagnac interferometer in which a left-circularly polarized beam from a He–Ne laser (λ = 633 nm) was divided at a nonpolarizing beam splitter into two beams that traversed the same rectangular path in opposite directions. A rotatable linear polarizer in one arm of the interferometer converted these two left-circularly polarized beams to linearly polarized beams. These two linearly polarized beams were recombined at the beam splitter and brought into a state of right-circular polarization by a circular analyzer placed at the output port of the interferometer. Rotation of the linear polarizer through an angle θ introduced a phase difference 4θ between the two beams. A simpler system using a single linear analyzer (a sheet polarizer) has also been described by Hariharan, Mujumdar and Ramachandran [1999]. The optical arrangement was, essentially, a Mach–Zehnder interferometer with an extra mirror in one arm. A circularly polarized laser beam was divided at the first (nonpolarizing) beam splitter into two beams, both of which underwent an odd number of reflections within the interferometer, so that, when they were incident on the second (nonpolarizing) beam splitter, they were in the same circularly polarized state. The transmitted beam emerged with its polarization unchanged, while the reflected beam had its polarization reversed (Berry [1987b]). As a result, the two beams leaving the second beam splitter had opposite circular polarizations. A linear analyzer brought these two beams into the same state of linear polarization. Rotation of the linear analyzer by an angle θ introduced an additional phase difference 2θ between the two beams. A system based on Young’s interference experiment, demonstrating the Pancharatnam phase arising purely from projections, has also been described by

4, § 6]

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Fig. 14. Optical system, based on Young’s interference experiment, used to demonstrate the Pancharatnam phase (Hariharan, Suresh and Mujumdar [1998]).

Hariharan, Suresh and Mujumdar [1998]. As shown in fig. 14, an expanded, linearly polarized beam from a He–Ne laser (λ = 633 nm) illuminated a pair of slits through two circular analyzers, RCA and LCA (50 µm thick films of two, different, cholesteric liquid-crystal materials), to yield, respectively, right- and left-circularly polarized states. Both these states were projected by a lens on to a linear analyzer LA to obtain the final, linearly polarized states. A magnified image of the interference pattern produced by these two states was formed on a screen by a second lens. Setting the linear analyzer LA at an azimuth angle θ to the vertical introduced a phase difference 2θ between the two states. In this interferometer, each state is produced by projecting the previous state on it. The two beams do not traverse any birefringent elements and do not undergo any reflections. Accordingly, the phase difference introduced between the two beams by a rotation of the linear analyzer can be regarded as a purely geometric phase.

§ 6. The Pancharatnam phase with white light Hariharan and Rao [1993] were the first to point out that, with white light, the behavior of the interference fringes due to a change in the Pancharatnam phase was very different from that due to a change in the optical path length (the dynamic phase). The reason for this is that the phase shift due to a change in the optical path length (the dynamic phase) is inversely proportional to the wavelength. On the other hand, the geometric phase is a topological phenomenon. As a result, the phase shift observed due to a change in the Pancharatnam phase is intrinsically independent of the wavelength.

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Quantitative measurements on the interference fringes obtained with white light were carried out by Hariharan, Larkin and Roy [1994] with an interferometer similar to that shown in fig. 9. A tungsten lamp was used as the source, and a CCD camera interfaced with a computer was used to view the interference fringes. Since the retardation produced by a single birefringent plate is inversely proportional to the wavelength, the quarter-wave plates QWP1 and QWP2 in the QHQ phase shifter were replaced by combinations of a half-wave plate and a quarter-wave plate, which yield an achromatic circular polarizer (Destriau and Prouteau [1949]), and the half-wave plate HWP was replaced by a sandwich of two quarter-wave plates and a half-wave plate, which is a good approximation to an achromatic half-wave retarder (HWR) (Hariharan and Malacara [1994]). With monochromatic illumination, rotation of the HWR resulted in a continuous movement of the interference fringes across the field of view. When the direction of rotation of the HWR was reversed, the fringes moved continuously in the opposite direction. With white light, the interference pattern seen without the phase shifter in the optical path consisted of a central black fringe, flanked on each side by a white fringe and a few colored fringes whose contrast decreased rapidly with their distance from the central fringe. With the phase shifter in the optical path, a rotation of the HWR by ±45◦ resulted in the central black fringe being replaced by a white fringe, while a rotation of the HWR by ±90◦ brought back the central black fringe. However, as can be seen from the profiles presented in fig. 15, showing the intensity along a line at right angles to the fringes, the positions of the zero-order fringe and the fringe envelope, in the field of view, did not change. This experiment confirms that the Pancharatnam phase does not depend on the wavelength. As a result, the effect of a change in the Pancharatnam phase equal to an integral multiple of 2π is to move the fringes formed at all wavelengths through an integral number of fringe spacings, so that the interference pattern returns to its original configuration. Baba and Shibayama [1997] have also described an experiment, using dispersed fringes produced by a Michelson interferometer, in which the introduction of a dynamic phase, by moving one of the mirrors, could be observed as a tilt of the fringes. On the other hand, the introduction of a geometric (Pancharatnam) phase, by a QHQ phase shifter placed at the output, produced a shift of the fringes which was independent of the wavelength. This result clearly demonstrates the difference between the dynamic phase and the Pancharatnam phase.

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Fig. 15. Intensity profiles across the interference patterns recorded with white light, for different angular settings θ of the HWR (Hariharan, Larkin and Roy [1994]).

§ 7. Achromatic phase shifters Several systems can be used to shift the phase of a light beam by means of a cyclic change in its state of polarization (the Pancharatnam phase). With a linearly polarized beam it is possible, as described earlier, to use a QHQ combination (a half-wave plate H mounted between two quarter-wave plates Q) as a phase shifter (Pancharatnam [1956], Martinelli and Vavassori [1990]). The two quarter-wave plates have their principal axes fixed at an azimuth of 45◦ , while the half-wave plate can be rotated. Rotation of the half-wave

178

The geometric phase

[4, § 7

plate through an angle θ shifts the phase of the linearly polarized output beam by 2θ . With a Michelson interferometer it is possible to use a combination of two quarter-wave plates inserted in the reference beam as a phase shifter (Crane [1969]). The quarter-wave plate next to the beam splitter (see fig. 7) is set with its principal axis at 45◦ to the incident polarization, and rotation of the second quarter-wave plate through an angle θ shifts the phase of the beam by 2θ . With two orthogonally polarized beams it is possible to use a simpler system, consisting of a quarter-wave plate with its principal axis at 45◦ followed by a rotatable polarizer, as a phase shifter (Hariharan and Sen [1960]).2 Rotation of the polarizer through an angle θ introduces a phase difference 2θ between the two beams. An early application of such systems was as frequency shifters for heterodyne interferometry. Some arrangements using rotating polarization components have been described by Sommargren [1975] and by Hu [1983]. Higher frequency shifts can be obtained by using multiple rotating retarders (Shagam and Wyant [1978]). A system using counter-rotating wave plates to obtain a higher frequency shift, with fewer components, has also been described by Kothiyal and Delisle [1984]. A detailed study of the frequency and mode spectrum of the output from such devices (Hils, Dultz and Martienssen [1999]) has shown that, in spite of the rapidly changing phase of the output, no higher-frequency components appear in the spectral distribution of the intensity, since phase nonlinearities are compensated for by intensity changes. A single detector sees a sinusoidal intensity variation, and only a phase-sensitive detector, such as an array of photodetectors, can observe nonlinear effects. The use of such systems as phase shifters in an interferometer with a polarizing beam splitter has been discussed by Kothiyal and Delisle [1985]. One arrangement which can be placed at the input of such an interferometer is shown in fig. 16(a). It consists of a rotating half-wave plate followed by a quarter-wave plate with its principal axis at 45◦ . Two phase shifters which can be used at the output are shown in figs. 16(b) and 16(c). The phase shifter shown in fig. 16(b) consists of a quarter-wave plate with its principal axis at 45◦ followed by a rotating polarizer. In fig. 16(c), we have a phase shifter with a rotating half-wave plate between a quarter-wave plate and a polarizer. It can be shown that the phase shift obtained with these systems is given by the relation φ = nθ , where θ is the change in the azimuth of the rotating compo2 This is essentially the system used in the Senarmont compensator, which is one of the earliest anticipations of the geometric phase (Jerrard [1948], Hariharan [1993c]).

4, § 7]

Achromatic phase shifters

(a)

(b)

179

(c)

Fig. 16. Some systems that can be used as phase shifters with two orthogonally polarized beams: H(rot.), rotating half-wave plate; Q, quarter-wave plate at an azimuth of 45◦ ; P, polarizer; P(rot.), rotating polarizer (Kothiyal and Delisle [1985]).

nent, and n has values of 4, 2 and 4 for the phase shifters shown in figs. 16(a), 16(b) and 16(c), respectively. A more detailed analysis (Kothiyal and Delisle [1986]) shows that errors of the retarder and the azimuth angles of the components of these phase shifters do not influence the measured values of the phase, up to the first order. Because the Pancharatnam phase is a geometric phase and, therefore, a topological phenomenon, it only depends on the solid angle subtended by the closed path on the Poincaré sphere. As a result, the phase shifts produced by these systems, even with simple retarders, are very nearly achromatic. In the case of a QHQ phase shifter, it has been shown that, even with simple retarders, the maximum deviation of the phase shift from its nominal value is less than ±6◦ over the whole range of visible wavelengths (Hariharan and Ciddor [1994]). A recent application of such a QHQ phase shifter has been in color digital holography. In this technique, a three-wavelength laser and a color CCD are used to record the interference patterns for all three wavelengths simultaneously. Three color images are reconstructed by a computer from four sets of phase-shifted interferograms and combined into a full-color image. The use of a phase shifter operating on the Pancharatnam phase ensures that the phase shifts introduced at all three wavelengths are very nearly the same (Kato, Yamaguchi and Matsumura [2002]). The residual variations with wavelength of the phase shift produced with such devices can be minimized if the simple retarders are replaced by achromatic retarders. One way to construct an achromatic retarder is by using a combination of two materials with differing dispersions of birefringence. Another way to obtain an achromatic retardation of 90◦ is by two successive total reflections in a Fresnel rhomb. Two Fresnel rhombs can be combined to obtain an achromatic retardation of 180◦ . It is also possible to replace the elements of a QHQ phase shifter by combinations of quarter-wave and half-wave plates of the same material to produce

180

The geometric phase

[4, § 7

an achromatic phase shifter. Two such systems have been described; in both, the two quarter-wave plates are replaced by a combination of a half-wave plate and a quarter-wave plate, which yields an achromatic circular polarizer (Destriau and Prouteau [1949]). In one design (Hariharan, Larkin and Roy [1994]) the half-wave plate is replaced by a sandwich of two quarter-wave plates and a half-wave plate, which is a good approximation to an achromatic half-wave retarder (Hariharan and Malacara [1994]); in another (Hariharan and Ciddor [1994]), the half-wave plate is replaced by an assembly of two of the achromatic circular polarizers described earlier (Destriau and Prouteau [1949]), mounted back to back, with the principal axes of the half-wave plates parallel. With the latter system, the deviations of the phase shift from its nominal value, over the range of wavelengths from 450 nm to 700 nm, are less than ±0.3◦ . As mentioned earlier, it is possible, with two orthogonally polarized beams, to use a simpler system, consisting of a quarter-wave plate with its principal axis at 45◦ followed by a rotatable polarizer, as a phase shifter (Hariharan and Sen [1960]). Helen, Kothiyal and Sirohi [1998] have shown that better results can be obtained by replacing the quarter-wave plate with an achromatic circular polarizer. In one arrangement, shown in fig. 17(a), the quarter-wave plate is replaced by a combination of a half-wave plate and a quarter-wave plate (Destriau and Prouteau [1949]). With this arrangement, the maximum deviation of the

Fig. 17. Achromatic phase shifter using a rotating polarizer with (a) a two-component achromatic circular polarizer and (b) a three-component achromatic circular polarizer (Helen, Kothiyal and Sirohi [1998]).

4, § 8]

Switchable achromatic phase shifters

181

phase shifts introduced between the two beams from their nominal values, over the wavelength range from 450 nm to 700 nm, is less than 4.1◦ . Even better results can be obtained with the combination shown in fig. 17(b) consisting of two half-wave plates and a quarter-wave plate (Pancharatnam [1955]). In this case, the maximum deviation of the phase shifts from their nominal values, over the wavelength range from 450 nm to 700 nm, is less than 1.4◦ . In addition, the amplitudes of the transmitted components show little change over the entire range of phase shifts and wavelengths.

§ 8. Switchable achromatic phase shifters As we have seen, a system consisting of a rotatable half-wave plate mounted between two fixed quarter-wave plates (a QHQ combination) can be used as an achromatic phase shifter with a linearly polarized beam. Such a QHQ combination can be modified to obtain a switchable, achromatic phase shifter by replacing the half-wave plate with a ferro-electric liquid-crystal (FLC) device. An FLC device can be regarded as a birefringent plate with a fixed retardation, whose principal axis can take one of two orientations depending on the polarity of an applied voltage. The angle θ through which the principal axis rotates depends on the particular FLC material (Saleh and Teich [1991]). Such an FLC device can be used as a binary phase shifter by adjusting its thickness to produce a retardation of half a wave and sandwiching it between two quarter-wave plates. Multiple levels of phase shifting can be produced by cascading a series of FLC devices between the two quarter-wave plates (Freeman, Brown and Walba [1992], Love and Bhandari [1994]). For a three-level phase shifter (Hariharan and Ciddor [1995]), two quarter-wave plates, QWP1 and QWP2, are set, as shown in fig. 18, with their principal axes at angles of +45◦ and −45◦ , respectively, to the plane of polarization of the input beam (azimuth 0◦ ). Two ferro-electric liquid-crystal devices (FLC1, FLC2) with nominal retardations of a half-wave, whose principal axes switch through an angle of 45◦ , when driven appropriately, are placed between QWP1 and QWP2. A linear polarizer P2 (azimuth 0◦ ) defines the output polarization. With this arrangement, phase shifts of 0◦ , 90◦ and 180◦ , or 0◦ , 90◦ and 270◦ , can be obtained. However, at wavelengths other than the design wavelength, the retardations produced by QWP1 and QWP2, as well as by FLC1 and FLC2, deviate from their nominal values. As a result, when the principal axes of FLC1 and FLC2 are switched to positions corresponding to nominal phase shifts of 90◦ and 180◦ , or 90◦ and 270◦ , the actual phase shifts exhibit residual variations

182

The geometric phase

[4, § 9

Fig. 18. Three-level, switchable, achromatic phase shifter using two FLC devices (Hariharan and Ciddor [1995]).

with the wavelength. The magnitude of these variations depends on the initial and final orientations of the principal axes of the FLCs. Hariharan and Ciddor [1999] have shown that these variations can be minimized by setting FLC1 to switch between azimuths of −22.5◦ and −67.5◦ , and FLC2 to switch between azimuths of 22.5◦ and 67.5◦ . The errors in the phase shift, at nominal values of the phase shift of 90◦ and 270◦ , are then less than ±11.5◦ over the range of wavelengths from 470 nm to 700 nm. With a Michelson interferometer it is possible to use an alternative system (Hariharan, Ciddor and Roy [2005]) in which each arm of the interferometer contains a fixed achromatic quarter-wave retarder with its principal axis at an angle of 45◦ to the plane of polarization of the beam, followed by an FLC device with a nominal retardation of a quarter wave and a switching angle of 45◦ . This arrangement produces switchable phase shifts of 0◦ and ±90◦ that are almost independent of the wavelength over the range of wavelengths from 450 nm to 700 nm.

§ 9. Polarization interferometers Polarization interferometers are now widely used in microscopy. Since the two beams traverse almost identical paths within the interferometer, a very stable interference pattern is obtained. In addition, it is easy to obtain interference with white light. However, because the two beams overlap, conventional phase-shifting techniques (such as a moving mirror) cannot be applied to make quantitative measurements. This problem can be overcome by using a phase shifter operating on the geometric (Pancharatnam) phase (Hariharan [1996]). An additional advantage is that, unlike a phase shift introduced by changing the optical path, the phase shift introduced by operating on the geometric (Pancharatnam) phase is almost

4, § 9]

Polarization interferometers

183

independent of the wavelength, so that accurate measurements of small optical path differences can be made even with white light. A simple modification that makes quantitative measurements possible with a conventional transmission or reflection differential-interference-contrast (Nomarski) microscope involves inserting a quarter-wave retarder oriented at 45◦ , just below the analyzer, in the two orthogonally polarized beams emerging from the second Wollaston prism. This retarder converts one beam into right-handed and the other into left-handed, circularly polarized light. When these two beams are made to interfere by the analyzer, a rotation of the analyzer through an angle θ introduces an additional phase difference of 2θ between the beams (Cogswell, Smith, Larkin and Hariharan [1997]). Four sets of intensity values recorded with additional phase shifts of 0◦ , 90◦ , 180◦ and 270◦ can then be used to calculate the original phase difference, modulo 2π, at any point in the image, using the simple relation tan φ =

I90 − I270 . I0 − I180

(9.1)

9.1. Two-wavelength interferometry With monochromatic light, problems arise at step changes in height greater than half a wavelength, because of ambiguities in the integral interference order. A simple way to track such changes in the integral interference order, for optical path differences up to a few micrometers, is by two-wavelength interferometry. In this technique, measurements are made, at each point on the test surface, of the fractional interference orders, ε1 and ε2 , for two wavelengths, λ1 and λ2 . It then follows that p = (m1 + ε1 )λ1 = (m2 + ε2 )λ2 ,

(9.2)

where p is the optical path difference at this point, and m1 and m2 are the integral interference orders for the two wavelengths, λ1 and λ2 , respectively. If the fractional interference orders, ε1 and ε2 , are known with a sufficient degree of accuracy, it is possible to obtain an unambiguous value for p over an extended range (Cheng and Wyant [1984], Creath [1987]). This technique can be implemented with a polarizing (Nomarski) microscope in which the illumination system has been modified to allow either a red He–Ne laser (λ1 = 633 nm) or a yellow He–Ne laser (λ2 = 594 nm) to be used as the source (Hariharan and Roy [1996]). The phase difference between the two orthogonally polarized beams leaving the microscope is varied by a simple QHQ phase shifter;

184

The geometric phase

[4, § 10

for plates with retardations of a half-wave and a quarter-wave at 633 nm, the deviations of the phase shift at 594 nm from its nominal value at 633 nm are less than ±0.5◦ . Four TV frames are acquired at each wavelength, corresponding to additional phase shifts of 0◦ , 90◦ , 180◦ and 270◦ . The phase differences, φ1 (x, y) and φ2 (x, y), for the two wavelengths at each point in the image, can be calculated directly (modulo 2π) from the two sets of intensity data. The repeat distance is 9.63 µm, permitting measurements of step changes in height up to ±4.81 µm.

9.2. Switchable achromatic phase shifters With a polarizing interference microscope, there are problems in replacing the QHQ phase shifter with a switchable phase shifter. This is because, with a switching angle θ , the two orthogonally polarized beams emerging from the interferometer experience phase shifts of +2θ and −2θ , respectively, so that the additional phase difference introduced between them is 4θ . With normal FLC devices, in which the principal axis switches through an angle of 45◦ , this would result in changes in the phase difference of ±180◦ , which would not be useful. This problem can be solved by using FLC materials with switching angles that differ significantly from 45◦ . Two FLC devices employing a material with a switching angle of 60◦ produce additional phase differences of ±240◦ , in which case we have I (+240) − I (−240) . tan φ = 31/2 (9.3) I (+240) + I (−240) − 2I (0) In this case, the best results are obtained by setting FLC1 to switch between azimuth angles of −15◦ and −75◦ , and FLC2 to switch between azimuth angles of 15◦ and 75◦ (Hariharan and Ciddor [1999]).

§ 10. White-light phase-shifting interferometry As mentioned earlier, a problem with interferometric profilers using monochromatic light is phase ambiguities arising at steps involving a change in the optical path difference greater than a wavelength. This problem can be eliminated by using white light. Because of the short coherence length of the illumination, the interference term is appreciable only over a very limited range of depths. As a result, a three-dimensional image can be extracted by scanning the object in depth and evaluating the degree of coherence (the fringe visibility) between corresponding

4, § 10]

White-light phase-shifting interferometry

185

pixels in the images of the object and reference planes (Davidson, Kaufman, Mazor and Cohen [1987], Kino and Chim [1990], Lee and Strand [1990]). This technique is known as low-coherence interference microscopy, or coherence-probe microscopy. We assume that the origin of coordinates is taken at the point on the z axis at which the two optical paths are equal, and the test surface is moved along the z axis in a series of steps of size z. With a broad-band source, the intensity at any point in the image plane, corresponding to a point on the object whose height is h, can then be written as 2π p + φ0 , I (z) = I1 + I2 + 2(I1 I2 )1/2 g(p) cos (10.1) λ¯ where I1 and I2 are the intensities of the two beams acting independently, g(p) is the coherence or fringe-visibility function (which corresponds to the ¯ + φ0 ] is a cosinusoidal envelope of the interference fringes), and cos[(2π/λ)p modulation. In eq. (10.1), λ¯ corresponds to the mean wavelength of the source, p = 2(z − h) is the difference in the lengths of the optical paths traversed by the two beams, and φ0 is the difference in the phase shifts on reflection at the beam splitter and the mirrors. Figure 19 shows the variations in intensity at a given point in the image as the object is scanned in depth along the z axis. Each such interference pattern can be processed to obtain the envelope of the intensity variations (the fringe-visibility function) and determine the peak amplitude of the intensity variations, as well as the location of this peak along the scanning (z) axis. The values of the peak amplitude correspond to an image of the test object, while the location of this peak along the scanning (z) axis yields the height of the surface at the corresponding point. One way to recover the fringe-visibility function from the sampled intensity data is by digital filtering in the frequency domain (Lee and Strand [1990]). This process involves two Fourier transforms (forward and inverse), along the z direction, for each pixel in the image. It is necessary, therefore, for the step size along the z axis to correspond to a change in the optical path difference that is less than a fourth of the shortest wavelength; typically, a step z around 50 nm is used. Consequently, this procedure requires a large amount of memory and processing time, though these requirements can be reduced to some extent, and good accuracy obtained, by modified sampling and processing techniques. An alternative approach involves shifting the phase of the reference wave by three or more known amounts at each position along the z axis and recording the corresponding values of the intensity; these intensity values can then be used to

186

The geometric phase

[4, § 10

Fig. 19. Output from an interferometric profiler as a function of the position of the object along the z axis.

evaluate the fringe visibility at that step. However, if these phase shifts are introduced by changing the optical path difference p between the beams (Dresel, Hausler and Venzke [1992]), the phase shift is inversely proportional to the wavelength. In addition, since p is changing, the value of the fringe-visibility function g(p) is not the same for all the readings. These problems can be avoided by using an achromatic phase shifter operating on the geometric (Pancharatnam) phase. The experimental arrangement used to demonstrate this method, shown schematically in fig. 20, consisted essentially of a Michelson interferometer illuminated with a collimated beam of white light (Hariharan and Roy [1994]). In this interferometer, a linearly polarized beam was divided at the polarizing beam splitter PBS into two orthogonally polarized beams, which were incident on the mirrors M1 and M2, respectively. After reflection at M1 and M2, these beams returned along their original paths to a second (nonpolarizing) beam splitter BS, which sent them through a polarizer P2 to the CCD array camera. The phase difference between the beams was varied by a system consisting of a rotating half-wave plate H between two fixed quarter-wave plates Q1 and Q2. Measurements were made by moving the mirror M1 along the z axis, by means of a piezoelectric translator (PZT), in steps of magnitude z = 0.66 µm (cor¯ over a range responding to changes in the optical path difference of about 2.4λ) of 4 µm centered approximately on the zero-order, white-light fringe. At each

4, § 10]


187

Fig. 20. Schematic of the experimental arrangement used to demonstrate the application of achromatic phase shifting to white-light interferometry (Hariharan and Roy [1994]).

step, four measurements were made of the intensity at the center of the fringe pattern, corresponding to additional phase differences of 0◦ , 90◦ , 180◦ and 270◦ . Since the additional phase differences introduced are the same for all wavelengths, the value of the fringe-visibility function at this point is given, apart from a normalizing factor which depends on the relative intensity of the two beams, by the same relation as for monochromatic light, 2[(I0 − I180 )2 + (I90 − I270 )2 ]1/2 (10.2) . I0 + I90 + I180 + I270 Figure 21 shows a typical curve obtained for the visibility of the interference fringes, at the center of the field, as a function of the position of the mirror M1 along the z axis. The solid curve represents the best-fit Gaussian. The value of z corresponding to the peak visibility of the interference fringes can be obtained directly from this curve. In addition, it has been shown (Hariharan and Roy [1995]) that, when the op¯ where λ¯ is the design wavelength of the tical path difference is less than ±λ, half-wave and quarter-wave plates, it is possible, with achromatic phase-shifting, g(p) =

188

The geometric phase

[4, § 10

Fig. 21. Values of the visibility of the interference fringes formed with white light, as a function of the position of the mirror M1 along the z axis (Hariharan and Roy [1994]).

to calculate φ(λ¯ ), the phase difference at this wavelength, from eq. (9.1) with an error less than 2π/180. Helen, Kothiyal and Sirohi [1999] have presented a detailed analysis of a whitelight interferometer using a phase shifter based on an achromatic circular polarizer and a rotating analyzer (Helen, Kothiyal and Sirohi [1998]). They found that the peak of the fringe-visibility function does not occur at zero optical path difference, but is shifted by a constant amount. However, this constant shift does not introduce any error in relative height measurements. Somervell and Barnes [2003] have used such an achromatic phase shifter in a system based on a Michelson-type polarization interferometer for measuring phase distributions with white light. The output beam from the interferometer was split into two separate beams, each of which passed through a phase shifter consisting of a Fresnel-rhomb quarter-wave retarder followed by a Glan–Thompson polarizer. The two phase shifters were adjusted to produce two fringe patterns in quadrature, which were imaged on two spatially separated regions of a CCD array. The phase at each point could then be recovered from the corresponding values of the intensity in the two quadrature fringe patterns and the incoherent sum of the intensities due to the two beams. Since the two fringe patterns are acquired in the same CCD frame, the system is insensitive to vibration noise and allows fast measurements of the phase.

4, § 10]


189

A similar achromatic phase shifter, consisting of an achromatic quarter-wave retarder and a rotating linear polarizer, has been used recently in a phase-stepping common-path radial-shearing interferometer for measurements on wavefronts from white-light sources (Barron, Somervell, Barnes and Cheung [2004]), as well as in a phase-shifting lateral-shearing Sagnac interferometer for accurate measurement of phase maps with broad-band light (Lo, Somervell and Barnes [2005]). A minor problem with a phase shifter placed at the output of an interferometer is that the interference pattern may undergo a lateral shift across the CCD array used to record it during the rotation of a retarder or a polarizer for phase shifting, resulting in errors. This problem does not arise with a phase shifter placed at the input of the interferometer. Helen, Kothiyal and Sirohi [2000] have shown how a combination of a rotating half-wave plate and a fixed quarter-wave plate placed at the input (Kothiyal and Delisle [1985]) can be used as a nearly achromatic phase shifter to measure the fringe-visibility function. This arrangement also has the advantage that it only uses two components against the three used in a QHQ phase shifter placed at the output.

10.1. Coherence-probe microscopy A computer-controlled coherence-probe microscope, which can rapidly and accurately map the shape of surfaces exhibiting steps and discontinuities, was developed by Roy, Cherel and Sheppard [1999]. This instrument used an optical system based on the Linnik interferometer and scanned the object in height. An achromatic phase shifter operating on the Pancharatnam phase was used to evaluate the fringe visibility directly, at each height setting, for an array of points covering the object. The location of the fringevisibility peak along the scanning axis, for each point on the object, gave the height of the object at the corresponding point. A later version of this instrument was built with high-numerical-aperture optics (Roy, Svahn, Cherel and Sheppard [2002]). In such a microscope, with Kohler illumination, only light from the object which has a phase front which matches that of the reference beam produces an appreciable interference signal. The result is that the visibility of the interference fringes is given by the product of the envelopes resulting from this correlation effect and the limited coherence of the illumination. The phase difference between the two orthogonally polarized beams was varied by an achromatic phase shifter consisting of a quarter-wave plate with its principal

190

The geometric phase

[4, § 10

axis at 45◦ and a linear analyzer which could be rotated by known amounts. The rotation of the analyzer was controlled by a computer through a stepper motor. Recently, this instrument has been modified to use a fast, switchable, achromatic phase shifter (Roy, Sheppard and Hariharan [2004]). The optical system of the modified instrument is shown schematically in fig. 22.

Fig. 22. Optical system of a coherence-probe microscope using a switchable, achromatic phase shifter operating on the Pancharatnam phase (Roy, Sheppard and Hariharan [2004]).

4, § 10]


191

A tungsten-halogen lamp (12 V, 100 W) was used as the source; a 3 mW He–Ne laser was also provided for finding the fringes. The linearly polarized beam transmitted by the polarizer was divided at the polarizing beam splitter into two orthogonally polarized beams, which were focused on to a reference mirror and the test surface by two, identical, infinity-tube-length, 40× microscope objectives with a numerical aperture of 0.75. After reflection at the reference mirror and the test surface, these beams returned along their original paths to a second beam splitter, which sent them through a second polarizer to the CCD camera. The phase difference between the beams was varied by a system consisting of two quarter-wave plates with their principal axes set at 45◦ , and a pair of FLCs whose principal axes could be switched through angles of ±51.5◦ , by applying appropriate voltages, to introduce phase differences of 0◦ and ±206◦ between the two beams. The value of the fringe-visibility function for each pixel could then be obtained from the corresponding three intensity values. Figure 23 shows the surface profile of an integrated circuit obtained with this system.

Fig. 23. Surface profile of an integrated circuit obtained with a coherence-probe microscope, using a switchable achromatic phase shifter operating on the Pancharatnam phase. The lateral dimensions of the specimen were 25 × 43 µm; its height was about 1 µm (Roy, Sheppard and Hariharan [2004]).

192

The geometric phase

[4, § 11

10.2. Spectrally resolved interferometry Another technique that can be used with white light to obtain unambiguous values of the optical path difference is spectrally resolved interferometry (Calatroni, Guerrero, Sainz and Escalona [1996], Sandoz, Tribillon and Perrin [1996]). In this technique, a spectroscope is used to analyze the light from each point on the interferogram. The optical path difference between the beams at this point can then be obtained from the intensity distribution in the resulting channeled spectrum. Much higher accuracy can be obtained by phase shifting, using an achromatic phase shifter operating on the Pancharatnam phase (Helen, Kothiyal and Sirohi [2001]). The values of the phase obtained at a number of pixels along the direction of dispersion (corresponding to a number of different wavelengths) yield a value for the surface height that is completely independent of the height of neighboring points and free from 2π phase ambiguities. Errors due to variations in the sensitivity of different pixels are avoided, since the intensity values used for calculating the phase difference at each wavelength are obtained from the same pixel. A resolution of 1 nm in height is possible. An advantage of this technique is that measurements have to be made only at one position of the object along the height axis. Since there is no need to translate the specimen along the height axis, the risk of errors due to mechanical vibration is reduced. However, each interferogram only yields a profile along a single line.

§ 11. Stellar interferometry Since the dimensions of a star are very small compared to its distance from the earth, the complex degree of coherence (µ12 ) between the light vibrations from the star reaching two points on the surface of the earth is given by the normalized Fourier transform of the intensity distribution over the stellar disc. The angular diameter of the star can, therefore, be obtained from measurements of the visibility of the fringes in an interferometer which samples the wave field at pairs of points separated by different distances (Michelson and Pease [1921]). With the development of modern detection, control and data-handling techniques, instruments designed to make measurements over baselines up to 640 m have been constructed (Tango and Twiss [1980]). In the Sydney University Stellar Interferometer (SUSI) (Davis [1984]), two siderostats, at the ends of a North– South baseline, direct light from the star, via two beam-reducing telescopes and an optical path-length compensator, to a beam combiner. Interference takes place between two pairs of nominally parallel wave fronts leaving the beam combiner.

4, § 11]

Stellar interferometry

193

Two photon-counting detectors measure the total flux in a narrow spectral band in the two complementary interference patterns. The signals from the two detectors are then proportional to (1 + |µ12 | cos φ) and (1 − |µ12 | cos φ), where φ, the phase difference between the beams, varies randomly with time, because of changes in the lengths of the optical path through the atmosphere. If, then, φ has a uniform circular distribution, the average of the square of the difference between the two signals is 2|µ12 |2 . This technique assumes that the visibility of the fringes is constant over the range of optical path differences introduced by the random variations in the lengths of the optical paths. To satisfy this condition, the spectral bandwidth has to be limited by a narrow-band filter, resulting in a serious loss of light. This problem can be overcome, and the use of broad-band illumination made possible, by holding the optical path difference at a value close to zero and using a phase shifter operating on the Pancharatnam phase to introduce phase shifts of ±90◦ that are almost independent of the wavelength (Tango and Davis [1996]). A problem then is that measurements have to be made in a time interval that is much shorter than the period of the atmospheric fluctuations, to avoid errors due to variations in the optical paths caused by atmospheric turbulence. However, very rapid, achromatic phase shifts can be obtained with a switchable phase shifter incorporating two FLC devices. As shown in fig. 24, the two linearly polarized beams incident on the beam combiner BC are brought into orthogonal states of polarization by the half-wave

Fig. 24. Optical system of a stellar interferometer using switchable, achromatic phase shifters operating on the Pancharatnam phase (Hariharan and Tango [1998]).

194

The geometric phase

[4, § 12

retarders HA and HB , so that it is possible to use a switchable phase shifter in each of the two output channels. If FLC devices with a switching angle of 60◦ are used (Hariharan and Tango [1998]), the visibility of the interference fringes can be calculated directly from three intensity measurements with phase shifts of 0◦ , +240◦ and −240◦ .

§ 12. Nulling interferometry A problem in trying to detect a planet near a star is that the star is 6–9 orders of magnitude brighter than the planet. Nulling interferometry is a way to reduce the light flux from the star, relative to its surroundings, by making the light from the star interfere destructively with itself (Bracewell [1978]). To do this, we need a system that ensures achromatic, destructive interference at the center of the field. This result can be achieved with a 180◦ rotational shear (Shao and Colavita [1992]). Some measurements of rejection ratios obtained in laboratory experiments with a rotational-shearing interferometer have been presented by Serabyn, Wallace, Hardy, Schmidtlin and Nguyen [1999], and the design of symmetrical optical systems for a nulling interferometer has been discussed by Serabyn and Colavita [2001].

12.1. Nulling using the spin-redirection phase A proposal for an achromatic nulling interferometer using the geometric (spinredirection) phase has been formulated by Tavrov, Bohr, Totzeck, Tiziani and Takeda [2002]. Their proposal envisaged an achromatic rotational-shearing system in which, as shown in fig. 25, the two beams in a stellar interferometer are directed along a plane at right angles to the plane of incidence before they are combined. This out-of-plane propagation results in geometric rotations of +90◦ and −90◦ , respectively, of the beams in the two arms, and the introduction of an additional phase difference of π between the two beams which is independent of the wavelength. With an on-axis source, the optical paths in the two arms are the same, resulting in destructive interference at all wavelengths. However, for an off-axis source, there is a nonzero optical path difference, so that the off-axis source can be detected. Experiments with a three-dimensional Sagnac interferometer, using the spinredirection phase to obtain achromatic nulling, were also described by Tavrov,

4, § 12]

Nulling interferometry

195

Fig. 25. Stellar nulling interferometer, showing out-of-plane geometry used to obtain achromatic nulling (Tavrov, Bohr, Totzeck, Tiziani and Takeda [2002]).

Bohr, Totzeck, Tiziani and Takeda [2002]. In this interferometer, the two beams traverse the same out-of-plane path in opposite directions, so that they undergo opposite geometric rotations of +90◦ and −90◦ , and a phase difference of π is introduced between them. In addition, since both beams undergo one reflection and one transmission at the beam splitter, they have exactly the same amplitude. Destructive interference was demonstrated at two wavelengths, 632.8 nm from a He–Ne laser and 441.6 nm from a He–Cd laser, confirming the achromatic properties of this scheme. In both cases, a doughnut-shaped interference pattern was obtained, with a dark area at the center. Calculations showed that, if the beam divergence could be held to less than 0.1◦ , deep nulling of the order of 10−6 could be obtained.

12.2. Nulling using the Pancharatnam phase In an alternative approach, Baba, Murakami and Ishigaki [2001] have proposed an achromatic nulling interferometer using the Pancharatnam phase. As shown in fig. 26, the light from a star, and a planet located near it, falls on two separate telescopes. For simplicity, we assume that the star is at the zenith. In this case, the optical path difference (OPD) for light from the star is zero, but

196

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[4, § 12

Fig. 26. Optical system of a stellar nulling interferometer with an achromatic phase shifter operating on the Pancharatnam phase (Baba, Murakami and Ishigaki [2001]).

there is a finite OPD for the light from the planet. When a phase difference of π is introduced between the two beams, by means of the QHQ phase shifter, the light from the star interferes with itself, destructively. However, light from the planet remains detectable. In an experiment to evaluate this system, the ends of two optical fibers illuminated, respectively, with light from a xenon lamp and a halogen lamp, were used to simulate a star and a planet. The light from the xenon lamp (the simulated star) was sent through a polarizer set at 45◦ , so that the s- and p-polarized components propagated without any OPD. The light from the halogen lamp (the simulated planet) was divided into s- and p-polarized components by a polarizing beam splitter and reflected back by two mirrors, so that an OPD could be introduced between the two components. Finally, the two sets of beams were combined by a beam splitter, so that the angular separation between the images of the

4, § 12]

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197

simulated star and planet could be adjusted by rotating this beam splitter. The s- and p-polarized beams from the two sources then passed through an achromatic QHQ phase shifter using Fresnel rhombs, and a polarizer, before reaching a CCD camera. Measurements with this system yielded a value for extinction of the image of the simulated star by a factor of 6 × 10−5 . Very recently, Murakami, Kato, Baba and Ishigaki [2004] have extended this idea to multiple telescopes. They have proposed a multielement nulling interferometer in which the phase of the light in each arm could be modulated independently. They have also demonstrated experimentally that, with separate phase shifters in each arm of such an interferometer, an extinction of 1 × 10−4 could be obtained.

Conclusions The wave function of a quantum system can exhibit a phase shift when the parameters of the system undergo a cyclic change. This phase shift is known as the geometric phase. Two manifestations of the geometric phase are commonly observed in optics – one due to a cyclic change in the direction of propagation of a beam (the spinredirection phase) and the other due to a cyclic change in the state of polarization of a beam (the Pancharatnam phase). Several experiments have confirmed that the magnitude and sign of these effects are those predicted by theory, and that the geometric phase and the phase change due to a change in the optical path (the dynamic phase) are additive. It has also been shown that these effects persist at the single-photon level. However, the results obtained with fourth-order interference using entangled photon pairs are very different from those expected from classical theory. Pairs of photons with parallel polarizations acquire twice the geometric phase of single photons and behave like single particles with spin 2. On the other hand, pairs of photons with orthogonal polarizations acquire geometric phases with opposite signs and behave like single particles with spin 0. These experiments have also led to some very interesting observations. One is that the geometric phase is unbounded. The unbounded nature of the Pancharatnam phase has been exploited in devices for phase control, where it eliminates the need for reset operations, and in frequency shifters. Another observation is that, unlike the dynamic phase, which is inversely proportional to the wavelength, the geometric phase is intrinsically independent of the wavelength. This result has

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[4

opened up new possibilities in optical interferometry. One application is in interference microscopy, where the use of white light and achromatic phase-shifting makes it possible to overcome the problem of phase ambiguities at discontinuities and steps involving a change in the optical path difference greater than a wavelength. Other applications are in stellar interferometry, and in nulling interferometry for detecting a planet orbiting a star.

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Chapter 5

Synchronization and communication with chaotic laser systems by

Atsushi Uchida Department of Electronics and Computer Systems, Takushoku University, 815-1 Tatemachi, Hachioji, Tokyo, 193-0985 Japan e-mail: [email protected]

Fabien Rogister Service d’Electromagnétisme et de Télécommunications, Faculté Polytechnique de Mons, 31 Boulevard Dolez, 7000 Mons, Belgium

Jordi García-Ojalvo Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Colom 11, 08222 Terrassa, Spain

Rajarshi Roy Department of Physics, Institute for Research in Electronics and Applied Physics, and Institute for Physical Science and Technology, University of Maryland, College Park, MA 20742, USA

ISSN: 0079-6638

DOI: 10.1016/S0079-6638(05)48005-1 203

Contents


205

§ 2. Synchronization of chaotic lasers . . . . . . . . . . . . . . . . . . .

220

§ 3. Communication with chaotic lasers . . . . . . . . . . . . . . . . . .

273

§ 4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

324

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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§ 1. Introduction The study of chaos and nonlinear dynamics has brought to light many interesting deterministic dynamical phenomena to observe and measure, that are highly irregular in time and space. Over the past three decades, such systems were discovered in every discipline and have led to a plethora of studies that charted routes to chaos and developed quantitative mathematical measurements for characterizing the various signatures of nonlinear dynamical systems. The emphasis in the future will surely be to discover novel and interesting applications of such systems and to decipher how they may already be employed by nature in biological contexts. The aim of our review is to present recent developments in nonlinear dynamics that have made an impact on research in optics, in the specific area of communication with optical waveforms that are chaotic and irregular in their spatiotemporal dynamics. Ever since the invention of the ruby laser (Townes [1999]), instabilities have been observed in laser operation (Nelson and Boyle [1962]). Over the years, optical scientists and engineers have discarded many laser systems as unusable due to the difficulty of obtaining stable light output. This is particularly true for lasers used for communication. Once it was realized that the unstable dynamics observed in many laser systems was not ‘noise’ but deterministic dynamics of a chaotic and aperiodic nature, a fundamentally different point of view arose. Can one take a laser that emits complex, chaotic waveforms and successfully encode and decode information? The key element necessary to answer this question in the affirmative was the discovery that it is possible to synchronize chaotic systems. The synchronization of periodic systems has been studied for centuries, and the development of radio communication originated from the synchronization of a carrier frequency sine wave between a transmitter and a receiver. This synchrony allows us to transmit messages via amplitude and frequency modulation and decode them with a suitably designed receiver synchronized to the carrier frequency. As we will examine in detail later, it is the possibility of synchronizing optical transmitter and receiver systems with complex, irregular dynamics that leads to communication of digital or analog information. 205

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1.1. Laser fluctuations and dynamics How was the connection between chaotic and nonlinear dynamical systems and lasers first made? In the early days of laser physics, the models for laser dynamics developed theoretically (Haken [1985], Sargent, Scully and Lamb [1972], Louisell [1973], Lax [1966]) had no place for chaotic behavior. The picture of the electric field emitted by a laser was often characterized by the phase-diffusion model. The electric field maintains an average amplitude and fluctuates in phase around the circle in the complex plane as shown in fig. 1 (Mooradian [1985]). The laser linewidth is generated primarily by the phase diffusion process, and not by the amplitude fluctuations. Both amplitude and phase fluctuations are generated by spontaneous emission in the laser models developed; in the absence of spontaneous emission, the laser field would be determined entirely by stimulated emission, and have negligible linewidth. The simplest laser model that explains this type of behavior is a Langevin equation of the type studied for many years in the theory of Brownian motion, dE (1.1) = aE − b|E|2 E + f (t), dt where E is the complex electric field and the parameters a (net gain) and b (saturation) govern the dynamical properties of the field, which is driven by the complex random noise term f (t) that represents spontaneous emission fluctuations (Haken [1985], Sargent, Scully and Lamb [1972]). The dynamics described by eq. (1.1)

Fig. 1. Concept of complex electric field. (Adapted from Mooradian [1985].)

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207

is simple – the laser field grows from random initial conditions (the spontaneous emission background) to a steady state where the amplitude and phase fluctuate in time, with an average amplitude and phase-diffusion rate that are determined by the parameters a and b and the strength of the correlation coefficient D of the noise source, f (t)∗ f (t ) = 2Dδ(t − t ). (1.2) The dynamics of the atomic or molecular active medium of the laser was contained in the values of these parameters, and the single equation was derived through a process of eliminating the variables that were necessary to describe the active medium evolution in more detail – the dipole moment of the medium and the number of inverted atoms/molecules per unit volume. It was apparent from the earliest days of laser development that some types of lasers, including the ruby laser, exhibited much more complex dynamics than contained in the simple model above (Nelson and Boyle [1962]). These observations of ‘irregular’ or ‘noisy’ dynamics were attributed to external noise sources, and extensive efforts were made by most optical scientists and engineers to eliminate them, or to use just those laser systems that were stable, for applications.

1.2. Chaotic dynamics in nonlinear systems: the Lorenz model As the theoretical basis for nonlinear dynamical behavior was developed, it became clear that more degrees of freedom than contained in the deterministic version of eq. (1.1) were necessary for chaotic dynamics to be present. The Poincaré–Bendixson theorem (Strogatz [1994]) shows indeed that at least three degrees of freedom are necessary for chaotic dynamics to exist. The paradigmatic model of chaos in the early 1970s was the Lorenz model (Lorenz [1963]), developed in the context of weather prediction. The model consists of a simple set of coupled ordinary differential equations describing in a reduced way, for numerical integration purposes, the spatiotemporal problem of fluid convection in a cell heated from below and maintained at a lower temperature on top: dx = σ (y − x), dt dy = −xz + rx − y, dt dz = xy − bz. dt

(1.3) (1.4) (1.5)

208


[5, § 1

Fig. 2. (a) Typical temporal waveform and (b) attractor of the Lorenz model; σ = 10, r = 28 and b = 83 .

The variables x, y and z represent properties of the convecting fluid flow, and temperature differences between the left- and right-hand sides of the cell of fluid and between its top and bottom, respectively. For certain ranges of parameters σ (Prandtl number), r (Rayleigh number), and b (aspect ratio of the cell), the Lorenz model showed a very remarkable feature of nonlinear coupled differential equations – the sensitivity to initial conditions for the variables x, y and z that resulted in the trajectories in phase-space diverging exponentially fast away from each other. Lorenz recognized this property as a very important signature of a new kind of dynamical behavior (Lorenz [1993]). The model consisted of just three equations with a few nonlinear terms, and had just enough nonlinearity and coupled degrees of freedom to generate dynamical behavior that was chaotic as seen in fig. 2(a), where the time series for just one variable is shown, and fig. 2(b), where the time evolution of all three variables is plotted in the shape of the Lorenz ‘butterfly’ attractor. Lorenz indelibly impressed on his audience the significance of chaotic dynamics through the title of his talk to the American Association for the Advancement of Science in 1972 – “Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in Texas?” (Lorenz [1993]).

1.3. Laser chaos: the Haken–Lorenz equations One of the most interesting interdisciplinary connections was made by Haken [1975]. He realized that the Maxwell–Bloch equations describing the time evolution of the atoms and field of a laser system could be put in the same form as the Lorenz equations through a set of transformations of variables and redefinition of

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parameters. These equations are now called the Haken–Lorenz equations: dx = σ (y − x), (1.6) dt dy = −xz + rx − (1 − iδ)y, (1.7) dt dz = Re x ∗ y − bz, (1.8) dt and describe the time evolution of the scaled electric field (x), induced dipole moment (y) and population inversion (z). The parameters b and σ consist of ratios of the decay rates for the population inversion and photons from the laser cavity to that of the induced dipole moment. The parameters r and δ represent the pump rate and detuning of the atomic resonance frequency from the optical frequency; i is the imaginary unit. A thorough treatment of the derivation of eqs. (1.6)–(1.8) is given by van Tartwijk and Agrawal [1998]. Such a set of three equations, with parameters that correspond to the situation where the decay rates for the field, dipole moment and inversion are all comparable, describes the dynamics of what is now known as a Class C laser, according to the terminology of Arecchi, Lippi, Puccioni and Tredicce [1984]. For parameters in a certain range, the dynamics displayed by this model is very similar to that of the Lorenz model, and much research followed the publication of Haken’s paper in 1975 to explore the connections between lasers and fluids. These connections have been investigated by many researchers, and a comprehensive account was published by Newell and Moloney [1992]. When there is a large difference in the decay rates, one may progressively eliminate the induced dipole moment equation and then the inversion equation, to finally obtain the single equation for the electric field, as in eq. (1.1). The model with two coupled equations for the dynamics of the field and the inversion is commonly referred to as the Class B laser system, and eq. (1.1) is the model for a Class A laser, in the terminology introduced by Arecchi, Lippi, Puccioni and Tredicce [1984]. The dynamics of laser systems has been reviewed by Weiss and Vilaseca [1991] and van Tartwijk and Agrawal [1998]. It may appear at first sight from our discussion of the Haken–Lorenz equations that only Class C lasers may be good candidates for generating chaotic waveforms for communications. Indeed, most Class B and Class A lasers that have been developed over the years have been operated under conditions and in configurations where they are stable in operation, and emit continuous-wave light suitable for a wide variety of applications. Alternatively, they have been used to generate periodic pulses either through mode-locking or Q-switching techniques. It is, however, possible to generate chaotic waveforms with Class B and Class A lasers under a wide variety of conditions easily realized experimentally.

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For Class B lasers, we have coupled dynamics for the electric field and the population inversion, and the time scale for exchange of energy between these two elements of the laser is characterized by the relaxation oscillation frequency. This frequency is determined by a combination of three parameters – the cavity decay rate for light, the pump rate above threshold and the decay rate for the upper lasing level (Siegman [1986]). One of the most common situations is that of optical or opto-electronic feedback. If a fraction of the laser output is fed back into the laser cavity with a time delay, Class B lasers display chaotic dynamics; this is very well known in the case of semiconductor lasers, and we will examine this situation in detail later in this chapter (Section 2.5.4). A second situation in which chaotic dynamics is generated by multimode lasers is through the presence of an intracavity nonlinear optical crystal. The ‘green problem’ provides an excellent illustrative example, where nondegenerate four-wave mixing at an intracavity frequency doubling crystal inside a green ND:YAG laser may cause an instability for certain parameter regimes. Third, modulation of cavity loss (or gain) can drive a laser into regimes of chaotic dynamics, for modulation frequencies close to the relaxation oscillation frequency for the laser, or its harmonics and subharmonics. This too, is well known for Class B lasers, and was the technique suggested by Colet and Roy [1994] to generate chaotic dynamics in an Nd:YAG laser for chaotic optical communications. A second major advance in the field of optical instabilities occurred when Ikeda [1979] considered the problem of light circulating in an optical cavity with a nonlinear absorber. Instead of writing down ordinary differential equations that described the time evolution of the optical field on time scales long compared to the round-trip time of the light in the cavity, he chose to write coupled delaydifferential equations for the light and population in the cavity (Ikeda [1979]). The light field changed with each passage around the cavity, and a delay equation with appropriate boundary conditions coupled to a differential equation for the population. He then derived, under suitable limiting conditions, a simple map from these coupled equations. This map, now known as the Ikeda map, led to much research into optical instabilities and the complex associated dynamics. It has become a paradigm for studies in bifurcation theory, delay equations, and dynamical instabilities in optical systems. The fiber laser models described in Section 2.5.3 (Williams and Roy [1996], Williams, García-Ojalvo and Roy [1997], Abarbanel and Kennel [1998]), the models for optoelectronic systems in Section 2.5.6 (Larger, Goedgebuer and Delorme [1998], Larger, Goedgebuer and Merolla [1998]), and the spatiotemporal communication model in Section 3.7 (García-Ojalvo and Roy [2001a, 2001b]) are based on ideas that incorporate the Ikeda instability.

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1.4. Characterization of chaotic dynamics: a mathematical toolkit When trying to identify the nonlinear dynamical characteristics of a system, one usually makes measurements in time of the physically accessible variables. In the case of a laser system, the most common variable one can measure is the total intensity of the light output. The induced dipole moment and the population inversion are not usually easily measured, and even the phase of the electric field requires more complex instrumentation and measurement techniques. The question then arises as to how to analyze the data and what information can be extracted from the measurements made as best possible. As a case in point, if we are able to measure only the variable x in the Lorenz system, what can we discover about the nature of the dynamics of the whole system of coupled variables? This was an important question in the early days of the development of nonlinear dynamics as a scientific discipline, and the answer is very elegant and beautiful. It came to be known as the method of phase-space reconstruction, and involved the use of a tool known as ‘embedding’ developed in the early 1980s by Takens [1981] and Packard, Crutchfield, Farmer and Shaw [1980]. The basic theorem tells us that for a multidimensional system, if we are able to observe a single scalar variable (such as the total laser intensity) at a sequence of times, we can reconstruct many important properties of the multivariate phase-space dynamics of the system by a time-delay embedding process. Practically, in the example of the Lorenz system we are considering, we may see in figs. 3(a) and 3(b) the consequence of plotting the time series measured for x at times t, t + τ , t + 2τ , t + 3τ , t + 4τ, . . . in the three-dimensional space reconstructed with the axes x(t), x(t + τ ), x(t + 2τ ). The values of τ determine the precise shape of the attractor that has been reconstructed, but it is clear that the same butterfly attractor is viewed in both figs. 3(a) and 3(b) in a topological sense. The important questions of what should be the dimensionality of the space

Fig. 3. Lorenz attractors embedded in the time-delayed phase-space with the axes of x(t), x(t + τ ), and x(t + 2τ ); (a) τ = 0.1, (b) τ = 0.2.

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that is necessary for reconstruction and the appropriate value for the time delay are discussed in detail in many texts, including those of Abarbanel [1996] and Kantz and Schreiber [1997]. Many other tools have been developed by scientists and engineers to extract information about the nature of a dynamical system. Prominent amongst these are the notion of the Poincaré section, which consists of plotting the values of certain variables at selected time intervals and for certain conditions, such as shown in fig. 4(a), which displays the section for the Lorenz system. The figure shows the intersection of the butterfly attractor with a plane parallel to the x–y plane when the trajectory goes upwards through it. If the dynamics is periodic, such a section would show only a fixed number of points. When chaotic dynamics occurs, the section consists of structured clouds of points. Another measure of chaotic dynamics (particularly its sensitive dependence on initial conditions) is shown in fig. 4(b), where the distance between two neighboring trajectories (x(t), y(t), z(t)) and (x (t), y (t), z (t)) in phase space is shown for the Lorenz system, as measured by 2 2 2 1/2 δ(t) = x(t) − x (t) + y(t) − y (t) + z(t) − z (t) = δ(0) exp(λt).

(1.9)

We see from fig. 4(b) that the distance grows exponentially on the average until it saturates at t ≈ 25. The rate of growth is characterized by the effective Lyapunov exponent for the system, denoted by λ. More generally, a comprehensive characterization of the dynamics is done through the computation of the Lyapunov spectrum for the system. Methods for doing these computations have been widely developed for both experimentally measured time series as well as model equa-

Fig. 4. (a) Poincaré section of the Lorenz attractor at z = 27. (b) Time evolution of the distance δ(t) between two trajectories in phase-space. Sensitive dependence on initial conditions is observed.

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tions for dynamical systems (Abarbanel [1996], Kantz and Schreiber [1997]). The computation of the Lyapunov spectrum allows one to compute the Lyapunov dimension of a system (Abarbanel [1996]), which is also related to the correlation dimension (Grasserger and Procaccia [1983]), or information dimension of a dynamical system (Hilborn [2000]). An example of the use of delay embedding and computation of correlation functions and correlation dimension for an ammonia laser that displays Lorenzlike chaos is shown in fig. 5 (Hübner, Abraham and Weiss [1989]). As the relevant parameter for this laser system is changed, the laser displays different types of dynamics, ranging from simple period doubling to chaotic dynamics of various types. Such ‘routes to chaos’ have been studied for many laser systems and there exists a large body of literature on this topic (Bergé, Pomeau and Vidal [1984]). The surprising and important aspect of the development of these mathematical tools for description of nonlinear systems is that one may obtain quantitative characterizations of multidimensional system dynamics through the observation and measurement of a single scalar variable of the system. This fundamental perspective has been carefully tested and verified on many systems and models, yet there remain many open questions that are still of considerable interest. The phasespace reconstruction process and the evaluation of measures that probe specific features of a dynamical system’s behavior, particularly when it is coupled to other systems, is a topic of much interest and ongoing research (Boccaletti, Valladares, Pecora, Geffert and Carroll [2002]).

1.5. Control and synchronization of chaos While much of the research in the 1980s focused on the identification of routes to chaos and the characterization of the dynamics of the system, the focus in the 1990s was on the possibility of controlling chaotic systems. The main idea behind the research on the control of chaos (Ott, Grebogi and Yorke [1990], Shinbrot, Ditto, Grebogi, Ott, Spano and Yorke [1992]) was to stabilize the unstable periodic orbits that were contained within the chaotic dynamics of a nonlinear system, or to stabilize the unstable steady state of the system. This type of research has led to many attempts to dynamically control laser systems that inherently display chaos. Two relatively recent efforts have aimed at the stabilization of lead salt lasers (Chin, Senesac, Blass and Hillman [1996]) and a solution of the ‘green problem’ in lasers with intracavity frequency doubling crystals (Ahlborn and Parlitz [2004]). The most amazing aspect of these novel schemes to control chaotic systems is that the perturbations necessary are vanishingly small as the system

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Fig. 5. Comparison of data sets for the various methods used to analyze the experimental data. Columns 1, 2, and 3 are for data at different parameter values. Row 1: pulse trains of characteristic data sets. Row 2: phase-space reconstructed portraits. Row 3: autocorrelation functions. Row 4: log–log (base e) plot of the correlation integral vs length scale r for embedding dimensions 1–20 (upper to lower). Row 5: slopes of the log–log plots of the correlation integral vs log(r) for embedding dimensions 1–20 (lower to upper). (From Hübner, Abraham and Weiss [1989].)

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approaches the dynamical state that is desired, whether it is periodic or steady. It is as if the very sensitivity of chaotic systems to perturbations allows one to control them with tiny perturbations. A second topic that evolved through the 1990s and is directly relevant to this review, is the problem of synchronization of chaotic systems. Though the synchronization of clocks (periodic systems) has been studied with great care over centuries, it was the surprising discovery of temporal synchronization between two chaotic systems that initiated the field of ‘chaotic communications’. At first, even the notion of synchronization of chaotic systems appears self-contradictory. How can two chaotic systems that will inevitably start from slightly different initial conditions ever be synchronized? We just looked at the exponentially fast divergence of phase-space trajectories from ever so slightly different initial conditions in Section 1.4, and the issue of sensitivity to initial conditions as one of the hallmarks of chaos. The crucial realization was that when two chaotic systems are coupled to each other in a suitable way, they exert a form of ‘control’ on each other, and it is possible for both systems to synchronize in their dynamics, even when they start from very different initial conditions. The first studies and demonstrations of synchronization of chaotic systems (Blekhman [1971], Fujisaka and Yamada [1983], Afraimovich, Verichev and Rabinovich [1986], Pecora and Carroll [1990]) were done on electronic circuits, and soon led to the question – could the chaotic dynamics of these systems be used for something practical? The possibility that chaotic lasers could be synchronized was examined very shortly after the appearance of Pecora and Carroll’s work, in an important numerical study by Winful and Rahman [1990] on a linear array of coupled semiconductor lasers. They showed that this could be achieved by mutually coupling nearest-neighbor lasers through overlap of their optical fields. Their work stimulated experiments a few years later on mutually coupled solid-state lasers (Roy and Thornburg [1994]) as described in Section 2.5.2. Independently, Sugawara, Tachikawa, Tsukamoto and Shimizu [1994] studied synchronization of one-way coupled CO2 lasers (see Section 2.5.1). These early studies demonstrated that synchronization of chaotic optical systems could provide a means for communications through free space or optical fibers, and motivated much of the research described in this chapter.

1.6. Private communications The idea of private or secure communications immediately arose as a possible application. The application of chaotic synchronization to secret communication

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Fig. 6. Schematics of private communications using chaos. (Adapted from Ditto and Pecora [1993].)

systems was suggested in early work by Pecora and Carroll [1990, 1991]. They discovered that a chaotic transmitter could consist of an electronic circuit that simulated the dynamics, for example, of the Lorenz model shown in fig. 6 (Ditto and Pecora [1993]). The message to be concealed, assumed small in magnitude, was added to the chaotic fluctuations, assumed to be much larger, of one of the variables (let us choose the z variable for this purpose) and transmitted to the receiver, while another chaotic variable (let us choose x) was separately transmitted. The receiver consisted of a subsystem of the circuits in the transmitter that generated the dynamics of the y and z variables, and was driven by the signal from the x variable of the transmitter. The receiver synchronized to the chaos of the transmitter if the conditional Lyapunov exponents for the systems were negative for the given operating parameters. One could then recover the message from the chaos through a subtraction at the receiver. Cuomo and Oppenheim [1993] (see also Cuomo, Oppenheim and Strogatz [1993]) introduced an elegant variation of the method above that did not require the separate transmission of a driving signal to the receiver (see Section 3.3). They showed that the receiver could actually synchronize to the chaotic dynamics of the transmitter even when a message was added to the chaotic driving signal from the transmitter. The synchronized output from the receiver was then used to

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subtract out the information from the transmitted signal. The synchronization was not perfect, and the message, treated as a perturbation of the chaotic signal, had to be small compared to the chaos. Development of techniques in which the message actually drives the chaotic transmitter system were made by Halle, Wu, Itoh and Chua [1993]. The synchronization between receiver and transmitter can be exact, so message recovery can be very accurate in principle. The experiments reported in this chapter are related in spirit to a method developed first in electronic systems by Volkovskii and Rul’kov [1993], who suggested the use of an openloop system in the receiver. A proposal to use modulated unstable periodic orbits (UPOs) for secure communications and multiplexing was made by Abarbanel and Linsay [1993]. Multiplexing would be possible by using different UPOs to carry different messages. A different, adaptive approach to synchronization and secure communications was introduced by Boccaletti, Farini and Arecchi [1997]. An alternate approach to chaotic communications with UPOs was developed by Hayes, Grebogi and Ott [1993], Hayes, Grebogi, Ott and Mark [1994]. They symbolically encoded digital information into UPOs of a chaotic system and used chaos-control methods to switch between different orbits. This approach does not attempt to provide any privacy to the information being transmitted.

1.7. Privacy in chaotic communications The issue of privacy, however, arises naturally in a discussion of chaotic communication and is an important motivation for chaotic communication research. In his pioneering paper, “Communication Theory of Secrecy Systems”, Claude Shannon discussed three aspects of secret communications systems: concealment, privacy, and encryption (Shannon [1949]; see also Hellman [1977], Welsh [1988]). These aspects apply to systems that use chaotic waveforms for communication and can be interpreted in that context. Concealment of the information occurs because the chaotic carrier or masking waveform is irregular and aperiodic; it is not obvious to an eavesdropper that an encoded message is being transmitted at all. Privacy in chaotic communication systems results from the fact that an eavesdropper must have the proper hardware and parameter settings in order to decode and recover the message. In conventional encryption techniques, a key is used to alter the symbols used for conveying information. The transmitter and receiver share the key so that the information can be recovered. In a chaotic communication system, a transmitter that generates a time-evolving chaotic waveform acts as a ‘dynamical key’ to transform the information symbols. The information can be recovered with a receiver possessing the same dynamical key, i.e. its configuration and parameter

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settings are matched to those of the transmitter. It is interesting to note that using a chaotic carrier to dynamically encode information does not preclude the use of more traditional digital encryption schemes as well. Dynamical encoding with a chaotic waveform can thus be considered as an additional layer of encryption. Two factors that are important to privacy considerations in chaotic communication systems are the dimensionality of the chaos and the effort required to obtain the necessary parameters for a matched receiver. Earlier work has shown that for certain chaotic communication techniques, particularly those involving addition of a message to the chaotic carrier, the message can be recovered from the transmitted signal by mathematically reconstructing the transmitter’s chaotic attractor if the chaos is low-dimensional (Short [1994, 1996], Pérez and Cerdeira [1995]). Higher-dimensional signals, especially those involving hyperchaotic dynamics, are likely to provide improved security. The number of parameters that have to be matched for information recovery and the precision with which they must be matched are important aspects of receiver design (Yoshimura [2004]). We will comment in Section 3.6.4 on the issue of security in the use of chaos in fiber-laser systems for communications. At this point we would like to emphasize that the security of communication techniques is a complex and involved issue. Indeed, we do not know of any systematic cryptographic approach that has been taken to examine the security of different chaotic communication systems. We regard this as a very important open problem for future analysis.

1.8. Communication with chaotic lasers Following the original implementation of chaotic communications in electronic circuits (Kocarev, Halle, Eckert, Chua and Parlitz [1992], Cuomo and Oppenheim [1993]), a suggestion to use optical systems was made by Colet and Roy [1994]. It was clear that one does not have access in laser systems to all the system variables in the way possible with electronic circuits, so a somewhat different conceptual approach was developed to synchronize chaotic lasers. One of the major motivations for using lasers was that optical chaotic systems offer the possibility of high-speed data transfer, as shown in early simulations of numerical models that include realistic operational characteristics of the transmitter, receiver and communication channel (Mirasso, Colet and García-Fernández [1996]). The basic questions that arose in the context of chaotic communication with lasers were simple. First, what types of lasers display chaotic dynamics on very fast time scales? Second, is it possible to synchronize the dynamics of lasers? Third, is it possible to transit and receive information with chaotic waveforms

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as carriers of the information? All these questions have been answered over the last decade, and two laser systems of interest in optical communications (namely, semiconductor and fiber lasers) have been shown to be good candidates for chaotic optical communications. Williams and Roy [1996] (see also Williams, García-Ojalvo and Roy [1997]) found experimentally that erbium-doped fiber lasers (EDFRLs) are capable of generating lightwaves that possess irregular and chaotic intensity fluctuations on nanosecond time scales. Coupled delay-differential equations based on the Ikeda model (Ikeda [1979], Ikeda, Daido and Akimoto [1980]) were used by these authors to explain their observations. Abarbanel and Kennel [1998] developed a more detailed delay-differential equation model (see also Abarbanel, Kennel, Buhl and Lewis [1999], Lewis, Abarbanel, Kennel, Buhl and Illing [2000]) for these EDFRL intensity dynamics, which show that the dynamics occurs on picosecond time scales. Numerical simulations of the model are able to reproduce many aspects of the experimentally observed dynamics. Several schemes for communication with chaotic waveforms were experimentally demonstrated in which the intensity fluctuations generated by an EDFRL were used either to mask or to carry a message (VanWiggeren and Roy [1998a, 1998b, 1999a]). More recent work by Zhang and Chu [2003] has numerically explored the limits for communication systems with EDFRLs and shown that bit rates of many Gbits/s can be achieved in these systems. Similar concepts were used by Goedgebuer, Larger and Porte [1998] to demonstrate optical communication using chaotic wavelength fluctuations in the output from a semiconductor laser. Optical chaotic communication was also demonstrated for semiconductor lasers with time-delayed optical feedback by Sivaprakasam and Shore [1999b], Fischer, Liu and Davis [2000], Kusumoto and Ohtsubo [2002]. For these lasers, Fischer, van Tartwijk, Levine, Elsäßer, Göbel and Lenstra [1996] demonstrated that the optical field fluctuates on subnanosecond time scales. Many researchers have been involved in developing new experimental techniques and numerical models, and the literature in this field has grown rapidly over the past five years. Currently, bit rates of multigigabits/s have been achieved experimentally for single channels (Tang and Liu [2001b], Argyris, Kanakidis, Bogris and Syvridis [2004]), and numerical simulations have explored the limits set by both transmitter/receiver devices and communications channels (Sánchez-Díaz, Mirasso, Colet and García-Fernández [1999], Kanakidis, Argyris and Syvridis [2003]). The purpose of this chapter is to present the main ideas and advances in this area, as well as to outline the major challenges and open questions that need to be answered.

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§ 2. Synchronization of chaotic lasers 2.1. Why should chaotic systems synchronize? One of the most remarkable characteristics of chaotic systems is their sensitive dependence on initial conditions. Given this feature, it may seem counterintuitive to expect two coupled chaotic devices (whose initial conditions are in general different) to synchronize, however similar their structures and parameters are. This contrasts with the intuitively clear synchronization routinely observed in periodic systems (Pikovsky, Rosenblum and Kurths [2001]). In spite of these naive expectations, and as we will see in what follows, coupled chaotic systems do synchronize their irregular dynamics starting from arbitrary initial conditions, provided coupling between them is large enough and the two devices are sufficiently similar (a thorough review of this topic has been written by Boccaletti, Kurths, Osipov, Valladares and Zhou [2002], see also Pikovsky, Rosenblum and Kurths [2001]). Historically, the concept of synchronization of chaotic systems was pointed out by Blekhman [1971, 1988] and Fujisaka and Yamada [1983]. Afraimovich, Verichev and Rabinovich [1986] observed synchronization of chaos (they called it stochastic synchronization) in a circuit implementation of coupled chaotic oscillators. A remarkable demonstration of chaos synchronization was carried out by Pecora and Carroll [1990]. In their proposal, a nonlinear system (drive) is divided arbitrarily in two subsystems, one of which is duplicated into a new subsystem acting as a response. These two subsystems are coupled unidirectionally through a common variable. Mathematically, this scheme can be represented by considering an autonomous n-dimensional dynamical system, du = f (u), dt

(2.1)

where u = (u1 , . . . , un ) is an n-dimensional vector variable describing the state of the system, which can be divided arbitrarily into two subsystems: dv = g(v, w), dt dw = h(v, w), dt

(2.2) (2.3)

with v = (u1 , . . . , um ) and w = (um+1 , . . . , un ) representing the reduced state vectors of the two subsystems, and g = (f1 (u), . . . , fm (u)) and h = (fm+1 (u), . . . , fn (u)) being the respective forces. One now creates a response

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system as a copy w of the w subsystem with the following dynamics: dw = h(v, w ). dt

(2.4)

Hence, the dynamics of the response subsystem w depends on the driving variable v. This corresponds to a unidirectional coupling between the drive and response subsystems. The stability of the synchronized solution w = w can be determined by analyzing the linear equation dξ = D w h(v, w)ξ , dt

(2.5)

where ξ = w − w and D w h is the Jacobian of the drive vector field with respect to w. The eigenvalues of this Jacobian are called the conditional Lyapunov exponents of the coupled subsystem, and their signs determine the stability of the synchronized solution: the subsystems w and w synchronize only if all the conditional Lyapunov exponents are negative (Pecora and Carroll [1990]). As a simple example, let us consider the Lorenz system: x˙ = σ (y − x),

(2.6)

y˙ = rx − y − xz,

(2.7)

z˙ = −bz + xy.

(2.8)

We can divide this set of equations in two subsystems given by the variables (y) and (x, z), and consider a response subsystem (x , z ) (VanWiggeren [2000]) with dynamics x˙ = σ (y − x ),

z˙ = −bz + x y

(2.9) (2.10)

so that y is the driving signal in this case. This coupling scheme is depicted schematically in fig. 7. The Jacobian D w h is in this case −σ 0 Dw h = (2.11) . y −b Hence, the two conditional Lyapunov exponents in this case (−σ , −b) are negative for positive σ and b and the synchronized solution is stable, as can be checked numerically (VanWiggeren [2000]). Pecora and Carroll [1990] used this argument to study the synchronization properties of other coupling schemes in the Lorenz system, and in the Rössler case. They also applied this idea to a pair of coupled chaotic circuits. They used a modified version of an electronic chaotic circuit by Newcomb and Sathyan [1983],

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Fig. 7. Schematic structure of a simple coupling scheme for synchronizing two Lorenz systems (see text).

Fig. 8. Oscilloscope traces of the response voltage x2 vs its drive counterpart voltage x2 for (a) equal circuit parameters and (b) circuit parameters different by 50%. (From Pecora and Carroll [1990].)

demonstrating synchronization between the chaotic dynamics of the drive and the response after a transient. Figure 8 shows oscilloscope traces of one of the vari-

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ables in the drive system versus its response counterpart for two different parameter values. Identical synchronization is observed at identical parameter values (fig. 8(a)), whereas a 50% mismatch in the circuit parameters distorts synchronization substantially (fig. 8(b)). The possibility for applications of chaos synchronization to communications was also pointed out.

2.2. Coupling schemes and synchronization types The scheme developed by Pecora and Carroll requires that a chaotic system can be separated into two subsystems, one of which has to be replicated into a response system that must have only negative conditional Lyapunov exponents when acted upon by one of the driver’s variables. While this ingenious scheme has been implemented in practice in electronic oscillators, it cannot be readily applied to optical systems, since it is basically impossible to separate the elements of a nonlinear optical system in that way. 2.2.1. Identical synchronization Another synchronization scheme, that better fits the capabilities of optical systems, was proposed by Pyragas, who extended his method of continuous control of chaos via feedback (Pyragas [1992]) to the synchronization of chaos (Pyragas [1993]). The block diagram of the method is presented in fig. 9 (see also fig. 11(a)). One of the drive variables is recorded in a memory, and the response system can be forced by using a small feedback perturbation of the difference between two variables for the drive and response. This can be represented schematically by dx = f (x, y), (2.12) dt dx (2.13) = f (x , y ) + kc [x − x ], dt where only the equations of the state variables directly involved in the coupling mechanism are shown. kc is the strength of the perturbation, x and x are the relevant variables of the drive (or transmitter) and response (or receiver) systems, with y and y representing the rest of variables of each system, and f is the nonlinear function that governs the dynamics of x and x . The perturbation has to be introduced into the response system as a negative feedback in terms of x (kc > 0). The important feature of this perturbation is that it vanishes when the transmitter and receiver signals coincide. For identical parameters of transmitter

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Fig. 9. Block diagram of the method proposed by Pyragas [1993].

Fig. 10. Segments of the recorded aperiodic output signal yap (t) and the dynamics of the output signal y(t) and the difference y(t) = yap − y for the nonautonomous Duffing oscillator. The arrows show the moment of switching on the perturbation. (From Pyragas [1993].)

and receiver, a synchronous solution of eqs. (2.12)–(2.13) exists in a mathematical sense. In other words, the synchronized state x = x (and y = y ) is a fixed point of the system (2.12)–(2.13), which can be stabilized by the coupling term for a sufficiently large strength kc . We refer to this type of synchronization as identical synchronization in the following sections. The expression complete synchronization is also used in the literature (Boccaletti, Kurths, Osipov, Valladares and Zhou [2002], Pikovsky, Rosenblum and Kurths [2001]). Pyragas numerically demonstrated this type of chaos synchronization in the nonautonomous Duffing oscillator (as shown in fig. 10), as well as in the Rössler and the Lorenz systems. Most studies of chaos synchronization in electronic circuits have relied on coupling through an operational amplifier as a buffer for unidirectional coupling, or

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through a resistor for mutual coupling. In both cases, the coupling signal is proportional to the difference between the voltages of the drive and response systems, which is identical to the Pyragas method. Synchronization of chaos has been experimentally observed in many different types of chaotic electronic circuits: Chua’s circuit (Chua, Kocarev, Eckert and Itoh [1992]), Anishchenko–Astakhov oscillator (Anishchenko, Vadivasova, Postnov and Safonova [1992]), Volkovskii– Rul’kov oscillator (Rul’kov, Volkovskii, Rodríguez-Lozano, Del Río and Velarde [1992]) and phase-locked loops (Endo and Chua [1991]). 2.2.2. Restricted type of generalized synchronization The Pyragas method is very general and applicable to most kinds of nonlinear dynamical systems. However, when one tries to apply that method to coherently coupled optical systems a problem arises, because it is not easy to make a coherent subtraction between two optical fields (i.e., a subtraction that includes their fast optical phases). For that reason, Pyragas coupling is frequently substituted in optical setups by simple injection (fig. 11(b)), which corresponds to replacing eq. (2.13) by dx = f (x , y ) + kc x. dt

(2.14)

In this case the completely synchronous solution x = x is no longer a fixed point of the system. This problem can be circumvented by introducing an extra (linear) cavity loss in the receiver, which replaces the term −kc x in eq. (2.13), so that the injection signal can be compensated by the extra loss (Colet and Roy [1994]). Therefore, a simple injection from the transmitter to the receiver can

Fig. 11. Coupling schemes for synchronization: (a) Pyragas method, (b) simple injection, (c) injection with feedback. TL, transmitter laser; RL, receiver laser.

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cause identical synchronization in optical systems under certain conditions, which make it equivalent to the Pyragas method. However, in many experimental implementations (for instance, when using semiconductor lasers as emitting/receiving devices), it is not possible to tune the receiver’s cavity loss, hence the simple injection scheme described in eq. (2.14) has to be used. In addition, it has been known experimentally and numerically that synchronization can still be achieved even in the absence of the fixed point x = x . This type of synchronization by simple optical injection has been observed in many semiconductor and solid-state lasers (see Section 2.5.4), yet the mechanism of synchronization has not been well understood. We refer to this regime as a restricted type of generalized synchronization in the following sections. This kind of synchronization may be associated with a nonlinear amplification of the driving signal in the receiver via injection locking, and is much more tolerant to parameter mismatch than the identical synchronization (see Section 2.5.4).

2.3. Synchronization in feedback systems There are other ways of recovering the symmetry between the transmitter and receiver equations, so that a completely synchronized solution exists. A common example consists in compensating the extra power injected into the receiver by including feedback in both systems, as shown in fig. 11(c). This setup can be represented schematically by dx (2.15) = f (x, y) + kt x(t − τt ), dt dx = f (x , y ) + kr x (t − τr ) + kc x(t − τc ), (2.16) dt where kt,r are the feedback rates of the transmitter and the receiver, kc is the coupling rate between the transmitter and the receiver, τt,r are the feedback delay times of the transmitter and the receiver, and τc is the coupling delay time from the transmitter to the receiver. We have replaced a subtraction of optical fields by an addition, which is readily attainable. In eqs. (2.15)–(2.16) we have made explicit the delay times associated with the two feedback loops, τt and τr , and with the coupling, τc , but the results that follow can also be obtained for instantaneous (i.e. zero-delay) feedbacks and coupling. A synchronization solution exists for eqs. (2.15)–(2.16), defined by x (t) = x(t), provided kt = kr + kc and τt = τr = τc . The stability of this solution depends on the particular dynamical system under consideration, but regions of

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stability in parameter space have indeed been observed in various systems. This type of synchronization corresponds to the identical synchronization described in Section 2.2.1. In the absence of receiver feedback (kr = 0), identical synchronization (x (t) = x(t)) can also be observed provided kt = kc and τt = τc . This is known as the open-loop configuration, as opposed to the closed-loop configuration defined by a nonzero kr . In general, the open-loop configuration is less sensitive to parameter mismatch than the closed-loop one, since the former does not require feedback in the receiver (and hence no tuning of feedback parameters in the receiver is needed at all). For the open-loop configuration, synchronization exists even if τc = τt . The synchronized state in this case is x (t) = x(t − τ ), where τ = τc − τt . For τt > τc (the feedback delay time of the transmitter is longer than the coupling delay time between the two lasers), τ < 0 and the receiver anticipates the behavior of the transmitter. This anticipative synchronization was predicted theoretically in general models of nonlinear systems with feedback by Voss [2000], reproduced numerically in a model of coupled semiconductor lasers with optical feedback by Ahlers, Parlitz and Lauterborn [1998], Masoller [2001], and observed experimentally by Liu, Takiguchi, Davis, Saito and Liu [2002], Tang and Liu [2003a] in different configurations. In spite of its seemingly counterintuitive character, this type of synchronization is just a particular dynamical state of coupled feedback systems with τt > τc . In the opposite case of τt < τc , retarded synchronization can be observed (Tang and Liu [2003a]). Identical synchronization x (t) = x(t) is a special case of the anticipative or retarded synchronization when τt = τc in time-delayed feedback and coupling systems. Similarly to the case without feedback described above, eqs. (2.15)–(2.16) can exhibit the restricted type of generalized synchronization described in Section 2.2.2 with a time delay of τc , i.e. x (t) = x(t − τc ). This response behavior can be interpreted as a driven waveform, since the time delay between x(t) and x (t) only depends on τc , not τt . The internal dynamics of the transmitter just generates the driving waveform and does not enter into the dynamics of the response system. Therefore, the restricted type of generalized synchronization can be distinguished from the identical synchronization (either synchronous, x (t) = x(t), or asynchronous, x (t) = x(t − τ )) by investigating the delay time between x(t) and x (t) (Liu, Davis, Takiguchi, Aida, Saito and Liu [2003]). It must be noted that the tolerance to parameter mismatch of this type of synchronization is much larger than that for identical synchronization in general (see Section 2.5.4).

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2.4. One-way versus mutual coupling In the previous subsection we have restricted ourselves to the situation in which the coupling signal travels unidirectionally from one laser to the other. Such a one-way coupling is well suited for communication applications, which connect two distantly separated chaotic lasers with well-defined transmitter and receiver roles, maintaining the original chaotic carrier generated by the transmitter free from the influence of the receiver’s dynamics. There are situations, however, where the coupling signal travels bidirectionally between the two lasers. This type of mutual coupling can be implemented ad hoc in face-to-face coupled lasers (Heil, Fischer, Elsäßer, Mulet and Mirasso [2001], Buldú, Vicente, Pérez, Mirasso, García-Ojalvo and Torrent [2002], Rogister and García-Ojalvo [2003]), but it also arises naturally in laser arrays, where each laser is coupled with its neighbors via the overlap between the evanescent tails of their respective electric fields, so that coupling is inherently mutual. The two cases of one-way and mutual couplings can be modeled by rewriting eqs. (2.12) and (2.14) as follows: dx = f (x, y) + kc1 x , dt dx = f (x , y ) + kc2 x, dt

(2.17) (2.18)

where kc1 is the coupling strength from the response to the drive, and kc2 is the coupling strength from the drive to the response. One-way coupling is the case when kc1 = 0 and kc2 = 0, whereas mutual coupling is the case when kc1 = 0 and kc2 = 0. In most optical systems, coupling can be achieved through the electrical field (or laser intensity) because the latter is easy to transmit between separate laser systems. From a historical viewpoint, synchronization of chaos in one-way coupled optical systems was predicted numerically in an Nd:YAG laser model (Colet and Roy [1994]), and observed experimentally in one-way coupled CO2 laser systems (Sugawara, Tachikawa, Tsukamoto and Shimizu [1994]). Also, chaos synchronization in mutually coupled lasers was investigated numerically in arrays of waveguide lasers coupled by means of their overlapping evanescent fields (Winful and Rahman [1990]), and experimentally reported in an array of two Nd:YAG lasers (Roy and Thornburg [1994]). The details of those works will be described in the following subsections. The comparison of characteristics between one-way and mutual coupling was also investigated numerically by Uchida, Ogawa and Kannari [1998]. The coupling strength required for synchronization is smaller in

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mutual coupling than in one-way coupling. Also, mutual coupling sometimes stabilizes chaotic oscillations, which is known as amplitude death (Herrero, Figueras, Rius, Pi and Orriols [2000], Kuntsevich and Pisarchik [2001]).

2.5. Examples of chaotic laser synchronization We now describe in detail examples where synchronization of chaotic dynamics between coupled lasers has been reported. 2.5.1. Gas lasers The investigation of the dynamics of gas lasers has a long history compared with other types of lasers. The first experimental observation of Lorenz–Haken chaos was reported in a highly pumped NH3 laser (Weiss and Brock [1986]). Since then, different types of bifurcation phenomena and chaotic behaviors were observed. The introduction of additional nonlinear components in gas lasers produces rich dynamics of temporal waveforms of laser intensity. Gas lasers provided one of the first experimental observations of synchronization of chaotic lasers, reported by Sugawara, Tachikawa, Tsukamoto and Shimizu [1994]. Their system consists of two carbon dioxide (CO2 ) lasers with gaseous saturable absorbers; one is the master laser and the other is the slave laser, as shown in fig. 12. An intracavity saturable absorber induces self-sustained pulsation in a single-mode oscillation, which is known as passive Q switching. This pulsation can become chaotic following a series of period-doubling bifurcations. Sugawara, Tachikawa, Tsukamoto and Shimizu [1994] demonstrated that the two

Fig. 12. Diagram of the experimental setup of Sugawara, Tachikawa, Tsukamoto and Shimizu [1994].

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chaotic pulsations are synchronized when the radiation from the master laser is unidirectionally injected into the saturable absorber of the slave laser. Figure 13 shows time sequences of the chaotic pulsations from the two lasers and correlation plots between the peak heights and between the peak intervals for different values of the input power of the master laser beam. When there is no coupling between the two systems, each laser exhibits chaotic pulsations as shown in fig. 13(a). In the correlation plots of figs. 13(a ) and 13(a ), data points are scattered twodimensionally in an erratic manner, indicating that the two lasers are pulsating independently. When the input power is raised, the chaotic pulsation of the slave laser is synchronized to the driving pulsation as seen in fig. 13(c). A linear relation between the corresponding peak heights and intervals is clearly observed in figs. 13(c ) and 13(c ). When the input power is further increased, the synchronization is destroyed, as shown in fig. 13(d). Coupled CO2 lasers with saturable absorber can be described by a three-level– two-level rate-equation model, as an extension of the Class B laser model, for the normalized photon density I , population density in the upper laser level M1 , population density in the lower laser level M2 , and the difference of the population density in the absorber levels N (Sugawara, Tachikawa, Tsukamoto and Shimizu [1994]): dIm = (M1,m − M2,m ) − Bm Nm − km Im , (2.19) dt dM1,m = Pm Mm − R10,m M1,m − (M1,m − M2,m )Im , (2.20) dt dM2,m = −R20,m M2,m + (M1,m − M2,m )Im , (2.21) dt dNm = −2bm Nm Im − rm (Nm − 1), (2.22) dt dIs = (M1,s − M2,s ) − Bs Ns − ks Is , (2.23) dt dM1,s = Ps Ms − R10,s M1,s − (M1,s − M2,s )Is , (2.24) dt dM2,s = −R20,s M2,s + (M1,s − M2,s )Is , (2.25) dt dNs = −2bs Ns (Is + CIm ) − rs (Ns − 1). (2.26) dt Here the subscripts “m” and “s” refer to the variables and parameters of the master and slave laser systems, respectively; B is the normalized rate of absorption in the passive medium, k is the cavity loss rate, P is the rate at which the upper laser level is pumped, R10 is the rate of vibrational relaxation from the upper laser level

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Fig. 13. Observed time sequences of the passive Q-switching pulsation of the master and slave lasers for different powers of the injected beam (a–d) together with the correlation plots between the peak heights (a –d ) and between the peak intervals (a –d ). (From Sugawara, Tachikawa, Tsukamoto and Shimizu [1994].)

to all other levels except the lower level, R20 is the rate of vibrational relaxation from the lower laser level to other levels, b is the normalized cross section of the absorption in the passive medium, and r is the rotational relaxation rate of the absorptive levels. The radiation of the master laser modulates the absorber’s population in the slave laser through the second term on the right-hand side of eq. (2.26), where C is the coupling coefficient. The two laser intensities interact with each other within the saturable absorber. The coupling is thus incoherent (i.e. independent of the optical phase). The dependence of the chaotic pulsations on the strength

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of the driving force (fig. 13) is qualitatively reproduced by a computer simulation based on this model. Several experiments and simulations have been carried out in configurations similar to fig. 12 (Liu, de Oliveira, Danailov and Rios Leite [1994], Sauer and Kaiser [1998], Cavalcante and Rios Leite [2003]). Another way of synchronizing two mutually coupled gas lasers is through a common intracavity saturable absorber as shown in fig. 14 (Barsella, Lepers, Dangoisse, Glorieux and Erneux [1999], Susa, Erneux, Barsella, Lepers, Dangoisse and Glorieux [2000]). The system is composed of two laser cavities with two separate active media (AM1 and AM2). The two lasers operate in orthogonal polarization states and allow the incoherent interaction to happen only in the saturable absorber cell (ABS), where the two beams are superimposed. This mutual coupling system allows lag synchronization between the two chaotic temporal waveforms (Barsella and Lepers [2002]). The synchronization between two modes within the same laser has been also reported (Tang, Dykstra and Heckenberg [1996]). Synchronization with a pre-recorded signal has been also demonstrated in gas lasers. A driving signal is stored in the memory of an arbitrary function generator. The arbitrary function generator produces an analog signal that has exactly the same waveform as the stored one. This analog signal is used to modulate in amplitude the RF driving signal of an acousto-optic modulator (AOM), which in return transfers this modulation signal to the intensity of the pump laser beam. The undiffracted beam from the AOM is used for pumping the gas laser. The driving signal can be the original chaotic signal (Tsukamoto, Tachikawa, Hirano, Kuga and Shimizu [1996]), a chaos generated by a different chaotic system (Tang, Dykstra, Hamilton and Heckenberg [1998]), a periodic signal (Allaria, Arecchi, Di Garbo and Meucci [2001]), or a noise signal (Zhou, Kurths, Allaria, Boccaletti, Meucci and Arecchi [2003]), corresponding to the observation of identical, generalized, phase, and noise-induced synchronization, respectively (see Section 2.6).

Fig. 14. Experimental setup for synchronization via a common saturable absorber. AM1, AM2, active media; ABS, absorber; G1, G2, diffraction gratings; CBW, coated Brewster window; M, mirror; OM, partly transparent output mirror. (From Barsella, Lepers, Dangoisse, Glorieux and Erneux [1999].)

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2.5.2. Solid-state lasers Solid-state lasers are excellent tools for testing the fundamental physics of synchronization of chaos in laser systems. The relaxation oscillation frequency of solid-state lasers ranges from kHz to MHz, much lower than that of semiconductor lasers, which lies around GHz. The low characteristic time scale of solid-state lasers allows us to experimentally detect chaotic temporal waveforms easily, without using high-speed detection equipment. From the dynamical point of view, the dynamics of solid-state lasers are relatively simple when compared with semiconductor lasers, because there is no coupling between the amplitude of the electric field and its optical phase, which in semiconductor lasers is represented by a linewidth enhancement factor or α parameter (see Section 2.5.4). Because of these advantages, many studies have been devoted to the synchronization of chaos in solid-state lasers, as described in this subsection. One of the first experimental observations of synchronization of chaotic lasers was reported in 1994 by using two neodymium-doped yttrium aluminum garnet (Nd:YAG) solid-state lasers (Roy and Thornburg [1994]). The system consisted of two Nd:YAG laser beams of wavelength 1.06 µm generated in the same crystal by two 514.5 nm pump beams of almost equal intensity obtained from an argon (Ar+ ) laser, as shown in the left panel of fig. 15. The spatial separation of the parallel pump beams is varied so that the mutual coupling between the Nd:YAG lasers can be provided by overlap of the intracavity laser fields. One or both lasers are driven chaotic by periodic modulation of their pump beams, and synchronized chaotic intensity fluctuations are observed in both cases when the lasers are sufficiently coupled. The synchronized nature of the chaotic lasers is evident in the right panel of fig. 15. Synchronization of the chaotic lasers persists stably over periods of tens of minutes, as long as the temperature and other environmental conditions are maintained constant. Further experiments with two coupled Nd:YAG lasers examined the parameter space of detuning and coupling strength, and also the role of colored noise in creating synchronized intensity bursts (Thornburg, Möller, Roy, Li, Carr and Erneux [1997]). Synchronization of chaos was subsequently reported in linear arrays of three Nd:YAG laser (Terry, Thornburg, DeShazer, VanWiggeren, Zhu, Ashwin and Roy [1999]) and three Nd:YVO4 lasers (Möller, Forsmann and Jansen [2000]). The synchronization dynamics of Nd:YAG laser arrays with an external feedback were also investigated (Uchida, Shimamura, Takahashi, Yoshimori and Kannari [2001a]). The dynamics of single-mode solid-state lasers can be described by a Class B laser model for the electrical field and population inversion. The rate equations for two spatially coupled single-mode solid-state lasers for the slowly varying, complex electric field amplitude Ei and real gain Gi of laser i are (Fabiny, Colet,

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Fig. 15. Left: experimental system for generating two spatially coupled chaotic Nd:YAG lasers. Right: (a) relative intensities of two strongly coupled lasers with the pump beam for laser 1 modulated by the AOM. Note the strong synchronization of the two laser intensities; (b) X–Y plot of the two laser intensities shown in (a). Note the strong linearity of this plot, indicating the synchronized nature of the time traces. (From Roy and Thornburg [1994].)

Roy and Lenstra [1993]) dE1 dt dG1 dt dE2 dt dG2 dt

1 (G1 − α1 )E1 + κE2 + iω1 E1 , τc 1 = p1 − G1 − G1 |E1 |2 , τf 1 = (G2 − α2 )E2 + κE1 + iω2 E2 , τc 1 p2 − G2 − G2 |E2 |2 . = τf =

(2.27) (2.28) (2.29) (2.30)

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In these equations, τc is the cavity round trip time, τf is the fluorescent time of the upper lasing level, p is the pumping coefficient, α is the cavity loss coefficient, ω is the detuning of the laser from a common cavity mode, and κ is the coupling coefficient; i is the imaginary unit. Chaos can be generated by using either pump modulation or loss modulation, i.e., p(t) = p0 (1 + b cos(Ωt)) or α(t) = α0 (1 + b cos(Ωt)), respectively, where b and Ω are the modulation amplitude and angular frequency. The terms with κE2 and κE1 in eqs. (2.27) and (2.29) represent the mutual coupling between the two lasers through spatial overlap of the electric fields, which is a coherent (optical-phase-dependent) coupling. Synchronization of chaos in coupled solid-state lasers has been observed by using this model (Terry, Thornburg, DeShazer, VanWiggeren, Zhu, Ashwin and Roy [1999], Kuske and Erneux [1997], Colet and Roy [1994]). Solid-state lasers that have a short cavity length (typically less than one millimeter) are specifically called microchip lasers. Due to their short cavity length, lasing occurs in a single mode or in a few longitudinal modes. It is thus easy to model the dynamics of microchip lasers precisely. Synchronization of spontaneous emission fluctuations in laterally coupled microchip arrays has been modelled by Serrat, Torrent, García-Ojalvo and Vilaseca [2001] (see also Serrat, Torrent, García-Ojalvo and Vilaseca [2002]) and experimentally by Serrat, Vilaseca, Bouwmans, Segard and Glorieux [2002]. Synchronization of mutually coupled chaotic lasers has been demonstrated in a microchip LiNdP4 O12 (LNP) laser array with self-mixing feedback modulation (Otsuka, Kawai, Hwong, Ko and Chern [2000]). In that case, a pumping beam from an Ar+ laser is divided into two beams and is focused on the input surface of the LNP crystal by a common focusing lens, as shown in the left panel of fig. 16. The coherent coupling between the two LNP lasers occurs through the overlap of their lasing fields, which have a spot size of 200 µm each. The two parallel beams from the LNP lasers are incident to and scattered from a turntable. The interference between the lasing and frequency-shifted feedback fields modulates the losses of the microcavity lasers, in what is known as self-mixing laser-Doppler-velocimetry feedback. The right panel of fig. 16 shows in (a) a plot of the correlation in amplitude between the two signals when the distance between the two beams is 1.5 mm. To examine the phase correlation of the chaotic pulsations, the time interval between the nth peak and the subsequent peak for laser 2 is plotted against that for laser 1 in fig. 16(b). In this case, asynchronous chaotic fluctuations in both amplitude and phase are apparent, and the two lasers are found to behave independently. When the separation between the two lasers is decreased, a mutual interaction appears and the phase fluctuations of the two lasers are squeezed while their amplitudes remain uncorrelated, just before the onset of chaos synchronization, as shown

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Fig. 16. Left: experimental setup of a LiNdP4 O12 (LNP) laser array subjected to Doppler-shifted light injections. Right: signal correlations of the LNP laser array: (a, b) weak coupling (two-beam distance 1.5 mm); (c, d) moderate coupling (0.85 mm); (e, f ) strong coupling (0.8 mm). Plots (a, c, e) are for amplitude correlation, and (b, d, f ) are for phase correlation. (From Otsuka, Kawai, Hwong, Ko and Chern [2000].)

in figs. 16(c) and 16(d). When the separation is decreased further, synchronized chaotic states, in which the two lasers exhibit chaotic pulsations with both strong amplitude and phase correlation, are obtained as shown in figs. 16(e) and 16(f ). For numerical simulation a globally coupled Class B laser array with incoherent feedback was investigated by Otsuka and Chern [1992]. Synchronization of chaos in one-way coupled microchip solid-state lasers has also been reported (Uchida, Ogawa, Shinozuka and Kannari [2000]). In that case, two neodymium-doped yttrium orthovanadate (Nd:YVO4 ) microchip lasers were used as laser sources (wavelength 1064 nm), as shown in the left panel of fig. 17. Chaotic outputs were obtained by modulating the pumping at frequencies of the order of the sustained relaxation oscillation frequency (a few MHz). A fraction of the master laser output is unidirectionally and coherently coupled to the slave laser cavity for chaos synchronization. The optical frequencies of the two microchip lasers are precisely controlled with thermoelectronic coolers in order to achieve injection locking, where the optical frequencies of two individual lasers are perfectly matched when the frequency difference is set within a certain injection-

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Fig. 17. Left: experimental setup for chaos synchronization in two Nd:YVO4 microchip lasers with pump modulation. BS, beam splitter; L, lens; M, mirror; VA, variable attenuator; LD, laser diode; MCL, Nd:YVO4 microchip laser; PL, Peltier device; IS, optical isolator; PD, photodiode; FC, fiber coupler; PM, pump modulation; λ/2 WP, λ/2 wave plate; F–P, Fabry–Pérot etalon. Right: experimentally obtained chaotic temporal wave forms and correlation plots for the two laser outputs: (a, b) without synchronization and (c, d) with synchronization. (From Uchida, Ogawa, Shinozuka and Kannari [2000].)

locking range. The right panel of fig. 17 shows the chaotic temporal waveforms and the correlation plots between the two laser outputs. In the absence of coupling, there is no correlation at all between the chaotic pulsations of the two lasers, as shown in figs. 17(a) and 17(b). When a fraction of the master laser output is injected into the slave laser cavity, injection locking is achieved between the two lasers and the chaotic oscillations are synchronized, as shown in fig. 17(c). The linear correlation between the two laser outputs, shown in fig. 17(d), confirms the existence of synchronization. This synchronization can be maintained for tens of hours, as long as injection locking of the two laser frequencies is preserved. This experiment shows that in certain situations the condition to achieve synchronization of chaos is almost equivalent to that for injection locking. In other words, synchronization of chaos in lasers can be based on the regeneration of the chaotic laser output through the mechanism of injection locking. Another coupling scheme of incoherent coupling via the difference of the two laser intensities but without using injection locking has been reported by Uchida, Matsuura, Kinugawa and Yoshimori [2002]. In solid-state lasers, spatial hole burning usually leads to laser emission in several longitudinal cavity modes (Mandel [1997]). The simple Class B laser model

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shown in eqs. (2.27)–(2.30) is not appropriate for describing the dynamics of multi-longitudinal-mode lasers. Tang, Statz and deMars [1963] introduced an excellent model describing cross-saturation of population inversion among modes due to the spatial hole-burning effect for multimode solid-state lasers (see also Mandel [1997], Otsuka [1999]). A standing wave for each electrical field amplitude is spatially distributed in the laser crystal, which causes a spatial grating of population inversion along the z-axis (propagation direction of laser light). The spatial distribution of the population inversion for an N -mode laser can be decomposed as follows (Pieroux, Mandel and Otsuka [1994]): 1 L n0 (t) = (2.31) ntotal (z, t) dz, L 0 2 L ntotal (z, t) cos(2ki z) dz, ni (t) = (2.32) L 0 where n0 and ni are the space-averaged and the first Fourier components (in space) of population inversion density for the ith mode (i = 1, 2, . . . , N ), ntotal (z, t) is the total population inversion density inside the cavity, L is the length of the laser cavity, and ki is the wave number of the electric field for the ith mode. The electric field of each laser mode is coupled to other modes through the spatially decomposed population inversions of n0 and ni . The Tang–Statz– deMars equations for two one-way-coupled N -mode lasers are described with n0 and ni as follows (Ogawa, Uchida, Shinozuka, Yoshimori and Kannari [2002]): N

nk,d dn0,d 2 = pd − n0,d − γk,d n0,d − , (2.33) Ek,d dt 2 k=1 N

dni,d 2 2 = γi,d n0,d Ei,d − ni,d 1 + (2.34) γk,d Ek,d , dt k=1

Kd dEi,d ni,d = γi,d n0,d − − 1 Ei,d , (2.35) dt 2 2 N

nk,r dn0,r 2 = pr − n0,r − Ek,r γk,r n0,r − , (2.36) dt 2 k=1 N

dni,r 2 2 = γi,r n0,r Ei,r − ni,r 1 + (2.37) γk,r Ek,r , dt k=1

Kr dEi,r ni,r = γi,r n0,r − − 1 Ei,r dt 2 2 Kr κi Ei,d cos(2πτ µi ) + (2.38) 2

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and dµi Kr Ei,d sin(2πτ µi ), = 2πτ νi − κi dt 2 Ei,r

(2.39)

where Ei is the real amplitude of the lasing electrical field for the ith mode (with the total electric field being Ei (t) = 12 {Ei (t) exp[i(2πνi t + µi (t))] + c.c.}), and µi = µi,d − µi,r is the optical-phase difference between the drive and response electrical fields for the ith mode. The subscripts “d” and “r” indicate the drive and response lasers, respectively; p is the pumping parameter scaled to the laser threshold, γ is the gain coefficient for the ith mode, K = τ/τp , with τ the upper-state lifetime and τp the photon lifetime in the laser cavity, κi is the coupling strength from the drive to the response laser, and νi = νi,d − νi,r is the detuning of the lasing frequency between the two lasers for the ith mode. Time is scaled by τ . Pump modulation can be used to generate chaos, i.e., p = p0 (1 + A cos(2πτf t)), where A and f are the modulation amplitude and frequency, respectively. Synchronization of multimode microchip solid-state lasers has been demonstrated with this model by Ogawa, Uchida, Shinozuka, Yoshimori and Kannari [2002]. In multimode lasers, synchronization of chaos requires frequency locking of all the corresponding longitudinal modes between two coherently coupled lasers. The power spectrum ratio of longitudinal modes needs to be matched between the drive and response lasers for accurate synchronization, which is a more severe condition for synchronization when compared with coherently coupled single-mode lasers (Uchida, Ogawa, Shinozuka and Kannari [2000]). 2.5.3. Fiber lasers In fiber lasers the optical gain is provided by rare-earth elements (such as erbium, neodymium and ytterbium) embedded in silica fiber. Under optical pumping, those atoms provide light amplification at a characteristic wavelength; in the particular case of erbium that wavelength is of the order of 1.55 µm, which lies within the spectral region of minimal loss of silica fibers. For that reason, erbiumdoped fiber amplifiers have been widely used since the mid-1990s in fiber-optics communication systems (Agrawal [1995], Desurvire [1994], Becker, Olsson and Simpson [1999]). In the presence of a feedback mechanism (either by using mirrors, or by closing the fiber on itself forming a fiber ring) laser emission can be obtained. The concurrence in fiber lasers of the inherent nonlinear character of both the optical fiber and the light amplification process leads to a rich variety of dynamical instabilities and nonlinear behavior (van Tartwijk and Agrawal [1998]). Erbium-doped fiber

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lasers, for instance, are very efficient generators of ultra-short pulses and solitons (Agrawal [1995]), and exhibit different types of complex nonlinear behavior, including bursting (DeShazer, García-Ojalvo and Roy [2003], DeShazer, Tighe, Kurths and Roy [2004]) and chaos (García-Ojalvo and Roy [1997], Abarbanel, Kennel, Buhl and Lewis [1999]). A second defining characteristic of fiber lasers is that, due to the waveguiding properties of optical fiber, their cavities can be very long (on the order of kilometers), orders of magnitude longer than other lasers. For that reason, the frequency separation between consecutive longitudinal modes is very small (on the order of MHz). Additionally, the amorphous character of the host medium leads to a very broad gain profile (on the order of tens of GHz). As a consequence, a large number of longitudinal cavity modes can experience gain and coexist inside the cavity, coupled through gain sharing. Hence, fiber lasers usually operate in a strongly multimode regime, and consequently their dynamics cannot be described in general by single-mode models, or by models containing a small number of coupled modes. For fiber lengths of tens of meters, the round-trip time taken by the light to travel once along the laser cavity is of the order of hundreds of nanoseconds. This time is much longer than the sampling time of today’s standard oscilloscopes, and therefore intracavity dynamics is straightforward to observe (and work with) in fiber lasers, at variance with other types of lasers. As a result, the dependence of the generated radiation field on the propagation direction cannot be neglected, as done usually in the mean-field approximations that lead to rate-equation descriptions of laser dynamics. In order to address the limitations (or rather, the lack of limitations) posed in the previous two paragraphs, a modeling approach based on delay-differential equations can be used (Williams and Roy [1996]). This approach requires neither single-mode nor mean-field approximations, making use instead of the boundary conditions affecting the laser light as it travels around the cavity (Ikeda [1979]). Introducing these conditions into the Maxwell–Bloch equations that govern the evolution of the electric field and population inversion, after the material polarization has been adiabatically eliminated, leads to the following delay-differential equation model (Williams, García-Ojalvo and Roy [1997]): E(t + τR ) = Einj exp(iωt) + RE(t) exp (β + iα)w(t) + iκ , (2.40) 2 [exp{Gw(t)} − 1] dw(t) , (2.41) = Q − 2γ w + 1 + E(t) dt G where E(t) is the complex envelope of the electric field, measured at a given reference point inside the cavity, and w(t) is the total population inversion of the

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nonlinear medium. The propagation round-trip time around the cavity is τR , and the fraction of light that remains in the cavity after one round trip is measured by the return coefficient R. The field acquires a phase κ after each round trip. Light of constant amplitude Einj is assumed to be injected into the cavity, with ω the detuning between the injected and the laser frequencies. Additionally, the gain medium is pumped at a rate Q. γ is the decay time of the atomic transition, α is a detuning parameter, and β and G are gain parameters. A limiting case of the model (2.40)–(2.41) is the well-known Ikeda map, introduced by Ikeda, Daido and Akimoto [1980] to provide an early example of optical chaos, and which has since become a canonical model of a chaotic system. Using this type of approach, Abarbanel and Kennel [1998] modeled the behavior of two coupled erbium-doped fiber ring lasers operating in a high-dimensional chaotic regime, showing that the chaotic dynamics of the two lasers becomes synchronized when a sufficient amount of light from the first laser is injected into the second. The process is represented schematically by Et (t + τR ) = f Et (t), wt (t) , (2.42) 2 dwt (t) = g Et (t) , wt (t) , (2.43) dt Er (t + τR ) = f cEt (t) + (1 − c)Er (t), wr (t) , (2.44) dwr (t) 2 (2.45) = g cEt (t) + (1 − c)Er (t) , wr (t) , dt where f and g represent the right-hand side terms of eqs. (2.40) and (2.41), respectively, and the subscripts “t” and “r” denote the corresponding variables of the transmitter and receiver lasers, respectively; c defines the fraction of power from the transmitter injected into the receiver; the same fraction is subtracted from the self-injected power in order to keep the total injected power constant. Under this condition, the equations for transmitter and receiver are the same when Et (t) = Er (t), indicating that identical synchronization is possible in this system (according to the classification given in Section 2.2.1). In the particular case c = 1 (open-loop configuration, no power of the receiver is reinjected into itself ), it can be shown in a straightforward way that the synchronized state is stable (Abarbanel and Kennel [1998]). Let us assume that the transmitted field is perturbed an amount ζ (t). From eq. (2.41), it is easily found that the difference between the population inversions of transmitter and receiver obeys the equation 2 [eG(wt −wr ) − 1] d[wt − wr ] . = −2γ wt − wr + Et (t) + ζ (t) eGwr dt G (2.46)

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Fig. 18. (a) Mean time to synchronization and (b) largest conditional Lyapunov exponent versus coupling strength c for the system (2.40)–(2.45), with Einj = 1, R = 0.9, κ = 0.4, β = 0, α = 6, ω = 0, G = 0.01, Q = 0 and γ = 1. (From Abarbanel and Kennel [1998].)

Since eA − 1 A for any real A, 2 d[wt − wr ] −2γ (wt − wr ) 1 + Et (t) + ζ (t) eGwr , dt

(2.47)

so that |wt (t) − wr (t)| tends to 0 faster than e−2γ t , and hence Et (t) tends to Er (t) as well. Synchronization is obtained also for c < 1. Figure 18 shows the mean time to synchronization (defined as the average time after which the difference between Et and Er decreases below a given small value) and the largest conditional Lyapunov exponent of the system, for increasing coupling strength c, as obtained by Abarbanel and Kennel [1998]. Both observables exhibit a transition from a desynchronized to a synchronized state as c increases. Other numerical investigations on synchronization of chaos in erbium-doped fiber-ring lasers have been reported by Luo, Tee and Chu [1998], Imai, Murakawa and Imoto [2003], Zhang and Chu [2004]. Experimental demonstrations of the synchronization between chaotic fiber lasers exist. The first experimental observation of this phenomenon was made by VanWiggeren and Roy [1998a], using the experimental setup shown in the left panel of fig. 19. The transmitter is an erbium-doped fiber ring laser, whose dynamical regime can be manipulated by means of an intracavity polarization controller. The laser is set to operate in a chaotic regime, from which 10% of the intracavity radiation is extracted via a 90/10 output coupler and transmitted to the receiver. 50% of that transmitted signal is injected into an erbium-doped fiber amplifier (EDFA) whose physical characteristics (fiber length, dopant concentration, and pump diode laser) are matched as close as possible to those of the transmitter EDFA. It should be noted that the receiver output is not reinjected back into itself,

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Fig. 19. Left: schematic experimental setup to observe synchronization of fiber-laser chaos. Right: experimental results showing the outputs of (A) transmitter and (C) receiver, (B) the phase-space reconstruction plot of the transmitted signal, and (D) the synchronization plot. (From VanWiggeren and Roy [1998a].)

i.e. the system operates in open loop. A second 90/10 coupler in the transmitter cavity allows the introduction of an external signal, whose role will be discussed in Section 3.6.4, in the context of chaotic communications. Plot A in the right panel of fig. 19 shows a sample time trace of the transmitted field in the absence of external signal. The phase-space reconstruction shown in plot B exhibits no lowdimensional structure of the dynamics displayed in plot A. The signal detected by photodiode B (see experimental setup in the left panel of fig. 19) after passing through the receiver’s EDFA is shown in plot C. This signal has been shifted in time an amount τ = 51 ns, equal to the time mismatch corresponding to the fiber length difference between the paths leading to photodiodes A and B. Visual inspection already indicates that the time series of the transmitter and receiver are very similar. The existence of synchronization is confirmed by plotting the output of the receiver (again shifted 51 ns) versus that of the transmitter; the resulting straight line with slope unity in this synchronization plot is a clear indicator of the occurrence of synchronization. The experimental setup described above is analogous to the open-loop version of the configuration modeled by Abarbanel and Kennel [1998], discussed earlier in this section. Similarly to that case, we can interpret the experimentally observed dynamics reported above to be identical synchronization (Section 2.2.1),

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since after compensating for fiber-length differences between the transmitter and the receiver, the latter can be expected to reproduce exactly the state of the former. Indeed, the plot shown in fig. 19(D) reveals a very good quality of synchronization. Other experimental realizations of chaos synchronization in coupled fiber lasers were reported on a time scale of hundreds of µs by Luo, Chu, Whitbread and Peng [2000], Kim, Lee and Kim [2001], Zhang, Chu, Lai and Chen [2005]. Further examples will be given in Section 3.6.4. 2.5.4. Semiconductor lasers Semiconductor lasers, also referred to as laser diodes, have several advantages over other types of lasers. Their size is very small (they are about 250 µm long for edge-emitting laser diodes and 1 µm long for vertical-cavity surface-emitting laser diodes, instead of tens of cm for gas lasers). They are easy to produce in large quantities and at relatively low prices. They are also very efficient, requiring low-power pumping. Owing to all these technological advantages, semiconductor lasers have become essential in fiber-based optical networks and optical data storage applications. On the other hand, semiconductor lasers are also very sensitive to external perturbations, due to the short photon lifetime in the cavity (a few picoseconds) compared with the carrier lifetime (a few nanoseconds) and, in the case of edgeemitting laser diodes, to the relatively low reflectivity of their facets (30% of intensity reflectance) or, in the case of vertical-cavity surface-emitting laser diodes, to their short round-trip time of light in the laser cavity. These perturbations, which are undesirable in most applications, lead to dynamical instabilities, such as chaos, that in turn can severely degrade the spectral and temporal performances of laser diodes. Much of the scientific literature about synchronization between semiconductor lasers deals with single-mode semiconductor lasers. These are generally characterized by three variables: the optical intensity and phase (or equivalently the real and imaginary parts of the complex electric field) and the electron–hole pair number (or carrier number) in the active layer of the lasers. The rate equations for the complex electric field E(t) and the carrier number N (t) in a solitary single-mode semiconductor laser are generally written as 1 + iα dE(t) GN N (t) − N0 − γp E(t), = dt 2 2 dN(t) = J − γs N (t) − GN N (t) − N0 E(t) , dt

(2.48) (2.49)

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where GN is the differential gain, N0 is the carrier number at transparency, α is the linewidth enhancement factor (α-parameter), γp is the photon loss rate, γs is the carrier loss rate, and J is the injection current density; i is the imaginary unit. E(t) is normalized in such a way that |E(t)|2 corresponds to the photon number (optical intensity) inside the laser cavity. In isolated laser diodes, the variable of optical phase is totally determined by the variable of carrier number. Since the occurrence of chaos is not possible in two-dimensional autonomous systems (i.e., two degrees of freedom), isolated laser diodes cannot exhibit chaotic motion as long as no other independent variable is added. By contrast, chaos can be observed when laser diodes are current-modulated (in which case they constitute two-dimensional nonautonomous systems), optically injected from another laser (in which case they constitute three-dimensional autonomous systems) or subjected to delayed feedback (in which case they become infinite-dimensional systems) (see van Tartwijk and Lenstra [1995], Ohtsubo [2002a, 2002b] and references therein). Coherent optical feedback may be caused by reflections of the laser output on a mirror in a laboratory experiment, on the reflective surface of a compact disk, or on a fiber facet in optical fiber transmission systems. When the laser diode is subject to optical feedback from a phase-conjugating mirror, the mirror inverts the phase of the field. Incoherent feedback can be achieved both optically and optoelectronically. Incoherent optical feedback occurs, for instance, when the polarized output field emitted by the laser is subjected to a π/2 polarization rotation before being reinjected into the laser cavity (Otsuka and Chern [1991], Chern, Otsuka and Ishiyama [1993]). To implement optoelectronic feedback, the output of the laser is detected by a high-speed photodetector, and the corresponding current is added to or deducted from the bias current of the laser diode (Tang and Liu [2001a]). We describe hereafter several synchronization schemes that implement some of the different ways to drive laser diodes into chaos, namely optical injection, coherent optical feedback, incoherent optical feedback and optoelectronic feedback. We present also the rate equations describing the schemes implementing coherent optical feedback and optoelectronic feedback, since those are the subject of much research (see also the special issues edited by Donati and Mirasso [2002], Larger and Goedgebuer [2004], Gavrielides, Lenstra, Simpson and Ohtsubo [2004] and the books by Ohtsubo [2005], Kane and Shore [2005]). Optical injection. Figure 20 shows the system that was investigated by Annovazzi-Lodi, Donati and Sciré [1996]. It is based on the synchronization method proposed by Pyragas [1993] (see Section 2.2.1). The transmitter and receiver laser diodes, which are ideally identical, are optically injected and driven to

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Fig. 20. Synchronization scheme for two optically-injected laser diodes. (From Annovazzi-Lodi, Donati and Sciré [1996].)

chaos by external isolated lasers. The output fields of the transmitter and receiver (E1 and E2 in the figure) are combined with a relative π phase shift and injected into the receiver semiconductor laser. This input acts as a control signal: it vanishes when the transmitter and receiver synchronize. By contrast, it contributes to resynchronizing the lasers when a small deviation between their respective dynamics occurs. It must be noticed that the coupling delay in the receiver feedback loop must be small with respect to the inverse of the chaotic waveform bandwidth and that the relative phase difference between the input fields E1 and E2 must not differ sensibly from π to allow synchronization (Annovazzi-Lodi, Donati and Sciré [1996]). The scheme that was numerically studied by Chen and Liu [2000] is depicted in fig. 21. In this scheme, an optical field is split into two parts. The first part drives the transmitter laser to chaos. The second part is amplified or attenuated and added to the output of the transmitter laser. The sum is injected into the receiver laser that synchronizes to the transmitter provided that both lasers’ parameters are identical, except for the photon decay rates. Indeed, the photon decay rate of the receiver laser must be adjusted in order to compensate for a photon excess since more photons are injected in the receiver laser than in the transmitter. This adjustment can be done by changing the coating on the laser facets. This scheme was experimentally investigated by Chen and Liu [2004]. They identified three synchronization regimes under different operating conditions (Chen and Liu [2004]). Coherent optical feedback. In the system depicted in fig. 22, the transmitter is driven into chaos by delayed, coherent optical feedback from a mirror. The re-

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Fig. 21. Open-loop synchronization scheme for two optically-injected laser diodes. (From Chen and Liu [2000].)

Fig. 22. Synchronization setup for two laser diodes subject to optical feedback.

ceiver, which is coupled to the transmitter through coherent optical injection, may or may not have an external mirror for optical feedback, which corresponds to a closed- or open-loop configuration, respectively (see Section 2.3). Synchronization is the most feasible in the open-loop configuration, otherwise the length of the external cavity of the receiver must closely match that of the transmitter in the closed-loop configuration (Vicente, Pérez and Mirasso [2002]). The equations that are generally used to describe the dynamics of this system are an extension of the equations that were proposed by Lang and Kobayashi [1980] for laser diodes subject to coherent optical feedback: dET (t) 1 + iαT = GN,T NT (t) − N0,T − γp,T ET (t) dt 2 + κT ET (t − τT ) exp(−iω0,T τT ), 2 dNT (t) = JT − γs,T NT (t) − GN,T NT (t) − N0,T ET (t) , dt 1 + iαR dER (t) = GN,R NR (t) − N0,R − γp,R ER (t) dt 2 + κR ER (t − τR ) exp(−iω0,R τR )

(2.50) (2.51)

+ σ ET (t − τc ) exp(−iω0,T τc + i∆t), (2.52) 2 dNR (t) (2.53) = JR − γs,R NR (t) − GN,R NR (t) − N0,R ER (t) , dt where the subscripts “T” and “R” stand for transmitter and receiver, respectively, ET,R (t) and NT,R (t) are the corresponding complex electric fields and carrier

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densities, and κT,R are the feedback rates of the transmitter and the receiver, where κR is zero for an open-loop configuration; σ is the coupling rate between the transmitter and the receiver, τT,R are the external-cavity round-trip times of the transmitter and the receiver, τc is the coupling delay time from the transmitter to the receiver, ω0,T and ω0,R are the angular optical frequencies of the isolated lasers at threshold, and ∆ is the angular optical frequency detuning between the transmitter and the receiver; i is the imaginary unit. The decay rates of photons and carriers are given by γp and γs , respectively. Both identical synchronization and the restricted type of generalized synchronization described in Sections 2.2 and 2.3 have been reported in this configuration. One of the first numerical predictions of chaos synchronization in this scheme corresponds to the restricted type of generalized synchronization (Mirasso, Colet and García-Fernández [1996]). The receiver output power can mimic the power of the transmitter with an amplification ratio A after a delay equal to the coupling delay time, ER (t)2 A · ET (t − τc )2 .

(2.54)

Transmitter and receiver output powers can match provided that the receiver photon loss rate is adjusted in order to compensate the injection from the transmitter. Figure 23 shows the temporal waveforms of the transmitter and the receiver and their correlation plots under the condition of identical synchronization and the restricted type of generalized synchronization. The quality of the restricted type of generalized synchronization is not as good as identical synchronization as shown in the correlation plot of the two lasers in figs. 23(c) and 23(e). Figure 24 shows the two-dimensional map of the synchronization error as functions of the optical frequency detuning and the coupling strength between the two lasers. The restricted type of generalized synchronization (shown as amplification area in fig. 24) can be obtained in wide ranges of coupling rates (typically at large coupling strengths) and frequency detunings, and for rather large mismatches between transmitter and receiver parameters. This type of synchronization is typically observed in parameter ranges where injection locking (locking of optical carrier frequency) is achieved, as shown in fig. 24. Due to the lack of mathematical understanding of this type of synchronization, a variety of different terms have been used in the literature: isochronous synchronization (Locquet, Masoller and Mirasso [2002], Vicente, Pérez and Mirasso [2002]), generalized synchronization (Buldú, García-Ojalvo and Torrent [2004], Lee, Paul, Pierce and Shore [2005]), time lag synchronization (Koryukin and Mandel [2002]), synchronization by nonlinear amplification (Murakami and Ohtsubo [2002]) and so on.

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Fig. 23. Examples of synchronization of chaotic waveforms. (a) Temporal chaotic waveform of the transmitter laser. (b) Temporal chaotic waveform of the receiver laser under identical synchronization. (c) Correlation plot between (a) and (b) for identical synchronization. (d) Temporal chaotic waveform of the receiver laser under the restricted type of generalized synchronization. (e) Correlation plot between (a) and (c) for the restricted type of generalized synchronization; τc = 0, τT = 1 ns. (From Murakami and Ohtsubo [2002].)

In experiments on chaos synchronization in semiconductor lasers with optical feedback, Sivaprakasam and Shore [1999a] reported the restricted type of generalized synchronization on millisecond time scales in a closed-loop configuration. Takiguchi, Fujino and Ohtsubo [1999] observed this type of synchronization in the low-frequency fluctuation regime. Fujino and Ohtsubo [2000] and Fischer, Liu and Davis [2000] demonstrated synchronization on nanosecond time scales in closed-loop and open-loop configurations, respectively. Figure 25 shows synchronized output intensity time series of both transmitter and receiver and the corresponding correlation plot as experimentally recorded by Fischer, Liu and Davis [2000] in an open-loop configuration. For the closed-loop configuration, the relative optical feedback phase strongly determines the quality of synchronization (Peil, Heil, Fischer and Elsäßer [2002]). The restricted type of generalized synchronization in multimode semiconductor lasers with optical feedback has also

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Fig. 24. Calculated synchronization error for variation in the frequency detuning and the coupling rate. Quality of synchronization is represented by using gray scale with synchronization error. ‘Complete synchronization area’ corresponds to identical synchronization area and ‘amplification area’ corresponds to the restricted type of generalized synchronization area described in Sections 2.2 and 2.3. The boundary denoted by the solid line represents the injection-locking area for constant-intensity injection into the receiver laser. (From Murakami and Ohtsubo [2002].)

Fig. 25. Left: synchronized output intensity time series of transmitter (upper trace) and receiver lasers (lower trace). Right: correlation plot of transmitter and receiver intensity. (From Fischer, Liu and Davis [2000].)

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been reported experimentally (Uchida, Liu, Fischer, Davis and Aida [2001]) and numerically (Koryukin and Mandel [2003], Buldú, García-Ojalvo and Torrent [2004]). As theoretically predicted by Ahlers, Parlitz and Lauterborn [1998], identical synchronization can also be achieved in the system depicted in fig. 22. Identical synchronization can take place when the sum of the feedback rate at the receiver and the coupling rate matches the feedback rate at the transmitter for the closedloop configuration, namely κT = κR + σ.

(2.55)

In this case, synchronization can be achieved as a mathematical solution of eqs. (2.50)–(2.53), i.e., ER (t)2 = ET (t − τc + τT )2 . (2.56) The time lag between the transmitter and receiver waveforms is τc − τT for identical synchronization (see Section 2.3). No synchronization error is found for identical synchronization under the parameter matching condition without noise, as shown in fig. 23(c). Liu, Takiguchi, Davis, Saito and Liu [2002] experimentally observed identical synchronization (also referred to as complete synchronization in the literature) in semiconductor lasers with optical feedback. They investigated that the time lag between the transmitter and receiver waveforms depends on τT to distinguish identical synchronization (eq. (2.56)) from the restricted type of generalized synchronization (eq. (2.54)), as shown in fig. 26. Identical synchronization requires precise matching of all the laser parameters between the transmitter and

Fig. 26. Experimentally measured time lag between the two laser outputs as a function of feedback delay time of the transmitter τT for identical synchronization. (From Liu, Takiguchi, Davis, Saito and Liu [2002].)

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the receiver. Identical synchronization is strongly sensitive to noise and frequency detuning between the lasers (see the complete synchronization area in fig. 24). In the special case of τT > τc , the receiver anticipates the injected signal coming from the transmitter with the anticipation time τT − τc as shown in eq. (2.56) (see also Section 2.3). Anticipative synchronization has been predicted numerically (Masoller [2001]) and observed experimentally in one-way coupled (Liu, Takiguchi, Davis, Saito and Liu [2002]) and mutually coupled (Sivaprakasam, Shahverdiev, Spencer and Shore [2001]) semiconductor lasers with optical feedback. The characteristics of the above-mentioned two types of synchronization have been intensively investigated by different groups (Liu, Chen, Liu, Davis and Aida [2000], Liu, Davis, Takiguchi, Aida, Saito and Liu [2003], Locquet, Rogister, Sciamanna, Mégret and Blondel [2001], Locquet, Masoller and Mirasso [2002], Locquet, Masoller, Mégret and Blondel [2002], Vicente, Pérez and Mirasso [2002], Koryukin and Mandel [2002], Murakami and Ohtsubo [2002], Uchida, Shibasaki, Nogawa and Yoshimori [2004], Buldú, García-Ojalvo and Torrent [2004], Lee, Paul, Pierce and Shore [2005]). Analytical studies on synchronization characteristics have also been reported by Revuelta, Mirasso, Colet and Pesquera [2002], Murakami [2002], Kouomou and Woafo [2003]. For other examples, anticorrelated synchronization was observed where the intensity of the response laser jumps up when the drive intensity drops in a low-frequency fluctuation regime. The correlation plot between the two lasers has a negative slope. This synchronization is referred to as inverse synchronization (Sivaprakasam, Pierce, Rees, Spencer and Shore [2001]) and antisynchronization (Wedekind and Parlitz [2002]). It has also been theoretically shown by Annovazzi-Lodi, Donati and Sciré [1997] and Murakami and Ohtsubo [2001] that synchronization of laser diodes subject to optical feedback can be achieved by implementing the continuous chaos control method proposed by Pyragas [1998] for systems with delay. Cascaded synchronization was reported for a system of three sequentially coupled semiconductor lasers (Sivaprakasam and Shore [2001]). Another approach was reported to analyze synchronization in asymmetric coupling configurations. Matus, Moloney and Kolesik [2003] proposed a full nonlinear partial differential equation model instead of the Lang–Kobayashi model, and calculated detailed characteristics of synchronization at different coupling configurations. They discussed the inherent limitations of the Lang–Kobayashi model related to the use of asymmetric devices and/or configurations. Synchronization in mutually coupled semiconductor lasers has also been investigated intensively. It was in that situation where the first numerical observa-

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Fig. 27. Intensity time series of two mutually coupled lasers. The lower trace shows the inverted time series. The coupling delay time indicated by τ (τc in eq. (2.52)) is 4.75 ns. (From Heil, Fischer, Elsäßer, Mulet and Mirasso [2001].)

tion of chaos synchronization in laser systems was reported by Winful and Rahman [1990]. Synchronization of periodic oscillations was observed in mutually coupled semiconductor lasers (Hohl, Gavrielides, Erneux and Kovanis [1997]). Heil, Fischer, Elsäßer, Mulet and Mirasso [2001] experimentally observed the restricted type of generalized synchronization with chaotic oscillations in mutually coupled semiconductor lasers in a low-frequency fluctuation regime. Mutual coupling induced chaotic oscillations on a nanosecond time scale. They observed that a symmetry breaking appears as a time lag corresponding to the coupling delay time τc between the two lasers as shown in fig. 27. This leader–laggard relationship in mutually coupled semiconductor lasers was also observed experimentally by Sivaprakasam, Spencer, Rees and Shore [2002], Sivaprakasam, Shahverdiev, Spencer and Shore [2001]. They investigated the role of detuning to determine which laser leads the other laser (Heil, Fischer, Elsäßer, Mulet and Mirasso [2001], Rees, Spencer, Pierce, Sivaprakasam and Shore [2003]). Note that they used the terms ‘anticipating’ and ‘lag’ synchronization to describe the leader–laggard relationship in mutually coupled lasers. These experimental observations were analytically interpreted by White, Matus and Moloney [2002]. In further studies, synchronization characteristics in mutually coupled semiconductor lasers with a short coupling delay time were investigated numerically by Rogister and García-Ojalvo [2003], Rogister and Blondel [2004], Yanchuk, Schneider and Recke [2004] and experimentally by Wünsche, Bauer, Kreissl, Ushakov, Korneyev, Henneberger, Wille, Erzgräber, Peil, Elsäßer and Fischer [2005]. Dynamics of synchronization in globally coupled semiconductor laser

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arrays was reported by García-Ojalvo, Casademont, Torrent, Mirasso and Sancho [1999], Kozyreff, Vladimirov and Mandel [2000, 2001]. Incoherent optical feedback. We now consider the situation in which the feedback and injected fields act on the carrier population in the diode active layers, but do not interact with the intracavity lasing fields. As a consequence, the phases of the feedback and injection fields do not intervene in the laser dynamics. For that reason, these schemes require no fine-tuning of the diode optical frequencies, a feature that can be considered an advantage for experimental realization, compared with other schemes based on coherent optical feedback and injection. A simple scheme for generating incoherent optical feedback proposed by Rogister, Locquet, Pieroux, Sciamanna, Deparis, Mégret and Blondel [2001], Rogister, Pieroux, Sciamanna, Mégret and Blondel [2002] is shown in fig. 28. In this setup, the linearly polarized output field of the transmitter laser first undergoes a π/2 polarization rotation through an external cavity formed by a Faraday rotator (FR) and a mirror. It is then split by a nonpolarizing beam splitter (BS) into two parts: one is fed back into the transmitter laser and the other is injected into the receiver laser. The polarization directions of feedback and injection fields are orthogonal to those of transmitter and receiver output fields, respectively. Therefore, the transmitter laser is subjected to incoherent optical feedback while the receiver laser is subjected to incoherent optical injection. A linear polarizer (LP) may be placed between the Faraday rotator and the mirror to prevent coherent feedback induced by a second round-trip in the external cavity after reflection on the transmitter laser front facet. The receiver is necessarily open-loop, because a feedback at the receiver would induce a beating with the coupling field.

Fig. 28. Synchronization setup with laser diodes subject to incoherent optical feedback and incoherent optical injection. (From Rogister, Locquet, Pieroux, Sciamanna, Deparis, Mégret and Blondel [2001].)

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Numerical simulations on an extension of a model investigated by Otsuka and Chern [1991] for semiconductor lasers subject to incoherent optical feedback predict identical synchronization between the two lasers (Rogister, Locquet, Pieroux, Sciamanna, Deparis, Mégret and Blondel [2001]). In an experiment with a similar setup, Sukow, Blackburn, Spain, Babcock, Bennett and Gavrielides [2004] have shown that the restricted type of generalized synchronization occurs with a time shift close to the coupling delay time between the transmitter and receiver outputs (see Section 2.2.2). A more complex dynamical model proposed by Heil, Uchida, Davis and Aida [2003] that takes into account both modes of polarization may be useful to describe the synchronization characteristics in coupled semiconductor lasers with polarization-rotated optical feedback. Optoelectronic feedback. Another way of obtaining incoherent feedback is via the injection current of the laser. An example is the experimental setup displayed in fig. 29, which was investigated by Tang and Liu [2001a, 2003b], Tang, Chen and Liu [2001], Abarbanel, Kennel, Illing, Tang, Chen and Liu [2001], Liu, Chen and Tang [2002]. The optical power emitted by the single-mode distributedfeedback (DFB) transmitter laser is detected by a photodiode and converted into an electric current, which is in turn reinjected into the laser. This optoelectronic feedback drives the laser to chaos, provided that the feedback delay is carefully adjusted. The receiver DFB laser is subject to its own feedback and coupled to the transmitter. This system can be modeled by the following simplified equations: dPT (t) = GN,T NT (t) − N0,T − γp,T PT (t), dt dNT (t) = JT + κT PT (t − τT ) − γs,T NT (t) dt − GN,T NT (t) − N0,T PT (t), dPR (t) = GN,R NR (t) − N0,R − γp,R PR (t), dt dNR (t) = JR + κR PR (t − τR ) + σ PT (t − τc ) − γs,R NR (t) dt − GN,R NR (t) − N0,R PR (t),

(2.57)

(2.58) (2.59)

(2.60)

where PT,R (t) = |ET,R (t − τc )|2 are the photon numbers in the transmitter and receiver lasers, respectively, and κT,R and σ are the feedback and coupling rates. Other parameters have been defined in eqs. (2.50)–(2.53). Identical synchronization can be achieved provided that the strength of the signal driving the receiver (composed of transmitted signal and self-feedback) is close to the feedback strength of the transmitter. Figure 30 shows time series of

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Fig. 29. Experimental setup for observing synchronization between two laser diodes subject to optoelectronic feedback. (From Tang and Liu [2001a].)

Fig. 30. Time series showing synchronization. (From Tang and Liu [2001a].)

the transmitter and receiver when the transmitter signal constitutes 80% of the driving signal. The two traces are almost identical. The synchronization quality was demonstrated to increase with the fraction of the receiver driving signal coming from the transmitter, and to be optimal when the receiver operates in the open-loop configuration (Tang and Liu [2001a]). Anticipated and retarded synchronization were also observed by changing the delay time of the feedback loop in the transmitter (Tang and Liu [2003a]). Synchronization of mutually coupled semiconductor lasers with optoelectronic feedback was reported by Tang, Vicente, Chiang, Mirasso and Liu [2004]. 2.5.5. Vertical-cavity surface-emitting lasers (VCSEL) Vertical-cavity surface-emitting semiconductor lasers (VCSEL) have become essential devices for optical communications and wireless local area networks. Because of their short cavity length (a few µm) VCSELs emit in a single-longitudinal mode. VCSELs have some advantages compared with edge-emitting conventional semiconductor lasers: lower threshold currents, more compactness, higher efficiency, larger modulation bandwidth; additionally they offer wafer-scale integrability for large array configurations. Typical VCSELs have two polarization modes and many transverse modes, what makes their dynamics very complicated (San Miguel, Feng and Moloney [1995]). However, the basic dynamics of

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VCSELs with optical feedback is very similar to that of edge-emitting semiconductor lasers, including for instance low-frequency fluctuations and coherence collapse (Masoller and Abraham [1999]). VCSELs are very sensitive to optical injection or optical feedback due to their short cavity length, in spite of the high reflectivity (>99%) of their facets. Synchronization of chaos in one-way coupled VCSELs has been reported numerically by Spencer, Mirasso, Colet and Shore [1998]. They used a travelingwave model in a strong optical-feedback regime, which induced chaotic oscillations in VCSELs. Synchronization of chaos was observed over a large range of parameters. Synchronization in multitransverse-mode VCSELs has also been numerically reported by Torre, Masoller and Shore [2004]. Experimental observation of chaos synchronization in VCSELs has been reported by Fujiwara, Takiguchi and Ohtsubo [2003]. Two stand-alone VCSELs were mutually coupled and chaotic dynamics were obtained in a low-frequency fluctuation regime. Two polarization modes were adjusted between the two VCSELs. Synchronization of chaos for x-polarization components was obtained as shown in fig. 31(a) when the wavelengths of the x-polarization modes were locked to each other by injection locking. However, synchronization was also observed between the y-polarization modes of the two VCSELs without locking the optical frequency, as shown in fig. 31(b). Since the two polarization modes are anticorrelated (anti-phase dynamics), synchronization of both modes can be achieved by locking either of the two polarization modes between two VCSELs. In a further development, Hong, Lee, Spencer and Shore [2004] have reported synchronization of chaos in one-way coupled VCSELs in a fully developed fast chaotic state (a few GHz). In that work, one of the two polarization modes (x-mode) was selected in the transmitter and the polarization direction was orthogonally rotated to couple into the y-polarization mode of the receiver. Synchronization was observed between the injected beam and the y-polarization of the receiver. This is the restricted type of generalized synchronization through injection locking, because the output of the receiver lags only by the coupling delay time between the two VCSELs (see Section 2.3). The injection power and the x-polarization component of the receiver also showed reasonable synchronization, although the gradient of the correlation plots was negative, which implies inverse synchronization (Sivaprakasam, Pierce, Rees, Spencer and Shore [2001], Wedekind and Parlitz [2002]). The occurrence of inverse synchronization in VCSELs can be explained by the antiphase dynamics between the two polarization modes in the receiver.

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Fig. 31. Time series of VCSEL outputs at synchronization: (a) x-polarization mode, (b) y-polarization mode. (From Fujiwara, Takiguchi and Ohtsubo [2003].)

2.5.6. Optoelectronic systems We have examined so far laser devices in which chaos is induced by the interplay between the lasing field and the population inversion (or carrier density) of the laser. Another method of inducing chaotic behavior by feedback exists. Specifically, chaotic behavior can be induced when the light emitted by a laser goes through an optoelectronic feedback loop containing a nonlinear optical device. A distinctive feature of these systems is the retardation time of the driving signal in the feedback loop, which is much longer than the time response of the systems. The chaotic dynamics is determined by the nonlinear optical device in the loop, not by the laser itself. The laser is thus treated as a linear component in this subsection. The dynamics of this type of setups can be described by a particular class of delay-differential equations where the feedback is modeled by a nonlinear function, whose transfer characteristic has at least a maximum or a minimum in the variable range. This class of systems was shown by Hopf, Kaplan, Gibbs and Shoemaker [1982] and Ikeda, Kondo and Akimoto [1982] to exhibit oscillating

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states with higher harmonics appearing successively in the course of transition toward developed chaos. For example, the model for two unidirectionally coupled optoelectronic systems with birefringent plates in an open-loop configuration that is presented in the following is (Larger, Goedgebuer and Delorme [1998], Larger, Goedgebuer and Merolla [1998]) dIt (t) = βt sin2 It (t − Tt ) − Φt , dt dIr (t) Ir (t) + τr = βr sin2 It (t − Tc ) − Φr , dt

It (t) + τt

(2.61) (2.62)

where the subscripts “t” and “r” indicate the transmitter and the receiver, respectively. I (t) is the dynamical variable of the system (i.e., laser intensity (Goedgebuer, Levy, Larger, Chen and Rhodes [2002]) or wavelength (Larger, Goedgebuer and Delorme [1998]), Tt is the delay time in the optoelectronic feedback loop of the transmitter, Tc is the coupling delay time between the two systems, τ is the response time of the system, β is the height of the nonlinear function in the optoelectronic feedback loop, and Φ is the initial phase of the nonlinear function. There is a mathematical synchronous solution of It (t) = Ir (t) in eqs. (2.61)–(2.62) under the parameter matching condition because of the symmetry of these two equations. Identical synchronization can be achieved with an adequate coupling strength between the transmitter and the receiver. Celka proposed two schemes for chaos synchronization where the nonlinear devices within the feedback loops were Mach–Zehnder (MZ) interferometers (Celka [1995, 1996]). Figure 32 depicts the open-loop scheme investigated by

Fig. 32. Experimental setup to observe chaos synchronization, implemented using Mach–Zehnder interferometers. (From Celka [1996].)

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Fig. 33. Experimental cryptosystem implemented with Mach–Zehnder modulators. (From Goedgebuer, Levy, Larger, Chen and Rhodes [2002].)

Celka [1996]. At the transmitter and receiver, the light emitted by continuouswave (CW) semiconductor lasers is modulated by integrated MZ interferometers. The output light of the transmitter interferometer is divided into two parts. One part is retarded in a fiber line and then detected by a photodiode. The photodiode voltage, which is proportional to the detected power, is applied on the RF electrode of the MZ interferometer and in turn modulates the light emitted by the laser. Synchronization is achieved when the second part of the transmitter interferometer output is injected with adequate strength in the open feedback loop of the receiver. A different setup implementing integrated MZ modulators was developed by Goedgebuer, Levy, Larger, Chen and Rhodes [2002] (see also Lee, Larger and Goedgebuer [2003], Kouomou, Colet, Larger and Gastaud [2005]). In this system, shown in fig. 33, the transmitter contains a laser diode operating in the linear part of its power–current curve and driven by a feedback signal. The electric current that results from the detection of the laser output is amplified and injected in an MZ modulator (electro-optic modulator (EOM) in the figure) that in turn nonlinearly modulates a CW optical source. The modulator output then feeds the laser. The receiver is similar to the transmitter, except that the feedback loop is open. The effect of parameter mismatch on synchronization quality was also studied in this system by Kouomou, Colet, Gastaud and Larger [2004].

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Fig. 34. Schematic representation of an optical cryptosystem based on synchronization between two optoelectronic wavelength oscillators. (From Goedgebuer, Larger and Porte [1998].)

In a different approach, Goedgebuer, Larger and Porte [1998] demonstrated an optical cryptosystem based on synchronization between two optoelectronic wavelength oscillators (see also Larger, Goedgebuer and Delorme [1998], Cuenot, Larger, Goedgebuer and Rhodes [2001]). In their system (see fig. 34), the transmitter and receiver are composed of a wavelength-tunable distributed-Braggreflector (DBR) semiconductor laser and a feedback loop containing an optical component that exhibits a nonlinearity in wavelength. The latter is implemented by a birefringent plate placed between two crossed linear polarizers. The variation range of the wavelength is small enough to ensure that the DBR lasers always oscillate in a single longitudinal mode. At the transmitter, the laser output passes through the nonlinear optical component. Its variable output power is converted into an electric current by a photodiode, and the current in turn is injected into the tunable-wavelength laser diode. As a consequence, the delayed feedback induces chaotic fluctuations in the laser wavelength for adequately chosen control parameters. The receiver is a replica of the transmitter, except that its feedback loop is open and is directly fed by the transmitter laser light. Liu and Davis [2000a], Liu, Davis and Aida [2001] studied a system using wavelength-tunable DBR lasers as well, but with a tuning range over many longitudinal modes. Additionally, they used optical band-pass filters covering several laser modes as nonlinear wavelength functions. The wavelength of each laser is a piecewise-continuous function of the current: a large change in current can induce mode hopping, even though the optical power remains nearly constant. Figure 35

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Fig. 35. Experimental setup for observing synchronized chaotic mode hopping in DBR lasers with optoelectric feedback. (From Liu and Davis [2000a].)

shows the corresponding synchronization scheme. At the transmitter, the light emitted by the laser goes through an optical tunable filter after retardation in a fiber line. The output power of the filter, which depends nonlinearly on the laser wavelength, is detected by a photodiode whose electric current is amplified and fed back into the DBR section of the laser diode. This feedback induces on–off intensity modulation of the individual modes with a chaotic pattern, the lasing intensity being concentrated in just one mode at any one time. Coupling between the transmitter and receiver is achieved by injecting the transmitter filter output into the receiver feedback loop, which is closed. The resulting driving signal (i.e. the transmitted signal plus the receiver’s own feedback) induces chaotic mode hopping in the receiver laser. Synchronization of chaotic mode hopping is possible when the coupling is strong enough. Figure 36 shows examples of synchronized chaotic mode hopping. High-quality synchronization during tens of hours was reported for optimal parameter matching.

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Fig. 36. (a) Synchronized waveforms of system total output and mode 1. (b) Synchronized waveforms of system total output and mode 4. (From Liu and Davis [2000a].)

2.6. Phase and generalized synchronization 2.6.1. Phase synchronization So far in this review, synchronization has been considered as a regime in which the intensities of two lasers have a well-defined relation at all times. But other forms of synchronizations exist. Consider, for instance, that the chaotic oscillations can be decomposed in terms of an amplitude and a phase. Under certain conditions, it can happen that the amplitudes of the chaotic oscillations of the two lasers are desynchronized, while a clear synchronization exists between the phases. This is called phase synchronization (Rosenblum, Pikovsky and Kurths [1996]). We note that ‘phase’ in this context indicates the phase of chaotic oscillations of laser intensity, not optical phase of the electrical field. We consider a fairly general means by which a phase may be associated with a real scalar signal such as the laser intensity I (t), following the work of DeShazer, Breban, Ott and Roy [2001], DeShazer, Breban, Ott and Roy [2004]. One can represent I (t) in terms of its Fourier transform Iˆ(ν): −1

I (t) = (2π)

∞ −∞

exp(iνt)Iˆ(ν) dν.

The variation of each Fourier component, Iˆ(ν) exp(iνt), is a complex number whose phase continually increases (decreases) with time for ν > 0 (ν < 0). Thus, one way to introduce a phase is to suppress the negative ν components by replacing Iˆ(ν) by 2θ (ν)Iˆ(ν) (where θ (ν) is the unit step function, θ (ν) = 1 for ν > 0 and θ (ν) = 0 for ν < 0). In this case, we obtain a superposition of rotating

264


complex numbers all of which have increasing phase, ∞ eiνt Iˆ(ν) dν. VA (t) = π−1

[5, § 2

(2.63)

0

Thus we may reasonably expect VA (t) to execute a rotation in the complex plane with continually increasing phase. The function VA (t) is Gabor’s analytic signal (Gabor [1946], Born and Wolf [1999]), which has recently been introduced for the purpose of the study of phase synchronization of chaos (Rosenblum, Pikovsky and Kurths [1996]). Noting that the inverse transform of 2θ (ν) is δ(t) + πi P 1t , we can express the analytic signal as VA (t) = I (t) + iIH (t) = I (t) +

1 i I (t) P , π t

(2.64)

where I (t) are related by the Hilbert transform: IH (t) = π−1 × ∞ IH(t) and P −∞ dt I (t )/(t − t ); denotes convolution; and P 1t is the principal part of 1/t. Writing VA (t) − VA = RA (t)eiΦA (t) ,

(2.65)

where RA (t) and ΦA (t) are real, and VA is the time average of VA (t), we call ΦA (t) the analytic phase (DeShazer, Breban, Ott and Roy [2001]). One can test phase synchronization by extracting the analytic phase from chaotic time series of laser intensity. Phase synchronization in lasers has been observed both between a chaotic oscillator and an external periodic signal, and between two or more chaotic oscillators coupled to each other. In the former case, phase synchronization can be interpreted as the phase locking of a single chaotic system with respect to an external periodic forcing (Allaria, Arecchi, Di Garbo and Meucci [2001]). The experiment was performed on a single-mode CO2 laser with an optoelectronic feedback proportional to the output intensity. A photon detector converts the laser output intensity into a voltage signal, which is fed back through an amplifier to an intracavity electro-optic modulator, in order to control the amount of cavity losses. As for the control parameter to be modulated, they chose either the bias voltage of the feedback amplifier or the pump of the gain medium. Different phase-locking regimes are shown in the left panel of fig. 37, together with the applied sinusoidal forcing at different modulation frequencies. As the external frequency is changed, various (p:q) phase locking states are observed. To provide a better understanding of phase synchronization, they explored the possible occurrence of phase slips. The right panel of fig. 37 shows the phase difference between the laser output intensity and the external modulation for different modulation frequencies within the synchronization domain corresponding to 1:1 locking. Departing from the perfect

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Fig. 37. Left: experimental time series for different phase synchronizations induced by a frequency-modulated control parameter. For comparison, time traces of the modulated control parameter are displayed in each case. The modulation frequencies are, respectively, (a) 1:1 at 1.6 kHz; (b) 1:2 at 0.7 kHz; (c) 1:3 at 0.5 kHz; (d) 2:1 at 2.6 kHz. Right: phase slips at different frequencies for the 20 mV amplitude. The dynamical system monotonically lags or leads in phase depending on whether the modulation frequency is above or below the perfect synchronization value (for which there are no slips) of 1.6 kHz. (From Allaria, Arecchi, Di Garbo and Meucci [2001].)

phase synchronization (zero phase slip) for the optimal modulation frequency, the slip rate increases as the edges of the 1:1 locking domain are approached. The same experimental laser system is used to observe the transition to phase synchronization (Boccaletti, Allaria, Meucci and Arecchi [2002]) and the enhancement of phase synchronization by noise (Zhou, Kurths, Allaria, Boccaletti, Meucci and Arecchi [2003]). Phase synchronization between coupled chaotic lasers has been examined experimentally for the case of a linear laser array by DeShazer, Breban, Ott and Roy [2001]. The chaotic system consists of three parallel, laterally coupled singlemode Nd:YAG laser arrays. Coupling through the electric fields of the individual beams exists only for adjacent pairs. Experimental intensity measurements are displayed in figs. 38(a)–38(c). The two outer lasers in the array (lasers 1 and 3) have nearly identical intensity fluctuations. However, no synchronization relationship is obvious between the center laser (laser 2) and the outer lasers (lasers

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Fig. 38. (a–c) Experimental intensity time series showing chaotic bursts of three Nd:YAG lasers evanescently coupled in a linear array. (d, e) Time series for the differences between the (d) analytic and (e) Gaussian filtered phases of the center and an outer laser. The solid line in (e) corresponding to a Gaussian filter is centered at 140 kHz, for the dotted line it is centered at 80 kHz; the dashed line is the phase difference with respect to a surrogate time series. (From DeShazer, Breban, Ott and Roy [2001].)

1 and 3), even though the center laser mediates the identical synchronization of the outer lasers. To test for interdependence between the time series of the outer and center lasers, an analytic phase and a Gaussian filtered phase are introduced (see details in the article by DeShazer, Breban, Ott and Roy [2001]). Phase synchronization between the outer and the center laser (lasers 1 and 2) is shown by plotting their relative phase versus time (figs. 38(d), 38(e)). Figure 38(d) shows the difference of the analytic signal phases for these lasers, which has a large range

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of variation (∼130 rotations). Phase synchronization is not discernible. Next, in fig. 38(e), the difference of the Gaussian filtered phase for these two lasers is plotted at different frequencies of the Gaussian filter. Synchronization of the side and central lasers in the frequency regime of 140 kHz is immediately apparent, since the flat portion of this plot extends across essentially the entire time of observation (solid line). Periods of phase synchronization and phase slipping are found in the less correlated frequency regime of 80 kHz (dotted line). No indication of synchronization is found when one of the component phases is replaced with a surrogate phase extracted from another experimental data set taken from this array under identical conditions (dashed line). These results illustrate that the detection of phase synchronization may require careful consideration of the nature of the time series measured. The time series considered in this experiment are of a distinctly nonstationary nature, and it is clearly advantageous to introduce a Gaussian filtered phase variable. One is then able to quantitatively assess phase synchronization for different frequency components of the dynamics. In a related development, McAllister, Meucci, DeShazer and Roy [2003] studied the competition between two distinct driving frequencies for phase synchronization of a chaotic Nd:YAG laser. Phase synchronization has also been observed experimentally in mutually injected Nd:YAG lasers by Volodchenko, Ivanov, Gong, Choi, Park and Kim [2001]. In that experiment, the laser output of each Nd:YAG laser is injected into the other laser cavity, and the intensity of the injection beam (coupling strength) is controlled with a linear polarizer (the two lasers emit linearly polarized light via Brewster windows). The left panel of fig. 39 shows the temporal behaviors of the two laser outputs for different coupling strengths. When the lasers are uncoupled, it is evident that the temporal behavior of one laser is different from that of the other (plot (a) in the left panel of fig. 39). However, when the two lasers are coupled, their temporal behaviors become more similar as the coupling strength increases (plots (b) and (c) in the left panel of fig. 39). When the lasers are fully coupled, the phases of the two chaotic laser outputs are locked with each other, as shown in plot (d) in the left panel of fig. 39. The right panel of fig. 39 shows the phase difference of the two laser outputs when the coupling strength is increased. As shown in plot (a) in the right panel of fig. 39, the phase difference of the two lasers increases and decreases irregularly for no coupling. As the coupling strength increases, the phase difference is locked at ±2nπ for a rather long time and jumps intermittently, as shown in plot (b) in the right panel of fig. 39. For strong coupling, the phase difference is locked within ±2π without jumps (plot (c), right panel of fig. 39), which implies phase synchronization. The transition from phase synchronization to identical synchronization has also been

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[5, § 2

Fig. 39. Left: temporal behavior of the two laser outputs depending on the coupling strength: the angles of the polarizer are (a) 90◦ , (b) 45◦ , (c) 15◦ , and (d) 0◦ . Right: temporal behavior of the phase difference: (a) nonsynchronous state when the angle of the polarizer is 90◦ (no coupling), (b) phase-jump state when the angle is 45◦ , and (c) phase synchronization when the angle is 15◦ . (From Volodchenko, Ivanov, Gong, Choi, Park and Kim [2001].)

observed as the coupling strength increases (Choi, Volodchenko, Rim, Kye, Kim, Park and Kim [2003]). Phase synchronization has also been observed experimentally for the lowfrequency fluctuations (LFFs) exhibited by unidirectionally coupled semiconductor lasers with external feedback (Wallace, Yu, Lu and Harrison [2000]), in three-mode dynamics in a LiNdP4 O12 microchip laser (Otsuka, Ohtomo, Yoshioka and Ko [2002]), and in a pump-modulated vertical-cavity surface-emitting laser (VCSEL) with noise signal (Barbay, Giacomelli, Lepri and Zavatta [2003]). Numerical calculations of phase synchronization in periodic and chaotic temporal waveforms have also been reported in optically injected semiconductor lasers (Lariontsev [2000]) and in a semiconductor laser array with delayed global coupling (Kozyreff, Vladimirov and Mandel [2000]). 2.6.2. Generalized synchronization Generalized synchronization is defined by the existence of a functional relation between the dynamics of drive and response: IR (t) = F ID (t) , where ID,R (t) are the laser intensities of the drive and response systems and F is the functional relation between them. Identical synchronization is a special case of generalized synchronization when F is the identity function. In general it is difficult to identify F directly from the time series ID,R (t), since the function itself

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can be very complicated. Generalized synchronization may be much more prevalent in nature than realized so far. It may have important applications in methods for noninvasive testing and monitoring of structures and materials, ranging from buildings to nanostructures, as well as in encoded communications systems (Terry and VanWiggeren [2001]). Generalized synchronization has been shown to exist through the predictability (Rulkov, Sushchik, Tsimring and Abarbanel [1995]) or the existence of a functional relationship (Brown [1998]) between the drive and response systems. These approaches are often difficult to implement in experimental measurements, due to the presence of noise and lack of precision in measurements. When replicas or duplicates of the response system are available, the auxiliary system method introduced by Abarbanel, Rulkov and Sushchik [1996] can be used for detecting generalized synchronization. In this method, two or more response systems are coupled with the drive system. If the response systems, starting from different initial conditions, display identical synchronization between them after transients have disappeared, one can conclude that the response signal is synchronized with the drive in a generalized way. The first experimental observation of generalized synchronization of chaos in laser systems was reported by Tang, Dykstra, Hamilton and Heckenberg [1998]. The chaotic system used in their experiment was an optically pumped singlemode NH3 laser in a ring-cavity configuration, shown in the left panel of fig. 40. The laser was operated in the parameter regime where it behaves similarly to the dynamics of the laser Lorenz–Haken equations. An acousto-optic modulator (AOM) is used to modulate the pump intensity of the laser. They either recorded a chaotic intensity waveform emitted by the laser or calculated a chaotic waveform from the Lorenz equations. This chaotic waveform was then stored in the memory of an arbitrary function generator, which produced an analog signal with exactly the same form as the stored one. This analog signal was then used to modulate the amplitude of the RF driving signal of the AOM, which in turn transferred this modulation signal to the intensity of the pump laser beam. The undiffracted beam from the AOM was used as the pump of the NH3 laser. The auxiliary system method (Abarbanel, Rulkov and Sushchik [1996]) was used to test generalized synchronization. Instead of using two response systems and comparing their dynamics, they repeatedly drove the chaotic laser under the same experimental conditions (with the same chaotic waveform), and compared the chaotic dynamics of the laser under each repetition. The right panel of fig. 40 shows temporal waveforms exhibiting generalized synchronization. The chaotic dynamics of the laser in response to a driving Lorenz-like chaotic signal, after a transient, is shown in fig. 40(a). These chaotic intensity evolutions are different from the driving signal,

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[5, § 2

Fig. 40. Left: schematic block diagram of an experimental setup showing generalized chaos synchronization. SG, signal generator; MOD, modulator; AMP, amplifier; AOM, acousto-optic modulator. Right: chaotic dynamics of the laser under the generalized synchronization of chaos to the driving signal. (a) Chaotic laser intensity evolution, (b) evolution of the chaotic driving signal. (From Tang, Dykstra, Hamilton and Heckenberg [1998].)

Fig. 41. Correlation plot of the laser dynamics under repeated drivings. t = 150 µs is the repetition period of driving signal. (a) Correlation plot of the driving signal, (b) correlation plot of the laser dynamics. (From Tang, Dykstra, Hamilton and Heckenberg [1998].)

shown in fig. 40(b). Figure 41 shows the comparison of the laser dynamics under repeated drivings. Figure 41(a) is the correlation plot of the repeatedly injected driving signals, and fig. 41(b) represent the corresponding outputs of the response laser. The chaotic dynamics of the laser repeats itself completely in successive events, showing that its dynamics is now completely controlled by the driving signal and insensitive to the noise and initial conditions. The dynamics shown in the right panel of fig. 40 is therefore an example of generalized synchronization of chaos. Tang, Dykstra, Hamilton and Heckenberg [1998] observed experimen-

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tally that the functional relation in a state of generalized synchronization is not fixed, but varies with the strength of the driving signal. When the driving strength is increased, the dynamical relationship between driving and response becomes simpler and simpler, eventually becoming a linear relation, which implies that the state of identical synchronization of chaos is approached. The examples of generalized synchronization discussed in the literature consist of situations where the drive and response systems are different from each other, or are based on the same system operated at different parameter values (Lewis, Abarbanel, Kennel, Buhl and Illing [2000]). One may expect that strongly coupled identical systems with similar parameter values will display identical synchronization, if they synchronize at all. However, generalized synchronization of chaos can occur with identical drive and response systems with similar parameter values. This was shown by Uchida, McAllister, Meucci and Roy [2003] (also McAllister, Uchida, Meucci and Roy [2004]), who used a two-longitudinal-mode Nd:YAG microchip laser as a laser source, as shown in the left panel of fig. 42. The total intensity of the laser output is detected by a photodiode and the voltage signal is fed back into an intracavity acousto-optic modulator (AOM) in the laser cavity, through an electronic low-pass filter with an amplifier. The loss of the laser cavity was modulated by the self-feedback signal through the AOM, which induced chaotic oscillations. Temporal waveforms of the laser output were measured by a digital oscilloscope and stored in a computer, for later use as a drive signal. In order to test for synchronization, the same laser was used as a response system, which ensured identical parameter settings between drive and response. The drive signal was sent to the AOM in the same laser cavity, using an arbitrary function generator connected with the computer. The original feedback loop was disconnected (dashed line in the left panel of fig. 42), i.e. the open-loop configuration (dotted line in the left panel of fig. 42) was used for the response system. The total intensity of the laser output was detected with the digital oscilloscope. Typical temporal waveforms of the drive and the response are shown in fig. 42(a). There is no obvious correlation between the drive and response outputs. Indeed, the correlation plot between the drive and response waveforms (fig. 42(b)) shows no evidence of identical synchronization. Then the drive signal was fed back into the response laser repeatedly, in order to apply the auxiliary system method (Abarbanel, Rulkov and Sushchik [1996]). Figure 42(a) shows two response outputs driven by the same drive signal at different times. The correlation plot between the two response outputs shows linear correlation, as shown in fig. 42(c). This implies that the response laser driven by the same drive signal always generates identical outputs, independent of initial conditions. Since the dynamics of the response laser are repeatable and reproducible, generalized

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[5, § 2

Fig. 42. Left: experimental setup of a diode-pumped Nd:YAG microchip laser with optoelectronic feedback. The dashed line corresponds to the closed-loop drive system, and the dotted line corresponds to the open-loop response system. AFG, arbitrary function generator; AOM, acousto-optic modulator; BS, beam splitter; COM, computer; F–P, Fabry–Perot interferometer; L, lens; LD, laser diode for pumping; LPF–A, low pass filter and amplifier; M, mirror; Nd:YAG, Nd:YAG laser crystal; OC, output coupler; OSC, digital oscilloscope; PD, photodetector. Right: (a) temporal waveforms of experimentally measured total intensity of the drive and two response systems; (b) correlation plots between the drive and response outputs; (c) correlation plots between the two response outputs. (b) and (c) are obtained from (a). (d) Temporal waveforms of the total intensity as obtained from numerical calculations. (From Uchida, McAllister, Meucci and Roy [2003].)

synchronization can be stably achieved in this system. Numerical results obtained from Tang–Statz–deMars equations (see Section 2.5.2) agree well with the experimental observations (fig. 42(d)). It turns out that modal dynamics is responsible for the occurrence of generalized synchronization in identical systems. Generalized synchronization has also been observed experimentally, in unidirectionally coupled He–Ne lasers with optical feedback (Uchida, Higa, Shiba, Yoshimori, Kuwashima and Iwasawa [2003]), and numerically in chaotic erbiumdoped fiber dual-ring lasers (Wang and Shen [2001]), by using the auxiliary system method. The studies of generalized synchronization lead to a more general concept of consistency (Uchida, McAllister and Roy [2004]), which is defined as the ability of a nonlinear dynamical system to produce identical response outputs after some transient period, when the system is driven by a repeated drive signal. Consistency of dynamics may be essential for information transmission in biological and physiological systems and for reproduction of spatiotemporal patterns in nature.

5, § 3]

Communication with chaotic lasers

273

§ 3. Communication with chaotic lasers 3.1. Introduction of chaotic communications Many communication systems in today’s technological world are based on the synchronization between periodic oscillators. In that scenario, tuning a receiver into synchronization with an emitter allows the recovery of information that was encoded (via frequency or amplitude modulation) in the periodic carrier. Can a chaotic carrier be used for the same purpose? An example is shown in fig. 43, which uses a coupling scheme that leads to synchronization, as described in Section 2.1. This system uses two communication channels: the first one is the coupling channel that leads to synchronization, and the second one contains the message added to another chaotic carrier. Upon synchronization between transmitter and receiver, z = z and the subtraction performed at the receiver allows recovery of the input message. The simple scheme shown in fig. 43 requires two communication channels to transmit a single signal, which is not very efficient. In what follows we review single-channel chaotic communication systems that have been described in the literature. Communications based on chaotic synchronization were already proposed by Pecora and Carroll [1990] in their seminal paper on chaos synchronization. The feasibility of this proposal was confirmed soon thereafter in electronic circuits by several authors (Kocarev, Halle, Eckert, Chua and Parlitz [1992], Oppenheim, Wornell, Isabelle and Cuomo [1992]). In particular, Cuomo and Oppenheim [1993] demonstrated two possible approaches to communications using one transmission channel, based on analog-circuit implementations of synchronized Lorenz oscillators, as described below.

Fig. 43. Two-channel communication system based on chaotic synchronization.

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3.2. Encoding and decoding techniques 3.2.1. Signal encoding A key issue when using chaos synchronization for communication purposes is the way in which the message to be transmitted is mixed with the chaotic carrier. The encoding techniques can be classified as follows: • External encoding: The message signal is externally added (or multiplied) to a chaotic carrier that is independent of the message. The message is recovered by subtracting (or dividing) the synchronized chaotic signal in the receiver from the transmitted signal. Synchronization may persist as long as the message is small enough to be considered a perturbation, since the synchronization manifold is stable, and thus it may not be affected by small enough perturbations. This type of external encoding is known in the literature as chaos masking. • Internal encoding: The message is introduced into the transmitter, either by modulating one of its parameters or by direct injection into its cavity. For instance, the message is added in a feedback loop of the transmitter and modulates the dynamics of the chaotic transmitter. This type of encoding is called in the literature chaos modulation. The modulation can be either digital or analog. When the modulation is digital and is used to switch between different synchronization states depending on the message, it is called chaos shift keying (CSK). 3.2.2. Signal decoding We now turn to the issue of decoding. We note that the quality of decoding depends on two factors: the accuracy of synchronization and the method of encoding. High-quality decoding requires a coupling setup that allows high-quality synchronization in the absence of a message. This is a necessary condition for high-quality decoding. Based on the essential limitation of the quality of decoding, the decoding schemes can be classified into two categories: • Perfect decoding: For chaos modulation, ‘perfect’ decoding can be achieved. The message modulates both the transmitter and the receiver in the same manner. Therefore, the dynamical equations of the two systems have symmetry and perfect synchronization can be maintained in the presence of the message. This leads to perfect decoding in principle (see Section 3.4). • Imperfect decoding: For chaos masking, communication errors may exist even if synchronization is perfectly achieved in the absence of message. The addition of the message perturbs the synchronization, and this synchronization error leads to a decoding error. To enhance the quality of decoding, the amplitude of the message signal must be much smaller than that of the chaotic

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carrier. Chaos shift keying makes use of the synchronization error as a method to detect the message. In this case, the difference between the synchronized and unsynchronized states determines the decoding errors. Larger differences increase the quality of decoding, but the difference must be small enough to prevent the detection of the message by eavesdroppers through direct detection of the transmission signal.

3.3. Examples of chaotic communication systems We now give specific examples of the different types of encoding and decoding listed above. This section deals with abstract implementations and electronic circuits, whereas further examples in optical systems will be given in the following sections. 3.3.1. Chaos shift keying One of the communication schemes proposed by Cuomo and Oppenheim [1993] is shown in fig. 44. The basic idea of this scheme is to modulate one of the parameters of the transmitter with the information-bearing waveform: x˙ = σ (y − x), y˙ = rx − y − 20xz, z˙ = −b m(t) z + 5xy,

x˙ = σ (y − x ),

(3.1)

(3.2)

z˙ = −b(0)z + 5xy .

(3.3)

y˙ = rx − y − 20xz ,

Note that the parameter b of the transmitter is modulated according to a digital message m(t), taking two different values for 0 and 1 bits (in the demonstration of Cuomo and Oppenheim [1993], b(0) = 4 and b(1) = 4.4). The emitter variable x is transmitted in the form of a chaotic drive signal towards the receiver, whose b parameter is fixed to the transmitter’s 0-bit value. In that way, at the receiver the coefficient modulation produces a synchronization error between the transmitted drive signal and the receiver’s regenerated x signal with an error signal amplitude that depends on the modulation. The modulation can then be detected from the synchronization error. Figure 45 shows the result for a digital message in the form of a square wave; it portrays the synchronization error at the output of the receiver. After low-pass filtering this signal, the original digital message is successfully recovered. In the configuration of fig. 44, the receiver consists of a single chaotic generator whose coding parameter is matched to one of the two parameters in the transmitter (the one corresponding to the 0 bit, for instance). Then, message recovery is

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Fig. 44. Chaotic communication system based on the synchronization of two Lorenz oscillators. The message is introduced via the modulation of one of the transmitter’s parameters. (Adapted from Cuomo and Oppenheim [1993].)

Fig. 45. Message recovery in the scheme of fig. 44: (a) input message; (b) unfiltered synchronization error; (c) low-pass filtered recovered wave form. (From Cuomo and Oppenheim [1993].)

accomplished by analyzing the synchronization error, which will be close to zero when the bit corresponds to the matching value of the coding parameter (for instance 0) and large for the other bit. This scheme, called chaos on–off keying,

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is a special case of chaos shift keying using a single chaos generator, which is switched on or off according to the binary message to be transmitted. 3.3.2. Chaos masking Another approach to chaotic communications, also proposed by Cuomo and Oppenheim [1993], is sketched in fig. 46. In this scheme, the information-bearing message signal is added to the chaotic carrier outside the transmitter: x˙ = σ (y − x ),

x˙ = σ (y − x),

(3.4)

y˙ = r(x + m) − y − 20(x + m)z ,

y˙ = rx − y − 20xz,

z˙ = −bz + 5(x + m)y .

z˙ = −bz + 5xy,

(3.5) (3.6)

Assuming that the receiver’s dynamics synchronizes with the original signal (i.e. the transmitter signal without message), information recovery follows from a simple subtraction x − x. This procedure works well when synchronization is not very sensitive to perturbations in the drive signal. In particular, the amplitude of the message must be small enough with respect to the carrier. Cuomo and Oppenheim [1993] demonstrated the performance of this system with a segment of speech signal. Figure 47 shows the original speech and the recovered speech signal. The speech signal is clearly recovered and is of reasonable quality in informal listening tests. 3.3.3. Chaos modulation As opposed to external encoding via chaos masking, in internal encoding (via both direct modulation and intracavity injection) the message is mixed nonlinearly with the chaotic carrier. In this encoding technique, the message does not need to be much weaker than the carrier. In particular, when this method is applied to feedback systems in such a way that the message is encoded inside the feedback loop, perfect message recovery can be achieved regardless of the amplitude of the message (see Section 3.4). An example is shown in fig. 48 for the Lorenz system (VanWiggeren [2000]). Let us now interpret the role of the y variable in the dynamics of x in a Lorenz oscillator as an instantaneous feedback. If we introduce an original message signal in this feedback loop by adding m(t) to y before making it act on x, we can build a receiver perfectly symmetric to the transmitter. Mathematically, x˙ = σ (s − x), y˙ = rx − y − xz, z˙ = −bz + xy,

x˙ = σ (s − x ),

(3.7)

y˙ = rx − y − x z ,

z˙ = −bz + x y ,

(3.8) (3.9)

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Fig. 46. Chaotic communication system based on the synchronization of two Lorenz oscillators. The message is added to the transmitter’s output. (Adapted from Cuomo and Oppenheim [1993].)

Fig. 47. Message recovery in the scheme of fig. 46: speech wave forms. (a) Original; (b) recovered. (From Cuomo and Oppenheim [1993].)

where s(t) = y(t) + m(t). The dynamics of the error variables ex = x − x , ey = y − y and ez = z − z are e˙x = −σ ex ,

(3.10)

e˙y = rex − ey − ex z − ez x,

(3.11)

e˙z = −bez + ex y + ey x.

(3.12)

From eq. (3.10) it is clear that ex → 0, since the eigenvalue σ > 0. Using this fact, the remaining eigenvalues (conditional Lyapunov exponents) for ey and ez are 1 −(b + 1) ± (b + 1)2 − 4 b + x 2 . λ2,3 = (3.13) 2 These exponents are negative for positive b, since x 2 > 0. Note that this result is independent of the message (in particular, of its amplitude).

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Fig. 48. Internal encoding via feedback in the Lorenz system.

Fig. 49. Schematic diagram for three encoding–decoding schemes using semiconductor lasers with optical feedback. CSK, chaos shift keying; CMS, chaos masking; ACM, additive chaos modulation. (From Liu, Chen and Tang [2002].)

The three encoding and decoding methods described above (chaos shift keying, chaos masking and chaos modulation) can be applied for chaotic optical systems. Chaotic masking and chaotic modulation have been experimentally implemented in fiber laser systems (VanWiggeren and Roy [1998b, 1999a]). All of the three schemes have been experimentally demonstrated in semiconductor laser systems (see Section 3.6.1). Figure 49 presents the three schemes applied to the same laser systems, as described by Liu, Chen and Tang [2002]. The same configuration can be used for the three schemes, except for the place where the message is encoded. Liu, Chen and Tang [2002] concluded that the best performance of bit error rate is achieved by using chaos modulation.

3.4. Decoding quality in chaos modulation Chaos modulation is the best of the three schemes described in Section 3.3 for decoding a message from chaos without any distortion, since in principle perfect decoding can be achieved by using this method. Let us explain the mathematical description of chaos modulation in this subsection. Chaos modulation can be applied for a chaotic system with self-feedback as shown in fig. 49. The dynamics

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of a transmitter system with a feedback loop can be described by dIT (t) = FT IT (t), IF (t) , (3.14) dt where IT (t) is the output (e.g., laser intensity) of the transmitter, IF (t) is the feedback signal, and FT is the nonlinear function of the transmitter dynamics. When a message m(t) is introduced in the feedback loop of the transmitter as shown in fig. 49, the dynamics of the laser is affected by the feedback signal with the message IF (t) = IT (t) + m(t): dIT (t) = FT IT (t), IT (t) + m(t) . (3.15) dt The signal IT (t) + m(t) is also transmitted through a transmission line and injected into the receiver, as shown in fig. 49. Assuming that the receiver is openloop, the dynamics of the receiver can thus be described as dIR (t) = FR IR (t), IT (t) + m(t) . (3.16) dt Equations (3.15) and (3.16) are identical when FT = FR and IT (t) = IR (t). Therefore, there exists a synchronous solution between IT (t) and IR (t) even in the presence of the message when two identical systems are used as transmitter and receiver. Synchronization between IT (t) and IR (t) can be achieved when the synchronous solution is stable. The message can be recovered by subtracting the synchronized signal IR (t) (= IT (t)) from the transmission signal IT (t) + m(t) without any errors in principle (perfect decoding). In the case of time-delayed feedback, when the delay time is much longer than the time scale of the oscillations of IT (t), IR (t), and m(t), the time delay must be compensated in the receiver between the synchronized signal and the transmission signal for decoding. In the presence of time delay in the feedback loop of the transmitter, eq. (3.15) is written as dIT (t) (3.17) = FT IT (t), IT (t − τ ) + m(t − τ ) , dt where τ is the delay time in the feedback loop. When the transmission signal IT (t) + m(t) is injected into the receiver directly, the dynamics of the receiver is written as dIR (t) (3.18) = FR IR (t), IT (t) + m(t) . dt Comparison of eqs. (3.18) and (3.17) shows that they are not identical, and synchronization IT (t) = IR (t) cannot be achieved. To obtain synchronization, exactly the same time delay τ must be introduced between the transmission signal

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IT (t) + m(t) and the output of the receiver IR (t) as dIR (t) = FR IR (t), IT (t − τ ) + m(t − τ ) . (3.19) dt In this case, a synchronous solution IT (t) = IR (t) exists between eqs. (3.17) and (3.19), and message decoding can be achieved perfectly with the same procedure as in the case of no time delay. When the message is not added but multiplies feedback signal of the transmitter, the message can also be recovered by dividing the transmitted signal by the synchronized signal. For chaos modulation, the quality of message decoding does not depend on the amplitude of the message. The message, however, must not be so large that it may completely change the dynamics of the original chaotic system. If the message amplitude is large, it may be easy for unauthorized users to detect the message by directly observing the transmission signal and by analyzing the change of the original dynamics.

3.5. Communications with chaotic electronic circuits Before turning to optical systems, and following the historical development of chaotic communication systems, we give a brief overview of the implementation of this technology in electronic circuits (for a more detailed introduction to this topic, see review papers and books by Kennedy, Rovatti and Setti [2000], Abel and Schwarz [2002], Kennedy and Kolumbán [2000], Dachselt and Schwarz [2001] and Kolumbán and Kennedy [2000]). The concepts described in this context are essentially applicable to chaotic optical systems. 3.5.1. Communications with synchronization The initial studies on communications with chaos were conducted primarily from a dynamical systems point of view, and dealt mainly with synchronization issues. Since the mid-1990s, researchers in chaos communications based on electronic circuits concentrated on the choice of methods and measures. Several communication schemes using chaotic electronic circuits were demonstrated experimentally almost at the same time as the study of Cuomo and Oppenheim [1993]. Notably, Kocarev, Halle, Eckert, Chua and Parlitz [1992] sent a sinusoidal waveform masked within a chaotic carrier generated by a Chua circuit (chaotic masking). They succeeded in recovering the message with a signal to noise ratio of 30–35 dB. Parlitz, Chua, Kocarev, Halle and Shang [1992] used two chaotic attractors generated by a Chua circuit at two parameter settings for the transmission

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of digital message: for each 0 in the binary sequence, they transmitted one of the variables of the first attractor, and for each 1 they transmitted the corresponding variable of the second attractor (chaos shift keying). To detect the synchronization state, the receiver’s parameters were matched with the transmitter’s parameters. The use of two different synchronization errors allowed them to decide which binary bits were being sent to the receiver. Every existing chaotic communication scheme described in Section 3.3 has been demonstrated in electronic circuits (Abel and Schwarz [2002]). As mentioned above, chaos masking was applied by Kocarev, Halle, Eckert, Chua and Parlitz [1992] and Cuomo and Oppenheim [1993]. Chaos modulation was demonstrated by Halle, Wu, Itoh and Chua [1993] and Volkovskii and Rul’kov [1993]. The chaos shift keying protocol was used in the seminal investigations of Parlitz, Chua, Kocarev, Halle and Shang [1992] and Cuomo and Oppenheim [1993]. Kolumbán, Kennedy and Kis [1997] used chaos on–off keying. Sushchik, Rulkov, Larson, Tsimring, Abarbanel, Yao and Volkovskii [2000] proposed the use of chaotic pulse-position modulation for ultrawide-bandwidth wireless communication systems. In this method, the information is encoded within a chaotic-pulse signal by using additional delays in the generated interpulse intervals obtained from a chaotic map, as shown in fig. 50. The position of each chaotic pulse is delayed (or not) according to the value of the binary bits. By matching the chaotic map of the decoder in the receiver to that of the encoder, the time of the next arriving pulse can be predicted. The digital bits can be obtained by comparing the pulse positions between the encoder and decoder. This method was applied to free-space laser communication over a turbulent channel by Rulkov, Vorontsov and Illing [2002]. 3.5.2. Communications without synchronization Additionally, a large body of work exists in electronic circuits concerning chaotic communications without chaos synchronization, such as differential chaos shift keying (Kolumbán, Vizvari, Schwarz and Abel [1996]). The basic idea of this scheme is shown in fig. 51. Every information bit to be transmitted is represented by two chaotic waveforms. The first sample waveform serves as a reference, while the second one carries the information. Bit 1 is sent by transmitting a reference signal provided by a chaos generator twice in succession, while for bit 0, the reference chaotic signal is transmitted, followed by an inverted copy of the same signal. The two sample waveforms are correlated in the receiver. A positive autocorrelation indicates that bit 1 has been received while a negative autocorrelation indicates bit 0. In this method, the robustness of digital signal transmission against channel noise is comparable to that of conventional modulation techniques.

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Fig. 50. (a) Illustration of the basics of chaotic pulse-position modulation schemes and chaotic pulse regenerator (CPRG) operation. (b) Diagram of the receiver for chaotic pulse-position modulation. (From Sushchik, Rulkov, Larson, Tsimring, Abarbanel, Yao and Volkovskii [2000].)

Fig. 51. (a) Transmitter for differential chaos shift keying. (b) Receiver for differential chaos shift keying. (From Abel and Schwarz [2002].)

Another method encodes the message into the symbolic dynamics of a chaotic generator (Hayes, Grebogi and Ott [1993], Hayes, Grebogi, Ott and Mark [1994]). Symbolic dynamics are obtained if the state space of the generator is completely partitioned into disjoint subsets, which are assigned to symbols. The sequence of symbols corresponding to a trajectory in the state space is the symbolic dynamics of that trajectory. They used a double scroll electrical oscillator which yielded a chaotic signal consisting of a seemingly random sequence of positive and negative peaks. When they associated a positive peak with 1 and a negative peak with 0, the signal yielded a binary sequence. They used small control perturbations to cause the signal to follow an orbit whose binary sequence represented the information they wished to communicate. Figure 52 shows a waveform encoded with the double scroll system. The authors represented each letter of the Latin alphabet by its five-bit binary representation (in terms of its location in the alphabet) to encode the word ‘chaos’. Methods of chaos control allow a chaos generator to be forced to a particular symbolic sequence for encoding a message. A method using symbolic dynamics has been implemented in electronic circuits for information transmission (Corron, Pethel and Myneni [2002]).

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Fig. 52. Controlled chaotic signal for the double scroll system encoding the word ‘chaos’. Each letter is shown at the top of the figure, along with its numerical position in the alphabet. Shown at the bottom are the corresponding binary code words. Extra bits (indicated by commas) are added to satisfy the constraints imposed by the grammar. (From Hayes, Grebogi and Ott [1993].)

Spread-spectrum communications might be an adequate direction for chaotic communications in electronic circuits. The key idea for the exploitation of spreadspectrum signals in communications is to increase the robustness against disturbances affecting narrow frequency ranges, i.e. multipath propagations and interfering signals. One of the promising applications in chaos communications is the use of chaos in direct-sequence code-division multiple access (DS–CDMA) systems (Kennedy, Rovatti and Setti [2000], Mazzini, Rovatti and Setti [2001]). Optimized spreading sequences for CDMA systems can be generated with a chaotic map. Their results show that chaos-based spreading leads to quantitative improvement of the ability to minimize multiple-access interference with respect to classical spreading methods.

3.6. Examples of communication with chaotic lasers We now turn to describing a representative subset of the many examples of communication systems based on chaotic lasers, concentrating on those situations where there has been an experimental demonstration of the technology (see also the special issues edited by Donati and Mirasso [2002], Larger and Goedgebuer [2004], Gavrielides, Lenstra, Simpson and Ohtsubo [2004] and the books by Ohtsubo [2005], Kane and Shore [2005]).

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3.6.1. Semiconductor lasers In recent years, much attention has been devoted to the application of synchronization of semiconductor lasers to chaos communication, since these lasers are the most popular optical sources in high-speed optical communication systems. We describe four configurations of semiconductor lasers for chaos communications in this subsection, following the classification of Section 2.5.4: optical injection, coherent optical feedback, incoherent optical feedback, and optoelectronic feedback systems. Optical injection. Annovazzi-Lodi, Donati and Sciré [1996] performed one of the first numerical studies on chaos communications using semiconductor lasers with optical injection. They used chaos shift keying and chaos masking as encoding and decoding schemes. For chaos shift keying, the pump current of a transmitter laser is modulated at two different values, corresponding to message bits 1 and 0. The pump currents of two receiver lasers have to be matched to one of the two current values in the transmitter. Only the receiver laser whose pump current corresponds to the transmitted bit will synchronize. The original digital bits can be recovered by detecting the error signals between the transmitted signal and one of the outputs of the two receivers, e2 , e3 . The left panel of fig. 53 shows the outputs e2 , e3 of the decoders for bits 0 and 1 and the transmitted digital signal. After a transient, e2 falls to zero when bit 0 is transmitted, while e3 falls to zero when bit 1 is transmitted. For chaos masking, a message bit sequence is added with the chaotic carrier in the transmitter and is sent to the receiver. The receiver

Fig. 53. Left: normalized error signals e2 , e3 showing demodulation of bits 0 and 1. Synchronization is reached after a transient. Right: reconstruction of the bit sequence by synchronization and subtraction of chaos. (From Annovazzi-Lodi, Donati and Sciré [1996].)

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laser synchronizes only with the chaotic component of the transmitted signal. Decoding is thus performed by taking the difference between the transmitted signal and the synchronized receiver output intensity. The right panel of fig. 53 shows the successfully reconstructed bit sequence after synchronization and subtraction of chaos. Chen and Liu [2000], Liu, Chen and Tang [2001] numerically demonstrated chaos communication via chaos modulation in semiconductor lasers with optical injection. Good-quality message recovery was achieved with random bits at a bit rate of 2.5 Gbits/s. Coherent optical feedback. Many studies have been devoted to communication schemes implementing semiconductor lasers with time-delayed feedback. These are of particular interest because time-delayed feedback can generate highdimensional chaos (Ahlers, Parlitz and Lauterborn [1998]). Indeed, the nonlinear dynamics techniques that have been proposed to unmask the messages encoded in chaos (Short [1994, 1996], Short and Parker [1998], Pérez and Cerdeira [1995]) are much less efficient in the case of high-dimensional chaos (Ding, Ding, Ditto, Gluckman, In, Peng, Spano and Yang [1997]). The first numerical studies on chaos communication with single-mode semiconductor lasers subject to coherent optical feedback were reported by Mirasso, Colet and García-Fernández [1996] and Annovazzi-Lodi, Donati and Sciré [1997]. Mirasso, Colet and García-Fernández [1996] externally added a chaotic carrier with an analog message signal in the transmitter and sent it to the receiver (chaos masking). The decoding process is based on the synchronization of the receiver output to the transmitter carrier field rather than to the transmitted signal with message, so that the message can be recovered in real time by comparing the transmitted signal with the synchronized receiver output. As shown in fig. 54, they demonstrated that a 4 Gbits/s message encoded through chaos masking can be decoded at the receiver despite the distortion introduced by the encoding of the message and the propagation through 200 km dispersion-shifted optical fiber. These results were also substantiated by Sánchez-Díaz, Mirasso, Colet and García-Fernández [1999]. Annovazzi-Lodi, Donati and Sciré [1997] numerically showed 50 Mbits/s digital data encoding and decoding by using the encryption method of chaos shift keying and synchronization based on continuous chaos control of two externalcavity semiconductor lasers. More recently, it was shown that the chaos shift keying encryption method can be performed with a similar configuration. Indeed, Mirasso, Mulet and Masoller [2002] reported that a message can be encoded at bit rates of the order of (but lower than) the relaxation oscillation frequency of the

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Fig. 54. (a) Typical laser output power from transmitter. (b) Transmitter output after encoding the signal. (c) Output of the receiver. (d) Encoded message. (e) Decoded message. (From Mirasso, Colet and García-Fernández [1996].)

semiconductor lasers by switching the injection current of the transmitter laser between two slightly different values. Heil, Mulet, Fischer, Mirasso, Peil, Colet and Elsäßer [2002] proposed an on–off phase shift keying method. In this method, the message is encoded by suitable discrete changes in the transmitter feedback phase with bit rates up to 100 Mbits/s. Liu, Chen, Liu, Davis and Aida [2001] numerically demonstrated chaos modulation in semiconductor lasers with optical feedback. A pseudo-random nonreturn-to-zero (NRZ) sequence with a bit rate of 2.5 Gbits/s is successfully recovered when identical synchronization is achieved. The bit error rate (BER)

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was evaluated and a very sensitive dependence of the BER on the frequency detuning between two lasers was demonstrated. For other developments based on numerical studies, White and Moloney [1999] proposed to apply chaos masking to multimode semiconductor laser diodes for multichannel communication. Buldú, García-Ojalvo and Torrent [2005] and Lee and Shore [2005] encoded a square-wave message in a multimode semiconductor laser of the transmitter and decoded the massage by using two single-mode semiconductor lasers in the receiver. Several experiments have demonstrated the feature of chaos masking in semiconductor lasers subject to optical feedback by using the restricted type of generalized synchronization (see Section 2.3). Early experiments were performed with a message signal at a frequency of a few kHz by Sivaprakasam and Shore [1999b, 2000]. The injection current of the semiconductor laser in the transmitter is modulated as a message as shown in the left panel of fig. 55. In this sense, this method can be classified as the internal encoding described in Section 3.2, although the term ‘chaos masking’ is used in the literature. The chaotic carrier with the modulation signal is sent to the receiver. The message is decoded by subtracting the synchronized chaotic signal in the receiver from the transmission signal consisting of the chaotic carrier and the message. The right panel of fig. 55 shows the transmission signal, synchronized receiver output, and the decoded message. Good recovery of the original message (a square waveform) was achieved. A sinusoidal waveform at GHz frequency was used as a message for highspeed chaos communications (Fischer, Liu and Davis [2000], Kusumoto and Ohtsubo [2002], Paul, Sivaprakasam, Spencer, Rees and Shore [2002], Lee, Paul, Sivaprakasam and Shore [2003]). The ability of the receiver laser to reproduce only the chaotic carrier of the transmitter and suppress the encoded message is crucial for chaos masking. The selective regeneration of the chaotic component from the transmitted signal was referred to as ‘chaos pass filtering’ by Fischer, Liu and Davis [2000]. Figure 56 illustrates the chaos pass filtering effect where the spectral peak corresponding to the message waveform (581.5 MHz) is suppressed in the receiver compared with that in the transmitter. Good message recovery is thus achieved by subtracting the receiver signal from the transmission signal. The chaos pass filtering effect is larger at lower frequencies of the message and decreases as the message frequency approaches the relaxation oscillation frequency of the laser. These frequency characteristics are similar to the response of steadystate injection-locked semiconductor lasers to small-signal modulation (Uchida, Liu and Davis [2003], Paul, Lee and Shore [2005]). Argyris, Kanakidis, Bogris and Syvridis [2004] experimentally investigated the performance of all-optical chaos communication systems with semiconduc-

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Fig. 55. Left: schematic diagram of the experimental arrangement. BS1–BS4, beamsplitters; PD1, PD2, photodetectors; OI1, OI2, optical isolators; M1, M2, mirrors; NDFs, neutral density filters; CA, coupling attenuator; CRO, digital oscilloscope. Top right: the chaotic transmitter output with message and the receiver output. Bottom right: decoded message corresponding to an encoded square-wave of frequency 9.5 kHz. (From Sivaprakasam and Shore [2000].)

tor lasers with optical feedback at high bit rate. Their experimental system is shown in fig. 57. Two DFB lasers from the same wafer with almost identical characteristics and operating at 1552.5 nm were selected as the transmitter and the receiver lasers. The chaotic carrier output of the transmitter is unidirectionally injected through optical fiber, after optical amplification (by EDFA) and filtering, into the receiver laser for synchronization. They compared the three encoding and decoding schemes (chaos masking, chaos modulation, and chaos shift keying)

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Fig. 56. RF spectra of (a) transmitter and (b) receiver semiconductor lasers with optical feedback. The transmitter laser’s pump current is modulated at a frequency of 581.5 MHz. The modulation peak is considerably weaker in the receiver output than in the transmitter output. (From Fischer, Liu and Davis [2000].)

and estimated bit error rate (BER) for all the schemes. A nonreturn-to-zero (NRZ) pseudorandom bit sequence of length 223 −1 was used as a message at 1.5 Gbits/s bit-rate. The left panel of fig. 58 shows the bit sequences of the original message, the encrypted signal into the chaotic carrier, and the decoded message for the chaos modulation scheme. The message recovery is successfully achieved after chaotic carrier subtraction and filtering. The right panel of fig. 58 shows BER measurements of both the chaotic carrier and the decoded message with respect to the injection power into the receiver for the chaos modulation scheme. By increasing the injection power, the synchronization improves resulting in better quality of the decoded message, as indicated by the reduced BER values. The optimum performance was measured for injection optical power of 0.23 mW, resulting in a BER value of 7 × 10−5 . Their experimental investigation indicates that chaos modulation gives the best results of the three encoding and decoding methods. Comprehensive numerical studies on the performance of different encoding and decoding schemes were also reported by Kanakidis, Argyris and

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Fig. 57. Experimental setup. PC, polarization controller; OI, optical isolator; 90/10–50/50, optical coupler; PD, photoreceiver. (A) chaos modulation, (B) chaos masking, (C) chaos shift keying. (From Argyris, Kanakidis, Bogris and Syvridis [2004].)

Fig. 58. Left: bit sequences of (a) the message, (b) the encrypted message into the chaotic carrier, and (c) the decoded message after chaotic carrier subtraction and filtering, for the chaos modulation method. Right: BER measurements of the encrypted message into the chaotic carrier and the decoded message, for the chaos modulation method, under different injection strengths. (From Argyris, Kanakidis, Bogris and Syvridis [2004].)

Syvridis [2003] (see also Argyris and Syvridis [2004], Bogris, Kanakidis, Argyris and Syvridis [2004], Kanakidis, Bogris, Argyris and Syvridis [2004]). Incoherent optical feedback. Rogister, Locquet, Pieroux, Sciamanna, Deparis, Mégret and Blondel [2001], Rogister, Pieroux, Sciamanna, Mégret and Blondel

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Fig. 59. (a) Encoded message at a bit rate of 250 Mbits/s. (b) Synchronization error after filtering. (From Rogister, Locquet, Pieroux, Sciamanna, Deparis, Mégret and Blondel [2001].)

[2002] numerically demonstrated chaos shift keying in semiconductor lasers with incoherent optical feedback. The setup is shown in fig. 28. The linearly polarized output field of the transmitter laser undergoes a 90◦ polarization rotation through an external cavity formed by a Faraday rotator (FR) and a mirror. It is then split into two parts by a beam splitter: One part is fed back into the transmitter laser, and the other part is injected into the receiver laser. An NRZ bit stream at a bit rate of 250 Mbits/s is encoded by modulating the injection current of the semiconductor laser in the transmitter. Message decoding is achieved by computing the normalized synchronization error. The synchronization error is low-pass filtered. Figure 59 shows successful recovery of the digital message. They found that parameter mismatches of only a few percent between the transmitter and receiver lasers lead to severe degradation of the synchronization quality, such that recovery of the message is no longer possible. However, this scheme requires no fine-tuning of the laser optical frequencies, unlike other schemes based on semiconductor lasers subject to coherent optical feedback. Optoelectronic feedback. Chaotic optical communication through synchronization of semiconductor lasers with optoelectronic feedback was experimentally demonstrated by Tang and Liu [2001b] (see also Liu, Chen and Tang [2001, 2002], Abarbanel, Kennel, Illing, Tang, Chen and Liu [2001]). In that experiment, the receiver laser was in an open-loop configuration and thus completely driven by the transmitter signal as shown in fig. 60. The message was encoded by means of

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Fig. 60. Schematic of the chaotic communication system with chaotic pulsing semiconductor lasers for encoding and decoding messages through chaos modulation. LDs, laser diodes; PDs, photodetectors; As, amplifiers. (From Tang and Liu [2001b].)

incoherent addition onto the output of the transmitter laser and hence it affected the dynamics of the transmitter and the receiver in the same way. It was recovered by subtraction of the receiver laser output (adequately shifted in time because of the delay in the transmitter feedback loop) from the received signal. A pulse stream with a 500 MHz repetition rate was successfully transmitted with no error bit detected within 6000 pulses, indicating a bit error rate less than 1.7 × 10−4 . The left panel of fig. 61 illustrates the recovery of the 500-MHz pulse stream. A pseudorandom nonreturn-to-zero bit sequence at the OC-48 standard bit rate of 2.5 Gbits/s was also transmitted through this system successfully, as shown in the right panel of fig. 61. Liu, Chen and Tang [2002] numerically investigated system performance of fig. 60 at the OC-192 standard bit rate of 10 Gbits/s for the three encoding– decoding schemes (chaos masking, chaos modulation, and chaos shift keying) in the presence of channel noise and internal laser noise. The left panel of fig. 62 shows the time series of the original message and the decoded messages for different encryption methods. Recovery of the high-bit-rate message is not possible for chaos shift keying (CSK in fig. 62), because the resynchronization time after a desynchronization burst has to be shorter than the bit duration for a following bit to be recoverable. For chaos masking (CMS in fig. 62), the errors in message recovery are primarily generated by the timing errors that are caused by the breaking of the symmetry between the transmitter and the receiver due to the presence of the message. The recovered message is thus contaminated by frequent spikes. The performance of additive chaos modulation (ACM in fig. 62) is the best among

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Fig. 61. Transmission of (left) a pulse stream with 500 MHz repetition rate and (right) a pseudorandom nonreturn-to-zero bit sequence at 2.5 Gbits/s. Both figures show, from top to bottom, time series of transmitted signal, receiver laser output, recovered signal, and encoded signal. (From Tang and Liu [2001b].)

Fig. 62. Left: time series of the decoded messages of the three different encryption schemes in the optoelectronic feedback system. CSK, chaos shift keying; CMS, chaos masking; ACM, additive chaos modulation. Right: bit error rate (BER) versus signal-to-noise ratio (SNR) for the three different encryption schemes in the optoelectronic feedback system. Solid curve: no laser noise. Dashed curve: laser linewidth 100 kHz for both transmitter and receiver. Dotted–dashed curve: laser linewidth 1 MHz. Dotted curve: laser linewidth 10 MHz. (From Liu, Chen and Tang [2002].)

the three encryption schemes. The error bits in the recovered message for chaos modulation are caused by synchronization error due to the channel noise and the laser noise. No desynchronization bursts are observed in the recovered message for chaos modulation. The system performance measured by BER as a function of channel signalto-noise ratio (SNR) for this optoelectronic feedback system is shown in the right panel of fig. 62. For chaos modulation, a BER lower than 10−5 can be obtained when the SNR is larger than 38 dB, whereas BER is always higher

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than 10−1 for chaos masking and chaos shift keying. The best performance is thus obtained for chaos modulation in the three encryption methods. Liu, Chen and Tang [2002] also compared the performance in message recovery of three different configurations of chaotic semiconductor lasers: optical injection, optical feedback, and optoelectronic feedback systems. The best performance is obtained with the optoelectronic feedback system. This is because the conversion from optical to electronic signals reduces the channel noise in optoelectronic feedback system. However, laser noise (laser linewidth) saturates the BER to a value higher than 10−3 even in the optoelectronic feedback system, as shown in ACM of the right panel of fig. 62, because the laser noise directly causes fluctuations in the intracavity laser field. 3.6.2. Vertical-cavity surface-emitting lasers Several encoding and decoding schemes using vertical-cavity surface-emitting lasers (VCSELs) for chaotic communications have been investigated numerically. The possibility of frequency multiplexing in multitransverse-mode VCSELs has been pointed out (Yu, Shum and Ngo [2001]). A polarization encoding method using VCSELs has also been proposed and numerically investigated (Sciré, Mulet, Mirasso, Danckaert and San Miguel [2003]). Lee, Hong and Shore [2004] experimentally demonstrated VCSEL-based chaotic communications with a 200-MHz sinusoidal waveform. Further studies are expected for chaos communications with VCSELs. 3.6.3. Optoelectronic systems The synchronization schemes based on delayed optoelectronic feedback described in Section 2.5.6 were originally developed already with the aim of implementing secure optical communications. In his early works, Celka proposed and numerically demonstrated message encoding by chaos shift keying (Celka [1995]) and chaos masking (Celka [1996]) with optoelectronic systems implementing Mach– Zehnder interferometers. Goedgebuer, Larger and Porte [1998] (see also Larger, Goedgebuer and Delorme [1998]) performed one of the first experimental demonstrations of chaos communication based on the synchronization scheme shown in fig. 34, using message encoding via chaos modulation. This system can generate chaos with high Lyapunov dimension. Figure 63 shows an example of transmission and decoding of a 4-kHz square waveform with this system. The figure shows that the message cannot be extracted by simple inspection of the transmitted signal without synchronization of the receiver. Chaos shift keying with this synchronization scheme was also investigated, and the time delay in the transmitter feedback

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(b)

Fig. 63. Transmission and decoding of a 4-kHz square waveform. (a) Chaotic encrypted wavelength emitted by the transmitter (top trace) and its FFT spectrum (bottom trace). The message is masked with a −10 dB signal-to-chaos ratio. (b) Decoded signal at the receiver output (top trace) and its FFT spectrum (bottom trace) showing the fundamental and the harmonics. The signal-to-noise ratio is 12 dB. (From Larger, Goedgebuer and Delorme [1998].)

loop was shown to be the switching parameter assuring the best conditions for data recovery (Cuenot, Larger, Goedgebuer and Rhodes [2001]). More recently, Goedgebuer, Levy, Larger, Chen and Rhodes [2002] (see also Lee, Larger and Goedgebuer [2003], Kouomou, Colet, Larger and Gastaud [2005]) reported digital data transmission with a cryptosystem based on Mach– Zehnder modulators as shown in fig. 33 in Section 2.5.6. A bit error rate of 2.4 × 10−3 was measured for message transmission at 100 Mbits/s through a 50-km-long standard single-mode fiber, but it is expected that lower BER could be achieved by improving the quality of the RF amplifiers and electrooptics drivers. Kouomou, Colet, Larger and Gastaud [2005] analyzed the influence of parameter mismatch between the transmitter and receiver systems on the bit error rate in the same system shown in fig. 33. Figure 64 shows the experimentally recorded BER as a function of SNR at the standard bit rates of OC-24 (1.24416 Gbits/s) and OC-48 (2.48832 Gbits/s). Liu, Davis and Aida [2001] showed that the synchronization of chaotic mode hopping in DBR lasers with delayed optoelectronic feedback can be advantageously applied to multiplexed data transmission (see also Davis, Liu and Aida [2001]). For this purpose, several messages modulate light sources operating at the different wavelengths of the transmitter laser diode. The resulting optical data signals are injected in the feedback loop of the transmitter. At the receiver, the transmitted signal and the receiver output are filtered by several pairs of narrow-

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Fig. 64. Experimental variations of BER as a function of SNR at the standard bit rates of OC-24 (1.24416 Gbits/s) and OC-48 (2.48832 Gbits/s). (From Kouomou, Colet, Larger and Gastaud [2005].)

band filters set at the transmitter wavelengths. The different messages can then be extracted by subtracting the corresponding pairs of detected signals. Figure 65 shows an example of a two-channel data transmission based on the synchronization of chaotic hopping between seven modes with a bit rate of 0.5 Mbits/s per channel. 3.6.4. Fiber lasers In their theoretical analysis of chaos synchronization of Ikeda ring laser systems (Section 2.5.3), Abarbanel and Kennel [1998] already examined the possibility of using that property for communication purposes. As shown in eq. (2.47), the states of two fiber lasers can become synchronized even when an external signal ζ (t) is added to the transmitter field Et (t). If we interpret the external signal ζ (t) as a message, we conclude that the receiver is able to filter the added signal, by synchronizing to the unperturbed field of the transmitter Et (t) instead of the total transmitted signal Et (t) + ζ (t) (i.e., chaos modulation). In that case, subtraction of the receiver signal from the total transmitted signal allows the recovery of the message. As an example of the feasibility of their proposal, Abarbanel and Kennel [1998] used their communication scheme to model the transmission of a segment of speech, both analogic and digital, at frequencies on the order of hundreds of MHz. Shortly after the work of Abarbanel and Kennel [1998], Luo and Chu [1998] presented a model of chaotic communications in fiber ring lasers based on a standard single-mode rate-equation description of the laser dynamics. This approach is justified by its being limited to long time scales (much longer than the cavity

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(a)

(b) Fig. 65. Experimental multiplexed data transmission based on synchronization of chaotic hopping between seven modes. (a) Dashed line: a portion of NRZ pseudo-random sequence encoded on a first wavelength; solid line: corresponding recovered signal at the receiver. (b) Dashed line: a portion of NRZ pseudo-random sequence encoded on a second wavelength; solid line: corresponding recovered signal at the receiver. (From Liu, Davis and Aida [2001].)

round-trip time, on the order of tens of microseconds). Chaotic dynamics in the fiber-laser model was induced by modulation of the cavity losses. The coupling scheme proposed by Luo and Chu [1998] consisted of the partial injection of the transmitter output into the receiver ring cavity, without compensation of the selfinjection to keep the total power reinjected into the receiver constant. For that reason, synchronization is not complete (and decoding is not perfect). Two types of internal encoding were analyzed: injection of the message into the transmitter cavity and chaos shift keying affecting the cavity-loss modulation frequency. In both cases successful transmission of square waves was reported. Synchronization

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Fig. 66. Message recovery in the experimental setup of fig. 19. (A) Difference between the signals of photodiodes A and B before filtering. (B) The same signal after low-pass filtering (solid line); the dashed line represents the original message (as detected by photodiode A when the transmitter EDFA is turned off ). (C) Low-pass-filtered transmitter signal. (From VanWiggeren and Roy [1998a].)

was found to be maintained for parameter mismatches between the transmitter and receiver cavities of 5% or more (Luo and Chu [1998]). Several experimental demonstrations of chaotic communications in fiber lasers have been reported. VanWiggeren and Roy [1998a] demonstrated one of the first experiments for chaos communications, as shown in the left panel of fig. 19. The possibility to produce chaos synchronization was discussed in Section 2.5.3. A small-amplitude, 10-MHz square-wave message is introduced into the transmitter’s cavity via an output coupler. Under these conditions synchronization between the receiver’s output signal and the transmitter’s unperturbed signal (i.e. without the message) is preserved, so that subtracting the signals from photodiodes A and B (fig. 19) yields the message. Figure 66 shows the result of such a subtraction. In the absence of low-pass filtering (plot A) not much trace of the original square-wave message is to be found, because photodiodes only record intensities, so that the resulting signal corresponds to |Et (t) − m(t)|2 − |Er (t)|2 = 2 Re(Et m) + |m(t)|2 , given that Er (t) = Et (t). Low-pass filtering that signal difference reproduces the message ‘intensity’ |m(t)|2 , since the typical frequency of the chaotic carrier fluctuations (hundreds of MHz) is much higher than the message frequency (10 MHz). Figure 66(B) shows the result of such low-pass filtering (solid line), and compares it with the original message (dashed line). The good quality of the message recovery is evident. When a similar filtering is applied to the raw transmitted signal (as detected by photodiode A), the result shows no trace of the original message (plot C in fig. 66). As we have seen, the transmission bit rate in the experimental implementation described above is limited by the requirement that the message frequency be sufficiently smaller than the characteristic frequency of the chaotic carrier (i.e. chaos masking), so that message recovery can be performed by low-pass filtering of the subtracted signals. In order to overcome this limitation, VanWiggeren and Roy [1998b] modified their original experimental setup in such a way that the message

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Fig. 67. A second experimental implementation of chaotic communications in coupled fiber lasers. Left: experimental setup. Right, top to bottom: transmitted signal, signal after filtering by the receiver, and result of dividing the two. (From VanWiggeren and Roy [1998b].)

was injected in the transmitter by acting upon an intracavity intensity modulator, as shown in the left panel of fig. 67. This type of parametric encoding via chaos modulation requires decoding by division, instead of subtraction. Therefore, no low-pass filtering of the decoded signal is required, and the frequency limitation discussed above disappears. Successful transmission of a 126-Mbits/s pseudorandom digital message is observed, as shown in the right panel of fig. 67. In this new experimental setup, the structure of the transmitter and receiver were modified by the addition of a second fiber loop, thereby introducing the requirement of multiple parameter matching for extraction of the message. As shown in fig. 68, accurate recovery of the message requires multiple matched parameters in the receiver. The geometrical configuration of the receiver must be the same as in the transmitter. The lengths of the fiber in the outer loop and the

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Fig. 68. Recovery of the message requires that certain parameters in the transmitter be matched in the receiver. (A) Recovery of the message with all appropriate parameters matched. (B–F) Attempts at recovery with just one parameter mismatch: (B) geometrical configuration in the receiver lacking the outer loop; (C) receiver lacking the main-line part; (D) extra meter of optical fiber in the outer loop; (E) time delay mismatch between photodiodes A and B of only 1 ns (±20 cm of optical fiber); (F) amplitude of the chaotic lightwave in outer loop of receiver too large when recombined with the chaotic lightwave in inner loop of receiver. (From VanWiggeren and Roy [1999a].)

time-delay between photodiode A and B must be matched fairly precisely. Finally, the relative power levels in the receiver must be properly matched to the power levels in the transmitter. This method suggests that more complicated geometries and systems requiring additional parameters for message recovery may also be possible to construct using an EDFRL as a basic element.

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A very interesting analysis of data from this laser system was performed by Geddes, Short and Black [1999]. Encoded waveforms and decoded data were provided to them by VanWiggeren and Roy [1999a], together with a detailed description of the configuration and parameters for the laser system. Geddes, Short and Black [1999] showed that in the parameter regime when the double-loop geometry resulted in stable steady-state operation of the laser output without message modulation, they could construct an accurate computational model and recover the message encoded in the transmitted signal. In this case, the laser dynamics was governed almost entirely by the modulation signal which echoed in the two loops, and nonlinear effects could be neglected to a good approximation. VanWiggeren and Roy [1999a] also performed experiments with message injection at wavelengths which were not resonant with the lasing wavelength of the EDFRL. Figure 69 shows signals measured by photodiodes A and B when the wavelength of the injected message is 1533.01 nm. In this case the EDFAs were pumped at about 85 mW, far above threshold. This resulted in an optical power in the ring of ∼9.1 dBm without any message injection. The injected message power was ∼−3.1 dBm. The subtraction of the traces in fig. 69(A) is seen in fig. 69(B). Once again, the same pattern of bits is obtained. The optical spectrum (fig. 69(C)) shows two distinct peaks. The first peak (1533 nm) corresponds to the message injection, whereas the second peak (1558 nm) corresponds to the natural lasing wavelength of the EDFRL. The very broad linewidth is characteristic of the EDFRL. The message light at 1533 nm stimulates the EDFRL to emit at the same wavelength, and the fraction of the light that remains in the ring continues to circulate, stimulating additional emission. Consequently, the light detected at 1533 nm consists of a combination of the message itself and chaotic light produced by the EDFRL. If a bandpass filter was used to isolate the message wavelength (1533.01 nm), it was observed that the message was well obscured by the chaotic laser light. Figure 70 shows one of these measurements. The sequence of bits is not visible even after isolating just the message wavelength. This experiment indicates that wavelength division multiplexing may be possible, while still using a chaotic waveform as carrier. In summary, the message wavelength can be varied around the natural lasing wavelength of the EDFRL and chaotic communication can still occur. Bit-rates of 125 Mbits/s and 250 Mbits/s (for a nonreturn to zero waveform) were demonstrated by VanWiggeren and Roy [1999a]. Taking full advantage of the large bandwidth available in the optical system would permit even faster rates, but these experiments were limited by the bandwidth of their photodiodes (125 MHz 3-dB roll-off ) and oscilloscope (1 GS/s). The method works

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Fig. 69. (A) Transmitted signal measured by photodiode A (thin curve) and the signal measured by photodiode B (thick curve). (B) Result of subtracting the thick curve from the thin curve. (C) Optical spectrum showing the lasing wavelengths of the EDFRL. Message injection is at ∼1533 nm. (From VanWiggeren and Roy [1999a].)

well even over long communication channels (∼35 km), and in both ordinary and dispersion-shifted fibers. A more recent experimental demonstration of highspeed optical communications with erbium-doped fiber lasers was by Luo, Chu and Liu [2000].

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Fig. 70. The transmitted signal after passing through a 1-nm bandpass filter at 1.533 µm. The chaotic light at this wavelength still masks the message. (From VanWiggeren and Roy [1999a].)

3.6.5. Solid-state lasers The first numerical study on communications with synchronized chaotic lasers was carried out in a solid-state Nd:YAG laser model by Colet and Roy [1994]. In that pioneer work, a spiky chaotic carrier was generated from a loss-modulated Nd:YAG laser. To encode a digital message on the chaotic carrier, bits are encoded in the pulses, with only a single bit per pulse. Intensity maxima of the chaotic pulses are either increased or decreased according to a message bit of 1 or 0, as shown in fig. 71. The encoded chaotic waveform is transmitted to the receiver laser through a communication channel. The receiver laser synchronizes to the original chaotic signal rather than to the transmitted signal, so that the message can be recovered in real time by subtraction of the original signal from the transmitted signal. The message signal has to be small enough to achieve synchronization between the receiver laser and the transmitted signal in the presence of a perturbation, which corresponds in this case to encoding a bit (i.e. chaos masking). A more extensive study on encoding and decoding messages with chaotic lasers was carried out by Alsing, Gavrielides, Kovanis, Roy and Thornburg [1997]. They proposed three types of encoding schemes: (1) external modulation of intensity maxima of chaotic pulses, (2) pump modulation with a periodic waveform, and (3) pump modulation with a quasiperiodic waveform. In the first scheme, the mes-

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Fig. 71. Illustration of encoding and decoding the bit sequence “11001”: (a) original and encoded time series displaced from the original time locations by 2 µs for clarity, (b) receiver output, (c) detected difference, (d) value of the integral of (c). (From Colet and Roy [1994].)

sage can be deciphered by plotting the intensity–interspike-interval return map of the transmitter laser alone. In the second and third schemes, the message can be decoded by driving the receiver laser with the output of the transmitter laser and extracting the message from an integrated intensity difference. An experimental demonstration on chaotic communications in solid-state lasers has been reported by Uchida, Yoshimori, Shinozuka, Ogawa and Kannari [2001]. In that work, two synchronized Nd:YVO4 microchip lasers were used as transmitter and receiver, as shown in the left panel of fig. 72. The chaotic output of the microchip laser in the transmitter is externally modulated with an acoustooptic modulator. A digital message is encoded in the chaotic carrier by turning the external modulation on and off. The accuracy of chaos synchronization in the receiver is changed by the external modulation. One can determine the transmitted binary signals from the accuracy of synchronization. The digital sequence clearly appears, and the binary bits can be recovered according to a certain threshold value, as shown in fig. 72(e). This method may be classified as a mixture of chaos masking and chaos shift keying (or chaos on–off keying), since the digital message is externally encoded, while it is decoded by using the synchronization error. A chaos-masking method has also been numerically demonstrated with a microchip solid-state laser model by Ogawa, Uchida, Shinozuka, Yoshimori and

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Fig. 72. Left: experimental setup for signal transmission based on the chaos on–off keying method: BS, beam splitter; L, lens; M, mirror; VA, variable attenuator; LD, laser diodes; MCL, Nd:YVO4 microchip laser; PL, Peltier device; IC, optical isolator; PD, photodiode; PM, pump modulation. Right: (a) temporal waveform of the transmitter output with a digital message; (b) temporal waveform of the synchronized receiver output; (c) encoded digital message; (d) absolute value of the difference between two normalized laser outputs of (a) and (b); (e) decoded temporal waveform with a low-pass filter from the signal in (d). Note the different scales of the vertical axes in (a), (b) and (d). (From Uchida, Yoshimori, Shinozuka, Ogawa and Kannari [2001].)

Kannari [2002]. They used an acousto-optic modulator to encode a message by modulating the intensity of a chaotic laser output. At the receiver, one detects both the transmission signal and the synchronized output of the receiver laser. The message can be recovered by dividing the synchronized outputs of the receiver by the transmission signal, that includes both the chaos and message components. The condition for achieving successful decoding was investigated as a function of the message frequency. It turns out that a message can be extracted from the chaotic carrier at frequencies lower than the relaxation oscillation frequency of the microchip solid-state laser.

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3.6.6. Gas lasers There are few reports of chaotic communications with gas lasers, in contrast with the many existing observations of synchronization phenomena in those systems, as described in Section 2.5.1. One example of a communication scheme with gas lasers is described in this subsection. A method of information encoding in homoclinic chaotic systems has been proposed by Mariño, Allaria, Meucci, Boccaletti and Arecchi [2003] using a CO2 laser with optoelectronic feedback. Phase synchronization of chaotic systems was used in order to encode a message within the interspike-interval sequences of the laser, operating in the homoclinic chaos regime (Allaria, Arecchi, Di Garbo and Meucci [2001]). The information was encoded only in the time intervals at which spikes occur, and did not affect any geometrical property of the chaotic flow, thus resulting in a better performance against additive noise contamination in the transmission channel. The modulating external signal was applied on the bias voltage of the feedback loop in the CO2 laser by using a waveform generator. The interspike interval can be controlled within the domain of phase synchronization. For a modulation signal in the form of a train of square waveforms, fig. 73(a) shows that the phase of the temporal series of the laser intensity follows the external signal almost perfectly, with the laser spikes adjusting in each orbit to the corresponding external signal. A binary message can be encoded experimentally in the return period of the intensity spikes as follows. An external pulsed signal modulating the laser intensity is synthesized with an interpulse frequency that changes after each pulse according to a uniform frequency distribution centered at the mean repetition frequency, f0 . If the value of the frequency is larger than f0 , a 1 bit is encoded in the external signal; if the value is lower than f0 , a 0 bit is encoded in the external signal. Figures 73(b) and 73(c) show a short fragment of the experimental laser signal and the corresponding bits obtained after decoding at the receiver. The information decoding process can be carried out by observing the instant at which the laser intensity is higher than a threshold level and determining the bit by checking the pulse intervals. No synchronization is necessary between the transmitter and receiver, and the message is not masked by chaos in this method. Since the information is contained entirely in the timing between spikes, channel distortions that affect the pulse shape will not influence the information transmission significantly. Mariño, Allaria, Meucci, Boccaletti and Arecchi [2003] claimed that the performance of the information transmission can be improved due to the fact that information is encoded in the interspike interval, rather than in the amplitude of the transmitted signal. Their approach is similar to the chaos pulse-position modulation method proposed in electronic circuits (see Section 3.5.1). As another

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Fig. 73. (a) Experimental temporal evolution of a CO2 laser in the homoclinic chaos regime, when the laser intensity is driven by an external pulsed signal. (b) Portion of the laser output intensity containing the encoding of the binary conversion of the letter “T”. (c) Binary decoding according to a threshold level set at 75% of the peak height. (From Mariño, Allaria, Meucci, Boccaletti and Arecchi [2003].)

example of chaos communications using a gas-laser model, masked signal communication between two chaotic CO2 lasers, described by rate equations, was also implemented using analog circuits by de Moraes, de Oliveira-Neto and Rios Leite [1997].

3.7. Spatiotemporal communication All communication systems discussed so far involve serial transmission of data through a single communication channel, using temporal chaotic signals as information carriers. A generalization of this approach to systems with spatial degrees of freedom would enable the use of spatiotemporal chaos for the parallel transfer of information, which would yield a substantial increase in channel capacity. However, in spite of early studies on coupled map lattices (Xiao, Hu and Qu [1996], see also more recent papers by Lü, Wang, Li, Tang, Kuang, Ye and Hu [2004], Wang, Zhan, Lai and Hu [2004]), multichannel chaotic communication systems have not been implemented so far. Implementing a parallel chaotic communication scheme in electronic systems would be a complex task (due mainly to the need for a comprehensive extended coupling between transmitter and receiver). Optical systems, on the other hand, provide a natural arena for the parallel transfer of information, as we review in what follows. Multichannel chaotic communications have been recently proposed in models of multimode semiconductor lasers with optical feedback by White and Moloney [1999] (see also Buldú, García-Ojalvo and Torrent [2004]). In this case, only variations of the electric field along its propagation direction are considered. Informa-

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tion is encoded in the different longitudinal cavity modes, giving rise to a technique for multiplexing (see Section 3.9). Spatiotemporal communication, on the other hand, utilizes the inherent large-scale parallelism of information transfer that is possible with broad-area optical wavefronts. As in the previous systems considered, spatiotemporal chaotic communications require the existence of synchronization between transmitter and receiver. Synchronization of spatiotemporal chaos has been investigated extensively in arrays of nonlinear oscillators (Kocarev and Parlitz [1996]) and in model partial differential equations (Amengual, Hernández-Garcá, Montagne and San Miguel [1997], Kocarev, Tasev and Parlitz [1997]). García-Ojalvo and Roy [2001a] have proposed a communication system based on the synchronization of the spatiotemporal chaos generated by a broad-area nonlinear optical cavity. The setup is shown schematically in fig. 74. Two optical ring cavities are unidirectionally coupled by a light beam extracted from the left ring (the transmitter) and partially injected into the right-hand one (the receiver). Each cavity contains a broad-area nonlinear absorbing medium, and is subject to a continuously injected plane wave Ai . Light diffraction is taken into account during propagation through the medium, in such a way that a nonuniform distribution of light in the plane transverse to the propagation direction appears. In fact, an infinite number of transverse modes can oscillate within the cavity. In the absence of a message, the transmitter is a standard nonlinear ring cavity, well known to exhibit temporal optical chaos (Ikeda [1979]). When transverse effects due to light diffraction are taken into account, a rich variety of spatiotemporal instabilities appear, including solitary waves (McLaughlin, Moloney and Newell [1983]) and spatiotemporal chaos (Sauer and Kaiser [1996], Le Berre, Patrascu, Ressayre and Tallet [1997]). This latter behavior is what we are interested in here, since such chaotic waveforms can be used as information carriers.

Fig. 74. Scheme for communicating spatiotemporal information using optical chaos. CM is a coupling mirror. (Adapted from García-Ojalvo and Roy [2001a].)

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The propagation of light through the nonlinear medium can be described by the following equation for the slowly varying complex envelope En ( x , z) of the electric field (assumed to be linearly polarized) in the nth passage through the resonator (García-Ojalvo and Roy [2001a]): ∂En ( i 2 x , z) α(1 + i∆) = ∇ En ( x , z) − En ( x , z). ∂z 2k 1 + 4|En |2

(3.20)

The first term on the right-hand side of eq. (3.20) describes diffraction, and the second saturable absorption. The propagation direction is denoted by z, whereas x is a vector in the plane orthogonal to the propagation direction. Equation (3.20) obeys the boundary condition √ En ( (3.21) x , 0) = T A + R exp(ikL)En−1 ( x , ), which corresponds to an infinite-dimensional map. z = 0 in eq. (3.21) denotes the input of the nonlinear medium, which has length . The total length of the cavity is L. Other parameters of the model are the absorption coefficient α of the medium, the detuning ∆ between the atomic transition and cavity resonance frequencies, the transmittivity T of the input mirror, and the total return coefficient R of the cavity (fraction of light intensity remaining in the cavity after one round-trip). The injected signal, with amplitude A and wavenumber k, is taken to be in resonance with a longitudinal cavity mode. Previous studies by Sauer and Kaiser [1996] have shown that for ∆ < 0, the model (3.20)–(3.21) exhibits irregular dynamics in both space and time for A large enough. An example of this regime is shown in fig. 75(a). This spatiotemporally chaotic behavior can become synchronized to that of a second cavity, also operating in a chaotic regime, coupled to the first one as shown in fig. 74. The coupling mechanism can be modeled by (1) En(1) ( x , 0) = F (1) En−1 ( x , ) , (2) (1) x , 0) = F (2) (1 − c)En−1 ( x , ) + cEn−1 ( x , ) , En(2) (

(3.22)

where the application F (i) represents the action of the map (3.21) in every roundtrip. The coupling coefficient c is given by the transmittivity of the coupling mirror CM (fig. 74). The superscripts “(1)” and “(2)” represent the transmitter and receiver, respectively. Junge and Parlitz [2000] have shown that local sensor coupling is sufficient to achieve synchronization of spatiotemporal chaos in model continuous equations. In the optical model shown in eqs. (3.20)–(3.22), however, the whole spatial domain can be coupled to the receiver in a natural way.

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Fig. 75. (a) Space–time representation of a spatiotemporal chaotic state. (b) Intensity difference between the transmitter and the receiver; coupling is switched on at n = 100. (c) Temporal evolution of the synchronization error (see text) for the case shown in plot (b). Parameters common to the two cavities are α = 100.0, ∆ = −10.0, R = 0.9, T = 0.1, k = 100.0, = 0.01, L = 0.015, A = 7.0. In (b), c = 0.4. (From García-Ojalvo and Roy [2001b].)

Figure 75(b) shows a space–time representation of the intensity difference between the transmitter and the receiver before and after coupling between the two systems is activated (at the 100th round-trip). The initially uncoupled systems evolve in time starting from arbitrary initial conditions, and after 100 roundtrips, when their unsynchronized chaotic dynamics is fully developed, coupling is switched on, which results in a rapid synchronization. The synchronization efficiency can be quantified by means of the spatially averaged synchronization error defined by Kocarev, Tasev and Parlitz [1997] 2 (1) 1 En ( en = x , ) − En(2) ( x , ) d x, S S

(3.23)

where S is the size of the system. The temporal evolution of this quantity is shown in fig. 75(c) for three values of the coupling coefficient c. The model (3.20)–(3.21) has been numerically integrated in a 1D lattice of 1000 cells of size dx = 0.1 spatial units, using a pseudospectral code for the propagation equation (3.20). The results indicate that, for large enough coupling coefficient c, the synchronization error decreases exponentially as soon as coupling is switched on, with a characteristic time that increases with c.

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In order to encode and decode information in space and time, one can modify the scheme of eqs. (3.22) according to fig. 74, which leads to (1) En(1) ( x , 0) = F (1) En−1 ( x , ) + Mn−1 ( x) , (1) (3.24) (2) En(2) ( x , 0) = F (2) (1 − c)En−1 ( x , ) + c En−1 ( x , ) + Mn−1 ( x) . Upon synchronization between transmitter and receiver, the message can be decoded by simply subtracting the transmitted signal and the one coming from the (1) (2) receiver: M˜ n ( x ) = En ( x , ) + Mn ( x ) − En ( x , ). For identical parameters of the transmitter and the receiver (the situation considered so far), it can be seen analytically in a straightforward way that, as the coupling coefficient c tends to 1, (1) (2) the difference |En − En | → 0 ∀ x , which corresponds to perfect synchronization, and hence to perfect message recovery. It should be noted that the message is not merely added to the chaotic carrier, but rather the former is driving the nonlinear transmitter itself (i.e., chaos modulation). Therefore, the amplitude of the message need not be much smaller than that of the chaotic signal to provide good masking of the information (see Section 3.4). Figure 76 shows an example of data encoded and decoded using the scheme described above, where a static 2D image has been transmitted in space and time with a coupling coefficient c = 0.7. The input image is shown at left, the real part of the transmitted signal (a snapshot of it, in this case) is in the middle, and the recovered data is at right. The maximum message amplitude is 0.01 (this value should be compared to the maximum intensity of the chaotic carrier, which oscillates between 1 and 10, approximately, for the parameters chosen). Simulations in this case were performed on a square array with 256 × 256 pixels of width dx = 1.0. The image is clearly recognizable even though the coupling coefficient is now as low as 0.7.

Fig. 76. Transmission of a 2D spatiotemporal static image: (left) input image; (mid) real part of the transmitted signal at a certain time; (right) recovered data. Parameters are those of fig. 75(a), plus c = 0.7. (From García-Ojalvo and Roy [2001a].)

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Figure 76 shows qualitatively that, even though coupling between transmitter and receiver is not complete, information varying in time and space can be successfully transmitted and recovered with the setup described in fig. 74. In order to have a quantitative measure of this effect, one can estimate the mutual information between the input and output message signals, and its dependence on several system parameters. To that end, one defines the entropy H of the message and the mutual information I between the original and recovered messages as Hinput = −

x

p(x) ln p(x),

I =−

x,y

p(x, y) ln

p(x)p(y) , (3.25) p(x, y)

where x and y are the values of M and M˜ discretized in space and time, p(x) and p(y) are their corresponding probability distributions, and p(x, y) is the joint probability. The entropy Houtput of the recovered message is defined analogously to Hinput . Note that, according to its definition, the mutual information is I = 0 for completely independent data sets, and I = Hinput when the two messages are identical. Figure 77(a) shows the value of the mutual information I , versus the coupling coefficient c, for the transmission of a 1D message (García-Ojalvo and Roy [2001a]). As c increases, I grows from 0 to perfect recovery, corresponding to the entropy of the input image, given by the horizontal dashed line in the figure. This result shows that, even though good synchronization appears for c 0.4, satisfactory message recovery requires coupling coefficients closer to unity. This can also be seen by examining the behavior of the entropy H of the recovered image, plotted as open squares in fig. 77(a): for values of c substantially smaller than 1, the entropy of the recovered data is appreciably larger than that of the input message, indicating a higher degree of randomness in the former. Finally,

Fig. 77. Information-theory characterization of 1D message transmission. Solid squares: mutual information I ; open diamonds: entropy H of the recovered data; horizontal dashed line: entropy of the original image. Open triangles are the values of I in the presence of noise (see text). Parameters are those of fig. 75. (From García-Ojalvo and Roy [2001a].)

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the behavior of the mutual information in the presence of noise is shown as open triangles in fig. 77(a). Uncorrelated, uniformly distributed noise is added continuously to the communication channel, with an amplitude 1% that of the message (García-Ojalvo and Roy [2001b]). The discussion so far has considered identical parameters between emitter and receiver. In a realistic implementation, however, parameter mismatches will exist between both devices. A systematic study of the synchronization error between mismatched cavities shows that the most sensitive parameter in this respect is the amplitude A of the injected signal (García-Ojalvo and Roy [2001a]). The data plotted in fig. 77(b) indicate that a slight mismatch in the value of A degrades message recovery, by leading to values of I much smaller than the entropy of the input message, and to a recovered message with substantially larger entropy than the original. For other examples, experimental observations of lag and generalized synchronization of spatiotemporal chaos in liquid-crystal spatial light modulators have been reported by Neubecker and Gütlich [2004] and Rogers, Kalra, Schroll, Uchida, Lathrop and Roy [2004], respectively. Using this synchronization for communication purposes is the next logical step.

3.8. Polarization encoding Information for communication can be encoded on the amplitude, frequency (phase) or polarization of a light wave. While the first two options have been explored extensively for decades (Agrawal [1997]), techniques to encode information on the polarization state of light have been developed only recently. Techniques in which the state-of-polarization (SOP) of light is used to carry information include multiplexing (Evangelides, Mollenauer, Gordon and Bergano [1992]) and polarization shift-keying (Betti, De Marchis and Iannone [1992], Benedetto, Gaudino and Poggiolini [1997]). 3.8.1. The concept of polarization encoding with nonlinear dynamical systems A different method of polarization encoding has been developed by VanWiggeren and Roy [2002]. In their scheme, there is no one-to-one correspondence between the SOP of the lightwave detected in the receiver and the value of the binary message bit that it carries. Instead, the binary message is used to modulate parameters of a dynamical laser system; in this case, an erbium-doped fiber ring laser. The modulation generates output light from the laser with fast, irregular polarization fluctuations. This light propagates through a communication channel to a suitable

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receiver, which is able to detect changes in the transmitter’s polarization dynamics caused by the message signal and ultimately recover the message from the irregular polarization fluctuations of the transmitted light. Concepts related to those introduced by VanWiggeren and Roy [1998a], Goedgebuer, Larger and Porte [1998] can be used to demonstrate communication using the irregular polarization state fluctuations of light output by an EDFRL. When VanWiggeren and Roy [1998a] began to study polarization fluctuations in the EDFRL output, they did not find any available instruments to measure polarization state fluctuations on a nanosecond time scale. To quantitatively specify the polarization state of light, one has to measure the Stokes parameters for the beam (Azzam and Bashara [1989]). A novel high-speed fiber-optic polarization analyzer was then constructed for this purpose by VanWiggeren and Roy [1999b]. 3.8.2. Experimental apparatus The experimental apparatus used to demonstrate optical communication with dynamically fluctuating polarization states is shown in fig. 78 (VanWiggeren and Roy [2002]). The transmitter consists of a unidirectional EDFRL with a mandreltype polarization controller (P.C.) and a phase modulator within the ring. The polarization controller consists of loops of fiber that can be twisted to alter their net birefringence. The phase modulator comprises a titanium in-diffused LiNbO3 strip waveguide. Electrodes on either side of the waveguide induce a difference in the index of refraction between the TE and TM modes of the waveguide. In this way, it can also be used to alter the net birefringence in the ring. The length of the ring is about 50 m, which corresponds to a round-trip time for light in the ring of roughly 240 ns. This time delay in circulation of the light makes the dynamics observed in this type of laser quite different from those of more conventional cavities. A 70/30 output coupler directs roughly 30% of the light in the coupler into a fiber-optic communication channel while the remaining 70% continues circulating around the ring. The communication channel, consisting of several meters of standard singlemode fiber, transports the light to a receiver comprising two branches. Such fiber does not maintain the polarization of the input light. Instead, due to random changes in birefringence along the length of the fiber, the polarization state of the input light evolves during its journey. The receiver is designed to divide the transmitted light into two branches. Light in the first branch of the receiver passes through a polarizer before being detected by photodiode A (125 MHz bandwidth). Light in the other branch passes through a polarization controller before it is incident on a polarizer. After passing through the polarizer, the light is measured by photodiode B (also 125 MHz bandwidth). Signals from these photodiodes are

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Fig. 78. Apparatus for communication with dynamically fluctuating states of polarization. The transmitter consists of an EDFA, a polarization controller and a phase modulator. A time delay is present between the two photodiodes that is equal to the round-trip time for the light in the transmitter cavity. (From VanWiggeren and Roy [2002].)

recorded by a digital oscilloscope at a 1 GS/s rate. A crucial element of receiver operation is the time delay between the signals arriving at the photodiodes, which is equal to the round-trip time of the light in the transmitter laser. It is this dual detection with time delay that allows us to differentially detect the dynamical changes that occur in the transmitter ring and recover the sequence of perturbations that constitute the digital message. 3.8.3. Analysis of operation The electric field amplitude of a lightwave located just before the output coupler in the transmitter EDFRL (see fig. 78) at time t is given by the vector field E(t). As light propagates around the ring, the net action of the birefringence of the single-mode fiber of the ring and other elements can be represented by a single

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unitary 2 × 2 Jones matrix J T with complex elements. Therefore, E(t + τR ) = J T E(t).

(3.26)

It is assumed that the light field is not influenced significantly by noise or nonlinearities in the fiber or amplifier in a single round-trip, and that the polarizationdependent loss is negligible. The elements of J T can be changed by varying the voltage applied to the phase modulator in the EDFRL. In these experiments on communication, a data generator converts a binary message into a two-level voltage signal that is applied to the phase modulator. Thus, J T can take on two different values, JT = J0

or J T = J 1 ,

(3.27)

depending on whether a 0 or a 1 bit is to be transmitted. In this way, the phase modulator drives the polarization dynamics of the EDFRL transmitter. A fraction (30% in this experiment) of the light in the transmitter ring is coupled into a communication channel, which can be either free space or fiber optic. As mentioned previously, a standard single-mode fiber communication channel was chosen in the experiment of fig. 78. There are variations of birefringence in this fiber which change the polarization state of the input light significantly even over short distances of a few meters. The effect of this birefringence is represented by a Jones matrix, J C which is assumed to be a unitary 2×2 matrix with complex elements. In the receiver, half of the light is directed toward photodiode B. Before it is measured by the photodiode, it propagates through fiber, a polarization controller, and a polarizer. The net effect of the birefringence in the fiber and the polarization controller can be represented by the unitary Jones matrix J B . The elements of J B are completely controllable (with the constraint that the matrix is unitary) using the polarization controller. The effect of the polarizer is represented using another Jones matrix, P , though this matrix is nonunitary. Thus, the light actually measured by the photodiode in arm B of the receiver can be written as a concatenation of Jones matrices, P J B J C E(t).

(3.28)

However, the length of arm B is shorter than that of arm A to ensure that photodiode A measures a component of E(t + τR ) at the same time that photodiode B measures a component of E(t). Said in another way, the light measured at photodiode A was output from the transmitter one round-trip time, τR , after the light that is simultaneously measured by photodiode B. Thus, the field measured by photodiode A is P J A J C E(t + τR ) = P J A J C J T E(t).

(3.29)

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As mentioned earlier, when a 0 bit is to be communicated, the transmitter’s Jones matrix, J T , is set equal to J 0 by applying the proper voltage to the phase modulator. When J T = J 0 , synchronization of the signals measured by photodiodes A and B can be observed, provided that J B is set properly using the polarization controller in that arm of the receiver. Synchronization in this situation occurs when J BJ C = J AJ CJ 0.

(3.30)

It can be shown mathematically that this condition can always be satisfied by a proper adjustment of J B . For this proper setting of J B , synchronization is lost when the voltage applied to the phase modulator is switched so that J T is set equal to J 1 for communication of a 1 bit. Thus, by applying a binary voltage to the phase modulator in the transmitter, the signals measured by the photodiodes can be made to either synchronize or lose synchronization. The receiver interprets synchronized photodiode signals as 0 bits and unsynchronized signals as 1 bits (i.e., chaos shift keying). 3.8.4. Robustness to birefringence perturbations Communication using this technique offers an unanticipated benefit compared with polarization shift keying. In fiber-optic communication channels, variations in temperature or stress in the channel induce fluctuations in the local birefringence of the channel that ultimately cause the channel Jones matrix, J C , to evolve in time. The evolution of J C in typical commercial channels has been observed over tenths of seconds. Using the polarization shift keying technique, this evolution leads to ambiguity at the receiver which is harmful to accurate message recovery. The receiver, in other words, cannot precisely determine the transmitted SOP from the received SOP. Complicated electronic tracking algorithms have been developed to compensate for this channel evolution in polarization-shift keying schemes. In the technique presented here, no such complicated algorithms are necessary. If the transmitter Jones matrix for a 0 bit is chosen to be equal to the identity matrix, it can be easily shown that a proper choice of J B can recover the message perfectly even when J C is evolving in time. In this way, information can be transmitted through a channel with fluctuating birefringence, and a receiver can recover the information without any compensation for the channel’s fluctuating birefringence. Conceptually, this is possible because, as mentioned above, there is no one-to-one correspondence between the polarization state of the transmitted lightwave and the value of the binary message that it carries. In the method pre-

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sented here, the message is encoded in the polarization dynamics (driven by the phase modulator) rather than in particular polarization states. 3.8.5. Experimental results Figure 79 shows experimental results demonstrating the communication technique described in Section 3.8.3 (VanWiggeren and Roy [2002]). The message,

Fig. 79. Demonstration of message recovery using dynamical fluctuations of SOP: (a) modulation voltage applied to the phase modulator; (b) transmitted signal measured by photodiode A; (c) time-delayed signal measured by photodiode B; (d) message recovered by subtraction of the data in (c) from that in (b). Loss of synchronization corresponds to 1 bits, while synchronization represents 0 bits. (From VanWiggeren and Roy [2002].)

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the binary voltage signal shown in fig. 79(a), drives the phase modulator as described above. It consists of a repeating sequence of 16 bits transmitted at 80 Mbits/s. At this rate, roughly 19 bits can be transmitted during the time light takes to complete one circuit around the ring. A completely random and nonrepeating sequence of bits can also be transmitted successfully, but the repetitive message signal provides some additional insights into the dynamics of the transmitter. Figure 79(b) shows the signal measured by photodiode A, IA , in the receiver. Despite the repeating nature of the message, the measured signal does not show the same periodicity. Figure 79(c) shows the signal IB measured simultaneously by photodiode B. As mentioned above, photodiode B measures the intensity, with a time delay, of a different polarization component of the same signal measured by photodiode A. A subtraction of IB from IA is shown in fig. 79(d). A 0 bit is interpreted when the subtracted signal is roughly 0 because this value corresponds to synchronization of the signals measured by the two photodiodes. A 1 bit is interpreted when the difference signal is nonzero. It is found that the transmitted message bits are accurately recovered by the receiver. Figure 80 shows the polarization fluctuations measured during transmission of the above message. The high-speed fiber-optic polarization analyzer constructed for this purpose by VanWiggeren and Roy [1999b] was used to make these observations and shows clearly how the polarization state of the light fluctuates on the Poincaré sphere as the message is transmitted. 3.8.6. Limitations on bit rate The bit rate is limited eventually only by the time scale of the polarization fluctuations which is of the order of a hundred femtoseconds or shorter. Measurements with a polarization analyzer clearly show the irregular polarization and intensity dynamics of the transmitted lightwaves. This technique demonstrates the use of vector properties of a dynamical system, driven by the message, for optical communication. A recent application of this technique to a model of VCSEL polarization dynamics is by Sciré, Mulet, Mirasso, Danckaert and San Miguel [2003]. The phase of the vectorial field is modulated. The total intensity of these lasers thus remains synchronized while the intensities in the polarization modes (de)synchronize following the phase modulation at a picosecond time scale. This technique allows for data transmission at high bit rates that are not limited by the relaxation oscillation frequency.

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Fig. 80. Polarization dynamics measured during message transmission: (a) polarization dynamics of the transmitter plotted on a Poincaré sphere; (b) degree of polarization (DOP) of the light output from the transmitter; (d–f ) normalized Stokes parameters showing fluctuations in the SOP of the transmitted light. (From VanWiggeren and Roy [2002].)

3.9. Multiplexing Although many schemes of chaotic communications have been proposed, as described in the previous sections, most of the studies on chaotic communications are limited to one-to-one communications using one pair of chaotic lasers. A scheme for multi-user communications using chaos may be important for more sophisticated communication applications (Tsimring and Sushchik [1996], Yoshimura [1999], Liu and Davis [2000b]).

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Several reports on multiplexed communications using optical chaos have been reported. Messages can be multiplexed on each mode of a multimode semiconductor laser (White and Moloney [1999], Buldú, García-Ojalvo and Torrent [2004]) and a multimode Nd:YAG laser (Wu and Zhu [2003]). Subcarrier multiplexing by chaotic multitone modulation was investigated in a self-pulsating laser diode by Juang, Huang, Liu, Wang, Hwang, Juang and Lin [2003]. Time-division multiplexing using spatiotemporal dynamics was reported by Joly, Derozier, Razdobreev and Bielawski [2003]. Paul, Sivaprakasam and Shore [2004] demonstrated experimentally dual-channel chaotic communications in external-cavity semiconductor lasers. Multiplexing in mode-hopping semiconductor lasers was experimentally achieved by Liu, Davis and Aida [2001], Davis, Liu and Aida [2001] (see Section 3.6.3). Chaotic waveforms could be useful as subcarriers in wavelength-division multiplexing (WDM) in multiuser optical communications. WDM using chaos as a subcarrier (called chaotic wavelength division multiplexing, CWDM) was demonstrated experimentally and numerically in two pairs of one-way coupled Nd:YVO4 microchip lasers by Matsuura, Uchida and Yoshimori [2004]. The experimental setup for the CWDM scheme is shown in the left panel of fig. 81. Four Nd:YVO4 microchip crystals were used, where two of the microchip lasers played the role of transmitters (referred to as T1 and T2) and the other two lasers were used as receivers (R1 and R2). Chaotic pulsations can be obtained in the setup shown in fig. 81 by sinusoidally modulating the injection current of the laser diodes pumping the transmitters. Two optical isolators were used to achieve one-way coupling. Two individual digital messages were encoded on the chaotic laser outputs of T1 and T2 by using two acousto-optic modulators (AOM) in front of the transmitters (chaos masking). The two laser beams of T1 and T2 with the two messages were then mixed at a beam splitter and were propagated through one transmission channel in free space. This combined signal was injected into the two laser cavities of R1 and R2 with precise temperature control. Synchronization can be achieved individually in two pairs of lasers by using injection locking (called dual synchronization (Uchida, Kinugawa, Matsuura and Yoshimori [2003])). Two wavelength filters were used to separate the two different wavelengths of T1 and T2. The individual transmission signal consisting of T1 or T2 with the corresponding message was detected by photodiodes and a digital oscilloscope, together with the synchronized chaotic waveform from the corresponding receiver. The message signal was recovered by subtracting the synchronized chaotic signal from the individual transmission signal and by applying a low-pass filter.

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Fig. 81. Left: experimental setup for chaotic wavelength division multiplexing using microchip lasers. AOM, acousto-optic modulator; BS, beam splitter; IS, optical isolator; L, lens; LD, laser diode; M, mirror; MCL, Nd:YVO4 microchip laser; PD, photodiode; PM, pump modulation; S, subtractor; VA, variable attenuator; WF, wavelength filter. Right: experimental results: (a, b) temporal waveforms of the transmitter output with the digital message and the synchronized receiver output; (c, d) temporal waveforms of the difference between the transmitter and receiver outputs shown in (a) and (b); (e, f ) temporal waveforms of the original message and the decoded message obtained by filtering the signals in (c) and (d) with a low-pass filter. (a, c, e) Transmitter 1, Message 1, and Receiver 1. (b, d, f ) Transmitter 2, Message 2, and Receiver 2. (From Matsuura, Uchida and Yoshimori [2004].)

The experimental temporal waveforms of the chaotic laser outputs and decoded message signals are shown in the right panel of fig. 81. The chaotic outputs of the transmitters (including messages) and the synchronized outputs of the receivers are displayed in the figs. 81(a) and 81(b). The message components of the square waveforms cannot be seen in these temporal waveforms. When the two laser outputs are normalized and one is subtracted from the other, the messages can be obtained as an envelope of chaotic oscillations, as shown in figs. 81(c) and 81(d). The original messages can be recovered by filtering the difference between the two outputs (figs. 81(c), 81(d)) with a low-pass filter, as shown in the lower traces of figs. 81(e) and 81(f ). Compared with the original square waveforms shown in the upper traces of figs. 81(e) and 81(f ), the digital sequences are successfully recovered and the binary bits can be detected by introducing a certain threshold value.

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§ 4. Summary 4.1. Motivations and achievements In this last section we briefly summarize achievements in communication that are based on the synchronization of chaotic optical systems. The last few years have witnessed intense activity in the field of chaotic optical communications, and there is active ongoing research at many laboratories around the world. Some major projects involving the collaboration of several laboratories and groups (the OCCULT – Optical chaotic communication with laser-diode transmitters – program in Europe and the MURI – Multi-University Research Initiative – group in the USA) have been engaged in research on chaotic communication for several years. The major breakthroughs accomplished so far can be summarized in the following points: • Synchronization of chaotic lasers has been proposed and demonstrated experimentally for several different types of lasers, including those most applicable to communication, such as semiconductor and fiber lasers. • Communication through the use of chaotic waveforms as carriers of information has been demonstrated by several groups, both in free space and through fiber channels. Information signals transmitted and received include digital and analog waveforms, and several different encoding and decoding schemes have been proposed and demonstrated. • Researchers have aimed at achieving higher bit rates (of 1–10 Gbits/s) with semiconductor and fiber laser systems (Liu, Chen and Tang [2002], Kusumoto and Ohtsubo [2002], Fischer, Liu and Davis [2000], Paul, Sivaprakasam, Spencer, Rees and Shore [2002], Luo, Chu and Liu [2000]). Generation and synchronization of chaotic oscillations at tens of GHz ranges has been reported by Takiguchi, Ohyagi and Ohtsubo [2003], Uchida, Heil, Liu, Davis and Aida [2003], Genin, Larger, Goedgebuer, Lee, Ferriére and Bavard [2004], showing the potential of high-speed chaotic communications. • Recent papers have examined the role of the fiber and free-space communication channels and respective constraints for chaotic communications (Zhang and Chu [2003], Kanakidis, Argyris and Syvridis [2003], Argyris, Kanakidis, Bogris and Syvridis [2004]), and the use of new devices such as VCSELs (Lee, Hong and Shore [2004]).

4.2. Short-term goals Several short-term goals have emerged from the research reviewed here:

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• The consensus appears to be that fundamental limitations to communication bit rates lie in the range of tens of Gbits/s, and the major issues to be studied in the future are bit error rates and error-control methods for different schemes of chaotic communication, since channel noise is unavoidable in any communication scheme. Low bit error rates are currently achieved and can be expected to be lowered in the near future. They do not represent a fundamental limitation of the technology. • A major area of research that will develop in the future will be that of multiplexed communication. Some efforts have already been made in this direction (see Section 3.9). A possible goal is to develop multiplexing techniques based on dynamical characteristics of signals, in addition to wavelength/frequency or time-division techniques. • The use of chaotic communication techniques to overcome perturbations of the communication channel is presently attracting attention (Rulkov, Vorontsov and Illing [2002], VanWiggeren and Roy [2002]). This could be a major area for advances using chaotic communication methods. • Compatibility between chaotic analog communications and conventional digital communications is an important issue to solve. • A remaining open question concerns the use of generalized synchronization for communication (Terry and VanWiggeren [2001]). Generalized synchronization has been demonstrated for laser systems (see Section 2.6.2) and spatiotemporal optical systems such as liquid-crystal spatial light modulators (see Section 3.7), but the utilization of this phenomenon for transmission and recovery of information remains to be demonstrated experimentally. • Developing a deeper understanding of security and privacy issues related to chaotic communications is necessary for applications in the field. Recent advances in this direction have been made by Lü, Wang, Li, Tang, Kuang, Ye and Hu [2004], Wang, Zhan, Lai and Hu [2004]. With regard to the last point, one of the original short-term motivations for research in chaotic communications arose from the ongoing need for methods of communication offering privacy and possibly security for information transfer. This motivation is not new; in fact methods for coding, or hiding, information have been developed for many centuries. The notion that one can hide information in a background of chaotic waveforms is therefore a very natural application originating from the new understanding of chaotic dynamical systems developed over the last two decades. It has, however, been very difficult to assess the security or privacy offered by any specific schemes for chaotic communication, electronic or optical. In particular, a meaningful comparison is still lacking between the security levels offered by chaotic encryption and those of standard software-

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based cryptography. The main reason for this is that the security measures introduced to quantify the efficiency of software-based encoding are too much tailored to the key-distribution process, which is something that chaotic communication schemes have not addressed so far. Moreover, chaotic waveforms provide an additional layer or level of privacy beyond any conventional key-distribution or encryption scheme that can be simultaneously part of the communication protocol. The natural evolution of research in this direction has thus been anecdotal to a great extent. Researchers have published papers with schemes for ‘secure’ communication without detailed consideration as to what the actual level of security for these methods might be; the titles of many of the papers quoted in this review provide examples of this approach. Others have claimed that chaotic communication schemes are not secure, with attention focused on very specific classes of models or techniques for chaotic communications (Pérez and Cerdeira [1995], Short [1994, 1996], Short and Parker [1998], Geddes, Short and Black [1999]). This cycle of making and breaking codes is perhaps unavoidable, and much can be learned from both these approaches, anecdotal as they are. It is hoped that eventually there will be comprehensive research on the security of methods of communication that use chaotic dynamical systems.

4.3. Longer-term perspective and open questions From our point of view, there is a longer-term perspective regarding chaotic communication that needs to be emphasized. In the development of traditional communication techniques, we have learned how to use sinusoidal waveforms as carriers of information, through the use of amplitude- or frequency-modulation techniques. Receivers for such communication use a single parameter to discriminate between different transmitters; this is the frequency or wavelength of the sine wave carrier. The receiver typically possesses a tuned circuit that synchronizes to the carrier frequency, and the modifications of the carrier sine waveform that are the information content of the message are then extracted by a suitable demodulation technique. This basic approach has been developed for a century with great success. It is also very clear that nature does not use such methods for communication, particularly in biological systems. If one asks how information is encoded in the electrical signals that propagate in biological systems, it becomes clear that nonsinusoidal, spiky waveforms are often used for conveying information. The need for highly precise, synchronized clocks that we use in modern communication systems is also avoided in biological systems.

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The basic question then becomes: can we discover alternate methods of communication that use irregular waveforms as carriers of information and explore such schemes to investigate new methods for communication that do not rely exclusively on sine waves and precise clocks, but which utilize more adaptive and versatile approaches to encoding and decoding information? We have seen that this question has already been answered positively (albeit to a very limited extent). Synchronization of chaotic dynamical systems offers us such possibilities. A multidimensional parameter space for transmitter–receiver synchronization with complex waveforms may allow the channel capacity for communications to be more fully utilized without the need for high-precision synchronization. New ways of multiplexing information based on the dynamical properties or characteristics of the carrier waveforms may emerge, and a fuller utilization of the degrees of freedom available in electromagnetic waves could be developed in the future. From the studies of synchronization reviewed here (see Section 3.6, for example), we found that a receiver can synchronize to a chaotic temporal waveform, given appropriate matching of parameters to the transmitter, and that there may be two or more parameters that need to be tuned for successful decoding of a message. Thus, complex temporal carrier waveforms can be used to transmit and receive information, and synchronization is a natural process for extraction of the information. In Sections 3.7 and 3.8 we found that this process of synchronization could be extended for optical systems to spatiotemporal waveforms and include vector dynamical variables such as polarization of light waves. These studies indicate that the degrees of freedom available for communication with electromagnetic waves may be more fully used, and exploring the dynamics of coupled systems is an important area for investigation for such applications. It is very natural to ask at all stages of such research – what is the possible advantage of such methods of communication over the more traditional methods that have been developed over the past century and found to be so effective and useful? While this is a very important question, we must recognize the need for research that attempts to discover qualitatively new ways of communicating information. Such attempts will hopefully lead to an understanding of basic concepts that underlie communication in biological systems, and eventually may lead to schemes that have useful applications in other specific contexts.

Acknowledgements We thank all our colleagues with whom we have, over the years, collaborated on the fascinating topic of synchronization and communications using optical chaos.

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R. Roy in particular recognizes the key contributions of K. Scott Thornburg, Jr. and Gregory D. VanWiggeren. A. Uchida acknowledges the Japan Society for the Promotion of Science for support. F. Rogister acknowledges support from the Fonds National de la Recherche Scientifique and the Interuniversity Attraction Pole program 5-18 of the Belgian Science Policy. J. García-Ojalvo is financially supported by the Ministerio de Educación y Ciencia (Spain, project BFM200307850), by the EC project OCCULT (IST-2000-29683) and by the Generalitat de Catalunya. R. Roy acknowledges support from the National Science Foundation and the Physics Division, the Office of Naval Research, USA.

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Author index for Volume 48 Arecchi, F.T., 209, 217, 232, 264, 265, 307, 308 Argyris, A., 219, 288, 290, 291, 324 Argyros, A., 22 Ashcon, J.B., 4, 18 Ashoka, V.S., 167 Ashwin, P., 233, 235 Astle, H.W., 4 Atkin, D.M., 3, 5 Aubry, S., 40, 90 Austin, R.H., 22 Azzam, R.M.A., 315

A Abarbanel, H.D.I., 210, 212, 213, 217, 219, 240–243, 255, 269, 271, 282, 283, 292, 297 Abdullaev, F.Kh., 38–40, 50, 51, 53, 55–57, 59–61, 68, 69, 78, 80–83, 89, 90, 93, 94 Abdullaev, S.S., 90 Abdumalikov, A.A., 51, 80 Abeeluck, A.K., 5 Abel, A., 281–283 Ablowitz, M.J., 89, 95, 96 Abraham, N.B., 213, 214, 257 Aceves, A.B., 50, 65, 66, 94 Afraimovich, V.S., 215, 220 Agarwal, G.S., 152 Agrawal, G.P., 50, 63, 70, 209, 239, 240, 314 Aharonov, Y., 151, 164 Ahlborn, A., 213 Ahlers, V., 227, 251, 286 Ahmad, F.R., 5 Aida, T., 227, 251, 252, 255, 261, 287, 296, 298, 322, 324 Akagi, Y., 77, 78 Akhmediev, N., 61 Akimoto, O., 219, 241, 258 Allan, D.C., 5 Allaria, E., 232, 264, 265, 307, 308 Alsing, P.M., 304 Amengual, A., 309 Anandan, J., 151, 164 Anderson, D., 47 Andreev, V.A., 168 Andreev, V.V., 168 Andrekson, P.A., 39, 78 Anishchenko, V.S., 225 Ankiewicz, A., 61 Annovazzi-Lodi, V., 245, 246, 252, 285, 286 Antaramian, A., 153, 154 Antonopoulos, G., 5 Aravind, P.K., 158

B Baba, N., 176, 195–197 Babcock, K.J., 255 Baek, Y., 39, 79 Baggett, J.C., 3 Baizakov, B.B., 38, 39, 51, 61, 68, 80, 93, 94 Baker, S.M., 38 Balakrishnan, V., 173 Bang, O., 40, 80, 83 Barbay, S., 268 Barkou, S.E., 3 Barnes, T.H., 188, 189 Barron, R.J., 189 Barsella, A., 232 Bartelt, H., 40, 90, 94 Barthelemy, B., 79 Bashara, N.M., 315 Baskar, S., 118 Bassett, I.M., 111, 121 Bauer, R.G., 51 Bauer, S., 253 Bauman, I., 39, 79 Bavard, X., 324 Becker, P.C., 239 Belardi, W., 3 Beloglazov, V.I., 7 Belostotsky, A.L., 118, 119 343

344

Author index for Volume 48

Ben-Aryeh, Y., 173 Benabid, F., 5 Benedetto, S., 314 Bennett, J.V., 255 Bennion, I., 38 Benoit, G., 6 Bergano, N.S., 38, 314 Bergé, P., 213 Berntson, A., 39, 69 Berry, M.V., 151, 153, 156, 157, 163, 167, 168, 173, 174 Betti, S., 314 Bhagavatula, V.A., 3, 4 Bhandari, R., 151, 158, 159, 161, 166–168, 173, 181 Bielawski, S., 322 Biener, G., 164–166 Biondini, G., 74 Birks, T.A., 3, 5–7, 12 Bischoff, S., 57, 59 Biswas, A., 38 Bjarklev, A., 3 Black, K., 302, 326 Blackburn, K.L., 255 Blass, W.E., 213 Blekhman, I.I., 215, 220 Bliokh, K.Yu., 152 Bliokh, Yu.P., 152 Blondel, M., 252–255, 291, 292 Blow, K.J., 38 Boccaletti, S., 213, 217, 220, 224, 232, 265, 307, 308 Bogris, A., 219, 288, 291, 324 Bohr, R., 194, 195 Bokor, N., 112–115, 117, 121–123, 125, 127–130, 133–136, 138–140, 146 Bolger, J.A., 3, 12 Bomzon, Z., 164, 165 Born, M., 73, 112, 137, 157, 264 Botten, L.C., 3 Bourliaguet, B., 79 Bouwmans, G., 235 Boyle, W.S., 205, 207 Bracewell, R.N., 194 Breban, R., 263–266 Brendel, J., 171, 172 Brentel, J., 77 Brock, J., 229 Broderick, N.G.R., 3 Broeng, J., 3

Bronski, J.C., 40, 51, 53, 55, 56, 60, 93 Brown, N., 169 Brown, R., 269 Brown, T.A., 181 Buhl, M., 219, 240, 271 Buldú, J.M., 228, 248, 251, 252, 288, 308, 322 Buryak, A.V., 39, 46 C Calatroni, J., 192 Cao, H., 22 Caputo, J.G., 38, 51, 68 Carr, T., 233 Carroll, T.L., 213, 215, 216, 220–222, 273 Casademont, J., 254 Cash, Jr. C.C., 143, 206, 215, 228, 233–235, 282, 283, 304, 314 Cavalcante, H.L.D.S., 232 Celka, P., 259, 260, 295 Cerdeira, H.A., 218, 286, 326 Chandrasekhar, S., 174 Chen, C.-C., 259, 260, 296 Chen, G.R., 244 Chen, H.F., 246, 247, 252, 255, 279, 286, 287, 292–295, 324 Chen, H.H., 38 Chen, Y., 38, 39, 76, 78, 167 Chen, Z., 94 Cheng, Y.-Y., 183 Cherel, L., 189 Chern, J.-L., 235, 236, 245, 255 Chertkov, M., 38, 39, 49, 51, 52, 56–58, 68, 70, 73 Cheung, D.C.L., 189 Cheyne, M.R., 155 Chiang, M.C., 256 Chiao, R.Y., 151–154, 168–170 Chim, S.S.C., 185 Chin, G., 213 Choi, M., 267, 268 Choi, S., 7 Chou, S.Y., 22 Christian, J., 151 Christiansen, P.L., 40, 80, 83, 91 Christodoulides, D.N., 94 Chu, P.L., 219, 242, 244, 297–299, 303, 324 Chua, L.O., 217, 218, 225, 273, 281, 282 Chung, Y., 38, 49, 51, 52, 56, 57 Chyba, T.H., 158–161

Author index for Volume 48 Ciddor, P.E., 179–182, 184 Clausen, C., 40, 80, 83 Cogswell, C.J., 183 Cohen, F., 185 Colavita, M.M., 194 Colet, P., 210, 218, 219, 225, 228, 233–235, 248, 252, 257, 260, 286, 287, 296, 297, 304, 305 Conroy, R.S., 29 Constantini, B., 79 Content, D., 143 Cooper-White, J., 3 Corron, N.J., 283 Cox, F., 22 Crane, R., 178 Crawford, P.R., 152 Creath, K., 183 Cregan, R.F., 5 Croft, T.D., 3, 4 Crutchfield, J.P., 211 Cuenot, J.-B., 261, 296 Cuomo, K.M., 216, 218, 273, 275–278, 281, 282 D Dachselt, F., 281 Daido, H., 219, 241 Danailov, M.B., 232 Danckaert, J., 295, 320 Dangoisse, D., 232 Darmanyan, S.A., 38, 40, 50, 80–83 Dasgupta, T., 161 Davidson, M., 185 Davidson, N., 112–115, 117, 121–123, 125, 127–130, 133–136, 138–140, 143–146 Davila, P., 143 Davis, J., 192, 193 Davis, P., 219, 227, 249–252, 255, 261–263, 287, 288, 290, 296, 298, 321, 322, 324 De Angelis, C., 79, 94 de Lignie, M.C., 76 De Marchis, G., 314 de Moraes, R.M., 308 de Oliveira, P.C., 232 de Oliveira-Neto, L.B., 308 De Vito, E., 158, 173 De-Xiu, H., 94 Del Río, E., 225 Delisle, C., 178, 179, 189

Delorme, F., 210, 259, 261, 295, 296 deMars, G., 238 Deparis, O., 254, 255, 291, 292 Derozier, D., 322 DeShazer, D.J., 233, 235, 240, 263–267 deSterke, C.M., 3 Destriau, M.G., 176, 180 Desurvire, E., 239 Di Garbo, A., 232, 264, 265, 307 DiGiovanni, D.J., 3 Di Trapani, P., 39, 46 Ding, E.-J., 286 Ding, M., 286 Ditto, W.L., 213, 216, 286 Doktorov, E.V., 38, 61, 62 Domachuk, P., 3, 7, 11, 26 Donati, S., 245, 246, 252, 284, 285, 286 Doran, N.J., 38 Dorn, R., 138 Dresel, T., 186 Duban, M., 142 Dukes, J.N., 158 Dultz, W., 155, 166–168, 171, 172, 178 Dyachenko, A., 38, 49, 51, 52, 56, 57 Dykstra, R., 232, 269, 270 E Eckert, K., 218, 225, 273, 281, 282 Eggleton, B.J., 3, 5, 7, 11, 12, 26 Elgin, J.N., 38, 56 Elsäßer, W., 219, 228, 249, 253, 287 Endo, T., 225 Erneux, T., 232, 233, 235, 253 Erzgräber, H., 253 Escalona, R., 192 Etrich, C., 39, 46 Evangelides, S.G., 38, 68, 314 F Fabiny, L., 233, 234 Fairchild, R.C., 109 Falkovich, D.E., 38, 56 Farini, A., 217 Farmer, J.D., 211 Fedotov, A.B., 7 Feng, Q., 256 Ferriére, R., 324 Fibich, G., 40, 92 Fienup, J.R., 109

345

346


Figueras, M., 229 Fink, Y., 6 Fischbach, M.A., 29 Fischer, I., 219, 228, 249–251, 253, 287, 288, 290, 324 Fisher, R.L., 118 Flytzanis, N., 38, 51 Forsmann, B., 233 Franco, P., 38 Frankena, H.J., 117 Frantzeskakis, D.J., 40 Freeman, M.O., 181 Friesem, A.A., 121–124, 143–146 Frins, E.M., 155, 167, 168 Fuerst, R.A., 79 Fujino, H., 249 Fujisaka, H., 215, 220 Fujiwara, N., 257, 258 Furusawa, K., 3 G Gabitov, I., 38, 39, 49, 51, 52, 56–58, 65, 68, 70, 73 Gabor, D., 264 Gaeta, A.L., 5 Gaididei, Yu.B., 40, 91 Galimzyanov, R.M., 40, 93 Gallagher, M.T., 5 Galtarossa, A., 76 Galvez, E.J., 152, 155 Ganga, K.M., 153, 154 Gao, Y., 167 García-Fernández, P., 218, 219, 248, 286, 287 García-Ojalvo, J., 210, 219, 228, 235, 240, 248, 251–254, 288, 308–314, 322 Garnier, J., 39, 40, 70–72, 85, 86, 89, 90, 93 Garstecki, P., 29 Gastaud, N., 260, 296, 297 Gattass, R., 4, 18 Gaudino, R., 314 Gavrielides, A.T., 245, 253, 255, 284, 304 Gaylord, T.K., 138 Geddes, J.B., 302, 326 Geffert, H.P., 213 Genin, É., 324 Giacomelli, G., 268 Gianello, G., 76 Gibbs, H.M., 258

Gisin, N., 73 Glorieux, P., 232, 235 Gluckman, B., 286 Göbel, E., 219 Gobel, W., 5 Goedgebuer, J.-P., 210, 219, 245, 259–261, 284, 295, 296, 315, 324 Gong, L., 167 Gong, S.-H., 267, 268 Gordon, G.B., 158 Gordon, J.P., 37, 38, 51, 56, 68, 74, 314 Grange, R., 142, 143 Grasserger, P., 213 Grebogi, C., 213, 217, 283, 284 Gredeskul, S., 40 Grillet, C., 3 Gripp, J., 50 Gu, M., 3, 7, 11, 26 Guerrero, A.L., 192 Gütlich, B., 314 H Hagan, D.J., 79 Haglin, P.J., 152 Haken, H., 206, 208 Hale, A., 3 Halle, K.S., 217, 218, 273, 281, 282 Hamilton, M.W., 232, 269, 270 Harada, T., 142 Harada, Y., 142, 143 Hardy, G.J., 194 Hariharan, P., 158, 161–163, 168, 169, 173–184, 186–188, 190, 191, 193, 194 Harrington, J.A., 7 Harrison, R.G., 268 Hart, S.D., 6 Hasegawa, A., 37, 50, 59, 77, 78, 84, 94 Hasman, E., 121, 143–146, 164–166 Haus, H.A., 37–39, 41, 56, 76, 78 Hausler, G., 186 Hayes, S., 217, 283, 284 He, S., 4, 18 Headley, C., 5 Heckenberg, N.R., 232, 269, 270 Hegedus, Z., 109 Heil, T., 228, 249, 253, 255, 287, 324 Helen, S.S., 180, 188, 189, 192 Hell, S., 138 Hellman, M.E., 217 Helmchen, F., 5

Author index for Volume 48 Henneberger, F., 253 Hensen, J.H., 57, 59 Hermann, D.S., 3 Hermansson, B., 19 Hernández-García, E., 309 Herrero, R., 229 Higa, K., 272 Hilborn, R.C., 213 Hillman, J.J., 213 Hils, B., 178 Hirano, T., 232 Hizanidis, K., 40 Hohl, A., 253 Holmes, C.D., 155 Holmes, P., 68 Hong, L., 94 Hong, Y., 257, 295, 324 Hopf, F.A., 258 Horikis, T.P., 38 Hu, G., 308, 325 Hu, H.Z., 178 Huang, S.T., 322 Huber, M.C.E., 143 Hübner, U., 213, 214 Hwang, T.M., 322 Hwong, S.-L., 235, 236 I Iannone, E., 314 Ikeda, K., 210, 219, 240, 241, 258, 309 Illing, L., 219, 255, 271, 282, 292, 325 Imai, Y., 242 Imoto, T., 242 In, V., 286 Isabelle, S.H., 273 Ishigaki, T., 195–197 Ishiyama, F., 245 Itoh, M., 217, 225, 282 Ivanov, V.N., 267, 268 Iwasawa, H., 272 J Jagoszewski, E., 116, 117, 120 Jannitti, E., 143 Jansen, M., 233 Jerrard, H.G., 157, 178 Jiao, H., 153, 154, 168 Joannopoulos, J.D., 6, 10

347

Joly, N., 322 Jones, C.K.R.T., 39, 65, 66, 70 Jordan, T.F., 152 Juang, C., 322 Juang, J., 322 Junge, L., 310 K Kaiser, F., 232, 309, 310 Kaiser, P., 4 Kallury, K.M.R., 30 Kalra, R., 314 Kanakidis, D., 219, 288, 290, 291, 324 Kanashov, A.A., 79 Kane, D., 245, 284 Kannari, F., 228, 233, 236–239, 305, 306 Kantz, H., 212, 213 Kaplan, D.L., 258 Karamzin, Y.N., 39, 79 Karlsson, M., 39, 77, 78 Karniadakis, G.E., 22 Karpman, V.I., 57, 98 Kath, W.L., 38, 48, 74 Kato, J., 179 Kato, Y., 197 Kaufman, K., 185 Kaup, D.J., 38, 49, 75, 77 Kawabata, T., 168 Kawai, R., 235, 236 Kennedy, M.P., 281, 284 Kennel, M.B., 210, 219, 240–243, 255, 271, 292, 297 Kerbage, C., 3 Kermene, V., 79 Khabibullaev, P.K., 38, 81 Kennedy, M.P., 282 Kim, C.-M., 267, 268 Kim, D.H., 244 Kim, G.U., 268 Kim, S., 244 Kimble, H.J., 161 Kino, G.S., 185 Kinugawa, S., 237, 322 Kis, G., 282 Kita, T., 142 Kitano, M., 153 Kivshar, Yu.S., 40, 80, 83 Klakoćar-Ciepacz, M., 116 Klein, A.G., 151, 163 Klein, S., 166, 167, 173

348


Kleiner, V., 164–166 Klyatzkin, V.I., 80, 91 Knapp, R., 40 Knight, J.C., 3, 5–7 Knox, F.M., 38 Ko, J.-Y., 235, 236, 268 Kobayashi, K., 247 Kobelke, J., 40, 90, 94 Kobyakov, A., 40, 50, 80, 82, 83 Kocarev, L., 218, 225, 273, 281, 282, 309, 311 Koch, K.W., 5 Koch, P.M., 155 Kodama, Y., 50, 59, 84, 94 Kohler, W., 50 Koike, M., 142, 143 Kolesik, M., 252 Kolokolov, I., 38, 49, 51, 52, 56, 57 Kolumbán, G., 281 282 Kondo, K., 258 Konorov, S.O., 7 Konotop, V.V., 81, 93 Kopidakis, G., 40, 90 Korneyev, N., 253 Koryukin, I.V., 248, 251, 252 Kothiyal, M.P., 178–180, 188, 189, 192 Kouomou, Y.C., 252, 260, 296, 297 Kovanis, V., 253, 304 Kozyreff, G., 254, 268 Kreissl, J., 253 Kritchman, E.M., 124 Kuang, J., 308, 325 Kuga, T., 232 Kuittinen, M., 138 Kuntsevich, B.F., 229 Kurths, J., 220, 224, 232, 240, 263–265 Kuske, R., 235 Kusumoto, K., 219, 288, 324 Kuten, I.S., 61, 62 Kutz, J.N., 68 Kuwashima, F., 272 Kuznetzov, E.A., 84 Kwiat, P.G., 169, 170 Kye, W.-H., 268 Kylemark, P., 78 L Labeke, D.V., 152 Ladoucer, F., 7

Laedke, E.W., 65 Lagasse, P.E., 117 Laget, M., 143 Lai, C.-H., 308, 325 Lai, R., 244 Lakoba, T.I., 38, 75, 77 Lakshmibala, S., 173 Lamb, W.E., 206 Landau, L.D., 53 Lang, R., 247 Large, M.C.J., 22 Larger, L., 210, 219, 245, 259–261, 284, 295–297, 315, 324 Lariontsev, E., 268 Larkin, K.G., 176, 177, 180, 183 Larsen, T.T., 3 Larson, L., 282, 283 Lathrop, D.P., 314 Lauterborn, W., 227, 251, 286 Lawrence, B.L., 79 Lax, M., 206 Le Berre, M., 309 Lebedev, V.V., 38, 49, 51, 52, 56, 57 Lederer, F., 39, 40, 46, 50, 80, 82, 83, 89, 90, 94 Lee, B.S., 185, 244 Lee, M.W., 248, 252, 257, 260, 288, 295, 296, 324 Lemaître, G.R., 142, 143 Lenstra, D., 219, 233, 234, 245, 284 Leon-Saval, S.G., 12 Leonov, A.S., 118, 119 Lepers, C., 232 Lepri, S., 268 Leuchs, G., 138 Levine, A.M., 219 Levrero, A., 158, 173 Levy, P., 259, 260, 296 Lewis, C.T., 219, 240, 271 Li, F., 112 Li, J., 78 Li, Q., 167 Li, R., 233 Li, X., 308, 325 Li, Y.W., 3 Lifshitz, E.M., 53 Lifu, G., 167 Lin, Q., 50 Lin, W.W., 322 Linsay, P.S., 217

Author index for Volume 48 Lippi, G.L., 209 Litchinitser, N.M., 5 Liu, C.Y., 322 Liu, H.F., 303, 324 Liu, J.M., 219, 227, 245–247, 251, 252, 255, 256, 279, 286, 287, 292–295, 324 Liu, Y., 219, 227, 232, 249–252, 261–263, 287, 288, 290, 296, 298, 321, 322, 324 Lizé, Y.K., 12 Lo, Y.H., 189 Locquet, A., 248, 252, 254, 255, 291, 292 Lorenz, E.N., 207, 208 Lou, J., 4, 18 Louisell, W.H., 206 Love, G.D., 167, 181 Love, J.D., 7 Lü, H., 308, 325 Lu, T., 112 Lu, W., 268 Luo, L., 242, 244 Luo, L.G., 297–299, 303, 324 Lushnikov, P.M., 39, 51, 58, 68, 70, 73 Luther, G.G., 94 M Ma, L.-H., 153 Macchesney, J.B., 3, 4 Mägi, E.C., 3, 12 Malacara, D., 176, 180 Malina, R.F., 142 Malomed, B.A., 39, 46, 47, 59, 69 Mamyshev, P.V., 50 Mamysheva, N., 69, 73 Manakov, S.V., 95, 97, 99, 100 Mandel, L., 158–161 Mandel, P., 237, 238, 248, 251, 252, 254, 268 Mangan, B.J., 5 Marcuse, D., 75 Mariño, I.P., 307, 308 Mark, A., 217, 283 Markovski, B., 151 Marom, E., 4 Martienssen, W., 171, 172, 178 Martinelli, M., 162, 166, 177 Maruta, A., 50 Masoller, C., 227, 248, 252, 257, 286 Mason, M.W., 12 Matsumoto, M., 77, 78

349

Matsumura, T., 179 Matsuura, T., 237, 322, 323 Matus, M., 252, 253 Maxwell, I., 4, 18 Mayers, B.T., 29 Mazor, I., 185 Mazur, E., 4, 18 Mazzini, G., 284 McAllister, R., 267, 271, 272 McCauley, D.G., 109 McGovern, M.E., 30 McKinney, W.R., 142 McLaughlin, D.W., 309 McPhedran, R.C., 3 Meade, R.D., 10 Mégret, P., 252, 254, 255, 291, 292 Melnikov, L.A., 51 Mendaš, I., 173 Menyuk, C.R., 38, 74, 75, 79 Merolla, J.-M., 210, 259 Meucci, R., 232, 264, 265, 267, 271, 272, 307, 308 Mezentsev, V.K., 65, 67 Michelson, A.A., 192 Midrio, M., 38 Mirasso, C.R., 218, 219, 228, 245, 247, 248, 252–254, 256, 257, 284, 286, 287, 295, 320 Mischall, R., 89 Mitrokhin, V.P., 7 Miyamoto, Y., 168 Moeser, J., 39, 51, 58, 68, 70, 73 Mogilevstev, D., 3 Moharam, M.G., 138 Mollenauer, L.F., 37, 38, 50, 69, 73, 74, 314 Möller, M., 233 Moloney, J.V., 209, 252, 253, 256, 288, 308, 309, 322 Monro, T.M., 3 Montagne, R., 309 Mooradian, A., 206 Moore, R.O., 39, 70 Morgan, J.S., 143 Mujumdar, S., 174, 175 Mulet, J., 228, 253, 286, 287, 295, 320 Müller, D., 5 Murakami, A., 248–250, 252 Murakami, N., 195–197 Murakawa, H., 242 Murbach, W.J., 109

350


Murty, M.V.R., 111 Musher, S.L., 65 Musslimani, Z., 89 Myneni, K., 283 N Nagel, H.G.J., 76 Nagel, S.R., 3, 4 Nakajima, K., 50 Namioka, T., 143 Nathel, R.H., 153, 154, 168 Navotny, D.V., 38, 39, 51, 61, 69, 78, 94 Nelson, D.F., 205, 207 Neubecker, R., 314 Neubelt, M.J., 50 Newcomb, R.W., 221 Newell, A.C., 209, 309 Ngo, N.Q., 295 Nguyen, H.C., 3, 7, 11, 26 Nguyen, H.T., 194 Nguyen, N., 3 Nimmerjahn, A., 5 Nityananda, R., 156, 173 Niv, A., 165, 166 Noda, H., 142, 143 Noda, J., 7 Nogawa, S., 252 Nolte, S., 40, 90, 94 Novikov, S., 95, 97 Nowak, J., 117 O O’Byrne, J.W., 168, 169 Ogawa, T., 153, 228, 236–239, 305, 306 Oh, K., 7 Ohashi, M., 50 Ohtomo, T., 268 Ohtsubo, J., 219, 245, 248–250, 252, 257, 258, 284, 288, 324 Ohyagi, K., 324 Okamoto, K., 7 Olsson, N.A., 239 Oppenheim, A.V., 216, 218, 273, 275–278, 281, 282 Orriols, G., 229 Osipov, G., 220, 224 Otsuka, K., 235, 236, 238, 245, 255, 268 Ott, E., 213, 217, 263–266, 283, 284 Ouzounov, D.G., 5

P Packard, N.H., 211 Palmer, C., 142 Pancharatnam, S., 152, 156, 165, 177, 181 Papanicolaou, G.C., 40, 50, 51, 53, 55, 56, 60, 92 Park, Y.-J., 267, 268 Parker, A.R., 109 Parker, A.T., 286, 326 Parlitz, U., 213, 218, 227, 251, 252, 257, 273, 281, 282, 286, 309–311 Patrascu, A.S., 309 Paul, J., 248, 252, 288, 322, 324 Paul, K.E., 29 Pease, F.G., 192 Pecora, L.M., 213, 215, 216, 220–222, 273 Peil, M., 249, 253, 287 Pellaux, J.P., 73 Peng, J.-H., 286 Peng, K.O., 117 Peng, R.F., 244 Pérez, G., 218, 286, 326 Pérez, T., 228, 247, 248, 252 Perrin, H., 192 Pertsch, T., 40, 90, 94 Peschel, T., 39, 46 Peschel, U., 39, 40, 46, 90, 94 Pesquera, L., 252 Pethel, S.D., 283 Pi, F., 229 Pierce, I., 248, 252, 253, 257 Pieroux, D., 238, 254, 255, 291, 292 Pikovsky, A.S., 220, 224, 263, 264 Pisarchik, A.N., 229 Pitaevskii, J.P., 95, 97 Poggiolini, P., 314 Polymilis, C., 40 Pomeau, Y., 213 Ponrathnam, S., 22 Poole, C.D., 76 Porte, H., 219, 261, 295, 315 Postnov, D.E., 225 Poutrina, E., 70 Prentiss, M., 29 Procaccia, I., 213 Prouteau, J., 176, 180 Puccioni, G.P., 209 Pujari, N.S., 22

Author index for Volume 48 Pyragas, K., 223, 224, 245, 252 Pysher, M.J., 152 Q Qu, L., 167 Qu, Z., 308 Quabis, S., 138 R Rabinovich, M.I., 215, 220 Rahman, L., 215, 228, 253 Rakecha, V.C., 158 Ramachandran, G.N., 157 Ramachandran, H., 174 Ramana, M.S., 167 Ramaseshan, S., 156, 157, 173 Ramaswami, R., 3 Rao, D.N., 175 Rarity, J.G., 7 Rashleigh, S.C., 74 Razdobreev, I., 322 Recke, L., 253 Rees, P., 252, 253, 257, 288, 324 Ressayre, E., 309 Revuelta, J., 252 Rhodes, W.T., 259–261, 296 Richards, B., 136 Richardson, D.J., 3 Rim, S., 268 Rios Leite, J.R., 232, 308 Ritter, J.E., 3, 4 Rius, J., 229 Roberts, P.J., 5 Robinson, P.A., 168, 169 Rodd, L.E., 3 Rodríguez-Lozano, A., 225 Rogers, E.A., 314 Rogister, F., 228, 252–255, 291, 292 Rosenblum, M.G., 220, 224, 263, 264 Rovatti, R., 281, 284 Rowland, H.A., 109 Roy, M., 161–163, 168, 169, 176, 177, 180, 182, 183, 186–191 Roy, R., 210, 215, 218, 219, 225, 228, 233–235, 240, 242, 243, 263–267, 271, 272, 279, 299–305, 309–316, 319–321, 325 Rozuvan, S.G., 153

351

Rubenchik, A.M., 79, 84, 94 Rudolph, D., 109, 111, 141 Rul’kov (Rulkov), N., 217, 225, 269, 271, 282, 283, 325 Russell, P.St.J., 3, 5–7, 12 Ryu, U.C., 7 S Safonova, M.A., 225 Saggese, S.J., 7 Sainz, C., 192 Saito, S., 227, 251, 252 Saleh, B.E.A., 181 Samuel, J., 151, 158, 159, 173, 174 San Miguel, M., 256, 295, 309, 320 Sánchez-Díaz, A., 219, 286 Sancho, J.M., 254 Sanders, B.C., 169 Sandoz, P., 192 Sargent, M., Jr., 206 Sasaki, Y., 7 Sathyan, S., 221 Sauer, M., 232, 309, 310 Scalora, M., 7 Scarin, P., 143 Schäfer (Schafer), T., 39, 65, 67, 70 Schiano, M., 76 Schiek, R., 39, 79 Schmahl, G., 109, 111, 141 Schmidt, E., 40, 80, 82 Schmidt-Hattenberger, C., 89 Schmidtlin, E.G.H., 194 Schmitzer, H., 166, 167, 172 Schneider, K.R., 253 Schreiber, T., 212, 213 Schroll, R.D., 314 Schuster, K., 40, 90, 94 Schwarz, W., 281–283 Sciamanna, M., 252, 254, 255, 291, 292 Sciré, A., 245, 246, 252, 285, 286, 295, 320 Scully, M.O., 206 Segard, B., 235 Segev, M., 94, 156 Segur, H., 95, 96 Sen, D., 178, 180 Senesac, L.R., 213 Serabyn, E., 194 Serrat, C., 235 Seshadri, S., 173 Setti, G., 281, 284

352


Shagam, R.N., 178 Shahverdiev, E.M., 252, 253 Shang, A., 281, 282 Shannon, C.E., 217 Shao, M., 194 Shapere, A., 151 Shapiro, E.G., 65, 68 Shariv, I., 143, 145, 146 Shaw, R.S., 211 Shcherbakov, A.V., 7 Shechnovich, V.S., 38 Shechter, R., 122, 123 Shen, J.-Q., 153 Shen, K., 272 Shen, M., 4, 18 Sheppard, C.J.R., 189–191 Shiba, T., 272 Shibasaki, N., 252 Shibayama, K., 176 Shimamura, Y., 233 Shimizu, T., 215, 228–232 Shin, W., 7 Shinbrot, T., 213 Shinozuka, M., 236–239, 305, 306 Shoemaker, R.L., 245, 258 Shore, K.A., 219, 248, 249, 252, 253, 257, 284, 288, 289, 295, 322, 324 Short, K.M., 218, 286, 302, 326 Shum, P., 295 Sidorov-Biryukov, D.A., 7 Siebert, K.J., 168, 172 Siegman, A.E., 210 Sigel, G.H., 7 Silcox, J., 5 Simon, B., 151 Simon, R., 152, 158–161 Simpson, C.E., 109 Simpson, J.R., 239 Simpson, T.B., 245, 284 Singh, K., 118 Singh, R.G., 158 Sirohi, R.S., 180, 188, 189, 192 Sivaprakasam, S., 219, 249, 252, 253, 257, 288, 289, 322, 324 Skibina, N.B., 7 Skryabin, D.V., 39, 46 Smeltink, J.W., 57, 59 Smith, K., 74 Smith, L.L., 155 Smith, N.F., 48

Smith, N.I., 183 Smith, N.J., 38 Sohler, W., 39, 79 Solomon, R., 156 Solov’ev, V.V., 57 Someda, C.G., 76 Somervell, A.R.D., 188, 189 Sommargren, G.E., 178 Sorensen, M.P., 57, 59 Spain, A.R., 255 Spano, M.L., 213, 286 Spatschek, K.H., 65, 67 Spencer, P.S., 252, 253, 257, 288, 324 Statz, H., 238 Steel, M.J., 3, 7, 11, 26 Stegeman, G.I., 39, 40, 79, 80, 82 Steinvurzel, P., 12 Stelzer, E.H.K., 138 Stewart, J.B., 155 Stolen, R.H., 37 Strand, T.C., 185 Straub, M., 3, 7, 11, 26 Strogatz, S.H., 207, 216 Sudarshan, E.C.G., 161 Sugawara, T., 215, 228–231 Sukhorukov, A.P., 39, 79 Sukow, D.W., 255 Sumetsky, M., 3, 7, 11, 26 Sunnerud, H., 39, 78 Suresh, K.A., 174, 175 Susa, I., 232 Sushchik, M.M., 269, 271, 282, 283, 321 Svahn, P., 189 Syvridis, D., 219, 288, 290, 291, 324 Sztul, H.I., 152, 155 T Tachikawa, M., 215, 228–232 Ta’eed, V.G., 3, 12 Takahashi, T., 233 Takeda, M., 168, 194, 195 Takens, F., 211 Takiguchi, Y., 227, 249, 251, 252, 257, 258, 324 Talatinian, A., 117, 120, 121 Tallet, A., 309 Tang, C.L., 238 Tang, D.Y., 232, 269, 270 Tang, G., 308, 325

Author index for Volume 48 Tang, S., 219, 227, 245, 255, 256, 279, 286, 292–295, 324 Tango, W.J., 192–194 Tappert, F., 37 Tasev, Z., 309, 311 Tateda, M., 50 Tatsuno, K., 116 Tavlove, A., 11, 31 Tavrov, A.V., 168, 194, 195 Tee, T.J., 242 Tegenfeldt, J.O., 22 Teich, M.C., 181 Temelkuran, B., 6 Terry, J.R., 233, 235, 269, 325 Tewari, S.P., 167 Thomas, M.G., 5 Thompson, M., 30 Thornburg, K.S., 215, 228, 233–235, 304 Tighe, B.P., 240 Tikhonov, E.A., 153 Timothy, J.G., 143 Tiwari, S.C., 152 Tiziani, H., 194, 195 Tomita, A., 151–153 Tondello, G., 143 Tong, L., 4, 18 Torner, L., 40, 79, 80, 82 Toroczkai, Z., 39, 51, 58, 68, 70, 73 Torre, M.S., 257 Torrent, M.C., 228, 235, 248, 251, 252, 254, 288, 308, 322 Torruellas, W.E., 79 Totzeck, M., 194, 195 Townes, C.H., 205 Tredicce, J.R., 209 Tribillon, G., 192 Trillo, S., 39, 46 Trout, C., 143 Tsimring, L.S., 269, 282, 283, 321 Tsukamoto, T., 215, 228–232 Tunnermann, A., 40, 90, 94 Turitsyn, S.K., 38, 56, 65–68, 94 Turunen, J., 138 Twiss, R.W., 192 U Uchida, A., 228, 233, 236–239, 251, 252, 255, 271, 272, 288, 305, 306, 314, 322–324

353

Ueda, T., 38 Ulm, M.H., 166 Umarov, B.A., 39, 78 Ushakov, O., 253 V Vadivasova, T.E., 225 Valladares, D.L., 213, 220, 224 van Deventer, M.O., 76 Van Stryland, E.W., 79 van Tartwijk, G.H.M., 209, 219, 239, 245 VanWiggeren, G.D., 219, 221, 233, 235, 242, 243, 269, 277 279, 299–304, 314–316, 319–321, 325 Vasquez, L., 81 Vavassori, P., 162, 166, 177 Velarde, M.G., 225 Venkataraman, N., 5 Venzke, H., 186 Verboven, P.E., 117 Vergelis, S.S., 38 Verichev, N.N., 215, 220 Vicente, R., 228, 247, 248, 252, 256 Vidal, C., 213 Vigoureux, J.-M., 152 Vilaseca, R., 209, 235 Vinitsky, V.I., 151 Vizvari, B., 282 Vladimirov, A.G., 254, 268 Volkovskii, A.R., 217, 225, 282, 283 Volodchenko, K.V., 267, 268 Von der Weid, J.P., 73 Vorontsov, M.A., 282, 325 Voss, H.U., 227 W Wabnitz, S., 38, 50 Wadsworth, W.J., 12 Wagh, A.G., 158 Wagner, R.E., 76 Wahiddin, M.R.B., 39, 78 Wai, P.K.A., 38, 74, 75 Walba, D.M., 181 Walker, K.L., 3, 4 Wallace, I., 268 Wallace, J.K., 194 Wanelik, K., 151 Wang, L.J., 158–161 Wang, R., 272

354


Wang, S., 308, 325 Wang, W.C., 322 Wang, X., 308, 325 Wang, Z., 79, 109 Wedekind, I., 252, 257 Wegener, M., 166 Wehner, M.U., 166 Wagener, J.L., 3 Weiss, C.O., 209, 213, 214, 229 Welford, W.T., 109, 111, 113, 115, 116, 121, 122, 124, 128, 135, 141, 142 Welsh, D., 217 Weng, Z., 112 Wereley, S., 3 Werner, S.A., 151, 163 Westbrook, P., 3 Whitbread, T., 244 White, J.K., 253, 288, 308, 322 White, T.P., 3 Whitesides, G.M., 29 Wilczek, F., 151 Wilkinson, S.R., 153, 154, 168 Wille, E., 253 Williams, Q.L., 210, 219, 240 Williams, R.E., 152 Wilson, M., 143 Windeler, R., 3 Winful, H.G., 215, 228, 253 Winn, J.N., 10 Winston, R., 111, 121, 122, 124, 128, 135 Wintner, E., 7 Woafo, P., 252 Wolf, E., 73, 112, 136, 137, 157, 264 Wolfe, D.B., 29 Wong, W.S., 41 Wornell, G.W., 273 Wood, A.P., 109 Wu, C.W., 217, 282 Wu, L., 322 Wu, Y.-S., 151, 152 Wünsche, H.-J., 253 Wyant, J.C., 178, 183 Wyrowski, F., 138

X Xiao, J.H., 308 Xie, C., 39, 69, 73, 78 Xie, Y., 112 Y Yabuzaki, T., 153 Yamada, T., 215, 220 Yamaguchi, I., 179 Yanchuk, S., 253 Yang, W., 286 Yannacopoulos, A.N., 40 Yao, K., 282, 283 Yariv, A., 4, 156 Ye, W., 308, 325 Yeh, R.P., 4 Yekutieli, G., 124 Yevick, D., 19 Yingli, C., 167 Yorke, J.A., 213 Yoshimori, S., 233, 237–239, 252, 272, 305, 306, 322, 323 Yoshimura, K., 218, 321 Yoshioka, A., 268 Yu, D., 268 Yu, S.F., 295 Z Zaiman, G., 80 Zajac, M., 117 Zak, J., 163 Zakharov, V.E., 84, 95, 97 Zavatta, A., 268 Zhan, M., 308, 325 Zhang, F., 219, 242, 244, 324 Zhang, Y., 109 Zhao, J., 112 Zharnitsky, V., 65, 66 Zheltikov, A.M., 7 Zhou, C.S., 220, 224, 232, 265 Zhu, S., 233, 235, 322

Subject index for Volume 48 A

– electronic circuits, communication with 281 – laser, communication with 218, 273–323 – – , synchronization of 220–272 – mode hopping 262 – optical communications 210 Chua circuit 281 colored noise 59 curved diffractive optical elements 109–138

Abbe’s sine condition 112, 113, 119, 135 achromatic phase shifter 177–181 – – – , switchable 181, 182, 184 acousto-optic modulator 232 adiabatic theorem of quantum mechanics 151 analytic signal 264 anisotropic crystal 156 Anderson localization 85, 90 aplanatic 112, 116 autocorrelation function 59, 61, 75

D data storage 111 Debye approximation 136 – integral 136 diffraction theory, vectorial 136

B Berry phase 158 bifurcation 210, 229 – , period-doubling 229 birefringence 38, 40, 42, 179, 315, 317 – parameters 75 Born approximation 81 Bose–Einstein condensate 37, 93 Bragg condition 125 – grating 24 – wavelength 24 Brownian motion 83, 206

E electro-optic modulator 260 Elgin–Gordon–Haus effect 61 – jitter 56–58, 62 F Faraday rotator 254, 292 fiber, hollow-core Bragg 6 – , optical solitons in 37, 38 – , photonic crystal 5 – , randomly birefringent 73–78 – , single-mode 30, 46 Fokker–Planck equation 38, 55, 72 Fresnel lens, curved refractive 124 Furutsu–Novikov formula 91

C chaos, control of 205 – , Lorenz–Haken 229 – masking 273, 277, 279, 281, 282, 288, 289, 293, 305 – modulation 277–280, 289, 293 – , route to 213 – shift keying 275, 281, 286, 290, 292, 306 – , synchronization of 213, 264, 297 chaotic attractor 218 – communications, privacy in 217 – dynamics 212, 257 – in nonlinear systems 207

G Gauss–Hermite expansion 67 Gel’fand–Levitan–Marchenko equation 96 – representation 38, 100 355

356

Subject index for Volume 48

geometric phase with photon pairs 171–173 – – – single photons 168–171 Glan–Thompson polarizer 188 Gordon–Haus jitter 38 grating, curved diffraction 109 – , holographic 132 – , spectroscopic 109 H Haken–Lorenz equations 208, 209, 269 Hamilton’s equations 54 Heterodyne interferometry 166 Hilbert transform 264 holography 109, 111 – , computer generated 110, 121 I imaging, aberration-free 111, 146 injection locking 237, 257 instability, dynamical 210 – , Ikeda 210 – , optical 210 interferometer, Hewlett–Packard 173 – , Linnik 189 – , microfluidic 29 – , polarization 182–184 inverse scattering transform 37, 48, 95 J Jones matrix 318 Jost coefficient 48 – function 95, 101 K Kaup perturbation technique 49, 51 Kepler potential 53 – problem, random 52–54, 60, 68, 93 Kerr coefficient 42, 43, 74 – effect 44 – – , optical 42, 43 – nonlinearity 61

laser, carbon dioxide 229 – chaos 208 – , Class A 209 – , Class B 209, 210, 230, 236, 237 – , Class C 209 – , communication with chaotic 284 – , coupled semiconductor 227 – , distributed-Bragg reflector semiconductor 261 – , distributed-feedback transmitter 255 – , erbium-doped fiber 239, 241, 242 – , fiber 297, 323 – fluctuations 206 – linewidth 206 – , semiconductor 244–256, 284, 286, 289, 291, 308, 322, 324 – , solid-state 233–239, 304, 305 – , vertical-cavity surface-emitting 256–258, 295 lithography 111 Lorenz ‘butterfly’ attractor 208 – equations 208 – model 207, 208, 216 Lyapunov exponent 212, 216 – – , conditional 221, 223 – spectrum 212, 213 M Mach–Zehnder interference 29 – interferometer 153, 154, 168, 174, 176, 178, 182, 259 Manakov system 37, 38, 76, 94, 99, 101, 102 – – perturbed 74 Markov process 72 Maxwell–Bloch equations 208, 240 Maxwell equations 11, 45 mean-field approximation 240 Michelson interferometer 160, 170–172 microscopy 111 multiplexing 321 N nulling interferometry 194–197

L

O

Lagrangian density 47 Lambertian source 111, 124, 125, 130, 131, 135 Langevin equation 206

optical fiber, hollow core 7 – – , microstructured 3, 4, 6–8, 12, 22 – signal processing 109 – spectrum analyzer 9, 19

Subject index for Volume 48 – tweezer 111 – trap 111 P Pancharatnam phase 152, 156–162, 164, 166–168, 171, 177, 179, 182, 186, 190–193, 195, 196 – – as a geometric phase 173–175 – – with white light 175, 176 Pancharatnam’s theorem 167, 168 – – in optical rotation 163 parametric amplification 173 paraxial rays 115 photonic crystal switch 3 – , two-dimensional 9 Poincaré–Bendixson theorem 207 Poincaré sphere 75, 156, 157, 159, 162, 166, 167, 173 polarization mode dispersion 38 Pyragas method 225, 226 Q Q switching 229 R Raman effect 61 ray optics 110 Rowland circle 141, 142 – geometry 141 – grating 109, 147

scanning electron micrograph 11 Schrödinger equation 43 – – , nonlinear 37, 40, 41, 44, 46, 48, 49, 51, 57, 61–65, 68, 72, 74, 78, 85, 86, 91, 92, 94–97 second harmonic field 44 – – wave 39 – law of thermodynamics 111 secure communications 215, 217 self-focusing 92 self-phase modulation 42 soliton, dispersion-managed 38, 39, 62, 77, 78 – in random medium 39, 40, 46, 56, 79 – , one-dimensional 39 – , optical 39, 90, 93, 94 – phase velocity 59 – , spatial 37, 83, 84, 86 – , Townes 41, 92 spontaneous emission fluctuations 206 squeezed state 152, 173 stellar interferometry 192–194 susceptibility, second-order 44 synchronization, communications without 282, 284 – in feedback systems 226 – of chaos 228, 239, 257 – , phase 263–268 – , stochastic 220 T Tang–Statz–deMars equations 238, 272

S Sagnac interferometer 161, 189, 194 saturable absorber 229–231

357

Y Young’s interference 174

Contents of previous volumes*

VOLUME 1 (1961) 1 2 3 4 5 6 7 8

The modern development of Hamiltonian optics, R.J. Pegis Wave optics and geometrical optics in optical design, K. Miyamoto The intensity distribution and total illumination of aberration-free diffraction images, R. Barakat Light and information, D. Gabor On basic analogies and principal differences between optical and electronic information, H. Wolter Interference color, H. Kubota Dynamic characteristics of visual processes, A. Fiorentini Modern alignment devices, A.C.S. Van Heel

1– 29 31– 66 67–108 109–153 155–210 211–251 253–288 289–329

VOLUME 2 (1963) 1 2 3 4 5 6

Ruling, testing and use of optical gratings for high-resolution spectroscopy, G.W. Stroke The metrological applications of diffraction gratings, J.M. Burch Diffusion through non-uniform media, R.G. Giovanelli Correction of optical images by compensation of aberrations and by spatial frequency filtering, J. Tsujiuchi Fluctuations of light beams, L. Mandel Methods for determining optical parameters of thin films, F. Abelès

1– 72 73–108 109–129 131–180 181–248 249–288

VOLUME 3 (1964) 1 2 3

The elements of radiative transfer, F. Kottler Apodisation, P. Jacquinot, B. Roizen-Dossier Matrix treatment of partial coherence, H. Gamo

1 2 3 4

Higher order aberration theory, J. Focke Applications of shearing interferometry, O. Bryngdahl Surface deterioration of optical glasses, K. Kinosita Optical constants of thin films, P. Rouard, P. Bousquet

1– 28 29–186 187–332

VOLUME 4 (1965)

* Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

359

1– 36 37– 83 85–143 145–197

360 5 6 7

Contents of previous volumes The Miyamoto–Wolf diffraction wave, A. Rubinowicz Aberration theory of gratings and grating mountings, W.T. Welford Diffraction at a black screen, Part I: Kirchhoff’s theory, F. Kottler

199–240 241–280 281–314

VOLUME 5 (1966) 1 2 3 4 5 6

Optical pumping, C. Cohen-Tannoudji, A. Kastler Non-linear optics, P.S. Pershan Two-beam interferometry, W.H. Steel Instruments for the measuring of optical transfer functions, K. Murata Light reflection from films of continuously varying refractive index, R. Jacobsson X-ray crystal-structure determination as a branch of physical optics, H. Lipson, C.A. Taylor 7 The wave of a moving classical electron, J. Picht

1– 81 83–144 145–197 199–245 247–286 287–350 351–370

VOLUME 6 (1967) 1 2 3 4 5 6 7 8

Recent advances in holography, E.N. Leith, J. Upatnieks Scattering of light by rough surfaces, P. Beckmann Measurement of the second order degree of coherence, M. Françon, S. Mallick Design of zoom lenses, K. Yamaji Some applications of lasers to interferometry, D.R. Herriot Experimental studies of intensity fluctuations in lasers, J.A. Armstrong, A.W. Smith Fourier spectroscopy, G.A. Vanasse, H. Sakai Diffraction at a black screen, Part II: electromagnetic theory, F. Kottler

1– 52 53– 69 71–104 105–170 171–209 211–257 259–330 331–377

VOLUME 7 (1969) 1 2 3 4 5 6 7

Multiple-beam interference and natural modes in open resonators, G. Koppelman Methods of synthesis for dielectric multilayer filters, E. Delano, R.J. Pegis Echoes at optical frequencies, I.D. Abella Image formation with partially coherent light, B.J. Thompson Quasi-classical theory of laser radiation, A.L. Mikaelian, M.L. Ter-Mikaelian The photographic image, S. Ooue Interaction of very intense light with free electrons, J.H. Eberly

1– 66 67–137 139–168 169–230 231–297 299–358 359–415

VOLUME 8 (1970) 1 2 3 4 5 6

Synthetic-aperture optics, J.W. Goodman The optical performance of the human eye, G.A. Fry Light beating spectroscopy, H.Z. Cummins, H.L. Swinney Multilayer antireflection coatings, A. Musset, A. Thelen Statistical properties of laser light, H. Risken Coherence theory of source-size compensation in interference microscopy, T. Yamamoto 7 Vision in communication, L. Levi 8 Theory of photoelectron counting, C.L. Mehta

1– 50 51–131 133–200 201–237 239–294 295–341 343–372 373–440

Contents of previous volumes

361

VOLUME 9 (1971) 1 2 3 4 5 6 7

Gas lasers and their application to precise length measurements, A.L. Bloom Picosecond laser pulses, A.J. Demaria Optical propagation through the turbulent atmosphere, J.W. Strohbehn Synthesis of optical birefringent networks, E.O. Ammann Mode locking in gas lasers, L. Allen, D.G.C. Jones Crystal optics with spatial dispersion, V.M. Agranovich, V.L. Ginzburg Applications of optical methods in the diffraction theory of elastic waves, K. Gniadek, J. Petykiewicz 8 Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions, B.R. Frieden

1– 30 31– 71 73–122 123–177 179–234 235–280 281–310 311–407

VOLUME 10 (1972) 1 2 3 4 5 6 7

Bandwidth compression of optical images, T.S. Huang The use of image tubes as shutters, R.W. Smith Tools of theoretical quantum optics, M.O. Scully, K.G. Whitney Field correctors for astronomical telescopes, C.G. Wynne Optical absorption strength of defects in insulators, D.Y. Smith, D.L. Dexter Elastooptic light modulation and deflection, E.K. Sittig Quantum detection theory, C.W. Helstrom

1– 44 45– 87 89–135 137–164 165–228 229–288 289–369

VOLUME 11 (1973) 1 2 3 4 5 6 7

Master equation methods in quantum optics, G.S. Agarwal Recent developments in far infrared spectroscopic techniques, H. Yoshinaga Interaction of light and acoustic surface waves, E.G. Lean Evanescent waves in optical imaging, O. Bryngdahl Production of electron probes using a field emission source, A.V. Crewe Hamiltonian theory of beam mode propagation, J.A. Arnaud Gradient index lenses, E.W. Marchand

1– 76 77–122 123–166 167–221 223–246 247–304 305–337

VOLUME 12 (1974) 1 2 3 4 5 6

Self-focusing, self-trapping, and self-phase modulation of laser beams, O. Svelto Self-induced transparency, R.E. Slusher Modulation techniques in spectrometry, M. Harwit, J.A. Decker Jr Interaction of light with monomolecular dye layers, K.H. Drexhage The phase transition concept and coherence in atomic emission, R. Graham Beam-foil spectroscopy, S. Bashkin

1– 51 53–100 101–162 163–232 233–286 287–344

VOLUME 13 (1976) 1

On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment, H.P. Baltes 2 The case for and against semiclassical radiation theory, L. Mandel 3 Objective and subjective spherical aberration measurements of the human eye, W.M. Rosenblum, J.L. Christensen 4 Interferometric testing of smooth surfaces, G. Schulz, J. Schwider

1– 25 27– 68 69– 91 93–167

362


5

Self-focusing of laser beams in plasmas and semiconductors, M.S. Sodha, A.K. Ghatak, V.K. Tripathi 6 Aplanatism and isoplanatism, W.T. Welford

169–265 267–292

VOLUME 14 (1976) 1 2 3 4 5 6 7

The statistics of speckle patterns, J.C. Dainty High-resolution techniques in optical astronomy, A. Labeyrie Relaxation phenomena in rare-earth luminescence, L.A. Riseberg, M.J. Weber The ultrafast optical Kerr shutter, M.A. Duguay Holographic diffraction gratings, G. Schmahl, D. Rudolph Photoemission, P.J. Vernier Optical fibre waveguides – a review, P.J.B. Clarricoats

1– 46 47– 87 89–159 161–193 195–244 245–325 327–402

VOLUME 15 (1977) 1 2 3 4 5

Theory of optical parametric amplification and oscillation, W. Brunner, H. Paul Optical properties of thin metal films, P. Rouard, A. Meessen Projection-type holography, T. Okoshi Quasi-optical techniques of radio astronomy, T.W. Cole Foundations of the macroscopic electromagnetic theory of dielectric media, J. Van Kranendonk, J.E. Sipe

1– 75 77–137 139–185 187–244 245–350

VOLUME 16 (1978) 1 2 3 4 5

Laser selective photophysics and photochemistry, V.S. Letokhov Recent advances in phase profiles generation, J.J. Clair, C.I. Abitbol Computer-generated holograms: techniques and applications, W.-H. Lee Speckle interferometry, A.E. Ennos Deformation invariant, space-variant optical pattern recognition, D. Casasent, D. Psaltis 6 Light emission from high-current surface-spark discharges, R.E. Beverly III 7 Semiclassical radiation theory within a quantum-mechanical framework, I.R. Senitzky

1– 69 71–117 119–232 233–288 289–356 357–411 413–448

VOLUME 17 (1980) 1 2 3

Heterodyne holographic interferometry, R. Dändliker Doppler-free multiphoton spectroscopy, E. Giacobino, B. Cagnac The mutual dependence between coherence properties of light and nonlinear optical processes, M. Schubert, B. Wilhelmi 4 Michelson stellar interferometry, W.J. Tango, R.Q. Twiss 5 Self-focusing media with variable index of refraction, A.L. Mikaelian

1– 84 85–161 163–238 239–277 279–345

VOLUME 18 (1980) 1 2

Graded index optical waveguides: a review, A. Ghatak, K. Thyagarajan Photocount statistics of radiation propagating through random and nonlinear media, J. Pe˘rina

1–126 127–203

Contents of previous volumes Strong fluctuations in light propagation in a randomly inhomogeneous medium, V.I. Tatarskii, V.U. Zavorotnyi 4 Catastrophe optics: morphologies of caustics and their diffraction patterns, M.V. Berry, C. Upstill

363

3

204–256 257–346

VOLUME 19 (1981) 1 2 3 4 5

Theory of intensity dependent resonance light scattering and resonance fluorescence, B.R. Mollow Surface and size effects on the light scattering spectra of solids, D.L. Mills, K.R. Subbaswamy Light scattering spectroscopy of surface electromagnetic waves in solids, S. Ushioda Principles of optical data-processing, H.J. Butterweck The effects of atmospheric turbulence in optical astronomy, F. Roddier

1– 43 45–137 139–210 211–280 281–376

VOLUME 20 (1983) 1 2 3 4 5

Some new optical designs for ultra-violet bidimensional detection of astronomical objects, G. Courtès, P. Cruvellier, M. Detaille Shaping and analysis of picosecond light pulses, C. Froehly, B. Colombeau, M. Vampouille Multi-photon scattering molecular spectroscopy, S. Kielich Colour holography, P. Hariharan Generation of tunable coherent vacuum-ultraviolet radiation, W. Jamroz, B.P. Stoicheff

1– 61 63–153 155–261 263–324 325–380

VOLUME 21 (1984) 1 2 3 4 5

Rigorous vector theories of diffraction gratings, D. Maystre Theory of optical bistability, L.A. Lugiato The Radon transform and its applications, H.H. Barrett Zone plate coded imaging: theory and applications, N.M. Ceglio, D.W. Sweeney Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity, J.C. Englund, R.R. Snapp, W.C. Schieve

1– 67 69–216 217–286 287–354 355–428

VOLUME 22 (1985) 1 2 3 4 5

Optical and electronic processing of medical images, D. Malacara Quantum fluctuations in vision, M.A. Bouman, W.A. Van De Grind, P. Zuidema Spectral and temporal fluctuations of broad-band laser radiation, A.V. Masalov Holographic methods of plasma diagnostics, G.V. Ostrovskaya, Yu.I. Ostrovsky Fringe formations in deformation and vibration measurements using laser light, I. Yamaguchi 6 Wave propagation in random media: a systems approach, R.L. Fante

1– 76 77–144 145–196 197–270 271–340 341–398

VOLUME 23 (1986) 1

Analytical techniques for multiple scattering from rough surfaces, J.A. DeSanto, G.S. Brown 2 Paraxial theory in optical design in terms of Gaussian brackets, K. Tanaka 3 Optical films produced by ion-based techniques, P.J. Martin, R.P. Netterfield

1– 62 63–111 113–182

364 4 5

Contents of previous volumes Electron holography, A. Tonomura Principles of optical processing with partially coherent light, F.T.S. Yu

183–220 221–275

VOLUME 24 (1987) 1 2 3 4 5

Micro Fresnel lenses, H. Nishihara, T. Suhara Dephasing-induced coherent phenomena, L. Rothberg Interferometry with lasers, P. Hariharan Unstable resonator modes, K.E. Oughstun Information processing with spatially incoherent light, I. Glaser

1– 37 39–101 103–164 165–387 389–509

VOLUME 25 (1988) 1

Dynamical instabilities and pulsations in lasers, N.B. Abraham, P. Mandel, L.M. Narducci 2 Coherence in semiconductor lasers, M. Ohtsu, T. Tako 3 Principles and design of optical arrays, Wang Shaomin, L. Ronchi 4 Aspheric surfaces, G. Schulz

1–190 191–278 279–348 349–415

VOLUME 26 (1988) 1 2 3 4 5

Photon bunching and antibunching, M.C. Teich, B.E.A. Saleh Nonlinear optics of liquid crystals, I.C. Khoo Single-longitudinal-mode semiconductor lasers, G.P. Agrawal Rays and caustics as physical objects, Yu.A. Kravtsov Phase-measurement interferometry techniques, K. Creath

1–104 105–161 163–225 227–348 349–393

VOLUME 27 (1989) 1 2 3 4

The self-imaging phenomenon and its applications, K. Patorski Axicons and meso-optical imaging devices, L.M. Soroko Nonimaging optics for flux concentration, I.M. Bassett, W.T. Welford, R. Winston Nonlinear wave propagation in planar structures, D. Mihalache, M. Bertolotti, C. Sibilia 5 Generalized holography with application to inverse scattering and inverse source problems, R.P. Porter

1–108 109–160 161–226 227–313 315–397

VOLUME 28 (1990) 1 2

Digital holography – computer-generated holograms, O. Bryngdahl, F. Wyrowski Quantum mechanical limit in optical precision measurement and communication, Y. Yamamoto, S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa 3 The quantum coherence properties of stimulated Raman scattering, M.G. Raymer, I.A. Walmsley 4 Advanced evaluation techniques in interferometry, J. Schwider 5 Quantum jumps, R.J. Cook

1– 86 87–179 181–270 271–359 361–416


365

VOLUME 29 (1991) 1 2

Optical waveguide diffraction gratings: coupling between guided modes, D.G. Hall Enhanced backscattering in optics, Yu.N. Barabanenkov, Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev 3 Generation and propagation of ultrashort optical pulses, I.P. Christov 4 Triple-correlation imaging in optical astronomy, G. Weigelt 5 Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics, C. Flytzanis, F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol

1– 63 65–197 199–291 293–319 321–411

VOLUME 30 (1992) 1

4 5

Quantum fluctuations in optical systems, S. Reynaud, A. Heidmann, E. Giacobino, C. Fabre Correlation holographic and speckle interferometry, Yu.I. Ostrovsky, V.P. Shchepinov Localization of waves in media with one-dimensional disorder, V.D. Freilikher, S.A. Gredeskul Theoretical foundation of optical-soliton concept in fibers, Y. Kodama, A. Hasegawa Cavity quantum optics and the quantum measurement process, P. Meystre

1 2 3 4 5 6

Atoms in strong fields: photoionization and chaos, P.W. Milonni, B. Sundaram Light diffraction by relief gratings: a macroscopic and microscopic view, E. Popov Optical amplifiers, N.K. Dutta, J.R. Simpson Adaptive multilayer optical networks, D. Psaltis, Y. Qiao Optical atoms, R.J.C. Spreeuw, J.P. Woerdman Theory of Compton free electron lasers, G. Dattoli, L. Giannessi, A. Renieri, A. Torre

2 3

1– 85 87–135 137–203 205–259 261–355

VOLUME 31 (1993) 1–137 139–187 189–226 227–261 263–319 321–412

VOLUME 32 (1993) 1 2 3 4

Guided-wave optics on silicon: physics, technology and status, B.P. Pal Optical neural networks: architecture, design and models, F.T.S. Yu The theory of optimal methods for localization of objects in pictures, L.P. Yaroslavsky Wave propagation theories in random media based on the path-integral approach, M.I. Charnotskii, J. Gozani, V.I. Tatarskii, V.U. Zavorotny 5 Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena, V.L. Ginzburg 6 Nonlinear processes in atoms and in weakly relativistic plasmas, G. Mainfray, C. Manus

1– 59 61–144 145–201 203–266 267–312 313–361

VOLUME 33 (1994) 1 2 3 4

The imbedding method in statistical boundary-value wave problems, V.I. Klyatskin Quantum statistics of dissipative nonlinear oscillators, V. Peˇrinová, A. Lukš Gap solitons, C.M. De Sterke, J.E. Sipe Direct spatial reconstruction of optical phase from phase-modulated images, V.I. Vlad, D. Malacara 5 Imaging through turbulence in the atmosphere, M.J. Beran, J. Oz-Vogt 6 Digital halftoning: synthesis of binary images, O. Bryngdahl, T. Scheermesser, F. Wyrowski

1–127 129–202 203–260 261–317 319–388 389–463

366

Contents of previous volumes VOLUME 34 (1995)

1 2 3 4 5

Quantum interference, superposition states of light, and nonclassical effects, V. Bužek, P.L. Knight Wave propagation in inhomogeneous media: phase-shift approach, L.P. Presnyakov The statistics of dynamic speckles, T. Okamoto, T. Asakura Scattering of light from multilayer systems with rough boundaries, I. Ohlídal, K. Navrátil, M. Ohlídal Random walk and diffusion-like models of photon migration in turbid media, A.H. Gandjbakhche, G.H. Weiss

1–158 159–181 183–248 249–331 333–402

VOLUME 35 (1996) 1 2 3 4 5 6

Transverse patterns in wide-aperture nonlinear optical systems, N.N. Rosanov Optical spectroscopy of single molecules in solids, M. Orrit, J. Bernard, R. Brown, B. Lounis Interferometric multispectral imaging, K. Itoh Interferometric methods for artwork diagnostics, D. Paoletti, G. Schirripa Spagnolo Coherent population trapping in laser spectroscopy, E. Arimondo Quantum phase properties of nonlinear optical phenomena, R. Tana´s, A. Miranowicz, Ts. Gantsog

1– 60 61–144 145–196 197–255 257–354 355–446

VOLUME 36 (1996) 1 2 3 4 5

Nonlinear propagation of strong laser pulses in chalcogenide glass films, V. Chumash, I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti Quantum phenomena in optical interferometry, P. Hariharan, B.C. Sanders Super-resolution by data inversion, M. Bertero, C. De Mol Radiative transfer: new aspects of the old theory, Yu.A. Kravtsov, L.A. Apresyan Photon wave function, I. Bialynicki-Birula

1– 47 49–128 129–178 179–244 245–294

VOLUME 37 (1997) 1 2 3 4 5 6

The Wigner distribution function in optics and optoelectronics, D. Dragoman Dispersion relations and phase retrieval in optical spectroscopy, K.-E. Peiponen, E.M. Vartiainen, T. Asakura Spectra of molecular scattering of light, I.L. Fabelinskii Soliton communication systems, R.-J. Essiambre, G.P. Agrawal Local fields in linear and nonlinear optics of mesoscopic systems, O. Keller Tunneling times and superluminality, R.Y. Chiao, A.M. Steinberg

1– 56 57– 94 95–184 185–256 257–343 345–405

VOLUME 38 (1998) 1 2 3

Nonlinear optics of stratified media, S. Dutta Gupta Optical aspects of interferometric gravitational-wave detectors, P. Hello Thermal properties of vertical-cavity surface-emitting semiconductor lasers, W. Nakwaski, M. Osiński 4 Fractional transformations in optics, A.W. Lohmann, D. Mendlovic, Z. Zalevsky 5 Pattern recognition with nonlinear techniques in the Fourier domain, B. Javidi, J.L. Horner 6 Free-space optical digital computing and interconnection, J. Jahns

1– 84 85–164 165–262 263–342 343–418 419–513


367

VOLUME 39 (1999) 1 2

Theory and applications of complex rays, Yu.A. Kravtsov, G.W. Forbes, A.A. Asatryan Homodyne detection and quantum-state reconstruction, D.-G. Welsch, W. Vogel, T. Opatrný 3 Scattering of light in the eikonal approximation, S.K. Sharma, D.J. Somerford 4 The orbital angular momentum of light, L. Allen, M.J. Padgett, M. Babiker 5 The optical Kerr effect and quantum optics in fibers, A. Sizmann, G. Leuchs

1– 62 63–211 213–290 291–372 373–469

VOLUME 40 (2000) 1 2 3 4

Polarimetric optical fibers and sensors, T.R. Woliński Digital optical computing, J. Tanida, Y. Ichioka Continuous measurements in quantum optics, V. Peˇrinová, A. Lukš Optical systems with improved resolving power, Z. Zalevsky, D. Mendlovic, A.W. Lohmann 5 Diffractive optics: electromagnetic approach, J. Turunen, M. Kuittinen, F. Wyrowski 6 Spectroscopy in polychromatic fields, Z. Ficek, H.S. Freedhoff

1– 75 77–114 115–269 271–341 343–388 389–441

VOLUME 41 (2000) 1 2 3 4 5 6 7

Nonlinear optics in microspheres, M.H. Fields, J. Popp, R.K. Chang Principles of optical disk data storage, J. Carriere, R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur Ellipsometry of thin film systems, I. Ohlídal, D. Franta Optical true-time delay control systems for wideband phased array antennas, R.T. Chen, Z. Fu Quantum statistics of nonlinear optical couplers, J. Peˇrina Jr, J. Peˇrina Quantum phase difference, phase measurements and Stokes operators, A. Luis, L.L. Sánchez-Soto Optical solitons in media with a quadratic nonlinearity, C. Etrich, F. Lederer, B.A. Malomed, T. Peschel, U. Peschel

1– 95 97–179 181–282 283–358 359–417 419–479 483–567

VOLUME 42 (2001) 1 2 3 4 5 6

Quanta and information, S.Ya. Kilin Optical solitons in periodic media with resonant and off-resonant nonlinearities, G. Kurizki, A.E. Kozhekin, T. Opatrný, B.A. Malomed Quantum Zeno and inverse quantum Zeno effects, P. Facchi, S. Pascazio Singular optics, M.S. Soskin, M.V. Vasnetsov Multi-photon quantum interferometry, G. Jaeger, A.V. Sergienko Transverse mode shaping and selection in laser resonators, R. Oron, N. Davidson, A.A. Friesem, E. Hasman

1– 91 93–146 147–217 219–276 277–324 325–386

VOLUME 43 (2002) 1 2 3

Active optics in modern large optical telescopes, L. Noethe Variational methods in nonlinear fiber optics and related fields, B.A. Malomed Optical works of L.V. Lorenz, O. Keller

1– 69 71–193 195–294

368


4

Canonical quantum description of light propagation in dielectric media, A. Lukš, V. Peˇrinová 5 Phase space correspondence between classical optics and quantum mechanics, D. Dragoman 6 “Slow” and “fast” light, R.W. Boyd, D.J. Gauthier 7 The fractional Fourier transform and some of its applications to optics, A. Torre

295–431 433–496 497–530 531–596

VOLUME 44 (2002) 1 2 3

Chaotic dynamics in semiconductor lasers with optical feedback, J. Ohtsubo Femtosecond pulses in optical fibers, F.G. Omenetto Instantaneous optics of ultrashort broadband pulses and rapidly varying media, A.B. Shvartsburg, G. Petite 4 Optical coherence tomography, A.F. Fercher, C.K. Hitzenberger 5 Modulational instability of electromagnetic waves in inhomogeneous and in discrete media, F.Kh. Abdullaev, S.A. Darmanyan, J. Garnier

1– 84 85–141 143–214 215–301 303–366

VOLUME 45 (2003) 1 2 3 4 5 6

Anamorphic beam shaping for laser and diffuse light, N. Davidson, N. Bokor Ultra-fast all-optical switching in optical networks, I. Glesk, B.C. Wang, L. Xu, V. Baby, P.R. Prucnal Generation of dark hollow beams and their applications, J. Yin, W. Gao, Y. Zhu Two-photon lasers, D.J. Gauthier Nonradiating sources and other “invisible” objects, G. Gbur Lasing in disordered media, H. Cao

1– 51 53–117 119–204 205–272 273–315 317–370

VOLUME 46 (2004) 1 2

Ultrafast solid-state lasers, U. Keller Multiple scattering of light from randomly rough surfaces, A.V. Shchegrov, A.A. Maradudin, E.R. Méndez 3 Laser-diode interferometry, Y. Ishii 4 Optical realizations of quantum teleportation, J. Gea-Banacloche 5 Intensity-field correlations of non-classical light, H.J. Carmichael, G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice

1–115 117–241 243–309 311–353 355–404

VOLUME 47 (2005) 1 2 3 4 5 6

Multistep parametric processes in nonlinear optics, S.M. Saltiel, A.A. Sukhorukov, Y.S. Kivshar Modes of wave-chaotic dielectric resonators, H.E. Türeci, H.G.L. Schwefel, Ph. Jacquod, A.D. Stone Nonlinear and quantum optics of atomic and molecular fields, C.P. Search, P. Meystre Space-variant polarization manipulation, E. Hasman, G. Biener, A. Niv, V. Kleiner Optical vortices and vortex solitons, A.S. Desyatnikov, Y.S. Kivshar, L.L. Torner Phase imaging and refractive index tomography for X-rays and visible rays, K. Iwata

1– 73 75–137 139–214 215–289 291–391 393–432

Cumulative index – Volumes 1–48* Abdullaev, F.Kh., S.A. Darmanyan, J. Garnier: Modulational instability of electromagnetic waves in inhomogeneous and in discrete media Abdullaev, F.Kh., J. Garnier: Optical solitons in random media Abelès, F.: Methods for determining optical parameters of thin films Abella, I.D.: Echoes at optical frequencies Abitbol, C.I., see Clair, J.J. Abraham, N.B., P. Mandel, L.M. Narducci: Dynamical instabilities and pulsations in lasers Agarwal, G.S.: Master equation methods in quantum optics Agranovich, V.M., V.L. Ginzburg: Crystal optics with spatial dispersion Agrawal, G.P.: Single-longitudinal-mode semiconductor lasers Agrawal, G.P., see Essiambre, R.-J. Allen, L., D.G.C. Jones: Mode locking in gas lasers Allen, L., M.J. Padgett, M. Babiker: The orbital angular momentum of light Ammann, E.O.: Synthesis of optical birefringent networks Anderson, R., see Carriere, J. Apresyan, L.A., see Kravtsov, Yu.A. Arimondo, E.: Coherent population trapping in laser spectroscopy Armstrong, J.A., A.W. Smith: Experimental studies of intensity fluctuations in lasers Arnaud, J.A.: Hamiltonian theory of beam mode propagation Asakura, T., see Okamoto, T. Asakura, T., see Peiponen, K.-E. Asatryan, A.A., see Kravtsov, Yu.A. Babiker, M., see Allen, L. Baby, V., see Glesk, I. Baltes, H.P.: On the validity of Kirchhoff’s law of heat radiation for a body in a nonequilibrium environment Barabanenkov, Yu.N., Yu.A. Kravtsov, V.D. Ozrin, A.I. Saichev: Enhanced backscattering in optics Barakat, R.: The intensity distribution and total illumination of aberration-free diffraction images Barrett, H.H.: The Radon transform and its applications Bashkin, S.: Beam-foil spectroscopy * Volumes I–XL were previously distinguished by roman rather than by arabic numerals.

369

44, 303 48, 35 2, 249 7, 139 16, 71 25, 1 11, 1 9, 235 26, 163 37, 185 9, 179 39, 291 9, 123 41, 97 36, 179 35, 257 6, 211 11, 247 34, 183 37, 57 39, 1 39, 291 45, 53 13,

1

29, 65 1, 67 21, 217 12, 287

370

Cumulative index – Volumes 1–48

Bassett, I.M., W.T. Welford, R. Winston: Nonimaging optics for flux concentration Beckmann, P.: Scattering of light by rough surfaces Beran, M.J., J. Oz-Vogt: Imaging through turbulence in the atmosphere Bernard, J., see Orrit, M. Berry, M.V., C. Upstill: Catastrophe optics: morphologies of caustics and their diffraction patterns Bertero, M., C. De Mol: Super-resolution by data inversion Bertolotti, M., see Chumash, V. Bertolotti, M., see Mihalache, D. Beverly III, R.E.: Light emission from high-current surface-spark discharges Bialynicki-Birula, I.: Photon wave function Biener, G.: see Hasman, E. Bloom, A.L.: Gas lasers and their application to precise length measurements Bokor, N., N. Davidson: Curved diffractive optical elements: Design and applications Bokor, N., see Davidson, N. Bouman, M.A., W.A. Van De Grind, P. Zuidema: Quantum fluctuations in vision Bousquet, P., see Rouard, P. Boyd, R.W., D.J. Gauthier: “Slow” and “fast” light Brown, G.S., see DeSanto, J.A. Brown, R., see Orrit, M. Brunner, W., H. Paul: Theory of optical parametric amplification and oscillation Bryngdahl, O.: Applications of shearing interferometry Bryngdahl, O.: Evanescent waves in optical imaging Bryngdahl, O., T. Scheermesser, F. Wyrowski: Digital halftoning: synthesis of binary images Bryngdahl, O., F. Wyrowski: Digital holography – computer-generated holograms Burch, J.M.: The metrological applications of diffraction gratings Butterweck, H.J.: Principles of optical data-processing Bužek, V., P.L. Knight: Quantum interference, superposition states of light, and nonclassical effects Cagnac, B., see Giacobino, E. Cao, H.: Lasing in disordered media Carmichael, H.J., G.T. Foster, L.A. Orozco, J.E. Reiner, P.R. Rice: Intensity-field correlations of non-classical light Carriere, J., R. Narayan, W.-H. Yeh, C. Peng, P. Khulbe, L. Li, R. Anderson, J. Choi, M. Mansuripur: Principles of optical disk data storage Casasent, D., D. Psaltis: Deformation invariant, space-variant optical pattern recognition Ceglio, N.M., D.W. Sweeney: Zone plate coded imaging: theory and applications Chang, R.K., see Fields, M.H. Charnotskii, M.I., J. Gozani, V.I. Tatarskii, V.U. Zavorotny: Wave propagation theories in random media based on the path-integral approach Chen, R.T., Z. Fu: Optical true-time delay control systems for wideband phased array antennas Chiao, R.Y., A.M. Steinberg: Tunneling times and superluminality

27, 161 6, 53 33, 319 35, 61 18, 257 36, 129 36, 1 27, 227 16, 357 36, 245 47, 215 9, 1 48, 107 45, 1 22, 77 4, 145 43, 497 23, 1 35, 61 15, 1 4, 37 11, 167 33, 389 28, 1 2, 73 19, 211 34,

1

17, 85 45, 317 46, 355 41, 97 16, 289 21, 287 41, 1 32, 203 41, 283 37, 345

Cumulative index – Volumes 1–48 Choi, J., see Carriere, J. Christensen, J.L., see Rosenblum, W.M. Christov, I.P.: Generation and propagation of ultrashort optical pulses Chumash, V., I. Cojocaru, E. Fazio, F. Michelotti, M. Bertolotti: Nonlinear propagation of strong laser pulses in chalcogenide glass films Clair, J.J., C.I. Abitbol: Recent advances in phase profiles generation Clarricoats, P.J.B.: Optical fibre waveguides – a review Cohen-Tannoudji, C., A. Kastler: Optical pumping Cojocaru, I., see Chumash, V. Cole, T.W.: Quasi-optical techniques of radio astronomy Colombeau, B., see Froehly, C. Cook, R.J.: Quantum jumps Courtès, G., P. Cruvellier, M. Detaille: Some new optical designs for ultra-violet bidimensional detection of astronomical objects Creath, K.: Phase-measurement interferometry techniques Crewe, A.V.: Production of electron probes using a field emission source Cruvellier, P., see Courtès, G. Cummins, H.Z., H.L. Swinney: Light beating spectroscopy Dainty, J.C.: The statistics of speckle patterns Dändliker, R.: Heterodyne holographic interferometry Darmanyan, S.A., see Abdullaev, F.Kh. Dattoli, G., L. Giannessi, A. Renieri, A. Torre: Theory of Compton free electron lasers Davidson, N., N. Bokor: Anamorphic beam shaping for laser and diffuse light Davidson, N., see Bokor N. Davidson, N., see Oron, R. Decker Jr, J.A., see Harwit, M. Delano, E., R.J. Pegis: Methods of synthesis for dielectric multilayer filters Demaria, A.J.: Picosecond laser pulses De Mol, C., see Bertero, M. DeSanto, J.A., G.S. Brown: Analytical techniques for multiple scattering from rough surfaces Desyatnikov, A.S., Y.S. Kivshar, L. Torner: Optical vortices and vortex solitons De Sterke, C.M., J.E. Sipe: Gap solitons Detaille, M., see Courtès, G. Dexter, D.L., see Smith, D.Y. Domachuk, P., see Eggleton, B.J. Dragoman, D.: The Wigner distribution function in optics and optoelectronics Dragoman, D.: Phase space correspondence between classical optics and quantum mechanics Drexhage, K.H.: Interaction of light with monomolecular dye layers Duguay, M.A.: The ultrafast optical Kerr shutter Dutta, N.K., J.R. Simpson: Optical amplifiers Dutta Gupta, S.: Nonlinear optics of stratified media

371 41, 97 13, 69 29, 199 36, 1 16, 71 14, 327 5, 1 36, 1 15, 187 20, 63 28, 361 20, 1 26, 349 11, 223 20, 1 8, 133 14, 1 17, 1 44, 303 31, 321 45, 1 48, 107 42, 325 12, 101 7, 67 9, 31 36, 129 23, 1 47, 291 33, 203 20, 1 10, 165 48, 1 37, 1 43, 433 12, 163 14, 161 31, 189 38, 1

372


Eberly, J.H.: Interaction of very intense light with free electrons Eggleton, B.J., P. Domachuk, C. Grillet, E.C. Mägi, H.C. Nguyen, P. Steinvurzel, M.J. Steel: Laboratory post-engineering of microstructured optical fibers Englund, J.C., R.R. Snapp, W.C. Schieve: Fluctuations, instabilities and chaos in the laser-driven nonlinear ring cavity Ennos, A.E.: Speckle interferometry Essiambre, R.-J., G.P. Agrawal: Soliton communication systems Etrich, C., F. Lederer, B.A. Malomed, T. Peschel, U. Peschel: Optical solitons in media with a quadratic nonlinearity Fabelinskii, I.L.: Spectra of molecular scattering of light Fabre, C., see Reynaud, S. Facchi, P., S. Pascazio: Quantum Zeno and inverse quantum Zeno effects Fante, R.L.: Wave propagation in random media: a systems approach Fazio, E., see Chumash, V. Fercher, A.F., C.K. Hitzenberger: Optical coherence tomography Ficek, Z., H.S. Freedhoff: Spectroscopy in polychromatic fields Fields, M.H., J. Popp, R.K. Chang: Nonlinear optics in microspheres Fiorentini, A.: Dynamic characteristics of visual processes Flytzanis, C., F. Hache, M.C. Klein, D. Ricard, Ph. Roussignol: Nonlinear optics in composite materials. 1. Semiconductor and metal crystallites in dielectrics Focke, J.: Higher order aberration theory Forbes, G.W., see Kravtsov, Yu.A. Foster, G.A., see Carmichael, H.J. Françon, M., S. Mallick: Measurement of the second order degree of coherence Franta, D., see Ohlídal, I. Freedhoff, H.S., see Ficek, Z. Freilikher, V.D., S.A. Gredeskul: Localization of waves in media with one-dimensional disorder Frieden, B.R.: Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions Friesem, A.A., see Oron, R. Froehly, C., B. Colombeau, M. Vampouille: Shaping and analysis of picosecond light pulses Fry, G.A.: The optical performance of the human eye Fu, Z., see Chen, R.T. Gabor, D.: Light and information Gamo, H.: Matrix treatment of partial coherence Gandjbakhche, A.H., G.H. Weiss: Random walk and diffusion-like models of photon migration in turbid media Gantsog, Ts., see Tana´s, R. Gao, W., see Yin, J. García-Ojalvo, J., see Uchida, A. Garnier, J., see Abdullaev, F.Kh.

7, 359 48,

1

21, 355 16, 233 37, 185 41, 483 37, 95 30, 1 42, 147 22, 341 36, 1 44, 215 40, 389 41, 1 1, 253 29, 321 4, 1 39, 1 46, 355 6, 71 41, 181 40, 389 30, 137 9, 311 42, 325 20, 63 8, 51 41, 283 1, 109 3, 187 34, 333 35, 355 45, 119 48, 203 44, 303

Cumulative index – Volumes 1–48 Garnier, J., see Abdullaev F.Kh. Gauthier, D.J.: Two-photon lasers Gauthier, D.J., see Boyd, R.W. Gbur, G.: Nonradiating sources and other “invisible” objects Gea-Banacloche, J.: Optical realizations of quantum teleportation Ghatak, A., K. Thyagarajan: Graded index optical waveguides: a review Ghatak, A.K., see Sodha, M.S. Giacobino, E., B. Cagnac: Doppler-free multiphoton spectroscopy Giacobino, E., see Reynaud, S. Giannessi, L., see Dattoli, G. Ginzburg, V.L.: Radiation by uniformly moving sources. Vavilov–Cherenkov effect, Doppler effect in a medium, transition radiation and associated phenomena Ginzburg, V.L., see Agranovich, V.M. Giovanelli, R.G.: Diffusion through non-uniform media Glaser, I.: Information processing with spatially incoherent light Glesk, I., B.C. Wang, L. Xu, V. Baby, P.R. Prucnal: Ultra-fast all-optical switching in optical networks Gniadek, K., J. Petykiewicz: Applications of optical methods in the diffraction theory of elastic waves Goodman, J.W.: Synthetic-aperture optics Gozani, J., see Charnotskii, M.I. Graham, R.: The phase transition concept and coherence in atomic emission Gredeskul, S.A., see Freilikher, V.D. Grillet, C., see Eggleton, B.J.

373 48, 35 45, 205 43, 497 45, 273 46, 311 18, 1 13, 169 17, 85 30, 1 31, 321 32, 267 9, 235 2, 109 24, 389 45, 53 9, 281 8, 1 32, 203 12, 233 30, 137 48, 1

Hache, F., see Flytzanis, C. Hall, D.G.: Optical waveguide diffraction gratings: coupling between guided modes Hariharan, P.: Colour holography Hariharan, P.: Interferometry with lasers Hariharan, P.: The geometric phase Hariharan, P., B.C. Sanders: Quantum phenomena in optical interferometry Harwit, M., J.A. Decker Jr: Modulation techniques in spectrometry Hasegawa, A., see Kodama, Y. Hasman, E., G. Biener, A. Niv, V. Kleiner: Space-variant polarization manipulation Hasman, E., see Oron, R. Heidmann, A., see Reynaud, S. Hello, P.: Optical aspects of interferometric gravitational-wave detectors Helstrom, C.W.: Quantum detection theory Herriot, D.R.: Some applications of lasers to interferometry Hitzenberger, C.K., see Fercher, A.F. Horner, J.L., see Javidi, B. Huang, T.S.: Bandwidth compression of optical images

29, 321 29, 1 20, 263 24, 103 48, 149 36, 49 12, 101 30, 205 47, 215 42, 325 30, 1 38, 85 10, 289 6, 171 44, 215 38, 343 10, 1

Ichioka, Y., see Tanida, J. Imoto, N., see Yamamoto, Y.

40, 77 28, 87

374


Ishii, Y.: Laser-diode interferometry Itoh, K.: Interferometric multispectral imaging Iwata, K.: Phase imaging and refractive index tomography for X-rays and visible rays

46, 243 35, 145 47, 393

Jacobsson, R.: Light reflection from films of continuously varying refractive index Jacquinot, P., B. Roizen-Dossier: Apodisation Jacquod, Ph., see Türeci, H.E. Jaeger, G., A.V. Sergienko: Multi-photon quantum interferometry Jahns, J.: Free-space optical digital computing and interconnection Jamroz, W., B.P. Stoicheff: Generation of tunable coherent vacuum-ultraviolet radiation Javidi, B., J.L. Horner: Pattern recognition with nonlinear techniques in the Fourier domain Jones, D.G.C., see Allen, L.

5, 247 3, 29 47, 75 42, 277 38, 419 20, 325

Kastler, A., see Cohen-Tannoudji, C. Keller, O.: Local fields in linear and nonlinear optics of mesoscopic systems Keller, O.: Optical works of L.V. Lorenz Keller, U.: Ultrafast solid-state lasers Khoo, I.C.: Nonlinear optics of liquid crystals Khulbe, P., see Carriere, J. Kielich, S.: Multi-photon scattering molecular spectroscopy Kilin, S.Ya.: Quanta and information Kinosita, K.: Surface deterioration of optical glasses Kitagawa, M., see Yamamoto, Y. Kivshar, Y.S., see Desyatnikov, A.S. Kivshar, Y.S., see Saltiel, S.M. Klein, M.C., see Flytzanis, C. Kleiner, V., see Hasman, E. Klyatskin, V.I.: The imbedding method in statistical boundary-value wave problems Knight, P.L., see Bužek, V. Kodama, Y., A. Hasegawa: Theoretical foundation of optical-soliton concept in fibers Koppelman, G.: Multiple-beam interference and natural modes in open resonators Kottler, F.: The elements of radiative transfer Kottler, F.: Diffraction at a black screen, Part I: Kirchhoff’s theory Kottler, F.: Diffraction at a black screen, Part II: electromagnetic theory Kozhekin, A.E., see Kurizki, G. Kravtsov, Yu.A.: Rays and caustics as physical objects Kravtsov, Yu.A., L.A. Apresyan: Radiative transfer: new aspects of the old theory Kravtsov, Yu.A., G.W. Forbes, A.A. Asatryan: Theory and applications of complex rays Kravtsov, Yu.A., see Barabanenkov, Yu.N. Kubota, H.: Interference color Kuittinen, M., see Turunen, J. Kurizki, G., A.E. Kozhekin, T. Opatrný, B.A. Malomed: Optical solitons in periodic media with resonant and off-resonant nonlinearities

38, 343 9, 179 5, 1 37, 257 43, 195 46, 1 26, 105 41, 97 20, 155 42, 1 4, 85 28, 87 47, 291 47, 1 29, 321 47, 215 33, 1 34, 1 30, 205 7, 1 3, 1 4, 281 6, 331 42, 93 26, 227 36, 179 39, 1 29, 65 1, 211 40, 343 42, 93


375

Labeyrie, A.: High-resolution techniques in optical astronomy Lean, E.G.: Interaction of light and acoustic surface waves Lederer, F., see Etrich, C. Lee, W.-H.: Computer-generated holograms: techniques and applications Leith, E.N., J. Upatnieks: Recent advances in holography Letokhov, V.S.: Laser selective photophysics and photochemistry Leuchs, G., see Sizmann, A. Levi, L.: Vision in communication Li, L., see Carriere, J. Lipson, H., C.A. Taylor: X-ray crystal-structure determination as a branch of physical optics Lohmann, A.W., D. Mendlovic, Z. Zalevsky: Fractional transformations in optics Lohmann, A.W., see Zalevsky, Z. Lounis, B., see Orrit, M. Lugiato, L.A.: Theory of optical bistability Luis, A., L.L. Sánchez-Soto: Quantum phase difference, phase measurements and Stokes operators Lukš, A., V. Peˇrinová: Canonical quantum description of light propagation in dielectric media Lukš, A., see Peˇrinová, V. Lukš, A., see Peˇrinová, V.

14, 47 11, 123 41, 483 16, 119 6, 1 16, 1 39, 373 8, 343 41, 97

43, 295 33, 129 40, 117

Machida, S., see Yamamoto, Y. Mägi, E.C., see Eggleton, B.J. Mainfray, G., C. Manus: Nonlinear processes in atoms and in weakly relativistic plasmas Malacara, D.: Optical and electronic processing of medical images Malacara, D., see Vlad, V.I. Mallick, S., see Françon, M. Malomed, B.A.: Variational methods in nonlinear fiber optics and related fields Malomed, B.A., see Etrich, C. Malomed, B.A., see Kurizki, G. Mandel, L.: Fluctuations of light beams Mandel, L.: The case for and against semiclassical radiation theory Mandel, P., see Abraham, N.B. Mansuripur, M., see Carriere, J. Manus, C., see Mainfray, G. Maradudin, A.A., see Shchegrov, A.V. Marchand, E.W.: Gradient index lenses Martin, P.J., R.P. Netterfield: Optical films produced by ion-based techniques Masalov, A.V.: Spectral and temporal fluctuations of broad-band laser radiation Maystre, D.: Rigorous vector theories of diffraction gratings Meessen, A., see Rouard, P. Mehta, C.L.: Theory of photoelectron counting Mendez, E.R., see Shchegrov, A.V.

28, 87 48, 1 32, 313 22, 1 33, 261 6, 71 43, 71 41, 483 42, 93 2, 181 13, 27 25, 1 41, 97 32, 313 46, 117 11, 305 23, 113 22, 145 21, 1 15, 77 8, 373 46, 117

5, 287 38, 263 40, 271 35, 61 21, 69 41, 419

376


Mendlovic, D., see Lohmann, A.W. Mendlovic, D., see Zalevsky, Z. Meystre, P.: Cavity quantum optics and the quantum measurement process Meystre, P., see Search, C.P. Michelotti, F., see Chumash, V. Mihalache, D., M. Bertolotti, C. Sibilia: Nonlinear wave propagation in planar structures Mikaelian, A.L.: Self-focusing media with variable index of refraction Mikaelian, A.L., M.L. Ter-Mikaelian: Quasi-classical theory of laser radiation Mills, D.L., K.R. Subbaswamy: Surface and size effects on the light scattering spectra of solids Milonni, P.W., B. Sundaram: Atoms in strong fields: photoionization and chaos Miranowicz, A., see Tana´s, R. Miyamoto, K.: Wave optics and geometrical optics in optical design Mollow, B.R.: Theory of intensity dependent resonance light scattering and resonance fluorescence Murata, K.: Instruments for the measuring of optical transfer functions Musset, A., A. Thelen: Multilayer antireflection coatings Nakwaski, W., M. Osiński: Thermal properties of vertical-cavity surface-emitting semiconductor lasers Narayan, R., see Carriere, J. Narducci, L.M., see Abraham, N.B. Navrátil, K., see Ohlídal, I. Netterfield, R.P., see Martin, P.J. Nguyen, H.C., see Eggleton, B.J. Nishihara, H., T. Suhara: Micro Fresnel lenses Niv, A., see Hasman, E. Noethe, L.: Active optics in modern large optical telescopes Ohlídal, I., D. Franta: Ellipsometry of thin film systems Ohlídal, I., K. Navrátil, M. Ohlídal: Scattering of light from multilayer systems with rough boundaries Ohlídal, M., see Ohlídal, I. Ohtsu, M., T. Tako: Coherence in semiconductor lasers Ohtsubo, J.: Chaotic dynamics in semiconductor lasers with optical feedback Okamoto, T., T. Asakura: The statistics of dynamic speckles Okoshi, T.: Projection-type holography Omenetto, F.G.: Femtosecond pulses in optical fibers Ooue, S.: The photographic image Opatrný, T., see Kurizki, G. Opatrný, T., see Welsch, D.-G. Oron, R., N. Davidson, A.A. Friesem, E. Hasman: Transverse mode shaping and selection in laser resonators Orozco, L.A., see Carmichael, H.J.

38, 263 40, 271 30, 261 47, 139 36, 1 27, 227 17, 279 7, 231 19, 45 31, 1 35, 355 1, 31 19, 1 5, 199 8, 201

38, 165 41, 97 25, 1 34, 249 23, 113 48, 1 24, 1 47, 215 43, 1 41, 181 34, 249 34, 249 25, 191 44, 1 34, 183 15, 139 44, 85 7, 299 42, 93 39, 63 42, 325 46, 355

Cumulative index – Volumes 1–48 Orrit, M., J. Bernard, R. Brown, B. Lounis: Optical spectroscopy of single molecules in solids Osiński, M., see Nakwaski, W. Ostrovskaya, G.V., Yu.I. Ostrovsky: Holographic methods of plasma diagnostics Ostrovsky, Yu.I., V.P. Shchepinov: Correlation holographic and speckle interferometry Ostrovsky, Yu.I., see Ostrovskaya, G.V. Oughstun, K.E.: Unstable resonator modes Oz-Vogt, J., see Beran, M.J. Ozrin, V.D., see Barabanenkov, Yu.N. Padgett, M.J., see Allen, L. Pal, B.P.: Guided-wave optics on silicon: physics, technology and status Paoletti, D., G. Schirripa Spagnolo: Interferometric methods for artwork diagnostics Pascazio, S., see Facchi, P. Patorski, K.: The self-imaging phenomenon and its applications Paul, H., see Brunner, W. Pegis, R.J.: The modern development of Hamiltonian optics Pegis, R.J., see Delano, E. Peiponen, K.-E., E.M. Vartiainen, T. Asakura: Dispersion relations and phase retrieval in optical spectroscopy Peng, C., see Carriere, J. Pe˘rina, J.: Photocount statistics of radiation propagating through random and nonlinear media Peˇrina, J., see Peˇrina Jr, J. Peˇrina Jr, J., J. Peˇrina: Quantum statistics of nonlinear optical couplers Peˇrinová, V., A. Lukš: Quantum statistics of dissipative nonlinear oscillators Peˇrinová, V., A. Lukš: Continuous measurements in quantum optics Peˇrinová, V., see Lukš, A. Pershan, P.S.: Non-linear optics Peschel, T., see Etrich, C. Peschel, U., see Etrich, C. Petite, G., see Shvartsburg, A.B. Petykiewicz, J., see Gniadek, K. Picht, J.: The wave of a moving classical electron Popov, E.: Light diffraction by relief gratings: a macroscopic and microscopic view Popp, J., see Fields, M.H. Porter, R.P.: Generalized holography with application to inverse scattering and inverse source problems Presnyakov, L.P.: Wave propagation in inhomogeneous media: phase-shift approach Prucnal, P.R., see Glesk, I. Psaltis, D., Y. Qiao: Adaptive multilayer optical networks Psaltis, D., see Casasent, D. Qiao, Y., see Psaltis, D.

377

35, 61 38, 165 22, 197 30, 87 22, 197 24, 165 33, 319 29, 65 39, 291 32, 1 35, 197 42, 147 27, 1 15, 1 1, 1 7, 67 37, 57 41, 97 18, 127 41, 359 41, 359 33, 129 40, 115 43, 295 5, 83 41, 483 41, 483 44, 143 9, 281 5, 351 31, 139 41, 1 27, 315 34, 159 45, 53 31, 227 16, 289 31, 227

378


Raymer, M.G., I.A. Walmsley: The quantum coherence properties of stimulated Raman scattering Reiner, J.E., see Carmichael, H.J. Renieri, A., see Dattoli, G. Reynaud, S., A. Heidmann, E. Giacobino, C. Fabre: Quantum fluctuations in optical systems Ricard, D., see Flytzanis, C. Rice, P.R., see Carmichael, H.J. Riseberg, L.A., M.J. Weber: Relaxation phenomena in rare-earth luminescence Risken, H.: Statistical properties of laser light Roddier, F.: The effects of atmospheric turbulence in optical astronomy Rogister, F., see Uchida, A. Roizen-Dossier, B., see Jacquinot, P. Ronchi, L., see Wang Shaomin Rosanov, N.N.: Transverse patterns in wide-aperture nonlinear optical systems Rosenblum, W.M., J.L. Christensen: Objective and subjective spherical aberration measurements of the human eye Rothberg, L.: Dephasing-induced coherent phenomena Rouard, P., P. Bousquet: Optical constants of thin films Rouard, P., A. Meessen: Optical properties of thin metal films Roussignol, Ph., see Flytzanis, C. Roy, R., see Uchida, A. Rubinowicz, A.: The Miyamoto–Wolf diffraction wave Rudolph, D., see Schmahl, G. Saichev, A.I., see Barabanenkov, Yu.N. Saito, S., see Yamamoto, Y. Sakai, H., see Vanasse, G.A. Saleh, B.E.A., see Teich, M.C. Saltiel, S.M., A.A. Sukhorukov, Y.S. Kivshar: Multistep parametric processes in nonlinear optics Sánchez-Soto, L.L., see Luis, A. Sanders, B.C., see Hariharan, P. Scheermesser, T., see Bryngdahl, O. Schieve, W.C., see Englund, J.C. Schirripa Spagnolo, G., see Paoletti, D. Schmahl, G., D. Rudolph: Holographic diffraction gratings Schubert, M., B. Wilhelmi: The mutual dependence between coherence properties of light and nonlinear optical processes Schulz, G.: Aspheric surfaces Schulz, G., J. Schwider: Interferometric testing of smooth surfaces Schwefel, H.G.L., see Türeci, H.E. Schwider, J.: Advanced evaluation techniques in interferometry Schwider, J., see Schulz, G. Scully, M.O., K.G. Whitney: Tools of theoretical quantum optics

28, 181 46, 355 31, 321 30, 1 29, 321 46, 355 14, 89 8, 239 19, 281 48, 203 3, 29 25, 279 35, 1 13, 69 24, 39 4, 145 15, 77 29, 321 48, 203 4, 199 14, 195 29, 65 28, 87 6, 259 26, 1 47, 1 41, 419 36, 49 33, 389 21, 355 35, 197 14, 195 17, 163 25, 349 13, 93 47, 75 28, 271 13, 93 10, 89


379

Search, C.P., P. Meystre: Nonlinear and quantum optics of atomic and molecular fields Senitzky, I.R.: Semiclassical radiation theory within a quantum-mechanical framework Sergienko, A.V., see Jaeger, G. Sharma, S.K., D.J. Somerford: Scattering of light in the eikonal approximation Shchegrov, A.V., A.A. Maradudin, E.R. Méndez: Multiple scattering of light from randomly rough surfaces Shchepinov, V.P., see Ostrovsky, Yu.I. Shvartsburg, A.B., G. Petite: Instantaneous optics of ultrashort broadband pulses and rapidly varying media Sibilia, C., see Mihalache, D. Simpson, J.R., see Dutta, N.K. Sipe, J.E., see De Sterke, C.M. Sipe, J.E., see Van Kranendonk, J. Sittig, E.K.: Elastooptic light modulation and deflection Sizmann, A., G. Leuchs: The optical Kerr effect and quantum optics in fibers Slusher, R.E.: Self-induced transparency Smith, A.W., see Armstrong, J.A. Smith, D.Y., D.L. Dexter: Optical absorption strength of defects in insulators Smith, R.W.: The use of image tubes as shutters Snapp, R.R., see Englund, J.C. Sodha, M.S., A.K. Ghatak, V.K. Tripathi: Self-focusing of laser beams in plasmas and semiconductors Somerford, D.J., see Sharma, S.K. Soroko, L.M.: Axicons and meso-optical imaging devices Soskin, M.S., M.V. Vasnetsov: Singular optics Spreeuw, R.J.C., J.P. Woerdman: Optical atoms Steel, M.J., see Eggleton, B.J. Steel, W.H.: Two-beam interferometry Steinberg, A.M., see Chiao, R.Y. Steinvurzel, P., see Eggleton, B.J. Stoicheff, B.P., see Jamroz, W. Stone, A.D., see Türeci, H.E. Strohbehn, J.W.: Optical propagation through the turbulent atmosphere Stroke, G.W.: Ruling, testing and use of optical gratings for high-resolution spectroscopy Subbaswamy, K.R., see Mills, D.L. Suhara, T., see Nishihara, H. Sukhorukov, A.A., see Saltiel, S.M. Sundaram, B., see Milonni, P.W. Svelto, O.: Self-focusing, self-trapping, and self-phase modulation of laser beams Sweeney, D.W., see Ceglio, N.M. Swinney, H.L., see Cummins, H.Z.

47, 139 16, 413 42, 277 39, 213

13, 169 39, 213 27, 109 42, 219 31, 263 48, 1 5, 145 37, 345 48, 1 20, 325 47, 75 9, 73 2, 1 19, 45 24, 1 47, 1 31, 1 12, 1 21, 287 8, 133

Tako, T., see Ohtsu, M. Tanaka, K.: Paraxial theory in optical design in terms of Gaussian brackets

25, 191 23, 63

46, 117 30, 87 44, 143 27, 227 31, 189 33, 203 15, 245 10, 229 39, 373 12, 53 6, 211 10, 165 10, 45 21, 355

380


Tana´s, R., A. Miranowicz, Ts. Gantsog: Quantum phase properties of nonlinear optical phenomena Tango, W.J., R.Q. Twiss: Michelson stellar interferometry Tanida, J., Y. Ichioka: Digital optical computing Tatarskii, V.I., V.U. Zavorotnyi: Strong fluctuations in light propagation in a randomly inhomogeneous medium Tatarskii, V.I., see Charnotskii, M.I. Taylor, C.A., see Lipson, H. Teich, M.C., B.E.A. Saleh: Photon bunching and antibunching Ter-Mikaelian, M.L., see Mikaelian, A.L. Thelen, A., see Musset, A. Thompson, B.J.: Image formation with partially coherent light Thyagarajan, K., see Ghatak, A. Tonomura, A.: Electron holography Torner, L., see Desyatnikov, A.S. Torre, A.: The fractional Fourier transform and some of its applications to optics Torre, A., see Dattoli, G. Tripathi, V.K., see Sodha, M.S. Tsujiuchi, J.: Correction of optical images by compensation of aberrations and by spatial frequency filtering Türeci, H.E., H.G.L. Schwefel, Ph. Jacquod, A.D. Stone: Modes of wave-chaotic dielectric resonators Turunen, J., M. Kuittinen, F. Wyrowski: Diffractive optics: electromagnetic approach Twiss, R.Q., see Tango, W.J.

47, 75 40, 343 17, 239

Uchida, A., F. Rogister, J. García-Ojalvo, R. Roy: Synchronization and communication with chaotic laser systems Upatnieks, J., see Leith, E.N. Upstill, C., see Berry, M.V. Ushioda, S.: Light scattering spectroscopy of surface electromagnetic waves in solids

48, 203 6, 1 18, 257 19, 139

Vampouille, M., see Froehly, C. Vanasse, G.A., H. Sakai: Fourier spectroscopy Van De Grind, W.A., see Bouman, M.A. Van Heel, A.C.S.: Modern alignment devices Van Kranendonk, J., J.E. Sipe: Foundations of the macroscopic electromagnetic theory of dielectric media Vartiainen, E.M., see Peiponen, K.-E. Vasnetsov, M.V., see Soskin, M.S. Vernier, P.J.: Photoemission Vlad, V.I., D. Malacara: Direct spatial reconstruction of optical phase from phasemodulated images Vogel, W., see Welsch, D.-G. Walmsley, I.A., see Raymer, M.G.

35, 355 17, 239 40, 77 18, 204 32, 203 5, 287 26, 1 7, 231 8, 201 7, 169 18, 1 23, 183 47, 291 43, 531 31, 321 13, 169 2, 131

20, 63 6, 259 22, 77 1, 289 15, 245 37, 57 42, 219 14, 245 33, 261 39, 63 28, 181

Cumulative index – Volumes 1–48 Wang, B.C., see Glesk, I. Wang Shaomin, L. Ronchi: Principles and design of optical arrays Weber, M.J., see Riseberg, L.A. Weigelt, G.: Triple-correlation imaging in optical astronomy Weiss, G.H., see Gandjbakhche, A.H. Welford, W.T.: Aberration theory of gratings and grating mountings Welford, W.T.: Aplanatism and isoplanatism Welford, W.T., see Bassett, I.M. Welsch, D.-G., W. Vogel, T. Opatrný: Homodyne detection and quantum-state reconstruction Whitney, K.G., see Scully, M.O. Wilhelmi, B., see Schubert, M. Winston, R., see Bassett, I.M. Woerdman, J.P., see Spreeuw, R.J.C. Woliński, T.R.: Polarimetric optical fibers and sensors Wolter, H.: On basic analogies and principal differences between optical and electronic information Wynne, C.G.: Field correctors for astronomical telescopes Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Bryngdahl, O. Wyrowski, F., see Turunen, J. Xu, L., see Glesk, I. Yamaguchi, I.: Fringe formations in deformation and vibration measurements using laser light Yamaji, K.: Design of zoom lenses Yamamoto, T.: Coherence theory of source-size compensation in interference microscopy Yamamoto, Y., S. Machida, S. Saito, N. Imoto, T. Yanagawa, M. Kitagawa: Quantum mechanical limit in optical precision measurement and communication Yanagawa, T., see Yamamoto, Y. Yaroslavsky, L.P.: The theory of optimal methods for localization of objects in pictures Yeh, W.-H., see Carriere, J. Yin, J., W. Gao, Y. Zhu: Generation of dark hollow beams and their applications Yoshinaga, H.: Recent developments in far infrared spectroscopic techniques Yu, F.T.S.: Principles of optical processing with partially coherent light Yu, F.T.S.: Optical neural networks: architecture, design and models Zalevsky, Z., D. Mendlovic, A.W. Lohmann: Optical systems with improved resolving power Zalevsky, Z., see Lohmann, A.W. Zavorotny, V.U., see Charnotskii, M.I. Zavorotnyi, V.U., see Tatarskii, V.I. Zhu, Y., see Yin, J. Zuidema, P., see Bouman, M.A.

381 45, 53 25, 279 14, 89 29, 293 34, 333 4, 241 13, 267 27, 161 39, 63 10, 89 17, 163 27, 161 31, 263 40, 1 1, 155 10, 137 28, 1 33, 389 40, 343 45, 53

22, 271 6, 105 8, 295 28, 87 28, 87 32, 145 41, 97 45, 119 11, 77 23, 221 32, 61

40, 271 38, 263 32, 203 18, 204 45, 119 22, 77

Progress in Optics, Volume 48

Progress in Optics, Volume 32

Progress in Optics, Volume 46

Progress In Optics Volume 8

Progress in Optics, Volume 47 (Progress in Optics)

Progress in Optics, Volume 26

Progress in Optics, Volume 55

Progress in Optics, Volume 52

Progress in Optics, Volume 51

Progress in Optics, Volume 53

Progress in Optics, Volume 27

Progress in Optics, Volume 28

Progress in Optics, Volume 36

Progress in Optics, Volume 40

Progress In Optics Volume 1

Progress in Optics, Volume 12

Progress in Optics, Volume 54

Progress in Optics, Volume 39

Progress in Optics, Volume 34

Progress in Optics, Volume 38

Progress in Optics, Volume 41

Progress in Optics, Volume 35

Progress in Optics, Volume 9

Progress in Optics, Volume 18

Progress in Optics, Volume 33

Progress in Optics : Volume 43

Progress in Optics, Volume 37

Progress in Optics, Volume 50

Progress in Optics Volume 11, Volume 11