Unlocking Dynamical Diversity: Optical Feedback Effects on Semiconductor Lasers

UNLOCKING DYNAMICAL DIVERSITY OPTICAL FEEDBACK EFFECTS ON SEMICONDUCTOR LASERS Edited by

Deborah M. Kane Macquarie University ~ Sydney, Australia

K. Alan Shore University of Wales, Bangor, UK

UNLOCKING DYNAMICAL DIVERSITY

UNLOCKING DYNAMICAL DIVERSITY OPTICAL FEEDBACK EFFECTS ON SEMICONDUCTOR LASERS Edited by

Deborah M. Kane Macquarie University ~ Sydney, Australia

K. Alan Shore University of Wales, Bangor, UK

Copyright © 2005

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone

(+44) 1243 779777

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We would like to dedicate this book to our families. DMK to her husband Robert Carman and children, Sarah and Geoffrey; KAS to his wife Lis Shore and children, Angharad and Rhodri.

Contents

List of Contributors Preface Acknowledgements 1 1.1

1.2 1.3 1.4 1.5 1.6

2 2.1 2.2 2.3 2.4

Introduction Deborah M. Kane and K. Alan Shore Semiconductor Laser Basics 1.1.1 Semiconductor Laser Materials and Output Wavelengths 1.1.2 Semiconductor Laser Structures 1.1.3 Semiconductor Laser Gain and Output Power versus Injection Current 1.1.4 Semiconductor Laser Relaxation Oscillations, Noise, Modulation and Linewidth Enhancement Factor Nonlinear Dynamical Systems Semiconductor Lasers with Optical Feedback Landmark Results: Theory and Experiment Overview of Feedback Response: Regimes I–V Outline of Applications References

Theoretical Analysis Paul Spencer, Paul Rees and Iestyn Pierce Introduction Basic Model: Single Mode Lasers with Weak Optical Feedback Steady State Analysis of the Lang–Kobayashi Equations Multimode Iterative Analysis of the Dynamics of Laser Diodes Subject to Optical Feedback 2.4.1 Dynamics of MultiMode Laser Diodes 2.4.2 Steady State Solutions 2.4.3 Comparison with Lang–Kobayashi Rate Equations

xi xiii xv 1 2 2 4 8 11 14 15 17 19 20 20

23 23 24 27 34 34 36 37

viii

Contents

2.5 Cavity Length Effects 2.5.1 Long External Cavities 2.5.2 Short External Cavities 2.6 Coupled Cavity Analysis 2.6.1 Theory 2.6.2 Comparison with LK Analysis 2.6.3 Typical Results 2.7 Conclusion References

38 39 40 44 45 49 49 53 53

3

55

3.1 3.2 3.3

3.4

3.5

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13

5

Generalized Optical Feedback: Theory Daan Lenstra, Gautam Vemuri and Mirvais Yousefi Varieties of Optical Feedback Compound-Cavity Analysis: Validity of Lang–Kobayashi Approach Filtered Optical Feedback 3.3.1 External Cavity Modes 3.3.2 Dynamics Phase-Conjugate Feedback 3.4.1 Steady State 3.4.2 Results of Stability Analysis for the Steady State 3.4.3 High-Frequency Oscillations Conclusion Acknowledgements Note References Experimental Observations A. Tom Gavrielides and David W. Sukow Introduction Experimental Apparatus Extremely Weak Feedback Effects – Regime I Very Weak Feedback Effects – Regime II Weak Feedback Effects – Regime III–IV Moderate Feedback Effects – Low Frequency Fluctuations Short Cavity Regime Double-Cavity Systems Multimode Effects Control Feedback and Modulation Phase Conjugate Feedback Conclusion References

Bifurcation Analysis of Lasers with Delay Bernd Krauskopf 5.1 Introduction 5.2 Bifurcation Theory of DDEs 5.2.1 The Phase Space of a DDE 5.2.2 Local Bifurcations of Steady States

55 58 62 64 67 70 71 72 77 78 79 79 79 81 81 85 88 89 93 104 114 117 119 122 125 133 139 140 147 147 149 149 151

Contents

5.3

5.4

5.5

5.6

6 6.1 6.2 6.3 6.4

7 7.1 7.2

7.3

7.4

7.5

ix

5.2.3 Local Bifurcations of Periodic Orbits 5.2.4 Unstable Manifolds and Global Bifurcations Numerical Methods 5.3.1 Simulation by Direct Numerical Integration 5.3.2 Numerical Continuation 5.3.3 Computation of 1D Unstable Manifolds Bifurcations in the COF Laser 5.4.1 Symmetry of the COF Laser Equation 5.4.2 External Cavity Modes 5.4.3 The Characteristic Equation of an ECM 5.4.4 Continuation Near Connecting Bridges 5.4.5 Global Bifurcations of ECMs Bifurcations in the PCF Laser 5.5.1 Symmetry of the PCF Laser Equation 5.5.2 Bifurcation Diagram Near the Locking Region 5.5.3 Bifurcations of ECMs 5.5.4 Break-up of a Torus and Crisis Bifurcation Conclusion Acknowledgements References

152 153 155 155 156 157 158 159 159 161 164 167 169 171 171 175 177 180 180 181

Chaos Synchronization Siva Sivaprakasam and Cristina Masoller Ottieri Introduction Synchronization of Unidirectionally Coupled Semiconductor Lasers Synchronization of Mutually Coupled Semiconductor Lasers Conclusion References

185

Laser Interferometry Guido Giuliani and Silvano Donati Introduction Laser Diode Feedback Interferometry: Theory and Basic Experiments 7.2.1 Interferometric Signal 7.2.2 Semiconductor Laser Types and Signal-to-Noise Ratio Application to Measurements 7.3.1 Displacement 7.3.2 Velocity 7.3.3 Vibration 7.3.4 Distance 7.3.5 Other Measurements and Developments Laser Diode Diagnostics Using Self-Mixing Techniques 7.4.1 SL Linewidth Measurement 7.4.2 SL Linewidth Enhancement Factor Measurement Conclusion Acknowledgements References

217

185 187 206 210 211

217 218 218 229 234 234 238 240 243 245 247 247 249 252 252 253

x

8

8.1

8.2

8.3

8.4

8.5

8.6

9 9.1 9.2 9.3 9.4 9.5 9.6 9.7

Contents

Single Frequency and Tunable Single Frequency Semiconductor Laser Systems Esa Jaatinen Introduction 8.1.1 Laser Mode Structure 8.1.2 Single Frequency Lasers 8.1.3 Frequency Tuning Effect of Frequency Filtering the Feedback for Robust Single Frequency Operation 8.2.1 Spectrally Sensitive Cavity Elements 8.2.2 Interferometrically Filtered Feedback 8.2.3 Diffraction Grating Feedback 8.2.4 Multicomponent Feedback Systems Tunable Semiconductor Laser System Designs and Operating Characteristics 8.3.1 External Cavity Diode Lasers 8.3.2 Tunable Monolithic Lasers Frequency Stabilization 8.4.1 Comparison of Relative Effects of Different Stabilization Techniques 8.4.2 Improving Passive Frequency Stability with Temperature Controls 8.4.3 Improving Passive Frequency Stability with Current Controls 8.4.4 Improving Passive Frequency Stability with Stable Resonators 8.4.5 Active Frequency Stabilization with Optical Feedback 8.4.6 Active Frequency Stabilization with Relative Frequency References 8.4.7 Active Frequency Stabilization with Absolute Frequency References Tunable Semiconductor Laser System Applications 8.5.1 Laser Absorption Spectroscopy 8.5.2 Saturated Absorption Spectroscopy 8.5.3 Optical Pumping 8.5.4 Raman Spectroscopy 8.5.5 Laser Cooling and Trapping of Particles 8.5.6 Optical Communications 8.5.7 Other Applications Conclusion References Chaotic Optical Communication Junji Ohtsubo and Peter Davis Introduction Communication Using Synchronized Laser Chaos Methods for Modulation and Recovery of Messages Mechanisms for Synchronization and Signal Recovery Parameter Sensitivity, Robustness and Security for Synchronized Chaos Communication Communication Bandwidth Conclusion Acknowledgements References

Index

257 257 258 261 262 262 263 264 268 270 272 272 277 284 285 287 287 288 288 289 291 292 293 295 297 297 298 299 301 301 301 307 307 308 310 317 322 325 330 331 331 335

List of Contributors

Peter Davis, Department 2, Adaptive Communications Research Laboratories, Hikaridai Seika-cho Souraku-gun, Kyoto, Japan. Silvano Donati, Professor, Dip. Electronica, Universita’ di Pavia, Italy. A. Tom Gavrielides, AFRL/DELO, Kirtland Air Force Base, USA. Guido Giuliani, Dip. Electronica, Universita’ di Pavia, Italy. Esa Jaatinen, School of Physical and Chemical Science, Queensland University of Technology, Brisbane, Australia. Deborah M. Kane, Associate Professor of Physics, Macquarie University, Sydney, Australia. Bernd Krauskopf, Professor, Department of Engineering and Mathematics, University of Bristol, UK. Daan Lenstra, Professor, Department of Physics and Astronomy, Vrije University, Amsterdam, Netherlands, also at COBRA Institute of Technical University, Eindhoven, Netherlands. Cristina Masoller Ottieri, Instituto de Física, Facultad de Ciencias, Montevideo, Uruguay. Junji Ohtsubo, Professor, Faculty of Engineering, Shizuoka University, Hamamatsu, Japan. Iestyn Pierce, School of Informatics, University of Wales, Bangor, Wales. Paul Rees, University of Wales, Swansea, Wales. K. Alan Shore, Professor of Electronic Engineering, School of Informatics, University of Wales, Bangor, Wales. Siva Sivaprakasam, Department of Physics, Indian Institute of Technology, Kanpur, India.

xii

List of Contributors

Paul Spencer, School of Informatics, University of Wales, Bangor, Wales. David W. Sukow, Assistant Professor, Department of Physics and Engineering, Washington and Lee University, USA. Gautam Vemuri, Indiana University, Purdue University, Indianapolis, USA. Mirvais Yousefi, Vrije University, Amsterdam, Netherlands.

Preface

Semiconductor lasers represent 99.8% of the world market for lasers, in terms of the number of units sold in a year (∼600 million units 2003/2004).1 In monetary terms they represent 63% of the world market (∼US$3.4 billion) reflecting their low unit cost compared to other lasers and laser systems commercially available. The high volume arises due to their use in optical storage systems; telecommunications; solid state laser pumping; medical therapeutics; and inspection, measurement and control. These are the five biggest application areas.2 Examples of high volume devices include the single wavelength distributed feedback (DFB) semiconductor lasers, with narrow linewidth, which are the standard sources used in the optical communications networks that now span and criss-cross the globe. The tens to hundreds of nanometre wavelength-gain-bandwidth of semiconductor lasers, combined with frequency filtered, strong optical feedback, lead to the tunable laser systems that are used in telecommunications, medical therapeutics, environmental sensing, and basic science investigations. The diversity of device structures and the diverse characteristics of the output of semiconductor lasers represent impressive achievements from the study of the basic science of semiconductor lasers carried out in parallel with developments in materials, fabrication and packaging technologies. This book illustrates this diversity in the context of telling the scientific story of the semiconductor laser with optical feedback. The book has been planned to provide the reader with a logically developed and reasonably complete coverage of the topic to date. The applications of these semiconductor lasers with optical feedback systems are driving rapid developments in both theoretical and experimental research. This book represents a timely synthesis of the background laser physics and dynamics in a broad context. The reader can expect a future book on this subject to be needed in the not too distant future, to explore further this rapidly developing area.

Notes 1 Laser Focus World (2004) 40, Jan., p. 75. 2 Laser Focus World (2004) 41, Feb., p. 71.

Acknowledgements

The editors (DMK & KAS) extend their appreciation to all the other contributors to this book. The book was formulated to develop the topic of semiconductor lasers subject to optical feedback in a manner which embraces theory, experiment, new developments and applications. As such, the contributors were invited to write a chapter to a preliminary brief which fitted into the overall plan for the book. The contributors, as leaders of research in the various areas covered, bring to the chapters their specialist knowledge and ability to give an overview of the various sub-fields in a manner that we, the editors, could not have done as well or as expertly, and certainly not to the deadline that was required. We are also grateful to the contributors for engaging in the discussion of their chapters, as they have been evolving, which we believe has resulted in valuable cross-fertilization on the basis of how theory will be read by experimentally oriented readers and vice versa. The contributors include Dr Peter Davis, Professor Silvano Donati, Dr Tom Gavrielides, Dr Guido Giuliani, Dr Esa Jaatinen, Professor Bernd Krauskopf, Professor Daan Lenstra, Dr Cristina Masoller, Professor Junji Ohtsubo, Dr Iestyn Pierce, Dr Paul Pees, Dr Siva Sivaprakasam, Dr Paul Spencer, Professor David Sukow, Professor Gautam Vemuri and Dr Mirvais Yousefi. As these contributors are resident in Australia, India, Italy, Japan, the Netherlands, the United Kingdom, the United States of America, and Uruguay, the book is very much an international collaboration by all involved and has been an enriching experience for the editors. We would also like to extend our warm thanks to all those at John Wiley & Sons, Ltd who have been involved with the publication of the book, at its various stages: Sophie Evans, Daniel Gill, Kathryn Sharples, Laura Kempster, and Sarah Hinton, and also to Susan Dunsmore, the copy editor, Sowmya Balaraman from Integra and all others who we have not known by name.

1 Introduction Deborah M. Kane and K. Alan Shore

The laser as a directed, high brightness, coherent source of light was a dream come true at its first demonstration in 1960. A jump of several orders of magnitude improvement towards the ideal of a single frequency, ‘zero’ linewidth, spatially coherent, plane wave source of light had been made. A major new research field in physics and engineering – the development of different types of lasers, aiming to cover that part of the electromagnetic radiation spectrum that can be called ‘light’, grew rapidly. The ‘solution without a problem’, as the laser was unsupportively described, soon became the light source of choice in so many applications, that as we write, the research field of laser applications is a far larger one than lasers. Indeed, much of laser development has been motivated by the significant markets for their end use. As laser physics and engineering grew, so did the knowledge that real lasers have outputs that are dynamically and spectrally diverse. The dynamics refer to output power variations in time and the spectrum means the time-averaged, optical-frequency-spectrum. The singlefrequency, frequency-stabilised dye laser or titanium sapphire laser, generating output with a sub-MHz linewidth [1] is quite different from a Kerr-lens-mode-locked titanium sapphire laser propagated through an optical fibre, generating femtosecond pulses with an optical frequency spectrum made up of a comb of mode-locked modes covering hundreds of nanometres, spaced by the pulse repetition frequency [2]. Almost any laser can be made to generate an output that is unstable in time by optical feedback of part of the output light back into the laser cavity. In some cases the resulting output can be shown to follow a well-defined route to deterministically chaotic output. Thus, the diversity of dynamic and spectral outputs available from lasers is high. Semiconductor lasers, as a subset of all lasers, represent a category in which a very broad range of the possible dynamic and spectral outputs obtainable from lasers can be achieved – from the chaotic to the narrow-linewidth, single-frequency, for example. The semiconductor laser has a history essentially as long as that of the laser itself, being first demonstrated in 1962. However, to a large extent, the development of the semiconductor laser has been quite separate from laser development more generally. This is primarily due to the knowledge, skills and infrastructure in semiconductor device fabrication being more Unlocking Dynamical Diversity Edited by Deborah M. Kane and K. Alan Shore © 2005 John Wiley & Sons, Ltd

2

Introduction

closely aligned to semiconductor physics and electronics than to the atomic and solid state physics, and optics that underpin most other laser systems. A large part of the semiconductor laser research community knows little about other types of lasers and laser systems and vice versa. However, the areas of overlap are growing. Incoherent diode laser arrays have become the pump source of choice for solid state lasers [3] and tunable diode laser systems based on the same design principles as other tunable, single frequency lasers have become common in spectroscopic, interferometric, sensing and telecommunications applications. Also, the range of wavelengths achievable with semiconductor lasers continues to grow with efficient and reliable blue and violet gallium nitride lasers now being commercially available [4] and 4–12 micron quantum cascade lasers have been used for high resolution spectroscopy. The future for semiconductor lasers and their application is bright, and it is strengthened by the diversity of possible modes of operation of the devices and systems, in addition to the broad range of device structures and material systems. The diversity of possible dynamic and spectral outputs from a semiconductor laser is well illustrated by the semiconductor laser subject to optical feedback. The feedback of part of the output light can be achieved using a mirror, a phase conjugate reflector or a diffraction grating. The latter will also frequency filter the optical feedback field. The coupled rate equations, or delay differential equations (DDEs), which describe these systems are among the classic examples of nonlinear science. Within the practically achievable parameter space (varying the strength of the optical feedback field, for example), sequences of bifurcations and transitions to chaos are seen. The nonlinear characteristics of these semiconductor laser systems are of interest both in their own right and in contrast to other biological, mechanical, hydrodynamical and electronic nonlinear systems. They also contrast with the nonlinear dynamics of other laser systems because of the high frequencies and small timescales (picoseconds) involved. The potential applicability, of chaotic semiconductor lasers, when synchronised in pairs and cascades of similar systems and devices, in secure optical communications has given a strong applications-based motivation for fully exploring the nature of chaotic outputs from semiconductor lasers with optical feedback. Optoelectronic feedback and optical injection (injection of light from a separate source) are also of interest. In the chapters which follow, the semiconductor laser, including all the device structures in a highly developed stage, are introduced and discussed in the context of their behaviour when subject to optical feedback. The key theory and theoretical results are presented so that the nonlinear science of these systems can be fully understood and appreciated. The full range of dynamic and spectral outputs from the systems that have been demonstrated experimentally are covered, as are the key applications in commercial systems, and systems with commercial potential. The subject of the book, semiconductor lasers with optical feedback, has been synthesized for audiences in laser physics, semiconductor lasers, and nonlinear science. As such, the commonality and complementarity of the usual language and perspectives of these three sub-disciplines have been presented for the reader.

1.1 SEMICONDUCTOR LASER BASICS 1.1.1 Semiconductor Laser Materials and Output Wavelengths Semiconductor lasers or laser diodes are the most widely used laser ever devised. They are normally pumped directly with an injection current. They are small, easily used devices which can be produced at low cost. Laser diodes are used in such everyday items as CD players

Semiconductor Laser Basics

3

and laser printers and are finding a host of new applications ranging from medical imaging to environmental sensing. The semiconductor laser is also the source which drives optical fibre communications. Indeed, it was this latter application which provided the motivation for progressing the development of semiconductor lasers. This started with simple structure prototype devices, made of relatively poor quality material, first operated pulsed, at liquid nitrogen temperatures [7–10]. It has progressed to the present sophisticated, versatile devices, made from high quality semiconductors, which are capable of reliable, long-lived, continuous wave (cw) operation in a range of environments. A specific aspect of the progress made in semiconductor lasers is the wide wavelength range covered by the sources using different semiconductor materials. Laser diode sources from the ultra-violet through to the mid and far infrared are available. The III–V semiconductor materials used for the devices are summarized in Table 1.1, along with some of their key physical properties. The materials are grouped as nitrides, arsenides, phosphides and antimonides which sequentially lead to longer wavelength devices when combined as ternaries (see Figure 1.1). Quaternaries combining the arsenides and phosphides are used Table 1.1 Key III–V Semiconductor materials used in semiconductor lasers which operate at room temperature and their key physical properties III–V Compounds

AlN GaN InN

Lattice Constant (Angstroms)

Electron (Conduction band) Effective Mass+

Heavy Hole (Valence band) Effective Mass+

Relative Dielectric Constant

Refractive Index (Near EG )

Band Gap (EG ) at ∼300 K(eV)

3
112# 4
982 3
190 5
185 3
545 5
703

0
40

3
53

8
5

2
15

6
28

0
20

0
80

8
9

2
5

3
425

0
11

1
63

15
3

2
9

1
20− 1
9$

AlAs∗ GaAs InAs

5
6611 5
6533 6
0584

0
146 0
067 0
022

0
76 0
45 0
40

10
1 13
1 15
1

3
2 3
4 3
5

2
168 1
42 0
354

AlP∗ GaP∗ InP

5
4635 5
4512 5
8686

0
83 0
82 0
08

0
70 0
60 0
56

9
8 11
1 12
4

3
0 3
37 3
4

2
45 2
26 1
35

AlSb∗ GaSb InSb

6
1355 6
0959 6
4794

0
33 0
041 0
014

0
47 0
27 0
34

12
0 15
7 16
8

3
5 3
9 3
5

1
63 0
70 0
175

Notes: Indirect Band Gap Compound, all others Direct Gap # Nitrides have a hexagonal (Wurtzite) crystal structure, hence two lattice constants, ao and co , c/a∼1.633 from theory for closest packed arrangement + Relative to the rest mass of an electron $ Earlier value of 0.70 is now regarded as incorrect, actual value measured varies in different materials ∗

4

Introduction

Ga1 – xInxN AlGaInP (MQW) Ga1 – xAlxAs Ga1 – xAlxInyP1 – y InAsSb/AlAsSb

0.1

1.0 Wavelength (µm)

Figure 1.1 Range of emission wavelengths possible with different III–V semiconductor material compositions.

for devices at the key telecommunications wavelengths of 1.3 and 1.55 m. In all the materials, lasers of shorter wavelength than that associated with the energy band gap of the bulk semiconductor material, a binary, ternary, or quaternary, can be obtained using quantum wells. The visible semiconductor lasers based on AlGaInP are quantum well (QW) devices, usually multiple QWs. Longer wavelength lasers (4–12 m) have been produced using intersub-band transitions in quantum cascade lasers [5]. Strain can also be used to vary the energy band gap of the active layer material in a semiconductor laser [11]. The II–VI semiconductor materials, such as ZnSe, ZnS and Znx Cd1−x S, have been used to develop short wavelength devices, and the lead salts such as PbSx Se1−x , Pbx Cd1−x Se, Pbx Ge1−x Te, Pbx Ge1−x Te, Pbx Sn1−x Se, and Pbx Sn1−x Te have been developed as tunable systems emitting in the 2–30 m range. These lasers operate at cryogenic temperatures. The study of optical feedback effects in semiconductor lasers has been primarily confined to room temperature systems and thus, these devices that operate at cryogenic temperatures will not be described further.

1.1.2 Semiconductor Laser Structures In the 40 years since their first demonstration the design of semiconductor lasers has undergone an almost continuous evolution. Laser action arises in laser diodes due to a recombination of charge carriers injected into semiconductor material using an electrical contact. In the first homojunction (p-n) lasers, optical gain was achieved in a volume essentially defined by the area of the electrical contact and a thickness determined by the charge diffusion and recombination processes. The major advance, which achieved laser diode operation at room temperature, was to utilise layers of dissimilar semiconductor materials – so-called heterostructures – to effect control over the thickness of the active volume. The double heterostructure semiconductor laser [12] is now regarded as the basic, standard semiconductor laser structure. This is shown in Figure 1.2. It achieves confinement of the injected carriers in the active layer region via potential barriers as indicated in the band structure for a forward biased ppn double heterostructure laser diode shown in Figure 1.3 [12, 13].

Semiconductor Laser Basics

5

1 µm 0.2 µm 1 µm

p++GaAs p+GaAIAs

p GaAs Active Layer

nGaAIAs n GaAs Substrate

Oxide Layer

ppn double heterostructure (DH)

120 K for near infrared GaAs/GaAlAs DH lasers and ∼150–180 K is determined for QW GaAs/GaAlAs lasers. The stimulated optical power inside the laser cavity increases linearly with the injection current according to: Pin = The external power, Pex is:

I − Ith  i h q

 1 ln I − Ith  R1 R2
 i h Pex = 1 q 2L + ln R1 R2

(1.8)




The external differential quantum efficiency, ex is: 
1 ln R1 R2
 i ex = 1 2L + ln R1 R2

(1.9)

(1.10)

The slope efficiency listed in the specification sheets of commercial semiconductor lasers is given in W/A and can be calculated from ex as:
 h (1.11)  = ex e Often it is the slope efficiency pertaining to the output from one of the output facets of the device that is of interest. For a laser with a large output coupling, the values associated with facets of R1 and R2 are given by [16]: √ 1 − R1  R2  1 =  √ √ √  R1 + R2 1 − R1 R2 (1.12) √ 1 − R2  R1  2 =  √ √ √  R1 + R2 1 − R1 R2

Semiconductor Laser Basics

11

Figure 1.7 Gain per unit length and output power as a function of the injection current. Source: Reproduced with permission from [17].

A typical output power versus injection current graph for a semiconductor laser is shown in Figure 1.7. Semiconductor lasers operate as LEDs (spontaneous emission predominantly) below the injection current threshold for lasing. Figure 1.7 indicates that this contribution due to spontaneous emission is large. It also contributes to noise in the laser output making semiconductor laser noisy compared to other lasers.

1.1.4 Semiconductor Laser Relaxation Oscillations, Noise, Modulation and Linewidth Enhancement Factor Other distinguishing features of semiconductor lasers, relative to most other lasers, that contribute to their dynamic timescales and diversity, and the extreme sensitivity to optical feedback include enhanced spontaneous emission; enhanced linewidth (due to spontaneous emission, and the strong coupling between variations in intracavity optical power and frequency, through the irradiance and carrier density dependent refractive index); high relaxation oscillation frequency, which scales as the square root of the injection current above threshold; a modulation bandwidth up to the relaxation oscillation frequency, and mixed FM and AM arising when the injection current is modulated, again, due to the coupling between power variation and frequency variation. All these topics are interrelated but they are dealt with separately, mostly, in standard texts on semiconductor lasers. Some key results and references are summarized here. Relaxation oscillation of the output power has been observed in most lasers and occurs with a time characteristically long compared to the laser cavity decay time, or the cavity round trip time. Typical periods of the relaxation oscillations in, for example, solid state lasers such as Nd:YAG are ∼ 0
1–1 s [14]. The basic mechanism is an interplay between the oscillation field in the resonator and the atomic/molecular/solid state inversion. An increase in the intracavity power leads to a reduction in the inversion due to the increased rate of stimulated emission. This in turn leads to a reduction in the power, and so on cyclically.

12

Introduction

In semiconductor lasers the competing timecales are very much shorter. The inversion can change dynamically on nanosecond timescales c and the photon lifetime p in the short, relatively low finesse semiconductor laser cavities is 1–3 picoseconds, typically. The relaxation oscillations in this case occur at GHz frequencies, fr . The relaxation oscillation frequency scales as the square root of the injection current for DH semiconductor lasers. The relaxation oscillations are damped with a rate fd which scales as fr 2 . 1 fr = 2




1 c p

1
2

I −1 Ith

1 2

(1.13)

Small signal modulation of a semiconductor laser is achieved by adding a small sinusoidal current of frequency f, to a dc injection current that operates the laser well above threshold. In this case the rate equations which describe the carrier density and photon density, with the modulation, can be linearised and solved. The general power modulation response, as the modulation frequency is varied, is given by the modulation transfer function (Eq. 1.14), the magnitude squared of which is shown in Figure 1.8. The power modulation follows the current modulation up to frequencies near the relaxation oscillation frequency, with a resonant response at a frequency close to, but slightly lower than the fR (Eq. 1.13). The modulation transfer function also depends on the damping constant, fd x = 2 fx . Hm  =

1


 im d im 2 1+ + r 2r

(1.14)

The modulation of the carrier density modulates both the power and the refractive index. This gives rise to frequency modulation which in semiconductor lasers dominates the power modulation. The ratio of the frequency modulation index to the power or irradiance modulation index is proportional to the linewidth enhancement factor, , which is given by

Figure 1.8 Log of the magnitude squared modulation transfer function for power/irradiance modulation as a function of (log circular) modulation frequency. The function broadens and flattens as fr and fd increase. Source: After [11].

Semiconductor Laser Basics

13

Equation (1.15). This quantity is the ratio of the change in the real part of the refractive index with carrier density to the change in the imaginary part of the refractive index with carrier density. It has typical values of 1–7 and the fact that it is usually larger than 1 has major implications for the dynamics as described here and the behaviour of semiconductor lasers with optical feedback, as discussed in the subsequent chapters of this book.   dn dN 4 dn dN   =− (1.15) ≡  dg dN dni dN The qualitative comparison of the FM and IM (irradiance modulation) response of a semiconductor laser is shown in Figure 1.9. The electric field (and hence the output power) of a semiconductor laser has fluctuations in amplitude and phase which result in power (irradiance) noise and phase/frequency noise. There are fundamental limits associated with spontaneous emission and statistical fluctuations in the carrier density. It can be shown that [18] the power spectral density function, SA f , (‘intensity’ noise) is given by:   2 f 2 + d 2 A20 f ST SA f  =   2 f 2 − fr2 2 + d 2 f 2 and the power spectral density of the frequency fluctuations is:   f ST 2 fr4 SF f  = 1+   2 f 2 − f 2 2 + d 2 f 2

(1.16)

(1.17)

r

Both these noise spectra broaden as the output power (injection current) increases. The fundamental laser linewidth, the Schawlow Townes linewidth, fST , from standard laser physics appears in these expressions. This also includes a spontaneous emission factor that is greater than 1 (typically 2.5–3) which takes account of the large spontaneous emission contribution. The laser line profile, and the associated linewidth, can be determined from the power spectral density of the electric field (an autocorrelation). The functional form is a

Figure 1.9 Qualitative comparison of the FM and PM/IM response of a semiconductor laser. Source: After [11].

14

Introduction

Figure 1.10 Line profile of a semiconductor laser – a Lorentzian with satellite peaks at the relaxation oscillation frequency.

Lorentzian with sidebands at the relaxation oscillation frequency as shown in Figure 1.10. These satellite peaks are suppressed in heavily damped semiconductor lasers. The linewidth is enhanced by the factor 1 + 2  compared to the Schawlow Townes linewidth.

1.2 NONLINEAR DYNAMICAL SYSTEMS The study of nonlinear systems has long antecedents, having its origins in studies pursued by Poincar´e in the nineteenth century and has experienced steady development – particularly within mathematical domains. The stimulus which has brought-more general awareness of the relevant mathematical techniques is the realization that chaos is a universal phenomenon which can arise in relatively simple nonlinear systems. The popular imagination has been caught by the application of chaos theory to weather forecasting as exemplified in the work of Lorenz. For laser specialists, a key event was the 1975 demonstration by Haken that equations of laser dynamics are isomorphic to the Lorenz equations. This observation stimulated significant experimental and theoretical analysis of routes to chaos in a variety of laser systems. In this way methods of nonlinear dynamical systems were absorbed into the mathematical capabilities of laser physicists. The need to study a number of practical issues had also caused several workers to use nonlinear systems theory to study, in particular, dynamical instabilities in laser diodes. Nonlinear dynamical systems theory is largely concerned with classifying the conditions under which systems undergo a qualitative change in dynamics. Such changes in dynamics are termed bifurcations. The simplest example is the process by which a system changes from a steady state to an oscillatory state – this is termed a Hopf bifurcation. A very familiar bifurcation sequence is one where a change in a nonlinear parameter causes a system to undergo successive doublings of the period of regular oscillations. The period-doubling bifurcation sequence is a classical ‘route to chaos’. Central to identifying the conditions for bifurcations is the need to define the stability of a given dynamical state. In essence, the general approach taken is to examine the effects of applying a small perturbation to the state of interest. That process allows linearization of the

Semiconductor Lasers with Optical Feedback

15

system dynamics and thus enables classical techniques such as Routh–Hurwitz analysis to be applied to test the system stability. Techniques of nonlinear system theory have been applied to several aspects of laser diode dynamics including studies of their response to direct-current modulation and external optical injection.

1.3 SEMICONDUCTOR LASERS WITH OPTICAL FEEDBACK The simplest experimental configuration in which a semiconductor laser is subject to optical feedback is illustrated in Figure 1.11. Here the optical feedback is derived from a plane mirror which reflects a part of the laser output back into the laser. It is the behaviour of this experimental configuration which is the attention of a substantial part of this book. It is perhaps somewhat surprising that this simple experimental arrangement should admit such detailed study as to require an entire book to describe the phenomena which thereby arise. It is then even more surprising to appreciate that the study of this arrangement is still far from complete. The present volume seeks to represent the current understanding of the behaviour of the laser in this configuration. However, this book also points to the need for further studies of important experimental and theoretical issues which remain unresolved. Moreover, this book also treats other scenarios where the behaviour of semiconductor lasers can be influenced by alternative forms of optical feedback, such as phase conjugate optical feedback. The importance of studying optical feedback effects in semiconductor lasers is due to the confluence of a number of significant features arising from both practical and theoretical considerations. The coincidence of all these factors in an experimental configuration of vital commercial importance provides a powerful impetus for gaining a detailed understanding of the laser behaviour in this configuration. It is the commercial application of semiconductor lasers in optical fibre communication systems which provided the primary practical motivation for studying the behaviour of semiconductor lasers subject to optical feedback. In optical fibre communication systems, information is impressed on light generated in laser diodes. The information-carrying light is transmitted, generally over quite long distances, by optical fibre waveguides. In order to operate successfully, such an optical fibre communication system must exhibit a number of basic features expected in all communications systems. In particular, the communication channel must be sufficiently free of extraneous signals – noise – to ensure accurate recovery of the transmitted information. It is precisely in this respect that the study of optical feedback effects on semiconductor lasers becomes of major importance. The unfortunate fact is that semiconductor lasers are highly prone to generating noise which may impair the operation of optical fibre communication systems. In part, this

R1

R2

R3

Figure 1.11 Semiconductor laser with facets with power reflectance R1 and R2 subject to optical feedback from an external mirror with reflectance R3 .

16

Introduction

noise-generating propensity arises from the basic physics of semiconductor lasers but more pertinently the process of coupling laser light into an optical fibre provides a further potential source of significant noise. The origin of this noise is the effect on the laser of reflections of very small amount of light from the ends of the optical fibre to which the light is being coupled. It is emphasized that such reflections are normally extremely small but it is now widely appreciated that they may have a profound effect on the behaviour of the laser. In many cases those effects are deleterious and specifically they may lead to a dramatic increase in the laser output noise. In turn, this will affect the performance of the optical fibre communication system. In the awareness of the dangers inherent in even very small back-reflections from the tips of optical fibres, the system designer can find very effective means for reducing these threats to successful optical communications. Specifically, laser diode transmitter modules may accommodate optical isolators to provide strong immunity of the lasers to optical feedback effects. A rather severe penalty is literally paid in adopting this approach: effective optical isolation is rather expensive and hence the commercial advantage of using low-cost laser diodes is offset. Nevertheless the approach is perfectly reasonable for applications – such as long-haul high-data-rate communication systems – where high returns on investment are expected. However, the approach becomes questionable when emphasis is given to deploying optical communication systems in a more general context where low-cost components are required. A constructive approach to this problem would be to design a semiconductor laser which is inherently immune to optical feedback effects. Such a challenge provides a fresh practical impetus for gaining a thorough understanding of optical feedback effects on laser diodes. Although the demands of optical fibre communication systems are quite properly seen as the initial practical reason for studying optical feedback effects, it is not the case that this exhausts the practical motivation for investigating these effects. Semiconductor lasers are increasingly the laser of choice for many practical applications in a wide range of technical areas spanning aeronautics, biotechnology, dentistry, environmental sensing, medicine, metrology, nanotechnology, security and transportation. In many, if not all, of these areas the demands are for versatile, low-cost, high-performance devices which operate in a reliable manner. The utilization of laser diodes in these contexts will often bring dangers of the impact of optical reflections – indeed, in some cases, the utilization will rely on optical feedback effects – and again will demand an understanding of the behaviour of the laser in these contexts. The provision of the underpinning knowledge required for the practical use of laser diodes is a compelling reason for sustaining research effort on optical feedback effects. Such studies also have legitimacy as an end in themselves. The basic physics of semiconductor lasers is a fascinating discipline whose vitality is sustained by the versatility and variety of laser diodes. There are rather profound reasons for enlarging the field of activity by also undertaking fundamental studies of the effects of optical feedback. Optical feedback may be used as a convenient tool for probing the basic physics of semiconductor laser operation. In this context, controlled optical feedback would be deliberately introduced and the response of the laser determined. A proper interpretation of the observed response relies on the availability of fundamental knowledge of optical feedback effects in laser diodes. Largely unexplored avenues for investigation would include elucidating ultrafast nanoscale dynamical processes in a variety of semiconductor lasers.

Landmark Results: Theory and Experiment

17

Optical feedback effects on laser diodes have a particularly important place in the fundamental study of optical chaos in particular, and of chaos, in general. Here the ease of operation of laser diodes makes them particularly attractive for experimental investigations. Counter-posed to their ease of use is the complexity of the basic physics of laser diodes. These features provide a potent combination which is expected to present significant scientific challenges for many years to come. Finally, it is important to draw attention to the opportunities which arise in the theoretical analysis of these effects. It is significant that a small but growing community of mathematicians has begun to appreciate the challenges inherent in studying laser diode optical feedback dynamics. It is foreseen that this will also remain a fertile area for activity for many years.

1.4 LANDMARK RESULTS: THEORY AND EXPERIMENT The response of semiconductor lasers to the effects of optical feedback has attracted interest for almost as long as semiconductor lasers have existed. In 1970 Broom et al. in Berne, Switzerland (where half a century earlier Einstein had worked on the basis concepts of lasers) reported the observation of dynamical effects arising in semiconductor lasers coupled to an external resonator [19]. Later Morikawa et al. discussed the appearance of oscillations in the output of semiconductor lasers subject to optical reflections [20]. A particular early focus for such investigations was provided by the recognition of the importance of optical feedback effects in determining the noise properties of lasers. Such considerations were particularly driven by the requirements of optical communications systems whose reliability would be expected to be compromised by noise penalties. This was the subject of theoretical investigations by Hirota and Suematsu [21]. Interestingly this paper stressed the effects of noise in analogue modulation systems. This paper pointed out that the behaviour of the laser depends upon the distance between the laser and the relevant external reflector. Specifically, the behaviour was classified as being either that of a ‘double cavity state’ or an ‘external injection state’ depending whether the distance to the external reflector was smaller or greater than the coherence length of the laser. On the basis of their analysis, the authors offered practical strategies for the reduction of noise effects – by driving the laser well above threshold – and also their possible elimination by the use of optical isolation. Nevertheless it was recognized in this paper that optical feedback induced noise would be expected to play a central role in many perceived applications of semiconductor lasers. Appreciation of the fundamental significance of optical feedback effects in understanding the behaviour of semiconductor lasers emerged following complementary studies which, paradoxically, indicated that the effects of optical feedback could actually be beneficial for semiconductor laser performance. The positive aspects of optical feedback were seen when consideration was given to the spectral properties of semiconductor lasers. The mechanisms for mode selection in semiconductor lasers represent a continuing area for fundamental investigation which has long antecedents in the pioneering work of Bogatov et al. [22]. External feedback was shown to enhance longitudinal mode selection and hence could be useful in narrowing the emission spectrum of semiconductor lasers [23]. The intimate connections between the dynamics and spectra of semiconductor lasers were identified in early work [24] and remain an ever-present theme in the development of semiconductor lasers.

18

Introduction

The contrasting effects which arise with variation of the distance between the laser and an external reflector were reported by Chinone et al. [25]. This work showed that enhancements of relaxation oscillations or induced self-sustained oscillations occurred with relatively large distances to the external mirrors while reduction in the relaxation oscillations was achieved when the external mirror was less than a few centimetres from the laser. It was in this context that an extremely important paper was prepared at NEC, Kawasaki, by Lang and Kobayashi [26]. In introducing their work, the authors of this paper allude to previous efforts to understand the variety of behaviour which had already been observed due to optical feedback effects on semiconductor lasers. In this discussion the authors observe that: ‘The compound cavity effects in conventional lasers have been well known. However, previous experimental observations of external feedback effects in semiconductor lasers did not appear to permit simple interpretation.’ With the benefits of hindsight, this may be regarded as a significant under-statement. In making their observations Lang and Kobayashi had very clear physical reasons for expecting rather complex behaviour to emerge from semiconductor lasers. They pointed out that the semiconductor lasers available to early experimentalists were susceptible to transverse mode instabilities and generally supported multi-longitudinal mode oscillation and hence ‘exhibited erratic behaviour even without external feedback’. Moreover, Lang and Kobayashi drew attention to aspects of the basic physics of semiconductor laser gain media which were likely to lead to complex behaviour under conditions of external feedback: broad gain spectrum; temperature dependence of material refractive index; carrier-density dependence of the refractive index. The paper includes experimental measurements of the behaviour of a semiconductor laser subject to optical reflections from a mirror a few centimetres from the laser. Among a number of interesting features of device behaviour the authors reported experimental observation of bistability and hysteresis in the light output versus drive current characteristics of the lasers. The Lang–Kobayashi paper has, however, become synonymous with the theoretical model, it attempts to describe the behaviour of single-mode semiconductor lasers subject to external optical feedback. Use of this model or its generalizations is ubiquitous in the subsequent literature. This widespread application of the model is explained rather simply in terms of its remarkable ability to describe almost all salient experimental features of laser diodes subject to optical feedback. This ability is underlined by the frequent citations of this model through the present volume as well as in the research literature. The appreciation of the subtleties inherent in the response of semiconductor lasers to external optical feedback led to increased and widespread efforts to record and understand the variety of dynamical and spectral features which could thereby be accessed in semiconductor lasers. The manner in which the single-mode linewidth of semiconductor lasers could be influenced by optical feedback was of particular interest in the context of developing coherent optical communications systems. Investigations showed that the laser linewidth could be both narrowed [27–30] and broadened [31, 32]. These seemingly contradictory responses were relatively easily understood as being a consequence of the sensitivity of the response to the phase of the reflected light. The noise properties of semiconductor lasers subject to optical feedback have remained a theme of considerable theoretical and practical interest [33–36]. More generally, much effort has been directed at measuring and modelling the impact of optical feedback on the dynamical behaviour of semiconductor lasers (for early work see, for example, [37–40].

Overview of Feedback Response: Regimes I–V

19

1.5 OVERVIEW OF FEEDBACK RESPONSE: REGIMES I–V Hirota and Suematsu [21] had pointed to the significant role played by the distance between the laser facet and the external mirror reflector in determining the nature of the response of semiconductor lasers to optical feedback. An experimental study of the so-called ‘regimes of feedback’, of semiconductor lasers was undertaken by Tkach and Chaprylyvy [41]. Here attention was paid to both the distance between the laser and the external reflector and the strength of the optical feedback. From this study a widely utilized terminology has emerged of five identified operating regimes which are conventionally labelled Regimes I to V. The regimes may be characterized by reference to either the dynamical or spectral properties of the laser when subject to appropriate feedback strengths. In Regime I, with weak optical feedback, the laser linewidth can be either narrowed or broadened depending upon the phase of the optical feedback; Regime II can be characterized by the appearance of longitudinal mode hopping; in regime III the laser becomes stable and locks to the mode with minimum single-mode linewidth; with increased feedback the linewidth of the laser broadens dramatically – a phenomenon referred to as ‘coherence collapse’ – this is Regime IV; for further increase in feedback strength into Regime V the laser enters a stable external cavity mode of operation. The laser facet needs to be AR coated to allow feedback levels for Regime V to be achieved. Rather extensive studies have been made of the five regimes of semiconductor laser operation (see, for example, [34]). Considerable effort has, in particular, been given to determining the nature of the laser dynamics in the coherence collapse regime first reported by Lenstra et al. [42]. Of specific interest for proposed applications of nonlinear dynamical effects in laser diodes was the relation between coherence collapse and chaotic dynamics which had been identified in external cavity lasers by Cho and Umeda [43]. The details of the theoretical models introduced here and the resultant predicted behaviour are covered in Chapters 2 and 3. In Chapter 2 the Lang–Kobayashi model for optical feedback in a single mode laser, and its further development by subsequent researchers, are fully described. The effect of the distance from the laser output facet to the external reflector on the dynamics is reviewed. An iterative model valid for multi-mode semiconductor lasers and arbitrarily large values of the optical feedback strength is introduced and the results are contrasted with the Lang–Kobayashi model within the parameter space that the LK model is valid. In Chapter 3 a theory of generalized optical feedback is developed. This approach uses the recent developments in techniques for solving delay differential equations. It is applied to the special cases of frequency filtered feedback and phase conjugate feedback. In Chapter 4 the enormous range of experimental studies of optical feedback, including phase conjugate and frequency filtered feedback, in standard, commercial, single stripe, semiconductor lasers is fully reviewed. Comparisons are made with the theoretical predictions in the cases covered. In Chapter 5 a bifurcation analysis approach from a more mathematical perspective is introduced. This shows that with the newly available mathematical tools it is possible to explore the nonlinear dynamics of the system of a semiconductor laser with optical feedback in a much more complete way within the very large parameter space available, by varying things like the feedback strength, injection current to the laser, etc. Such analyses are expected to lead to new predictions of useful and interesting dynamic behaviours in the near future.

20

Introduction

Chapter 6 introduces the synchronization of two similar chaotic semiconductor lasers with optical feedback. This is one of the synchronized chaos sources being researched for possible application in private optical communication systems. This application is described more fully in Chapter 9.

1.6 OUTLINE OF APPLICATIONS It is a salutary observation that novel applications for laser diodes subject to optical feedback were actually identified very early in the study of this behaviour. It is perhaps unsurprising that one of the earliest proposals for such a novel application emanated from one of the key figures in this activity: Kobayashi – who suggested the use of optical feedback to reduce waveform distortion in laser modulation [44]. Similar early inventiveness was displayed in the proposal for using a laser diode as both a source and a detector [45]. Chapters 7 to 9 cover the key systems for applications, and the applications themselves, of semiconductor lasers with optical feedback. In Chapter 7 the development of self-mixing interferometry which can give sub-nanometre sensitivity in pathlength measurements, is reviewed. In Chapter 8 the development of tunable, single frequency semiconductor lasers and systems are reviewed, along with the enormous range of applications in sensing and optical communications. The appearance of chaotic dynamics in semiconductor laser subject to optical feedback has been identified as a means of developing private communications systems exploiting this behaviour. Chapter 9 deals with this topic. With increased understanding of the behaviour of semiconductor lasers subject to optical feedback it is to be expected that further applications will be found for this configuration. Major areas where applications have and continue to be identified include sensors and spectroscopy. The former area can exploit the changes in the optical emission intensity which occur in lasers subject to optical feedback. In the latter area the more subtle changes in the spectra of semiconductor lasers can be put to good effect. As the range of semiconductor lasers expands, so it can be envisaged that there will follow an expansion in the applications where the response of those lasers to optical feedback is utilised to good effect. One of the functions of this book is to provide the fundamental knowledge which will stimulate the broadening of the applications base of semiconductor laser subject to optical feedback.

REFERENCES [1] MBR110, Single Frequency ti:sapphire Laser, Coherent Laser Group http://www.coherent.com/Downloads/ MBR110_DS.pdf [2] S.A. Diddams, D.J. Jones, J. Ye, T. Cundiff, J.L. Hall, J.K. Ranka, R.S. Windeler, R. Holzwarth, T. Udem and T.W. Hansch, Phys. Rev. Lett., 84, 5102–5105 (2000). [3] G.P.A. Malcolm and A.I. Ferguson, ‘Diode-pumped solid state lasers’, Contemporary Physics, 32, 305–19 (1991). [4] Nichia Corporation, Laser Diodes http://www.nichia.co.jp/product/laser.html [5] F. Capasso, C. Gmachl, D.L. Sivco and A.Y. Cho, ‘Quantum cascade lasers’, Physics Today, 55, May, 34–40 (2002). [6] S. Donati and C.R. Mirasso (eds) Feature section on optical chaos and applications to cryptography, IEEE J. Quant. Electron, 38, 1138–1184 (2002). [7] M.I. Nathan, W.P. Dumke and G. Burns, ‘Stimulated emission of radiation from GaAs p-n junctions’, Appl. Phys. Lett., 1, 62–64 (1962).

References

21

[8] R.N. Hall, G.H. Fenner, J.D. Kingsley, T.J. Soltys and R.D. Carlson, ‘Coherent (visible) light emission from GaAs junctions’, Phys. Rev. Lett., 9, 366–368 (1962). [9] N. Holonyak and S.F. Bevacqua, ‘Coherent (visible) light emission from Ga(As1−x Px ) junctions’, Appl. Phys. Lett., 1, 82–84 (1962). [10] T.M. Quist, R.H. Rediker, R.J. Keyes, W.E. Krag, B. Lax, A.L. McWorther and H.J. Zeiger, ‘Semiconductor maser of GaAs’, Appl. Phys. Lett., 1, 91–93 (1962). [11] L.A. Coldren and S.W. Corzine, Diode Lasers and Photonic Integrated Circuits, Wiley, New York (1995). [12] H.C. Casey and M.B. Panish, Heterostructure Lasers, Academic Press, Orlando (1978). [13] K.A. Jones, Introduction to Optical Electronics, John Wiley & Sons, New York (1987). [14] A. Yariv, Quantum Electronics, 2nd edn, John Wiley & Sons, New York (1975). [15] J.T. Verdeyen, Laser Electronics, 2nd edn, Prentice Hall, New Jersey (1989). [16] K. Petermann, Laser Diode Modulation and Noise, Kluwer Academic Publishers, Dordrecht (1988). [17] J. Buus, Single Frequency Semiconductor Laser, SPIE Press, Bellingham (1991). [18] T. Okoshi and K. Kikuchi, Coherent Optical Fibre Communication, D. Reidel Pub. Co., Tokyo (1988). [19] R.F. Broom, E. Mohn, C. Risch and R. Salathe, ‘Microwave self-modulation of diode lasers coupled to an external cavity’, IEEE J. Quant. Electron., QE-6, 328–334 (1970). [20] T. Morikawa, Y. Mitsuhashi, J. Shimoda and Y. Kojima, ‘Return beam induced oscillations in self-coupled semiconductor lasers’, Electron. Lett., 12, 435–436 (1976). [21] O. Hirota and Y. Suematsu, ‘Noise properties of injection lasers due to reflected waves’, IEEE J. Quant. Electron., QE-15, 142–149 (1979). [22] A.P. Bogatov, P.G. Eliseev, L.P. Ivanov, A.S. Logginov, M.A. Manko and K.Y. Sentarov, ‘Study of the single-mode injection laser’, IEEE J. Quant. Electron., QE-9, 392–398 (1973). [23] C. Voumard, R. Salathe and H.P. Weber, ‘Resonance amplifier model describing diode lasers coupled to short external resonators’, Appl. Phys., 12, 369–378 (1977). [24] C. Risch and C. Voumard, ‘Self-pulsation in the output intensity and spectrum of GaAs-AlGaAs cw diode lasers coupled to a frequency selective external optical cavity’, J. Appl. Phys., 48, 2083–2085 (1977). [25] N. Chinone, K. Aiki and R. Ito, ‘Stabilization of semiconductor laser outputs by a mirror close to a laser facet’, Appl. Phys. Lett., 33, 990–992 (1978). [26] R. Lang and K. Kobayashi, ‘External optical feedback effects on semiconductor laser properties’, IEEE J. Quant. Electron., QE-16, 347–355 (1980). [27] K. Kikuchi and T. Okoshi, ‘Simple formula giving spectrum narrowing ratio of semiconductor laser output obtained by optical feedback’, Electron. Letts., 18, 10–11 (1982). [28] R. Wyatt and W.J. Devlin, ‘10 kHz linewidth 1.5 m InGaAsP external cavity laser with 55 nm tuning range’, Electron. Letts., 19, 110–112 (1982). [29] E. Patzak, H. Olesen, A. Sadimura, S. Saito and T. Mukai, ‘Spectral linewidth reduction in semiconductor lasers by an external cavity with weak optical feedback’, Electron. Letts., 19, 938–939 (1983). [30] G.P. Agrawal, ‘Line narrowing in a single-mode injection laser due to external optical feedback’, IEEE J. Quant. Electron., QE-20, 468–471 (1984). [31] R.O. Miles, A. Dandridge, A.B. Tveten, H.F. Taylor and T.G. Giallorenzi, ‘Feedback induced line broadening in CW channel substrate planar laser diodes’, Appl. Phys. Lett., 37, 990–992 (1980). [32] L. Goldberg, H.F. Taylor, A. Dandridge, J.F. Weller and R.O. Miles, ‘Spectral characteristics of semiconductor lasers with optical feedback’, IEEE J. Quant. Electron., QE-18, 555–563 (1982). [33] P. Spano, S. Piazzola and M. Tamburrini, ‘Theory of noise in semiconductor lasers in the presence of optical feedback’, IEEE J. Quant. Electron., QE-20, 350–357 (1984). [34] N. Schunk and K. Petermann, ‘Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback’, IEEE J. Quant. Electron., QE-24, 1242–1247 (1988). [35] N. Schunk and K. Petermann, ‘Measured feedback-induced intensity noise for 1.3 m DFB laser diodes’, Electron. Letts., 25, 63–64 (1989). [36] Z. Xing-Sha and Y. Peida, ‘Intensity noise of semiconductor lasers in the presence of arbitrary optical feedback’, Electron. Letts., 25, 445–446 (1989). [37] B. Tromborg, J.H. Osmundsen and H. Olesen, ‘Stability analysis for a semiconductor laser in an external cavity’, IEEE J. Quant. Electron., QE-20, 1023–1032 (1984). [38] B. Tromborg, H. Olesen, X. Pan and S. Saito, ‘Transmission line description of optical feedback and injection locking for Fabry-Perot and DFB lasers’, IEEE J. Quant. Electron., QE-23, 1875–1889 (1986). [39] H. Olesen, J.H. Osmundsen and B. Tromborg, ‘Nonlinear dynamics and spectral behaviour for an external cavity semiconductor laser’, IEEE J. Quant. Electron., QE-22, 762–773 (1986).

22

Introduction

[40] C.H. Henry and R.F. Kazarinov, ‘Instability of semiconductor lasers due to optical feedback from distant reflectors’, IEEE J. Quant. Electron., QE-22, 294–301 (1986). [41] R.W. Tkach and A.R. Chraplyvy, ‘Regimes of feedback effects in 1.5 m distributed feedback lasers’, IEEE J. Lightwave Technology, LT-4, 1655–1661 (1986). [42] D. Lenstra, B.H. Verbeek and A.J. den Boef, ‘Coherence collapse in single mode semiconductor lasers due to optical feedback’, IEEE J. Quant. Electron., QE-21, 674–679 (1985). [43] Y. Cho and T. Umeda, ‘Observation of chaos in a semiconductor laser with delayed feedback’, Opt. Commun., 59, 131–136 (1986). [44] K. Kobayashi, ‘Improvements in direct pulse code modulation of semiconductor lasers by optical feedback’, Trans IECE Japan, E59, 8–14 (1976). [45] Y. Mitsuhashi, T. Morikawa, K. Sakurai, A. Seko, and J. Shimada, ‘Self-coupled optical pickup’, Opt. Commun., 17, 95–97 (1976).

2 Theoretical Analysis Paul Spencer, Paul Rees and Iestyn Pierce

2.1 INTRODUCTION The first semiconductor laser diodes were demonstrated in 1962, a development that triggered a revolution in optics. Laser diodes have helped to advance several areas, not least of which are nonlinear optics and the study of nonlinear dynamics. These developments in turn have been commercialized and coherent light sources are now common both in the home and in industry. The wholehearted adoption of optical fibre communication by the telecommunication industry has seen a dramatic reduction in the cost of telephone calls and data transmission. The semiconductor laser has played a key role in these changes. Due to their technological importance it is not surprising that a huge research effort has been devoted to studying their fundamental and applied physical properties. There are several striking differences between the attributes of semiconductor lasers and those of conventional lasers, not least is their size: a typical helium-neon laser, familiar to most from supermarket checkouts, has a length of the order of a quarter of a metre, whereas the typical length of a laser diode is of the order of a quarter of a millimetre. In addition to their small size, semiconductor lasers are relatively easy to make and are very reliable. Laser diodes also differ from conventional lasers in their openness: while conventional lasers only emit a small fraction of the total light intensity into the outside world (1–5%) and are therefore considered ‘closed’, semiconductor lasers are in comparison wide open, as they allow about 70% of the light intensity to escape. The increased efficiency of light coupling out of the laser cavity is of course tempered by the fact that laser diodes are consequently much more sensitive to the external environment than conventional lasers. Laser diodes respond very strongly to external stimuli, which for most applications is highly undesirable. However, from the fundamental physics perspective this sensitivity is extremely interesting and the fact that the dynamic response of laser diodes is highly nonlinear, and can quite easily exhibit chaotic dynamics, has meant that nonlinear dynamicists and engineers have joined forces in attempting to understand the dynamics of semiconductor lasers.

Unlocking Dynamical Diversity Edited by Deborah M. Kane and K. Alan Shore © 2005 John Wiley & Sons, Ltd

24

Theoretical Analysis

The use of lasers in nonlinear dynamics was initiated by Haken in 1975 [1], who showed that there is a strong mathematical resemblance between the dynamics of some types of lasers and turbulent fluid flow seen in the seminal paper by Lorenz on the unpredictability of atmospheric dynamics–Chaos [2]. Since then, numerous types of lasers have been used to investigate nonlinear dynamics. In 1980 Lang and Kobayashi published a paper [3] on the dynamics of laser diodes exposed to external optical feedback that can be considered to have laid the foundations for the current investigations in this area. Two review articles covering a subset of the material in this chapter can be found in [4, 5]. We begin this chapter by introducing the well-known Lang–Kobayashi equations which describe the effect of weak-to-moderate optical feedback in a single-mode laser diode. We then show how the steady state solutions for the electric field and carrier density are obtained. From this steady state analysis we show how laser diodes are destabilized by optical feedback. We next consider the multimode iterative model which is valid for arbitrary levels of optical feedback through the inclusion of multiple cavity round-trips. This model is compared with the Lang–Kobayashi analysis in the limit of one round-trip and one longitudinal mode. Next, we examine the influence of the cavity round-trip time on the stability of the system. First an asymptotic analysis for long cavities is presented, and then the case of short cavities is examined. The final model we consider is the multimode, coupled cavity model, which is valid for all feedback strengths. In contrast to the previous models which are perturbations about the solitary laser operating point, this technique views the system as a whole and determines the modes of the laser-external cavity system.

2.2 BASIC MODEL: SINGLE MODE LASERS WITH WEAK OPTICAL FEEDBACK The first attempt at explaining the effect of weak optical feedback on laser diodes was made by Lang and Kobayashi in 1980 [3]. Their model describes the evolution of the electric field in a single mode laser diode in the rate-equation limit and includes the influence of the optical feedback through a time-delayed field term with coupling strength
. Their model is entirely deterministic and was originally applied to short external cavities (of the order of a few centimetres). The electric field in the compound cavity is assumed to be of the form of the product of an envelope function E(t) modulated by the rapidly oscillating optical field ei0 t, with an assumed centre-frequency 0 . The rate equation for the complex electric field in the compound cavity is then given by: 
d E t ei0 t = iN n + 21 G n − 0  E t ei0 t +
E ˜ t −  ei0 t− (2.1) dt  where N n is the longitudinal mode frequency, given by, N = Nc l. The functional dependence of the mode frequency on the carrier density n is included by accounting for the variation of the laser diode refractive index, , with carrier density due to plasma loading. In the expression for the mode frequency, l is the length of the laser diode, c is the velocity of light and N is an integer. The first term in the E-field equation therefore accounts for a possible difference between the assumed optical centre frequency 0 and the instantaneous frequency of the light in the cavity. At a fixed carrier density this

Basic Model: Single Mode Lasers with Weak Optical Feedback

25

term therefore applies a simple phase shift of eiN t to the field envelope Et, which in the frequency domain corresponds to a frequency translation of N . With a time-varying carrier density, the first term leads to so-called phase-amplitude coupling, where changes in the amplitude of the field cause changes in the carrier density which in turn lead to changes in the instantaneous frequency of the laser. We shall see later that this phase-amplitude coupling can be more explicitly accounted for through the use of the ‘alpha factor’. The second term in the E-field equation accounts for amplification through stimulated emission, quantified by the optical gain term Gn, and for attenuation due to the cavity losses, 0 . The final term in the E-field equation is the important addition made by Lang and Kobayashi: the inclusion of the effect of the feedback of a time-delayed proportion of the optical field. The coupling coefficient
˜ is given by:  c 1 − R2  R3 /R2
=
˜ = 2l in

(2.2)

and the transit time of the light through the external cavity is . Since the rate equation only includes the effect of a single round trip through the external cavity, this model is inherently limited to the case of weak optical feedback (i.e. small values of
) where the contribution to the E-field rate-equation from the second and subsequent round trips can be ignored. To obtain the feedback coefficient given in Equation (2.2) we must first determine the effective reflectivity of the laser facet subject to the optical feedback. Figure 2.1 shows the external cavity configuration under consideration. If the field incident on the right laser diode facet (reflectivity R2 ) is given by, Ei ei0 t , then providing we only consider one reflection of the field in the external cavity, the total reflected field from that facet is given by:    R2 + 1 − R2  R3 ei0  Ei ei0 t (2.3) Er ei0 t = This external cavity configuration can be thought of as a laser diode with an effective facet reflectivity,    Er = reff = R2 1 + ae−i0  Ei

(2.4)

 where a = 1 − R2  R3 R2 The total cavity loss  of the laser with the external cavity is given by:  c   c   = w − ln reff R1 = 0 − ln 1 + ae−i0  l l

R1

R2

R3

Figure 2.1 Schematic of an external cavity configuration.

(2.5)

26

Theoretical Analysis

where 0 is the cavity loss of the laser diode without feedback (w being the waveguide loss of the cavity). If we compare this loss term with Equation (2.1), then we can equate: 1 1 Eei0 t = 0 Eei0 t +
E ˜ t −  ei0 t− 2 2

(2.6)

and substituting for  we obtain:   c ln 1 + ae−i0  =
e ˜ −i0  2l which for values of a
1 gives:


˜ = ca 2l

(2.7)

(2.8)

The rate-equation for the E-field is coupled with another rate equation for the carrier density, n: J d n = − n − G n E2 + dt ed

(2.9)

The first term of this rate-equation describes the loss of carriers due to spontaneous emission and other loss processes, where is the inverse of the carrier lifetime. The second term describes carrier depletion through stimulated emission, where Gn is the optical gain. The third term describes the increase in carrier density due to electrical injection where J is the injection current density, e is the electronic charge and d is the active layer thickness. In Lang and Kobayashi’s paper, the authors show how changes in the carrier density modify the refractive index of the semiconductor, which in turn alters the resonant frequency of the laser diode. This effect was later included explicitly in the Lang and Kobayashi equations by the use of the alpha factor. In order to include the effect of the carrier density change on the resonant frequency, a linear approximation for the gain and refractive index is used. If we consider small variations n around the value for the carrier density at threshold of the solitary laser, nth , then  n = th + n

 n

using this approximation the resonant frequency can be expressed as:

  N = th − th n th n

(2.10)

(2.11)

The gain Gn can be similarly approximated: G n = Gth + n

G n

(2.12)

At threshold, the gain in the solitary laser is equal to the cavity losses of the diode allowing us to obtain an expression for n:  G

n = G n − 0  (2.13) n

Steady State Analysis of the Lang–Kobayashi Equations

27

Substituting this expression into Equation (2.11) we obtain: N = th +

Gn − 0  2

(2.14)

where: 2 = − th th



  G n n

(2.15)

substituting N relative to the solitary laser frequency at threshold (N − th ) into Equation (2.1) gives:
dE t = 1/2 1 + i  Gn n t − nth  E t + E t −  e−i0  dt in

(2.16a)

It should be noted that in the literature an alternative form of this equation is occasionally employed:
dE t = 1/2 1 − i  Gn n t − nth  E t + E t −  ei0  dt in

(2.16b)

this different form being a consequence of the exact definition of the propagating electromagnetic field (Equation 2.16b arises from the assumption that the field is varying as e−it ). In this way, the alpha factor, popularized by Henry [6] as a convenient phenomenological way of describing the strong amplitude and phase coupling that occurs in semiconductors, is included explicitly in the Lang–Kobayashi equations.

2.3 STEADY STATE ANALYSIS OF THE LANG–KOBAYASHI EQUATIONS By defining Et in terms of a slowly varying amplitude and phase: E t = E0 t ei t

(2.17)

the complex E-field equation can then be rewritten as two purely real equations for the amplitude and phase: dE0 t = 1/2Gn n t − nth  E0 t + 
/ in  E0 t −  cos 0 t + t dt d t
E t −  = Gn n t − nth  + 0 sin 0 t + t dt 2 in E0 t

(2.18a) (2.18b)

where, t = t − t − , and nth is the threshold carrier density of the solitary laser. Several groups have revised and extended various aspects of the original analysis [7–17] and shown that a steady state analysis of the equations again gives rise to a coupled set of

28

Theoretical Analysis

equations that describe how the threshold carrier density (and therefore the threshold gain) and the lasing frequency are effected by external optical feedback. The steady state solutions are found by substituting E0 t = E0 t − = Es  nt = ns and t = s t = s −0 t into Equation (2.18) and setting the derivatives to zero, thus: 0 − s =


 1 + 2 sin s  + arctan  in

ns = nth −

2
cos s  Gn in

(2.19a) (2.19b)

The individual solutions to the coupled equations (Equation 2.19a,b) are usually referred to as either a mode or an anti-mode of the external cavity. Constructive and destructive interference between the laser field and the reflected field gives rise to these so-called modes and anti-modes of the system. These modes are also commonly referred to as compoundcavity modes of the system. The introduction of the alpha factor has a significant effect on the values of these steady state or fixed point solutions. It is evident from the second expression given above, Equation (2.19b), that the threshold carrier density and hence the threshold gain can be significantly modified by the external optical feedback. For a fixed feedback rate, the threshold carrier density, and hence the optical power, can be dramatically increased or decreased by varying the length of the external cavity. The photon density is given by: Ps = Es 2 =

1 Gn ns − n0 



J n − s ed s

(2.20)

The transcendental Equation (2.19a) is plotted in Figure 2.2 for the case when all the parameters are kept constant except for the optical feedback rate. Increasing the feedback rate is seen to increase the number of solutions, which correspond to where the curves intersect the horizontal line. The solutions are formed in pairs, one mode and one anti-mode. Equation (2.19a) allows us to find all the possible compound cavity modes that the system can support, if we now differentiate Equation (2.19a) with respect to s we can determine whether the curvature near each solution is positive giving a local minimum (modes), or negative giving a saddle point (anti-modes). Hence the modes satisfy the following condition [7]: do
  = 1+ 1 + 2 coss  + arctan  > 0 ds in

(2.21)

It is obviously important to be able to visualize these fixed-point solutions in a convenient way. The usual method employed is shown in Figure 2.3. The modes are denoted by stars and the anti-modes by triangles. Two of the modes, the minimum linewidth mode and the maximum gain mode, are of particular importance and will be discussed in detail later. The number of possible lasing modes is determined by Equation (2.19b), however, it is convenient to recast this equation in the following form:

s  = −C sin  s  + 0 

(2.22)

Steady State Analysis of the Lang–Kobayashi Equations

29

0.02

0.01

0.0

–0.01

–0.02 –2

–1

0 (ωs – ω0)

1

2

Figure 2.2 Round trip phase change versus optical frequency for three external reflectivities. Graphical solution to Equation (2.19a) for two feedback strengths r3 = 002 (solid line) and r3 = 004 (dotted line). Steady state solutions correspond to zeroes of the round-trip phase change. Changing the external cavity length changes the period of the oscillation.

8400 maximum gain mode 8200

external cavity mode

Ps 8000 antimode 7800 solitary 7600 –30

–20

–10

0 ∆ωsτ (rad)

10

20

30

Figure 2.3 Location of the modes (stars) and anti-modes (triangles) in the (Ps  s ) plane. Source: A.M. Levine, G.H.M. van Tartwijk, D. Lenstra and T. Erneux, ‘Diode lasers with optical feedback: stability of the maximum gain mode’, Phys. Rev. A, 52, R3436–R4339 (1995).

30

Theoretical Analysis

from which it is clear that the two dimensionless parameters C and 0 control the number of modes. The external phase is given by: 0 = 0  + arctan  

(2.23)

and the dimensionless feedback parameter is given by:
 C=  1 + 2 in

(2.24)

Acket et al. [18] were the first to show that the (C, 0 ) parameter space separates into regions containing different numbers of modes (Figure 2.4). The curves that define each region obey the following relationship.   (2.25) 0 = 2k + 1  ± cos−1 1/C ∓ sin cos−1 1/C where k is an integer and C and 0 are dimensionless parameters. Clearly, the line C = 1 is a special case: when C < 1, the laser has only one frequency to choose from, which is only slightly shifted from the solitary laser frequency. As the feedback level is increased, C increases and additional modes are created in pairs. The lines given by Equation (2.25) actually indicate the locations of saddle-point bifurcations [4]. An alternative method of plotting the fixed points has been suggested by Mork [9], whereby, after a suitable variable substitution, all the modes sit on the circumference of a circle, whose radius is proportional to the optical feedback strength (Figure 2.5). The two normalized variables used to produce Figure 2.5 are: y = in Gn n − ns  /2

(2.26a)

x + y = in s − 0 

(2.26b)

10 VII 8

V

V

C

6 III

4

III I

2

0 –1.0

–0.5

0.0

0.5 1.0 ϕ0/π [rad]

1.5

2.0

Figure 2.4 The number of modes are shown in the (C, 0 ) parameter space. Source: G.H.M. von Tartwijk and D. Lenstra, Quant. Semiclassical Optics, 7, 87 (1995).

Steady State Analysis of the Lang–Kobayashi Equations

31

0.02

y

0.01

0.00

–0.01

–0.02 –0.02

–0.01

0.00 x

0.01

0.02

Figure 2.5 Modes and anti-modes are plotted on a normalized plane in the complex x y plane. Two feedback strengths are shown. The anti-modes are all in the hatched section on the left-hand side of the figure. Source: J. Mork, B. Tromborg and J. Mark, ‘Chaos in semiconductor lasers with optical feedback: theory and experiment’, IEEE J. Quantum Electron., 28, 93–108 (1992).

A mode may, however, be dynamically unstable when subject to a small perturbation. This may cause the system to rapidly diverge away from the stable point. In order to assess the stability of a mode, a small-signal linear stability analysis can be performed. Here a small perturbation from the mode’s stationary position is assumed to cause the system to oscillate around its original position in the form xt = x0 texp(st), where xt may represent nt Et, or  t. The parameter ‘s’ is complex and is written as s =  ± iR , where  is the damping factor and R is the angular relaxation oscillation frequency. In order for a mode to be dynamically stable, a negative damping factor is required. The process used to determine whether a particular mode is robust to small perturbation will now be discussed. A small signal analysis on the LK equations yields rate equations for nt Et, and  t, these are then subjected to a Laplace transform. After some algebraic manipulation, a transcendental equation for the system determinant, Ds, can be obtained whose roots, s, are sought. Ds is given below: Ds = s3 + a2 s2 + a1 s + a0

(2.27)

where the coefficients are: a0 =

1 2
+ A coss  R in

(2.28a)

a1 =


2 A2 2
+A coss  + 2R R in in2

(2.28b)

a2 =

1
2 A2
+ A 2R coss  − sins  R in2 in

(2.28c)

32

Theoretical Analysis

and A = 1 − exp−s 

1 1 = + Gn P s  R sp

and

2R =

Gn Ps ph

Approximate expressions for the curves where Ds has zeros on the imaginary axis have been derived by several groups. For example, Tromborg et al. [7] have shown that the following stability condition must be satisfied for a damped relaxation oscillation:    2   in  2 (2.29) −1 y < x + 2 R R 2R sin2  / 2 for all values of  that satisfy: 

 2 − 2R R =  cot  / 2

(2.30)

Each of the solutions of Equation (2.30) yields a stability boundary line given by Equation (2.29). The shaded area in Figure 2.6 denotes the unstable region. Notice that for very short external cavities, the system is always stable, regardless of feedback strength. The dynamics of a narrow linewidth single longitudinal mode laser subject to very weak feedback can be described using a ‘thermodynamic’ potential model, which is based on ignoring ALL the power fluctuations [19–21]. One consequence of these calculations is the assertion that the most stable external cavity mode is the one in the deepest potential well, namely, the minimum linewidth mode [22]. Of course, this conclusion is only valid when the relaxation oscillations are heavily damped. The work of Tromborg et al. [8], and Ritter and Haug [23], has shown that the relaxation oscillation becomes destabilised at quite low feedback levels. Subsequent work by Levine et al. [24] on the maximum gain mode and minimum linewidth mode has indicated that in many cases the maximum gain mode will 0.010

Feedback level κ

0.008 0.006 0.004 0.002 0.000 0

1

2 3 ωRτ/(2π)

4

5

Figure 2.6 Minimum feedback levels required reaching the corresponding stability boundaries versus external round trip time. The shaded area denotes the unstable region. Source: J. Mork, B. Tromborg and J. Mark, ‘Chaos in semiconductor lasers with optical feedback: theory and experiment’, IEEE J. Quantum Electron., 28, 93–108 (1992).

Steady State Analysis of the Lang–Kobayashi Equations

33

be the only stable external cavity mode. Levine et al. state that their extensive asymptotic analysis, based on the ratio of the photon and carrier lifetimes, indicates that there is always a feedback phase condition where an optimised maximum gain mode is stable for arbitrarily large feedback levels. The feedback phase for these two modes are:

s  = − 

or

s  = 0 mod 2

Maximum gain mode

and

s  = 0

or

s  = − arctan  

Minimum linewidth mode

It can be shown that the minimum feedback level that destabilizes the system (via a Hopf bifurcation) for a mode with a frequency s is, [23, 24]: H =

1/s + Gn P0 B 1 − cos R 

(2.31)

where:  B = − 1 + 2 cos s  − arctan 

(2.32)

The significance of the parameter B has been studied by Levine et al. [24]. They maintain that only modes where B > 0 can be destabilized by increased feedback. One such mode is the minimum linewidth mode, where: 2 − 1 Bmlw = √ 1 + 2

(2.33)

Hence this mode is easily destabilized, and for large C values, all adjacent modes will also be unstable. In the case of the maximum gain mode we find that: Bmgm = −1

(2.34)

and thus this mode cannot be destabilized by increased feedback. Modes in the vicinity of the maximum gain mode are also likely to be stable. Consequently, for any value of C, by careful control of the phase of the external cavity it should be possible to adjust the system so that there exists a stable mode at the maximum gain mode detuning. For other feedback phases, the highest power mode may not be stable against relaxation oscillations. For very large values of C, however, there will be a range of stable modes near the maximum gain mode, these modes satisfy the following condition: − arctan 1/  < s  < 0

(2.35)

However, if the noise is large enough, be it from spontaneous emission or mechanical vibration, the system is likely to hop from the stable mode to unstable mode(s).

34

Theoretical Analysis

2.4 MULTIMODE ITERATIVE ANALYSIS OF THE DYNAMICS OF LASER DIODES SUBJECT TO OPTICAL FEEDBACK For low feedback levels the standard Lang–Kobayashi rate equations [3] have been shown to successfully describe the behaviour of laser diodes. These rate equations require that the dynamic change in the complex propagation constant, k, is small, that is, kl
1, where l is the laser diode cavity length. This condition is violated for the coherence collapse state at moderate to high feedback levels. Analysis using Lang–Kobayashi type rate equations has included work on multimode operation up to and including coherence collapse state [25]. Rate equations have also been used to study the properties of modulated laser diodes subject to weak optical feedback in both the multimode [23] and single mode [26, 27] cases. However, much less work has been focused on predicting the effects of strong optical feedback on the properties of both CW and modulated laser diodes. The presence of several different characteristic times within the Lang–Kobayashi rate equations means that the equations form a set of stiff differential equations, and are thus problematical to solve numerically. Enhancing the standard rate equations to include multimode effects and multiple external cavity reflections further increases the numerical difficulties. To ensure numerical stability, very short time steps have to be used and consequently the simulations are computationally intensive and are thus time-consuming. The need to use very short time steps does not mean that the equations encompass ultrafast dynamics, but instead reflect the numerical difficulties associated with solving stiff differential equations. An alternative method for modelling optical feedback in laser diodes can be obtained using an iterative approach based on a travelling wave description. Such an approach has been developed by Sporleder [28] and extended by Mork [29]. The principal advantages of this approach are, first, that the technique can be applied to higher levels of optical feedback and, second, it can easily be extended to include multimode laser cavity operation. Also unlike the rate equation approach, the iterative technique explicitly includes multiple external cavity round trips. The iterative nature of the travelling wave description also ensures that the formalism is computational efficient, as time steps equal to the laser cavity round-trip time are used.

2.4.1 Dynamics of MultiMode Laser Diodes A schematic of the situation being considered, that of a laser diode operating in an external cavity, is shown in Figure 2.7. The iterative scheme used to analyse the system shown in Figure 2.7 is based on a perturbation approach. The unperturbed state is assumed to be that of a solitary laser and the perturbation is provided by optical feedback and noise. A reference plane is defined just inside the right-hand laser facet at z = 0. A+ t is the right-moving field passing the laser facet of field reflectivity, r2 . A− (t − ) is the left-moving field re-entering the laser diode and  is the external cavity round-trip time. This approach is based upon the work of Mork [29]. The iterative single mode equations may be extended to include multiple longitudinal modes by assuming that the angular mode frequencies of the solitary laser are: m = 0 + m 

m = 0 ±1     ±N

(2.36)

Multimode Iterative Analysis of Laser Diodes Dynamics

35

r3

r2

r1

τ

τin z=0

Figure 2.7 Schematic diagram of a laser diode subject to optical feedback. The field reflectivities of the laser and external cavity are denoted by r1 , r2 and r3 respectively.

where  = 2in is the approximate longitudinal mode spacing. Here in is the diode cavity round-trip time. The total electric field at z = 0 is taken as:  E + t = A+ (2.37) m t exp im t m

The right-moving slowly varying envelope function of each longitudinal mode A+ m t is calculated at steps of the laser diode’s internal round-trip delay, in , and is given by:     + 2 t  −r2 r3 q−1 A+ +  = G A+ in tot r2 Am t + r3 1 − r2 m m t − q q=1



× exp −im q + in F t where:

  2   1 + i  Gn n − nth  + G  − p in 1  Gtot = exp  r2 2

(2.38)

(2.39)

Here r2 and r3 are the amplitude reflectivities of the laser facet and external reflecting surface, respectively. nth is the threshold carrier density of the solitary laser with feedback. A Langevin noise term, Ft, is used to model spontaneous emission noise. The spontaneous emission spectrum has been taken to be mode independent. The local gain G n is assumed to have a parabolic frequency dependence, as well as the usual carrier density dependence. G  n = Gn n − n0  +

2 G   − p 2

(2.40)

 Here Gn is the differential gain, n0 is the carrier density at transparency, G = 2 G 2 is a constant, and the peak of the gain shifts with carrier density as: 

p = 0 + n n − nth 

(2.41)

where n = 2 0 n2 is a constant. For the solitary laser it is assumed the gain peak coincides with one of the longitudinal modes. The optical frequency m t of each mode is given by: m t = m +

d m t dt

(2.42)

36

Theoretical Analysis

where m t is the phase of the slowly varying envelope function of each mode, Am t. The carrier rate equation can be incorporated into an iterative scheme by using a secondorder Taylor expansion: n t + in   n t + in

dn in2 d2 n + dt 2 dt2

(2.43)

The first-order derivative is simply the well-known standard carrier rate equation and the second order is found by differentiation. However, the stimulated emission term has to be modified to account for the presence of multiple modes. The carrier rate equation in the multimode case becomes: 2   I n dn  = − − G m  n A+ m t dt eV s m = ±N

(2.44)

where N is the number of longitudinal modes being considered.

2.4.2 Steady State Solutions The steady state solutions can be deduced for the single longitudinal mode case, (m = 0), by considering the resonance condition: rR rL = 1 where rL and rR are effective reflectivities, given by:    2 1 n    + i  G in − n + G −  n th  s 1  exp −i  − 0  in  rL = exp  r2 2    −r2 r3 q−1 exp −i0 qext  rR = r2 + r3 1 − r22

(2.45)

(2.46a)

(2.46b)

q=1

= rR  exp i∠r R  By taking the logarithm of Equation (2.45) and then separating the real and imaginary parts, the following two conditions must be satisfied before lasing can be achieved:

1 rR  ⇒ ln (2.47a) + Gn n − nth  in = 0 Re ln rR rL  = 0 r2 2 1 Im ln rR rL  = ±2 ⇒ ∠rR + Gn n − nth  in −  − 0  in = ±2 2

(2.47b)

The right effective reflectivity, rR , simplifies considerably if an infinite number of reflections are considered: rR =

r2 + r3 exp −i0 ext  1 + r2 r3 exp −i0 ext 

(2.48)

Multimode Iterative Analysis of Laser Diodes Dynamics

The case of finite external reflections is only slightly more complicated:

  exp −i0 ext  1 − −r2 r3 q exp −iq0 ext  rR = r2 + r3 1 − r22 1 + r2 r3 exp −i0 ext 

37

(2.49)

where q reflections have been considered. In the limit of one external reflection, the same expressions as those obtained for the steady state solutions of the LK model are obtained, as will be seen in the next subsection.

2.4.3 Comparison with Lang–Kobayashi Rate Equations We will now show that in the limit of weak feedback (single external reflection) and single longitudinal mode operation the iterative model reduces to the Lang–Kobayashi model. Using the aforementioned assumptions, Equation (2.38) simplifies to:    A+ t + in  = G r2 A+ t + r3 1 − r22 A+ t −  exp −i (2.50) where the round-trip gain, G, (neglecting the frequency dependence), is:   1 1 + i  Gn n − nth  in G = exp r2 2

(2.51)

Expanding the exponential function and only keeping the first two terms, gives: G

1 + 1 + i  Gn n − nth  in /2 r2

(2.52)

Substituting this back into the simplified iterative equation and rearranging:   1 + i  Gn n − nth  A+ t + in  − A+ t 1 + i  Gn n − nth  1 A t + = + in 2 in 2 ×
A+ t −  exp −i

(2.53)

In the limit of in → 0 the right-hand side reduces to dA/dt and the terms multiplying the delayed feedback, simplify to 1/in , and hence we obtain the Lang–Kobayashi rate equation: dA+ t
= 1/2 1 + i  Gn n − nth  A t + A+ t −  exp −i dt in

(2.54)

This demonstrates that for weak feedback levels the two models will predict the same behaviour. As the feedback increases, and the carriers begin to fluctuate more significantly, differences between the two models will begin to appear. This is purely a consequence of the way the round-trip gain is incorporated into the two models. The iterative model was derived using a travelling-wave approach and predicts the optical field just inside the facet facing the external cavity. It also admits larger variations in the gain. These effects can clearly be seen in Figure 2.8 which show the steady state solution obtained using the LK model (circles) and the iterative model (crosses). For low to moderate feedback levels the two models are in close agreement (Figure 2.8(a)). At higher feedback levels there are significant differences; the iterative model generally predicts more modes and the modes are shifted to more positive frequencies and also have lower powers (Figure 2.8(b)).

38

Theoretical Analysis

Figure 2.8(a) Steady state solutions for the case of a moderate feedback level r3 = 002 when  = 1ns, = 35, and ph = 15 ps. The LK solutions are denoted by circles and those of the iterative model by crosses. Here P0 is the solitary laser power.

Figure 2.8(b) Steady state solutions for the case of a high feedback level r3 = 02 when  = 1ns, = 35, and ph = 15 ps. The LK solutions are denoted by circles and those of the iterative model by crosses. Here P0 is the solitary laser power.

2.5 CAVITY LENGTH EFFECTS The previous section mentions in passing that under certain conditions a laser diode with a short external cavity can be unconditionally stable. It was also seen in Figure 2.6, which reported the work of Tromborg et al., that the maximum feedback strength for stable operation depended on the cavity length. As the cavity length is increased in Figure 2.6, it is observed

Cavity Length Effects

39

that the maximum feedback strength approached an asymptotic value for long cavities (shown as a dotted line). This variation in behaviour with cavity length justifies the distinction between ‘short’ and ‘long’ external cavities. The following sub-sections examine these two cases, beginning with long external cavities.

2.5.1 Long External Cavities In an attempt to distil the results of several numerical studies of the stability of external cavity laser diodes, Helms and Petermann [30] derived an analytic expression for the critical feedback strength from the modulation current to optical power transfer function. The expression was derived under the assumption of a long external cavity, which Helms and Petermann define as being a cavity that satisfies the inequality r  1 where r is the solitary laser relaxation oscillation angular frequency and  is the cavity round-trip delay. The starting point for the analysis is the small-signal transfer function of a single mode laser diode with optical feedback: HK j = 1 − Kj

Hj 1 − Kj Hj

(2.55)

In the expression above Hj is the solitary laser transfer function, assumed to be of the standard second-order form: Hj =

j r

2

1 j + +1 d

(2.56)

where d is the damping frequency of the solitary laser modulation response and r is the resonant frequency of the response. The modification to the solitary laser response due to weak optical feedback is given by the term: √ kc 1 + 2 Kj = − (2.57) 1 − exp −j j where  is the external cavity round trip time and , the linewidth enhancement factor, is assumed to be greater than 1 in deriving this expression. The strength of the feedback term is accounted for by the coefficient kc , which for a Fabry–Perot laser diode is related to the external cavity reflectivity by the expression:   2  1 − R2   Rext (2.58) kc = √ in  2 R2  where R2 is the facet reflectivity of the solitary laser, in is the laser diode internal round-trip time and Rext is the external cavity reflectivity. As stated above, the expression for Kj is correct for weak feedback, which in practice means values of Rext of the order 10−3 or less. Examination of Equation (2.55) reveals that the system will be unstable to a small perturbation (it is important to realize that Equation (2.55) is the small-signal transfer

40

Theoretical Analysis

function) if an unstable pole occurs. If we assume that r < d , then Hj has its maximum near r and an unstable pole occurs when: Kj H j = 1

(2.59)

It is at this point in the analysis that the long cavity assumption is made, since if it is assumed that r  1, then the term: 1 − exp −j in Kj (which has a maximum whenever  is an odd multiple of ) will lead to a maximum in Kj near r . Effectively, the condition  1 means that the exponential term oscillates rapidly, ensuring that Kj exhibits a local maximum near the maximum of Hj, which occurs for  ≈ r . With the long cavity assumption in place, it is then simple to derive, from Equation (2.59), the following expression for the critical feedback coefficient: kc critical =

1 2r √ 2d 1 + 2

(2.60)

This expression is seen to depend only on the solitary laser response and on the linewidth enhancement factor. There is no explicit dependence on the external cavity length, since this value for kc critical is the long-cavity asymptotic value. In validating this expression against numerical simulations of external cavity laser diodes Helms and Petermann observed that the expression in Equation (2.60) deviated from the numerical results for low values of the linewidth enhancement factor. They then extended their expression for the critical feedback coefficient by replacing the term √

1 1 + 2

with an empirically determined -dependence to yield the expression: √ 2r 1 + 2 kc critical = 2d 2

(2.61)

Figure 2.9 shows the variation of critical feedback with linewidth enhancement factor described by Equations (2.60) (which deviates from the true behaviour for low values of ) and (2.61) and compares them with numerically determined values of the critical feedback coefficient. The improved agreement of Equation (2.61) with the numerically determined results is clear from Figure 2.9.

2.5.2 Short External Cavities A significant part of the work undertaken to date on the influence of external optical feedback has been concerned with external cavities whose lengths are several orders of magnitude greater than that of the laser diode. Despite early work by Petermann and co-workers, [10, 22, 31–33] that showed that smaller external cavities exhibit markedly different effects to those

Cavity Length Effects

41

Critical feedback level fext,c

10–2

10–3

10–4

1

2 3 4 5 Linewidth enhancement factor α

6

Figure 2.9 Variation of critical feedback strength with linewidth enhancement factor. Solid line: numerically determined critical feedback strength; dashed line: from Equation (2.60); dash-dotted line: from Equation (2.61). Source: J. Helms and K. Petermann, ‘A simple analytic expression for the stable operation range of laser diodes with optical feedback’, IEEE Journal of Quantum Electronics, 26, 5, 833–6 (May 1990).

observed in long external cavities, little further work in this area has been reported. Schunk and Petermann theoretically predicted that there is an external cavity length below which coherence collapse should not be observed [22]. In a later paper Tager and Petermann also theoretically predicted that narrow bands of high-frequency oscillations should be observed for certain feedback levels and cavity lengths [31]. Tager and Elenkrig have investigated the modulation response of short external cavities [33]. Work on frequency modulation in external cavities has recently again highlighted that there is a significant difference between long and short external cavities [34]. Petermann has previously defined a short cavity as one where the product of the relaxation oscillation frequency and the external round-trip time is less than unity [22, 31]. A slightly broader definition of what constitutes a short cavity will be used here, where a short cavity is defined as one where the development of coherence collapse is a function of external cavity phase. In long cavities, sub-wavelength changes in the cavity do not influence the dynamics of the system once coherence collapse has occurred. The importance of the product of the relaxation oscillation frequency and the external roundtrip time can clearly be seen in Figure 2.10, which was generated [22] using the Lang– Kobayashi equations. The boundary between stable and unstable operation only starts to exhibit any dependence on the external cavity length when the period of the relaxation oscillation is longer than the external roundtrip time. Work by Tager and Elenkrig [33] has shown that mode beating effects between external cavity modes, mediated via modulation of the carriers at intermode frequencies, must be considered in the short cavity limit. Tager and Elenkrig revised the small signal steady state analysis undertaken by Tromborg et al. [7] and found that additional terms, discarded in the original analysis, become critically important in short cavities.

42

Theoretical Analysis

τext fr (fext = 0) 1

0.1

10

IV Feedback fraction fext

10–2 C=1

unstable "coherence-collapse" regime

10–3

10–4

10–2

10–1

100 τext

101 [ns]

Figure 2.10 Feedback strength versus external round-trip time in the short cavity limit. The dashed line corresponds to a feedback coefficient C = 1. Source: N. Schunk and K. Petermann, ‘Stability analysis for laser diodes with short external cavities’, IEEE Phot. Technol. Lett., 1, 49–51 (1989).

In a later paper co-authored by Tager and Petermann [31], further numerical calculations were undertaken and bands of high-frequency (>20 GHz) self-oscillation were observed for certain feedback levels and external cavity lengths, shown in Figure 2.11. In all these calculations the external phase was always tuned to ensure that the condition for minimum linewidth mode was satisfied. Tager and Petermann briefly considered other external cavity phases and their numerical calculations showed that various types of dynamical behaviour could be induced, including coherence collapse, shown in Figure 2.12. Later work by Jones et al. [35] has extensively investigated the effect that the external cavity phase has on the dynamics, in the short external cavity limit. The chaotic dynamics of the short cavity limit were found to be restricted to islands in the feedback strength and cavity length phase plane. Increasing the external cavity length caused these islands to coalesce until one continuous unstable region is obtained, indicating that the system is now insensitive to phase. The observed behaviour has been explained in terms of the stability of the modes that the system can support. This transition from several isolated unstable regions sensitive to phase variations to one continuous regime independent of phase was studied theoretically using a two-step process. The modes of the system were first found, for all possible external phase values and each of the feedback levels considered. Then the small signal stability of each of these modes was determined. A detailed study of the nature and evolution of these unstable regions, or islands, for different external cavity lengths has been undertaken and this work has revealed some interesting features. The shaded areas in Figure 2.13 represent the unstable islands. Each of the panels has been calculated for a particular external cavity length, ranging from 4 mm to 18 mm. Increasing the external cavity

Cavity Length Effects

43

υRτ

0.175

0.075

0.275

stable

feedback fext

0.20 high-frequency oscillations

0.15

0.10

b

0.05 coherence–collapse 0.00

a 15

25 35 45 EC roundtrip time τ (ps)

55

Figure 2.11 Boundaries of the various unstable regimes as the feedback strength and external round trip time are varied. Source: A.A. Tager and K. Petermann, ‘High frequency oscillations and self-mode locking in short external cavity laser diodes’, IEEE J. Quant. Electron., 30 1553–61 (1994).

feedback fext

0.3

high-frequency oscillations

0.2

0.1 coherence–collapse 0.0 –180

–120

–60

0

60

120

180

detuning φa (deg.)

Figure 2.12 Boundaries of the various unstable regimes as the feedback strength and external phase are varied. Source: A.A. Tager and K. Petermann, ‘High frequency oscillations and self-mode locking in short external cavity laser diodes’, IEEE J. Quant. Electron., 30 1553–61 (1994).

length squashes vertically and broadens horizontally the shape of the unstable region and hence the separation between islands reduces, as can be seen in Figure 2.13. This process continues until at some point all the islands merge and there is a band of feedback levels for which the system is unstable regardless of external phase value. The islands of unstable dynamics are formed when either the only mode that the system can

44

Theoretical Analysis

0.35 (a) 4 mm

(b) 12 mm

0.16

0.30

0.14

0.25 Reflectivity r3

Reflectivity r3

0.18

0.12 0.10 0.08 0.06

0.15 0.10

0.04 0.01

0.20

0.05 0.01 –2

–1

0 1 Phase pi

2

–2

–1

0 1 Phase pi

2

0.25

0.35

(d) 30 mm

(c) 18 mm 0.30 0.20 Reflectivity r3

Reflectivity r3

0.25 0.20 0.15

0.15

0.10

0.10 0.05 0.05 0.01

0.01 –2

–1

0 Phase pi

1

2

–2

–1

0 Phase pi

1

2

Figure 2.13 Stability of the system for various feedback strengths and external cavity phases. The hatched area denotes the unstable regions. Source: R.J. Jones, P.S. Spencer, J. Lawrence and D.M. Kane, ‘Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback’, IEE Proc. Optoelectronics, 148, 7–12 (2001).

support becomes unstable or the multiple modes that do exist are stable for different ranges of the external phase and these ranges do not overlap. In this latter case, the system is stable for only certain phase values and the optical frequency of the mode(s) no longer varies in a linear manner as the external phase is varied. A second distinct type of stable operation was also observed. The system in this case supports multiple modes, which are stable for different ranges of the external phase.

2.6 COUPLED CAVITY ANALYSIS Most attempts at the theoretical analysis of laser diode systems subject to external optical feedback have made the assumption of a single longitudinal laser mode. The Lang–Kobayashi model [3] is probably the most widely used single-mode analysis of feedback effects in laser

Coupled Cavity Analysis

45

diodes. The Lang–Kobayashi method is based on a perturbation approximation, where the external feedback acts as a perturbation to normal solitary laser operation. Such single mode analyses are at odds with the majority of the reported experimental results, which were obtained with multimode (Fabry–Perot) laser diodes. In a few cases multimode phenomena have been included in modelling laser diodes with optical feedback by extending the Lang–Kobayashi analysis to multiple modes, either in a rate-equation model [25] or in an iterative scheme, as has been previously described. In both the multimode Lang–Kobayashi approach and the iterative scheme, the system is assumed to have equally spaced longitudinal modes whose wavelengths are each perturbed slightly by the optical feedback. This requires an a priori assumption about the number of longitudinal modes and their wavelengths before any simulations are undertaken. Coupled cavity analysis [36] accounts for the effect of optical feedback in a fundamentally different way, where the system is described as a coupled cavity system in a method that draws on the early theory of Lamb, Lang and Scully on the whole-universe mode theory of lasers [37, 38]. That theory was originally proposed in order to explain the narrow linewidth of lasers, where laser behaviour is modelled accounting for the effect of coupling between the Fox–Li [39] modes of the Fabry–Perot laser cavity and an external cavity that models the effect of power leakage into the universe. The original work concentrated on the limiting case of an infinitely long external cavity (i.e. no back-reflection in a finite time). In that case the laser line was shown to be very narrow due to the locking together of the many modes of the universe to a single frequency. In contrast, the coupled cavity analysis is for the case when the frequency locking does not occur due to the finite length of the external cavity, and the resulting output power spectrum shows distinct modes. The time evolution of the mode wavelengths and amplitudes is calculated by self-consistently solving for the wavelengths of the coupled-cavity longitudinal modes and the optical susceptibility. In contrast to the Lang–Kobayashi approach, no initial assumptions about the number of cavity modes or of their wavelengths are required.

2.6.1 Theory The system is modelled as a coupled-cavity with a semiconductor gain region coupled with an air-filled external cavity, as illustrated in Figure 2.14. The structure is bounded by two semi-infinite regions, with air to the left of z0 and a region with refractive index 3 to the

Air

Gain

η0 = 1

η1 = 3.3

η2 = 1

κ0 = 0

κ1 < 0

κ2 = 0

Z0

Reflector η2 + √R η3 = ———— η2 − √R κ3 = 0

Air Cavity

Z1

Z2

Figure 2.14 Laser diode with external reflector as a coupled-cavity system. Refractive index for each section is denoted by , gain by
. Negative values of
indicate a region in gain. Source: Pierce, P. Rees and P.S. Spencer, ‘Multimode dynamics in laser diodes with optical feedback’, Phys. Rev. A, 61, 053801 (2000).

46

Theoretical Analysis

right of the external cavity at z2 . The value of 3 is chosen to yield an identical reflectivity to the proposed external reflector:  1 + Rext  3  = (2.62)  1 − Rext  where Rext is the external cavity reflectivity. It is important to note that Rext can be a function of frequency, , as suggested by the expression above for the frequency-dependent refractive index of the reflector region. The boundary between the semiconductor gain region and the external cavity is z1 . The two outer regions in Figure 2.14 are the universe into which optical power leaks in this model. The inclusion of arbitrary external reflectivities in the model, through Equation (2.62), allows the coupled cavity method to be extended to include the effect of frequency-selective feedback. Further extensions to the model could be made to cater for such device structures as distributed-feedback (DFB) lasers or vertical cavity surface-emitting lasers (VCSELs), through suitable modifications to the matrix analysis presented later in this section. The longitudinal modes of the coupled system are calculated using the method of Ebeling and Coldren [40] self-consistently with a determination of the optical susceptibility. The analysis must include the variation of the gain and refractive index with the carrier density in order to describe the dynamic behaviour of the system. The Ebeling–Coldren analysis calculates the wavelength and threshold gain of each of the possible modes of the combination of laser and external cavity. This calculation is undertaken under the assumption of sinusoidally varying electromagnetic fields propagating in both directions in each section of the system depicted in Figure 2.14 whose electric field component satisfies the wave equation: d 2 Em 2 + km Em = 0 dz2

(2.63)

The optical properties of each section are described by the complex propagation constants: km =

  + i
m  c m

(2.64)

where m is the refractive index of the mth section and
m is related to the material gain, gm (in reciprocal metres) by the expression:
m = −

c gm  2

(2.65)

Each section is assigned a backward-propagating field amplitude Am and a forwardpropagating field amplitude Bm , so that the total field Em in each section is given by Em = Am e−ikm z + Bm eikm z

(2.66)

By matching the field amplitudes at the interfaces situated at positions z1 , z2 and z3 , and accounting for the optical gain
m in each section, it is possible to write an equation that relates the field amplitudes in the left-most air section A0 and B0 to the amplitudes in the right-most reflector section A3 and B3 via the complex propagation constants. The only propagation constant that is variable is that of the laser gain section, k1 . This may change during the simulations because the refractive index of the gain section, 1 , depends

Coupled Cavity Analysis

47

on the instantaneous value of the carrier density. The carrier density itself varies with time in accordance with the carrier rate equation, to be described below. At each time step the refractive index, 1 , is obtained from the current value of the optical susceptibility. The resulting equation interlinking the field amplitudes at either end of the system is best written in matrix form: A3 A0 = Q z2  Q z1  Q z0  (2.67) B3 B0 where each matrix Q is of the form:   k m+1 + km ikm+1 −km zm km+1 − km ikm+1 +km zm e e   2km+1 2km+1  Q zm  =   km+1 − km −i k +k z km+1 + km −i k −k z    m+1 m m m+1 m m e e 2km+1 2km+1

(2.68)

Since we know that there should be no incoming electromagnetic fields, the amplitudes B0 and A3 are set to zero. Under this condition, Equation (2.67) can then be solved numerically for the complex propagation constant of the gain section k1 . This is most easily accomplished by writing Equation (2.46) in the following form: 0 A0 (2.69) =Q B3 0 where: Q ≡ Q z2  Q z1  Q z0  This equation stipulates the following condition on k1 the complex propagation constant of the gain section (where A0 has been set to unity with no loss of generality): Q11 = 0

(2.70)

or #   eik3 z2 k3 + k0  e−ik0 z2 −z1  k0 + k1 2 e−ik1 z1 − k1 − k0 2 eik1 z1 8k3 k0 k1 $   − 2i k3 − k0  k02 − k12 eik0 z2 −z1  sin k1 z1  = 0

(2.71)

The values of k1 that satisfy Equation (2.71) give the frequency, mode , and threshold gain gth of each possible laser mode: c Re k1  1

(2.72a)

gth = −2Im k1 

(2.72b)

mode =

where it is emphasized that while 1 is treated as a constant at any particular time, it actually varies with the laser carrier density.

48

Theoretical Analysis

The self-consistent calculation thus accounts for the mode-pulling effects ignored in [40] by including the effect of the carrier-induced change in the gain section refractive index on the instantaneous mode wavelengths. The largest proportion of the computational effort in performing a coupled-cavity simulation is spent solving Equation (2.71). For very long external cavities (with a corresponding large number of external cavity modes supported), this becomes prohibitive. The coupled cavity approach detailed above is valid when the feedback into the laser device is coherent, as is the case for the majority of interesting phenomena. Like the Lang–Kobayashi and iterative methods, the coupled cavity analysis also limits the model to dynamics that occur on timescales longer than the cavity round-trip time. When finding the frequency and threshold gain of each mode it is not necessary to make an initial assumption of the number of modes that will be found: standard numerical methods for the solution of Equation (2.71) can find all the possible modes within the gain window of the device. This ensures that all the longitudinal modes of the system, both lasing and non-lasing modes, are accounted for correctly. The mode calculation is carried out alongside a system of rate equations: one for the carrier density, n, in the gain section:  Jspon J n dn = − − vg g l Sl − dt wq wq  e l

(2.73)

and m equations for the photon densities S in each of the m coupled-cavity modes: Jspon Sl dSl = vg g l  Sl +  − dt wq l

(2.74)

In this system of rate equations J is the drive current, w is the width of the quantum well gain region, q is the elementary charge, Jspon is the equivalent spontaneous emission current,  is the spontaneous emission coupling factor, vg is the modal group velocity,  is the confinement factor (assumed to be the same for all modes) and e is the carrier lifetime. The gain terms, g, in the rate equations are also calculated from the optical susceptibility at every time step. The lifetime, l , is calculated for each mode via the definition: 1 1 1 = mir + scatt l l l

(2.75)

where the scattering losses  scatt are the same for each mode, but the mirror lifetimes are variable and calculated from the expression: lmir =

energy stored in system energy lost per second

(2.76)

which yields, for the system in Figure 2.14 the expression: %z1 z0

%z2 U1 z dz+ U2 z dz z1

lmir = &

 0 2 0 A0 2 + 3 B3 2 0

(2.77)

Coupled Cavity Analysis

where the cycle-averaged energy densities U1 z and U2 z are given by: ' ( U1 z = 20 12 A1 2 e−gth z + B1 2 egth z ' ( U2 z = 20 A2 2 + B2 2

49

(2.78a) (2.78b)

On evaluating the integrals we find the following expression for the mirror lifetimes: ' ( 12 A1 2 1 − e−gth z1  + B1 2 egth z1 − 1 ' ( lmir = (2.79) 2cgth 0 A0 2 + 3 B3 2 The threshold gain gth for each mode is already known, and the intermediate field amplitudes A1 , A2 , B1 , B2 and B3 are found by applying the matrices Q z1 , Q z2  and Q z3  in turn. Strictly, these amplitudes have normalized units since the initial condition A0 = 1 was chosen in solving Equation (2.69), but the expression for the mirror lifetime involves only ratios of amplitudes, rendering any normalization immaterial.

2.6.2 Comparison with LK Analysis It is interesting to compare the coupled cavity model with the Lang–Kobayashi analysis. The Lang–Kobayashi method is based on a perturbation about the solitary laser case; therefore the operating wavelength is assumed to be known in advance, and a correction to that wavelength is calculated by the perturbation analysis. Basing the model on a perturbation analysis limits its applicability to relatively weak feedback, in order to keep the perturbation small. When the Lang–Kobayashi model is extended to multimode operation, similar presuppositions are made about the existence and positions of the modes – a centre wavelength is chosen and satellite modes, spaced by the reciprocal of the cavity round-trip time, are assumed to exist on either side of the centre wavelength. Again, corrections to the wavelength of each mode are made by the perturbation analysis. In contrast, the coupled cavity method makes no assumptions about the existence or position of the laser modes. The mode positions and threshold gains are calculated from Equation (2.71) at each time step. The multimode extension to the Lang–Kobayashi model also typically assigns a single lifetime to all the modes, while each mode calculated from this analysis has its own lifetime, calculated selfconsistently with the mode frequency. The key to the rich variety of possible phenomena predicted by the coupled cavity model is the effect of the variation in mode lifetime calculated above: different modes can have different lifetimes and each mode lifetime can also change with time. When the mode lifetimes change, there is a corresponding variation in the field amplitudes in the coupled cavity which can further affect the lifetimes themselves. This feedback mechanism can give rise to chaotic dynamics in conditions when neither of the sections that make up the coupled-cavity system dominates.

2.6.3 Typical Results To illustrate the method we will carry out some numerical simulations for some representative external cavity reflectivities. Power spectra for the output laser light are calculated by forming

50

Theoretical Analysis

a histogram of the instantaneous mode wavelengths over an interval of time. The resulting spectra are therefore ‘average spectra’, displaying the system’s behaviour over the whole of the histogram time-interval. All spectra are averaged over 1ns – a value that corresponds to the bandwidth of a typical photo-detector – with the histogramming process beginning after the initial turn-on transient has decayed. In the results that follow, the carrier density and wavelength dependence of the optical gain and refractive index (i.e. the optical susceptibility) are evaluated using a ‘manybody’ calculation [41–43]. The many-body calculation includes effects such as bandgap renormalisation and Coulomb enhancement which are significant in semiconductor laser materials. It should be noted that a parabolic gain approximation, with an associated linewidth enhancement factor, could be used in place of the many-body calculation. Using such a parabolic gain dispersion curve leads to parabolic features in the resulting output spectrum [43] while the many-body calculation yields spectra that have features more like those observed in practical laser diodes. The coupled-cavity model is independent of the detailed susceptibility calculation carried out, but for the results to be useful, the susceptibility must vary with the carrier density since it is the coupling between the mode frequencies, threshold gains and carrier density that leads to the range of dynamical behaviours observed. 2.6.3.1 ‘Solitary laser’ operation

Power (arb. units)

By setting the refractive index of the reflector region to a value close to unity, it is possible to approach the limit of solitary laser behaviour. With a very low external reflectivity, Rext ≡ −80 dB the coupled cavity model reproduces solitary laser operation. After the decay of the relaxation oscillation transient the mode spectrum portrayed in Figure 2.15 is observed. In the absence of appreciable optical feedback the system is seen to exhibit stable behaviour, with the mode envelope reflecting the steady state gain spectrum. (If a parabolic gain dispersion curve had been chosen, the mode envelope would also resemble a parabola.)

665.0

667.5

670.0 Wavelength (nm)

672.5

675.0

Figure 2.15 Power spectrum of solitary laser Rext = −80 dB, averaged over 1 ns. Source: I. Pierce, P. Rees and P.S. Spencer, ‘Multimode dynamics in laser diodes with optical feedback’, Phys. Rev. A, 61, 053801 (2000).

Coupled Cavity Analysis

51

In this regime all the mode lifetimes are similar in value and the modal gain profile is the dominant effect in determining the amplitude distribution of the modes. The stability of the mode lifetimes yields corresponding stability in the carrier density which reduces the carrier-induced change in the refractive index. The corresponding reduced mode-pulling further reinforces the stability of the mode lifetimes. Once the initial transients have died down, the system therefore operates in a stable regime, as expected for a solitary laser. 2.6.3.2 External cavity operation Figure 2.16 shows the power spectrum of the system when the external feedback level is set at the much higher level of Rext ≡ −6dB. This corresponds to the case where the external cavity dominates the behaviour of the system. Although each mode is narrow, which implies stable operation with little wavelength variation over time, the amplitude distribution between the modes is markedly different from the solitary laser case. In this external cavity regime the mode with the longest lifetime dominates at the expense of the other, shorter lifetime modes, which only exist at very low relative amplitudes. The long-lifetime mode corresponds to a wavelength where the overlap between the notional Fabry–Perot modes of the laser chip and the longitudinal modes of the compound cavity is maximum. This strong single-mode behaviour (coupled with tunability through variation of the cavity length) is exactly what makes external cavity laser systems useful in practice. It should be emphasized that the disparity of the mode lifetimes is sufficient to cause single-mode operation. There is no requirement to assume that a single mode should dominate. The wavelength of the dominant mode is also decided without any a priori assumptions. 2.6.3.3 Chaotic operation

Power (arb. units)

The system behaves very differently when the feedback level is set between the two extremes described above. At feedback levels that correspond to Regime IV in Tkach and Chraplyvy’s

670

671

672 673 Wavelength (nm)

674

675

Figure 2.16 Spectrum of external cavity laser Rext = −6 dB. Source: I. Pierce, P. Rees and P.S. Spencer, ‘Multimode dynamics in laser diodes with optical feedback’, Phys. Rev. A, 61, 053801, (2000).

Theoretical Analysis

Power (arb. units)

Power (arb. units)

52

662.5

670.00

665.0

667.5

670.25 670.50 670.75 Wavelength (nm)

670.0 672.5 Wavelength (nm)

675.0

671.00

677.5

680.0

Figure 2.17 Spectrum during coherence collapse Rext ≡ −28 dB. Inset shows close-up in range 670–671 nm displaying greatly broadened modes, consistent with the concept of coherence collapse. Source: I. Pierce, P. Rees and P.S. Spencer, ‘Multimode dynamics in laser diodes with optical feedback’, Phys. Rev. A, 61, 053801 (2000).

classification scheme [44], the system dynamics are chaotic. Figure 2.17 shows the output power spectrum for the case Rext ≡ − 28 dB. The optical modes are barely discernible – the system seems to emit over a broad spectrum. The inset to Figure 2.17 illustrates that, in fact, each mode has broadened significantly when compared with solitary-laser or external-cavity operation. This is consistent with the concept of coherence collapse, where the temporal coherence length of the laser mode is drastically reduced as a result of the broadening of the modes. This catastrophic mode-broadening is a result of the interplay between the fields in the different sections of the coupled cavity, neither of which dominates. Changes in the carrier density cause corresponding changes in the wavelength of the modes and in the mode lifetimes. The changing mode lifetimes cause further changes in the carrier density and, in the absence of the stabilizing influence of a dominant cavity, lead to limit cycles and eventually fully chaotic dynamics. This route to chaos via time-varying mode lifetimes is a complementary approach to the usual timedelayed field term; its strength is that the regimes of behaviour of the system are determined purely by the variations of the mode lifetimes. At the two extremes of feedback level – very high feedback and negligible feedback – the model correctly predicts stable behaviour in the external cavity and solitary laser regimes. In the former, the similarity of the mode lifetimes means that the mode profile is determined by the gain spectrum, while in the latter the non-uniform lifetime distribution leads to the dominance of one external cavity mode. In the coherence collapse regime the model correctly predicts deterministic chaotic dynamics. During coherence collapse the mode lifetimes vary with time, causing corresponding changes in the carrier density that eventually feed back as further changes in the lifetime, leading to instability and chaos. The coupled cavity model differs from the Lang–Kobayashi analysis by calculating the mode wavelengths and

References

53

lifetimes from considerations of the forward and backward propagating waves in the coupled cavity rather than calculating a perturbation to a pre-supposed operating wavelength. The model is also valid for higher feedback levels than the Lang–Kobayashi method, correctly predicting stable, single-mode operation in the external cavity operating regime. In all cases the dynamical behaviour and the amplitude distribution of the mode spectrum are completely determined by the level of external optical feedback – no a priori assumptions are made about the expected regime of operation or about the expected emission wavelengths.

2.7 CONCLUSION In this chapter we have reviewed three models for laser diodes subject to optical feedback. We first considered the Lang–Kobayashi equations which are valid in the weak-to-moderate feedback regime. A steady state analysis of these equations shows that the laser can be destabilized due to optical feedback. We then considered a model which is valid for arbitrary levels of optical feedback, known as the iterative model. Due to its reduced computational complexity when compared with the Lang–Kobayashi model, the iterative technique is easily extended to multimode operation. The iterative model was compared with the Lang–Kobayashi model. For low feedback levels the two models agree, but as the feedback level is increased, the effect of the multiple round trips plays an increasingly important role and the results of the two models diverge. The chapter concludes by discussing a third model for optical feedback in diode lasers, namely, the coupled-cavity model. While the model is valid for arbitrary values of feedback strength and self-consistently includes multimode effects, it is practically limited to short external cavities due to its increased computational complexity.

REFERENCES [1] H. Haken, Phys Rev A, 53, 77 (1975). [2] E.V. Lorenz, J. Atmos. Sci., 20, 130 (1963). [3] R. Lang and K. Kobayashi, ‘External optical feedback effects on semiconductor injection lasers laser properties’, IEEE J. Quant. Electron. QE-16, 347 (1980). [4] G.H.M. van Tartwijk and D. Lenstra, Quant. Semiclassical Optics, 7, 87 (1995). [5] G.H.M. van Tartwijk and G.P. Agrawal, ‘Laser instabilities: a modern perspective’, Prog. Quant. Electron., 22, 43–122 (1998). [6] C.H. Henry, IEEE J. Quant. Electron., QE-18, 259 (1982). [7] B. Tromborg, J.H. Osmundsen and H. Olesen, ‘Stability analysis for a semiconductor laser in an external cavity’, IEEE J. Quant. Electron., 20, 1023–1032 (1984). [8] J. Mork, J. Mark and B. Tromborg, ‘Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback’, Phys. Rev., 65, 1999–2002 (1990). [9] J. Mork, B. Tromborg and J. Mark, ‘Chaos in semiconductor lasers with optical feedback: theory and experiment’, IEEE J. Quant. Electron., 28, 93–108 (1992). [10] K. Petermann, Laser Diode Modulation and Noise, Dordrechtm, Kluwer Academic (1988). [11] D. Lenstra, B.H. Verbeek, and A.J. den Boef, ‘Coherence collapse in single-mode semiconductor lasers due to optical feedback’, IEEE J. Quant. Electron., 21, 674–679 (1985). [12] P. Besnard, B. Meziane and G. M. Stephan, ‘Feedback phenomena in a semiconductor laser induced by distant reflectors’, IEEE J. Quant. Electron., QE-29, 1271–1284 (1993). [13] C.H. Henry and R.F. Kazarinov, ‘Instability of semiconductor lasers due to optical feedback from distant reflectors’, QE-22, 294–301 (1996). [14] D. Lenstra, B.H. Verbeek and A.J. den Boef, ‘Coherence collapse in single-mode semiconductor lasers due to optical feedback’, IEEE J. Quant. Electron., 21, 674–679 (1985).

54

Theoretical Analysis

[15] H. Li, J. Ye and J.G. McInerney, ‘Detailed analysis of coherence collapse in semiconductor lasers’, IEEE J. Quant. Electron., 29, 2421–2432 (1993). [16] Y. Liu, N. Kikuchi and J. Ohtsubo, ‘Controlling dynamical behaviour of a semiconductor laser with external optical feedback’, Physical Rev. E, 51, 2697–2700 (1995). [17] N. Kikuchi, Y. Liu and J. Ohtsubo, ‘Chaos control and noise suppression in external-cavity semiconductor lasers’, IEEE J. Quant. Electron., 33, 56–65 (1997). [18] G.A. Acket, D. Lenstra, B.H. Verbeek, and A.J. den Boef ‘The influence of feedback intensity on longitudinal mode properties and optical noise in index-guided semiconductor lasers’, IEEE J. Quant. Electron., 20, 1163–1169 (1984). [19] J. Mork, M. Semkow and B. Tromborg, ‘Measurement and theory of mode hopping in external cavity lasers’, Electron. Lett., 26, 609–610 (1990). [20] J. Mork and B. Tromborg, ‘The mechanism of mode selection for an external cavity laser’, IEEE Phot. Technol. Lett., 2, 21–23 (1990). [21] D. Lenstra, Opt. Commun., 81, 209 (1991). [22] N. Schunk and K. Petermann, ‘Stability analysis for laser diodes with short external cavities’, IEEE Phot. Technol. Lett., 1, 49–51 (1989). [23] A. Ritter and H. Haug, J. Opt. Soc. Am. B, 10, 130 (1993); J. Opt. Soc. Am. B, 10, 144 (1993). [24] A.M. Levine, G.H.M. van Tartwijk, D. Lenstra and T. Erneux, ‘Diode lasers with optical feedback: stability of the maximum gain mode’, Phys. Rev. A, 52, R3436–R4339 (1995). [25] A.T. Ryan, G.P. Agrawal, G.R. Gray and E.C. Gage, IEEE J. Quant. Electron., QE-30, 668, (1994). [26] L.N. Langley and K.A. Shore, IEEE Phot. Technol. Lett., 4, 1207 (1992). [27] L.N. Langley and K.A. Shore, IEEE J. Light. Tech., 11, 434 (1993). [28] F. Sporleder, Proceedings of the URSI International Symposium on Elecromagnetic Theory, Brussels, Belgium (1983). [29] J. Mork, ‘Nonlinear dynamics and stochastic behaviour of semiconductor lasers’, PhD thesis, Report No. S48, Technical University of Denmark (1989). [30] J. Helms and K. Petermann, ‘A simple analytic expression for the stable operation range of laser diodes with optical feedback,’ IEEE J. Quant. Electron., 26, 833–836 (1990). [31] A.A. Tager and K. Petermann, ‘High frequency oscillations and self-mode locking in short external cavity laser diodes’, IEEE J. Quant. Electron., 30, 1553–1561 (1994). [32] K. Petermann, ‘External optical feedback phenomena in semiconductor lasers’, IEEE J. Quant. Electron., 1, 480–489 (1995). [33] A.A. Tager and B.B. Elenkrig, ‘Stability regimes and high frequency modulation of laser diodes with short external cavity’, IEEE J. Quant. Electron., 29, 2886–2890 (1993). [34] P.S. Spencer, D.M. Kane and K.A. Shore, ‘Coupled-cavity effects in FM semiconductor lasers’, IEEE J. Light. Tech., 17, 1072–1078 (1999). [35] R.J. Jones, P.S. Spencer, J. Lawrence and D.M. Kane, ‘Influence of external cavity length on the coherence collapse regime in laser diodes subject to optical feedback’, IEE Proc. Optoelectron., 148, 7–12 (2001). [36] I. Pierce, P. Rees and P.S. Spencer, ‘Multimode dynamics in laser diodes with optical feedback’, Phys. Rev. A, 61, 053801 (2000). [37] M.B. Spencer and W.E. Lamb, Phys. Rev. A, 5, 884 (1972). [38] R. Lang, M.O. Scully and W.E. Lamb, Phys. Rev. A, 7, 1788 (1972). [39] A.G. Fox and T. Li, Bell Syst. Tech. J., 40, 453 (1961). [40] K.J. Ebeling and L.A. Coldren, J. Appl. Phys., 54, 2962 (1983). [41] H. Haug and S.W. Koch, Quantum Theory of the Optical and Electrical Properties of Semiconductors, World Scientific, Singapore (1990). [42] F.P. Logue et al., Phys. Rev. B, 54, 16417 (1996). [43] J.K. White and J.V. Moloney, Phys. Rev. A, 59, 2422 (1999). [44] R.W. Tkach and A.R. Chraplyvy, ‘Regimes of feedback effects in 1.5 m distributed feedback lasers’, IEEE J. Lightwave Tech., LT-4, 1655 (1986).

3 Generalized Optical Feedback Theory Daan Lenstra, Gautam Vemuri and Mirvais Yousefi

3.1 VARIETIES OF OPTICAL FEEDBACK Optical feedback into a semiconductor laser is often an inevitable consequence of any application in which the laser is used. Such feedback can occur from optical components in the path of the light, from the edges of an optical fibre, or from the windows of a vapour cell in spectroscopy experiments. So far, this is no different from any other laser, yet what makes semiconductor lasers so much more sensitive to feedback than most other lasers? The answer is that the semiconductor laser owes its feedback sensitivity to the combination of three different factors, i.e. its very high intrinsic gain, its unusually high chirping effect (i.e. intensity variations produce frequency variations) and its relaxation oscillation. In any case, since the first cw-operating semiconductor lasers became available in the 1970s, publications have been reporting on their remarkable feedback sensitivity. The feedback light, upon injection into the laser cavity, is superimposed on the existing intra-cavity field, and will interfere with this existing field. Clearly, the phase difference between the feedback field and the intra-cavity field determines whether the total field inside the laser cavity is amplified or suppressed. Usually, a simple linear relationship exists between the feedback field and the total output field from the laser. Note that the feedback field is time-delayed due to the time it takes the light to reflect from a distant object before being re-injected into the laser. In addition to the above-mentioned, more intrinsic factors, it is this time delay that completes the ingredients responsible for the enormous variety of dynamical effects that may arise in semiconductor lasers with delayed optical feedback. In a generalized conception of optical feedback, any functional relationship between the feedback and the emitted field can be considered, although we do confine ourselves to situations where the feedback light is created by some external device. This is shown in Figure 3.1 where box L represents the laser whose dynamics we want to study and box F Unlocking Dynamical Diversity Edited by Deborah M. Kane and K. Alan Shore © 2005 John Wiley & Sons, Ltd

56

Generalized Optical Feedback

E(t) L

F F(t)

Figure 3.1 The light E
t emitted by laser L is transformed by external device F into F
t and sent back to L.

represents the external device that produces the feedback light. The emitted laser light field is represented by E
tei0 t + cc and the feedback field by F
tei0 t + cc, where 0 is a conveniently chosen optical frequency. Typically, this is the solitary laser frequency, i.e. the central laser optical frequency in the absence of feedback. The complex quantities E and F are then slowly time-dependent field amplitudes, that is, slow compared to the rapid oscillations with the optical frequency. In our generalized feedback approach, the feedback field F is given by some external-device-specific functional, i.e.: F =

E

(3.1)

In case of a linearly responding device, the most general form of
is:

E =


t

dt f
t − tE
t 

(3.2)

−

Here, f represents the response function of the external device. It is possible that the device itself develops new dynamics driven by E and this would mean that the function f in Equation (3.2) becomes dependent on E as well. We are then dealing with nonlinearly responding devices such as when the external device is another laser. Certain phaseconjugating materials provide another well-known type of nonlinear response. To cover such cases in our general theory, to be developed in the next sections, it is sometimes sufficient to replace in Equation (3.2) the field E by its complex conjugate, thus staying well within the simple framework of Equation (3.2). If the external system is a simple Lorentzian frequency filter with central frequency m and a half-width at half-maximum (HWHM) , the response function is given by: f
t = e−t+i
m −0 t

(3.3)

As is usual in a dynamical systems description, we will reformulate the feedback light contribution in terms of a first-order differential equation in time. For the simple filter given by Equation (3.3) it is straightforward to derive, by differentiating Equation (3.2), and using Equation (3.3), the following rate equation: F˙
t = E
t − e−i0  −  + i
m − 0 F
t

(3.4)

Here we have taken into account the possibility of a delay in time, , such as due to an external optical path length. In case of a more complicated, non-Lorentzian linearly responding filter, one can generally decompose the response function as a sum of exponential functions of the type Equation (3.3).

Varieties of Optical Feedback

57

This means that, instead of one single rate equation (3.4), one now has to deal with a collection of rate equations: F˙ j
t = j cj E
t − e−i0  − j + i
j − 0 Fj
t

(3.5)

In these equations, cj is the coefficient indicating the relative strength of the jth oscillator with damping rate j and centre frequency j . Following the Lang–Kobayashi formulation [1] (see also Chapters 1 and 2), external optical feedback is described by a rate equation with the following structure: ˙ sol + F
t E˙ = E

(3.6)

i.e. the solitary rate of change of the laser field E is augmented by the feedback field F , where is the coupling rate or feedback strength, usually a control parameter. The total feedback field is given by: F
t =



Fj
t

(3.7)

j

Here, each Fj satisfies a rate equation of the type Equation (3.5). This almost concludes the set-up of our generalized feedback theory in terms of coupling to a set of linear filters. There are a few cases that deserve special attention here. First, we will derive the case of conventional optical feedback (COF), i.e. feedback without frequency filter. In that case we should, in Equation (3.4), consider the limit where  approaches infinity. In that case the two terms on the right-hand side proportional to  will dominate the other term and Equation (3.4) will have the solution: F
t = E
t − e−i0 

(3.8)

When substituted in Equation (3.6), Equation (3.8) yields the usual formulation according to Lang–Kobayashi [1] for conventional delayed optical feedback, see Chapter 2. Second, the very narrow filter, i.e. the case  → 0. In this case the feedback field will be nearly monochromatic with frequency equal to the filter centre frequency, and appreciable only if the laser emits at that same frequency. Hence, this case will show similarity with a laser with external optical injection, a system that thanks to its reduced dimensionality (3D instead of infinite dimensionality for the delay system, see Chapter 5), has been thoroughly analysed and its bifurcation structure much better understood [2]. Third, another important situation that arises in the context of the optical feedback induced dynamics is phase-conjugate feedback from a medium with finite response time and where the feedback field is detuned from the solitary laser. A complete derivation of the feedback term for phase conjugate feedback based on four-wave mixing can be found in [3]. We will not repeat that derivation here, but rather state the result for the feedback field F , which can be cast in the form of a damped oscillator driven by the complex conjugate of the laser field emitted one external delay time  earlier, i.e. F˙ = E ∗
t − e2i 0
t−/2 −
 − i 0 F

(3.9)

58

Generalized Optical Feedback

Here, 0 is the detuning frequency between the four-wave mixing pump beam p and the reference frequency 0 , i.e. 0 = p − 0 . The phasor e2i 0
t−/2 after E ∗
t −  expresses the fact that the phase conjugating process occurs with respect to the pump frequency p . The damping  is due to the finite response time of the phase-conjugating mirror, which in turn is determined by the underlying physical process that creates the phase conjugation. In most cases this is related to the typical relaxation times of the charged particles in the medium that interact with the light, i.e. nanoseconds for atomic gases and picoseconds for semiconductors. Certain nonlinear crystals are known to have macroscopically long response times on the order of seconds. In any case, if the phase-conjugating process is fast,  will be large and the solution to Equation (3.9) will read, similar to Equation (3.8): F
t = E ∗
t − e2i 0
t−/2

(3.10)

On the other hand, when the phase-conjugating mirror is very slow, it will behave as a narrow filter as well, producing appreciable reflections only if the laser emits at the pump frequency. As in the above-discussed narrow-filter case, one may expect here also similarities with external optical injection at frequency p .

3.2 COMPOUND-CAVITY ANALYSIS: VALIDITY OF LANG–KOBAYASHI APPROACH In a Fabry–Perot type diode laser subject to delayed optical feedback, a sharp transition from internal to external mode operation occurs upon increasing the feedback strength. This transition could mark the onset of multi-longitudinal mode operation, while most theoretical descriptions are based on single-mode analysis, notably the Lang–Kobayashi rate equations, which are derived for a laser operating in a single longitudinal mode [1], see also Chapter 2. The transition can be analysed without explicit knowledge of the actual properties of the gain medium and, in line with intuition, occurs when the reflectivity of the external mirror equals the reflectivity of the facet facing the external cavity. This was shown by Tromborg et al. [4] for conventional optical feedback, but we will show in this section how this result can be extended to include generalized optical feedback. We model the compound cavity as a one-dimensional system, as sketched in Figure 3.2. A uniform active medium with (complex) refractive index n1 is located in 0 < x < l, whereas the external mirror with reflection coefficient r
 is placed at x = −L. We consider continuous-wave type solutions of the form:   Ej
x t = eit Aj e−ikj x + Bj eikj x + cc (3.11)

r(ω) X=–L

B0 exp(ik0x)

B1 exp(ik1x)

A0 exp(–ik0x)

A1 exp(–ik1x) X=0

A2 exp(–ik0x) X=l

Figure 3.2 The compound-cavity configuration. The external cavity (vacuum) system extends from x = −L to x = 0. The laser active medium is uniform, has refractive index n1 and extends from x = 0 to x = l. The external mirror in x = −L has reflection coefficient r
.

Compound-Cavity Analysis: Validity of Lang–Kobayashi Approach

59

Here, j = 0 for −L < x < 0, j = 1 for 0 < x < l and j = 2 for x > l, B2 = 0 and we apply the usual connection rules at the interfaces x = 0 and x = l, that is, continuity of the field and its derivative at x = 0 and finally, at x = −L: A0 e−ik0 L = r
B0 eik0 L

(3.12)

We obtain a homogeneous set of 5 linear equations for the 5 field amplitudes A0 , B0 , A1 , B1 and A2 , which have nontrivial solutions only if the coefficient determinant vanishes. This leads to the condition: e2ik1 l =

r
e2ik0 L + rF r
e2ik0 L + rF−1

(3.13)

where rF is the Fresnel reflection coefficient of the active medium facets [5]: rF =

n1 − 1 n1 + 1

(3.14)

Equation (3.13) is the central result for our discussion of the transition from internal to external mode operation. It is the frequency determining equation, since both k0 and k1 are related to the laser frequency  through their respective dispersion relations, which for k0 is the trivial relation  = ck0 , but which for k1 depends on the details of gain and dispersion in the active medium. Roughly speaking, we have  = ck1 /n1 , with n1 the refractive index. Strictly speaking, the Fresnel reflection coefficient rF depends on , although, for the usual edge emitting semiconductors with index values in the range n1 ∼ 32 to 3.6 in practice, rF will have a value around 0.55. In order to discuss the solutions to Equation (3.13) for the frequency , we draw in Figure 3.3(a) the contour in the complex plane of the right-hand side of Equation (3.13) versus  with  varying in an interval within the filter profile, given by a Lorentzian with HWHM  and centre frequency m , i.e.: r
 =

 r  − i
 − m  max

(3.15)

In this example we consider for clarity a case where the external modes (determined by 2k0 L) are three times denser than the solitary laser modes (determined by 2k1 l). In reality, there may be 100 or more external cavity modes per solitary laser mode. The solid black line in Figure 3.3(a) corresponds to the low-frequency side of the filter and the solid grey line to the high-frequency side. The spiralling curve is traversed in the clockwise direction when  increases; the radius is larger when  gets closer to the filter centre frequency. Since the maximum reflectivity rmax is taken 1.25 times larger than rF , the loops will encompass the origin for  near the filter centre. In Figure 3.3(b) we draw the loci of corresponding solutions to (13) for 2k1 l in the complex plane. We note that the solutions for 2k1 l have modulo 2 multiplicity, but we consider only the physically relevant solutions for which k1 ∼ n1 k0 . It is clear that as long as the spiralling curve of Figure 3.3(a) does not encompass the origin, the locus for 2k1 l spirals around the solitary laser solutions, i.e. the ones given by r = rmax = 0, or: 2k1 lsolitary = 2 n + 2i ln rF−1

(3.16)

60

Generalized Optical Feedback

(a)

(b)

Figure 3.3 (a) The right-hand side of Equation (3.13) versus  as a contour in the complex plane, with  varying in an interval centred around the filter and encompassing 7 solitary laser longitudinal modes. (b) The loci for 2k1 l of corresponding solutions to Equation (3.13) in the complex plane, with the solitary laser modes indicated by ‘x’. We note that the solutions for 2k1 l have modulo 2 multiplicity, but we consider only the physically relevant solutions for which k1 ∼ n1 k0 . Here,  was varied in a discrete and equidistant manner, so that each solitary laser mode has roughly 1000 points. The parameters are n = 36; l = 3010−4 m, L = 32410−3 m, m = 0, and  = 87266 109 rad/s, corresponding to one longitudinal mode spacing. The maximum reflectivity at the filter centre frequency equals rmax = 125rF , where rF = 057. In the filter wings, the compound cavity modes are grouped within isolated islands, where each island belongs unambiguously to a given longitudinal laser mode, a feature that no longer holds for the modes in the central area, where each point may belong to any one of the three longitudinal laser modes.

Compound-Cavity Analysis: Validity of Lang–Kobayashi Approach

61

where the integer n is the longitudinal mode number. This leads to the islands of solutions that are isolated regions in the complex plane, each of which shrinks to the solitary laser point x in its interior when the external reflectivity vanishes. This means that each island is topologically connected to a unique solitary laser solution. However, the situation in the middle area of solutions in Figure 3.3(b) is different. This region has several solitary laser points in its interior, and a certain given solution to Equation (3.13) is no longer connected to one solitary laser point Equation (3.16). Since this extended area corresponds to the frequency range where the effective reflectivity exceeds the facet reflectivity, a sufficient condition to have isolated solitary mode islands everywhere, is that: rmax < rF

(3.17)

i.e. the maximum external reflectivity should not exceed the reflectivity of the opposing facet. The condition (3.17), derived for the Lorentzian filter, holds for any filter if rmax is the maximum reflectivity of that filter. Physically this means that under condition (3.17) the modes of the laser with feedback, i.e. the compound-cavity modes, can be unambiguously assigned to solitary laser modes, i.e. the longitudinal modes. If, on the other hand, the maximum external reflectivity is larger than the facet reflectivity, it may happen that the origin is encompassed by the contour given by the right-hand side of Equation (3.13), in which case the solution for 2k1 l as a function of  will describe an extended contour in the complex plane, and no longer be topologically connected to one unique solitary laser mode (3.16). For conventional optical feedback, criterion (3.16) reads rext < rF and this was interpreted by Tromborg et al. [4] as the condition for which a single-longitudinal mode description of optical feedback would be consistent. Later, Duarte et al. [6] used similar arguments to discuss the abrupt transition from solitary-laser longitudinal-mode operation to externalmode operation, which they called the metamorphosis of the spectrum. However, they find for conventional optical feedback a value for the maximally allowed external reflectivity of 0.0108 for single-longitudinal mode description, i.e. much lower than the above derived value rF ∼ 055. The explanation for this difference may be that Duarte et al. considered a solitary laser for which the facet facing the external feedback was anti-reflection coated. The above-given derivation extends the analysis to generalized optical feedback and we can conclude, following the same argument as Tromborg et al. [4], that so long as the maximum external reflectivity does not exceed the facet reflectivity, a consistent description based on single-longitudinal mode operation can be given. Moreover, in case of sufficiently narrow filtering, it may happen that for a certain special detuning the contour of the right-hand side of Equation (3.13) will not wind around the origin and the solution remains connected to one unique longitudinal solitary laser mode, even when Equation (3.17) may not be satisfied. When extended loci occur, the interpretation of the compound-cavity mode being a perturbed free-running laser mode no longer holds. In that case, the compound-cavity mode must be regarded as a perturbed external cavity mode. To conclude this section, we have studied the CW-operation modes for a compound laser-cavity system and analysed the transition from a regime of external reflectivity where these modes are perturbed longitudinal modes of the free-running laser to a regime where these modes are perturbed external-cavity modes. The transition occurs when the external reflectivity exceeds the Fresnel reflectivity of the corresponding laser facet. Clearly there is

62

Generalized Optical Feedback

no reason to doubt the consistency of a single-longitudinal mode Lang–Kobayashi model approach as long as the external reflectivity is small compared to the Fresnel reflectivity of the laser facet. In this respect, an external reflection coefficient r = 01 is already considered strong feedback, but the compound-cavity modes are situated on a closed locus around a unique longitudinal laser mode. In terms of the feedback rate parameter the Lang– Kobayashi validity condition can be expressed as (with rF ∼ 055): =


1 − rF2  r < 1012 r s−1 in rF

(3.18)

The above-stated conclusion is relevant for the discussion of low-frequency fluctuations (LFF), where it is sometimes claimed that multi-longitudinal laser mode operation is an essential feature of the LFF-type dynamics [7]. Indeed, there have been experimental observations of such multimode operation [8], and it has also been shown that certain multimode analyses are capable of reproducing such LFF-type dynamics [9]. But from that one should not conclude that single-mode Lang–Kobayashi theory gives an inadequate description of LFF. In any case, the results from this section confirm that the Lang–Kobayashi theory is certainly consistent.

3.3 FILTERED OPTICAL FEEDBACK The treatment in this section follows that in Yousefi et al. [10]. Whereas early studies on optical feedback in semiconductor lasers dealt with conventional optical feedback (COF), more recently filtered optical feedback (FOF) has become a topic of interest, since it offers a potential to control the laser dynamics via two external parameters, viz. the spectral width of the filter and its detuning from the solitary laser frequency. The recent experiments by Goedgebuer and co-workers have demonstrated a number of dynamical effects that arise from FOF in semiconductor lasers, including period doubling bifurcations and chaotic encryption via chaos synchronization [27]. The experiments of Tredicce and co-workers have studied low-frequency fluctuations behaviour in semiconductor lasers subject to FOF [8]. Lenstra and co-workers [12, 29] and, very recently, Gauthier and co-workers [30] have reported experimental and theoretical results on FOF-induced dynamics, when a Fabry–Perot and Mach-Zehnder filter, respectively, are used. In each of these works, the emphasis has been on the dynamics that arise from FOF, and how these differ from the effects observed under COF. The complex dynamical behaviour arising in lasers with feedback is rooted in the undamping of the intrinsic relaxation oscillations in the laser, the relatively large self-phasemodulation property of semiconductor lasers (which manifests itself as a coupling between amplitude and phase of the laser field and is expressed via the -parameter) and the feedback delay. Filters not only provide a mechanism for controlling the influence of relaxation oscillations on the dynamical response, but they also introduce an externally controllable nonlinearity of the device. In the dynamical-systems approach, the filter can be viewed as a mechanism for restricting the phase space that is available to the feedback-laser system. During COF, the dynamics of the system is determined, among other factors, by the number of different external-cavity modes (ECM). Introducing a filter not only decreases the number of ECMs, but also moves

Filtered Optical Feedback

63

them around in phase space (that is, determined by the plane defined by solitary laser frequency and the laser frequency in the presence of feedback), which can lead to some unexpected dynamics. The configuration under study consists of a diode laser and an external cavity with a frequency filter, as sketched in Figure 3.4. To describe this system we employ a singlelongitudinal mode model for the laser, while the filter is described by a Lorentzian [11, 12]. The electric component of the electromagnetic field in the laser is represented by E
t = E
texp{i0 t}+ c.c., where 0 is the operation frequency of the laser in the absence of feedback (to be referred to as the ‘solitary laser’), while E
t is the (complex) slowly varying amplitude. The model equations read: 1 ˙ =
1 + i n
tE
t + F
t E
t 2 F˙
t = E
t −  exp
−i0  +
if − F
t n
t ˙ = J − Jthr −

n
t − 0 + n
t E
t2 T

(3.19) (3.20) (3.21)

Here is the linewidth enhancement factor,  the gain coefficient, T the spontaneous recombination time and 0 the photon decay rate from the solitary laser cavity. The feedback rate is given by Equation (3.19) and refers to maximum feedback, i.e. at the filter centre frequency; F
t is the (complex) field amplitude re-entering from the external cavity, n
t describes the inversion, or the number of electron-hole pairs relative to their value at solitary laser operation,  is the half-width at half-maximum (HWHM) of the filter, f the centre frequency of the filter relative to the solitary laser frequency 0 , J = I/e is the normalized pump rate, I the pump current, Jthr the pump rate at the solitary laser threshold of the specific longitudinal mode and thr the corresponding threshold frequency. To account for the variation of the solitary laser frequency, 0 , with the pump current, we set: 0 = thr − k
J − Jthr 

(3.22)

where k > 0 is an empirical constant of proportionality. All other parameters are described √ in Table 3.1. Writing E
t ≡ P
texpi
t where 
t and P
t are the slowly varying

Figure 3.4 Sketch of set-up studied for filtered optical feedback. The diode laser DL emits light, which passes through a beam splitter, an optical isolator and a frequency filter. The filtered light is fed back into the diode laser. The optical isolator makes the external ring unidirectional.

64

Generalized Optical Feedback

Table 3.1 The parameter values used in the simulations Quantity

Symbol

Linewidth enhancement factor Feedback rate External Cavity roundtrip time Differential gain coefficient Photon decay rate Carrier life time Threshold pump rate Pump rate Pump current induced frequency shift

  0 T Jthr J k

Value 2 11.18 109 s−1 3 ns 2.142 104 s−1 3.57 1011 s−1 0.167 ns 1.4 1017 s−1 ∼1.4Jthr 3.58 GHz/mA

phase and power, the normalization is such that P equals the number of photons inside the laser. Consistent with this normalization, n represents the number of electron-hole pairs in the active layer.

3.3.1 External Cavity Modes The simplest modes of operation are those corresponding to single-frequency light emission. In dynamical systems language these modes are derived from the fixed points, which are defined as solutions of Equations (3.19)–(3.21) with time-independent amplitude and single fixed frequency, i.e. the solutions of the form1 E
t =



Ps expis t  F
t = Qs expi
s t + s 

(3.23)

n
t = ns

(3.25)

(3.24)

where Ps  Qs ≥ 0 s  s  ns are real and time-independent quantities. After inserting Equations (3.24)–(3.26) into Equations (3.19)–(3.21), five equations for the quantities are derived, from which the following closed transcendental equation for the phase difference s can be derived: s  = −Ceff sins  + 0  + arctan
 − arctan
s − f /

(3.26)

where Ceff = 

√  1 + 2 2 +
s − f 2

(3.27)

Equation (3.26) is solved numerically for the operation frequencies si of the fixed points, where after the corresponding power and inversion levels Psi  Qsi  si  nsi can be

Filtered Optical Feedback

65

calculated. A comparison with COF shows that the number of fixed points for FOF is always smaller than for √ COF. This is a direct consequence of the fact that (see Equation (3.27)) Ceff ≤ C ≡  1 + 2 , the effective feedback strength for COF. In Figure 3.5 the fixed-point frequencies calculated from Equation (3.26) are plotted versus the solitary laser frequency 0 . A continuous, multi-valued and oscillating curve is observed. For this case the filter centre frequency is at −321 GHz with respect to thr and the filter HWHM  = 2 GHz. Here, the pump rate J is varied within an interval J ∈ Jmin  Jmax , where the centre value of this interval is 14 Jthr . The intersections of the curve with a vertical line give the fixed points at that particular value of 0 . The three insets correspond to such vertical intersections as indicated and show the fixed point values in the
 P-plane for some fixed value of the pump current. Here  is the feedback round-trip phase difference, i.e.  ≡ 
t − 
t − . For a fixed point  equals s , where s is the corresponding frequency shift. The round-trip phase difference governs the interference between the feedback light and the laser light. The fixed points corresponding to constructive interference should be identified with the usual modes; the fixed points of destructive interference are called anti-modes. In frequency, anti-modes always lie between

Figure 3.5 Compound-cavity mode frequency versus solitary mode frequency in the presence of filtered optical feedback. In (a, b, c) the fixed points are depicted in the power (arbitrary units) versus phase difference plane for three different solitary laser frequencies as indicated. The filter HWHM equals 2 GHz. In (a, b, c) the + indicates the solitary laser fixed point. Source: Reproduced from: M. Yousefi, D. Lenstra, A. Fischer, Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback, IEE Proc. Optoelectronics 37, 233–237 (2001), with permission from IOP©2001.

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Generalized Optical Feedback

modes. Fixed points appear and disappear pairwise in saddle node bifurcations, i.e. one of them, the anti-mode, is an unstable saddle point. Generally, moving upwards along a vertical line, the first intersection is a mode, followed by an anti-mode, then again a mode, etc. and the last intersection is a mode again. Stability analysis can tell us whether a mode is stable, but anti-modes are always unstable. In Figure 3.5 the amplitude of oscillations clearly reflects the Lorentzian filter profile. Inset (c) shows two distinct islands of fixed points, i.e. a global ‘bistability’, induced by the filter. Roughly speaking, the system here can choose between one of two operational states; either adjust its operation frequency in response to the maximum feedback, or ignore the filter resonance and operate as a laser with weak feedback. Note that the filter-resonance ECMs have higher intensity than the weak-feedback ECMs. This is a consequence of the phase-amplitude coupling described by [13], which implies higher intensities on red shifts from the laser operation frequency. The opposite effect is seen in inset (a), where the islands of ECMs are just about to separate, but now the weak-feedback ECMs have higher intensity. In the next section the dynamical behaviour of a semiconductor laser due to FOF will be studied. In this respect, knowledge of the fixed points may provide hints of where attractor orbits are likely to be found and how complex the attractors may get. This can be achieved through a combination of studying the temporal behaviour of the laser light intensity, as well as the phase space dynamics of the laser. Formally, our delay system is infinite-dimensional, whether the feedback is filtered or not [14]. In practice, however, when looking at the attractors, the dynamics of the system seem to take place effectively in some finite-dimensional phase space. This appears to be related to the complexity of the dynamics. We therefore propose to introduce the number of fixed points Cplx as a quantitative measure for the complexity √of the system. In case of COF this quantity fluctuates around a value close to ≈
/  1 + 2 with jumps of at most ±2 when the pump current is varied. Figure 3.6 shows the function Cplx versus solitary laser frequency for the same FOF case as Figure 3.5. Far away  from the filter centre frequency, Cplx is a decreasing function with increasing detuning 0 − f , but since the number of fixed points is higher in the middle flanks of the filter profile than at the centre, it is a twin peaked function of the detuning. From this we expect that the dynamics will in general be higher in complexity at the filter

Figure 3.6 The measure for complexity Cplx versus detuning for the same case as in Figure 3.5.

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67

mid-flank, just when the two islands of fixed points become connected, than at the filter centre. This agrees with intuition, which tells us that the filter-induced nonlinearity is largest where the slope of the filter profile is maximal.

3.3.2 Dynamics Formally, the full phase space of our delay system is infinite dimensional, and so it should be realized that our analysis concerns projections onto a finite-dimensional phase sub-space. Specifically, the physical quantities, viz. power P, inversion n and phase difference , span a three-dimensional phase sub-space to which we will restrict most of our analysis. Numerical integrations of Equations (3.19)–(3.21) are performed using a modified Runge-Kutta method of fourth order [15]. The parameter values are given in Table 3.1. In order to allow a quick overview of all the different types of dynamics that may occur during a full scan of the solitary laser frequency as in Figure 3.5, we have used a representation very similar to the Poincaré section that works so well for finite-dimensional systems. We intersect the trajectory with a plane  in
P n  sub-space. This plane is constructed in such a manner that all fixed points are almost on it, at least as close as possible. Since the number of fixed points changes with the detuning and the filter width,  changes with 0 , which sometimes leads to artificial discontinuities in the trace of the attractor, as for example occurs in the limit cycle indicated by g in Figure 3.7. Our main results are summarized in the rather complicated Figure 3.7. In fact, this figure comprises all the information that one could extract in an experiment where one scans the solitary laser frequency through the filter profile, similar to the experiment in [11]. In such an experiment J should be scanned slowly enough through this interval for adiabatic response. In Figure 3.7 the filter width is set equal to  = 2 GHz. This figure should be viewed together with Figure 3.8 where a sample of the dynamics is shown in the
 P-plane together with some corresponding representative time series of the power for the attractor indicated. The fixed point surrounded by a box indicates stable CW-operation at that fixed point. Note that the vertical scales differ in magnitude and so does the range for the time series in the subplots of Figure 3.8. The horizontal axis in Figure 3.7 gives the solitary-laser frequency relative to threshold (see text around Equation (3.22)), while the vertical axis gives the instantaneous operating frequency with respect to threshold, 0 +  (t)/. The curly line gives the fixed points (as in Figure 3.5), the black dots indicate the instantaneous frequencies at the intersections with , while the grey lines indicate the time-averaged value of (t)/, taken over 50 external round trips. The results in Figures 3.4 and 3.5 pertain to a situation of the laser biased high above threshold, that is J ∼ 14Jthr . Note that at this bias the system would be in the coherence collapse regime in case of COF. The line shape of a diode laser in unstable operation i.e. when the relaxation oscillations are undamped, consists of a main peak and two side peaks located ∼ ± RO away in frequency from the main peak. Here RO is the relaxation oscillation angular frequency. As the solitary laser frequency is detuned ∼ ± RO from the centre of the filter, the side peaks can ‘leak’ back into the laser diode, i.e. the light contained within the side peaks can fall within the filter profile. At these detunings most of the direct feedback from the main peak is suppressed and, in the absence of relaxation oscillations, the light re-entering the laser diode corresponds to very low feedback strengths. However, when the laser does have some RO-dynamics, side peaks will exist and the effective feedback strength increases, thus sustaining the RO. This is

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Figure 3.7 Bifurcation diagram summarizing the different dynamics that occur when the solitary laser frequency is tuned from below the filter centre frequency to above. Vertical scale gives the instantaneous frequency of the laser (relative to solitary threshold) in the presence of filtered feedback; horizontal scale gives the solitary laser frequency (relative to threshold). The grey curly line indicates the corresponding fixed points (as in Figure 3.5). The black portions of the curly line indicate stable cw operation on that fixed point. Black dots indicate instantaneous frequencies at the intersections with  (see text). Black dots organized in regular structures indicate limit cycle attractors; irregular structures indicate chaotic dynamics. Discontinuities in certain limit cycles, e.g. in (g), are artificial; they are implied by the Poincaré-section technique employed (see text). The dot-dashed vertical line indicates the filter centre frequency and the filter HWHM is 2 GHz. Source: Reproduced from: M. Yousefi, D. Lenstra, and A. Fischer, Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback, IEE Proc. -Optoelectronics 37, 233–237 (2001), with permission from IOP ©2001.

what happens at detuning–27 GHz, roughly 5 GHz
∼ RO  above the filter centre, indicated by (a) in Figure 3.7, Here, a RO-based chaotic dynamics is observed (see Figure 3.8(a)), but also RO-based limit cycles occur at slightly different detuning as seen in Figure 3.8(b). Surprisingly, a similar structure of RO-induced dynamics when the laser is detuned one RO-frequency below the filter centre (i.e. at −37 GHz) does not occur or is much less pronounced. The explanation for this apparent asymmetry rests on the amplitude-phase coupling mechanism expressed by the -parameter. According to this, laser operation at frequencies lower than solitary frequency generally goes with higher output intensities, while up-shifted operation frequencies go with lower intensities. Under the tuning condition of case (a) in Figure 3.7, the laser would increase its output intensity if it could shift its frequency down to the filter centre. But since a stable filter-induced ECM is not yet available, this is not a real option. Instead, by developing RO-dynamics the laser realizes an alternative way of

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Figure 3.8 Phase portraits for various attractors of Figure 3.7 are shown in the
 P-plane together with corresponding representative time series of the power for the attractor indicated (by a letter and/or a thin arrow). Fixed points are indicated by stars (not visible in chaotic attractors); a box indicates stable CW-operation on the enclosed fixed point. Note that the vertical scales differ in magnitude and so do the horizontal scales for time and phase difference. The bold horizontal two-sided arrow corresponds to the round-trip time , the bold horizontal line to the RO period. Source: Reproduced from M. Yousefi, D. Lenstra, and A. Fischer, Control of nonlinear dynamics of a semiconductor laser with filtered optical feedback, IEE Proc. -Optoelectronics 37, 233–237 (2001), with permission from IOP©2001.

profiting from the filter. On the other side of the filter, filter-induced dynamics would not be favourable anyway, since now highest intensity is reached close to solitary laser frequencies. A cascade of bifurcations, some of which can be identified as Hopf-bifurcations, occurs as the solitary laser frequency comes closer to the flank of the filter (c). Note that this limit cycle has RO-periodicity. Meanwhile one of the fixed points in the centre of the filter changes its stability and a state of CW operation at that frequency becomes available to the system (d). Just before the fixed-point islands become connected, the system resides on a chaotic attractor indicated by (e). Then a region of multi-stability follows, where several quasi-periodic oscillation states and limit cycles exist simultaneously (Figure 3.8(f)). The time scales of the dynamics in this region is usually a mixture of the RO time scale and the external cavity round trip time. In the two 3 long time series in (Figure 3.8(f)) the multi-period limit cycles are shown. The fast oscillations are on the RO-time scale while the period of the limit cycle itself is . This multi-stable behaviour continues as the solitary laser frequency approaches the centre of the filter. Here is co-existence among several CW-states and a limit cycle oscillating

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at the RO-frequency and arising from a Hopf bifurcation at −315 GHz (see Figure 3.8(g) where a -long segment of the time series of the limit cycle is also shown). As the solitary laser frequency moves down further away from the centre of the filter (positive detuning), the number of multi-stable states decreases and chaotic attractors become more frequent, of which the one around −34 GHz is the most pronounced (h). A sample of this attractor is shown in Figure 3.8(h). After this attractor only CW-operation occurs on the stable fixed points (ECMs) in the solitary laser island (i). The specific fixed point of operation depends on the initial conditions.

3.4 PHASE-CONJUGATE FEEDBACK In this section the single-mode semiconductor laser with delayed optical feedback from a phase-conjugating mirror (PCM) is considered. Such a mirror reverses the phase of the incident wave and, as a consequence, the wave fronts of forward and backward travelling waves coincide everywhere. This property gives optical feedback from a PCM advantages over conventional optical feedback (COF) from a normal mirror. First of all, the alignment of the external system to the laser is automatically fulfilled with PCM and, second, PCF may give rise to linewidth narrowing [16] and noise reduction [17, 18]. We will focus on a PCM operating on the principle of phase conjugation by four-wave mixing (FWM). For a detailed discussion of this process and the derivation of the rate equations, the reader is referred to [3, 19, 26]. Here we will confine ourselves to the outline and some general remarks, which will help understand the rate equations better. In the four-wave mixing configuration, one needs a medium with a nonlinearity associated with a sizeable third-order susceptibility. Then, incident on this medium are two counter-propagating pump waves with the same optical frequency, p , (the pump frequency), see Figure 3.9. This pump frequency is an important reference frequency of the mirror and it will be referred to as the frequency of the PCM. A third beam, which is not co-linear with the pump beams, incident on the pumped material will now create, as a consequence of parametric interaction

1 s c

2

Figure 3.9 Four-wave mixing configuration. Beams 1 and 2 pump the nonlinear medium. Signal beam s gives rise to its phase-conjugated beam c. In the optical feedback situation, beam s is the laser output beam, while beam c will be fed back into the laser.

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71

through the third-order polarization response of the medium, a reflected beam, travelling precisely in the opposite direction. This reflected beam is the phase-conjugated version of the incident beam with respect to the PCM frequency, i.e. if the incident field is expressed as A exp
ip t + iA
t then the reflected field is given by rpcf A exp
ip t − iA
t, where rpcf is the (complex) phase-conjugate reflection coefficient and A is a slowly varying phase, slow with respect to the typical response time of the PCM. For details about the mechanism of phase conjugation by FWM, we refer the reader to the book by Yariv [26]. As mentioned in Section 3.1, the feedback term in the rate equation associated with phase-conjugate reflection can be cast in the form of a damped harmonic oscillator driven by the conjugate of the delayed laser field, see Equation (3.19). We can therefore cast the rate equations for the single-longitudinal-mode semiconductor laser with phase-conjugated optical feedback (PCF) in the form: 1 ˙ =
1 + i n
tE
t + F
t E
t 2 F˙
t = E ∗
t − e2i 0
t−/2 −
 − i 0 F
t n
t ˙ = J − n
t/T − 0 + n
t E
t

2

(3.28) (3.29) (3.30)

Most symbols in these equations have the same meaning as in Equations (3.19) to (3.21).  is the inverse response time of the phase-conjugating mirror and 0 its detuning relative to the solitary laser frequency (see below Equation (3.9));  the external delay time; J the pump rate relative to its threshold value; T the electron-hole life time for spontaneous recombination (both radiative and non-radiative); and 0 is the inverse photon life time in the laser cavity. The feedback rate can be expressed as [see also (19)] =


1 − rF2  r in rF p

(3.31)

where the facet reflectivity rF is given by (14), in is the laser internal round-trip time, and rp is the phase-conjugate reflection coefficient at maximum reflection condition, i.e. at the pump-beam frequency p .

3.4.1 Steady State The steady state solutions to Equations (3.28)–(3.30) are of the form: E
t = Es eis t+iE  F
t = Fs eis t+iF  n
t = ns

(3.32)

with Es  Fs  s  ns time independent. Substitution of Equation (3.32) in Equations (3.28)– (3.30) then yields: s = 0  F = −E 

0 = − 1 + 2 sinarctan
 + 2E  ns = −

2 cos
2E  

Es2 = Fs2 =

J − ns /T J +
2 /T cos
2E  = 0 + ns 0 − 2 cos
2E 

(3.33) (3.34) (3.35) (3.36)

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Clearly, laser locking to the pump frequency of the phase conjugate mirror characterizes the steady state. Note that the phase E is with respect to the phase of the pump field. No other steady state is possible but the one given by Equations (3.33)–(3.36). If this steady state is unstable, the laser output will show dynamics. In this section we will be concerned exclusively with the stability of the steady state Equation (3.32) given by Equations (3.33)– (3.36). The bifurcation structure associated with such dynamics is investigated in Chapter 5 for the special case of zero detuning and instantaneous response, 0 = 0  → . From Equation (3.34) it follows that the steady state can only exist as long as the locking condition,  (3.37)  0  ≤ 1 + 2 is satisfied. This means that the solitary laser frequency should not deviate too much from the phase-conjugate pump frequency. Details of the analysis can be found in [19]. Here, the essential results will be presented and discussed. We have applied a linear stability analysis, meaning that a small deviation from the steady state is considered as an initial state of the laser, after which the rate equations are linearized with respect to the steady state. Thus, a linear set of coupled rate equations with delay is obtained, the solution of which can be analysed using the Laplace-transform method. The resulting characteristic equation for the Laplace variable s, is written as D
s = 0, where D is a combination of a polynomial of degree 5 and exponentials exp(−s) and exp(−2s). All stable solutions of the characteristic equation have negative real part for the steady state solution. If at least one root has a non-negative real part, then the steady state is not stable. The roots of the characteristic equation are investigated using the ‘principle of the argument’, which is based on a theorem from the theory of complex functions of one complex variable [20]. This principle has previously been applied to the problem of conventional feedback in, for example, [21]. A detailed description of the method and its application to our problem of phase-conjugate feedback can be found in [2, 19]. Here we will present results of the stability analysis, mostly organized in terms of stability diagrams, in which two relevant parameters of the system are varied in the plane spanned by these parameters, and the line is indicated where a transition from stable to unstable occurs. This transition is in most cases characterized by one root moving through the imaginary axis without changing the imaginary part. Hence, if the system destabilizes through a Hopf-bifurcation, the imaginary part is the oscillation frequency of the corresponding limit cycle.

3.4.2 Results of Stability Analysis for the Steady State The treatment here follows closely that of Van der Graaf et al. [19]. The stability analysis has been performed numerically using the parameter values listed in Table 3.2 and we consider two different cases for the operating current, 5% and 50% above threshold current. For the case of zero detuning ( 0 = 0) and 5% above threshold, Figure 3.10 shows the stability diagram when and  are varied for several different values of the mirror response time −1 . Focusing first on  → , we see that for very small feedback rates the laser is stable for all values of , but for increasing strength of feedback it becomes unstable. This corresponds to undamping of the relaxation oscillation: note the periodic modulation of the stability-edge curve, indicating lowest stability when the RO matches an external roundtrip resonance, that

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73

Table 3.2 The parameter values used in the calculations Quantity Linewidth enhancement factor Differential gain coefficient Photon decay rate Carrier lifetime Threshold pump rate Pump rate

Symbol

 0 T Jthr J

Value 3 1.19 103 s−1 7.2595 1011 s−1 2.0 ns 3.87 1017 s−1 1.05 Jthr

tm = Λ-1 = 0 100 ps 400 ps

Figure 3.10 Feedback rate at which the laser becomes unstable, versus the cavity round-trip time, , normalized to the relaxation oscillation period. The laser is stable for low feedback and becomes unstable upon increasing . The stability boundary depends periodically on . Source: Reproduced from: W.A. van der Graaf, L. Pesquera, and D. Lenstra, Stability and noise properties of diode lasers with phase conjugate feedback, IEEE J. Quant. Electron., 37, 562–573 (2001), with permission from IEEE©2001.

is, when R /2 is an integer. This is different from COF, where it has been reported that the highest stability occur at these resonances [22, 23], which may be due to the phase shift that occurs on reflection from a normal mirror and which is absent on phase-conjugate reflection. Comparing in Figure 3.10 the stability boundaries for an instantaneous mirror (solid line) with that for a slow mirror (dashed and dotted lines), we see that the stability enhances somewhat with increasing response time. This is a consequence of the filtering property of the PCM. More striking is the shifted location of the stability peaks, which in view of the delay time in the mirror, resembles a situation of larger external roundtrip length. The effective round-trip length is not sharply defined, which reduces the quality of the resonance for small . Since the relative importance of this increases when  gets smaller, this may explain why the peak at R /2 = 05 is lower for  = 25 109 s−1 than for shorter response times. We also investigate the behaviour at moderate feedback. The result is plotted in Figure 3.11 (note the different scale along the horizontal axis, compared to Figure 3.10). The stability of

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Generalized Optical Feedback

Figure 3.11 We show quantities along the axes that are identical to Figure 3.10, but allow for higher feedback rates. Due to the sluggishness of the mirror the laser becomes stable again for higher values of . Source: Reproduced from: W.A. van der Graaf, L. Pesquera, and D. Lenstra, Stability and noise properties of diode lasers with phase conjugate feedback, IEEE J. Quant. Electron., 37 562–573 (2001), with permission from IEEE©2001.

the laser is enormously enhanced by the sluggish mirror: for an instantaneous mirror there is only a small region for short cavities where stability is found at higher values of the feedback rate, whereas over the whole range of  indicated in the figure, the PCF laser is stable at higher feedback for mirror response times −1 of 100 and 400 ps. This recovery of stable operation was anticipated by DeTienne et al. [24]. In numerical simulations they found that the standard deviation of the output power becomes very small for reflectivities on the order of 4%. Our stability analysis confirms that their finding corresponds to stable steady-state operation of the laser. Moreover, in the instability region DeTienne et al. found pulsations for which the frequency increases with . In Section 3.4.3 we will consider this resonance frequency in more detail and derive an expression for it. The structure in the higher stability boundaries of Figure 3.11, i.e. the upper dashed and dotted lines, is caused by this frequency matching an external roundtrip resonance. Therefore this is analogous to the oscillations in Figure 3.10. Also in a laser with filtered optical feedback we expect a sluggish mirror, i.e. with finite filter width, such as from a distant Bragg-grating, to stabilize the system for higher feedback. The influence of the detuning 0 on the stability is shown in Figures 3.12 and 3.13. For weak feedback we see a narrow band of stable operation, but this band widens for higher feedback and finally we find a large region of stable laser output, which is consistent with Figure 3.11. Notice that for an instantaneous mirror we do not find this large region of stability. The general shapes of the curves for finite tm = −1 resemble the stability diagram of a diode laser with external optical injection, see [2]. Also the low-feedback part of the stability diagram with R /2 = 05 is very similar to the corresponding part of the injection laser stability diagram; see [1]. Again we see here the destabilizing influence of the RO matching an external roundtrip resonance (see Figure 3.12). The stability properties of the laser with phase conjugate feedback also depend on the pump current. This is shown in Figures 3.14 and 3.15 for the same parameters as Figures 3.10

Phase-Conjugate Feedback

75

Figure 3.12 Stability of the PCF laser as a function of and the pump detuning 0 . The dash-dotted curve corresponds to an instantaneous mirror at RO /2 = 05. All other curves have tm = −1 = 100 ps. The solid line indicates the locking range. To the left of this line no steady-state solutions exist. Source: Reproduced from: W.A. van der Graaf, L. Pesquera, and D. Lenstra, Stability and noise properties of diode lasers with phase conjugate feedback, IEEE J. Quant. Electron., 37 562–573 (2001), with permission from IEEE©2001.

Figure 3.13 A close-up of the region near the origin of Figure 3.12. Source: Reproduced from: W.A. van der Graaf, L. Pesquera, and D. Lenstra, Stability and noise properties of diode lasers with phase conjugate feedback, IEEE J. Quant. Electron., 37 562–573 (2001), with permission from IEEE©2001.

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Generalized Optical Feedback

Figure 3.14 Stability diagram of a PCF laser pumped 50% above threshold for low feedback rates. Source: Reproduced from W.A. van der Graaf, L. Pesquera, and D. Lenstra, Stability and noise properties of diode lasers with phase conjugate feedback, IEEE J. Quant. Electron., 37 562–573 (2001), with permission from IEEE©2001.

Figure 3.15 Stability diagram for a laser pumped 50% above threshold for moderate levels of feedback. Source: Reproduced from W.A. van der Graaf, L. Pesquera and D. Lenstra Stability and noise properties of diode lasers with phase conjugate feedback, IEEE J. Quant. Electron., 37 562–573 (2001), with permission from IEEE©2001.

and 3.11, except that the current is now 50% above threshold, instead of 5%. There are two major changes: due to the higher RO frequency there are more oscillations in the same range of , and the stability at moderate feedback sets in at a higher feedback strength. All stability diagrams were checked by direct numerical integration of the rate equations (3.28)–(3.30) at several points of the diagrams, and no discrepancy between the two methods was found. We also mention the work by Bochove [25] who has analyzed the configuration of a diode laser with PCF by numerically solving the roots of the system

Phase-Conjugate Feedback

77

determinant. He reports stable behaviour regardless of the feedback strength for certain values of the ‘phase’ in a laser with an instantaneous mirror. This phase can be kept constant when increasing the feedback, by proper variation of the frequency of the mirror pump beam. This probably corresponds to the channel of stable laser operation as shown in Figure 3.12. However, due to the different representation, his results are sometimes difficult to compare. In this section, we have presented results from an exact stability analysis of a semiconductor laser with phase-conjugate feedback, in the sense that we have given an exact expression, which has been solved numerically. In the weak-feedback part of the
 -plane, the periodic modulation of the stability-edge curve is due to the effective external delay time being an integer multiple of the relaxation oscillation period. A finite mirror response time tends to stabilize the system: the low-feedback stability-edge curve shifts upwards with increasing tm = −1 . Upon further increase of the feedback rate, the PCF laser becomes stable again if the mirror has a finite response time. This is in sharp contrast to the unstable behaviour of a laser with an instantaneously responding mirror with the same amount of feedback. The stability areas for low and moderate feedback are not two distinct regimes, but they are connected in the ( , , 0 ) space, which is apparent from the ( , 0 ) slice shown in Figure 3.12.

3.4.3 High-Frequency Oscillations In numerical simulations of a laser with a 0.4 ns response-time PCM, DeTienne et al. [24] found limit-cycle pulsations with frequencies substantially exceeding the relaxation oscillation frequency. They also found numerically that the oscillation frequency depends almost linearly on the feedback rate. Such behaviour is interesting, since it may indicate potential improvement of the speed of the laser, which normally is bounded by the relaxation oscillation frequency. Indeed, numerical calculations of the frequency noise spectrum by Van der Graaf et al. [19] show that in the strong-feedback region with ≥ RO , where RO is the solitary-laser relaxation oscillation angular frequency, the resonance frequency peak no longer lies at RO , but has shifted to a higher frequency. A good feeling for the origin of this new resonance can be obtained by first realizing that the feedback rate defines a new time scale in the laser-with-feedback system, which for external reflectivities of only 1% may already bring us in the above-mentioned regime ≥ RO . Second, the laser frequency is to a first approximation given by Equation (3.29),  ≈ n/2, √ while at zero detuning, one can derive from Equations (3.34) and (3.35), n ≈ 2 /
 1 + 2 . Combining these two relations, we find:  1 + 2 (3.38)  ≈ and this confirms the reported linear dependence of the pulsation frequency on . Even the dependence on is very limited. Remarkably, no other parameters are involved in Equation (3.38), which was in fact also observed by [19]. A more precise, but also more complicated, derivation of the resonance frequency can be obtained by analyzing the roots of the characteristic equation involved in the stability investigation (see at the end of Section 3.2.2). This leads for large values of to the same result as Equation (3.38). Figure 3.16 shows, for  = 0, the imaginary part (solid curve) of the eigenvalue relevant for the stability as a function of . For zero feedback, the resonance

78

Generalized Optical Feedback

Figure 3.16 The eigenvalue corresponding to the relaxation oscillation as a function of the feedback rate (solid curve). The dashed line gives the asymptotic high-feedback rate limit (29). Source: Reproduced from: W.A. van der Graaf, L. Pesquera, and D. Lenstra, Stability and noise properties of diode lasers with phase conjugate feedback, IEEE J. Quant. Electron., 37, 562–573 (2001), with permission from IEEE©2001.

frequency coincides with the relaxation oscillation frequency, while for increasing feedback it rapidly approaches the asymptotic result (3.38) indicated by the dashed line. More detailed analysis shows that the pulsating behaviour of the laser and the resonance in the noise spectra in the stable regime have a common origin, which is the detuning between the eigen-frequency of the compound system and the actual lasing frequency as posed by the mirror pump beams.

3.5 CONCLUSION It is well known that the presence of noise in a deterministic dynamical system will change the location of the bifurcation points in parameter space but, more importantly, it may even change the structure of phase space resulting in different attractors. Therefore, if one is to accurately compare the results of rate equations models with experimental scenarios, the ‘correct’ metamorphosis of the deterministic system should be employed. In the case of filtered optical feedback, such a study has recently been published [29] and we expect many more studies to follow. We have treated several varieties of optical feedback in quite some detail and the question now is what to expect further from generalized optical feedback. A common aspect of all cases considered is the presence of time delay. In general, dynamical systems with time delay are hard to deal with, mainly because of the high, formally even infinite, dimensionality [14]. The mathematical framework and corresponding tools for systematic bifurcation analysis are currently not as advanced as for low-dimensional dynamical systems. Fortunately, enormous progress has been made recently in developing bifurcation continuation methods for delaydifferential equations [30], similar to the very successful methods for low-dimensional systems

References

79

that have been available for several years (see Chapter 5). We therefore expect that important progress will be made in the near future on systematic analysis of global bifurcation structures for lasers with delayed optical feedback. The insight to be gained will be essential for applications such as optical signal processing, chaotic optical communication [31], and phase space trajectory steering techniques [32].

ACKNOWLEDGEMENTS MY and DL acknowledge financial support from the EU TMR-Network FALCON (ERB 4061 PL97-0131) and the Belgian IAP-18 Network ‘PHOTON’. MY also acknowledges grant and project support by FWO-Vlaanderen. GV acknowledges partial support from the US National Science Foundation.

NOTE 1 The word fixed point is used here in a more general meaning, than the usual definition of strictly time-independent solutions.

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

R. Lang and K. Kobayashi, IEEE J.Quant. Electron., 16, 347–355 (1980). S. Wieczorek, B. Krauskopf and D. Lenstra, Optics Commun., 172, 279–295 (1999). W.A. van der Graaf, thesis, Vrije Universiteit, Amsterdam, 1–152 (1997). B. Tromborg, H. Olesen, X. Pan and S. Saito, IEEE J. Quant. Electron., 23 1875 (1987). J.D. Jackson, Classical Electrodynamics, Wiley, New York (1975). A.A. Duarte and H.G. Solari, Phys. Rev., A 58, 614 (1998). T.W. Carr, D. Pieroux and P. Mandel, Phys. Rev., A 63 033817 (2001). G. Huyet, S. Balle, M. Guidici, C. Green, G. Giacomelli and J.R. Tredicce, Optics Commun., 149, 341–347 (1998). P.S. Spencer and K.A. Shore, Quantum Semiclass. Opt., 9, 819–830 (1997); T.W. Carr, D. Pieroux and P. Mandel, Phys. Rev., A 63, 033817 (2001). M. Yousefi, D. Lenstra, G. Vemuri and A.P.A. Fischer, IEE Proc. Optoelectron., 148, 233–237 (2001). M. Yousefi and D. Lenstra, IEEE J. Quant. Electron., 35, 970–976 (1999). A.P.A. Fischer, O.K. Andersen, M. Yousefi and D. Lenstra, IEEE J. Quant. Electron., 36, 375–384 (2000). T. Heil, I. Fischer and W. Elsässer, Phys. Rev., A 60, 634–641 (1999). S.M. Verduyn Lunel and B. Krauskopf, in B. Krauskopf, D. Lenstra (eds) ‘Fundamental issues of nonlinear laser dynamics’, American Institute of Physics (AIP) Conference Proceedings, 548, 66–86 (2000). W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge (1994). K. Vahala, K. Kyuma, A. Yariv, S. Kwong, M. Cronin-Colomb and K.Y. Lau, Appl. Phys. Lett., 49, 1563–1565 (1986). G.P. Agrawal and G.R. Gray, Phys. Rev., A 46, 5890–5898 (1992). L. Petersen, U. Gliese and T.N. Nielsen, IEEE J. Quant. Electron., 30, 2526–2533 (1994). W.A. van der Graaf, L. Pesquera and D. Lenstra, IEEE J. Quant. Electron., 37, 562–573 (2001). J.J. DiStefano, A.R. Stubberud and I.J. Williams, Feedback and Control Systems, McGraw-Hill, Singapore (1967), pp. 187–223. J.S. Cohen, R.R. Drenten and B.H. Verbeek, IEEE J. Quant. Electron., 24, 1989–1995 (1988). J. Moerk, B. Tromborg and J. Mark, IEEE J. Quant. Electron., 28, 93–108 (1992). A. Ritter and H. Haug, J. Opt. Soc. Am. B 10, 130–144 (1993). D.H. DeTienne, G.R. Gray, G.P. Agrawal and D. Lenstra, IEEE J. Quant. Electron., 33, 838–844 (1997). E. Bochove, Phys. Rev., A 55 3891–3899 (1997).

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[26] A. Yariv, Quantum Electronics, John Wiley & Sons, New York (1988). [27] J.P. Goedgebuer, L. Larger and H. Porte, Phys. Rev. Lett., 80, 2249 (1998); J.P. Goedgebuer, L. Larger, H. Porte and F. Delorme, Phys. Rev., E57, 2795 (1998). [28] G. Huyet, S. Balle, M. Giudici, C. Green, G. Giacomelli and J.R. Tredicce, Optics Commun., 149, 341 (1998). [29] M. Yousefi, D. Lenstra and G. Vemuri, Phys. Rev., E67, 046213 (2003). [30] D.J. Gauthier, private communication. [31] K. Engelborghs, T. Luzyanina and G. Samaey; DDE-BIFTOOL v2.00: a Matlab package for bifurcation analysis of delay differential equations. Tech. Rep. TW-330. Department of Computer Science, K.U. Leuven, Belgium, 2000. http://www.cs.kuleuven.ac.be/∼koen/delay/ddebiftool.shtml [32] S. Donati and C.R. Mirasso (eds) Feature section on optical chaos and applications to cryptography, IEEE J. Quant. Electron., 38 (9) Sept., 1138–1184 (2002). [33] G.L. Lippi, S. Barland, N. Dokhane, F. Monsieur, P.A. Porta, H. Grassi and L.M. Hoffer, J. Opt. B: Quant. Semiclass. Opt., 2 375–381 (2000).

4 Experimental Observations A. Tom Gavrielides and David W. Sukow

4.1 INTRODUCTION Within a few years after the successful demonstration of stimulated emission in GaAs p-n junctions by several groups [1]–[4], it was discovered that semiconductor lasers are very susceptible to stray reflections that are reintroduced into the laser cavity. Such reflections can originate from fiber facets or any other optical elements introduced into the light path. As a result, Faraday isolators are used extensively to reduce back reflections by as much as 60 db. Early studies seem to indicate that external feedback results in an excess noise occurring at frequencies corresponding to harmonics of the external cavity round-trip time [5, 6]. If the external cavity length is of the order 1 meter, noise peaks appear at gigahertz frequencies and are referred to as ‘high frequency noise’. Additionally, ‘low frequency’ noise dominates at frequencies less than 100 MHz and appears to be proportional to the external cavity length. As will be discussed later in more detail, many effects caused by external feedback greatly depend on experimental conditions. While very weak feedback can induce line narrowing or line broadening, depending on the feedback phase, higher reflections can result in dramatic line broadening termed coherence collapse [7], in which the linewidth of the laser spans tens of gigahertz. Even more surprisingly, very high feedback levels result in pronounced line narrowing, suppression of excess noise, an effect extensively used to produce narrowband tunable single-mode lasers which are very useful in coherent communications, spectroscopy and a host of other applications. The problem of semiconductor lasers with optical feedback is a topic of great technological importance. Communication and data storage technologies are among the most prominent applications in which delayed optical feedback is encountered. Feedback-induced instabilities are particularly problematic in technologies that require high signal-to-noise ratios such as videodisk players [8], where even low-level laser noise results in degraded picture quality. Fiber optic communication systems can be limited by unwanted optical feedback arising at fiber facets and junctions [9, 10]. Although an isolator can address this problem, elimination of the isolator is desirable because it leads to simplified packaging and reduced cost [11]. Unlocking Dynamical Diversity Edited by Deborah M. Kane and K. Alan Shore © 2005 John Wiley & Sons, Ltd

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Experimental Observations

Alternatively, fibers with cleaved angled facets can be used to alleviate such reflections. Methods of reducing or suppressing the relative intensity noise (RIN) are therefore an important line of inquiry. Such laser diode noise both in the < 20 KHz control frequency and, in the video band (4–13 MHz), can adversely effect the signal-to-noise ratio. Reduction of noise levels in the region of importance can be significantly achieved by imposing 100% modulation-depth, 1 GHz signal on the current of the diode laser. This high frequency current superposition method has been shown to be effective and can be optimized for RIN reduction [12]. While external optical feedback represents a problem in some situations, it also has several beneficial effects that can improve laser performance. Very weak and very strong feedback both can result in line narrowing and noise suppression. This, coupled with the large gain bandwidth of the semiconductor laser, can produce a highly tunable, narrow line source that finds many uses in spectroscopy, metrology, and telecommunications. Indeed, the wide range of applications has created a market for such devices, leading to their easy commercial availability. This topic is covered in more depth in Chapter 8. A sample of the developments involving the beneficial properties of external cavity feedback illustrates the range of uses. A widely tunable semiconductor optical amplifier designed with a multiple quantum well structure has been realized, spanning a wavelength range from 1300 to 1540 nm, to take advantage of broadband fibers [13]. In a broad stripe laser diode, angle-selective optical feedback has been used to create narrow-lobe, near diffraction-limited emission, thus increasing the useful output power of the device [14]. Another interesting demonstration has been to subject a broad area laser to injection from a single-mode external cavity laser, in which case the spectral characteristics of the master laser were transferred efficiently to the slave oscillator [15]. A novel application is high-precision displacement measurements, in which the laser is directed at a reflective, movable target (which forms the external cavity) at a distance of the order of meters. The displacement of the target from an initial position is recovered interferometrically without sign ambiguity [16]. In Chapters 7–9 systems and applications of lasers in external cavities are described in more detail. Semiconductor lasers belong to class B lasers and are usually very well described by two equations, one for the complex field and another for the inversion. The fast decay times T2 of the polarization, of the order of 200 fsec, as determined mostly from carrierphonon dephasing collisions, make it possible to adiabatically eliminate the polarization. The remaining time scales are the carrier lifetime that appears in the inversion equation, and is of the order of a few nanoseconds, and the photon lifetime of a few picoseconds. The fast photon lifetime is mainly the result of the short cavity of the diode lasers. The additional feature that distinguishes semiconductor lasers is what is known as the linewidth enhancement factor that couples the inversion to the phase of the electric field and it is defined by:
=−

G/N n/N

(4.1)

the ratio of the change of the inversion with carrier density divided by the change of the index of refraction with carrier density. At the rate equation level the usual assumption is to take the linewidth enhancement factor to be constant. However, care is taken to operate the laser consistently in the same regime in which the linewidth enhancement factor was

Introduction

83

originally determined. Corrections to its dependence to the carrier density can be included in more consistent treatments. The result is a dramatic broadening of the optical lineshape above the usual Schawlow Townes [17] limit by a factor of 1 +
2 . Since most lasers have
∼ 4 − 6, this effect can be readily observed [18]. Therefore, solitary semiconductor lasers can be most uninteresting in dynamical terms, since they simply possess a steady state as determined by the amplitude of the field and the inversion. The phase is naturally determined also, however, it is a dependent quantity and does not enter in to the dynamics. However, this ceases to be the case when external influences are introduced and additional terms appear in the field equation or the inversion. These could appear as a result of injection of radiation from another laser, reflection from external optics, feedback from a phase conjugator, injection current modulation, in addition to optical feedback, etc. The reinjection of the optical radiation can also be modulated by a phase modulator inserted in the external cavity, or frequency shifted by an acousto-optic modulator before reinjection. All these effects lead to an additional coupling of the phase of the laser to the intensity and can produce a variety of dynamical outputs. In addition, the delayed nature of the reinjected field enlarges an already complicated dynamical structure and leads to so many interesting phenomena, that, almost thirty years after the first observations of instabilities in these systems, they still continue to be actively investigated. In this chapter we will concentrate on experimental observations of the dynamics of emitted radiation of semiconductor lasers in an external cavity. The dynamics of such devices will be outlined as a function of feedback levels starting progressively from weak feedback at which the basic local bifurcations take place, to moderate feedback at which global bifurcations, and what is known as the low frequency fluctuations begin to appear. Finally, we will briefly examine strong feedback and the transition from coherence collapse to single mode narrow-line emission. This can be very nicely summarized by the diagram [19] shown in Figure 4.1. At the lowest feedback levels, Regime I, the laser is operating on a single external cavity mode that emerges from the solitary laser mode, which has been narrowed or broadened depending on the phase of the feedback. In Regime II and at a feedback level that depends on the distance from the external reflector, the mode appears to split into two modes, an effect arising from rapid mode hopping. Noise-induced hopping between two external cavity modes is the underlying reason for this behavior. The transition into Regime II has been analyzed in [20] and it is found to correspond to multiple solutions of the steady state equation that determines the frequency of the laser. This condition is given by C = 1 where
C = c 1 +
2

(4.2)

and c = 2L/c is the external cavity round trip time and  the effective feedback from the external cavity. In Regime III the laser restabilizes in a single external cavity mode (the lowest linewidth mode) with constant power. As the level of optical feedback is increased, and independently of the length of the external cavity, the laser system undergoes a transition to a chaotic state known as coherence collapse and enters Regime IV. This is characterized by a dramatically broadened optical and noise spectra and contains many external cavity modes. The low current injection regime near the solitary laser threshold is also known as the low frequency fluctuation regime, so named from the irregular and slow power drop off events. The route

84

Experimental Observations

Figure 4.1 Regimes of operation of a laser with external feedback. Source: After [19], © 1986 IEEE.

to this chaotic state is characterized by a series of bifurcations which will be carefully outlined and detailed in Section 4.4, ‘Weak Feedback Effects’. The low frequency fluctuation regime will be introduced and its dynamical state discussed in Section 4.6 ‘Moderate Optical Feedback’. Still further increase in the optical feedback level results in a transition to another single mode, constant intensity, and narrow linewidth regime (Regime V) when the diode laser facet facing the mirror has been anti-reflection coated. This regime cannot be reached when diodes with uncoated facets are used. Regime V, in which the laser is operating on a steady state, will not be discussed as it is rather uninteresting from the dynamical viewpoint. In the next few sections the most important effects in each of the feedback regimes will be discussed with emphasis on the sequence of bifurcations as a function of the feedback strength. In addition, some particular experimental effects that occur in short external cavities and also in double cavity systems will be outlined (Sections 4.7 and 4.8). One section will be devoted to multimode effects (Section 4.9). In this section the external cavity feedback phenomena under single or multimode operation of the laser will be contrasted. In the Control section (Section 4.10) we discuss several experimental procedures to alleviate the chaotic behavior of the laser in the external cavity but still retain all the operational characteristics of the laser. The following section reviews the experimental literature of the semiconductor laser in an external cavity with the addition of current modulation of the laser. Indeed, one such technique is to take advantage of modulation to open loop stabilize the external cavity laser. The penultimate section of this chapter discusses the effects and phenomena observed

Experimental Apparatus

85

in experiments when the feedback from an external mirror is replaced with a phase conjugate reflection from a number of possible phase conjugators.

4.2 EXPERIMENTAL APPARATUS To facilitate descriptions of the results obtained in experimental systems, it is useful to consider the standard elements of experimental design used in research on semiconductor lasers with delayed optical feedback. Despite the complexity of the dynamics produced in optical feedback systems, the typical experimental apparatus used is relatively simple. The main elements are: the laser diode and its power and stabilization electronics, an external cavity formed by a distant reflector and a variety of intracavity elements, and an array of detection and diagnostic instruments to measure the dynamics. Experimentally accessible parameters are typically the laser pump current and the optical feedback strength. Other parameters that may be designed to be adjustable are the polarization state, cavity length, laser temperature, and feedback frequency. A standard experimental schematic is shown in Figure 4.2. The laser device itself can be as varied as fabrication technology allows, in terms of materials and architecture: edge-emitting lasers, VCSELs, and broad area lasers; multiplequantum well, channel substrate planar, index-guided, gain-guided, DFB, and so forth. Regardless of specific type, the laser is typically set up in a temperature-controlled mount, with stability to ±001 K  easily achieved. A temperature controller is necessary because thermal variability causes changes in the lasing frequency, due both to shifts in the semiconductor gain bandwidth as well as thermal expansion that changes the spacing of the diode mirror facets, and thus alters the eigenmodes. The laser is driven by a low-noise current supply: an electronic system or a simple battery. If current modulation is required, an oscillating source can be coupled with the DC drive current using a bias-tee.

Figure 4.2 Schematic of a standard experimental apparatus for a laser with an external cavity.

86

Experimental Observations

The external cavity contains a variety of components in addition to the end mirror. A collimating lens with large numerical aperture is needed to capture and collimate the strongly diverging beam that emerges from the laser. This can be a microscope objective, GRIN rod, or other short-focal length lens [21]. The collimated beam spreads into the cavity and strikes the end mirror, which must be precisely aligned to optimally retroreflect the beam back to the laser [22]. A variety of intracavity elements can be employed depending on the needs of the experiment. Feedback strength can be controlled by attenuating the beam, which can be accomplished with crossed linear polarizers, a /4 plate and a linear polarizer, or a variable neutral density filter. Frequency-selective feedback is often desired, typically for studies involving a single longitudinal mode of the solitary laser as modeled by the LK equations. In this case, an intracavity etalon or diffraction grating may be inserted to restrict the feedback frequency to a single mode, if the bandwidth of the etalon or grating is sufficiently smaller than the longitudinal mode spacing of nominally 150 GHz. The apparatus described so far constitutes the dynamical system itself: a powered, stabilized laser with an appropriate external cavity to provide delayed optical feedback with the desired properties. To observe and characterize the dynamics created in the system, a wide range of detection devices can be employed. If the end mirror of the external cavity is a partial mirror, the beam that passes through can be used for detection. If the end mirror is highly reflective, other ways to obtain a detection beam are to capture the beam emerging from the other facet of the laser (not shown) or to insert a beamsplitter into the external cavity and detect the beams thus diverted (although this will alter the feedback strength as well). Regardless, the beam or beams to be detected must be isolated from the dynamical system to prevent back reflections from the detection devices from altering the dynamics that one wished to observe. Faraday isolators are effective for this purpose, providing isolation of up to 60 dB. The beam, thus isolated, then can be directed to any or all of several instruments. Time-domain measurements of the laser intensity are gathered by directing the beam to a high-speed photodetector and oscilloscope, or directly to a streak camera. Frequencydomain measurements are made using a photodetector and RF analyzer, or with an optical spectrum analyzer to observe the FM spectrum. Bandwidth considerations are important when selecting a photodetector or amplifier. Semiconductor laser characteristic frequencies are of the order of several GHz, requiring a large detector or amplifier bandwidth. However, to accurately detect some dynamical states such as LFF, the low-end cutoff frequency must be simultaneously quite small, of the order of tens of MHz, to accurately capture the extremely wide range of timescales present. Although the system described above demonstrates the most common elements employed in experimental studies, it is by no means inclusive. Other measurements have also been used where appropriate, such as CCD imagery or frequency noise spectroscopy [23, 24]. An important element of experimental work is to connect the accessible, measurable system parameters with theoretical models. Recalling the Lang–Kobayashi equations:   dE 1 1 E + E t − c  e−i 0 c = 1 + i
 g N − N0  − dt 2 p

(4.3)

dN N = J − − g N − N0  E2 dt e

(4.4)

Experimental Apparatus

87

The feedback rate  is an important quantity, and is often used as a bifurcation parameter. The expression for  in terms of mirror reflectivities is: 

Rext  = f 1 − R2  R2

1 2

c 2l

(4.5)

where c is the external cavity roundtrip time, Rext is the power reflectivity of the external reflector, R2 is the power reflectivity of the front laser facet, l is the optical path length of the laser cavity and f is the fraction of the reflected field which couples back into the lasing mode. The coupling fraction f is difficult to measure directly in experiments, and so an alternate method of deducing  is useful. To characterize experimental feedback rates to the laser, one measures the reduction in the effective threshold current in the presence of feedback. This is illustrated in Figure 4.3, which shows sample light-current curves for a solitary semiconductor laser, and the same laser with optical feedback [25]. It is evident that the current threshold is reduced. In practice, then, the quantity related to  that is most easily measured experimentally is the fractional threshold reduction I, which is related to the threshold current with feedback Ith and the solitary threshold current Ithsol as:  sol  Ith − Ith

I = Ithsol

(4.6)

Figure 4.3 Light-current curve of the solitary laser, and of the laser with external feedback. Source: After [25] © 1993 IEEE.

88

Experimental Observations

The fractional threshold reduction I is directly proportional to :

I = 

p 1 + gN0 p

(4.7)

where g is the optical gain and N0 is the transparency carrier density.

4.3 EXTREMELY WEAK FEEDBACK EFFECTS – REGIME I It was found that in this regime even extremely small feedback from an external reflector can affect the laser rather strongly. For example, for a 1 mW laser power, feedback corresponding to a power of 10 pW can have important effects. To understand the source of this, we can look at the steady state solutions of Equations (4.3)–(4.4). Assuming the form for the electric field E = As e−i 0 − s t 

N = Ns

(4.8)

where As  Ns are the steady state amplitude of the electric field and the inversion. In addition, 0 is the frequency of the solitary laser and s is the frequency of the external cavity mode, and it is feedback dependent, as we will shortly see. Substituting these expressions in Equations (4.3)–(4.4), we find Ns = − cos s 

(4.9)

 s − 0  = −C sin s  + tan−1 


(4.10)

√ where C =  1 +
2 . Clearly the last equation is a transcendental equation that can have multiple solutions s for the same value of C. Indeed, this occurs at C = 1. Regime I is clearly the regime for which a single external cavity mode exists and it is for feedback levels that satisfy C < 1. Additionally, the effect of the feedback on the linewidth of the laser can be calculated by including spontaneous emission Langevin noise sources in Equations (4.3)– (4.4) [26].

=

0

1 + C cos s  + tan−1 


(4.11)

where 0 is the linewidth of the laser without external feedback. Clearly, the linewidth of the laser now can broaden or become narrow depending on the relative phase of the feedback. This can be clearly seen in the spectra shown in Figure 4.4 [19] for a 15 m DFB laser. The first trace corresponds to the case of the solitary laser i.e. no feedback. The second trace Figure 4.4(b) is with a feedback power level of −80 dB and with the phase adjusted for maximum narrowing, and the last trace Figure 4.4(c) is with the phase adjusted for maximum broadening.

Very Weak Feedback Effects – Regime II

89

Figure 4.4 Optical spectra of a laser with an external cavity of 40 cm. (a) No feedback; (b) −80 dB feedback, in phase; (c) −80 dB feedback, out of phase. Source: After [19] ©1986 IEEE.

4.4 VERY WEAK FEEDBACK EFFECTS – REGIME II As the feedback strength is further increased, we move into Regime II. Clearly more than one solution is possible for the steady state for C > 1. When there are multiple solutions, the laser appears to operate at the frequency which results in the lowest linewidth. When, therefore, there are two solutions of the lowest linewidth, the line appears to be split. This will happen when the phase of the feedback is adjusted to be 0  =  − tan−1 
 as can be readily seen from Equation (4.11). This apparent splitting of the laser line is the result of the laser rapidly hopping between the two available modes. These two modes correspond to the mode that is a continuation of the solitary mode and the next mode that is born through a saddle-node bifurcation. In general, the modes will shift frequency toward the red as the feedback ratio is increased, the inversion will become more negative and the power will increase. The birth of another mode through a saddle-node bifurcation also gives rise to a hysteresis phenomenon as a function of the feedback phase but of fixed feedback strength. This hysteresis can be seen in the output power of the laser as well as the frequency as the feedback phase is scanned through 2. This bistability was studied experimentally and theoretically using Equations (4.3)–(4.4) in [27]. The apparent splitting is illustrated in the sequence of spectra shown in Figure 4.5. Indeed, this effect is clearly seen by comparing the spectra of the laser without feedback Figure 4.5(a), the laser with a weak feedback, −62 dB, in phase, Figure 4.5(b), and the laser at the same feedback level but with the relative feedback phase out of phase Figure 4.5(c). In addition, it can be shown that the splitting depends on the amount of feedback and it can be easily calculated from the solutions of Equation (4.10). Indeed, the agreement between experimentally determined splitting versus feedback denoted by the filled circles, and the calculated solid line in Figure 4.6 is impressive [19] for a 15 m DFB laser. The mode hopping phenomenon is very interesting because it combines the external cavity dynamics and noise-induced phenomena. Since the external cavity is in general long and the frequency separation between modes is small, there is always a finite probability that the laser will jump from one available mode into another. To study such effects, the Lang–Kobayashi

90

Experimental Observations

Figure 4.5 Optical spectra of a laser with an external cavity of 40 cm. (a) No feedback; (b) −62 dB feedback, in phase; (c) −62 dB feedback, out of phase. Source: After [19] © 1986 IEEE.

Figure 4.6 Mode separation frequency of the modes in Figure 4.5(c) as a function of feedback rate. Filled circles are experimental points and solid line is obtained from the steady state solutions of the Lang–Kobayashi equations. Source: After [19] © 1986 IEEE.

Very Weak Feedback Effects – Regime II

91

equation must be modified by the inclusion of stochastic spontaneous emission terms. However, due to the existence of strong nonlinearities, the equations are difficult to solve except by numerical techniques. An approximate model [28]–[29] constructed from the Lang–Kobayashi equations by retaining only the equation for the phase difference of the electric field t = t − t −  and assuming that the fluctuations of the amplitude and inversion can be ignored, gives: dV  d t =− + Ft dt d 

(4.12) 

where Ft represents the Langevin noise term with correlation < FtFt  >= 2 0   t − t , where 0 is the linewidth of the solitary laser. The potential is given by: V  =


1 V 2 −  1 +
2 cos 0  + tan−1 
 +  2

(4.13)

and it can be interpreted as a particle moving with coordinate  in a potential. The potential has maxima and minima at / = s − 0 , the minima corresponding to the stable modes and the maxima to the unstable saddles. In the absence of noise, the laser will settle in one of these depending on the initial conditions. However, in the presence of noise the laser will have a finite probability of being driven over the barrier and into the adjacent minimum of the potential. The mean transition time from one mode to another can be determined from the first passage time [30]–[31]. T=




2 

 Vmin Vmax

e V/2 0 

(4.14)

where Vmin , and Vmax are the curvatures at the potential maximum and minimum and V is the height of the barrier. Experimentally the first passage time and its dependence on the feedback strength, or on the phase of the feedback, can be determined by monitoring the power of the laser with a fast oscilloscope. Since the two modes have different powers, and of course different frequencies, the modes can be identified by the intensity levels. Since noise is an important ingredient, and strongly affects the first passage time, it is of considerable importance to drive the laser with an ultralow-noise current source and for it to be temperature stabilized [32]. Figure 4.7 shows the experimentally obtained evolution of the oscillation frequency of a 1.3 m DFB laser and the corresponding potential wells obtained from Equation (4.13) for three different values of the phase. The feedback strength is −457 dB and for the lower trace the relative phase is set to 0  + tan−1 
 ∼ 0. The trace shows mode hopping between three modes 250 MHz apart corresponding to the external cavity round-trip frequency. The absolute minimum of the potential is at the solitary laser frequency, and the laser infrequently can jump to either of the two wells adjacent to the main one. The top trace is obtained for a relative phase 0  + tan−1 
 ∼  where the solitary laser frequency lies exactly between the minima. Here the laser can visit the two wells with equal frequency. The middle trace is taken for 0  + tan−1 
 ∼ 14 and shows the intermediate situation. The experimental data in Figure 4.8 show the frequency evolution as a function of time for different feedback levels with the feedback phase adjusted for maximum hopping rate to 0  + tan−1 
 ∼ .

92

Experimental Observations

Figure 4.7 The laser frequency as a function of time, for an injection current I = 28 mA. Lower trace is for a relative phase of 0  + tan−1 
 ∼ 0, middle trace 0  + tan−1 
 ∼ 14, upper trace 0  + tan−1 
 ∼ . Source: After [28] © 1990 IEEE.

Figure 4.8 The laser frequency as a function of time, for a relative phase of 0  + tan−1 a ∼  and injection current of I = 28 mA. Lower trace is for a feedback rate of  = −464 dB, middle  = −457 dB, and upper  = −450 dB. Source: After [28] ©1990 IEEE.

The transition time increases considerably as a function of feedback, as can be seen from the three traces: upper −45 dB, middle −457 dB, lower −464 dB. This is in complete agreement with the model, since the barrier increases with . From such measurements the first passage time can be computed as the average time spent by the laser in one of the longitudinal modes before it hops to the adjacent mode as a function of feedback. The relative transition time is shown in Figure 4.9 along with the prediction of Equation (4.14) as a function of feedback and for three different pumping levels. The solid lines are the theoretical calculations and the solid squares are the experimental data for injection current I = 24 mA, open triangles for I = 28 mA, and filled circles for I = 32 mA. The agreement with the

Weak Feedback Effects – Regime III–IV

93

Figure 4.9 The transition time as a function of feedback strength for a relative phase of 0  +  −1 
 ∼ . Solid lines are the theoretically computed curves and the data points shown by solid squares are for I = 24 mA, open triangle I = 28 mA, and solid circles I = 32 mA. Source: After [28] ©1990 IEEE.

theoretical curves is fairly good showing some unexplained discrepancies at intermediate feedback levels, but very accurate at higher feedback levels. The roll-off at high feedback levels is due to nonlinear effects that take over very quickly and dominate the process as the modes can become unstable with the increase of the feedback rate [32]. Indeed, experiments in [32] show that the transition time deviates significantly from the exponential growth at larger feedback levels in contrast to the predictions of the simple theory.

4.5 WEAK FEEDBACK EFFECTS – REGIME III–IV In this section we will consider the series of bifurcations encountered and the increasingly more complex dynamics of the laser as the feedback is increased. These effects correspond to the transition between Regime III and into Regime IV as discussed previously and shown in Figure 4.1. In general, the laser will be lasing in a steady state with constant intensity and with a frequency that is down-shifted relative to the solitary laser frequency. This frequency depends on the amount of feedback, however, because the bifurcations take place over a rather small range of external feedback, the shift is not immediately obvious. Several authors have investigated experimentally the route into chaos. The system has two frequencies available, the relaxation frequency which can become undamped and the frequency of the external cavity fc = c/2L. Numerically it has been found that for relatively long cavities, and with high pumping, the laser undergoes a Hopf bifurcation to a periodic oscillation that has a frequency close to the relaxation frequency [34]–[36]. Secondary Hopf bifurcations to quasiperiodicity take place with a frequency close to the external cavity frequency [37]. Two frequency locking may or may not be observed depending on whether the two frequencies are commensurate or not. Figure 4.10 shows a bifurcation diagram computed from the Lang–Kobayashi equations Equations (4.3) and (4.4).

94

Experimental Observations

Figure 4.10 Bifurcation diagram of the inversion as a function of feedback strength  for an injection current of 13 × Jth and the external delay is 1 n sec. Source: After [40] ©1992 IEEE.

The pumping of the laser is set at 13 × Ith and the external delay is 1 ns. The bifurcation parameter is the feedback strength and only the intersection points of the normalized carrier density corresponding to traversal through the Poincar´e plane in the direction of decreasing field amplitude are recorded [40] in Figure 4.10(a). The same information is contained in the bifurcation diagram in terms of the phase delay in Figure 4.10(b). In addition, Figure 4.11 shows the Poincar´e section of the various attractors at fixed feedback levels in the inversion and phase delay plane. Figure 4.12 shows the time series at selected feedback ratios. The feedback strength for a laser with a Fabry–Perot cavity laser is determined from  Rext (4.15)  = 1 − R2  R2 and 2 is the power reflected from the external cavity relative to the power from the laser mirror. This is the same quantity that is represented in Equation (4.5) but normalized to the solitary laser cavity round-trip time. Additionally, the coupling fraction into the diode is assumed to be unity. For a DFB laser diode  becomes complex and it has to be determined from a rather complicated nonlinear expression involving the details of the grating [38, 39]. However, here  is assumed to be positive and real, and any possible phase can be included in the rather arbitrary phase 0 .

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Figure 4.11 Poincar´e section of the various attractors at fixed feedback levels in the inversion and phase delay plane for the same conditions as in Figure 4.10 [40]. (a)  = 4 × 10−3 , (b)  = 8 × 10−3 , (c)  = 872 × 10−3 , (d)  = 116 × 10−2 , (e)  = 12138 × 10−2 , (f)  = 12279 × 10−2 . Source: After [40] ©1992 IEEE.

There are no intersections with the Poincar´e plane for  < 24 × 10−3 , indicating the solution is a stable fixed point as can be seen in Figure 4.10. However, at  = 24 × 10−3 the relaxation oscillations become undamped and a limit cycle is born with frequency of 2.38 GHz. Figure 4.11(a) records a single point at the Poincar´e section and Figure 4.12(a) confirms the existence of the limit cycle at 4 × 10−3 . The frequency of this limit cycle is very close to the relaxation frequency of the solitary laser. At  = 74 × 10−3 the limit cycle bifurcates to a torus with two incommensurate frequencies. The second frequency is slightly lower than the external cavity frequency 1/c . This is confirmed in Figure 4.11(b) in which the Poincar´e section shows a trajectory that executes a motion on a torus. Further increasing the feedback only slightly to 8 × 10−3 the two frequencies lock in 1:3 ratio, as can be inferred from Figure 4.11(c) and from the

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Figure 4.12 Time series of the electric field for the same parameters as in Figure 4.10 for a feedback rate, (a)  = 4 × 10−3 , (b)  = 8 × 10−3 , (c)  = 116 × 10−2 , and (d)  = 15 × 10−2 . Source: After [40] ©1992 IEEE.

time series of Figure 4.12(b). Increasing further the feedback rate to 116 × 10−2 a subharmonic bifurcation occurs and components at a frequency of ext /2 appear. This can be clearly seen in the section of Figure 4.11(d) but the confirmation is in the time series of Figure 4.12(c). For  = 121 × 10−2 the limit cycle of order six bifurcates to a torus of order six Figure 4.11(e) and finally for  = 122 × 10−2 the torus bifurcates into a chaotic attractor. This is evident by the multisheet appearance of the Poincar´e section Figure 4.13 which is a blow-up of the region delineated by the square outlined in Figure 4.11(f). Experiments were conducted with a 1.3 m distributed feedback semiconductor laser from Fujitsu [40]. The feedback level was controlled with a variable attenuator inserted in the external cavity. The external cavity length was 16 cm corresponding to a delay of  = 107 n sec. Since the purpose of this experiment was to follow the basic bifurcation of a single external cavity mode to chaos, the length of the cavity was chosen to be rather short to avoid complications arising from multiple attractors with different relaxation frequencies. In general, the relaxation frequency of semiconductor lasers is in the GHz regime and it becomes very difficult to measure time series at such frequencies directly; note that recent improvements to oscilloscopes are making it possible. For example for a bias current of I/Ith = 187 the relaxation frequency of the laser was 4 GHz. To obtain some idea of the basic bifurcations, an RF-heterodyne technique can be employed. This technique uses a local oscillator with a frequency close to the frequency of interest, in this case the frequency of the relaxation oscillation. Therefore, a local oscillator

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Figure 4.13 Poincar´e section of the chaotic attractor at  = 122 × 10−2 . Parameter values are the same as in Figure 4.10. Source: After [40] ©1992 IEEE.

with a frequency of 4088 GHz was chosen. The light from the diode laser is detected with a photodiode and its current is mixed with the signal from the local oscillator. The output of the nonlinear mixer contains the frequency difference of the signal with respect to the local oscillator and it can be studied with a digital oscilloscope. Phase portraits are obtained by splitting the signal after the mixer and delaying one arm relative to the other by one quarter of the oscillation period. These phase portraits are shown in Figure 4.14. For very low feedback strength  = −45 dB the laser is stable and is operating in the minimum linewidth mode. The phase portrait shows a significant noise component which can be mainly attributed to spontaneous emission noise. At  = −43 dB a noisy limit cycle is clearly evident and grows as the feedback increases to  = −39 dB. Indeed, the amplitude of the oscillation of the intensity becomes larger as the feedback is increased. The limit cycle has a frequency of 42 GHz, and it is very close to the relaxation frequency of the solitary laser. As the feedback

Figure 4.14 Experimental phase portraits of various attractors for increasing feedback strength. The injection current is I = 183 × Ith , and the external cavity round-trip time 1.0 n sec. Source: After [40] ©1992 IEEE.

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strength is further increased to  = −34 dB and to  = −31 dB, the limit cycle gets distorted and the portraits become more and more complicated. Their structure clearly cannot be explained as being due to random noise and the existence of some more complex dynamics must be assumed. However, only frequencies in a small passband around the relaxation frequencies are retained by the experimental procedure. Therefore, since the majority of the frequencies are rejected, the resulting information can present a distorted and limited picture. The above picture can be confirmed from an examination of the intensity spectra as a function of feedback strength, a sample of which is shown in Figure 4.15(a). The two basic peaks R and C correspond to the relaxation frequency and to the external cavity frequency respectively. This example demonstrates a frequency locked state in which r / ext = 5. Additionally noisy bumps appear halfway between the strong peaks and could be explained as a period doubling of this locked state as was seen previously in the numerical bifurcation sequence. Strong evidence of the quasiperiodic route into chaos is presented in Figure 4.15(b) in which the ratio R / ext is plotted as a function of the feedback level as determined from the RF spectra. In region I there is only one frequency, the relaxation frequency and it corresponds to the limit cycle. Moving into region II the secondary Hopf bifurcation takes place and the C peak

Figure 4.15 (a) RF spectra of the laser at the quasiperiodic regime. The peak designated R corresponds to the relaxation frequency of the laser and peak C corresponds to the external cavity frequency. (b) Frequency ratio R / ext in the quasiperiodic regime as a function of feedback level. Injection current is I = 184 × Ith and the external cavity round trip time is 1.07 n sec. Source: After [40] ©1992 IEEE.

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becomes stronger and narrower. In Regime III the ratio of the frequencies becomes constant at a value of 5 indicating that frequency locking has taken place. Finally, in Regime IV the peaks broaden considerably indicating that the laser is exhibiting chaotic behavior. Additional experiments lend support to this picture. In this case the laser used was a GaAs/AlGaAs double-heterostructure junction laser diode (Hitachi HLP1400 with built-in lateral index guiding via the channeled substrate planar structure). The optical spectrum of the laser diode remained in the single longitudinal mode of the solitary laser throughout the experiments. A series of spectra were taken for two conditions of the bias current and the external cavity length. For the first series the current was fixed at I/Ith = 159 with Lext = 155 cm and the series of noise spectra in Figure 4.16, at increasing feedback strength, clearly show a quasiperiodic bifurcation sequence to chaos. The first Hopf bifurcation, indicated by R , appears at the relaxation frequency, followed by a quasiperiodic bifurcation at the external frequency. The lines successively broaden as the feedback is increased and the noise floor grows as the feedback is increased indicating complex and perhaps chaotic dynamics. Figure 4.17 shows a sequence of noise and optical spectra for I/Ith = 138 with Lext = 9 cm. In this sequence after the primary Hopf bifurcation at the relaxation frequency in Figure 4.17(b) a bifurcation at the external cavity frequency takes place. Here clearly the two frequencies are locked at a 1:3 ratio in Figure 4.17(c). Further bifurcations take place at a period doubling of the locked state, and peaks at c /2, c /4 appear sequentially with increasing feedback in Figure 4.17(d) and 4.17(e). Similar information can be gleaned from the optical spectra which show asymmetric sidebands relative to the lasing frequency at the same frequencies as the noise spectra. The coherence collapse state is dramatically revealed at the highest feedback rate in Figure 4.17(g) at which the linewidth of the laser has broadened significantly to span several tens of gigahertz. Very similar experiments were carried out earlier [43] to investigate and analyze the phenomenon of coherence collapse. The experimental observations were limited to the

Figure 4.16 RF spectra obtained at I/Ith = 159 and with an external cavity of Lext = 155 cm. Relative feedback strength is (a)  = 0, (b)  = 1 (arb. units), (c)  = 25 (arb. units), (d)  = 32 (arb. units), (e)  = 41 (arb. units). Source: After [41] ©1993 IEEE.

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Figure 4.17 RF (column 1) and optical (column 2) spectra obtained at I/Ith = 138 and with an external cavity of Lext = 9 cm. Relative feedback strength is (a)  = 0, (b)  = 1 (arb. units), (c)  = 20 (arb. units), (d)  = 217 (arb. units), (e)  = 22 (arb. units), (f)  = 221, (g)  = 32 (arb. units). Source: After [41] ©1993 IEEE.

changes apparent in the optical spectra as a function of feedback. The spectra clearly show a quasiperiodic route into chaos involving first the relaxation frequency followed by a torus bifurcation at the external cavity frequency, very similar to those seen in Figure 4.17, column 2. At higher feedback rates the laser enters the coherence collapse state identified by a few low features residing on a broad noise level. Based on the experimental spectra, and

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coupled with extensive numerical calculations of optical spectra from the Lang–Kobayashi equations, it was suggested that the phenomenon of coherence collapse represents a region of deterministic chaos. This is confirmed by the quasiperiodic route into chaos, a route well known in other deterministic systems [44]–[46]. An interesting experimental investigation of the dynamics leading to the coherence collapse was undertaken still earlier by measuring the visibility of the laser. This effectively tells us the coherence of the radiation and it is precisely such measurements that foretold the broadening of the linewidth that eventually led to the collapse of the coherence of the laser [47]–[48]. The visibility is simply measured by employing a Michelson interferometer. The experimental set-up is shown in Figure 4.18. It consists of an index-guided CSP-type laser diode that is temperature-stabilized and is operated in a single solitary laser longitudinal mode. The feedback rate is controlled by either using a polarization filter or inserting neutral density filters in the external cavity. The amount of feedback was calibrated using the method proposed in [27]. To avoid reflection back into the cavity from the Michelson interferometer, a diaphragm and a corner cube are used in each of the two arms. The fringe visibility was computed from the maxima and minima irradiance of the fringe pattern as: V=

Imax − Imin Imax + Imin

(4.16)

It was measured as a function of the length mismatch between the two arms of the interferometer. Two specific measurements of the visibility are shown in Figure 4.19 for a laser operating with output power of 6 mW and with an external cavity of 8 cm. The first,

Figure 4.18 Experimental set-up to measure the visibility of a semiconductor laser in an external cavity. The external cavity of length Lext consists of two lenses L1 , and L2 and an external mirror M. The Mickelson interferometer is composed of the beam splitter BS2 and two corner cubes CC1 and CC2 . Source: After [47] ©1985 IEEE.

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Figure 4.19 Measured visibility curves versus arm length difference for a laser operating at 6 mW with an external cavity of 8 cm, at (a) feedback  = 56 × 109 sec−1 , (b)  = 24 × 109 sec−1 . Source: After [47] ©1985 IEEE.

Figure 4.19(a), corresponds to a feedback rate of k = 56 × 109 sec−1 and it is well within the coherence collapse regime. The second is for k = 24 × 109 sec−1 and it corresponds to the undamping of the relaxation oscillations. There are several features that can be seen in this figure. Both visibilities contain a broad central region on which several small spikes can be discerned. From the spacing between these spikes, 1.04 mm, which corresponds to the optical path of the solitary laser cavity, it is determined that these spikes are due to several non-lasing or low-power lasing longitudinal modes. The broad peak in Figure 4.19(b) located at 34 mm corresponds to a frequency of 4.4 GHz and it is due to the relaxation frequency of the laser. This can be confirmed either by measuring the location of the peak as a function of pumping current, or by measuring the relaxation frequency of the solitary laser. In the first case the square root dependence on the pumping reveals that the frequency is the relaxation frequency. This is in agreement with the previous discussion on the route to coherence collapse, where it was shown that the laser undergoes a bifurcation from steady state to a limit cycle with frequency close to the relaxation frequency of the solitary laser. It is also clear that in Figure 4.19(a) the broad central maximum is considerably narrower than that of Figure 4.19(b), indicating that the coherence is lower. Theoretical analysis of these results can be performed using the Lang–Kobayashi equations. In general, the equations are solved for the steady state solution and then are linearized about this solution as described in Chapter 2. For weak fluctuations only the phase difference t − t −  is important and enters in the correlation function that determines the visibility. The mean square phase fluctuations can be solved self-consistently after a number of judicious approximations and the correlation function is determined unambiguously. The resulting correlation function is in good qualitative agreement with the experimentally measured visibility, especially in the central broad region. For small feedback levels, the additional peak at the relaxation

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frequency is also reproduced, however, it tends to remain for higher values of feedback albeit with progressively decreasing height. A rather stronger conclusion regarding the origin of the coherence collapse state is that since spontaneous emission noise was left out of the calculations, the theory is completely deterministic. Therefore, it can be concluded that the coherence collapsed state is predominately deterministic in origin. This in itself is important because long time series of these fluctuations have not been obtained experimentally due to instrumentation limitations, and then analyzed to ascertain whether it is stochastic or deterministic by the use of several algorithms designed for this purpose [46]. Recent development in oscilloscope technology will make such measurements possible in the future. As was pointed out earlier, the diode laser with an external cavity system can possess a number of attractors which coexist at the same feedback strength. Such attractors can be steady states, limit cycles, torus and chaotic attractors. In particular, two limit cycles can coexist, as was found in [49]. A rather curious optical spectrum is shown in Figure 4.20 in which the sidebands at the relaxation frequency appear to be split. The two split sidebands B, C are located are 436, and 4.79 GHz and are separated by 0.43 GHz that corresponds to an external cavity round trip frequency. For this particular experiment the external round trip time was 2 n sec. The location of both peaks shows the characteristic square root dependence on the pumping current and therefore can be identified as being related to the relaxation frequency of A. The spectrum therefore appears to exhibit a competition between two relaxation frequencies. Indeed, the spectra were taken with a planar Fabry–Perot of 12 GHz FSR and show the time average behavior of the two attractors. In order to understand what is going on, time-resolved spectra need to be acquired. To this end, two Fabry– Perot interferometers of 12 GHz FSR were used as to filter each set at a center frequency corresponding to the peaks A, B or C. The upper two traces in Figure 4.21 are obtained when the two interferometers are set to B and C. Clearly the two traces are anticorrelated showing that the laser randomly jumps between the two peaks. The lower two traces are obtained when the two interferometers are set at the center line A and the sideband B. It indicates that the strength of the signal at the central peak also changes as the laser jumps randomly between the two spectra lines. However, the position of the central peak in frequency remains constant, indicating both sidebands belong to the same external cavity mode. In summary, it

Figure 4.20 Optical spectrum of the laser in an external cavity of 2 n sec and biased at I = 201Jth . The frequency of the solitary laser is at 0 GHz. Source: After [40] ©1992 IEEE.

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Figure 4.21 Time dependence of the peak C and B, upper trace, and time dependence of the peaks A and B, lower trace. Source: After [40] ©1992 IEEE.

appears that the spectrum is a time-averaged spectrum of two limit cycles of two different relaxation frequencies, both limit cycles emerging from the same external cavity mode. In terms of dynamical systems, a possible explanation of such behavior can be found in the LK model. A stability analysis of this model yields all of the possible steady states consisting of modes and antimodes emerging either from the solitary laser mode or through saddle-node bifurcations [34, 50, 35, 51]. Focusing on the primary mode that is stable for weak external feedback, a stability analysis can be undertaken to determine at which point the mode loses stability and the dynamical mechanism that induces such instabilities. Analytical calculations have determined that the mode loses stability through a Hopf bifurcation into a limit cycle with a frequency close to the relaxation frequency of the solitary laser. Furthermore, after the Hopf bifurcation, the steady state is naturally unstable but nevertheless accessible. In particular, since the stability equations turn out to have multiple solutions [52], a secondary but unstable Hopf bifurcation can take place that emerges from the unstable steady state. The frequency of this unstable Hopf bifurcation is equal to the r − ext as calculated from the imaginary part of the stability eigenvalue. Furthermore, when the two Hopf bifurcations, one stable and the other unstable, are close together in feedback space, the two bifurcations can interact strongly and as a result of such interaction the second Hopf bifurcation can restablilize. Thus, the laser at specific feedback level can have two possible limit cycles available with two different frequencies, both emerging from the same external cavity mode [53].

4.6 MODERATE FEEDBACK EFFECTS – LOW FREQUENCY FLUCTUATIONS Strong feedback is considered to occur when the external feedback mirror reflectivity is comparable or greater than the laser facet reflectivity. Thus very high feedback levels can be attained by applying high-quality anti-reflection coatings to one of the laser facets. Reflectivities of 2−3% can be routinely obtained in this manner.

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As the feedback rate is increased in a laser with an external cavity of the order of a few tens of centimeter, and the laser is typically pumped close to solitary laser threshold, a phenomenon dubbed Low Frequency Fluctuations (LFF) dominates the intensity of the laser. Spontaneous self-pulsations appear, in which the intensity of the laser seems to drop off abruptly and intermittently to a very low value before it again climbs up to what appears to be a steady state intensity. This phenomenon was seen as early as 1977 by Risch and Voumard [54] in a multimode diode laser in a frequency selective external cavity. The pulsations or rather the dropoff events in the intensity appear to be at very slow time scales compared to intrinsic time scales of the relaxation frequency or external cavity round-trip times. This frequency is apparent at the low frequency region of the noise spectrum at about 3–30 MHz and the frequency increases as the pumping of the laser is increased in a linear fashion rather than the square root dependence exhibited by the relaxation frequency of the laser. An indication that some new dynamical phenomenon is taking place was recognized very early by analyzing the light-current curve of the laser under external feedback [55, 25]. Figure 4.3, for example, shows such a curve in which the output power of the laser is plotted as a function of input injection current. The solitary laser curve exhibits a threshold at about Ith = 51 mA, with an appreciable round off due to spontaneous emission. Further, the curve is characteristically linear as a function of current. In contrast, the light-current curve with the laser under external feedback shows the characteristic decrease of the threshold related to the strength of the feedback and in addition a change in the slope at a current close to the solitary laser threshold known as the kink. It is in this regime of feedback and pumping that the laser exhibits the rich dynamical behavior called LFF. In many cases multiple kinks as a function of increasing pumping can become apparent, behavior that envisions even richer dynamics [55]. A characteristic intensity time series illustrating low frequency fluctuations [56]–[58] and the noise spectrum of the laser is shown in Figure 4.22. The dropout events appear to occur randomly about a steady state. The noise spectrum shows a broad feature around the external cavity frequency and a similar feature at low frequencies of the order of MHz is attributed

Figure 4.22 (a) Noise spectra and (b) time series of the optical power of the laser. The laser is pumped 1.5% below solitary laser threshold. Source: After [60] ©1991 IEEE.

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to the low frequency fluctuations. For this experiment the external cavity frequency was 352 MHz and it corresponds to an external cavity of 1.2 m. It was found that the frequency of the dropout events was increasing as the feedback decreased, however, it also increases linearly as the pumping current of the laser increases. Several possible interpretations have been suggested to account for the laser behavior. One of the earliest interpretations suggests that the laser is undergoing a time inverted type II intermittency [59], a purely deterministic effect. It is also possible to explain such behavior in terms of bistability between two steady states, one at high intensity, the other at very low intensity. Simplified models based on a diode laser subject to optical injection were constructed. Based on the relative amount of pumping and including noise, it was shown that the laser would show random dropout events in the intensity followed by a stepwise climb back to high intensity levels [61]–[62]. It was also noticed, that omitting noise in the full equations, the LFF persisted confirming the notion that they are taking place on a chaotic attractor. Another rather successful model attributed the drop-off events to a peculiar instability of the steady state [63]. In particular, it was argued that even though the maximum gain mode and the external cavity modes close to the maximum gain mode (MGM) or equivalently maximum intensity mode, appeared to be stable in the usual linear stability approximation, higher-order contributions from quadratic terms can modify the stability arguments in the presence of spontaneous emission noise. The authors were successful in deriving an equation that described the fluctuations of the laser about the steady state in terms of a particle in a quadratic-cubic potential well. Such potentials display a minimum at the MGM and a barrier assures its stability in the absence of noise. However, in the presence of noise and under favorable conditions of feedback and pumping current, spontaneous emission noise can produce finite probabilities for the system to overcome the barrier and drop down the hill to the zero intensity level. Naturally, after the drop-off event, due to high inversion the laser must return to the high intensity equilibrium. The theory nicely determines the transition probability and the average transition time between dropout events as a function of feedback and pumping current [63]. An alternate explanation was suggested by Sano [64] and elaborated by van Tartwijk [65]. It was shown by extensive numerical calculations that LFF can be attributed to a chaotic itinerancy in which the trajectory winds its way up towards the maximum gain mode by visiting the chaotic ruins of several of the external cavity modes. A representative trajectory is shown in Figure 4.23 in which a portion of an LFF trajectory is shown in the inversion nt, and phase difference t = t − t −  plane. Figure 4.23 shows only a snapshot of the trajectory close to the maximum gain mode that is situated at the most negative frequency and inversion region. Several windings of the trajectory in the neighborhood of the chaotic ruins of the external modes can be seen. The arrows indicate the sense of the trajectory and its drift towards the MGM. In general, the motion of the trajectory is unidirectional, however, several mode slips or reverses in the drift towards the MGM can occur, especially as it reaches high intensities. The amount of time spent around each of the chaotic ruins of the external modes appears to be random. Eventually this wandering is interrupted when the trajectory has a collision with one of the antimodes located in the neighborhood of the MGM and following its unstable manifold moves towards zero intensity as the arrows indicate. This collision is the characteristic of a dropout event, and appears to occur at random times. Since the frequency of the laser is also proportional to t as the trajectory moves to higher intensities, the frequency of the laser shifts towards the red. An abrupt blue shift occurs during the drop-off

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Figure 4.23 Numerically calculated trajectory in the inversion v.s. phase-difference space. The pumping current is set at 2% above threshold,
= 5 and C = 510. Source: After [66].

event, resetting the frequency of the laser to its solitary value before this sequence is randomly repeated. A typical time series slice of the intensity is shown in Figure 4.24. The intensity is composed of short ∼ psec pulses. Experimental verification is shown in Figure 4.25 for a laser biased 3% above the solitary laser threshold. The well-know LFF behavior is clearly seen in Figure 4.25(a) with fast intensity drop-off events occurring randomly at an average time of about 50 n sec. This time series was obtained by low-pass

Figure 4.24 Time slice of the photon number as a function of time in units of external cavity round trip time. The pumping current is set at 2% above threshold,
= 5 and C = 510 and external cavity round trip time  = 10 n sec. Source: After [65] ©1995 IEEE.

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Figure 4.25 (a) Times series taken obtained with a low-pass filter, (b) streak camera measurement. Source: After [66].

filtering the signal with a cut-off frequency of 1 GHz and therefore any fast dynamics have been obscured. Figure 4.25(b) shows a 6.6 n sec time slice of the intensity taken with a single shot streak camera for the same operating conditions as in Figure 4.25(a). These measurements provide the appropriate time resolution to detect the fast dynamics predicted in [65]. They show that the main features were hidden when conventional measurement techniques were used. The time slice shows definite pulsations with 100% modulation and a pulse to pulse time of 0.2–1 ns. The pulsations are irregular in intensity and in temporal position. Very similar results were obtained throughout the LFF regime and even for larger currents at which the LFF dropout events are not observable. Even though the theory of [63] is couched in terms of the laser being in a stable MGM rather than exhibiting very fast pulsations, as was shown in [66], nevertheless the theory could be thought of as applying to the average behavior of the laser between dropout events. In particular, as was shown numerically in [64], between drop-off events, the laser spends most of the time visiting the modes close to the MGM and remains in this neighborhood for a long time. Therefore the theory could still apply in the average sense, implying some sort of average over the short time behavior. Several careful experiments were carried out to attempt to validate the theory by investigating the predicted statistical behavior of the drop-off events and their dependence on pumping and feedback [67]. A large number of experimentally recorded long time series in which the number of dropout events number to several thousands have been recorded and analyzed to obtain the probability-density functions (PDF).

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This was done for several injection current strengths, three of which are shown in Figure 4.26. Along with the experimental results shown in diamonds on the dashed curves, the numerically computed PDF are shown as stars on a solid curve. All the PDF show a characteristic exponential decay for long times and a dead time for short times. The dead time is particularly important as it signifies that the laser requires some time to drift towards the MGM before it can collide with its closest antimode that produces the dropout event. It is also clear that the average drop-off period decreases as the pumping level is increased. The numerical PDF have the same features as the experimental results and show the similar dependence on the pumping current. The characteristic exponent for the drop-off events can be obtained from a plot of the average drop-off period v.s. injection current shown in Figure 4.27. From a log-log plot the exponent is determined from the linear fit as < T >= −1 with  = I/Ith − 1. Both data sets in the LFF regime, numerical and experimental show the

Figure 4.26 Power density functions of the time between dropout events for three injection currents, (a) I = 098Jth , (b) I = 0104Jth , and (c) I = 108Jth . Diamonds on the dashed curve are the experimental results, and stars on a solid curve denote the numerically computed data. Source: After [67].

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Figure 4.27 The average time between dropout events v.s. normalized injection current plotted on a log-log scale. Diamonds and dashed curve are experimental data, stars and solid curve are numerical data and the dashed-dotted curve is computed from the Henry and Kazarinov model [63]. Source: After [67].

same scaling law. The fit based on the model of [63] shows substantial deviations at the low injection current regime. The same scaling behavior of the average time between dropout events, the exponential decay and the dead time have also been observed in experiments using multimode, Fabry–Perot semiconductor lasers [68]–[69]. The evolution into the LFF regime can be investigated experimentally by using an external cavity of intermediate length and operating the laser close to the solitary laser threshold. [70] uses a laser diode operating at about 780 nm (SDL 5301) and pumped near threshold at Ith = 252 mA. In Figure 4.28 a series of optical spectra are shown as a function of measured threshold reduction. For zero feedback (trace (a)), the laser operates on a single line at the solitary laser frequency that is located at 0 GHz. Since the laser is operating near threshold,

Figure 4.28 Experimental bifurcation sequence if a laser pumped close to threshold for intermediate external cavity lengths. Traces show the sequence of optical spectra as a function of threshold reduction due to the increase of the external feedback strength. The laser is operating stably in traces (b), (d), (f), (h), (j), and (l). In traces (c), (e), (g), (i) and (k) the laser is unstable. Source: After [70].

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the intensity is very weak and the line can be barely distinguished in the figure. As the feedback is increased to 36% threshold reduction (trace (b)), the laser shifts to a new external cavity mode located at about 1 GHz to the longer wavelength side. For higher feedback strength (trace (c)), a sideband appears close to the next external cavity frequency indicating the laser is now unstable. However, the signal strength is too weak to discern the dynamical state of the laser from the time series. A similar sequence occurs as the laser stabilizes in mode 3, shown in trace (d), at a threshold reduction of 4.4%. Further increasing the feedback, the mode becomes unstable, showing undamped relaxation oscillation sidebands close to the external cavity frequencies. Increasing the feedback further, trace (e) induces broadening of the spectrum, the center line decreases and broadens, and the sidebands become stronger. This dynamical state is known as the coherence collapse state and is typical of the dropout events observed. This type of bifurcation sequence continues with the laser stabilizing on successive external cavity modes, which in turn become unstable, with the laser entering another LFF regime only to restabilize on the next external cavity mode as the feedback strength is slowly increased. The typical LFF regime that the laser enters for a feedback strength corresponding to a threshold reduction of 6.5%, after the fourth external cavity mode becomes unstable, is shown in Figure 4.29. In the time series Figure 4.29(a) the drop-off events are clearly

Figure 4.29 Experimentally obtained time series (a) after the fourth external cavity mode destabilizes for a threshold reduction of 65%. The corresponding optical spectrum and noise spectrum are shown in (b) and (c) respectively. Source: After [70].

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visible, occurring randomly with an average period of about 25 n sec. The optical spectra are shown in Figure 4.29(b), and the noise spectra in addition to the sequence of external cavity mode peaks in Figure 4.29(c), show the typical LFF at low frequencies. This bifurcation sequence of stability and instability followed by LFF was also demonstrated numerally using the Lang–Kobayashi equations in [70]. The transitions between stable single mode operation on the MGM and LFF are not in general abrupt, but usually follow a region in which the two coexist. Time series show regimes in which the laser alternates randomly between stable emission and regimes in which LFF are present. In other cases the laser can exhibit long LFF transients before it finally settles on the MGM and stable emission. Such a transition from LFF into the stable MGM is shown in Figure 4.30. The inset shows the optical spectrum of the laser under stable emission obtained with a scanning Fabry–Perot interferometer. For this particular experiment a Hitachi (HLP 1400) diode laser was used. This laser has been used extensively in nonlinear studies of external optical feedback. The regions of stable emission, regions of coexistence between stable emission and LFF, and regions of coherence collapse depend strongly on the value of the amplitude-phase coupling as expressed by the linewidth enhancement factor
. A series of experiments were carried out to delineate such regimes as a function
. This factor was controlled by tuning the emission wavelength of the laser away from the maximum of the solitary laser gain by inserting an etalon in the external cavity [72]. An example of such a diagram is shown in Figure 4.31 in which the various regimes are outlined in the injection current versus feedback plane. The LFF regime is the region bounded by the solid line. The coexistence between LFF and stable emission is embedded in the LFF regime and corresponds to the dark region. Finally, the region bounded by the dashed line and the solid line corresponds to the transition regime between LFF and fully developed coherence collapse. The crosses on the solid line denote the feedback-reduced laser threshold. On crossing this line from below the laser enters the stable emission regime V, which can be reached on a narrow current

Figure 4.30 Time series showing the transition from LFF to stable emission on the MGM. Inset shows the optical spectrum obtained with a scanning Fabry–Perot interferometer. Source: After [71].

Moderate Feedback Effects – Low Frequency Fluctuations

113

Figure 4.31 Stability diagram in the injection current versus feedback for a laser subject to optical feedback. The diagram shows the regions of stable emission, coexistence of LFF and stable emission, LFF, and coherence collapse. Source: After [71].

interval close to the laser threshold. Similar diagrams are obtained as the value of
changes. In general, with reduced values of
, the coexistence regime is enlarged, and the stability of the MGM is increased. The earlier literature on LFF contains some provocative experimental results and observations. Under certain circumstances it was found that the dropout events can become very regular, producing a narrow resonance in the noise spectrum, which manifests as a self-modulation of the intensity [25]. Time series of such events appears as a low frequency limit oscillation under low-pass detection and it is not clear at this time what is the underlying physical mechanism. Such a time series of regular LFF is shown in Figure 4.32, along with the RF spectrum. The data were obtained with a 20 mW Toshiba TOLD9140 index guided InGaAlP diode laser emitting at 693 nm. Indeed, the time series appears to be that of a noisy, distorted limit cycle. From the RF spectrum it is clear that the LFF frequency located at the low frequency part of the spectrum and indicated by an arrow exhibits a narrow resonance with sharp multiple peaks denoting higher harmonics. However, the spectrum still contains a large number of external cavity frequencies indicating that what appears to be a noisy limit cycle is indeed LFF events containing strong pulsations under a very regular envelope. Measurements with high frequency resolution have indeed resolved the pulsations and revealed that the limit cycle is composed of short pulses much as was seen in Figure 4.25. In general, regular LFF are found only in a very limited region of pumping current, feedback strength and cavity length, and the laser must be carefully aligned before they attain long time stability. It appears that the inherent noise in diode lasers tends to destabilize and destroy the mechanism that produces this regularity in the LFF. Recent results suggest that such regular LFF are very common and stable in the short cavity regimes, and are easily captured over a large region of injection current and feedback strength [74].

114

Experimental Observations

Figure 4.32 (a) Time series of the regular LFF and (b) the RF spectrum. Source: After [73].

The wealth of bifurcations and the abundance of complex dynamics that have been observed experimentally have been conveniently summarized in the Introduction to this chapter and are depicted in Figure 4.1. The characterized behavior at all feedback regimes was experimentally described for a number of different semiconductor laser types and under different experimental situations. Substantially the same behavior, the basic bifurcations and their dynamics were observed under multimode operation as well as when operated with frequency selected elements in the external cavity. Similar dynamical behavior was observed in DFB lasers in an external cavity operating on a single longitudinal mode as well as in Fabry–Perot lasers under multimode operation. In the next few sections we will examine in detail not only specific external cavity arrangements, but will also focus on experimental situations between multimode or single-mode behavior.

4.7 SHORT CAVITY REGIME The case of optical feedback from short external cavities has several unusual features and deserves special attention. When a short external cavity is used, the product ext remains small even for weak feedback, and so the number of external cavity modes and antimodes

Short Cavity Regime

115

also is relatively small. A working definition of a ‘short’ cavity then is one for which the number of ECMs is small (∼10 or less) and the external cavity frequency is greater than the relaxation oscillation frequency. The reduced complexity of the system can make it easier to observe certain effects that are obscured in the presence of many external cavity modes, thereby providing additional insights into the dynamics. Even without the guidance of a full theoretical understanding based on nonlinear dynamics, experiments in the early 1980s observed many interesting effects. In [56], a detailed P-I characteristic curve was measured for a GaAlAs/GaAs Plano-Convex Waveguide (PCW) single-mode laser with feedback from an external cavity of length L = 24 cm. Associated RF power spectra were recorded as well, leading to a relationship between the LFF peak frequency and the solitary laser relaxation oscillation frequency. Where external cavities of tens of cm lead to the well-known kink in the P-I curve when pumped near solitary laser threshold current Ith , as in Figure 4.3, this experiment observed a distinct undulation in the curve as shown in Figure 4.33. They observed that in the valleys of the P-I curve the LFF peak power was maximized and the LFF peak frequency minimized; the opposite was found at the maxima of the P-I curve. Later work has rediscovered the short cavity regime and several interesting effects have been observed there. Numerical studies have identified high-frequency oscillations (tens of GHz) in very short cavity (≤ 5 mm) systems [76], arising from self-locking of two external cavity modes. In [77], experimental and numerical studies investigated the effects of small changes (of the order of optical wavelengths) in the external cavity length in a short cavity. Repetitive sequences of stable, periodic, quasiperiodic and chaotic dynamics were observed, recurring with every half-wavelength variation in cavity length. The laser used in these experiments was a single-mode CSP laser (Hitachi HL7801E). Figure 4.34 displays a sequence of experimental optical spectra as a function of small changes in external cavity length with a nominal length of L = 42 cm (the laser was operating at 30% above solitary current threshold, and the external cavity reflectivity Rext = 2%). This figure also highlights an interesting and important feature of short cavity systems: they display qualitatively significant changes with variations in the feedback phase. This stands in contrast to longer

Figure 4.33 Light-current characteristic of a laser without and with feedback from an external cavity of 24 cm. Source: After [56].

116

Experimental Observations

Figure 4.34 Optical spectra as a function small variations in the cavity length. Offsets are from an external cavity of 4.2 cm and external feedback is set to 2%. Source: After [77] ©1998 IEEE.

cavity systems, whose dynamics are relatively insensitive to feedback phase due to the large numbers of ECMs present. More recently, investigations have considered how other phenomena change in the short cavity regime. In [78], short cavities were found to change the characteristics of LFF dynamics and the underlying short pulsations. Among the findings were that the erratic short pulse train of long-cavity LFF is transformed into regular pulses occurring at the external cavity frequency. Similarly, the low-frequency (power dropout) phenomena change from an irregular into a regular global trajectory along several ECMs. These phenomena are illustrated in Figure 4.35. A Hitachi HLP1400 laser was driven at I = 115Ith in a L = 32 cm cavity, and the time-series output measured with a streak camera. Short pulses appear at regular intervals of the external cavity round-trip time ext . These short pulses are modulated by a slower envelope to form ‘regular pulse packages (RPP’s)’ which appear to be the short-cavity equivalent of the regular LFF power dropouts as discussed in the previous

Double-Cavity Systems

117

Figure 4.35 Streak camera measurements of the intensity time series. The external cavity is 32 cm and the injection current is 115Ith . Source: After [78].

section. These features are supported by RF power spectra that confirm the interpretation of the time-domain data.

4.8 DOUBLE-CAVITY SYSTEMS Another subset of experiments worthy of discussion has involved the use of two sources of delayed optical feedback. In such a configuration, the laser’s output beam is typically split with a non-polarizing beamsplitter, and the resulting beams strike different retroreflecting mirrors, thereby creating a double external cavity. The configuration resembles an interferometer, although the cavity lengths are generally not varied. In double-feedback configurations each cavity can be controlled fully independently, in terms of the cavity lengths, feedback rate, polarization or frequency properties, and so forth. One might suspect that if one source of delayed feedback produces complex dynamics, two sources would be even more complex. This notion led to an experimental realization of highdimensional optical chaos in [79] using a double-feedback system. This work emphasized the fundamental nonlinear dynamics of the delayed system, examining the system in terms of the Ikeda scenario. The laser used in these experiments was a Hitachi HLP1400 with an antireflection-coated front facet, which was placed in a T-shaped cavity with two arms, both of which could be varied in length from 30 cm to 1.5 m. However, this experiment set the delay time in one arm to be equal to twice the delay in the other. This allowed the second cavity to be treated as a means to increase the nonlinearity of the feedback. Two timeseries of the intensity are shown in Figure 4.36, and they show switching between attractor ruins, i.e. chaotic itinerancy, which is also a feature of single-cavity systems. Figure 4.36(a) illustrates one such transition from dynamics dominated by the shorter cavity fundamental frequency to the third harmonic; Figure 4.36(b) shows a more complex trajectory on the global attractor which also includes evidence of the fifth harmonic. Time-series analysis of long data sets (32 K points) showed correlation dimensions of up to approximately seven, confirming the high-dimensionality of the chaos in this system. Interestingly, the addition of a second delayed feedback does not always enhance the complexity of the system’s dynamics, and in some cases will stabilize previously chaotic or complex behavior. The counterintuitive idea of using optical feedback to stabilize coherence collapse was considered in [80], and this concept was analyzed and extended to

118

Experimental Observations

Figure 4.36 (a) Chaotic times series from a double cavity feedback system. (a) transition from the fundamental frequency to the 3rd harmonic, and (b) dynamics on a global attractor with transitions from the 3rd harmonic to 5th harmonic followed by a transition to the fundamental. Source: After [79].

LFF dynamics. Particular attention was paid to the effects of the second feedback on the ECMs [81], and stabilization of LFF with this technique was realized experimentally [82]. This will be further elaborated in Section 4.10, Control. Further extensions of double feedback work examined not only the stabilization properties but the interesting dynamics between regions of stabilization as well [83]. An example of these results is illustrated in Figure 4.37, which shows a sequence of optical spectra measured by a scanning Fabry–Perot cavity with a FSR of 1100 GHz and a finesse >13 000. The experimental system consisted of a SDL-5401 laser driven at I = 099Ith subject to double cavity feedback with cavity lengths of L1 = 19 cm and L2 = 16 cm. The feedback from the first cavity was fixed to create a threshold reduction I1 = 240%, and the second feedback of strength I2 was varied as a bifurcation parameter. As is evident from the spectra, the system began in a mildly unstable state (with no second feedback), then progressed through stability, periodic oscillations of several GHz, quasiperiodic oscillations, LFF-like complex dynamics, and then restabilization as the feedback increased to I2 = 574% This sequence, in conjunction with numerical and analytical findings, helped identify these dynamics as originating from a mixed-mode state. Furthermore, this sequence tended to repeat at even higher feedback rates, and also showed evidence of coexisting states of like type (two high-frequency periodic states) or unlike types (one periodic, one quasiperiodic). The use of double and multiple feedback systems (in the strong feedback regime) is one way of obtaining single frequency operation as discussed in Chapter 8.

Multimode Effects

119

Figure 4.37 Optical spectra as a function of feedback from the second cavity, denoted by the threshold reduction I2 . Threshold reduction from the first cavity was fixed at I1 = 24%. The laser is operating stably in traces (b) and (g), exhibits periodic oscillations in (c) and (d), quasiperiodic in (e), and exhibits LFF in (f). Source: After [83].

4.9 MULTIMODE EFFECTS The Lang–Kobayashi equations in their basic form describe single-longitudinal mode operation. However, it is observed that most edge-emitting diode lasers tend to run in multiple optical modes in the presence of feedback from a flat mirror, even if it would normally operate in a single mode in the absence of feedback at the same pump current level. The addition of a multiple longitudinal mode operation increases the complexity of the system, and creates many questions regarding the mode dynamics and interactions, and their relationship to dynamical states and instabilities. Multimode experimental investigations have generally followed two main approaches: to measure statistical properties of the system, or to examine the mode interactions based on time-resolved data. Statistical measurements have been made on a variety of properties displayed by multimode LFF systems, from the statistics of the fast pulsing intensities, to the times (‘interspike intervals’) between power dropout events. Interspike intervals are quite accessible experimentally and can be measured using an oscilloscope with a bandwidth of only a few hundred MHz. In [86] statistical measurements were made as a function of pumping current in a multimode system using a SDL-5401 laser in an external cavity of length L = 71 cm. Mean dropout times were measured and shown to decrease smoothly and rapidly with increasing pumping. The statistical distribution of dropout times was also measured, and showed several general features: for very short dropout intervals, the probability was near zero, then rose quickly to a most probable interval, then fell off exponentially (examples of such distributions are shown in Section 4.11 on Feedback and Modulation, and also in Section 4.6, Moderate Feedback Effects Low Frequency Fluctuations). Furthermore, the probability distributions acquired additional structure at stronger pumping. While the mean times agreed reasonably with predictions based on the Henry and Kazarinov model [63], the zero probability at short times was better interpreted in terms of bistability as discussed by Mork [61].

120

Experimental Observations

Other insights have been gained by comparing dropout statistics in a multimode system to a comparable single-mode system, and analyzing the results in terms of the LK model. Among other findings in [84], it was observed that the dropouts in single-mode LFF tended to occur roughly three times more frequently than if an otherwise identical system were operating in a multimode condition. Fast pulsing dynamics have also been measured statistically to compare single-mode and multimode behavior in the LFF and coherence collapse regimes. These experiments typically demand high-bandwidth detection apparatus, such as a digital sampling oscilloscope or streak camera. In one such experiment [85], a Hitachi HLP-1400 laser with a solitary threshold of 53 mA was used in a cavity of length L = 22 cm which provided feedback sufficient to produce a threshold reduction of 10%. A digital sampling scope was used to measure the probability distribution of the laser intensity. These statistical samples of the fast dynamics showed maxima near the average laser intensity, with exponential tails extending to three times the average. These data are shown in Figure 4.38, plotting the intensity probability versus the normalized injection current; the different symbols correspond to four different current levels ranging from 70 to 99 mA, but all distributions appear quite similar. Time-resolved probability measurements following dropouts were also made, and showed an initially deterministic recovery, followed by a strongly modulated transient stage which then relaxed to a lower level of fluctuations. These data were not well described by the single-mode LK model. Subsequent experiments were performed in similar systems. In [86], two different experimental systems were investigated, and were used to compare single-mode and multimode behavior in the LFF regime. A randomly-triggered single-shot streak camera sampled the intensity of another Hitachi HLP-1400 laser in a 45 cm cavity, and a communications signal analyzer with a 20 GHz sampling head measured the fast dynamics of a Sharp LT015MD laser in a 22 cm cavity. Both systems normally display multimode characteristics in the LFF regime, but by inserting an intracavity etalon to restrict the feedback to one supported laser mode, single-mode operation was also achieved and measured. In all these experiments, the intensity distributions showed a peak at low intensities with exponential tails extending to several times the average, indicating pulsating behavior. In these experiments, single-mode and multimode statistics were virtually indistinguishable,

Figure 4.38 Probability distribution of the laser intensity for pumping currents of 70, 80, 90, and 99 mA. The intensity axis is normalized to the average intensity. Source: After [85].

Multimode Effects

121

and also agreed well with numerical results from the LK model. In the multimode extension of the LK model, the longitudinal mode fields were coupled by cross- and self-saturation terms in the carrier dynamics. However, it was also observed that gain saturation effects and the coexistence of LFF with stable emission could easily shift these distributions to show peaks near the average intensity. Other experiments have addressed facets of multimode dynamics beyond purely statistical measurements. As stated previously, standard diode lasers with optical feedback typically run with multiple longitudinal modes even though the solitary laser would be single-mode under the same circumstances. Experiments therefore will often restrict the feedback to a single mode, to improve agreement with the standard LK model. Therefore, the ability of a grating or etalon to fully restrict a laser to a single longitudinal mode, and thus the extent to which the basic LK formulation is adequate, is a topic of interest. In these studies, the longitudinal mode to which the feedback is tuned is called the ‘selected mode’ whereas the others are ‘free modes’. It has been found numerically [82] and experimentally [88] that the laser is generally restricted to oscillate in the selected mode, however, in the LFF regime there is an intensity burst in the free modes immediately following a power dropout event until the selected mode becomes the dominant mode again. These dynamics naturally cannot be captured by a single-mode LK model. Furthermore, in [88] it was also observed that the post-dropout recovery of the total power was faster than the recovery of the powers of individual modes, implying that antiphase dynamics were at work. The question of inphase and antiphase dynamics has become important in developing a full understanding of multimode dynamics, as will be discussed further below. If no attempt is made to restrict the modes (i.e. if a flat mirror is used to form the external cavity, and if the laser is not a DFB laser), the complex mode dynamics and interactions become an interesting subject of inquiry. These mode–mode interactions and their relationship to LFF dynamics have been investigated through time-resolved studies of the individual and total intensities of the modes. Due to the very fast dynamics involved, to conduct such experiments requires very high-bandwidth detection instrumentation such as a streak camera or a transient digitizing scope of several GHz bandwidth. One such experiment is described in [89], in which a Hitachi HL6314MG laser emitting at about 630 nm with a mode spacing of 135 GHz and threshold current Ith = 24 mA is used in a cavity with a partially-reflective mirror placed L = 22 cm distant, which yields a threshold reduction of 8.3%. By placing a grating monochromator in front of a 40 GHz streak camera, the beam that struck the detector was dispersed in wavelength, allowing simultaneous timeresolved detection of the modes. One important result of this data is that all modal intensities dropped simultaneously below detectable levels during a LFF dropout, after which several modes grew and began to exchange energy. In addition, it was noted that the individual modes tended to pulse out of phase with other modes most of the time. However, if several modes synchronized and began to pulse in phase, it was observed that this phenomenon was a precursor to the total intensity power dropout. These experimental results have been taken into account in formulating a multimode extension to the LK model which accounts for the carrier grating created by the standing wave optical field and admits the possibility of antiphase behavior between laser modes [90] (in contrast to the phenomenological model studied in [86] in which it was assumed that carrier diffusions wash out such gratings). Interestingly, further experimental work along these lines indicates that the full picture is still more complex and continues to warrant further investigation. The details of the

122

Experimental Observations

antiphase dynamics were studied in [91], and it was found that both in-phase and antiphase components play a role. These results were obtained using an Anritsu SD3F513T laser operated at 50% above solitary threshold in an external cavity with a round-trip time of 3.8 n sec . Under these conditions the laser operated with 18 distinct longitudinal modes. The dynamics of the total intensity were observed, as well as those of individual modes and subgroups of 3, 10, and 18 modes, which were isolated using a bandwidth-variable opticalwavelength filter. Power spectral densities (PSD) were calculated, considering the in-phase and incoherent sums of modes, which were compared with the total intensities for various mode sets. Two key deductions were made from these data. At high frequencies (near the relaxation oscillation frequency, ∼ 24 GHz), the PSD peak heights of the total intensity were the same as those of the in-phase sum, indicating that modal components at these frequencies are completely in phase. However, at lower frequencies (particularly 0. Again, it is important not to confuse the fixed value x0 in physical space with the constant function q0 over the delay interval, which is a point in phase space. The stability of q0 is given by the linearization of Equation (5.1) around q0 (acting on a point q ∈ ): DFq0  ∗ qt = A1 q0  ∗ qt + A2 q0  ∗ qt −


(5.2)

The linearization is the generalization of the Jacobian of an ODE to the setting of DDEs. The matrices A1 and A2 are given as
Fu v 

A1 q0    = and
u x0 x0 ∗  ∗


Fu v 

A2 q0    =
v x0 x0 ∗  ∗

(5.3)

where u v ∈
n were used as the arguments of Fu v . The stability of Equation (5.1) near q0 can be found by considering the eigenvalues of the n × n matrix

q0  ∗   = I − A1 q0  ∗  − A2 q0  ∗ e−


(5.4)

which are given as the roots of the characteristic equation: det q0  ∗   = 0

(5.5)

Due to the transcendental nature of Equation (5.5), there are infinitely many eigenvalues. This makes the study of DDEs much harder than that of ODEs. However, a DDE with a fixed delay has the crucial property that the eigenvalues of Equation (5.4) are discrete. This means that there are no accumulation points of eigenvalues in the complex plane, and that there is no essential spectrum (a continuous part of the spectrum). What is more, for any fixed  > 0, there are always finitely many eigenvalues with real part larger than . For the special case that  = 0, one gets the result that there are always only finitely many unstable eigendirections (associated with eigenvalues with real part greater than zero), for further details see [5]. A steady state is called hyperbolic if there are no eigenvalues that have zero real part. A hyperbolic steady state is either attracting or a saddle point, the latter having infinitely many attracting and finitely many repelling eigendirections. When the parameter  is changed, eigenvalues can cross the imaginary axis of the complex plane, leading to changes in the stability of the equilibrium. Because the eigenvalues are isolated, when a single parameter is changed, there are only two generic or typical possibilities

152

Bifurcation Analysis of Lasers with Delay

(for a DDE without any extra symmetry properties), giving rise to the two local bifurcations of steady states of codimension one: 1 A single real eigenvalue goes through zero. This is the case of a saddle-node bifurcation in which two equilibria are born. 2 A complex conjugate pair of eigenvalues move across the imaginary axis of the complex plane. This is the case of a Hopf bifurcation from which a small periodic orbit bifurcates. In order to check that one is dealing with either of these bifurcations, one has to check the so-called genericity conditions, which is difficult in practice as it involves computing the normal form on a centre manifold. However, the good news is that these conditions are typically satisfied. If they are not satisfied, then one is dealing with a degeneracy, such as a bifurcation of higher codimension. The Hopf bifurcation may be supercritical or subcritical, depending on whether the bifurcating periodic orbit is attracting or not. This is easy enough to detect in practice, so that tedious computations involving third-order terms can generally be avoided.

5.2.3 Local Bifurcations of Periodic Orbits A periodic orbit is a solution  such that T q = q for some period T > 0 and all q ∈ . In projection onto the physical space
n , the periodic orbit  is a closed curve. As for ODEs, the Poincaré map P is defined as the return to a suitable section  transverse to . In what follows a section  ⊂
n is fixed and  denotes the space of points in  with headpoints in . The Poincaré map P is defined as P    →  

q → tq q

(5.6)

where tq > 0 is the return time to . Note that it is always possible to find a section  (locally) transverse to a periodic orbit , and a Poincaré map P so that q ∈  with q0 ∈  is a fixed point under P. (Note that T q = q.) This is illustrated in Figure 5.2(a). The stability of the periodic orbit  is given by its Floquet multipliers. They can be found as the eigenvalues of the linearization DPq of the Poincaré map P around the associated fixed point q ∈  . (Note that there is always the extra trivial Floquet multiplier at +1, corresponding to the eigendirection along .) To find DPq, one needs to compute the linear variational equation around , which, in general, has T-periodic coefficients [5]. The linearization DPq has a spectrum that consists of countably many eigenvalues (the Floquet multipliers). It has the origin of the complex plane as its only accumulation point. In other words, much like in the situation for equilibria, there is only a finite number of Floquet multipliers outside a circle of fixed radius r > 0. For the special case of r = 1, one gets the result that there is always only a finite number of unstable Floquet multipliers. A periodic orbit  is called hyperbolic if there are no Floquet multipliers on the unit circle (except for the trivial one). A hyperbolic periodic orbit is either attracting if all the Floquet multipliers are inside the unit circle or of saddle type with infinitely many attracting and finitely many repelling eigendirections. When the parameter  is changed, Floquet multipliers can cross the unit circle of the complex plane, leading to changes in the stability of the periodic orbit . Because the Floquet multipliers are isolated (away from the origin of the complex plane), when a single

Bifurcation Theory of DDEs

153

(a)

(b)

768

768 ∑ q

765 N

765 N

q(0) 762

q(0) Eu(Γ)

762

Γ

759 2

759 2 Ex

0 –2

2

0

–2 Ey

Ex

0 –2

2

0

–2 Ey

Figure 5.2 A periodic orbit  of the DDE and a fixed point q (boldface) of a suitable Poincaré map P (a), and the linear unstable eigenspace E u  (b) in the projection onto physical space; the example is of Equations (5.23) and (5.24) for I = 651mA and 
= 19266 where  = Ey = 0 .

parameter is changed, there are exactly three generic or typical possibilities (for a DDE without any extra symmetry properties), giving rise to the local bifurcations of periodic orbits of codimension one: 1 A single real Floquet multiplier goes through +1. This is the case of a saddle-node bifurcation of limit cycles in which two periodic orbits are born. 2 A single real Floquet multiplier goes through −1. This is the case of a period-doubling bifurcation in which a new periodic orbit of period 2T bifurcates. 3 A complex conjugate pair of Floquet multipliers moves through the unit circle of the complex plane at e2i . Provided that  = 0 21  13  23  41  43 then this is the case of a torus (or Neimark-Sacker) bifurcation, in which an invariant torus is born. (The conditions on  are to avoid the more complicated strong resonances; see, for example, [38, 45] for details.) In order to check that one is dealing with any of these bifurcations, one has to check the respective genericity conditions. Again, these conditions are typically satisfied, which is good because it is even harder to verify them for periodic orbits. The reason for this is that the Poincaré map P and its linearization cannot be computed explicitly, but must be computed numerically. The period-doubling bifurcation and the torus bifurcation may be supercritical or subcritical, depending on whether the bifurcating object is attracting or of saddle type. This is usually apparent from the bifurcation scenario being considered, so that it is generally not necessary to compute the respective normal form coefficients, which involve higher-order terms.

5.2.4 Unstable Manifolds and Global Bifurcations Associated with a saddle steady state q0 , there are a finite number of unstable eigenfunctions of the linearization around q0 , which span the linear unstable eigenspace E u q0 . The local

154

Bifurcation Analysis of Lasers with Delay

u unstable manifold Wloc q0  is the set of points q that can be integrated backwards, never leave a small neighbourhood V of q0 under t for t < 0, and are such that t q → q0 u as t → −. As is the case for ODEs, Wloc q0  is tangent to E u q0  at q0 and of the same dimension (given by the number of unstable eigenvalues). The (global) unstable manifold u W u q0  is defined as the globalization of Wloc q0 : u q0  and t > 0

W u q0  = t p  p ∈ Wloc

The stable manifold W s q0  is defined similarly as the set of points that converge to q0 under t in positive time, but it is always infinite-dimensional since there are infinitely many stable eigenvalues of the linearization around q0 . An important special case is that of a one-dimensional unstable manifold W u q0 , because then W u q0  is a smooth 1D curve in projection onto the physical space
n . In particular, the manifold W u q0  has two branches, one on either side of W s q0 , which can be computed efficiently by integrating from two initial conditions near q0 on E u q0 . If W u q0  lies in W s q0 , then it forms a codimension-one homoclinic connection. If W u q0  lies in W s q1  for a different saddle point q1 ∈  with a one-dimensional unstable manifold, then it forms a codimension-one heteroclinic connection between q0 and q1 . At the connection, which occurs for some fixed value ∗ of the parameter , there is a solution xt of Equation (5.1) such that the limits lim xt = q0 

t→−

lim xt = q01

t→+

(5.7)

exist. Homoclinic and heteroclinic connections are important for the overall organization of phase space. As is explained in Section 5.5.2, they can be found and followed by continuation [53]. A periodic orbit  of saddle-type comes with unstable manifolds as well. In what follows, the case is considered that  has a single unstable Floquet multiplier, where there is a single unstable eigenfunction spanning the linear unstable eigenspace E u  . In projection onto physical space E u   forms a one-parameter family of directions along . This is illustrated in Figure 5.2(b). Again, consider the associated saddle fixed point q ∈  of a suitable Poincaré map P to a u q of q as the set of all points p ∈  section  and define the local unstable manifold Wloc that can be iterated backwards under P, never leave a small neighbourhood V of q, and are u q is tangent to E u q (which can be such that P l p → q as l → −. The manifold Wloc u derived from E ) and of the same dimension, in our case dimension one. The (global) u q defined by unstable manifold W u q is the globalization of Wloc u q and l > 0

W u q = P l p  p ∈ Wloc

The stable manifold W s q is defined similarly as the set of points that converge to q under P in positive time, and it is always infinite-dimensional [32]. In projection onto the physical space
n the unstable manifold W u q forms a complicated two-dimensional object. However, the set of its headpoints in , which is called its trace, is a 1D curve that is smooth (except possibly at isolated points due to the projection), very much like the 1D unstable manifold of a saddle point of a finite-dimensional map. The difference

Numerical Methods

155

in the case of DDEs is that the trace W u q ∩  may have self-intersections, which is a clear reminder of the fact that W u q lives in an infinite-dimensional phase space. As for ODEs, the study of 1D unstable manifolds of saddle points of the Poincaré map is a powerful tool when one wants to study transitions to chaos. With the method explained in Section 5.3.3 they can now be computed; this is demonstrated in Section 5.5.4 with the example of the break-up of an invariant torus in the PCF laser [27].

5.3 NUMERICAL METHODS This is an exciting time to study lasers with delay. Powerful new numerical techniques are now becoming available, similar to those that have been used for many years and with great success for the study of finite-dimensional dynamical systems defined by ODEs. Most importantly, the continuation package DDE-BIFTOOL [9] allows one to find and follow steady states and periodic orbits irrespective of their stability, a great advantage over mere simulation which only finds stable objects. DDE-BIFTOOL can detect and follow codimension-one bifurcations. This package is still under development and lasers with feedback have emerged as arguably the most important motivation and application for recent additions to its capabilities [27, 28]. Building on the computation of saddle periodic orbits to obtain the necessary starting data, a method for computing 1D unstable manifolds was developed in [40]. These numerical methods are now reviewed in quite some technical detail, because they are still not widely known in the field of laser physics. The reader is again invited to look ahead at Sections 5.4 and 5.5 to see how they can be used to analyze the dynamics of lasers with delay.

5.3.1 Simulation by Direct Numerical Integration The most basic operation is to integrate a DDE to simulate the dynamics. After transients have died away, the dynamics will settle down to some attractor of the DDE. (Here it is assumed that the DDE is dissipative, as is the case for laser systems). A DDE with a fixed delay is best integrated with a fixed time-step method, where the time step is chosen as an integer fraction
/M of the delay time
. This means that the history interval −
 0 is discretized into M equal subintervals. A single integration step then produces a numerical approximation of the evolution operator 
/M over the time step
/M. As was explained in Section 5.2.1, one needs to initialize and keep the entire (discretization of the) history array during a computation. In the simplest case one could use forward Euler integration. Other more involved and efficient integration routines, such as Runge-Kutta integrators, require storing a few more elements in the history array as input to the integrator. In the simulations in Section 5.5 an Adams-Bashforth fourth-order multistep method was used. This requires storing a list up to and including the vector q−
− 3 M
, that is, three extra elements. Numerical simulation has been used for many years to study the dynamics of lasers with delay. Typically, one real variable, for example, the inversion at a given value of the electric field, is plotted after transients died down for a range of a control parameter. In this way, one gets a one-parameter bifurcation plot of how the attractor in the system changes with the parameter. This approach can be found virtually everywhere, for examples,

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Bifurcation Analysis of Lasers with Delay

in [16, 21, 39, 48], in Chapters 3 and 4, and in Figures 5.10(a) and 5.15(a). It provides a good way of getting an impression of the attracting dynamics of the laser.

5.3.2 Numerical Continuation The idea of numerical continuation is to find an object of interest and then follow it in suitable parameters. The big advantage is that this object can be followed irrespective of its stability. This means that information can be obtained also about objects that are not attracting, which is a big advantage over numerical simulation. Mathematically, continuation is set up in such a way that there are L − 1 equations for L unknown variables, so that the solution space is geometrically a one-dimensional curve in
L . Once a first point on this solution curve is found, it is possible to follow or continue it in the parameters. This is usually done with the method called pseudo-arclength continuation, which involves a prediction step in the direction tangent to the curve and a Newton correction step in the space perpendicular to this tangent. During a computation the stepsize along the branch is adapted in response to the success of the Newton correction. The idea of continuation can be used to follow branches of many implicitly defined problems, including bifurcations. It is implemented, for example, in the much used continuation package AUTO [6] for (among other things) the bifurcation analysis of ODEs. It must be stressed that the technique of numerically solving appropriately formulated bifurcation equations by continuation is firmly established (with proofs of convergence) in the field of numerical analysis. Admittedly, using numerical continuation requires the user to know a good deal more about bifurcation theory than is needed for numerical simulation. The reward is a very powerful tool that, if used properly (that is, with some experience and the necessary checks), allows one to unravel even complicated dynamics in great detail and with great accuracy. See, for example, [56] for more general background reading in the context of ODEs. Here we concentrate on the numerical continuation in DDEs. The recently developed continuation package DDE-BIFTOOL [9] consists of Matlab routines for the continuation and bifurcation analysis of steady states and periodic solutions. DDE-BIFTOOL solves a large system that is obtained by discretizing the delay interval −
 0 and uses pseudo-arclength continuation to follow solution branches. This package detects and follows saddle-node and Hopf bifurcations of steady states, and detects the local codimension-one bifurcations of periodic orbits, namely saddle-node bifurcations of limit cycles, period-doubling bifurcations and torus bifurcations. (The continuation of bifurcations of periodic orbits is presently not implemented.) A steady state is represented by the value of the parameter , the steady state position x0 and the eigenvalues i of this steady state. These eigenvalues i are found as the (appropriate number) of right-most roots of the characteristic equation Equation (5.4), which are corrected using Newton iterations. A saddle-node bifurcation is detected and represented by a nullvector of the matrix 0 in Equation (5.4), and a Hopf bifurcation by the complex null vector of i for a corresponding frequency . Periodic solutions are represented by a suitable boundary value problem, which is solved by orthogonal collocation, which is a piecewise polynomial representation of the solution. Specifically, a periodic solution is represented by the value of the parameter , the period T and a profile x∗ t/T on a mesh over the (time-scaled) interval [0,1]. The stability information

Numerical Methods

157

for periodic orbits is obtained by computing a finite number of the largest Floquet multipliers (for example, all those outside a suitable radius r > 0). The software package DDE-BIFTOOL is able to switch to and continue emanating branches of periodic orbits at bifurcation points (such as a Hopf bifurcation), and continue codimension-one bifurcations of steady states in two parameters. DDE-BIFTOOL can be extended to monitor other test functions that might be of interest to users. An example is the detection and continuation of a neutral saddle-focus point in [28]. A recent addition to DDE-BIFTOOL is the computation of connecting orbits and their continuation in two parameters [53]. This algorithm uses what are known as projection boundary conditions (appropriately adapted to the context of DDEs) and is a natural extension of the method that was implemented for the computation of connecting orbits of ODEs in the HomCont [3] extension of AUTO. The computation of a connecting orbit can be started from a nearby periodic orbit with a sufficiently large period. As will be illustrated with the examples in Sections 5.4 and 5.5 the usefulness of DDEBIFTOOL for the bifurcation study of lasers with delay, and of DDEs in general, can hardly be exaggerated. For more details on the inner workings of DDE-BIFTOOL, see [7, 8, 9, 53].

5.3.3 Computation of 1D Unstable Manifolds Suppose that q0 is a steady state with one unstable eigenvalue, that is, with a one-dimensional linear unstable eigenspace E u q0 . As was explained in the previous section, q0 can be found by numerical continuation. The unstable eigenspace is spanned by a (generalized) vector v  −
 0 →
n of linear directions along q0 . This vector v can be found by an iterative approach or directly with DDE-BIFTOOL; see [27, 40]. Both branches of the 1D unstable manifold W u q0  can then be computed by integrating from the initial conditions q0 ± v for a suitably small  > 0, that is, from initial conditions (approximately) on E u q0  close to q0 . It is much more challenging to compute the 1D unstable manifold of a saddle fixed point q ∈  of a Poincaré map (associated with a saddle periodic orbit ). The 1D linear unstable eigenspace E u   can again be found as a vector v by an iterative approach. The computation of v using DDE-BIFTOOL has recently been developed [29]. This allows us to find as starting data a first point q ± v ∈  at a small distance  from q along E u   (whose headpoint lies in ), see Figure 5.2(b). The idea is now to ‘grow’ a branch of W u q step by step until a sufficiently long piece has been computed. In fact, any of the known algorithms for computing 1D unstable manifolds of saddle points of finite-dimensional maps can be generalized to the setting of DDEs by working on points in  , instead of on points in a finite-dimensional space. However, points in  are computationally expensive to work with because they are represented by M × n data arrays (where n is the dimension of the physical space
n ). It is therefore imperative to compute as few points as necessary to achieve a prescribed accuracy of the computation. For this reason the generalization of the growth method in [42, 43] was implemented. All relevant operations, such as the iteration of the Poincaré map and interpolation, were interpreted and implemented as operations on points in  , or rather on their discretizations as data arrays. A sketch of the algorithm is presented here for completeness; see [40] for a detailed explanation. A branch of the 1D unstable manifold W u q is represented by a list p1  p2      pk of points in  . Linear interpolation is used between consecutive list points. The algorithm proceeds in steps by computing one new point at a time, hence, growing the

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Bifurcation Analysis of Lasers with Delay

list representing the branch. To find the next point pk+1 in the list one searches for a point in the part of W u q that was already computed (this requires interpolation between mesh points) that maps to a circle of a predetermined radius k around the last point pk . Because of the invariance of W u q, this new point pk+1 lies on W u q in good approximation. The distance k is adapted during the computation depending on the curvature of the trace of the manifold W u q ∩  in the section . A computation stops after a prescribed arclength distance of the trace W u q ∩  has been reached or when convergence to an attracting fixed point is detected. The only way of checking the accuracy of a computation in practice is to compare the results of successive computations with increased accuracy. (It is impossible to derive a priori bounds of any global manifold computation. To test the algorithm, it was also compared with the method of fundamental domain iteration, which does not involve interpolation, an important contributor to the overall error; see [40].) The method works reliably, provided that sufficiently small error bounds are chosen. Clearly, the longer and more complicated the manifold becomes, the more careful one has to be. The examples in Section 5.5.4 show that the method can be used successfully to study global bifurcations in a laser with delay.

5.4 BIFURCATIONS IN THE COF LASER A semiconductor laser receiving optical feedback from a mirror placed at a fixed distance L is the classic example of a laser subject to delay. One also speaks of a laser with conventional optical feedback (COF) or an external cavity laser. This laser system is physically straightforward, technologically relevant and has received a lot of attention, both experimentally and in terms of mathematical modelling. There is an extensive body of literature on the subject; see, for example, the recent surveys [14, 16, 60, 61] and Chapters 1,2 and 4 for more background material and further references. There is a well-established rate equation model of the COF laser, the so-called Lang– Kobayashi (LK) equations [46]. They can be written in the dimensionless form dE = 1 + iNtEt + e−iCp Et −
 dt dN = P − Nt − 1 + 2NtEt2 T dt

(5.8) (5.9)

for the complex electric field E = Ex + iEy and the inversion N . The parameters are the pump current P, the ratio of decay times T , the linewidth-enhancement factor , the feedback strength , the delay time
= 2L/c (where c is the speed of light), and the feedback phase Cp . In fact, Cp = 0
, where 0 is the optical frequency. However, Cp is generally seen as an independent parameter because it can be changed by miniscule changes of, for example, the length of the external cavity (using a PZT) or the pump current [33, 35]. The LK equations are based on two main modelling assumptions, namely that the laser operates in single mode and that the feedback is not too strong. This allows Equations (5.8) and (5.9) to describe only one mode and to take into account a single round-trip in the external cavity. Implicitly, this also means that the external cavity length L should be larger than the length of the laser itself; in most experiments it is of the order of centimetres to metres. While one should be aware of the underlying modelling assumptions, the LK

Bifurcations in the COF Laser

159

equations are known to describe many experimentally observed dynamics with amazing accuracy; see again Chapter 4 for more examples. Complicated dynamics in the LK equations have been studied mainly by direct numerical simulation. However, when advanced numerical methods for DDEs recently became available, the LK equations emerged as a natural first application. The following sections review some recent work on the bifurcation analysis of the COF laser as modelled by the LK equations.

5.4.1 Symmetry of the COF Laser Equation The LK equations are a DDE of the form Equation (5.1) and their physical space is the three-dimensional E N -space. So it seems that one can immediately apply the tools of bifurcation theory introduced in Section 5.3 to Equations (5.8) and (5.9). However, this is not the case, because the LK equations feature an important symmetry: they are invariant under the continuous symmetry group S 1 of all rotations of the complex E-plane. This symmetry group can be written as S 1 = c ∈ c = 1 and an element c ∈ S 1 acts on a trajectory E N t of Equations (5.8) and (5.9) by c  E N t = cE N t

(5.10)

This means that the rotation over of any angle  = argc of a trajectory is again a trajectory! In other words, a trajectory is either S 1 -symmetric itself, or it is not isolated but comes as a rotational family parametrized by the elements of S 1 . This S 1 -symmetry is not an artefact of the LK equations, but is also present in other rate equation models of COF lasers; see [44] for details. It is a consequence of two properties: Equation (5.8) is linear in E, and E enters Equation (5.9) only as its modulus E. The continuous symmetry group S 1 of the LK equations cannot be divided out. However, as is explained below, bifurcation techniques can still be brought to bear on the system after appropriate modifications. Let us start with some thoughts about the physical space. If one writes E in polar coordinates as Et = Rteit , then the action of c ∈ S 1 on E is that of adding argc to t. It is sufficient to consider the intersection (also called the trace) of all S 1 -images of a trajectory E N t with the two-dimensional half-plane E N  Ey = 0 and Ex ≥ 0 . This half-plane can simply be identified with the R N -plane, which emerges as a reduced physical space. In practice, to obtain the trace all one needs to do is plot N versus R. This is only a useful projection; the -dynamics is important and cannot be neglected.

5.4.2 External Cavity Modes The most basic solution of Equations (5.8) and (5.9) is the trivial solution E Nt ≡ 0 P (where P is the pump current). If it is stable, then the laser is in its off-state. The laser threshold is given by the loss of stability of the trivial solution. In Equations (5.8) and (5.9), this occurs in a Hopf bifurcation. (When the electric field equation (5.8) is written in polar coordinates the threshold is given by a pitchfork bifurcation, and when Equation (5.8) is written in terms of the power E2 it is given by a transcritical bifurcation.) In the Hopf bifurcation a small periodic orbit is born. The laser does produce constant output, as is

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Bifurcation Analysis of Lasers with Delay

expected just above threshold (when this periodic orbit is stable), because it turns out that this periodic orbit is of a special form with constant R and N . Indeed, it is one of the external cavity modes that are introduced now. An external cavity mode (ECM), also referred to as a continuous wave solution or CW-state, is a solution of Equations (5.8) and (5.9) of the form: E N t = Rs eis t  Ns 

(5.11)

for constant Rs , s and Ns . In other words, an ECM is a perfectly circular periodic orbit in E N -space of period 2/s . In projection onto the R N -plane an ECM is a single point; in some papers ECMs are referred to as fixed points. By writing Equations (5.8) and (5.9) in polar coordinates and inserting (5.11) as an ansatz, one obtains: R2s =

P − Ns 1 + 2Ns

(5.12)

s = − coss
+ Cp  + sins
+ Cp 

(5.13)

Ns = − coss
+ Cp 

(5.14)

where it is assumed that Rs = 0 (and Ns = − 21 ). These equations have the structure, that once s is known, Ns and then Rs can be found. However, Equation (5.13) is transcendental and cannot be solved explicitly. It can be rewritten as s = −

K sins
+ Cp + arctan


(5.15)

√ where K = 
2 + 1 (also called the effective feedback strength). In other words, the solutions of Equation (5.15) are the intersection of a sine function with amplitude K
with the diagonal (the graph of the function fws  = ws ). This already shows that there is always at least one solution of Equation (5.13). When  is increased, more solutions are created in pairs, which already shows that new ECMs are born in saddle-node bifurcations. In particular, one ECM (the one with lower Ns ) of a newly-born pair is initially attracting, while the other is a saddle. Furthermore, as is expected already from Equations (5.8) and (5.9), the solution set of Equation (5.15) is 2-periodic in the feedback phase Cp . It is important to realize that, in order for a solution of Equations (5.12)–(5.14) to be physically relevant, the radius Rs must be positive. Since the pump current P is positive, this implies that 1 − < Ns ≤ P 2

(5.16)

It is well known that the ECMs lie on an ellipse in a suitable projection, for example in the s  N-plane as is shown in Figure 5.3(a). In projection onto the R N -plane all ECMs lie on a straight line; see Figure 5.3(b). The ECMs on the lower part of the ellipse (that are initially stable) are often referred to as modes, and those on the upper part as anti-modes. The ellipse is entirely in the physically relevant region if condition (5.16) is satisfied. Otherwise, only part of it corresponds to physical ECMs. By changing Cp over several cycles of 2,

Bifurcations in the COF Laser

161

6 (a)

(b)

6

N 5

4 2π

N

5

(c)

Cp

π SN 0 –100

SN

–50

0

ωs

50

4

100

2.0

2.5

3.0

R

Figure 5.3 An ellipse of ECMs (a) in the s  N -plane and the same ECMs in the R N -plane (b); modes are circles and anti-modes crosses, the black dot is a stable ECM, also called the maximum gain mode. The ECMs can be continued as one branch in the 2-periodic parameter Cp (c); the ECMs in (a) and (b) correspond to the value of Cp = 43  indicated by the dashed line in (c). The LK equations and the other parameters are as in [35].

an individual ECM can be followed from its emergence in a saddle-node bifurcation to its disappearance in another saddle-node bifurcation. This is shown in Figure 5.3(c) in the s  Cp -plane; the ellipse in Figure 5.3(a) for a fixed value of Cp = 43  corresponds to a cross section through this image. These results on the ECMs are now classical; see Chapter 2. The solutions of Equations (5.12)–(5.14) can be found, for example, by Newton’s method. However, the following sections stress the point of view of bifurcation theory. Indeed, Equations (5.12) – (5.14) are implicit equations whose solutions, the ECMs, can best be computed by starting at a first ECM and then continuing it in parameters.

5.4.3 The Characteristic Equation of an ECM The next step is to consider and compute the stability of the ECMs in order to find and continue their bifurcations. To do this one needs to compute the linear variational equation around an ECM, because the ECMs are after all periodic orbits, not steady states. Following the procedure in [64] and writing Equations (5.8) and (5.9) in polar coordinates, they take the form:   Rt d  t  = FRt t Nt Rt −
 t −
 (5.17) dt Nt

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Bifurcation Analysis of Lasers with Delay

The function F is defined as: 

 xz + v cosw − y − Cp  v     Fx y z v w =  z +  x sinw − y − Cp     1 P − z − 1 + 2zx2  T

(5.18)

where x y z v w denote its five arguments. As for a steady state, the linearization of F can be written in the form of Equation (5.2) as DFqt = A1 qt + A2 qt −


(5.19)

The matrices A1 and A2 can be computed as (compare Equation (5.3))   −Rs sins
+ Cp  Rs Ns     sins
+ Cp  − coss
+ Cp      A1 =  Rs    2 1 − 1 + 2Ns Rs 0 − 1 + 2R2s  T T

(5.20)

and 

−Ns

Rs sins
+ Cp  0



   − sins
+ Cp   coss
+ Cp  0  A2 =    Rs 0 0 0

(5.21)

The block form of A2 indicates that the N -equation Equation (5.9) of the LK equations does not have a delay term. The matrix

 = I − A1 − A2 e−
gives rise to the characteristic equation as det  = 0. As was mentioned in Section 5.2.3, the variational equation has T-periodic coefficients in general. However, neither A1 nor A2 have periodic coefficients. Therefore, the characteristic equation is autonomous, as one would expect from a steady state. This is a direct consequence of the S 1 -symmetry of the ECMs. The characteristic equation can be calculated as: 0 = 3 1 + 1 + 2R2s  − 2Ns 1 − e−
2 T 2R2s 2 1 + 2Ns  + 2 1 − e−
2 + coss
+ Cp 1 + 2R2s 1 − e−
 + T T 2 + 1 + 2R2s 1 − e−
2 T 2R2s 1 + 2Ns coss
+ Cp  −  sins
+ Cp 1 − e−
 + (5.22) T

Bifurcations in the COF Laser

163

where Rs , s and Ns of the ECM under consideration are given by Equations (5.12) – (5.14). (We note that Equation (5.22) corrects some minor errors in (the derivation of) the characteristic equation presented in [64].) Equation (5.22) is an implicit, transcendental equation that contains all stability information of the ECMs. Note that  = 0 is always an eigenvalue (corresponding to the trivial Floquet multiplier of the ECM). If Equation (5.22) has an additional eigenvalue  = 0 then a saddlenode bifurcation takes place. If it has a complex pair of  with zero real part, then an ECM undergoes a Hopf bifurcation. The respective solution branches can best be found by continuation in the spirit of bifurcation theory. Linear stability analysis of ECMs, that is, computing regions of different numbers of ECMs and the curve of the first Hopf bifurcation, is a well-established technique that has been applied to different types of lasers (given by sets of laser parameters in Equations (5.8) and (5.9)). Figure 5.4, reproduced from the survey paper [61], shows such regions in the Cp  -plane. A bifurcation analysis of the characteristic equation in the regime of moderate delays was recently performed by Wolfrum and Turaev in their paper [70]. They pay particular attention to the condition that there are two ECMs with the same inversion Ns but different s . This is called the Petermann–Tager condition, because Petermann and Tager observed that near such a point one can find oscillations of the laser at a frequency that is given by the difference of the two ECMs involved. This was later confirmed, by asymptotic methods and by numerical continuation with DDE-BIFTOOL [13, 31, 50, 51]; these continuation results are discussed in the next section. In the  -plane shown in Figure 5.5 (the Cp  
-plane in our

Figure 5.4 Regions in the 0  C-plane (the Cp  -plane in our notation) with an indication of the maximal number of ECM that may occur (roman numerals). The saddle-node bifurcation curves are dashed and the Hopf bifurcation curve is a solid curve. (The Hopf curve has two parts, which match up by identifying 0 with 2; the dotted line is the lower boundary for possible multiple solutions.) Source: Reproduced from G.H.M. Van Tartwijk and D. Lenstra, Quantum Semiclass. Opt., 7, 87–143 (1995) ©1995 by Institute of Physics Publishing.

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Bifurcation Analysis of Lasers with Delay

1.4 PT 1.2 1.0 MD

η

FH

0.8 0.6 0.4

DQS

0.2 0.0 1.0

1.5

φ

2.0

2.5

3.0

Figure 5.5 Numerically computed Hopf curve (thick curve) in the  -plane (the Cp  
-plane in our notation) and the asymptotic approximation (dashed curves) in the limit of small  =
/T . Source: Reproduced from M. Wolfrum and D. Turaev, Opt. Commun., 212 (2002) 127–138 ©1995 by Elsevier Science.

notation) a curve of Petermann–Tager points (denoted PT) emerges from a mode degeneracy point MD on the saddle-node curve. Apart from the oscillations associated with the curve PT, there is also a second type of oscillations with a frequency close to the relaxation oscillation frequency of the solitary laser; the authors call it dispersive self Q-switching (DQS) pulsations due to their similarity with pulsations in two-section DFB lasers. The Hopf bifurcation condition is derived from the characteristic equation (the equivalent of Equation (5.22)) and analyzed by scaling methods in the limit of small  =
/T . Good agreement is found between these theoretical results and a numerical continuation of the Hopf bifurcation condition; see Figure 5.5 where the DQS Hopf curve is in good agreement with the asymptotic dashed curves emerging from the point MD of mode degeneracy. A full bifurcation analysis of the characteristic equation Equation (5.22) has not been performed. The next step of presenting a geometrical picture of the bifurcation set in Cp   P-space of all bifurcations of ECMs is the topic of an ongoing project. This bifurcation set can be seen as the ‘ECM-backbone’ of the COF laser, and it will be presented elsewhere.

5.4.4 Continuation Near Connecting Bridges The numerical continuation study of the LK equations by Haegeman et al. in [31] is a good representation of the state-of-the-art in numerical bifurcation analysis of the COF laser. In this section (a number of) their results are reviewed. The authors are considering the LK equations in the exact same form as Equations (5.8) and (5.9) but use o
instead of Cp .

Bifurcations in the COF Laser

165

The starting point of their analysis is the realization that when different branches of ECMs cross (as a function of ) and have the same inversion Ns , then there should be periodic orbits with a frequency given by the frequency difference of the involved ECMs. This was the observation of Petermann and Tager mentioned in the previous section. As was shown in [51], in the limit of small  = 1/T this gives rise to mixed mode solutions for the electric field of the form Et = A1 expi 1 − Cp t + A2 expi 2 − Cp t where 1 and 2 are the frequencies of the two ECMs and A1 and A2 are their amplitudes. As a consequence, the power E2 oscillates at the frequency  1 − 2 . These mixed mode solutions survive in the case of nonzero  and give rise to what are called bifurcation bridges in [31]. They are in fact branches of tori connecting Hopf bifurcation points on different branches of ECMs. In order to study this phenomenon by continuation with DDE-BIFTOOL one first needs to take into account the S 1 -symmetry of the LK equations. The reason is that DDE-BIFTOOL, like any continuation software, requires isolated solutions. This was achieved in [31] by introducing a new phase parameter b by setting Et = Ateibt where At is also complex. With this ansatz it is possible to remove the indeterminacy due to the symmetry group by fixing the phase of At and solving for the associated unkown value bs of the new variable b. Then an ECM corresponds to an equilibrium of this extended system with the same expression for its stability (because the -direction is neutral at an ECM). What is more, quasiperiodic solutions (that is, tori) can now be continued as periodic solutions of this extended system. This makes it possible to compute the tori that form the bridges between different branches of ECMs as periodic orbits; see [31] for details. Figure 5.6(a) shows a number of branches of ECMs that are born in saddle-node bifurcations (seen as the left-hand fold points). As was mentioned earlier, one of the ECMs (the mode) is initially stable and the other (the anti-mode) is not. Both the modes and the anti-modes undergo Hopf bifurcations. Branches that cross each other are connected by bridges of periodic orbits (or rather tori); Figure 5.6(b) and (c) shows two enlargements. The periodic orbits generating the bridges generally undergo further bifurcations, either perioddoubling or torus bifurcations, and this generally leads to regions of chaotic dynamics [48] (not shown in the figure). The paper [31] also studies the dependence of the connecting bridges on the parameters  and Cp , showing that they may ‘rupture’, meaning that the connection is lost in further bifurcations. Shown here are only their results of the dependence on the feedback phase Cp , because they fit nicely into our earlier discussion. Figure 5.7(a) shows curves of saddle-node bifurcations (solid) and of Hopf bifurcations (dashed) bounding regions of stability of ECMs in the Cp  -plane. From the point p2 two curves of pitchfork bifurcations emerge. At this bifurcation the radius of an ECM goes to zero (which corresponds to a Hopf bifurcation of the trivial equilibrium). Figure 5.7(b) shows bifurcation curves of periodic orbits in the Cp  -plane. In the grey region the periodic solution corresponding to the first bifurcation bridge is stable. It is bounded by the Hopf bifurcation curve from Figure 5.7(a) and by curves of period-doubling bifurcations. Note how Figures 5.4 and 5.5 capture the respective aspects of Figures 5.7(a) and (b).

166

Bifurcation Analysis of Lasers with Delay

0.06 max(|E|)

0.10

0.06

0.05 0.04

0.03

0

1

2

3

0.09

0.04 max(|E|)

max(|E|)

0.08

0.02

0.08 0.07 0.06

0

0

1

2

3

4

κ

5

6

3

7

4

×10

–3

5 κ

6 7 ×10–3

Figure 5.6 Bifurcation bridges connecting different branches of ECMs of the LK equations; dashed curves correspond to unstable solutions, the two right panels are enlargements. Source: Reproduced from B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux, and T. Erneux, Phys. Rev. E 66 (2002) 046216 ©2002 by the American Physical Society.

3.0

×10–3

×10–3 1.8

2.5

1.6 1.4

2.0

1.2 κ

κ

1.5 1.0

0.8

p2

0.6 0.4

0.5 0.0

1.0

p1 –3

–2

–1

0 ω0τ (a)

1

0.2 2

3

0

2

ω0τ

4

6

8

(b)

Figure 5.7 Panel (a) shows a bifurcation diagram in the Cp  -plane (note that 0
= Cp ) of ECMs of the LK equations; shown are curves of saddle-node bifurcations (solid), Hopf bifurcations (dashed) and pitchfork bifurcations (dot dashed). Panel (b) shows a bifurcation diagram in the Cp  -plane of periodic orbits of the LK equations; shown are curves of saddle-node bifurcations of periodic orbits (solid), Hopf bifurcations (dashed), period-doubling bifurcations (dot dashed) and torus bifurcations (dotted), and Cp is not taken modulo 2 Source: Reproduced from B. Haegeman, K. Engelborghs, D. Roose, D. Pieroux, and T. Erneux, Phys. Rev. E 66 (2002) 046216 ©2002 by the American Physical Society.

Bifurcations in the COF Laser

167

5.4.5 Global Bifurcations of ECMs A periodic orbit born in a Hopf bifurcation of an ECM first ‘surrounds’ only this ECM, a notion that makes sense because a Hopf bifurcation is an inherently two-dimensional phenomenon: close to the Hopf bifurcation the bifurcating periodic orbit is (in very good approximation) a nice round circle in a plane. (More technically, the Hopf bifurcation takes place in a two-dimensional centre manifold.) As  is increased, there appear periodic orbits ‘surrounding’ several ECMs (in certain projections). The process of going from one situation to the other is sometimes called glueing, and it generally involves homoclinic bifurcations. Indeed, it has been known for some time that homoclinic bifurcations occur in the LK equations. An example of a homoclinic orbit of an ECM with a one-dimensional unstable manifold and a two-dimensional strongest stable direction with complex conjugate eigenvalues (a saddle-focus) is shown in Figure 5.8(a) and (b). The ECM involved in the homoclinic

(a)

(b)

5

Im(E) 0 10

–5

N 10 8

N

8 6

6 –5 0

Re(E) 20

(c)

0

5

R

5

(d)

Im(E) 0 0

–20 0 –25

N

N –40

–50 –70 –80 –20 0

Re(E)

0 20

10

20

R

Figure 5.8 A homoclinic orbit to a saddle focus ECM in E N -space (a) and in the R N -plane (b), and an example of complicated dynamics surrounding several ECMs in E N -space (c) and in the R N -plane (d). The parameters are  = 10,  = 255 × 10−3 , A = 496 × 10−7 ,
= 7143, and  = 07313236 × 10−3 in panels (a) and (b), and  = 56 × 10−3 in panels (c) and (d) for the LK equations in the form given in [64].

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Bifurcation Analysis of Lasers with Delay

bifurcation is an anti-mode shortly after its emergence in a saddle-node bifurcation (so that the unstable manifold is indeed one-dimensional). The periodic orbit is born in a Hopf bifurcation of the ECM that it surrounds in Figure 5.8(b). The homoclinic orbit was found by simulation as a large-period periodic orbit. Also in [31] the authors find homoclinic orbits (some also of saddle-focus type) by continuing periodic orbits to large period, and they show that the ‘rupture’ of bifurcation bridges is due to homoclinic bifurcations. The main challenge is to study the sequence of global bifurcations, homoclinic and heteroclinic, that lead to complicated dynamics with the aim to characterize the exact nature of the well-known low frequency fluctuations (LFF), see Chapters 2 and 4. An example of complicated dynamics is shown in Figure 5.8(c) and (d). The attractor, which seems to be chaotic, ‘surrounds’ a number of ECMs. However, in terms of complexity this dynamics is still quite far from full-blown LFF when the laser has hundreds or even thousands of ECMs. Neverthetheless, even the exact bifurcation transition from Figure 5.8(b) to (d) is not understood. This section on the COF laser now finishes with some recent results [35] on how the study of 1D unstable manifolds of ECMs can cast some light on the nature of bifurcations as Cp is changed in the case of a short cavity (of the order of a few centimetres) as studied recently in [33], see also Chapter 4. Figure 5.9 shows antimodes (denoted by +) and modes (denoted by ) and one branch of the 1D unstable manifold of one of the anti-modes. This branch was computed by integrating from an initial condition close to the ECM along its linear unstable eigendirection. (Because of the S 1 -symmetry, this can be done as for a steady state 6

N

(a)

(b)

(c)

(d)

(e)

(f)

5

4 6

N

5

4 0

2

4 R

6

0

2

4 R

6

0

2

4

6

R

Figure 5.9 One branch of the 1D unstable manifold of a saddle ECM (anti-mode) in the short cavity regime, shown in panels (a) to (f) for decreasing values of Cp in projection onto the R N -plane. From (a) to (f) Cp takes the values 2498, 2348, 2113, 2073, 1956, and 1893, where the LK equations and the other parameters are as in [35].

Bifurcations in the PCF Laser

169

as was explained in Section 5.3.3.) The branch initially ends up at a stable ECM (a), and then at increasingly complicated attractors as Cp is increased, namely a periodic orbit (b), a torus (c) and chaotic attractors of increasing sizes; see panels (d) to (f). The dynamics on the attractor in Figure 5.9(f) is in fact very regular and has been called pulse packages [33]. An important feature of the branch is a mechanism of reinjection: the branch first visits a region of higher inversion N and then quickly extends all the way to the region of low N (or high gain). This produces an initial pulse in the laser power. There appear to be a number of global bifurcations in which the attractor increases in size, ‘surrounding’ more and more ECMs. The pulse packages themselves become more regular, which is seemingly due to the appearance of a quite narrow ‘channel of reinjection’ through which the trajectory travels at the beginning (producing the first and largest pulse) of each pulse package. This forces the pulse packages to be more alike until they finally disappear in a sudden transition back to constant laser output. The scenario in Figure 5.9 agrees well with experimental measurements reported in [33]; see [35] for details.

5.5 BIFURCATIONS IN THE PCF LASER Semiconductor lasers with phase-conjugate feedback (PCF) from a phase conjugating mirror (PCM) have received considerable attention because they have a number of positive properties. Due to the reversal of the light front in the PCM, the reflected wave travels back along the same path as the incident wave, meaning that alignment of the mirror is less of an issue. Furthermore, perturbations on the way to the PCM are undone on the way back, resulting in potentially more stable operation. The main technical difficulty is realizing a PCM. Phase conjugation can be achieved by a number of nonlinear optical processes, including stimulated Brillouin scattering, backward stimulated Raman scattering and three-wave down conversion [17]. Probably the most common process is degenerate four-wave mixing in which three input waves mix to produce a fourth output wave. Two of the input waves are counter-propagating pump waves of a given frequency, for example, from additional semiconductor lasers. The incident wave is the third wave, that enters the medium at any angle to the pump waves and is coupled to the pump waves through a third-order susceptibility  3 . This results in a fourth wave, which is phase-conjugated to the incident wave, see Chapter 3 for more introductory information. Phase conjugate feedback may produce a highly focused beam [17], which is of considerable advantage when stable output is desired. The main application is mode locking [20] and phase locking, where PCF has been shown to reduce the laser noise considerably [1, 21, 61] However, many other dynamical regimes have been identified, including periodic and quasiperiodic output, as well as chaotic dynamics, see, for example, [1, 17, 21, 39]. As is the case with the COF laser, the PCF laser can be modelled by a well-established rate equation for the evolution of the slowly varying complex electric field Et and the population inversion Nt [21, 39], which can be written in the form:

 1 dE 1 = Et (5.23) −iGN Nt − Nsol  + Gt − dt 2
p +E ∗ t −
 exp2it −
/2 + iPCM  I Nt dN = − − GtEt2 dt q
e

(5.24)

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Bifurcation Analysis of Lasers with Delay

In Equations (5.23) and (5.24), nonlinear gain is included as Gt = GN Nt − N0  1 − Pt, where  = 357 × 10−8 is the nonlinear gain coefficient and Pt = Et2 is the intensity. Parameter values are set to realistic values corresponding to a Ga-Al-As semiconductor laser [21, 39] as summarized in Table 5.1, where Nsol = N0 + 1/GN
p . As is common in the field [21, 39], the constant phase shift at the PCM and the detuning parameter were both set to zero, so that the feedback term in Equations (5.23) and (5.24) reduces to E ∗ t −
. The main bifurcation parameter is the effective feedback strength 
(a dimensionless parameter), and in two-parameter investigations also the pump current I (in units of ampere) is varied. The modelling assumptions here are, as for the LK equations, that the laser operates in single mode and the feedback is weak (because again only a single round trip in the external cavity is taken into account). However, there is the further assumption that the PCM is so fast that the phase-conjugation can be modelled as instantaneous. The effect of non-instanteneous PCF has been considered in [4, 26, 59]. As the interaction time in the PCM is increased from zero (the instantaneous case), the dynamics of the PCF laser is affected first for larger and then for lower and lower values of 
. For the moderate values of 
near and just above the locking region (of steady state operation; see Section 5.5.2) of the PCF laser used in the results below, the dynamics is essentially unaffected when the interaction time in the PCM does not exceed 0.1 ns, which appears to be achievable with semiconductor PCMs, see [4] and also Chapter 3. The general picture of the dynamics that emerged from earlier studies by numerical simulation is that there are windows where the PCF laser has stable periodic orbits, also called external cavity modes (ECMs) of the PCF laser [1, 21, 39]. When the laser operates at an ECM, then it produces periodic output at a frequency that is very close to a multiple of the external round-trip frequency. Different ECMs are separated from each other by ‘bubbles’ of more complicated dynamics, including regions of chaos. The transitions through these bubbles can be interpreted as a competition in the laser between neighbouring ECMs, see [21, 39] for more details. The PCF laser is a second laser system that has been studied recently with new methods from bifurcation theory [24, 25, 27, 28], and this is summarized in the next sections. The question is how the laser changes its behaviour as parameters are changed. The main bifurcation parameters are the effective feedback strength 
and the pump current I. Table 5.1 Parameter values of the PCF laser as modelled by Equations (5.23) and (5.24) Symbol

Parameter

Value

 GN
p q
e N0
PCM 

line-width enhancement factor optical gain photon lifetime magnitude of the electron charge electron lifetime transparency electron number external cavity round-trip time constant phase shift at PCM detuning parameter

3.0 1190s−1 14 ps 16 × 10−19 C 2 ns 164 × 108 2/3 ns 0.0 0.0

Bifurcations in the PCF Laser

171

A detailed two-parameter bifurcation diagram near the locking region is presented in the 
 I-plane. Furthermore, it is shown how unstable manifold computations can be used to understand a transition to chaos via the break-up of a torus.

5.5.1 Symmetry of the PCF Laser Equations In contrast to the LK equations, Equations (5.23) and (5.24) are symmetric under the discrete 2 -symmetry given by the transformation E N  → −E N . In other words, the symmetry group is generated by a rotation over  of the complex E-plane, which is physically a phase shift of the electric field by . A consequence of this symmetry is that any attractor (or other invariant set) is either symmetric, or has a symmetric counterpart. This allows for symmetry-breaking and symmetry-restoring bifurcations; see [27, 39]. For example, when a symmetric periodic orbit has a Floquet multiplier +1 then it may undergo a symmetrybreaking bifurcation instead of a saddle-node bifurcation. Furthermore, a symmetric periodic orbit cannot undergo a period-doubling bifurcation [45]. Because the symmetry group of the PCF laser is discrete, solutions are isolated (that is, they do not occur as solution families as for the COF laser). This means that DDE-BIFTOOL can be used without any modification to find and continue equilibria and periodic orbits.

5.5.2 Bifurcation Diagram Near the Locking Region As was mentioned, an important motivation for using PCF is to achieve stable locked laser output. This type of dynamics occurs in the laser locking region. The question arises of what happens at the boundaries of the locking region. The bifurcation diagram in Figure 5.10(a) was obtained by numerical simulation and it shows the normalized value of the inversion Nˆ = N/Nsol − 1 × 103 whenever the intensity P crosses its average value in the positive direction. The locking region is the region without any points, while a small number of points corresponds to a periodic orbit and a large number of points to chaotic dynamics. The dynamics for 
∈ 00000 02953 was computed for increasing 
and that for 
∈ 07487 09004 for decreasing 
. This is due to small hysteresis loops caused by bistabilities at the boundaries of the locking region that are discussed below. Figure 5.10(b) was obtained by numerical continuation with the package DDE-BIFTOOL. Plotted is the real part Ex of the steady states, and  maxEx  − minEx  (offset by the Ex -value of the steady states at the Hopf point) for periodic orbits. Attracting solutions are drawn as solid curves and unstable ones as dashed curves. Figure 5.10(b) gives more information than Figure 5.10(a) in terms of the bifurcations taking place, because also followed are unstable solutions and their bifurcations. In terms of bifurcations of the stable equilibrium, the boundaries of the locking region are given by the saddle-node bifurcation SN at 
≈ 02794 and the Hopf bifurcation H at 
≈ 07487. There is a saddle-focus heteroclinic (or Shilnikov) bifurcation SFH, where the periodic orbit for small values of 
(the continuation of the free-running laser solution) is destroyed. This bifurcation takes place at 
≈ 02953, that is, just after SN for increasing 
, leading to a bistability between the locked state and a large periodic orbit. Similarily, there is a saddle-node bifurcation of limit cycles SL creating two pairs of symmetric periodic solutions at 
≈ 07247, that is, before the Hopf bifurcation H. So at the right boundary of the locking region there is also a small region of bistability, this time between the locked state and a small periodic orbit.

172

Bifurcation Analysis of Lasers with Delay

(a)

6

PD1

4 Nˆ

SFH 2

SL

PD2

0 2 0.0 (b)

0.2

0.4

6

0.6

0.8

1.0

SFH PD2

5

PD1

.

4 SL

3 SN

2 1 0.0

0.2

H

0.4

0.6

0.8

1.0

κτ Figure 5.10 The bifurcation diagram of the PCF laser Equations (5.23) and (5.24) near the locking region, obtained by simulation showing normalised inversion Nˆ versus the feedback strength 
(a), and computed by continuation with DDE-BIFTOOL showing a normalized amplitude versus 
(b). The pump current was set to I = 651 mA. Source: Reproduced from K. Green and B. Krauskopf, Phys. Rev. E 66 (2002) 073207 ©2002 by the American Physical Society.

This stable periodic orbit then undergoes successive period-doublings, leading to a region of chaotic dynamics; see Figure 5.10(a). More details of this transition and phase portraits in E N -space (obtained by computing the 1D unstable manifolds of the saddle equilibria) can be found in [24]. How typical is the transition in Figure 5.10 when one considers different values of the pump current I? Or in other words, what is the bifurcation diagram in the two-dimensional 
 I -plane? As was summarized in Section 5.3, the continuation of the bifurcations SN, H and SFH in the 
 I -plane is possible with DDE-BIFTOOL, while the bifurcations SL, PD and T can at present only be detected but not continued. Figure 5.11 shows the resulting 2D bifurcation diagram in the 
 I-plane near the locking region, where the computations were started from the bifurcation points in Figure 5.10(b). The locking region is bounded on the left by the curve SN of saddle-node bifurcations and on the right by the (supercritical) left-most parts of the curves H1 and H2 of Hopf bifurcations. (The curves H1 and H2 are drawn dark when they are supercritical and lighter when they are subcritical.) The lower boundary of the locking region is given by a curve PF of pitchfork bifurcations. When this curve is crossed, the locked solution emerges from the trivial steady state as a pair of non-symmetric stable steady states, which constitutes the laser threshold in the PCF laser due to the 2 -symmetry of Equations (5.23) and (5.24).

Bifurcations in the PCF Laser

173

0.072

H2

SN

H1 TP SFH

0.068

I

DH

SL

→

PD SNH

0.064

T H2

H1 PF 0.060

0

1

κτ

2

3

Figure 5.11 Bifurcation diagram in 
 I-space near the locking region of the PCF laser Equations (5.23) and (5.24). The arrow on the vertical axis corresponds to the one-parameter continuation for I = 651 mA in Figure 5.10.

Inside the locking region there is a curve SFH, the continuation of the saddle-focus heteroclinic orbit. This curve has been computed with the recent extension to DDE-BITOOL introduced in [53], the heteroclinic orbits along SFH are shown in Figure 5.12. This curve SFH starts on the curve SN at a codimension-two noncentral saddle-node heteroclinic point SNH at 
 I ≈ 0225 006433, and it ends at what is known as a T-point [18], indicated by TP, at 
 I ≈ 2177 0070394. (The point SNH is a saddle-node homoclinic bifurcation [45] when dividing out the 2 -symmetry of Equations (5.23) and (5.24).) Below SNH the saddle-node bifurcation along SN takes place on a periodic orbit. The T-point is at the centre of a spiral of the curve SFH, and at TP the heteroclinic connections between the two non-symmetric steady states are destroyed; see [18]. As can be seen in Figure 5.12(d), this happens because the heteroclinic connection passes closer and closer to the trivial equilibrium, eventually leading to two separate connections, one between a first non-symmetric equilibrium and the trivial equilibrium and a second one from the trivial equilibrium to the symmetric counterpart of the first non-symmetric equilibrium; see [28] for details. In the region above the curve SFH and between SN and H2 there is a bistability between a large periodic orbit and a pair of non-symmetric steady states, in agreement with our earlier discussion of the left locking boundary. The bistability on the right-hand boundary of the locking region mentioned earlier occurs in the small region bounded by the curves H1 and SL. (Note that SL cannot be continued and was instead found at the values of I marked by the crosses.)

174

Bifurcation Analysis of Lasers with Delay

2 (a1)

(a2)

7.655

(a3) 7.655

Ex

N

0

N 7.645

7.645

7.635 –2

0

τ

7.635

48.3

0

τ

2

48.3

Ey

0

2

0 –2

Ex

–2

3 (b1)

(b2)

7.7

(b3) 7.70

Ex 0

N

7.66 N

7.66 7.62

7.62

7.58 –3

0

τ

7.58

5.18

0

τ

Ey

5.18

(c2)

5 (c1)

3

3

0

0 –3 –3

Ex

(c3)

7.8 7.8 Ex

N 7.7

0

N 7.7 7.6

7.6 –5

10

5

5 0

τ

4.68

0

(d1)

τ

4.68

0 Ey

8.65 (d2)

0 –5

–5

Ex

(d3) 8.65

Ex

8.25

0

8.25

N

N

–10

7.85

–20

7.45

7.85 7.45

0

τ

11.4

15 15

0

τ

11.4

0 Ey

0 –15

–15

Ex

Figure 5.12 Heteroclinic orbits along the curve SFH; from (a) to (d) 
 I takes the values 02533 006467, 15835 006987, 21141 007050, and 21767 007039. Panel (d) shows the heteroclinic orbit at the point TP in Figure 5.11. The first and second columns shows the Ex -profile and the N -profile over the time interval that was used in the computation, and the third column is a projection onto E N -space.

Bifurcations in the PCF Laser

175

The point DH at 
 I ≈ 0893 006589 where the curves H1 and H2 cross is a further codimension-two point called a double-Hopf bifurcation. At this point there is an equilibrium with two pairs of complex eigenvalues on the imaginary axis and it is known that the system can bifurcate to a number of invariant objects, including two-dimensional tori, see, for example, [30, 45]. It appears that near DH there is a switch from a period-doubling route to chaos (as observed in Figure 5.10(a)) to a route to chaos via the break-up of a torus. This is evidenced by the points PD and T that were found in the region of 
< 15. As is shown in Figure 5.13, the periodic orbits bifurcating from the Hopf curve H2 undergo further bifurcations in the region of 
> 2. The situation is quite complicated. There is a region with a cusp bounded by saddle-node bifurcations of limit cycles SL. Along the curve SL a pair of periodic orbits bifurcates. Furthermore, there are points PD of period-doubling and T of torus bifurcations that appear to form nice curves. In the regions between the curves SL and T one finds extra stable periodic orbits. However, it is not entirely clear which points belong to which curve and where they start and end up. It is known that the different curves can interact at special points, such as 1:1 and 1:2 resonance points. This is not uncommon in laser systems; see, for example, [67]. The exact structure of these bifurcations will only be revealed when the continuation of the respective curves becomes available.

5.5.3 Bifurcations of ECMs External cavity modes of the PCF laser are genuine periodic orbits of Equations (5.23) and (5.24). There are no analytic expressions for the ECMs or their stability, but they have

0.071 TP

SFH

I 0.069

SL PD T 0.067 2.0

2.5

3.0

κτ Figure 5.13 Bifurcation diagrams in 
 I-space of periodic orbits originating from the Hopf curve H2 in Figure 5.11.

176

Bifurcation Analysis of Lasers with Delay

been found and followed numerically; see [25] for details. Figure 5.14 shows the result of a bifurcation study with DDE-BIFTOOL in the parameter range 
∈ 00 80. Panel (a) shows branches of periodic orbits; plotted is the period T along these branches and boldfaced parts of curve indicate that the periodic orbit is stable. The main branches are of symmetric periodic orbits and the three boldfaced parts correspond to the first three ECMs; the continuation was started from these ECMs, which were found by integration. As a consequence of the 2 -symmetry of Equations (5.23) and (5.24) a symmetric periodic orbit of period T corresponds to periodic output of the laser of period T/2 [37]. Note that ECM1

4

(a) × SL + SB PD + T ×

3 H1 ECM1 2 T H2

ECM2 1

H4

H3 ECM3

2.0 (b) H1 H

2

. 1.5

H3 H4

1.0

0

2

4

κτ

6

8

Figure 5.14 The ECMs of the PCF laser are connected to the steady state via curves of Hopf bifurcations. Panel (a) shows the continuation of periodic orbits, plotted as the period T versus 
; panel (b) shows the continuation of the steady state, plotted as a normalised amplitude versus 
; the pump current is I = 651 mA.

Bifurcations in the PCF Laser

177

to ECM3 correspond to power oscillations in the ranges 1.21–1.40 GHz, 2.51–2.67 GHz and 4.03–4.20 GHz, respectively, which is roughly at integer multiples of the external cavity frequency of c/L of the PCF laser [39]. Secondary branches of non-symmetric periodic orbits bifurcate at symmetry-breaking bifurcations. These branches end up at Hopf bifurcations, denoted H1 to H4 . Figure 5.14(b) shows the branch of the non-symmetric steady states, which is also shown in Figure 5.10(b) in a smaller 
range. On this branch the Hopf bifurcations H1 to H4 are marked. The dashed vertical lines show that the points H1 to H4 are indeed the same points in both panels. Note that the points H1 and H2 are on the curves H1 and H2 in Figure 5.11 at the indicated value of I = 651mA. Figure 5.14 shows that the ECMs are connected to the branch of steady state by branches of periodic orbits. In [11] this phenomenon was confirmed analytically in the singular limit of large T =
e /
p of Equations (5.23) and (5.24). There is in fact a complicated web of other branches of periodic orbits, some of which end up at homoclinic bifurcation points; see [25] for details.

5.5.4 Break-up of a Torus and Crisis Bifurcation There is quite a complex bifurcation scenario of a transition to chaos via the break-up of an invariant torus in the PCF laser; for more details see [27]. It occurs in the range of 
∈ 2300 2800 as is shown in Figure 5.15 with bifurcation diagrams obtained by simulation (a) and by numerical continuation of the locked periodic orbit (b). Figure 5.15(a) shows a stable periodic solution, namely the external cavity mode ECM1 from Figure 5.14(a), that loses its stability at 
≈ 2307. The dynamics on the bifurcating attracting torus is initially quasiperiodic or periodic with extremely high frequency. At 
≈ 2441 there is a saddle-node bifurcation of limit cycles SL where the dynamics on the torus becomes locked, as is evidenced by the five distinct branches in the bifurcation diagram. At 
≈ 2556 the locked periodic orbit itself loses its stability and bifurcates to a torus. This new torus corresponds to quasiperiodic modulations of the laser output, and is shaped like a ‘hose’ wrapped around the remainder of the initial locked solution. At 
≈ 2570 this new torus suddenly is replaced by a much larger chaotic attractor in a bifurcation that does not show any hysteresis. This is quite a spectacular change in the laser’s output: a sudden transition from a periodic oscillation of the power to a chaotic signal. Note that a similar scenario has been found in simulations of the LK equations in [48]. To shed more light on the bifurcations involved, Figure 5.15(b) shows a bifurcation diagram of the period T against 
, obtained by continuation with DDE-BIFTOOL that was started from the stable locked periodic solution. The oval branch S1 of symmetric periodic solutions is born and destroyed at the maximum and minimum values of 
in two saddlenode bifurcations of limit cycles SL. As is indicated by boldfacing, there is a stable locked solution in the interval 
∈ 2441 2556, the right endpoint of which is indeed a torus bifurcation. There are two more bifurcations on S1, namely the two symmetry-breaking bifurcations denoted by SB. These two bifurcation points are connected by the branch N1 of non-symmetric periodic orbits. There are further bifurcations along N1, which we will not discuss in detail here. Note, however, that this branch N1 features a window of stability between the points SL and T in the range 
∈ 2530 2543; for details, see [27].

178

Bifurcation Analysis of Lasers with Delay

(a)

T

SL

2 T

Nˆ 0

–2 2.3 11.1

2.4

T SB

2.8

PD

SL

× SL + SB

2.3

2.7

PD SL T

2.4

SB

N1 PD

2.5

SL

N2

T PD

SL 10.7

2.6

(b) PD T

T 10.9

2.5

S1 SL

2.6

2.7

2.8

κτ Figure 5.15 The bifurcation diagram of the PCF laser Equations (5.23) and (5.24) near the break-up of a torus, obtained by simulation showing normalized inversion Nˆ versus the feedback strength 
(a), and computed by continuation with DDE-BIFTOOL showing the period T versus 
(b). The pump current was set to I = 651 mA. Source: Reproduced from K. Green and B. Krauskopf, Physica D, 173(1–2) (2002) 114–129 ©2002 by Elsevier Science.

While the bifurcation diagrams in Figure 5.15 give a clear indication that one is dealing with a transition to chaos via the break-up of a torus, it is crucial to look at what happens to the torus itself when one wants to understand the exact nature of the sudden transition to a large chaotic attractor. To this end Figure 5.16 shows the 1D unstable manifolds of the saddle periodic orbit in the locking region (which corresponds to the lower part of the branch S1 in Figure 5.15(b)). This orbit intersects a suitable plane  ≡ N = 7620 × 108 in four clusters of five points, of which one cluster corresponding to one of the four intersections of the torus with  is shown. Each of the five points comes with its history of length
and is a fixed point of the Poincaré return map defined by . The 1D unstable manifolds of these five points, which are indicated by crosses + in Figure 5.16, were computed with the method explained in Section 5.3.3. It is important to remember that shown is a two-dimensional projection of an infinite-dimensional system; in particular, branches of 1D unstable manifolds are allowed to intersect each other. Figure 5.16 reveals that the torus is smooth immediately after locking (b) and resembles the unlocked torus (a), as is to be expected from general theory. Indeed, all branches of the

Bifurcations in the PCF Laser

179

(a)

(b)

(c)

(d)

(e)

(f)

Ey

Ey

Ex

Ex

Ex

Figure 5.16 Break-up of the torus of the PCF laser (see text for details); from (a) to (f) 
takes the values 2.400, 2.445, 2.450, 2.480, 2.500 and 2.531; the region shown is E ∈ −300 300 × 140 300 and I = 651 mA. Source: Reproduced from K. Green and B. Krauskopf, Physica D, 173 (1–2), 114–129 (2002) ©2002 by Elsevier Science.

manifolds connect smoothly at the attracting periodic points (×). At 
≈ 24502 the torus loses its smoothness because the branches now start to spiral into the attractors (c). This is due to them developing two complex conjugate Floquet multipliers, as was confirmed with DDE-BIFTOOL. In terms of the laser’s dynamics, this corresponds to additional damped oscillations as the laser settles down to periodic output. As 
is increased further, the, 1D unstable manifolds make longer and longer excursions before converging to the attractors, becoming increasingly folded and stretched in the process; see panels (d) and (e). This means that there are increasingly complicated transients in the system. Finally, for 
= 2531 the 1D unstable manifolds cover a large part of the upper part of the E-plane (f). At 
≈ 2570 there is a sudden transition to chaos with a discontinuous change in the size or shape of the attractor. This is indicative of a crisis bifurcation [22, 23, 52]. At the crisis bifurcation there is a rearrangement of stable and unstable manifolds of suitable saddle points. Figures 5.17(a) and (b) show for one (of the five) saddle points one branch of the 1D unstable manifold just before and just after the crisis bifurcation. For 
= 2569 (a) the branch ends up, after a long transient, at an attracting invariant circle, the intersection of the attracting hose-like torus with . Just after the crisis bifurcation at 
= 2571 (b) the 1D unstable manifold accumulates on a large chaotic attractor. Notice that the initial piece of the manifold that was computed does not change much. In Figure 5.17(c) the attracting invariant circle that exists before the crisis bifurcation is overlayed (in black) on top of the chaotic attractor (in grey) that exists after the crisis bifurcation. This shows that one is indeed dealing with a crisis bifurcation, in which the small hose-like torus suddenly and discontinuously changes in size to become a much larger chaotic attractor.

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Bifurcation Analysis of Lasers with Delay

Ey

(a) Ex

(b) Ex

(c) Ex

Figure 5.17 Crisis bifurcation responsible for a sudden transition to chaos in the PCF laser (see text for details); the region shown is E ∈ −95 105 × 235 275 and I = 651 mA. Source: Reproduced from K. Green and B. Krauskopf, Physica D, 173(1–2), 114–129 (2002) ©2002 by Elsevier Science.

5.6 CONCLUSION Bifurcation analysis is a powerful tool to unravel the dynamics and bifurcations of lasers with delay as modelled by delay differential equations. In particular, transitions to chaos can now be studied in unprecedented detail. This has been demonstrated here with the review of recent work on the COF and PCF lasers. While this work consists largely of theoretical studies of the respective rate equations, there is good reason to believe that bifurcation analysis will be increasingly helpful in the explanation and prediction of experimental results. A first example of a qualitative interpretation of an experiment of a laser with delay (COF in the short cavity regime) can be found in [35]. It is an important challenge for future work to provide qualitative and quantitative connections between theoretical results on the dynamics of lasers with delay and the physics of actual laser devices in experiments. This chapter is an attempt to sketch the present state-of-the-art of a young and developing field. Recent results not discussed in detail here can be found in, for example, [12, 54, 55, 65]. At present more work on the bifurcation analysis of a number of laser systems featuring delay is in progress, including lasers with COF and PCF, and mutually coupled lasers. The list of publications in the field is likely to grow quite rapidly in the coming years. Honesty demands to mention again that bifurcation theory for DDEs and the associated numerical methods are not straightforward to learn. To help the interested reader, this chapter has summarized the relevant theory on DDEs from the point of view of applying it to lasers with delay. Furthermore, the numerical methods were explained in more detail than appears to be necessary to the reader who is mainly interested in the results of their application to laser systems. While it may be useful to skip over the more technical sections in a first reading, this chapter also aims at providing a self-contained review of the underlying theory, that can act as a possible entry point to the even more technical mathematical literature. Regardless of the background of the reader, the author hopes that this chapter has managed to convey the possibilities and the excitement of the bifurcation analysis of lasers with delay!

ACKNOWLEDGEMENTS My sincere thanks go to Kirk Green for his input and many valuable suggestions on a draft of this chapter, to Vivi Rottsch¨afer for collaborating on the computations in Sections 5.4.2 and 5.4.3, and to Matthias Wolfrum and Bart Haegeman for providing electronic originals

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of Figures 5.5 and Figures 5.6–5.7, respectively. Furthermore I thank my other colleagues with whom I had the pleasure of working on the dynamics and bifurcations of delay lasers over the years. Finally, I gratefully acknowledge the support of my research by an Advanced Research Fellowship grant from the Engineering and Physical Sciences Research Council (EPSRC).

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6 Chaos Synchronization Siva Sivaprakasam and Cristina Masoller Ottieri

6.1 INTRODUCTION Chaos is one of the most appealing topics for researchers from various subject backgrounds having non-linear dynamics as a common interest. This is due, primarily, to its significance as a tool which one can use to make sense of a complex world. Examples of chaos include weather patterns, biological systems, fluid turbulence, mobile communications, traffic, population dynamics, and astrophysics – to name just a few. Synchronized chaotic systems have received a lot of attention due to their potential use for secure communications. Synchronized semiconductor lasers are especially attractive due to their potential use in all-optical communication systems. In this book, chaos generation in semiconductor lasers with optical feedback is discussed in Chapter 2 as one of the regimes of operation of these systems. The use of chaotic semiconductor lasers in all-optical secure communication systems is discussed in Chapter 9. Historically, synchronization in non-chaotic systems was realized before that of chaotic systems. Christiaan Huygens first observed anti-phase synchronization of two pendulum clocks, with a common frame, in 1665. This was the subject of some of the earliest deliberations of the Royal Society. Huygens found that the pendulum clocks swung at exactly the same frequency and 180° out of phase. When he disturbed one pendulum the antiphase state was restored within half an hour and pendulum clocks remained synchronized indefinitely, thereafter, if left undisturbed. He found that synchronization did not occur when the clocks were separated beyond a certain distance, or oscillated in mutually perpendicular planes. Huygens deduced that the crucial interaction came from very small movements of the common frame supporting the two clocks. He also provided a physical explanation for how the frame motion set up the anti-phase motion [1]. The synchronization of chaotic systems is a subject with a long history that has been a ‘hot topic’ in the past ten years. Fujisaka and Yamada [2–5] did early work on synchronization of chaotic systems, but it was not until the work of Pecora and Carroll [6, 7] that the subject received a significant amount of attention. The term ‘chaotic synchronization’ refers to a Unlocking Dynamical Diversity Edited by Deborah M. Kane and K. Alan Shore © 2005 John Wiley & Sons, Ltd

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variety of phenomena in which chaotic systems adjust a given property of their motion to a common behaviour due to a coupling or to a driving force [8]. The systems might be identical or different, the coupling might be unidirectional (master–slave or drive–response coupling) or bi-directional (mutual coupling) and the driving force might be deterministic or stochastic. Intrinsic interest in synchronization phenomena from a dynamical systems and chaos theory point of view, and practical applications in secure communications [9], have spurred a wide range of research studies. Two systems are coupled unidirectionally if the dynamics of one system (called master or drive) affects the dynamics of the other (called slave or response), while the dynamics of the slave does not affect the dynamics of the master. The fact that two unidirectionally coupled chaotic systems can be used in a secure communication scheme was first shown by Cuomo and Oppenheim [10, 11], who built a circuit version of the Lorenz equations and showed the possibility of using this system to transmit a small speech signal. While some years ago the word ‘chaos’ had a negative connotation in applied research, now a days researchers are studying techniques for taking advantage of chaotic dynamics instead of trying to avoid it. The use of chaos synchronization in secure communication systems is a good example of this modern point of view. Chaos synchronization includes a number of different phenomena such as (1) completely identical oscillations in the coupled systems (complete synchronization); (2) frequency locking ( frequency synchronization); (3) phase locking between the two systems while their amplitudes remain uncorrelated ( phase synchronization); (4) the output of one system is correlated with the output of the other system but it lags in time (lag synchronization); and (5) the outputs of the two systems are functionally related (generalized synchronization). The latter usually occurs in real experiments of chaos synchronization, in which no complete synchronization is observed due to parameter mismatches. In each case, one can separate cases of full and partial synchronization and, thus, several measures have been proposed to determine the degree of synchronization. (these measures include correlation function, mutual information, synchronization diagrams, etc.) A recent and complete review of chaotic synchronization phenomena can be found in [8]. In the field of lasers, the first experimental demonstrations of synchronization of two chaotic lasers were performed using Nd: YAG [12] and CO2 lasers [13, 14]. In the scheme used by Roy and Thornburg [12], one or both of the lasers were driven into chaos by periodic modulation of their pump beams. Sugawara et al. [13] demonstrated synchronization of two chaotic passive Q-switched lasers by modulating the saturable absorber in the cavity of one laser with the output of the other laser. Liu et al. [14] used two passive, optically-coupled, Q-switched lasers, and the amount of coupling used induced transitions from unsynchronized periodic oscillations to synchronized chaotic ones. Colet and Roy [15] were the first to propose a scheme for encoding data within a chaotic carrier from a loss-modulated solidstate laser, and the first experimental demonstration of chaotic communication with an optical system (an erbium-doped fibre ring laser) was done in 1998 by Van Wiggeren and Roy [16–18]. In the past decade research from various groups, both theoretical and experimental, has focused on understanding synchronization phenomena in unidirectionally coupled lasers, its potential for use in secure communications, and its dependence on various laser parameters [19–103]. Synchronization phenomena in spatially extended systems (such as broad-area nonlinear optical cavities, laser arrays, 1D chains and 2D lattices of coupled dynamical systems, neural

Synchronization of Unidirectionally Coupled Semiconductor Lasers

187

networks, etc.), have also received a lot of attention. Local and global coupling have been considered and two different synchronization regimes have been found [8]. Either all elements of the ensemble display the same behaviour for any initial conditions (full synchronization) or the ensemble splits into groups of mutually synchronized elements (cluster synchronization). In the field of lasers, White and Moloney [94] proposed a practical application of synchronization of spatio-temporal chaos. They demonstrated the successful multiplexing of random bit sequences between transmitter/receiver semiconductor lasers modelled by coupled nonlinear partial differential equations. They reported the synchronization of spatiotemporal chaos through a single scalar complex variable (the total electric field). The separate messages were encoded into individual channels (the longitudinal modes) by injecting weak signals at the relevant mode frequencies, thus enabling the transmission of multiple messages through a single channel. Another application of spatiotemporal chaos synchronization was proposed by García-Ojalvo and Roy [95, 96], who suggested a model system (a broad-area nonlinear optical ring cavity that exhibits spatiotemporal chaos) that allows parallelism of information transfer using an optical chaotic carrier waveform. The synchronization of laser arrays has also been the object of extensive investigation [97–101]. In this chapter we review synchronization of chaotic semiconductor lasers and discuss various types of synchronization in both, unidirectionally and mutually coupled lasers. In Section 6.2 we present numerical and experimental results of synchronization on unidirectional or master–slave configuration. In Section 6.3 we discuss results of the synchronization of mutually coupled semiconductor lasers. In Section 6.4 we present our conclusions and personal perspectives on the research field, at present and in the near future.

6.2 SYNCHRONIZATION OF UNIDIRECTIONALLY COUPLED SEMICONDUCTOR LASERS Two lasers are coupled unidirectionally if the dynamics of one laser (called master or transmitter) influences the dynamics of the other (called slave or receiver), and the master laser is isolated from the slave laser, so that the dynamics of the slave does not affect the dynamics of the master. A typical experimental setup for unidirectionally coupled externalcavity lasers is shown in Figure 6.1. Mirasso, Colet and García-Fernández [20] were the first to show theoretically that chaotic semiconductor lasers with optical feedback, unidirectionally coupled, can be synchronized and used in encoded communications systems. In their scheme a message was encoded in the chaotic output of the master laser, and it was transmitted to the slave laser using an optical fibre. The slave laser was assumed to operate under similar conditions as the master laser. The message could be decoded by comparing the chaotic input and output of the slave laser. Since then, a lot of theoretical and experimental studies have focused on understanding the synchronization properties of unidirectionally coupled chaotic semiconductor lasers [21, 24, 25, 30–32, 34, 36–38, 40–48, 50, 52, 53, 57–61, 63–70, 72, 74–80, 82–86, 88–93]. Synchronization of power dropouts in the low-frequency fluctuations regime

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NDF

BS1

M1

ML

OI1 PD1

BS2

CRO CA BS4

NDF

M2

SL BS3

OI2

PD2 CRO

Figure 6.1 Experimental set-up used in [37]: ML – Master Laser; SL – Slave Laser; BS1, BS2, BS3, BS4 – Beam Splitters; PD1, PD2 – Photo Detectors; OI – Optical Isolators; M1, M2 – Mirrors; NDF – Neutral Density Filter; CA – Coupling Attenuator; CRO – Digital Oscilloscope.

(LFF, see Chapter 2) was first shown numerically by Ahlers et al. [34]. The theoretical studies were based on simple rate equations for the complex electric fields and carrier densities in the lasers, which are extensions of the well-known Lang–Kobayashi equations [102] (see Chapter 2). These equations are: 
 • 1 + i
1 Em = (6.1) Em t + m Em t −  exp −im  + Dm m t Gm t − 2 pm •

Nm =

•

Es =

Jm Nm t − − Gm t
Em t
2 e nm


1 + i
1 Es t + s Es t −  exp −is  Gs t − 2 ps  + Em t − c  exp −i m c + t + Ds s t •

Ns =

Js Ns t − − Gs t
Es t
2 e ns

(6.2)

(6.3)

(6.4)

Synchronization of Unidirectionally Coupled Semiconductor Lasers

189

Here, the indices m and s refer to the master and slave lasers respectively. Em  Es are the slowly varying complex fields, and Nm  Ns are the normalized carrier densities. The equations are written in the reference frame where the complex optical fields of the lasers are given by Em t exp im t, Es t exp is t, where m , s are the optical frequencies of the solitary lasers. The external optical feedback in each laser is described by the parameters m and s , which are the feedback levels of the master and slave lasers respectively and  is the delay time in the external cavity. The term Em t − c  exp −i m c + t in the right-hand-side of Equation (6.3) accounts for the light injected from the master laser to the slave laser, is the injection rate, c is the time for light to travel from the master laser to the slave laser and  = m − s is the frequency detuning between the lasers. m t and s t are independent complex Gaussian white noises that represent the effects of spontaneous emission and spontaneous recombination. D measures the noise intensity. The other parameters p is the photon lifetime,
is the linewidth enhancement  are as follows: 

factor, G = N 1 +
E
2 is the optical gain, is the gain saturation coefficient,  is the phase accumulation after one round-trip in the external cavity, J is the injection current density, e is the electronic charge and n is the carrier lifetime. This model does not include multiple reflections in the external cavity, and therefore it is valid for weak feedback levels. It is assumed that the mirrors are positioned such that the external cavity length is the same for both lasers. It is also assumed that the optical field does not experience any distortion during its propagation from the master to the slave laser. In experiments of synchronization of chaotic external-cavity lasers, two configurations for the slave laser have been considered. In the first configuration, the slave laser is subjected to both, optical feedback from an external reflector and optical injection from the master laser (we will refer to this as closed-loop configuration). In the second configuration, the slave laser is subjected only to optical injection from the master laser (we will refer to this as open-loop configuration). In the theoretical model, the open-loop configuration corresponds to setting s = 0. Sivaprakasam and Shore [37] were the first to demonstrate experimentally the synchronization of two chaotic external-cavity semiconductor lasers. The experimental arrangement is shown schematically in Figure 6.1. Two laser diodes (APL-830 of linewidth 200 MHZ) were driven by ultra-low noise current sources (ILX-Lightwave LDX-3620) and their temperature was controlled using thermo-electric controllers (ILX-Lightwave LDT5412) to a precision of 0.01 K. Both lasers were subjected to optical feedback from external mirrors (M1 and M2) and the feedback strength was controlled using a continuously variable neutral density filter (NDF1). The length of the external cavity is 25 cm in both the cases. The optical isolators (OFR-IO-5-NIR-HP) ensure the lasers were free from unwanted back reflection. The typical isolation is –41 dB. Isolator (OI1) ensured the master laser was isolated from the slave laser. The coupling attenuator (CA) enables the percentage of master power fed into slave laser to be controlled. PD1 and PD2 were two identical fast photodetectors (EG&G - FFD040B) with a response time of 2.5 ns. The output of the master laser is coupled to the photodetector (PD1) by the beamsplitters BS1 and BS2. Beamsplitter (BS3) acts as the coupling element between the master and slave. Beamsplitter (BS4) couples the slave output to the photodetector (PD2). The photodetector output signals are stored in a digital storage oscilloscope (Fluke Combiscope PM3394B, 200MHz) and then acquired by a PC. The master laser and slave lasers were driven into chaos by application of appropriate optical feedback from the external cavity mirrors. The optical feedback for the master and

190

Chaos Synchronization

slave led to a reduction in their threshold currents of 7.0% and 9.7% respectively. (The freerunning threshold current Ith  values were 57 mA and 53 mA respectively for the master and slave laser.) The temperature and current were adjusted to ensure that the master (2666  C, 1136Ith ) and slave (2792  C, 1035Ith ) operate at the same wavelength. A small percentage (3%) of the master optical output is injected into the slave and this led to synchronization of the slave to the master. In Figure 6.2(a), the time evolution of the master and slave lasers is shown. The upper trace (master laser) is shifted vertically for clarity. The slave laser actually lags behind the master laser by a time equal to the time for light to travel between the two lasers c , but this cannot be noticed in Figure 6.2(a) due to the different time scales: c is of the order of nanoseconds, and the time-resolution is of the order of microseconds. The synchronization can conveniently be illustrated by using a synchronization diagram, which consists of a plot of the master intensity vs. the slave intensity. If the two lasers were perfectly synchronized, the synchronization diagram will be a straight line with a positive gradient.

(a)

Output Power (arb. units)

0.070 0.065 0.060 0.055 0.050 0.045 0.040 0.0

0.1

0.2

0.3

0.4

Time (ms) (b)

Slave Output (arb. units)

0.050 0.048 0.046 0.044 0.042 0.040 0.056

0.060

0.064

0.068

0.072

Master output (arb. units)

Figure 6.2 (a) Time traces of the master (upper) and slave (lower) laser output, the time traces are shifted vertically for clarity, (b) the corresponding synchronization diagram [37], (c) Synchronized LFFs for master (ML) and slave (SL) lasers, (d) one shot of enlarged LFF waveforms corresponding to (c) [41].

Synchronization of Unidirectionally Coupled Semiconductor Lasers

191

Output Power (a.u.)

(c)

ML 1000 Time (ns)

2000

1000 Time (ns)

2000

Output Power (a.u.)

0

SL 0

Output Power [a.u.]

(d)

ML 0

100

200

300

200

300

Output Power [a.u.]

Time (ns)

SL 0

100 Time (ns)

Figure 6.2 (continued )

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Chaos Synchronization

Less than perfect synchronization leads to a broadening of the diagram. Figure 6.2(b) shows the synchronization diagram corresponding to Figure 6.2(a). The experimental synchronization of LFF power dropouts was later done by Ohtsubo and co-workers [41] and the time traces are shown in Figure 6.2. In Figure 6.2(c) the upper and lower time trace correspond to the master and slave laser output respectively. The enlarged waveforms of laser outputs are shown in Figure 6.2(d). The frequency detuning between the master and slave lasers is a critical parameter affecting synchronization. The frequency detuning is defined as the master laser frequency minus the slave laser frequency,  = m − s . In the experiments of [104] the detuning was varied by modifying the slave laser bias current (a change in the bias current by 1 mA shifts the frequency by 1 GHz, and positive or negative detuning arise depending upon whether the slave laser bias current was increased or decreased). Synchronization plots for four different values of detuning are shown in Figure 6.3. Figure 6.3(a) shows degradation in synchronization for a positive detuning ( = 6 GHz). Good synchronization is shown in Figure 6.3(b), where the gradient is positive. As the detuning is made negative, the synchronization plot starts to branch with a portion of negative gradient as shown in Figure 6.3(c) ( = −3 GHz). As the detuning is further increased to −6 GHz the newly developed branch with a negative gradient dominates and the branch with a positive gradient disappears. This is shown in Figure 6.3(d). The appearance of a negative gradient in the synchronization diagram was termed ‘inverse synchronization’. The output intensities of the lasers, for two different detunings (corresponding to normal and inverse synchronization), are shown in Figure 6.4. These time traces serve as direct time-domain evidence of the inverse behaviour seen in synchronization diagrams of Figure 6.3. Inverse synchronization was modelled theoretically in [104] based on non-resonant coupling between the master and slave lasers. It was argued that when the injected light is detuned from the slave laser mode wavelength, the coupling into the slave lasing mode is very weak and, therefore, the effects of non-resonant amplification dominate. Even though the injected light is non-resonant with the slave laser cavity mode, it is still amplified through the stimulated emission process. The effect of the non-resonant amplification of light (that does not couple into the slave laser mode) is a reduction of the carriers in the device. Thus, fewer carriers take part in the gain process for the slave laser, reducing its optical output. Wedekind and Parlitz [105] also observed the regime of inverse synchronization (they refer to it as anti-synchronization) and proposed a different theoretical explanation, based on polarization effects due to the Faraday isolator used for the unidirectional coupling. Thus, more theoretical and experimental work is needed in order to clarify the physical mechanisms underlying inverse synchronization phenomena. From a theoretical point of view the synchronization of two identical lasers in a closed-loop configuration leads to two qualitatively different synchronization regimes [66, 74, 77, 86]. One regime occurs when the master and slave lasers are subjected to the same optical feedback strength:  m = s 

(6.5)

and the slave laser is subjected to strong enough optical injection (roughly speaking, an injection strength corresponding to the injection locking region of a laser under CW

Synchronization of Unidirectionally Coupled Semiconductor Lasers

193

optical injection). In this regime the slave laser output at time t synchronizes with the injected field at time t, which, taking into account the travel time from the master to the slave, c , is the field of the master laser at time t−c . The intensities are related by: Is t = a Im t − t

(6.6)

where t = c and a is a proportionality constant. Analytic conditions for the occurrence of this type of synchronization were given by Revuelta et al. [75]. A different regime (found numerically by Ahlers et al. [34]) occurs when: (1) the master and slave lasers have the same amount of total external injection, and (2) the master injection strength is strong enough. Condition (1) implies that the master feedback strength is equal to the sum of the slave feedback strength and the optical coupling strength:  m = s +

(6.7)

Condition (2) implies that the dynamics induced by the optical injection from the master laser dominates the dynamics induced by the external cavity of the slave laser (roughly speaking, > s ). In this regime the output of the slave laser synchronizes with the output of the master laser as: Is t = Im t − t

(6.8)

t = c − 

(6.9)

where:

A qualitative experimental verification of the condition (6.7) was done in [50]. The output power levels of the master and slave lasers at different current levels were measured when the lasers were synchronized in their output intensity. The feedback power to the transmitter (Pfm ) and receiver (Pfs ) lasers and the transmitter power coupled to the receiver laser (Pc ) were measured. The quantities Pfm , Pfs , and Pc are proportional to m , s , and respectively. A plot of Pfm vs. (Pfs + Pc ) is shown in Figure 6.5. The linear dependence between Pfm and (Pfs + Pc ) provides a qualitative the verification of Equation (6.7). The experiment was done using very slow photodetectors, hence it was not possible to verify the lag time of Equation (6.9). When c < , t = c −  < 0 and the slave laser intensity anticipates the master laser intensity. Masoller [55] has shown that this is a particular case of the anticipating synchronization regime discovered by Voss [103]. Figure 6.6 displays numerical solutions corresponding to anticipated synchronization [55]. In Figures 6.6(a)–(c), the master laser operates in the LFF regime: the intensity suddenly drops toward zero and then recovers gradually, only to drop out again after a random time interval. The intensity dropouts are actually the envelope of a series of fast, picosecond pulses. In Figure 6.6(a) the laser parameters are identical, and the noise level is zero. For large enough coupling strength, , and for an adequate slave feedback level (s = m − ), the slave laser anticipates by  − c (5 ns) the output of the master laser. The dotted line

194

Chaos Synchronization

7.0 16

(a) 6.5

(b)

14 12

6.0

10 5.5

8 6

Slave laser output (arb. units)

5.0

4 4.5

2

4.0

5

10

15

20

0

5

10

15

20

25

18 14

(c)

(d)

16

12

14

10

12 10

8 8 6

6

4

4 2

2 0

1

2

3

0

0

1

2

3

4

Master laser output (arb. units)

Figure 6.3 Synchronization plots for transmitter-receiver detuning of (a) +6 GHz, (b) 0 GHz, (c) −3 GHz and (d) −6 GHz and with the transmitter laser operating at a low-frequency fluctuation regime [104].

plots Im t +  − c  − Is t, and proves that after a transient time, the slave laser is perfectly synchronized to the future chaotic output of the master laser. The physical origin of this behaviour can be understood by looking at the simultaneous turn-on of the master and slave lasers, Figure 6.6(b). The lasers emit the first intensity pulse at approximately the same time (because they are identical, and the initial conditions are that both lasers are off at t = 0). The master laser emits a train of pulses at the relaxationoscillation period, before relaxing to the solitary steady state. This train of pulses interferes with the steady state, when it returns from the external mirror, at time  after the emission of the first pulse. A fraction of the master intensity is transmitted to the slave laser, and the train of pulses interferes with the slave laser emission, at time c after the emission of the first pulse. Therefore, if the coupling is strong enough, the slave laser will respond in

Synchronization of Unidirectionally Coupled Semiconductor Lasers

195

(b)

Output (arb. units)

(a)

0

200

400

600

800

1000

0

200

400

600

800

1000

Time (ns)

Figure 6.4 Time traces of the master (upper traces) and slave (lower traces) lasers for detunings (a) −6 GHz and (b) 0 GHz. The master laser operates in the low-frequency fluctuation regime [104].

a similar manner as the master laser, only that it will do it at time c , while the master laser will do it at time . The simultaneous turn-on of the lasers allows an understanding of the mechanism of anticipated synchronization, but the lasers also synchronize if they are turned-on independently. In Figure 6.6(c) there are small parameter mismatches between the lasers, and a small amount of noise. The lasers are not perfectly synchronized, and because Js is slightly lower than Jm  Im t +  − c  − Is t, fluctuates about a mean value different from zero. Bursts of desynchronization are observed when Im drops to zero. Figure 6.6(d) displays solutions corresponding to anticipated synchronization when the master laser operates in the coherence collapse regime (which occurs for larger injection current, see Chapter 2). In [55] the degree of synchronization and the lag time between the lasers were quantified with the similarity function, defined as: 

Im t + o  − Is t 2 S 0  =  Im2  Is2  2

 (6.10)

If Im (t) and Is (t) are independent time series, S0  = 0 for all 0 . If the lasers are synchronized such that Is (t) = Im t +  − c ), S − 0  = 0. Figure 6.7(a) shows the similarity function when there is perfect anticipated synchronization. S(0 ) presents a sharp minimum at 0 =  − c . There are also additional minima at 0 = n( − c ) (with n integer), which arise from

196

Chaos Synchronization

0.50 0.45 0.40 Pfm (mW)

0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 0.0

0.5

1.0

1.5

2.0

2.5

Pfs + Pc (mW)

Figure 6.5 Pfm vs (Pfs + Pc  under conditions of synchronization [50]. Pfm is the feedback power to the master laser, Pfs is the feedback power to the slave laser and Pc is the power coupled to the slave laser. The size of the squares correspond to a power variation of about 0.01 mW. Maintenance of the synchronization requires current fluctuations less than 0.25 mA, which would produce a change of approximately 0.01mW in the feedback power.

time correlations of the master intensity. When there is noise and parameter mismatches, Figure 6.7(b), S(0 ) has a similar shape, but the minima are shallower. The dependence of the lag time with the delay time given by Equation (6.9) was verified experimentally by Liu et al. [82]. The experimental arrangement used is shown in Figure 6.8(a). The laser diodes are two similar single-mode DFB laser diodes (NELNLK1555) driven with a low-noise high precision injection current, and temperature controlled. At the injection current Ib = 114 mA (1.5 Ith ) the laser wavelength is 1537.17 nm with a linewidth of about 4 MHz. The intensity variations of the laser outputs are detected by 6 GHz bandwidth photodetectors (New Focus 1514LF) and observed on digital oscilloscope (Tektronix TDS694C) with a 3 GHz bandwidth and a 10 Gps (giga bits per second) sampling rate, as well as on a RF spectrum analyzer (Advantest R#267) with an 8 GHz bandwidth. The light output from the right facet (high-reflection (HR) coated, reflection >95%) of the master laser is fed back to the left facet (antireflection (AR) coated, reflection C1 . It can be observed that the dark region where good anticipating synchronization occurs is narrower than in the isochronous case. The two regimes of synchronization also exhibit different sensitivity to the noise level. This sensitivity is illustrated in Figure 6.11, which displays the synchronization diagrams: master

Synchronization of Unidirectionally Coupled Semiconductor Lasers

201

output intensity, Im , vs. slave output intensity, Is . In Figure 6.11(a) the feedback levels are s = m and Im is shifted in time by −c ; In Figure 6.11(b) the feedback and injection levels are related by + s = m and Im is shifted in time by  − c . The noise level in both cases is the same, but the points in Figure 6.11(a) are clearly concentrated along a straight line, whereas in Figure 6.11(b) they are more scattered owing to noise-induced large bursts of desynchronization. It has been shown numerically that by varying certain parameters a transition from anticipated to isochronous synchronization can be observed [77, 86]. An example of such transition is illustrated in Figure 6.12, which shows the master laser and slave laser intensities (averaged in time to simulate the typical bandwidth of the detectors used in experiments). Figure 6.12(a) displays Im (t–c ), while Figures 6.12(b), 6.12(c) and 6.12(d) display Is (t) for different injection rates, . In Figure 6.12(b) = m and Is t anticipates the injected intensity Im t − c  by an anticipation time  (= 1 ns). In Figure 6.12(c) is slightly larger

C1

(a)

1.0

50

0.9

45

0.8 40

0.7 0.6

Injection Rate η (ns–1)

35

0.5 30

0.4

25

0.3 0.2

20

0.1 0.0

15

–0.1 10

–0.2 –0.3

5

–0.4 0 –30

–20

–10

0

10

20

30

Frequency Detuning fm – fs (GHz)

Figure 6.9 Comparison between isochronous synchronization and CW injection-locking when the slave laser is an external-cavity laser [86]. (a) Correlation coefficient C1 as a function of the frequency detuning and the injection rate when the slave is an external cavity laser subjected to chaotic injection from the master laser. m = s = 10 ns−1 . (b) Correlation coefficient C1 as a function of the frequency detuning and the optical coupling strength, when the slave is an external cavity laser subjected to CW injection from the master laser. s = 10 ns−1 

202

Chaos Synchronization

C1

(b)

1.0

50

0.9

45

0.8 40

0.7 0.6

Injection Rate η (ns–1)

35

0.5 30

0.4 0.3

25

0.2 20

0.1 0.0

15

–0.1 10

–0.2 –0.3

5

–0.4 0 –30

–20

–10 0 10 Frequency Detuning fm – fs (GHz)

20

30

Figure 6.9 (continued )

than m and isochronous synchronization occurs. The time traces shown in Figures 6.12(a) and 6.12(c) are most of the time equal, the main difference being a less pronounced dropout in the intensity of the slave laser. In Figure 6.12(d) is slightly less than m and synchronization disappears: the time traces shown in Figures 6.12(a) and 6.12(d) are completely different. Unidirectionally coupled external cavity semiconductor lasers operating on the LFF regime can also exhibit intermittent synchronization. Wallace et al. [52] have shown that the outputs of master and slave lasers are synchronized in their sudden dropouts, but remain uncorrelated in the process of power recovery. This is because during the recovery process the intensity of the master laser is very low and the optical injection from the master to the slave laser is not strong enough to govern the dynamics of the slave laser. Synchronizing several lasers becomes important when the lasers are used in practical communication systems. Sivaprakasam and Shore experimentally demonstrated the possibility of synchronizing three chaotic external cavity diode lasers in a cascade scheme [72]. The experimental set-up is shown in Figure 6.13. The master laser and both the slave lasers are driven to LFF by application of appropriate feedback from the external cavity. The effective external reflectivities for the master, slave-1 and slave-2 are 16 × 10−3 , 50 × 10−4 , and 40 × 10−3 respectively. The free running threshold current (Ith ) values are 56 mA, 49 mA and 55 mA respectively for master, slave-1 and slave-2 laser. The

Synchronization of Unidirectionally Coupled Semiconductor Lasers

203

temperature and current are adjusted to ensure that the master (2566  C, 117Ith ) and slave-1 (2635  C, 120Ith ) operate at the same wavelength. A small percentage (15%) of the master optical output power is fed to the slave-1, which leads to the synchronization of slave-1 to the master. 6.5 percentage of the slave-1 output power is fed to the slave-2 laser. The temperature and operating current of the slave-2 (2696  C, 115Ith ) are adjusted so as to obtain synchronization between the slave-1 and slave-2 lasers. The process of synchronizing slave-2 and slave-1 laser does not affect the synchronization between the slave-1 and master laser. Hence, all the three lasers are synchronized. The time evolution of the intensities of the three lasers is shown in Figure 6.14: trace (a), (b) and (c) are the output intensities of master, slave (1) and slave (2) lasers respectively. It is noticeable from this figure that the output of slave laser (1) lags behind master laser, and slave laser (2) lags behind the slave laser (1).

(a)

C1

50

1.0 0.9

45 0.8 40

0.7 0.6

Injection Rate η (ns–1)

35

0.5 30

0.4 0.3

25

0.2 20

0.1 0.0

15

–0.1 10

–0.2 –0.3

5

–0.4 0 –30

–20

–10 0 10 Frequency Detuning fm – fs (GHz)

20

30

Figure 6.10 Comparison between isochronous synchronization, CW injection-locking, and anticipated synchronization when the slave laser is a solitary laser s = 0 [86]. (a) Correlation coefficient C1 as a function of the frequency detuning and the injection rate, when the slave laser is subjected to chaotic injection from the master laser. m = 10 ns−1 . (b) Correlation coefficient C1 as a function of the frequency detuning and the optical coupling strength, when the slave laser is subjected to CW injection from the master laser. (c) Correlation coefficient C2 as a function of the frequency detuning and the optical coupling strength, when the slave laser is subjected to chaotic injection from the master laser.

204

Chaos Synchronization

(b)

C1

50

1.0 0.9

45

0.8 40

0.7 0.6

Injection Rate η (ns–1)

35

0.5 30

0.4

25

0.3 0.2

20

0.1 0.0

15

–0.1 10

–0.2

5

–0.3

0 –30

–0.4 –20

–10

0

10

20

30

Frequency Detuning fm – fs (GHz) C2

(c) 50

1.0 0.9

45

0.8 40

0.7 0.6

Injection Rate η (ns–1)

35

0.5 30

0.4

25

0.3 0.2

20

0.1 0.0

15

–0.1 10

–0.2 –0.3

5

–0.4 0

–30

–20

–10

0

10

Frequency Detuning fm – fs (GHz)

Figure 6.10 (continued )

20

30

Synchronization of Unidirectionally Coupled Semiconductor Lasers

205

Figure 6.11 Effect of noise in the isochronous and anticipated synchronization regimes [74]. (a) Im t − c ) is plotted against Is t. The parameters are m = s = 20 ns−1 , = 50 ns−1 ,  = 0, D = 002 ns−1 . C1 = 099 (b) Im t +  − c  is plotted against Is t. The parameters are m = 20 ns−1 , s = 0, = 20 ns−1 ,  = 0, D = 002 ns−1 . C2 = 094

Im(t – τc)

(a) 1 0.08 0.06 (b) Is(t)

1.00 0.08 0.06 (c) Is(t)

1.00 0.08 0.06 (d) Is(t)

1.00 0.08 0.06 0

5

10

15

20 t (ns)

25

30

35

40

Figure 6.12 Transitions between anticipated and isochronous synchronization are illustrated by plotting the time traces of the time-averaged master and slave intensities [86]. (a) Intensity of the master laser (lagged c in time) for m = 10 ns−1 . Intensity of the slave laser for (b) = m; (c) = 12 ns−1 ; (d) = 9 ns−1 .

206

Chaos Synchronization

NDF1

BS1 ML

M1 OI1 OI2 PD1 BS2 CA1

NDF2 BS4

SL1

M2 BS3

OI3 OI4 PD2 BS5 CA2

BS7

SL2

M3 BS6 OI5

NDF3

PD3

Figure 6.13 Schematic diagram of the experiment of cascade synchronization in Ref. [72]: ML – Master laser; SL1 – Slave laser-1; SL2 – Slave laser-2; M1– M3 – external cavity mirrors; NDF’s – neutral density filters, BS1–BS7 – Beam splitters, PD1–PD3 – Photodetectors, OI1–OI5 – Optical isolators, CA1 and CA2 – Coupling attenuator.

6.3 SYNCHRONIZATION OF MUTUALLY COUPLED SEMICONDUCTOR LASERS A lot of research has focused on mutually coupled lasers, either subjected only to mutual injection, or subjected to mutual injection and optical feedback from an external mirror [106–117]. When the lasers are subjected only to their mutual optical injection and have dissimilar intensities, their coupling strength may be asymmetric, and in this case, Hohl et al. [106] found that the coupled lasers exhibit a form of synchronization which is characterized by small oscillations in one laser, and large oscillations in the other. Heil et al. [109] have found that two mutually coupled lasers may exhibit subnanosecond synchronized chaotic dynamics, in which a leading laser synchronizes its lagging counterpart.

Synchronization of Mutually Coupled Semiconductor Lasers

207

(a)

Optical output (arb. units)

(b)

(c)

0

100

200 300 Time (nS)

400

500

Figure 6.14 Cascade synchronization [72]. Time-series plots of (a) master laser, (b) slave laser-1 and (c) slave laser-2. The time traces have been shifted in y-axis for clarity.

The time lag between the intensities is equal to the time for light to travel from one laser to the other. This effect was interpreted as a spontaneous symmetry-breaking phenomenon. Fujino and Ohtsubo also observed the synchronization of the chaotic outputs from mutually injected lasers [111]. It was shown that the synchronization mechanism was not based on complete chaos synchronization but on injection-locking phenomena (which the authors refer to as generalized chaos synchronization). A recent theoretical study on synchronization of mutually coupled multimode semiconductor lasers shows that the two lasers can be synchronized within each laser-cavity mode, while the synchronization across different cavity modes is significantly weaker [112].

208

Chaos Synchronization

The rate equations commonly used in the literature to model mutually coupled single-mode lasers are 
• 1 + j
1 = t − (6.13) E12 t + 12 E21 t − c  G E 12 12 2 p12 •

N12 =

 2 J12 N12 t − G12 t E12 t − e n12

(6.14)

where E1 and E2 are the complex optical fields of the two lasers, N1 and N2 are the carrier densities in the two lasers, and the other parameters have the same meaning as in the previous section. These equations were derived in [116], in the limit of small mutual coupling, starting from Maxwell equations, supplemented with adequate boundary conditions. In the experiments of [110] a different mutual coupling configuration was used. The experimental set-up is shown in Figure 6.15. Two single-mode (side mode suppression ratio equal to −20 dB), Fabry–Perot laser diodes emitting at 830 nm were used, but only one is subjected to optical feedback from an external reflector (this laser will be called externalcavity laser, or laser 1, while the laser without external feedback will be called solitary laser, or laser 2). The laser operating temperatures were stabilized using thermo-electric elements

NDF

BS1

M1

ML

BS2

OI1 PD1

OSA OI2 CA

M2

SL BS3

OI3

PD2

Figure 6.15 Experimental setup used in Ref. [110]. ML – external-cavity laser, SL – solitary laser, BS1–3, Beam Splitters, NDF – Neutral density filter, OI1–3, Optical Isolators, M1–2, Mirrors, CA – Coupling Attenuator, PD1–2, Photodetectors, OSA – Optical spectrum analyzer.

Synchronization of Mutually Coupled Semiconductor Lasers

209

Output (arb. units)

and controllers to a precision of 0.01 K. The output of each laser was coupled to a fast photodetector (Newport - AD-70xr) and monitored using a digital oscilloscope (LeCroyLC564A). The optical isolators ensure that no feedback from the photodetectors reaches the laser diodes. Laser 1 (laser‘2) was biased at 1.08 (1.04) times the free-running threshold. The time of flight between the two lasers was 3.5 ns. Laser 1 was operated in an external cavity configuration with external reflectivity 175 × 10−3 , which drives the laser into the LFF regime (when the external mirror was removed, no power dropouts were observed). Beamsplitters BS1 and BS3 coupled the transmitter laser output to the receiver laser. The coupling attenuator (CA) was used to control the amount of light coupled between the lasers (the percentage of laser 1 output power reaching laser 2 and vice versa is 0.14% throughout the experiment). It is noted that, when the optical feedback from the external cavity mirror is not applied, the lasers did not show any LFF dynamics. Figure 6.16 shows the time traces of the intensities of laser 1 and laser 2. It can be seen that close to 250 ns and 450 ns the output of laser 2 drops and recovers ahead of the output of laser 1. Hence, laser 2 is leading laser 1 by an ‘anticipation time’, A = 35 ns, which was measured with the digital oscilloscope. The measurement of anticipation time was confirmed by studying the quality of synchronization between the laser 1 and 2 outputs. The laser 2 output was plotted against laser 1 output so as to obtain the synchronization plot. The synchronization plot was then least square fitted to a straight line and the slope (m) and its variation (m) was calculated. The inverse of the variation (1/m) represents the quality of the synchronization (SQ ). Good synchronization would be indicated by m = 1 and low variation (m) implying high synchronization quality. On the other hand, poor synchronization would give a relatively large variation (m) and hence a low synchronization

200

300

400

500

Time (ns)

Figure 6.16 Time traces of the external-cavity laser (upper) and solitary laser (lower) outputs [110]. The vertical lines identify that the solitary laser is ahead of the external-cavity laser. The external cavity round trip time is 13.5 ns.

210

Chaos Synchronization

τ = 13.5 ns τ = 6.7 ns

Synchronization Quality (SQ) (Normalised units)

1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65

0

1

2

3

4

5

Transmitter laser time shift (τs), (ns)

Figure 6.17 Synchronization quality as a function of the time-shift in external-cavity laser output for two external cavity round trip times,  = 135 ns (
) and 6.7 ns (×) [110].

quality. Figure 6.17 shows the dependence of the (normalized) synchronization quality (SQ ) on the laser 1 time shift S , where S is the time by which the laser 1 output is shifted relative to the laser 2 output. It is shown for two external-cavity delay times. It can be seen that the synchronization quality, in both cases, shows a sharp maximum at S = 35 ns, which is the flight time from one laser to the other. This indicates that laser 2 is leading laser 1 by 3.5 ns irrespective of the external-cavity round-trip time. Thus, in mutual coupling a regime of anticipating synchronization was found in which the anticipation time does not depend on the external-cavity round-trip time of the laser 1. Anticipating synchronization depends on the operating wavelength of both lasers. Once the lasers are synchronized, both the lasers emit at a common wavelength denoted by sys and it is found that sys < 1 (where 1 is the central operating wavelength of laser 1). The operating temperature of one of the lasers is modified by a fraction of a degree such sys < 1 , and hence rendering laser 2 to lag behind laser 1 [114].

6.4 CONCLUSION The aim of this chapter was to give an overview of the field of synchronization of chaotic semiconductor lasers, from the theoretical and experimental points of view. In Section 6.2 we discussed synchronization of unidirectionally coupled lasers (in master–slave or driver-response configuration). We considered the case in which the slave laser has its own feedback (closed-loop scheme) and the case in which it is a solitary laser (openloop scheme). We discussed the two well-known synchronization regimes: isochronous, which has been interpreted as an injection-locking-type phenomenon that requires the frequency detuning between the lasers to be within a certain injection-locking range, and anticipated synchronization, which requires perfect matching of the optical frequencies and all the internal laser parameters. In terms of the well-known synchronization regimes of chaotic systems, anticipated synchronization is identified with complete synchronization, and isochronous synchronization is identified with generalized synchronization. We reviewed

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recent, exciting developments such as inverse synchronization, which is a phenomenon not yet fully understood. In Section 6.3 we discussed synchronization of mutually coupled lasers, in a face-to-face configuration, and in the case when one of the lasers is also subjected to its own feedback from an external mirror. We have chosen to discuss only the synchronization of edge-emitting lasers and not of vertical-cavity surface-emitting lasers, because, in spite of the fact that there are several theoretical studies that indicate the possibility of synchronizing VCSELs, no experimental demonstration has been done to the best of our knowledge. We hope that future developments of technology towards secure all-optical communications based on chaotic synchronized systems will clarify relevant issues such as the physical mechanisms underlying complete and generalized synchronizations.

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7 Laser Interferometry Guido Giuliani and Silvano Donati

7.1 INTRODUCTION Optical feedback in semiconductor lasers (SLs) leads to very interesting complex dynamics, rich in unexpected new phenomena that have been discussed in earlier chapters. The study of optical feedback on SLs has led to a huge amount of knowledge on these devices. It also provides the basis for new practical applications, such as chaotic signal transmission, coherent echo detection, and measurement of physical quantities through self–mixing interferometry. The latter is the subject of this chapter. Laser interferometry [1] is a well-established technique, widely used in industrial and laboratory environments to measure displacement, velocity (of both solid targets and fluids), vibration and distance. Applications have flourished in the fields of mechanical metrology, machine-tools control, profilometry, velocimetry and vibrometry. Usual techniques rely on an external interferometer, i.e. an optical transducer made of lens, prisms and mirrors, which is read out using laser light or white light. This is the case for the well-known Michelson and Mach-Zehnder interferometers. More recently, a new interferometric sensing technique has appeared [2], based on the optical mixing of the field within the laser cavity with the weak field back-reflected or back-scattered by a remote target, thus giving rise to a modulation of both the amplitude and the frequency of the lasing field. This technique is called self-mixing, feedback or inducedmodulation interferometry. In this approach, the laser source acts as a sensitive detector for the pathlength 2kL (where k = 2
/, and L is target distance) travelled by the light to the target and back. It exploits the so-called injection-detection [3]. First demonstrations of this principle used gas lasers to detect the Doppler shift caused by a moving remote reflector [2]. Key developments were the experimental demonstration of the first complete self-mixing interferometer/vibrometer [4], and the use of a semiconductor laser as the source/detector [5]. These developments paved the way for a number of practical applications that utilize the well-known advantages of SLs.

Unlocking Dynamical Diversity Edited by Deborah M. Kane and K. Alan Shore © 2005 John Wiley & Sons, Ltd

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Remote sensing applications based on the self-mixing effect in low-cost commercial Fabry–Perot (FP) SLs have been appearing in the scientific literature since 1986 [5–6]. The feasibility of velocity, distance and displacement measurements has been demonstrated [5–7]. Advantages of the semiconductor laser self–mixing sensing scheme are listed below: 1 There is no need for an optical interferometer to be set up external to the laser source. This results in a very simple and self-aligned set-up, made up of the SL, a focusing lens and the target under measure. 2 There may be no need for an additional photodetector, because the interferometric signal can mostly be derived from the monitor photodiode already available in the SL package. 3 The sensitivity of the scheme is very high, because it realizes a coherent detection of the returned field, in the quantum detection regime [1]. 4 Sub-nanometer sensitivity in pathlength measurement is easily achieved. 5 Operation is readily extended to targets with rough diffusive surfaces. 6 The information is carried by the laser beam. Thus, it can be picked up everywhere along the optical path, and also at the remote target location. From a theoretical point of view, the analysis of feedback interferometry relies on the well-known rate–equation model developed by Lang and Kobayashi [8], that has been fully discussed in Chapter 2. Although optical feedback in SLs has been studied for more than twenty years from both theoretical and experimental viewpoints, it is only in recent years that experimental set-ups oriented to interferometric sensing and suitable theoretical analyses have been devised [7, 9]. In fact, earlier and pioneering works focused mostly on classifications of different feedback regimes, optical spectrum modification, and RF noise characteristics of back-injected SLs [10, 11, 12, 13]. For sensing applications the analysis concentrates on the low frequency modulation of the power emitted by the SL, as a function of the back-reflected field phase. Effort has concentrated on identifying optical feedback regimes which are practically useful for interferometry, and the thorough characterization of the interferometric signal.

7.2 LASER DIODE FEEDBACK INTERFEROMETRY: THEORY AND BASIC EXPERIMENTS In this section it will be shown how the system composed of a semiconductor laser and a remote reflector or diffuser can be regarded as an interferometric measuring apparatus [14]. The interferometric signal can be extracted from the small modulation that is superimposed on the emitted power, which is a periodic function of the back-reflected field phase. A detailed analysis of signal characteristics, as a function of optical feedback strength, target distance, and semiconductor laser parameters will be provided.

7.2.1 Interferometric Signal The basic interferometric scheme is depicted in Figure 7.1, that refers to a Fabry–Perot (FP) semiconductor laser. This configuration is equivalent to a three-mirror cavity, where P0 is the emitted power, Pr = P0 /A is the power back-diffused or back-reflected by the remote target, and A > 1 is the power attenuation of the external cavity, given by A = 1/Rext , where Rext is power reflectivity of the remote target.

Laser Diode Feedback Interferometry

219

L P0 P0

MONITOR PD

REMOTE TARGET

Pr = P0 /A

ηPD SL

Rext

R2

Figure 7.1 Basic self-mixing configuration based on a Fabry–Perot SL.

FM sin[2kL(t)] E0

Er · ei2kL(t)

AM cos[2kL(t)]

Figure 7.2 Simple interpretation for injection-detection performed by a laser source. E0 : lasing field; Er : back-reflected or back-scattered field.

A simple interpretation for injection-detection can be devised referring to Figure 7.2: the small back-reflected field phasor Er re-enters the laser cavity and it adds to the lasing field phasor E0 [3]. The optical phase of Er is t = 2kLt, where k = 2
/ and Lt is the distance from the SL to the remote target. The lasing field amplitude and frequency are modulated by the term  = 2kL, namely, the FM term is sin(2kL), and the AM term is cos2kL. This detection scheme very closely resembles the well-known homodyne detection at radio frequencies. From the two quadrature signals, AM and FM, the interferometric phase  = 2kL can be in principle retrieved without ambiguity. Thus, the incremental measurement of target displacement from the initial position L1 to position L2 is possible from: L2 − L1 = 2 − 1 /2k [4]. The simple treatment illustrated above applies conveniently to gas and solid-state lasers; however, the analysis of a Fabry–Perot semiconductor laser feedback interferometer requires some amendments. The first reason for this is that in SLs the frequency modulation term (FM) cannot be detected by heterodyning, because of the large linewidth of semiconductor laser sources, which is typically between 1 and 30 MHz. This implies that in the self–mixing configuration only one interferometric channel is available in practice, as opposed to a He–Ne source used for injection detection [4], or to conventional external interferometers that provide two interferometric signals which are in quadrature. The second reason more deeply involves SL physics, because the intrinsic nonlinear nature of the semiconductor active medium (in which both the optical gain and refractive index are coupled to the effective optical power in the cavity through their dependence on injected carrier density) leads to an amplitude modulation term that can be very different from the usual cosine function. An analysis of the optical feedback interferometer can be performed by using the standard Lang and Kobayashi equations [8]. Let us first summarize the effects of the optical delayed feedback from the remote target. Back-reflected light interferes with the light already present in the laser cavity and, depending on the delay and on the phase of the back-reflected light, the SL threshold condition is varied. Thus, the emitted power changes as the pump current is held constant while, for example, the target is being displaced. The change in threshold

220

Laser Interferometry

implies a change in the actual SL carrier density; as a consequence, also the wavelength emitted by the SL subject to backreflections is slightly varied [11]. The following simple expression for the power emitted by the SL subject to feedback can be then assumed: P = P0 1 + m · F

(7.1)

where P0 is the power emitted by the unperturbed SL, m is a modulation index and F is a periodic function of the interferometric phase  = 2kL of period 2
, where k = 2
/. The modulation index m and the shape of the function F depend on the so-called feedback parameter C (after [11]): √  · L · 1 + 2 C= (7.2) l·n where is the SL linewidth enhancement factor, L is the remote target distance, l is the SL cavity length, n is the cavity refractive index and the factor  is given by:


1−R 1−R  = √ · √ 2 = Rext √ 2 R2 R2 A

(7.3)

where A is total optical power attenuation in the external cavity, Rext is the power reflectivity of the remote target and R2 is the SL output facet power reflectivity (see Figure 7.1). The coefficient ≤ 1 was introduced in [11] to account for possible mode overlap mismatch between the back-reflected light and the lasing mode; typical values are 0.1– 0.8. The value of the feedback parameter C depends on the amount of feedback and, interestingly, also on target distance L. The C parameter is of great importance in sensing applications because it allows discrimination between different feedback regimes, which in turn lead to different values for the modulation coefficient m, and also different shapes of the interferometric signal waveform F. A variation of the interferometric phase  can be trivially generated by a displacement L of the target, obtaining:  = 2
/ · L, and also by a modulation of the SL wavelength (e.g. through injected current modulation), yielding:  = −4
L/2  · . Let us now present the analytical description based on the steady-state solution of the Lang–Kobayashi equations [7, 9, 11]. It should be emphasized that a steady state solution analysis is necessary in order to gain insight into the operation of the feedback interferometer. Previous studies have focused their attention on the coherent detection of the Doppler-shifted light back-scattered by a target in motion, and disregarded the possibility of operation as an interferometer. The Lang–Kobayashi equations describe the dynamics of a single mode semiconductor laser subject to delayed optical feedback from a distant reflector [8]:   dEt 1 1  = Et + Et −  cos 0 + t − t −  Gn Nt − N0  − dt 2 p c  Et −  dt 1 = Gn Nt − Nth  − sin 0 + t − t −  dt 2 L Et dNt Nt = Ggen − − Gn Nt − N0 E 2 t dt e

(7.4)

Laser Diode Feedback Interferometry

221

The meaning of the symbols used is as follows: Et t

0 Gn Nt N0 Nth p e c Ggen 

slowly varying envelope of the electric field, normalized so that E 2 t corresponds to photon density S m−3 ; electric field phase; angular oscillation frequency of the unperturbed SL; modal gain coefficient (typical value: Gn = 8 × 10−13 m3 s−1 ); spatially averaged carrier density; carrier density at transparency (typical value: N0 = 14 × 1024 m−3 ); carrier density at threshold (typical value: Nth = 23 × 1024 m−3 ); photon lifetime within the cavity; a typical value for a Fabry–Perot SL is: p = 16 ps, and the following equation holds: 1/ p = Gn × Nth − N0 ; carrier lifetime (typical value: e = 2 ns); SL cavity round–trip time c = 2 · l · n/c (typical value: c = 8 ps); external cavity round–trip time, given by = 2L/c electrical pumping term, given by Ggen = J/qd with J injection current density,  internal quantum efficiency, d active layer thickness, q electron charge; linewidth enhancement factor fraction of the back-reflected field that efficiently couples with the lasing mode, given by (7.3)

The scope of the analysis is to find steady state solutions for the system composed by the semiconductor laser and remote reflector, in terms of the emitted power P, as given by Equation (7.1). Stationary solutions are found by letting the time derivative of the first and third equations of the set (7.4) equal zero. Since the instantaneous optical frequency is given by t = 0 + dt/dt, the term dt/dt represents the instantaneous frequency deviation, and we can take t =  F − 0 t where F = F  ) is the instantaneous angular oscillation frequency of the SL with feedback. By substitution into the first equation of the set (7.4), we obtain: NF   = Nth −

2 cos F   ·  Gn c

(7.5)

which shows that optical feedback causes carrier density modulation within the SL. Substitution of Equation (7.5) into the phase equation yields:

F   = 0 −

  · cos F   ·  + sin F   ·  c

(7.6)

that corresponds to the zero round-trip phase condition [10]. Using the feedback parameter defined in Equation (7.2), we obtain the two equivalent expressions: C  · cos F   ·  + sin F   · 

F   · = 0 · − √ 1 + 2

F   · = 0 · − C · sin F   · + a tan 

(7.7a) (7.7b)

222

Laser Interferometry

Equations (7.7a,b) represent the core of the steady state behaviour of the SL with feedback, since they give solutions for the instantaneous angular frequency of the emitted light, which in turn determines the emitted power through Equation (7.5). The importance of the feedback parameter C lies in the fact that when C < 1 Equations (7.7) have only one solution, while for C > 1 multiple solutions exist [7, 10, 11], the system can be bistable or multi-stable, and a stability analysis is required to determine effective stable operating conditions of the SL [11]. To determine the shape of the power modulation function F, we substitute Equation (7.5) into the carrier density equation, obtaining:

P ∝ EF2   =

 Ggen − NF   e Gn NF   − N0 

 =

p e

 Ggen e − Nth + 1−

2 cos F   ·  Gn c

(7.8)

2 p cos F   ·  c

where EF indicates that the solution is calculated in the presence of feedback. The above expression can be linearized under the assumption   c /2 p (i.e.   25 for the values used, yielding a corresponding value for C in excess of 102 for a target distance L = 01m, obtaining:     2 p p 2 Ggen e − Nth + cos F    1 + cos F    P ∝ EF2   ≈ e Gn c c      2 p p  Ggen e − Nth 1 + cos F    (7.9) ≈ e c The above assumption requires that the optical feedback strength be sufficiently weak. As the power emitted in the absence of feedback is given by   P0 ∝ E02 = p Ggen − Nth / e

(7.10)

referring to Equation (7.1) we can determine the modulation index m and the function F : m=

2 p 2 p 1 − R2 2 p
2 · p · c 1−R = ·√ · √ = · Rext √ 2 = C · √ c c c R2 R2 L · 1 + 2 A 

 0 · F = cos F  · 2
c

(7.11)

 (7.12)

where the dependence of F on the interferometric phase  has been made explicit using the relation = )/(2
c). To find the shape of the self-mixing interferometric waveform F one has to first solve one of Equations (7.7) to find the dependence of the angular frequency of the SL with feedback on the interferometric phase , and then use Equation (7.12). Figure 7.3 reports numerically calculated solutions for F /0 =  F / 0 =  F − 0 / 0 and F as a function of the interferometric phase  for different values of the feedback parameter C. For C = 01 both the instantaneous frequency and the emitted power are represented by a sine function, similar to the behaviour of gas lasers [4]. For C = 07 the two

Laser Diode Feedback Interferometry

223

functions are distorted. For C = 1 the distortion becomes even more evident, and periodically both functions exhibit points with infinite slope. The condition C = 1 is a critical point for the system, because for C > 1 a bifurcation appears, and for certain values of the phase the system has three solutions. This is illustrated by the curves with C = 3: within each period there are two points with infinite slope. A stability analysis is performed to determine which branches of the obtained solutions are stable [11] and to identify nonphysical operating points for the system. In [11] it is also demonstrated analytically that for C > 46, a new bifurcation

(a)

(b)

Figure 7.3 Calculated dependence of the function F and the quantity F /0 (relative frequency deviation of the SL with feedback) on the back-reflected field phase  for different values of the C parameter. Linewidth enhancement factor = 5 L = 1 m (a) C = 01 modulation index m = 188 × 10−5 ; (b) C = 07 m = 131×10−4 ; (c) C = 1 m = 188×10−4 ; (d) C = 3, m = 564×10−4 ; (e) C = 6 m = 112 × 10−3 . In (d), the segment XY is unstable. In (d) and (e) HYST represents the hysteresis width.

224

Laser Interferometry

(c)

φHYST Y′

X

W X′ Y

(d) 4

×10–7

c=6 α=5

∆ν/ν0

2 0 –2 –4

0

5

φHYST

10

1.0 Y′′

F(Φ)

0.5

20

15

20

X X′

0.0

Y′

–0.5 –1.0

15

Y 0

5

X′′

10 Φ (rad) (e)

Figure 7.3 (continued)

Laser Diode Feedback Interferometry

225

takes place, and for certain values of the phase five solutions exist, as illustrated by the example with C = 6 in Figure 7.3. The shape of the function F has some dependence on the value of linewidth enhancement factor , but overall behaviour is not affected, especially for what concerns the hysteresis width. It should be noticed that, as illustrated in previous chapters, the allowed values for the actual optical emission frequency of the SL with feedback correspond to longitudinal modes of the external cavity. It is interesting to determine the actual waveforms of the interferometric signal for a practical case, in which the remote target is put into vibration and the interferometric phase changes sinusoidally with time. Simulated waveforms are reported in Figure 7.4, and experimental traces are reported in Figure 7.5. The dependence of signal amplitude and shape on external cavity optical power attenuation A can be appreciated both in the simulated and experimental traces. The signal has a periodicity of 2
in phase, i.e. a complete fringe appears as the target is displaced by an amount /2. Examples with C ≤ 1 (Figures 7.3(a),(b), 7.4(b), 7.5(a)) show a continuous signal. As the value of C approaches unity, the signal gets asymmetric, and discrimination of the direction of motion of the target is possible. When C = 1 (Figures 7.3(c), 7.4(c), 7.5(b)), the signal has a clearly asymmetry and for a specific phase value it has infinite slope. Cases with C > 1 deserve more attention. For example, for C = 3 (see Figure 7.3(d)) the branch between points X and Y is unstable. Suppose the system is in point W: as the interferometric phase is increased, the point moves along the curve up to point X, where it suddenly jumps down to point X that is located on the adjacent stable branch. Conversely, if the system is in X and the phase is decreased, point Y is reached, and subsequently an upper jump to point Y occurs. Thus, when C > 1 the interferometric signal is discontinuous, exhibiting step-like transitions each time a 2
phase variation occurs (corresponding to /2 target displacement); this is illustrated in Figures 7.4(d) and 7.5(c). For C = 6 (see Figure 7.3(e)), when the system reaches point X, there exist two possible evolutions, because the system can jump down to two distinct stable points, namely X and X . Similarly, when point Y is reached with decreasing phase, the system can jump up to points Y or Y . It has been observed that experimental results obtained for feedback regimes with C > 46 strongly depend on the type of SL under test. Figure 7.6 illustrates such phenomena, and reports experimental traces obtained from different SLs. For the first SL (Figure 7.6(a), Hitachi HL 7851 g, Fabry–Perot, emitting at 780 nm), the jumps always occur to the nearest branch (i.e., from X to X and from Y to Y ); for the second SL (Figure 7.6(b), Mitsubishi ML2701, Fabry–Perot, emitting at 850 nm), the jumps can randomly occurr to points X or X (and Y and Y ), and some fringes actually correspond to 4
phase variation, i.e. a target displacement corresponding to . It is at present unknown which SL parameters determine one behaviour or the other. It is likely that this phenomenon is strictly related to SL dynamics, because it is influenced by the stability of the different external cavity modes. The occurrence of the signal with step-like discontinuities (C > 1) is very interesting for interferometric applications, because the sign of these transitions is related to the target direction of motion, which can be reconstructed without ambiguity using only a single interferometric channel (see Section 7.3.1). As the generation of the self-mixing signal involves a change in both SL carrier and photon density, the characteristic time scale of the fast switching is comparable to the relaxation oscillation frequency, i.e. it is in the subnanosecond range.

226

Laser Interferometry

(a) Displacement (µ m)

2 1 0 –1 –2

(b) Self-Mixing Signal (µ A)

2

0 ×10–3

0.5

1.0

1.5

C = 0.01 A = 5.23e + 010

1 0 –1 –2

0

0.5

1.0

1.5

1.0

1.5

1.0

1.5

Time (ms)

(c) Self-Mixing Signal (µ A)

C = 1 A = 5.23e + 006 0.2 0.1 0.0 –0.1 –0.2

0

0.5

(d) Self-Mixing Signal (µ A)

C = 3 A = 5.81e + 005 0.4 0.2 0.0 –0.2 –0.4

0

0.5 Time (ms)

Figure 7.4 Calculated self-mixing signal for a sine target displacement. (a) target displacement. (b–d) self-mixing signal. (b) C = 0001 (c) C = 1 (d) C = 3 Total optical attenuation A is also shown on for each graph.

Self-Mixing Signal [2 mV/div]

(a) Time [200 µs/div]

Self-Mixing Self-Mixing Signal [20 mV/div] Signal [10 mV/div]

227

Target Displacement [1.2 µm/div]

Laser Diode Feedback Interferometry

(b)

(c) Time [200 µs/div]

Figure 7.5 Experimental self-mixing signal waveforms obtained for different values of the feedback parameter C, obtained by adjusting the total optical attenuation A (i.e., the effective target reflectivity). Upper-left trace: loudspeaker drive signal at 657 Hz, 1.2 m/div. (a) C  1, A ≈ 2 × 108 ; (b) C ≈ 1 A ≈ 8 × 106 ; (c) C > 1 A ≈ 4 × 105 . Source: Adapted from [14].

(a)

(b)

Figure 7.6 Illustration of the behaviour of different SLs for high feedback level (C > 46). Upper trace: target (loudspeaker) drive signal; lower trace: self-mixing signal. (a) SL Hitachi HL 7851 g, Fabry–Perot, emitting at 780 nm. (b) SL Mitsubishi ML2701, Fabry–Perot, emitting at 850 nm. For SL (b), some fringes correspond to a target displacement of  instead of /2, in a random fashion.

228

Laser Interferometry

As shown in Figures 7.3, 7.4 and 7.5, for C > 1 hysteresis appears in the self-mixing signal, as a consequence of the bistability of the system. The amount of hysteresis can be properly represented by the width HYST of the bistability region measured in terms of the phase (see Figure 7.3). An analytical expression for the hysteresis width can be determined from Equation (7.7) by finding the phase values for which the derivative of the angular frequency of the SL with feedback d /d is infinite [11]. It is obtained as: HYST = 2 ×

√

C 2 − 1 + arccos

   1 − −
rad C

(7.13)

Interestingly, the hysteresis width only depends on the feedback coefficient C, and it does not depend on SL parameters, such as the linewidth enhancement factor . The hysteresis width is plotted in Figure 7.7 versus C, showing an almost linear dependence for C > 3 Considering the characteristics of the interferometric signal, different feedback regimes can be classified as follows: • C  1 – very weak feedback regime. The function F is a cosine (as in gas lasers and conventional √ external interferometers), and the modulation index m is inversely proportional to A, consistent with Equation (7.11). • 01 < C < 1 weak feedback regime. The function F gets distorted,√showing a non-symmetrical shape; the modulation index m is inversely proportional to A.

0.6

16

12

0.4 φ HYST [rad]

Hysteresis Width - φ HYST [rad]

14

0.5

0.3 0.2 0.1

10

0.0 1.0

1.1

1.2

1.3

1.4

1.5

C

8 6 4 2 0 1

2

3

4

5 6 7 Feedback Coefficient - C

8

9

10

Figure 7.7 Calculated dependence of the hysterisis width HYST of the self-mixing waveform on the feedback parameter C for C > 1.

Laser Diode Feedback Interferometry

229

• 1 < C < 46 – moderate feedback regime. The interferometric signal becomes sawtooth√ like and exhibits hysteresis. The modulation index m still increases for decreasing A, but experimental observations show that the peak-to-peak signal amplitude saturates [7]. • C > 46 – strong feedback regime. The function F may become five-valued. It has been experimentally demonstrated that not all the specimens of FP SLs tested remain in the self-mixing regime. In some cases the SL enters the mode-hopping regime and interferometric measurements are not possible due to the instability of the signal. Operating conditions that can be useful for interferometric measurements lie within Regimes I and II of the classification made by Tkach and Chraplyvy [12]. In fact, considering a conventional Fabry–Perot or DFB SL, with a remote target placed at L ≈ 1 m, the boundary between Regimes I and II is trivially found for C = 1, and the boundary between Regimes II and III corresponds to C ≈ 30. The asymmetry in the shape of function F when C > 05 allows a clear discrimination of the target direction of motion. This is a crucial point, because this peculiar characteristic of self-mixing in SLs makes non-ambiguous interferometric displacement measurement possible even using a single interferometric channel (e.g. the amplitude modulation). There is no need to have two quadrature signals, as is the case for conventional interferometry. Also, the modulation coefficient m is such that its value, in the moderate feedback regime, is in the range 0.005–0.05. This is adequate for any kind of subsequent signal processing. The expression for the feedback coefficient C (7.2) shows that in order to achieve operation in the moderate feedback regime, the optical feedback strength (described by  is obviously crucial, but an important role is also played by the ratio L/l between the remote target distance and laser cavity length.

7.2.2 Semiconductor Laser Types and Signal-to-Noise Ratio For general applications of the self-mixing sensing configuration it is interesting to analyze the dependence of signal amplitude on the parameters l (SL cavity length), L (target distance), consider different types of SLs. As the and remote target reflectivity Rext , and also to √ modulation index m is inversely proportional to A, it is useful to evaluate the minimum allowable target reflectivity Rext that ensures a good signal-to-noise ratio (SNR) and to determine the limiting operation conditions for sensing applications. Starting from the expression for the modulation coefficient m given by Equation (7.11), the peak amplitude of the photocurrent self-mixing signal SI can be calculated as follows: SI = pd

2 p

p c1 − R2  q q q P0 · m = pd P0 · = pd P0 · √ √ h h c h l · n · A · R2

(7.14)

where pd = co · q is the product of monitor photodiode coupling efficiency co and quantum efficiency q . Expression (7.14) is valid as long as the condition C ≤ 1 is fulfilled, because experimental observations report a saturation of the signal amplitude for larger optical feedback values [7]. Figure 7.8 sketches the self-mixing signal amplitude as a function of external cavity attenuation for varying l and L, also accounting for the latter feature. For the same emitted power, long-cavity lasers yield a smaller signal and they enter the moderate feedback regime (sawtooth-like signal) for a smaller value of external cavity attenuation

230

Laser Interferometry

INCREASING TARGET DISTANCE L

Signal Amplitude (dB)

0 –10 –20

INCREASING LASER CAVITY LENGTH l

–30 –40 –50 –60 1

10–2

10–4

10–6

10–8

10–10

External Mirror Reflectivity - Rext 0

20

40 60 80 Optical Attenuation - A (dB)

100

Figure 7.8 Schematic graph reporting self-mixing signal amplitude as a function of remote target reflectivity Rext and external cavity power attenuation A, showing the effect of varying target distance (thick line) and SL cavity length (thin line). Thick line indicates that the condition C > 1 is met, hence the self-mixing signal is sawtooth-like.

(i.e. larger target reflectivity). Also, it is shown that large values of target distance L allow entry to the moderate feedback regime, even for small values of target reflectivity, with the drawback of a smaller signal amplitude. Figure 7.9 reports calculated self-mixing photocurrent signal amplitude as a function of external cavity attenuation A and target reflectivity Rext for different SL sources. Target distance is L = 1 m, and all lasers have the following common parameters:  = 850 nm P0 = 10 m W (except for the VCSEL, that emits 3 mW), pd = 0016 = 5 = 05 Other parameters are specified for each laser type below. In Figure 7.9 the condition C ≥ 1 is represented by thick curve. • F–P (Fabry–Perot). l = 350 m; p = 16 ps; R2 = 035. • DFB (Distributed Feedback Laser). For these sources an equivalent output mirror reflectivity R2 has been determined based on the value of the DFB · l product, where DFB is the grating coupling factor. Intuitively, a larger DFB · l product corresponds to a grating with a higher reflectivity efficiency, which corresponds to a higher equivalent output mirror reflectivity R2 and decreased sensitivity to feedback. Petermann [10] illustrates how the DFB · l product and the equivalent output mirror reflectivity R2 relate to each other. Here we consider two different DFB lasers, namely: DFB #1. l = 350 m; p = 134 ps; L = 2 R2 equivalent = 025 DFB #2. l = 350 m; p = 225 ps; L = 5 R2 equivalent = 087 • ECL (External Cavity Laser) #1. A Hybrid Distributed Bragg Reflector Laser (H-DBR) is considered, which is made of a SL with an anti-reflection coated facet that is butt-coupled to an optical fibre with a Bragg grating written onto it [15]. The laser output is taken from

Laser Diode Feedback Interferometry

231

Self-Mixing Signal [µ A]

100 10–1

DFB1 DFB2

B = 1 MHz

10–2

ECL1

VCSEL

B = 10 kHz ECL2

10–3

F–P

ECL3

B = 100 Hz

10–4 B = 1 Hz

NOISE LEVEL

10–5 100

10–2

10–4

10–6

10–8

10–10

10–12

10–14

104 106 108 1010 1012 External Cavity Power Attenuation A

1014

Remote Target Reflectivity Rext 100

102

Figure 7.9 Calculated self-mixing signal amplitude as a function of remote target reflectivity Rext and external cavity power attenuation A for different laser sources. Thick lines indicate the regime for which C ≥ 1. Common laser sources parameters:  = 850 nm, P0 = 10 mW (except for the VCSEL, that emits 3 mW), pd = 0016 = 5 = 05 L = 1 m. Other parameters are specified in the main text. Horizontal lines represent noise level for specific values of the measurement bandwidth, reported on the graph (dashed horizontal lines refer to VCSEL). Source: Adapted from [14].

the fibre end, while the output from the uncoated rear facet is coupled to the monitor photodiode. Laser parameters are: l = 001 m; p = 19 ps; R2 = 035 • ECL #2. A different H-DBR laser is considered, with l = 01 m; p = 19 ps; R2 = 035 • ECL #3. This is made of a SL with an anti-reflection coated facet whose emitted light is collimated onto a bulk grating in Littrow configuration. Laser parameters are: l = 01 m; p = 25 ps; R2 = 08 • VCSEL (Vertical Cavity Surface Emitting Laser). The parameters are: l = 12 m p = 14 ps; R2 = 0992 P0 = 3 m W. To assess the attainable SNR, it is assumed that the measurement is limited by semiconductor laser RIN (relative intensity noise), and the photocurrent noise can be expressed as: In2 = 2qpd

q FP B h 0

(7.15)

where B is the measurement bandwidth and F is the excess noise factor with respect to an ideal SL RIN (typ. F = 2). As explained in [10], SL RIN is not degraded, relatively to the so-called feedback Regimes I and II [12] that correspond to useful operating conditions for sensing applications. Calculated noise levels are also plotted in Figure 7.9 for different values of the measurement bandwidth (1 Hz, 100 Hz, 10 kHz, 1 MHz). It is deduced that large values of the external cavity optical attenuations can be tolerated. The high sensitivity

232

Laser Interferometry

0.15

60 SNR (dB)

Self-Mixing Signal Amplitude (mA)

of the self-mixing scheme is achieved thanks to the coherent detection principle on which it is based. From Figure 7.9 it can be noted that the largest self-mixing signal is obtained by F–P lasers, and by DFB lasers with small grating coupling factor. These lasers are best suited for operation in the moderate feedback regime with hysteresis C > 1, because they can tolerate higher attenuation (up to A = 107 ), and this allows proper operation even on low reflectance or rough targets. A good quality self-mixing signal can be obtained from nearly any type of singlelongitudinal mode SL for which the side-mode suppression ratio is larger than, say, 10–15 dB. We successfully tested SLs with Fabry–Perot, DFB, VCSEL, and ECL structures, with emission wavelengths ranging from the visible (635 nm) to the third communication window (1550 nm). Visible F–P lasers are particularly interesting for obvious reasons. However, we noticed that these lasers generally have a poorer side-mode suppression ratio and have the tendency to exhibit mode–hopping as the moderate feedback regime is approached. A thorough characterization of different SL specimen has been carried out to identify those best suited for self-mixing sensing applications. The self-mixing effect has been demonstrated in VCSELs [16], that are attractive for their low cost and potential planar integration into a matrix of sensors. It is interesting to assess the optimal operation condition for the self-mixing detection scheme with respect to injected SL current. This can be deduced from Figure 7.10, that reports calculated self-mixing signal amplitude and SNR when C = 1 for an F–P laser with 20 mA threshold. Just at threshold, the SNR exhibits a maximum; then, it rapidly decreases to a minimum and it increases again with injected current. It could be argued that operation at or just above threshold is very favourable. However, in practice, this operating point might be unsuitable for two reasons. First, achieving shot-noise-limited operation using an op-amp transimpedance amplifier at such reduced power levels requires a very large feedback resistance, thus limiting the available bandwidth. Second, proper self-mixing operation can generally be achieved when the SL operates on a single longitudinal mode or, at least, it exhibits a side–mode suppression larger than 10–15 dB. This condition is in general not

0.10 0.05 0.00 20

30 40 50 Laser Current (mA) (a)

60

55 50 45 20

30 40 50 Laser Current (mA)

60

(b)

Figure 7.10 Calculated self-mixing signal amplitude (a) and SNR (b) versus injected current for a Fabry–Perot laser with 20 mA threshold current and 38.4% differential efficiency (emitted power at 60 mA is 11.2 mW). Shot-noise-limited detection is assumed, with B = 100 Hz, C = 1, L = 1 m, pd = 0016 Source: Adapted from [14].

Laser Diode Feedback Interferometry

233

satisfied if the F–P laser is operated just above threshold. Hence, from Figure 7.10 it is concluded that optimum operation is achieved for high injected current. As far as SNR is concerned, we have already shown that the self-mixing configuration can attain a high sensitivity because it is a kind of homodyne (i.e. coherent) detection. It is worthwhile carrying out a comparison of its performance to that of the conventional coherent detection scheme, commonly used in interferometric applications. To this purpose, let us consider the general coherent detection scheme depicted in Figure 7.11, and calculate the q P0 Pr , where q is photodiode quantum efficiency, and Pr = useful signal as: Is = q h 4 P0 /A is the power received from the target, with P0 emitted power and A optical attenuation q P0 B. along measurement path. The quantum-limited noise term is: In2 = 2qI0 B = 2qq h 4 Thus, the signal-to-noise ratio for the conventional coherent detection, (S/N )COH , is given by: S/N COH =

P P /A Is2 = q r = q 0 In2 2hB 2hB

(7.16)

For the self-mixing case we consider the same emitted power P0 , and the SNR can be calculated from Equations (7.14) and (7.15) as: S/N SM =

2P0 /A 2 2 p2 1 − R2 2 Is2 · =   q co In2 2hB c2 R2

(7.17)

The ratio of the SNRs obtained for the two configurations is: 2 2 p2 1 − R2 2 S/N SM = co = 0476 · co = 953 × 10−3 = −202 dB S/N COH c2 R2

(7.18)

where the following typical values have been used: = 05 p = 16 ps; c = 8 ps; R2 = 035 co = 002. It can be concluded that the SNR for the self-mixing configuration is indeed worse than that of conventional interferometry, but most of the SNR penalty is due to the poor collection efficiency of the monitor photodiode. This is the only price one has to pay for the great reduction in complexity offered by the self-mixing approach with respect to conventional sensing schemes. MIRROR P0

Pr = P0/A

LASER 50/50 BS PHOTODIODE

Figure 7.11 General interferometric coherent detection scheme used to compare the SNR attainable for this configuration to that of self-mixing scheme.

234

Laser Interferometry

7.3 APPLICATION TO MEASUREMENTS In the previous section it was demonstrated that a system comprising a SL and a remote reflecting/diffusing target acts as an interferometer. Based on this principle, several schemes for measuring mechanical quantities have been developed and reported in the scientific literature, since the mid-1980s. In the following, application to the measurement of displacement, velocity, distance and vibration is illustrated, as well as other more specific measuring schemes [1].

7.3.1 Displacement For the basic application of displacement measurement, the SL is driven by a constant current, and the output power is directed onto the target through a conjugating lens. The target can be reflective (mirror), retroreflective (corner-cube or 3M Scotchlite™ reflective paper [17]) or diffusive (rough surface). Depending on the target type, an optical attenuator may be inserted along the light path to avoid excessive optical feedback, a precaution that is generally required for reflective and retroreflective targets. The easiest way to build a displacement sensor is to operate the SL in the moderate feedback regime (C > 1), so that the self-mixing signal is sawtooth-like and bistability shows up. Fast transitions are observed each time the target is displaced by /2, and the sign of the fast transitions depends on the target direction of motion (see Figures 7.4 and 7.5). Target displacement can thus be retrieved with /2 resolution (ie ≈ 325 nm when working with a visible SL emitting around 630 nm) without sign ambiguity. In fact, by performing an analogue derivative of the self–mixing signal, and by counting the occurrence of negative and positive pulses, the up and down counts are obtained, as shown by the block scheme of Figure 7.12(a) [7]. Figure 7.12(b) reports an experimental self-mixing signal for a vibrating target and its derivative, showing upward and downward pulses. The inset of Figure 7.12(b) shows a pulse duration of 200 ns, obtained using fast operational amplifiers. By this approach, displacement of a retroreflective target has been successfully measured over 1 m distance with an allowed maximum speed of 0.4 m/s, solely limited by electronics bandwidth. The maximum target distance is limited by SL coherence length: it can reach 7–8 m when using a moderate power, 780 nm SL, a unit intended for CD pick-ups [18]. In [19] it is demonstrated that, by interfacing the interferometer to a PC, it is possible to compensate for temperature wavelength drift of the SL, hence obtaining an overall precision of 5 m over 1 m displacement. Starting from the basic set-up shown in Figure 7.12(a), improvements of distance measurement apparatus can be made in two directions: (1) increasing the resolution; and (2) allowing operation on diffusive surfaces. 7.3.1.1 Increase of interferometer resolution The resolution can be improved by two different techniques: (1) linearization of the interferometric waveform F when C > 1 20, or its exact numerical inversion when C < 1 21; and (2) generation of fast periodic modulation of the interferometer phase and subsequent synchronous sampling of the self-mixing signal (phase-shifting method) [22, 23].

Application to Measurements

235

Trans-Z Amplifier

Derivative

Monitor PD Up

Pulse Shaper

Up-Down Counter

Polarity Discrimination

Pulse Shaper Down

Display (a) T

200 ns

Ch1

200 mV

Ch2 50.0 mV

M 100 µs Α

Ext

132 mV

(b)

Figure 7.12 (a) Electronic signal processing block scheme for fringe-counting displacement interferometer. (b) Upper trace: experimental self-mixing signal obtained for a sinusoidal target displacement of 3.3 m p-p amplitude and 1 kHz frequency; lower trace: analogue derivative of selfmixing signal, showing up/down pulses. Time scale: 100 s/div. The inset shows a detail of the derivative pulses, with 200 ns FWHM. Source: Adapted from [14].

When the feedback parameter is C > 1 and the self-mixing signal is sawtooth-like (see Figures 7.4 and 7.5), an accuracy better that /2 can be achieved by linearization of the interefrometric fringe, e.g., the function F is approximated by an ideal sawtooth. This approach can be conveniently implemented using a real-time digital signal processing scheme based on a FPGA (Field Programmable Gate Array). Residual inaccuracy is caused by the non-linearity of the actual self-mixing waveform, and the overall resolution/accuracy is

236

Laser Interferometry

improved by a factor ≈ 6 with respect to conventional fringe–counting technique. A resolution of 65 nm has been achieved using a 780 nm SL [20]. When C < 1, resolution improvement can be attained by sampling the self-mixing signal, and performing off-line signal processing to invert the function F, so as to exactly reconstruct the target displacement L [21]. This can be performed using the following formulae:


 C 2 Lt = · Ft + 1 − F t + m · 2
arccos Ft + √ 4
1 + 2 (7.19a) dFt dLt · < 0, and dt dt


 C Lt = · Ft − 1 − F 2 t − arccos Ft + √ 4
1 + 2

+m + 1 · 2
(7.19b)

valid for

dFt dLt · >0 valid for dt dt In the above expressions, the integer m has to be increased (respectively, decreased) by one for dLt/dt > 0 (respectively, dLt/dt < 0) in correspondence with each 2
crossing of the phase term in braces. The underlying principle is that the function F can be inverted for either its increasing or decreasing branch using one of the above formulas. This requires that the derivative sign of the actual displacement is known. Although the latter condition looks like a loss of generality for the method, it actually works in most cases, because one may guess the sign of the derivative from F t and after computing Lt check and correct the estimates as necessary. So, in most cases ambiguity is removed without errors when the reconstructed displacement is continuous and band-limited. To perform the above reconstruction, the value of the C parameter must be determined. This can indeed be done experimentally by evaluating the asymmetry of the interferometric waveform F [21]. However, this may prevent this method from being applicable in the case of diffusive targets, because speckle effects can cause variations of the feedback strength during the measurement. In [21] it is also shown that a 15% error in the estimation of the linewidth enhancement factor results in reconstruction errors smaller than a few tens of nanometers. Experimental results [21] reported a displacement reconstruction error around 50 nm, with an increase in accuracy of a factor ≈10 with respect to the fringe-counting method. The phase-shifting method [23] relies on fast modulation (in the hundreds of kHz range) of the interferometric phase by an external lithium niobate crystal phase modulator. The signal is sampled syncronously with the phase dither, and real-time signal processing allows the target displacement to be reconstructed with /12 resolution (corresponding to 65 nm). This method still works well for small optical feedback C < 1, it allows real-time measurement, and it does not require a preliminary characterization of the self–mixing waveform F. An improvement might utilize direct current modulation of the SL to achieve the wavelength shift, rather than the use of an expensive external phase modulator.

Application to Measurements

237

7.3.1.2 Operation on diffusive targets Operation of conventional displacement measuring interferometers requires a reflective target and accurate alignment procedure [1]. Typically, the target is a corner-cube mounted on the moving object under test. To improve the ease of use of these instruments, direct operation on a diffusing, rough surface, as found in the normal workshop environment, would be desirable, eliminating the need to keep optical surfaces clean and improving usability. This chance is actually offered by the self-mixing interferometric configuration, because it is intrinsically self-aligned and it is effective even for the case of very small optical backreflections (see Figure 7.9). However, working on diffusive targets generally implies practical limitations due to speckle-pattern effects [24], especially for the case of target displacements larger than a few millimetres, because the speckle distribution may change randomly, thus causing signal fading. This problem is obviously common also to conventional interferometric techniques, which require appropriate consideration for these applications [1]. The self-mixing approach, in conjunction with an appropriate ‘bright’-speckle tracking system, has been demonstrated as a solution to this problem. It was the first interferometer capable of working satisfactorily even on a rough surface [25]. The method employed to avoid amplitude signal fading is based on a slight change of the laser spot position on the target in the transversal direction [25]. The spot movement is obtained by means of a pair of piezo-actuators, holding the focusing lens, that allow control of the deflection angle of the laser beam (the experimental arrangement is shown in Figure 7.13). The piezo-actuators are driven by two square waves in quadrature, so that the spot position follows a square path on the target. The length of the sides of the square is set to be much less than the spot size (few micrometres). A closed-loop control circuit reads the amplitude of the self-mixing signal (which depends on the strength of optical feedback, see Equation (7.11)) and actively changes the DC bias voltage of the piezo-actuators so as to obtain a transversal translation of the SL spot in the direction of a ‘bright’ speckle. By this arrangement, the interferometer is dynamically locked to the local maximum of a bright speckle. Experimental results [25] confirm the effectiveness of the speckle-tracking method applied to a fringe-counting displacement interferometer designed to work in the moderate feedback regime C > 1 with a white paper target. When the speckle tracking system is disabled, there is a 10% probability of getting a self-mixing signal with a value of the C parameter smaller than unity, and frequently the displacement measurement is misread.

laser

monitor photodiode

lens

laser spot path

target

piezo-actuators

Figure 7.13 Experimental optical head arrangement for the speckle-tracking self-mixing interferometer. Two small piezo-actuators transversally move the lens, and the laser spot draws a square path on target. Source: Adapted from [14].

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Laser Interferometry

As the speckle-tracking system is turned on, the probability of faulty operation drops to below 0.5%, and actual displacement measurements can be successfully performed over a 0.5 m target travel distance with a negligible loss of signal. 7.3.1.3 Comparison with He-Ne Michelson interferometer Although the self-mixing technique has proved very effective for displacement measurements, one aspect still remains for which conventional displacement interferometers based on He-Ne lasers are superior to those based on SLs. This is precision. In fact, a stabilized He-Ne laser has a long-term wavelength stability of 10−6 , a figure that a temperature-controlled FP SL can match only in the short term. Moreover, for SLs a tight control against mode hopping has to be performed choosing the ‘right’ temperaturecurrent drive pair, because the uncertainty corresponding to longitudinal mode spacing is around 10−4 . However, we have demonstrated experimentally that a visible emission DFB SL performed satisfactorily in the self-mixing regime. This laser, if suitably temperaturecontrolled, could obtain a very good long-term wavelength stabilty, estimated at 10−5 [26].

7.3.2 Velocity Self–mixing sensors have been deployed to measure the velocity of solid targets and fluids. Good performance has been achieved with compact and low cost experimental arrangements. The measurement of the speed of a solid target can be trivially performed by differentiating the displacement signal Lt obtained as described in Section 7.3.1, but alternative approaches have been designed for velocity measurement. Actually, velocity measurements were performed, using the self-mixing effect, well before non-ambiguous, interferometric, displacement measurement was demonstrated [5, 6, 27]. This probably happened because the ‘velocity’ signal can be more directly interpreted as being the result of coherent mixing, within the SL cavity, of the lasing field and the Dopplershifted light back-scattered by the remote target. In fact, for a target moving with constant velocity v, the self-mixing photocurrent signal can be written as: ISM t = I0 1 + m · F t = I0 1 + m · F 2k · Lt = I0 1 + m · F 2
· 2v/c · t

(7.20)

where the term within the square parentheses is the well-known Doppler frequency shift (recall that F is a 2
-periodic function). Similarly to the displacement measurement, the asymmetry of the function F for moderate optical feedback C ≥ 05 allows recovery of the sign of the velocity. Of great interest is the measurement of the speed of a rotating or translating diffusive target, such as a disc, a drum, or, as encountered in paper or metal milling plants, a slab that moves linearly. These situations are summarized in Figure 7.14. These cases can be classified as a rough diffusive surface moving with a velocity that is not parallel to SL beam. In this case, the Doppler shift description still applies, and the measured speed is the component parallel to the light wave-vector. Inaccuracies may arise due to speckle effects, which are of particular importance for this case, since the portion of the target illuminated by the

Application to Measurements

239

BEAM

BEAM LD

LD

LENS

LENS

(a)

(b)

(c)

Figure 7.14 Arrangements for measurement of the speed of rotating targets such as: (a) drum; (b) disc; and (c) transversally moving diffusive slab.

SL light beam is continuously changing, and the self-mixing signal is randomly amplitudemodulated. Figure 7.15(a) shows an experimental time-domain optical-power waveform for the case of a rotating disc [28]. It is clearly seen that the self-mixing amplitude is strongly altered by speckle effects, and difficulties may arise in the reconstruction of the motion of the target. Appropriate signal processing techniques are used to filter out the speckle–related signal components. A simple analysis based on the use of real-time, Fast Fourier Transform (FFT) techniques, is shown in Figure 7.15(b). This demonstrates that it is possible to determine the Doppler beat frequency fb , and to calculate the component of the speed of the target parallel to the laser beam axis. Random intensity modulations due to speckle cause inaccuracies in the value of the instantaneous velocity. The error can be reduced by increasing the time integration constant of the FFT signal processing. A set of experimental results for target velocities up to 200 km/h (i.e. 55.5 m/s) are shown in Figure 7.16. The error increases for decreasing speed, and a maximum relative error of 5% is found [28]. Velocity measurements performed using the ‘Doppler frequency’ approach are effective as long as the angle between the SL beam and the normal to the moving surface is larger than zero. Interestingly, in a number of papers [29, 30], it has been demontrated that measurement of the velocity of a translating surface is possible also for normal incidence. This can be performed by analysing the frequency of the occurence of ‘speckle grains’ that are detected through the amplitude of the self-mixing signal. The reciprocal of the autocorrelation time of speckle signals is shown to have a linear relationship with surface velocity. This can be measured with an accuracy better than 3%. A possible limitation of this technique is the dependence of sensor calibration on the type of surface under test. Velocity measurements by the self-mixing technique have been performed on moving, scattering fluids. Of particular interest, is the medical application to blood flow analysis, both in vitro and in vivo. This approach generally makes use of a SL coupled to an optical fibre, to

240

Laser Interferometry

5.00νs / 20.0νs / Sng1 Pat STOP

A1 400m v/ Optical power Pc

A1 20 µs Zoom

A1 5 µs (a) fft(A1)

2.44 MHz 1.22 MHz

Sng1 Pat STOP

Beat frequency fb

0

1.22

2.44

Frequency (MHz) (b)

Figure 7.15 (a) Time-domain self-mixing signal for velocity measurement on a rotating diffusing disc. (b) FFT spectrum of the signal. The Doppler beat frequency fb = 146 MHz corresponds to 0.56 m/s speed. Source: Adapted from [14].

facilitate access to biological vessels [31]. Another approach based on speckle-correlation has been applied to in vitro and in vivo blood flow measurement, obtaining different self-mixing signal autocorrelation functions for different blood flow regimes [32]

7.3.3 Vibration Vibration measurements involve detection of zero-mean displacements of solid rough surfaces, with vibration frequencies from DC up to a few tens of MHz. This field is

Application to Measurements

241

Error 5% 4% 3% 0 km/h 200 km/h

50 km/h

100 km/h

150 km/h

200 km/h

100 km/h Target Speed

150 km/h

200 km/h

Measured Speed

150 km/h 100 km/h 50 km/h 0 km/h 0 km/h

50 km/h

Figure 7.16 Experimental calibration of velocity measurement on a rough white paper surface Source: Adapted from [14].

strictly related to acoustic noise measurements and mechanics. Optical non-contact vibration sensing techniques are generally called laser vibrometry. The most widely used approach is the so-called LDV (Laser Doppler Velocimetry) technique that is based on He-Ne lasers and conventional Michelson or Mach-Zehnder interferometers. Several commercial LDV instruments and systems have been successfully used in a variety of scientific and industrial applications such as modal analysis, vibration and noise testing, and characterization of loudspeakers and piezoceramic transducers [33, 34]. The self-mixing scheme has proved to be efficient for this application [35, 36, 37, 38]. Measurement of vibrations of amplitude much larger than /2 can be performed by applying the fringe counting technique described in Section 7.3.1, so that the periodic traget displacement is reconstructed [37]. Besides this, a direct self-mixing signal analysis can be performed in the frequency domain using FFT [38]. Irrespective of the optical feedback strength, the maximum frequency contained in the self-mixing signal for the case of a target vibrating at a frequency f0 with amplitude L0 is proportional to the product f0 ·L0 . Hence, spectral analysis of the self-mixing signal can supply information about the vibration frequency f0 (corresponding to the fundamental self-mixing signal frequency) and its amplitude, with an accuracy that can be slightly better than the SL wavelength. An example of direct FFT vibration measurement is shown in Figure 7.17. When smaller vibrations are to be measured, or a resolution much better than /2 is required, a different approach based on a closed-loop technique can be used. The principle, shown in Figure 7.18(a), relies on operation in the moderate feedback regime (with C > 1 and sawtooth-like interferometric signal) and on locking of the interferometer phase to the half-fringe position. By means of a suitable feedback loop acting on the SL wavelength, environmental low-frequency phase fluctuations can be cancelled out, and vibrations of amplitude smaller than /2 can be linearly transduced into an electrical signal. By adopting an additive active phase-tracking system, the maximum measurable vibration amplitude can be extended up to a few hundreds microns [36]. The operating principle relies on wavelength

Laser Interferometry

Self-Mixing Signal [V]

242

0.06 0.04 0.02 0.00 –0.02 –0.04 0

1

2

3

4 Time [s]

5

20

25

6

7

8 ×10

–3

Signal Spectrum [dB]

–30 –40 –50 –60 –70 –80

0

5

10

15

30

35

40

Frequency [kHz]

Figure 7.17 Measurement of the vibration of a diffusive target through FFT analysis. Upper trace: time domain self-mixing signal for a sine vibration with 140 Hz frequency and 7.86 m p–p amplitude. Lower trace: FFT spectrum of the above signal, showing a peak at 7.85 kHz. The peak frequency is proportional to the product of target vibration frequency and amplitude. SL wavelength: 786 nm.

modulation of the SL by injection current modulation. The active phase-tracking system is designed so that a constant number of wavelengths are contained in the path from the SL to the target. For example, if the target moves away from the SL, the wavelength is increased, so that the number of wavelengths is kept constant. This task is performed by a suitably designed electronic feedback loop (whose block scheme is shown in Figure 7.18(b)) that provides an output signal that is an analogue replica of the target displacement. Examples of vibrations measured by such a system are shown in Figure 7.19 for the case of a loudspeaker membrane actuated by a function generator. In this case the target has a rough diffusive surface. Upper traces of Figures 7.19(a) and 7.19(b) show the loudspeaker drive signal for the case of a 3 Hz triangle wave and 10 Hz square wave respectively; lower traces show the output signals obtained from the self-mixing vibrometer together with an appropriate displacement scale reference. The output signal is a replica of target displacement, and in Figure 7.19(b) the loudspeaker resonance reveals itself in form of damped oscillations. Also, from Figures 7.19(a) and 7.19(b) it is confirmed that the above explained approach is capable of linear measurement also for vibration amplitudes much larger than the wavelength. The ultimate sensitivity for vibration (i.e displacement) measurement is set by the quantum noise associated with the detected photocurrent signal. This can be expressed [3] in terms of noise equivalent displacement (NED) as: NED = /2
/(SNR) where SNR is the signal-to-noise ratio of the self-mixing signal (see Equation (7.17)). The typical

Application to Measurements

243

P PSM

P0 t

β

2π Φ = 2kL

λ /2 L

L

t (a) TARGET DISPLACEMENT

ELECTRONIC FEEDBACK LOOP MONITOR PHOTODIODE

TRANS-Z AMPLIFIER

LASER DIODE

VOLTAGE-CONTROLLED CURRENT SOURCE LP FILTER

VIBROMETER OUTPUT SIGNAL A

(b)

Figure 7.18 (a) Principle of linear measurement of target vibrations in the moderate feedback regime by locking the interferometer phase to the half-fringe position (light grey, mid-section). The vertical axis represents the power emitted by the SL; the horizontal axes represent interferometric phase and target displacement (b) Block diagram for the electronic loop of a self-mixing vibrometer. Source: Adapted from [36].

√ NED experimentally obtained by the prototype self-mixing vibrometer is 10 pm/ Hz [36]. A remarkable figure indeed.

7.3.4 Distance The absolute distance of a stationary target can be measured using the self-mixing scheme by modulating the SL emission wavelength with a triangular waveform using injection

244

Laser Interferometry

T

20 V

1

2

4 µm

Ch 1

10.0 V

Ch 2 20.0 mV

M 100 ms A

Ext

3.00 mV

(a) T

20 V

1

10 µm

2

Ch 1 10.0 V

Ch 2

50.0 mV

M 40.0 ms A Ext

3.00 mV

(b)

Figure 7.19 Examples of vibration measurements performed using the self-mixing vibrometer [36]. The target is a loudspeaker with a black paper surface. Upper traces: loudspeaker drive signal; lower traces: vibrometer output signal, with 20 mV/m responsivity. (a) 3 Hz triangular wave; (b) 10 Hz square wave, damped resonance oscillations of the loudspeaker are clearly visible. Source: Adapted from [36].

current modulation. The photodetected current shows a triangular modulation due to the SL power variation, and a self-mixing interferometric signal is superimposed upon it. The self-mixing signal is generated because the wave-number k = 2
/ is changed by the amount k = −2
/2 , where  is peak-to-peak wavelength variation [39, 40, 41]. By counting the number, N , of interferometric fringes occurring for a known wavelength variation , the target distance, s, can be retrieved from: s=

2 ·N 2 · 

(7.21)

A typical example of self-mixing waveform for this case is shown in Figure 7.20, together with its analogue derivative, showing the pulses to be counted [40]. The resolution of the

Application to Measurements

245

Figure 7.20 Self-mixing signal for absolute distance measurement, obtained for a 0.8 mA peak-to-peak triangular current modulation in a F-P SL. The analogue derivative of the signal, with the pulses to be counted, is also shown. Source: Adapted from [14].

counting method is limited by the quantization error, and it is given by: serr = ±2 /2. The latter formula reveals that measurement accuracy can be improved by increasing the extent of the wavelength modulation. Unfortunately, continuous thermal wavelength tuning by injection current is typically limited to about 0.1 nm by longitudinal mode-hopping in F–P SLs. Hence, by using a conventional low-cost F–P the attainable accuracy cannot be reduced below 1 mm. This performance can be improved by using a multi-electrode DBR SL [42]. Accuracy improvement can be obtained by adjusting the injection current wavelength sweeping waveform with a pre-distorted signal to achieve a linear wavelength sweep, thus avoiding the non-linearities caused by thermal effects. When a good linear wavelength sweep is generated, the distance can be measured by evaluation of the so-called ‘beat frequencies’ of the self-mixing fringes, using the formula: s=

2 · fup + fdown  4d/dt

(7.22)

where fup refers to the fringe frequency of the raising ramp and fdown to the decreasing one. By using this more refined method the resolution can be improved up to 0.1 mm.

7.3.5 Other Measurements and Developments A large variety of clever remote sensing applications, based on the self-mixing approach, have been demonstrated, and most of these use SLs. A partial summary is reported here. Self-mixing has been applied to the characterization of MEMS (Micro-Electro-MechanicalSystems) silicon devices such as accelerometers or gyroscopes, to support or complement electrical techniques [43, 44, 45]. The laser beam is aimed at microscopic, suspended masses

246

Laser Interferometry

that are caused to vibrate by electrostatic forces. In this application, the ability of the selfmixing sensor to detect the vibration of a rough surface with holes, having a void ratio of 50%, is amazing, as well as the possibility of measuring small object through a vacuum glass shield. MEMS characterization by self-mixing can compete with the more expensive technique based on the LDV approach [33]. A novel hybrid micromechanical gyroscope with self-mixing interferometric readout has been proposed [46]. The self-mixing signal obtained from a vibrating diffusive MEMS suspended mass located very close to the SL (≈1 mm distance) can be used to measure the rotation rate, relying on Coriolis force. Limit sensitivity of ≈10/h comparable to that of all-electrical MEMS can be achieved. The measurement of optical isolators placed into a SL package has also been demonstrated using self-mixing [47]. The linear dependence of the modulation index m (see Equation (7.11)) on the external target reflectivity Rext can be exploited to assess the amount of optical isolation provided by optical isolators, once these devices have been included into the SL package. This technique allows for a sensitivity around 80 dB. Similarly to the previous application, the self-mixing effect in a hybrid-ECL (External Cavity Laser) has been used to measure the optical return loss of fibre optics components [15], with a sensitivity around 90 dB. Self-mixing has also been applied to the task of classification of rough surfaces. This technique relies on the fact that, for normal incidence, when a rough surface is displaced transversally at constant speed, speckle pattern grains properties (statistics, autocorrelation, etc.) are correlated with the surface texture, roughness and reflectivity. By using a neural network, successful classification in more than 92% of cases has been reported for eight different surfaces [48]. Recently, an optical mouse for applications in cellular phones and laptop computers has been demonstrated based on the self-mixing effect [49]. This sensor prototype allows the discrimination of the direction of motion of a fingertip moving on the sensor surface, and its commercial deployment is likely to take place in the near future. Studies of the possibility of operating the SL in the high-feedback regime [50] and/or using multimode SLs have been carried out [51]. It is demonstrated that the self-mixing effect can be observed also for the case of multi-longitudinal mode lasers. In particular, the occurence of mode-jumps that are synchronous with the phase of the backreflected field can be of help in increasing the resolution of the fringe-counting displacement interferometer. Velocity measurement through self-mixing effect has also been demonstated using microchip solid-state lasers instead of SLs [52]. The Doppler-shifted backscattering from a rotating rough surface is detected through the self-mixing effect generated into a diode-laserpumped LiNdP4 O12 microchip laser. For this source, the target direction of motion cannot be discriminated through the self-mixing waveform because it is not asymmetric. This happens because solid-state lasers have = 0, contrary to SLs that have > 0. However, a good sensitivity has been obtained, and direction of motion discrimination can be added by using an external acousto-optical modulator [52]. Angle measurement of a remote, reflective surface has been demonstrated when the Sl operates in the strong feedback regime corresponding to the so-called coherence collapse [53]. As the power reflectivity of the remote surface approaches 0.01, the SL enters the coherence collapse regime, and the laser threshold depends strongly on the reflectivity of the surface. If the surface is tilted, optical feedback decreases, as well as the emitted power.

Laser Diode Diagnostics Using Self-Mixing Techniques

247

By exploiting this effect, the angle of the remote surface can be measued with an accuracy of 5 × 10−7 rad.

7.4 LASER DIODE DIAGNOSTICS USING SELF-MIXING TECHNIQUES Besides applications aimed at the measurement of properties and mechanical quantities of remote objects/targets, the self-mixing technique can be useful to gain further insight into some relevant operating characteristics and parameters of the SLs themselves, such as linewidth and linewidth enhancement factor, that can be measured with a good accuracy still retaining the intrinsic simplicity of the self–mixing sensing scheme.

7.4.1 SL Linewidth Measurement The SL linewidth can be evaluated from measurements of the phase noise of the selfmixing interferometer performed for different target distances, because the phase noise is proportional to both SL linewidth and target distance L. Linewidth estimation by phase noise measurement in interferometers is a well-known technique [54]. The selfmixing scheme, in the moderate feedback regime (feedback parameter C > 1, sawtooth-like waveform), offers two practical advantages. First, due to the fast switching that arises due to bistability, the phase noise can be evaluated by simple time-domain measurements instead of spectral density analysis in the radio frequency (RF) range. Second, a compact set-up can be used, with overall length much shorter than the SL coherence length, in contrast to the other well-known, non-RF method, called fringe visibility analysis [55]. When light from a laser source enters an interferometer, the fluctuation of the laser frequency (represented by the linewidth ) generates phase noise. For a self-mixing interferometer the RMS phase noise is obtained as: 
4
02 L2 + L20  2

2 = c

(7.23)

where 0 is SL mean emission frequency, and the target distance has a deterministic value L0 to which a random fluctuation L is added. If phase noise is measured for different values of
    4
L0
target distance L0 so that L20  2  02 L2 , a linear dependence 2 ≈  2

c
is obtained, the slope of the curve 2 vs. L0 is proportional to the SL linewidth. So, SL linewidth can be recovered from subsequent measurement of the RMS phase noise performed at different target distances. The sawtooth-like self-mixing signal when there is hysteresis allows an easy and accurate measurement of phase noise, as is clarified by observing the fast switching occurring between two specified fringes on an oscilloscope. The effect of phase noise is such that switching times corresponding to successive observations of the same fringes have a randomness, and there is a statistical distribution of switching instants around the most probable value (the one that would be observed in absence of phase noise). This is shown in Figure 7.21. Now, the hysteresis in the power-phase characteristic of the self-mixing waveform prevents the occurrence of multiple switching, with opposite sign, for a single period of target oscillation.

248

Laser Interferometry

(a) TARGET DISPLACEMENT

SELF-MIXING SIGNAL

1 ms

(b)

20 µs

Figure 7.21 Illustration of SL linewidth–induced phase noise of the self-mixing interferometer. (a) Upper trace: loudspeaker (target) drive signal. Lower trace: self–mixing signal waveform in the moderate feedback regime. (b) Zoomed frame reporting the superposition of subsequent singlesweep acquisitions of the self-mixing signal corresponding to a specific fringe. The switching time randomness caused by phase noise can be clearly observed. Horizontal scale: 20 s/div, corresponding to 0.50 rad/div when converted into interferometric phase. Source: Adapted from [18].

The phase noise variance, with nearly Gaussian statistics in practice, can be evaluated by acquisition of the fringe switching times, and by subsequent statistical data analysis, yielding a value of the RMS phase noise. Figure 7.22 reports the measurement of the RMS phase noise as a function of target distance L0 for different SL types. For small distances, mechanical fluctuations of the set-up can be significant, but for longer distances the theoretical linear dependence is obtained. The linewidth is estimated from the slope of the fitted curves. From Figure 7.22 it is deduced that the linewidth for the Mitsubishi ML2701 SL (850 nm Fabry–Perot) is 14.4 MHz for 25 mA injection current and 12.7 MHz for 30 mA. For another SL (SDL SDL5401, 800 nm Fabry–Perot) the linewidth is 7.8 MHz for 40 mA injection current, 4.65 MHz for 50 mA, and 3.5 MHz for 60 mA. The linewidth values measured using the self-mixing technique show the expected decrease for increasing injection current, and they are in good agreement with the ones obtained by the self-heterodyne method [56] for the same SLs. This confirms that the optical feedback from the remote reflector causes no practical relevant perturbations on the LD linewidth. This fact suggests that further theoretical investigation is needed, because previous simulation results [10, 57] show that the actual

Laser Diode Diagnostics Using Self-Mixing Techniques

249

Figure 7.22 Results for SL linewidth evaluation through interferometer phase noise measurement. The graph reports measured RMS phase noise as a function of target distance L0 for different SLs and varying injected currents. Empty symbols refer to SL#1: Mitsubishi ML2701, 850 nm Fabry–Perot; linewidth: 14.4 MHz @ 25 mA, 12.7 MHz @ 30 mA. Full symbols refer to SL#2: SDL SDL5401, 800 nm Fabry– Perot; linewidth: 7.8 MHz @ 40 mA, 4.65 MHz @ 50 mA, 3.5 MHz @ 60 mA. Source: Adapted from [18].

linewidth of the SL, in feedback regimes I and II, is expected to vary by more than one order of magnitude, depending on the specific value of the back-reflected field phase.

7.4.2 SL Linewidth Enhancement Factor Measurement It is clear from Equations (7.7), (7.8), (7.9) that the shape of the self-mixing interferometric waveform F depends on both the feedback parameter C and on the linewidth enhancement factor . The dependence of self-mixing waveform on is illustrated by numerical simulations reported in Figure 7.23. The value of the linewidth enhancement factor can be obtained using a simple self-mixing experimental arrangement, by carrying out two steps: (1) evaluation of the actual value of the feedback parameter C; and (2) identification of a specific feature of the self-mixing signal that can be easily and conveniently related to . A good starting point for the evaluation of C is represented by Equation (7.13) and Figure 7.7, that report the extent of the self-mixing signal hysteresis as a function of C. As the hysteresis width HYST is measured in terms of the interferometric phase, the actual value of HYST can be retrieved from the experiment when a triangular phase variation is applied to the interferometer, and the time intervals T1 and T2 are measured (see Figure 7.24(a)). The hysteresis value is obtained as: HYST = 2
·

T2 − T1 T

(7.24)

250

Laser Interferometry

Figure 7.23 Numerical simulations of the self-mixing signal showing the effect of varying linewidth enhancement factor . (a) Feedback parameter C = 07; = 05 (solid line), = 1 (dashed line), = 4 (dash-dotted line), = 8 (dotted line). (b) Feedback parameter C = 3; = 05 (solid line), = 1 (dashed line), = 4 (dash-dotted line), = 8 (solid line).

where T is the fringe period, also shown in Figure 7.24(a). Once HYST is known, the actual value of the feedback coefficient C can be obtained from Equation (7.13) with an accuracy better than 10%. A specific feature of the self-mixing waveform F that allows to estimate is the time interval T3 (also shown in Figure 7.24(a)) normalized to T . With reference to Figure 7.24(b), time interval T3 corresponds to the length of segment AB, where points A and A are those

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251

zero line

T1

T3

T

T2

(a) 1.0 0.8 0.6 0.4

F(φ)

0.2 A’

0.0

A

B

–0.2 –0.4 –0.6 –0.8 –1 –1

0

1

2 φ (rad) (b)

3

4

5

Figure 7.24 (a)Experimental self-mixing signal suitable for the evaluation of linewidth enhancement factor . T is fringe period. T1 and T2 allows to evaluate the feedback parameter C. T3 allows to evaluate . Obtained values are: C = 16 = 29. (b) Explanation of the meaning of points A, A’, B.

for which the derivative dF/d is infinite, and in point B we have F = 0. The relation between the AB segment and parameters and C can be analytically found as [58]:   √   1 1 3 AB 2 =
− C · cosarctan  + arctan  − arccos (7.25) − C −1 2
2
2 C The relationship (7.25) is plotted in Figure 7.25 for different values of C. It is shown that, from the experimental measurement of T3 /T = AB/2
, and of the C parameter, the linewidth enhancement factor can be obtained. The accuracy on is better for smaller values, because the curves of Figure 7.25 have a smaller slope. Preliminary experimental results have shown that the linewidth enhancement factor can be estimated with 10% accuracy [58],

252

Laser Interferometry

8

Linewidth Enhancement Factor α

7 2.1 1.8 1.5 1.2

6 5 4 3 2 1 0

0

0.1

0.2

0.3

0.3

0.5

AB/2π

Figure 7.25 Graph used to evaluate , once the value of the C parameter and the quantity T3 /T = AB/2
have been measured experimentally (see Figure 7.24(b)). Different curves refer to different values of the feedback parameter C, shown on the graph.

a figure that is consistent with those obtained by other methods based on optical injection locking [59–61]. The present method has, however, the advantage of a simpler experimental set-up, that does not require optical spectrum nor RF spectrum analyses.

7.5 CONCLUSION Semiconductor laser interferometric techniques based on optical feedback (the so-called self-mixing interferometry) have been analysed and described. It has been demonstrated that optical feedback in SLs is not always detrimental. Rather, when the feedback strength is properly controlled, the way is open to perform interferometric measurements with high accuracy. Peculiar characteristics of this measurement approach are simplicity and low cost, accompanied by high sensitivity and a versatility that is uncommon to other conventional methods. Self-mixing interferometry has been successfully applied to mechanical metrology (measurement of displacement, vibration, velocity, distance) and to a variety of other tasks. The achieved results are encouraging, and it can be foreseen that commercial developments of this technique will soon appear.

ACKNOWLEDGEMENTS G. Giuliani wishes to thank Michele Norgia, Sabina Merlo, Valerio Annovazzi–Lodi, Thierry Bosch and N¨oel Servagent for fruitful discussions and collaboration. Part of the research results described herein were achieved within the EU Brite-Euram SELMIX Project.

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[30] S.K. Ozdemir, S. Ito, S. Shinohara, H. Yoshida and M. Sumi, ‘Correlation-based speckle velocimeter with self-mixing interference in a semiconductor laser’, Appl. Opt., 38, 6859–6865 (1999). [31] F.F.M. de Mul, M.H. Koelink, A.L. Weijers, J.G. Aarnoudse, R. Graaff, and A.C.M. Dassel, ‘Self-mixing laser-Doppler velocimetry of liquid flow and blood perfusion of tissue’, Appl. Opt., 31, (27), 5844–5851 (1992). [32] S.K. Ozdemir, S. Shinohara, S. Takamiya and H. Yoshida, ‘Noninvasive blood flow measurement using speckle signal from a self-mixing laser diode: in vitro and in vivo experiments’, Opt. Eng., 39 (9), 2574–2580 (2000). [33] L.E. Drain, The Laser Doppler Technique, John Wiley & Sons, New York (1980). [34] P. Castellini, G.M. Revel and E.P. Tomasini, ‘Laser doppler vibrometry: a review of advances and applications’, The Shock and Vibration Digest, 30 (6), 443–456 (1998). [35] G. Giuliani, S. Donati and L. Monti, ‘Self-mixing laser diode vibrometer with wide dynamic range’, SPIE Proceedings 4827, in E.P. Tomasini, (ed.), Fifth International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, pp. 353–362, (2002). [36] G. Giuliani, S. Bozzi-Pietra and S. Donati, ‘Self-mixing laser diode vibrometer’, Meas. Sci. Technol., 4, S283–S294 (2002). [37] N. Servagent, T. Bosch and M. Lescure, ‘A laser displacement sensor using the self-mixing effect for modal analysis and defect detection’, IEEE Trans. Instrum. Meas., 46, 847–850 (1997). [38] L. Scalise, ‘Self-mixing feedback laser Doppler vibrometry’, in E.P. Tomasini (ed.), Fifth International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, Proceedings of SPIE, Vol. 4827, pp. 374–384 (2002). [39] S. Shinohara, H. Yoshida, H. Ikeda, K. Nishide and M. Sumi, ‘Compact and high-precision range finder with wide dynamic range and its application’, IEEE Trans. Instrum. Meas., 41, 40–44 (1992). [40] F. Gouaux, N. Servagent and T. Bosch, ‘Absolute distance measurement with an optical feedback interferometer’, Appl. Opt., 37 (28), 6684–6689 (1998). [41] E. Gagnon and J-F. Rivest, ‘Laser range imaging using the self–mixing in a laser diode’, IEEE Trans. Intrum. Meas., 48, 693–699 (1999). [42] G. Mourat, N. Servagent and T. Bosch, ‘Distance measurements using the self-mixing effect in a 3-electrode DBR laser diode’, Opt. Eng., 39, (3), 738–743 (2000). [43] V. Annovazzi–Lodi, S. Merlo and M. Norgia, ‘Measurement on a micromachined silicon gyroscope by feedback interferometry’, IEEE/ASME Trans. Mech., 6 (1), 1–6 (2001). [44] V. Annovazzi–Lodi, S. Merlo and M. Norgia, ‘Characterization of silicon microstructures by feedback interferometry’, J. Opt. A: Pure and Appl.Opt., 4 (6), S311–S317 (2002). [45] V. Annovazzi–Lodi, S. Merlo and M. Norgia, ‘Comparison of capacitive and feedback-interferometric measurements on MEMS’, IEEE J. Microelectromech. Syst., 10 (3), 327–335 (2001). [46] M. Norgia and S. Donati, ‘A hybrid opto-mechanical gyroscope with injection-interferometer readout’, El. Lett., 37 (12), 756–758 (2001). [47] S. Donati and M. Sorel, ‘A phase-modulated feedback method to test optical isolators assembled into the laser package’, IEEE Phot. Techn. Lett., 8, 405–407 (1996). [48] S.K. Ozdemir, S. Shinohara, S. Ito and H. Yoshida, ‘Compact optical instrument for surface classification using self-mixing interference in a laser diode’, Opt. Eng., 40 (1), 38–43 (2001). [49] M. Liess, G. Weijers, C. Heinks, A. van der Horst, A. Rommers, R. Duijve and G. Mimnagh, ‘A miniaturized multidirectional optical motion sensor and input device based on laser self-mixing’, Meas. Sci. Technol., 13, 2001–2006 (2002). [50] Y. Yu, H. Ye and J. Yao, ‘Analysis for the self-mixing interference effects in a laser diode at high optical feedback levels’, J. Opt. A: Pure and Appl.Opt, 5, 117–122 (2003). [51] M. Ruiz-Llata, H. Lamela and J.I. Santos, ‘The self-mixing technique for vibration measurements in a multimode laser diode’, Proc. IEEE-LEOS ODIMAP III, Pavia, (2001). [52] K. Otsuka, R. Kawai, Y. Asakawa and T. Fukazawa, ‘Highly sensitive self-mixing measurement of Brillouin scattering with a laser-diode-pumped microchip LiNdP4 O12 laser’, Opt. Lett., 24 (24), 1862–1864 (1999). [53] G. Giuliani, S. Donati, M. Passerini and T. Bosch, ‘Angle measurement by injection detection in a laser diode’, Opt. Eng., 40 (1), 95–99, (2001). [54] J.A. Armstrong, ‘Theory of interferometric analysis of laser phase noise’, J. Opt. Soc. Am., 56, 1024–1031 (1965). [55] D. Derickson (ed.), Fiber Optic Test and Measurement, Prentice Hall, Englewood Cliffs, NJ (1997).

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8 Single Frequency and Tunable Single Frequency Semiconductor Laser Systems Esa Jaatinen

8.1 INTRODUCTION Much of this book is devoted to the nonlinear dynamic behaviour that results when optical feedback from an ‘external reflector’ is introduced into a semiconductor laser resonator. Feedback response such as line narrowing, coherence collapse and chaotic behaviour have been thoroughly discussed in Chapters 1–4. To a large extent, it is the level of external feedback with respect to the diode facet reflectivity, that determines the nature of the response of the laser, and can be used to characterize the feedback regime of the system. This chapter will look at diode laser systems that belong to Regime V of the classifications discussed in Chapter 1–diode laser systems that have strong levels of external feedback in comparison to the reflection from the diode facet. The focus will be on laser system designs that force robust and reproducible single frequency emission. As for all lasers, it is the interplay between the gain of the semiconductor and the losses in the cavity that dictates the spectral content of the emission of the laser. Therefore, reliable single frequency output is possible if external feedback that is suitably frequency selective is introduced to the diode laser resonator. Obtaining single frequency output from a laser by selectively increasing cavity losses with frequency filtering is not a new concept. Filtering elements such as Fabry–Perot etalons and birefringent filters are used in the cavities of lasers such as the Ti:Sapphire ring laser to obtain single frequency output. A typical example is shown in Figure 8.1 where a sophisticated cavity design can deliver cw output in the NIR that has an instantaneous linewidth of 10 MHz that can be tuned over some 100 nm.

Unlocking Dynamical Diversity Edited by Deborah M. Kane and K. Alan Shore © 2005 John Wiley & Sons, Ltd

258

Single Frequency and Tunable Laser Systems

Pump beam

Ti:Sapphire Crystal

Birefringent filter PZT mounted mirror

Output Etalon

Brewster Plates

Figure 8.1 Schematic of a ‘typical’ ring cavity Ti:Sapphire laser showing some of the intra-cavity components that are required to deliver single frequency cw output that is tunable without mode-hops.

Diode lasers with frequency selective external feedback can deliver performance specifications similar to the tunable single frequency Ti:Sapphire laser. However, due to the high gain of the semiconductor and its nonlinear response to feedback, single frequency behaviour can be obtained from a much simpler and more compact cavity design than that of ‘equivalent’ Ti:Sapphire systems. For example, feedback from one external diffraction grating is sufficient to produce output from a diode laser resonator that has a linewidth and tuning range similar to the Ti:Sapphire system shown in Figure 8.1. Single frequency behaviour also occurs in diode laser cavity designs that use external feedback from interferometers rather than diffraction gratings. The relative merits of both types of filtered feedback will be discussed in this chapter. Also, the advantages of monolithic cavity designs, where the external feedback reflector is fabricated on the same substrate as the diode laser resonator, will be contrasted with designs where feedback results from elements that are external to the diode laser substrate. Issues affecting the tunability of the more commonly encountered single frequency diode laser cavity designs will be addressed as will factors that influence the frequency stability of the laser. In the final section of this chapter some applications of single frequency diode lasers will be discussed. These can be broadly classified as spectroscopic or communications oriented with each application placing its own set of demands on the performance requirements of the diode laser.

8.1.1 Laser Mode Structure Most of the earliest diode lasers employed a Fabry–Perot resonator and the design is still in wide use today [1]. This is partly due to the ease of fabrication of this type of diode laser making mass production inexpensive. Figure 8.2 shows the basic layout of a Fabry–Perot diode laser. The end mirrors of the resonator are obtained by simply cleaving the crystal at the two ends and reflection coating the facets. Lasing will still occur even if the quality of the end mirrors and their alignment are poor because of the high gain of diode lasers [1, 2]. The Fabry–Perot resonator does not support lasing at all frequencies under the gain curve because it acts as a filter that transmits certain frequencies without loss. The total amplitude of the field transmitted by a Fabry–Perot is the summation of the many partial reflections

Introduction

259

100 µ m

R2 ( <
X − X


(9.10)

and the signal error is defined by:

=


X − Y
 
Y


(9.11)

It is assumed that where necessary the waveforms are normalized with respect to amplitude and appropriately adjusted for time lag. We will now briefly present examples of each of these three schemes. First, we consider an example of data transmission based on CSK. Figure 9.4 shows results obtained in numerical

Methods for Modulation and Recovery of Messages

313

Figure 9.4 Chaotic communication based on ON/OFF CSK. (a) Chaotic oscillation in the transmitter without a message. (b) Transmitted signal of chaotic oscillation together with a message. (c) Synchronized chaotic signal in the receiver. (d) Decoded message after filtering. The bit rate of the message is 2 Gbps. Source: After Mirasso et al. ©2002 IEEE.

simulations by Mirasso et al. [37]. Figure 9.4(a) is a typical chaotic waveform without a message and Figure 9.4(b) is a chaotic waveform which contains a 2 Gbps NRZ message. The shift-key modulation is achieved by modulation of the injection current. The direct injection current was 2.93Ith (Ith being the threshold injection current) and the modulation amplitude was 0.13Ith . The relaxation oscillation frequency of the solitary laser is 4 GHz at this injection current. Though it is not clearly observable from this figure, the receiver output in Figure 9.4(c) synchronizes with that of the transmitter in Figure 9.4(b) at ON states of the message, while it has a different chaotic oscillation at OFF states. The message is recovered by subtracting the reproduced chaotic waveform from the transmitted signal – a constant signal is obtained in the synchronized state, while chaotic bursts appear in the non-synchronized state. The result of the decoding after low-pass filtering is shown in Figure 9.4(d). The time required to recover synchronization after switching, which depends on system configuration and parameters, limits the achievable rate of data transmission. Figure 9.5 shows the degradation of the accuracy of message recovery as the rate of data transmissions is increased(1, 2, and 3 Gbps from (a) to (c)). The synchronization recovery time will be discussed again in Section 9.6. Next we present an example of data transmission using CMA. This scheme was demonstrated numerically by S´anches-D´iaz et al. [41]. They described the transmission of a 4 Gbps NRZ signal in a closed-loop type system. The message was a small perturbation of

314

Chaotic Optical Communication

Figure 9.5 Recovered message in ON/OFF CSK for different bit rates of 1, 2, and 3 Gbps from (a) to (c), respectively. Source: After Mirasso et al. ©2002 IEEE.

order 0.5% of the average amplitude of the chaotic oscillation in the transmitter, and it was imposed by directly modulating the injection current. They also considered the nonlinear effects on the signal due to transmission through optical fiber between the transmitter and the receiver. The transmission through the optical fiber, including effects of dispersion and nonlinearity, is described by the following nonlinear Schr¨odinger equation [18]: i

E i 1 2 E = − f E + 2 2 − 
E
2 E z 2 2 T

(9.12)

where Ez T is the slowly varying complex field, z is the propagation distance, and T is the time measured in the reference frame moving at the group velocity.  is the nonlinear parameter that takes into account the optical Kerr effect, f is the fibre loss, and 2 is the second-order dispersion parameter. Figures 9.6 and 9.7 show their results. Figure 9.6 shows (a) a chaotic oscillation from the transmitter, (b) the binary bit-sequence used for message signal, (c) the chaotic waveform containing the message, (d) the synchronized chaotic oscillation generated in the receiver laser, (e) the decoded signal, and (f) a low-pass filtered version of the decoded signal. The message signal m was extracted by comparison of the received signal Eext t∝ Em t with the regenerated chaotic signal Er t, as  m t =


Eext t
2 −1
Er
2

(9.13)

The results showed that excellent chaos synchronization and message recovery could be attained after propagation through 50 km of optical fibre. Figure 9.7 shows the performance,

Methods for Modulation and Recovery of Messages

PM [mW]

(a)

315

14.0 12.0 10.0 8.0 6.0 4.0 2.0

(b) Message

0.1 0.0 –0.1 PM with Message [mW]

14.0 12.0 10.0 8.0 6.0 4.0 2.0

PS [mW]

(c)

14.0 12.0 10.0 8.0 6.0 4.0 2.0

(d)

Decoded Message

(e) 0.1 0.0 –0.1

Recovered Message

(f) 0.1 0.0 –0.1 0

2

4

6 t [ns]

8

10

12

Figure 9.6 Numerical example of chaos communications based on CMA through optical fiber transmission of 50 km. (a) Chaotic output from the transmitter. (b) Encoded message sequence at a bit rate of 4 Gbps. (c) Transmitter output including the message. (d) Output of the receiver. (e) Decoded message. (f) Decoded message after filtering. Source: After Sánches-Díaz, et al. ©1999 IEEE.

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(a)

14 12 10 8 6 4 2

(b)

0.04

0.04

0.02

0.02

0.00

0.00

–0.02

–0.02

–0.04

–0.04

0.04

0.04

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14 12 10 8 6 4 2

(c)

14 12 10 8 6 4 2

(d)

Recovered message

0.00 Eye diagram

Ps (mw)

14 12 10 8 6 4 2

–0.02 –0.04 0.04 0.02

2

4

6

8 10 12 14 PM (mw)

–0.04 0.04 0.02

0.00

0.00

–0.02

–0.02

–0.04

–0.04

0.04

0.04

0.02

0.02

0.00

0.00

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–0.02

–0.04 0

0.00 –0.02

–0.04 0

100

200 300 t (ps)

400

500

0

2

4

6 8 t (ns)

10

12

Figure 9.7 Performance of chaotic data transmission in CMA through optical fiber transmission. Fiber lengths are 50, 100, 150, and 200 km from (a) to (d), respectively. The message is modulated at a bit rate of 2 Gbps. Left column; correlation plots of the transmitter and receiver lasers without a message. Center column; eye diagram of the recovered message after decoding and filtering. Right column; encoded and recovered message. Source: After S´anches-D´ıaz et al. ©1999 IEEE.

for modulation at 2 Gbps, after transmission over different lengths of optical fibre: (a) 50, (b) 100, (c) 150, and (d) 200 km, respectively. The synchronization and the quality of the decoding become worse after longer lengths of fibre transmission, due to the accumulation of the effects of dispersion and nonlinearity during propagation. Finally, we present an example of the CMO method. This scheme requires the achievement of complete synchronization and hence is the most difficult to realize experimentally [42]. The possibility of transmission of a NRZ pseudo-random binary sequence at 2.5 Gbps using CMO was demonstrated by numerical simulation [39]. In this example it is assumed that an electro-optical modulator is inserted into the optical feedback loop, and used to modulate the feedback signal as mCMO t = 1 +  for ‘1’ and mCMO t = 1 −  for ‘0’, where  < 1 is the modulation depth. Figure 9.8 shows (a) the dependence of the synchronization error on the amplitude of the embedded signal , and (b) the dependence of the BER on the frequency of the detuning between the transmitter and receiver lasers, both with noise added (triangles) and without noise (circles). The results show good synchronization (low ) for a wide range of relative modulation amplitudes and sensitive dependence of the BER on the detuning, which are strong features of the CMO method using complete synchronization.

Mechanisms for Synchronization and Signal Recovery

317

60 50

σ (%)

40 30 20 10 0

0

0.2

0.4

0.6

χ

0.8

1.0

(a) 0 –1

log(BER)

–2 –3 –4 –5 –6 0.0001

0.001

0.01

0.1

1

10

100

1000

Ω/2π (MHz) (b)

Figure 9.8 Calculated synchronization error and bit error rate (BER) for the normalized dimensionless bias injection currents of 0.67, the normalized feedback and injection ratios of 0.01. (a) Synchronization error versus modulation amplitude . (b) BER versus frequency detuning at the modulation factor of  = 015. Black dots: no noise, triangles: noise at SNR = 40 dB.

9.4 MECHANISMS FOR SYNCHRONIZATION AND SIGNAL RECOVERY In the following sections we review work that has been done on analyzing and characterizing the mechanisms for synchronization of the chaotic receiver laser and the recovery of the message from the chaotic carrier. From the symmetry of the dynamical equations in the theoretical formulation, we can see that in the case of CMO there exists an exactly synchronous solution even in the presence of the message. This symmetry property allows an exact synchronous solution even for large amplitude modulation in the CMO case. However, this symmetry is not shared by the CMA

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scheme. If there is an exact synchronization state which is stable (i.e., the synchronization state is an attractor in the dynamical sense) before messages are added, then we can expect that when a small amplitude message is added, the message will be suppressed in the receiver due to the attracting properties of the synchronization state. However, we cannot easily predict theoretically what will happen when the modulation amplitude is large or when there is no exact synchronization state. Nevertheless, experiments and numerical analysis have shown that good chaotic synchronization is possible in certain regimes even when the conditions for exact synchronization are not satisfied, and that message recovery is possible in CMA schemes even with quite large amplitude messages. As explained in Chapter 6, there is more than one type of synchronization in systems with delayed feedback. Correspondingly, there is more than one regime of operation of the receiver laser where signal recovery is possible. Figure 9.9 shows possible regions of chaos synchronization in the phase space of the frequency detuning between the two lasers and the optical injection ratio [43]. There are two different regimes of synchronization corresponding to the two types of synchronization; complete chaos synchronization (for detuning close to zero) and generalized synchronization (for a large range of detunings at high optical injection levels). In Figure 9.9, solid lines show the boundary between the injection-locking and the non-locking areas when the signal from the transmitter has a constant intensity. For a constant optical injection from the transmitter laser, injection-locking states exist when the condition  √ 1 + 2 Sm (9.14)

≤ 2 Ss

10

Generalized synchronization area

R (%)

1.0

0.1

Error [%] 40~ 32 Stable injection 24 locking region

0.01

Unstable injection locking region

16 0.001

Complete synchronization area

8 0 –10

–5

0 ∆f [GHz]

5

10

Figure 9.9 Numerical results of the areas for complete and injection-locking chaos synchronization in the phase space of the frequency detuning (horizontal axis) and the optical injection rate (vertical axis). The system is CMA and open-loop. The synchronization error is represented by grey level.

Mechanisms for Synchronization and Signal Recovery

319

is satisfied. Here,  is the frequency detuning and Sm and Ss are the photon numbers (intensities) in the transmitter and receiver lasers, respectively. In Figure 9.9, the region inside the symmetrical solid curves shows the above condition. In parts of this region the steady injection locking state exists but is unstable. The boundary of stability of the steady injection locking state is shown by the dashed curve in the figure. (The laser parameters for the transmitter and the receiver are assumed to be the same.) The dark area in Figure 9.9 shows the region where good chaos synchronization is attained. The area of complete synchronization is situated in the very narrow region near zero detuning and small optical injection ratio. In this region, chaotic oscillations are usually observed even for constant optical injection. On the other hand, an area of good generalized synchronization is also found at higher injection ratios, within the region where injection-locking is stable for constant optical injection, and it is realized over a broad range of frequency detuning. An important difference between the different types of synchronization is the time lag in the synchronized output with respect to the injected transmitter signal. It is essential to allow for the correct time lag when comparing the injected signal and the receiver laser output signal to recover messages. When complete chaos synchronization is realized, the receiver laser has a chaotic signal which is in advance of the injection signal that drives it, and the time shift between the two chaotic waveforms is: ER t = ET t + t

(9.15)

where t =  − t . This scheme is called anticipating synchronization [17, 18] as the receiver can anticipate the injected signal. For the case of generalized synchronization, the time shift between the two waveforms is given by ER t = ET t − t 

(9.16)

i.e., the receiver laser follows the injected signal [17, 18]. It is essential to allow for these timing differences when comparing the received signal and the synchronized laser output signal to successfully recover the message. Now, let us show an example from an experiment which demonstrates the basic mechanism behind the recovery of a message that has been added to a chaotic carrier by the CMA method. Figure 9.10(a) shows examples of chaotic oscillations of intensity of light output from the transmitter and receiver lasers when a message is included in the transmitter output [26]. The message was a sinusoidal wave with a frequency of 1.5 GHz imposed by modulating the injection current of the transmitter laser. The modulation depth was −14 dB of the averaged chaotic intensity, and the intrinsic relaxation oscillation frequency of the solitary laser was about 4 GHz. The two chaotic waveforms appear to be synchronized in spite of the presence of the message. The synchronization is confirmed by the correlation plot in Figure 9.10(b). The correlation value (Equation (9.10)) in this case is about 0.86. Figure 9.11 shows the corresponding RF spectra. Besides the broad spectral peaks of the external cavity mode and its higher harmonics, a sharp spectral peak for the message at 1.5 GHz is clearly visible in the transmitter spectrum. The receiver spectrum bears resemblance to that of the transmitter except there is no distinct spectral component corresponding to the message. This effect has been called ‘chaos-pass-filtering’ [24], and results from the tendency of the receiver laser to re-generate only the intrinsic chaotic oscillations similar to those in the transmitter. This is

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(a)

Output Power (a.u)

Transmitter

Receiver

0

10

20

30

Time (ns)

Receiver Output Power (a.u)

(b)

Transmitter Output Power (a.u)

Figure 9.10 Experimental example of data transmission in closed-loop CMA system. (a) Transmitter signal with a message of 1.5 GHz sinusoidal wave and synchronized waveform in the receiver, and (b) X-Y correlation plot for the waveforms of (a). The correlation value is 0.86. The bias injection currents for the transmitter and receiver lasers are J = 150Jth and 156Jth , respectively. The feedback fraction in the transmitter system is 3.75% (intensity) and the optical injection is 6.54%.  is 2.3 ns.

the key mechanism for recovering the message as we can now extract the message from the chaotic carrier by simply subtracting the chaotic signal reproduced in the receiver laser from the transmitted signal. Looking at the spectrum in Figure 9.11(a), it might be suspected that the message may be extracted by filtering the waveform with a narrow bandpass filter at the message frequency. Figure 9.12 shows the filtered waveforms for (a) the extracted message signal, (b) the transmitted signal, and (c) the receiver output, all obtained using a narrow bandpass

Mechanisms for Synchronization and Signal Recovery

321

0

(a)

Output Power (dB)

–20

–40

–60

–80 0

1

2

3

4

Frequency (GHz)

0 (b)

Output Power (dB)

–20

–40

–60

–80 0

1

2

3

4

Frequency (GHz)

Figure 9.11 RF spectra corresponding to Figure 9.10. Spectra of transmitter (a) and receiver (b) outputs.

filter of bandwidth of 200 MHz centered at the message frequency of 1.5 GHz. The message was extracted by a simple subtraction of the receiver output from that of the transmitter. The extracted message shows an excellent sinusoidal oscillation which is almost the same as the original message. But the filtered waveforms for the transmitted signal and the receiver output are not good harmonic signals and they are not even in-phase with the message signal. This demonstrates the difficulty of extracting the message by only band-pass filtering the received signal. Finally, we note that the message can be better hidden than in this example by reducing the amplitude of the modulation.

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Output Power (a.u)

(a)

(b)

(c)

0

1

2

3

4

5

Time (ns)

Figure 9.12 Effects of filtering for signals in Figure 9.11(a) with a narrow band-pass filter of ± 100MHz centred at the message frequency of 1.5 GHz. (a) Decoded message, (b) transmitter, and (c) receiver outputs.

9.5 PARAMETER SENSITIVITY, ROBUSTNESS AND SECURITY FOR SYNCHRONIZED CHAOS COMMUNICATION The successful demonstrations of synchronization and data transmission in experiments have proved that these schemes are robust to some degree, but there is much still to be done in terms of evaluating the robustness and practical trade-offs with the security based on parameter sensitivity. A system based on the use of synchronization for replication will necessarily have some allowance for parameter mismatches. The smaller this allowance, the greater the security can be if the parameters are used as secret keys. On the other hand, the sensitivity of the lasers to changes in parameter may cause difficulties for stable operation. In this section we consider the issue of parameter mismatches with respect to chaos synchronization. But first we make a comment on the issue of security. Another aspect to the security of messages hidden in chaotic waveforms is the difficulty of separating the message from the chaotic carrier by analyzing recorded waveforms. It has been demonstrated that systems with low-dimensional chaos are not secure for data transmissions, in the sense that a low dimensional attractor is easily reconstructed from time series data, and system parameters are easily estimated from this attractor [44, 45]. Decoding without knowing key parameters becomes more difficult with the increase of the dimension of the chaotic dynamics [46]. The system of a semiconductor laser with optical feedback, as described by the theoretical model, is embedded in an infinite dimensional system due to the delay. However, the actual dimension of the dynamics is typically much lower, being restricted by the intrinsic response times in the laser. Quantitative analysis of the dimension of the synchronized chaos and the degree of security remains an important challenge for future study. Here we limit our discussion to the issue of the matching of the receiver laser for message recovery by synchronization.

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323

Now, let us consider in more detail the dependence of synchronization on parameter mismatch. Figure 9.13 shows the dependence of synchronization errors on parameter mismatches for the same systems as in Figure 9.9 [43]. Figure 9.13(a) is the case for complete chaos synchronization. In this case, synchronization is attained for a very small range of parameter mismatches and the accuracy is lost with small parameter mismatches. On the other hand, the allowance for the parameter mismatches is rather large for the case of the generalized synchronization, as shown in Figure 9.13(b). In this figure, the frequency detuning was assumed zero and the optical injection ratio was 0.04 % of the intensity. The accuracy of the synchronization is worse than that of the complete chaos synchronization. However, it gradually decreases with the increase of the parameter mismatches. It is also 100

σ %

Error (%)

10

1

: τp : τe :g :α

(a) 0.1 –5

–4

–3

–2

–1 0 1 2 Parameter Mismatch (%)

3

4

5

100

σ %

Error (%)

10

1

: τp : τe :g :α (b)

0.1 –50

–40

–30

–20

–10 0 10 20 Parameter Mismatch (%)

30

40

50

Figure 9.13 Calculated synchronization errors in CMA system as a function of mismatches of the laser device parameters. (a) Complete and (b) injection-locking chaos synchronization. The system is open-loop. p : photon life time, e : carrier life time, g: gain, : linewidth enhancement factor.

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noted that the best synchronization is not always attained at zero parameter mismatches. Also the error rates have non-symmetrical dependence on the parameter mismatches. This is considered to be a result of a non-symmetrical gain profile or non-symmetrical  parameter characteristic of semiconductor lasers. Clearly, the effects of the parameter mismatches are quite different for the respective synchronization schemes. The conditions for complete chaos synchronization are more constrained. Hence the complete chaos synchronization state is less robust with respect to parameter variations, but offers the possibility of higher degree of security based on the use of parameter values as secret keys for message decoding. Also note that the error can be much lower for complete chaos synchronization. The fidelity of the chaos synchronization in a CMA scheme is degraded by the inclusion of a message in a transmitted signal, and the degree of degradation depends on the amplitude and the frequency of the message signal. Figure 9.14 shows the results of a numerical analysis of the accuracy of complete chaos synchronization in the presence of a message [17]. The dependence of accuracy on bias injection current is shown. Figure 9.14(a) is a bifurcation

Output Power (mW)

(a) 4

2

0 1.3

1.4 Injection Current (×Ith)

1.5

Correlation Coefficient

1.0 (b)

0.5

0

1.3

1.4

1.5

Injection Current (×Ith)

Figure 9.14 Injection current parameter dependence of chaotic oscillation and synchronization error in CMA system (a) Chaotic bifurcations with variation of the bias injection current and (b) synchronization errors in the presence of a message. The system is closed-loop. The laser is directly modulated by a 500 MHz sinusoidal wave modulation of the injection current. The modulation depth is 2% and  is 2 ns. The solid and broken lines in (b) are the correlations without and with message, respectively.

Communication Bandwidth

325

diagram showing the oscillations of peak values of the output of the transmitter laser as a function of the bias injection current. The transmitter laser shows chaotic behaviour throughout this range of the injection current except for a small window. The degree of the synchronization in a CMA scheme is shown in Figure 9.14(b). Without a message (solid line), the correlation coefficient is almost unity throughout the injection current range and the receiver laser shows excellent synchronization with the transmitter laser. With a message in the form of a 500 MHz sinusoidal modulation to the injection current, the calculated correlation coefficient is shown as a dotted line. The depth for the injection current modulation is about 2% (the resulting optical intensity modulation corresponds to about 4% of the averaged chaotic intensity fluctuations). The correlation coefficient in the presence of a message is always less than that without a message, but there is a range where adequate synchronization and message recovery is possible. The use of electro-optic modulation, rather than injection current modulation, in CMA is expected to give better synchronization and recovery. The discussions here have focused on schemes using chaos generated by semiconductor lasers with coherent optical feedback. In these schemes the optical feedback and synchronization signals are coherently coupled with the internal laser field. Hence the detuning of the frequencies between the two lasers plays a crucial role for the synchronization. However, in practice it is not easy to control the frequencies for lasers and a drift of the frequencies due to the external perturbations, such as a temperature drift, can easily destroy the synchronization. The fine dependence on detuning in coherently coupled lasers can be an advantage from the point of view of security, but causes stability problems in practice. We note that chaotic systems based on incoherent optical feedback and injection have also been proposed [28]. Chaotic synchronization and data transmissions can be realized using similar architectures as described for coherent feedback except for the use of polarization optics (such as quarter wave plates and polarizers) in the optical paths to make the optical feedback and the optical injection signal incoherent with respect to the lasing light. These systems do not have sensitive dependence on wavelength differences between transmitter and receiver lasers. Another closely related scheme involves semiconductor lasers with optoelectronic feedback – feedback modulation of the injection current by a signal proportional to the output optical intensity. Tang et al. have demonstrated transmission of low error NRZ bit-sequences at 2.5 Gbps (OC-48 standard data rate for communications) [16], and Liu et al. have performed numerical experiments of data transmission at 10 Gbps to directly compare different schemes for generating chaotic carriers – opto-electronic feedback, optical feedback and optical injection [29]. Since the channel noise is not injected directly via the optical field in the case of the opto-electronic feedback system, it has an advantage with respect to robustness against channel noise within the frequency range where analog electronic circuits are available.

9.6 COMMUNICATION BANDWIDTH The dependence of the fidelity of recovered signals on the frequency of modulation is an important property determining the communication bandwidth of the system [17, 47]. Figure 9.15 indicates the dependence of the accuracy of message recovery on the message modulation frequency for the same system as that in Figure 9.14. In the numerical calculation, the bias injection current was set to be 1.42Ith where good synchronization was attained

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Error (%)

40

30

20

10 0

0.5 1.0 Message Frequency (GHz)

1.5

Figure 9.15 Dependence of error of recovered signal on message frequency. The relaxation oscillation frequency (carrier frequency) of the laser is 2.6 GHz. The conditions are the same as those in Figure 9.14.

for message frequencies less than 500 MHz. The relaxation oscillation frequency of the modeled laser was 2.6 GHz at the operating bias injection current. For low frequency modulation (