Beam Shaping and Control with Nonlinear Optics (NATO Science Series: B:)

Beam Shaping and Control with Nonlinear Optics Edited by

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Beam Shaping and Control with Nonlinear Optics

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Series B: Physics

Beam Shaping and Control with Nonlinear Optics Edited by

F . Kajzar Commissariat a l’Energie Atomique Gif-sur-Yvette, France and

R. Reinisch Institut National Polytechnique de Grenoble Grenoble, France

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PREFACE

The field of nonlinear optics, which has undergone a very rapid development since the discovery of lasers in the early sixties, continues to be an active and rapidly developing research area. The interest is mainly due to the potential applications of nonlinear optics: directly in telecommunications for high rate data transmission, image processing and recognition or indirectly from the possibility of obtaining large wavelength range tuneable lasers for applications in industry, medicine, biology, data storage and retrieval, etc. New phenomena and materials continue to appear regularly, renewing the field. This has proven to be especially true over the last five years. New materials such as organics have been developed with very large second- and third-order nonlinear optical responses. Important developments in the areas of photorefractivity, all optical phenomena, frequency conversion and electro-optics have been observed. In parallel, a number of new phenomena have been reported, some of them challenging the previously held concepts. For example, solitons based on second-order nonlinearities have been observed in photorefractive materials and frequency doubling crystals, destroying the perception that third order nonlinearities are required for their generation and propagation. New ways of creating and manipulating nonlinear optical materials have been developed. An example is the creation of highly nonlinear (second-order active) polymers by static electric field, photo-assisted or all-optical poling. Nonlinear optics involves, by definition, the product of electromagnetic fields. As a consequence, it leads to the beam control. This includes amplitude or phase modulation, generation of new laser frequencies and altering the propagation of beams either in space or time. Different nonlinear optical interactions and mechanisms lead to a large variety of functions. The time thus seemed appropriate to us to bring all these new developments into focus in a summer school format. This was the main objective of the NATO Advanced Science Institute held in Cargese (Corsica, France) August 4–16, 1997. A good understanding of nonlinear optical phenomena and their dependence on wavelength, electronic structure, structural properties, etc. requires not only an excellent knowledge of the basic laws of physics, governing the nonlinear optical phenomena, but also a good knowledge of materials and the laws governing the interaction between light beams in different forms of matter. The lectures given at the school covered the following topics: nonlinear optical phenomena and their applications, temporal and spatial solitons, third-order effects, organic materials, organic and inorganic multiple quantum wells, hybrid excitons and microcavity effects, cascading effects and applications, parametric processes, applications of optical parametric oscillators and second harmonic generators, the latest developments in nonlinear magneto-optics, new techniques for molecule orientation, ato-optics, light-induced kinetic effects in gases, light upconversion to the blue, nonlinear waveguiding optics, photorefractive effects and photorefractive solitons, χ (2) spatial solitons. The subjects covered by the school underline the importance of the ever improving fundamental research and continuing technological developments. We hope that this book will contribute to the dissemination of

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the theoretical and experimental results concerning this fascinating field of all-optical interactions. Organization of the school would be impossible without financial support. We are highly indebted to its main sponsor: the NATO Scientific Affairs Division. The financial contributions from other sources such as Centre National de la Recherche Scientifique, Direction des Systemes de Forces et de la Prospective de la DGA, Institut National Polytechnique de Grenoble, LETI-Saclay, and Centre National d’Etude des Telecommunications were also very helpful and we would like to thank these organizations for their support. We would like also to acknowledge the Scientific Committee members V. M. Agranovich, G. Assanto, C. Flytzanis, S. Kryszewski and G. Stegeman for their suggestions and help in the organization of the school. Thanks are also due to Ms. Amelie Kajzar for her assistance in the organizational tasks and in the preparation of these proceedings. Finally, many thanks are due to the staff of the Institut Scientifique de Cargese for its efficiency and kindness from which we benefited during the school, and more generally to all the lecturers and students for their contribution in making this meeting very pleasant and successful. François Kajzar Raymond Reinisch Saclay and Grenoble

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CONTENTS

Introduction to Nonlinear Optics: A Selected Overview . . . . . . . . . . . . . . . . . . . . . . . . 1 G. I. Stegeman Introduction to Ultrafast and Cumulative Nonlinear Absorption and Nonlinear Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 E. W. Van Stryland From Dipolar Molecular Engineering to Multipolar Photonic Engineering in Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 J. Zyss and S. Brasselet Molecule Orientation Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 F. Kajzar and J.-M. Nunzi Nonlinear Pulse Propagation along Quantum Well in a Semiconductor Microcavity . . 1 3 3 V. M. Agranovich, A. M. Kamchatnov, H. Benisty, and C. Weisbuch Some Aspects of the Theory of Light Induced Kinetic Effects in Gases . . . . . . . . . . . 1 4 9 S. Kryszewski Temporal and Spatial Solitons: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 8 3 A. Boardman, P. Bontemps, T. Koutoupes, and K. Xie Spatial Solitons in Quadratic Nonlinear Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 9 L. Torner Photorefractive Spatial Solitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 9 M. Segev, B. Crosignani, P. Di Porto, M.-F. Shih, Z. Chen, M. Mitchell, and G. Salamo Sub-Cycle Pulses and Field Solitons: Near- and Sub-Femtosecond EM-Bubbles . . . . . 2 9 1 A. Kaplan, S. F. Straub, and P. L. Shkolnikov Nonlinear Waveguiding Optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 9 R. Reinisch Quadratic Cascading: Effects and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 1 G. Assanto

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Nonlinear Optical Frequency Conversion: Material Requirements, Engineered Materials, and Quasi-Phasematching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 M. Fejer Low-Power Short Wavelength Coherent Sources: Technologies and Applications . . . . 407 D. Ostrowsky Artificial Mesoscopic Materials for Nonlinear Optics . . . . . . . . . . . . . . . . . . . . . . . . . . 427 C. Flytzanis Contributors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .465 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

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INTRODUCTION TO NONLINEAR OPTICS: A SELECTED OVERVIEW George I. Stegeman C.R.E.O.L., University of Central Florida 4000 Central Florida Blvd., Orlando, FL 32816-2700, USA

INTRODUCTION Historical Perspective Nonlinear optics (NLO) has enjoyed great success as a discipline for over 30 years now. Although it was a relative newcomer to nonlinear wave sciences, other examples being nonlinear fluid dynamics, acoustics, plasmas etc, it has contributed many new phenomena. Since its inception in 1962, nonlinear optics has passed through many phases and different topics have been “hot” at any given time. 1,2 One of the fascinating features of the nonlinear optics field is its regenerative power to develop new topics over the years. The first ten years witnessed demonstration of many of the fundamental interactions such as second harmonic generation (SHG), sum and difference frequency generation, stimulated Raman, Brillouin and Rayleigh scattering, self-focusing etc. And many more interesting phenomena were predicted theoretically, some having to wait two decades before experimental confirmation was forthcoming. This “novelty” trend continued into the second decade with the development of multiple nonlinear spectroscopies and their applications to materials science, phase conjugation, bistability leading to concepts of all-optical signal processing, the beginnings of nonlinear optics in fibers, etc. The third phase, from about the mid 1980s to the present has had its own highlights such as the development of nonlinear guided wave optics, especially in fibers where a whole spectrum of new propagation effects and light induced noncentrosymmetric effects were found, the exciting development of efficient, widely tunable sources through optical parametric oscillators, temporal solitons and their potential for longhaul communications, a surprising variety of spatial solitons, terahertz sources, femtosecond pulses, generation of tens of higher harmonics in gases etc. One of the exciting recent developments is the blurring of the roles of second and third order nonlinear optics, namely the creation of second order nonlinear effects via third order nonlinearities, and the use of second order effects to mimic third order phenomena. Many of these topics will be discussed in this book. The key to applications of nonlinear optics is, has been and always will be the availability of appropriate materials. The initial stages of the field which focused on demonstrating and understanding new effects utilized the materials available at that time. For

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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example, for second harmonic generation, materials developed for piezoelectric applications which also require non-centrosymmetric media were used first. In the case of third order, much of the early work was done with liquids. Ultimately, the search for better materials was driven by the realization that in order for any applications to be practical, nonlinear optics had to move forward from the era in which high power lasers were almost exclusively needed to observe nonlinear phenomena. Compact semiconductor lasers with 100s of mW power levels drove the need for sub-watt nonlinear optics. The search for better materials gained momentum in the mid to late 1970s and continues unabated to the present day. In the case of second order materials, there have been multiple goals including doubling into the UV region of the spectrum, widely tunable sources via parametric interactions, inexpensive sources in the blue, etc. The exciting concept of all-optical processing has fueled third order nonlinear optics for many years. Formalism Traditionally nonlinear optics has been discussed in terms of the nonlinear polarization induced in a nonlinear medium by the mixing of one or more intense electromagnetic waves.2,3 Typically multiple beams with different frequencies are incident onto a nonlinear medium, either modifying the linear optical properties of the medium or leading to the generation of new waves at new frequencies. As a matter of notation, the incident fields E(r,t) of frequency ω i and wavevector ki for propagation along the z-axis can be written in the form:

(1)

For plane waves, ei is the electric field unit vector, fi (x,y) = 1 and a i (z), the slowly varying (complex) amplitude, is normalized so that | a i (z) | ²is the intensity in units of W/cm² . When the interacting beams are of finite extent, for example in a waveguide, the f i (xy) describe the transverse field profiles. The nonlinear polarization

(2) induced by the mixing of the optical fields has the general form:²

(3)

where E(r,t) is the total electric field in the medium and χ(n) is the n’th order nonlinearity. Here the r - r i allow for a spatially non-local response, e.g. carrier diffusion and t - t i allow for a polarization field at time t to be generated by fields at an earlier time t i , for example due to a finite carrier recombination time. For a total field of the form E(r,t) = ∑ ½ E(r;ω i ) exp[i(ω it - k i r)] + c.c., i.e. an expansion in terms of its Fourier components, the induced polarization can be written as a Taylor’s expansion in the Fourier components of the mixing fields,² 2

(4)

Here k p = k1 + k 2 + k 3 + … is the wavevector of the induced polarization, with the number of terms determined by the order of the nonlinearity. Furthermore if the complex conjugate term participates in the mixing, the corresponding field term E( ω i ) in the field product appears as the complex conjugate, i.e. E*(ω i ). The polarization field, and any electromagnetic field that it subsequently generates, oscillates at the frequency ω = ω1 ± ω 2 ± ω 3 ± … where the minus sign corresponds again to the conjugate terms, if appropriate. The Di (ω ), the degeneracy factors unique to each nonlinear interaction, essentially “count” the number of equivalent terms which contribute to the polarization field for a given set of input fields.² For example, D3 ( ω ) takes the values 1, 3, 6 and 6 for third harmonic generation (THG), an intensity-dependent refractive index (n 2 ), electric field induced second harmonic generation (EFISH) and degenerate four wave mixing (DFWM) respectively. The nonlinear polarization source term now drives electromagnetic fields at the frequency ω . This part of the nonlinear optics problem is a relatively straightforward application of Maxwell’s equations, leading to the usual polarization driven wave equation. These fields can be both radiating and non-radiating in nature. Usually, only the radiative fields are of interest because they can under certain conditions grow to be usefully large. This will be clear with subsequent specific examples. It is important to note that the well-known polarization expansion, given above, is not complete and was originally meant to describe nonlinear phenomena which involve purely electronic nonlinearities. As such, the crystal symmetry (or lack of it) plays a key role in determining which tensor components are zero, and which are non-zero. However, in the modern context this polarization expansion is used to describe any and all processes which involve optically induced nonlinearities, including charge excitation in semiconductors, thermal effects etc. It is important to note that the physics of the nonlinearity, in concert with the crystal symmetry determines the non-zero coefficients and their inter-relationships.² Higher order terms can be and sometimes are included, but to date only the fifth order term χ(5) (- ω; ω1 , ω2 , ω 3 , ω 4 ) in the limit ω = ω 1 = - ω2 = ω 3 = -ω4 has proven to be important. It occurs as a correction to an intensity-dependent refractive index which itself is proportional to χ(3) (- ω; ω1 , -ω 1, ω1 ). Furthermore, additional terms are typically added to include interactions with other types of excitations such as magnetic or acoustic waves, polaritons etc. As implied by their arguments, the nonlinear susceptibility coefficients all undergo dispersion, i.e. are frequency dependent.² There is resonant enhancement of the coefficients inherent in the spectral content of the nonlinear coefficients, the exact form depending on the physics of the interaction of the electromagnetic fields with matter. For example, it can reflect the interaction of radiation with multi energy level molecules, virtual and real transitions of electrons in semiconductors etc. Near such resonances the χ(n) are complex, far from such resonances they are real. This resonant enhancement can be very useful. For example, nonlinear spectroscopy such as multi photon absorption accesses transitions which are not one photon active, for example two photon states in centrosymmetric molecules. Furthermore, the large nonlinearities can be used for bistable logic elements. A common oversight made in nonlinear optics involves the dispersion of nonlinearities with frequency. It is incorrect to assume that the nonlinearity χ(3) measured with a specific technique at some frequency is the same as the value used in a different interaction with different frequency inputs. Consider a simple case of dispersion frequently derived in textbooks, χ (2) based on the anharmonic electron oscillator model.² Specifically,

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(5)

where the constant K contains the details of the anharmonic forces, etc. The key point is that even this simple susceptibility contains two resonance denominators which produce sizeable dispersion and large enhancements. Until recently, the roles played by χ(2) and χ (3) were well defined. Second order phenomena were used for frequency conversion, usually second harmonic generation, sum and difference frequency mixing, optical parametric oscillators etc.² Also in this class is the electro-optic effect which has been investigated from the 1970s for modulators. On the other hand, third order phenomena are the origin of many effects. For example, the intensity-dependent refractive index derived from χ(3) (- ω:ω,- ω,ω), has been applied to alloptical signal processing, spatial and temporal solitons (discussed in later chapters).² Other interesting phenomena such as phase conjugation, electric field induced second harmonic generation, and nonlinear spectroscopies such Raman gain, coherent Anti-Stokes Raman scattering etc. have all been investigated.. One of the most interesting developments of the last five years is the use of χ(2) t o mimic effects well known in χ (3) , and vice-versa. For example, optically induced electric fields lead to charge migration and permanent second order nonlinearities for second harmonic generation in glass fibers.5 Conversely, the strong energy exchange between a fundamental and its second harmonic leads to nonlinear phase shifts, and even to solitons.6 This area will be explored in subsequent chapters.

SECOND ORDER NONLINEAR PHENOMENA Second Harmonic Generation It is a useful exercise to work through the formalism of second harmonic generation in detail. It is typical of a wide spectrum coherent interactions.² The total incident field is (6) which corresponds to Type I SHG with one fundamental input field and one harmonic field. The dominant terms in the field product Ej E k are: (7) which give the following nonlinear polarization terms (8a) (8b) These terms are then substituted into the polarization driven wave equation (9)

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which gives, for example at the frequency ω,

(10) Note that it has been assumed that the so-called slowly varying phase and amplitude approximation.² Since the fields are eigenmodes of the wave which gives equation, (11)

with ∆ k = k 3 - 2k 1 . A similar analysis gives the second coupled mode equation at 2 ω : (12)

with where the ei are the electric field unit vectors. Furthermore, Kleinman symmetry was assumed so that deff (- ω ) = deff (-2 ω). Assuming the simplest case of negligible depletion of the fundamental, (13)

There are a number of parameters which are important for second harmonic generation. 1. SHG Figure of Merit: the smaller of Here α is the dominant loss mechanism, usually at the second harmonic frequency. 2. Phase-matching condition: ∆φ = 0 This is a generalization to the case where there are where two orthogonally polarized fundamental beams with wavevectors k 1 ( ω ) and k2 ( ω). 3. Minimize beam cross-sectional area A (which can be achieved in waveguides) 4. Minimize absorption (because it can lead to the thermal detuning of the phase-matching condition and reduce the effective length of the medium) 5. Environmental and chemical long term stability Good mechanical properties (for polishing) 6. 7. Material homogeneity .......... 8. SHG Phase-Matching: Bulk Media The dispersion of refractive index with frequency in the wavelength range 300nm < λ < 2000nm makes it essentially impossible to phase-match second harmonic generation in bulk crystals with co-polarized fundamental and second harmonic beams without the artificial introduction of additional dispersion, for example through a grating.² In general, the refractive index decreases with increasing wavelength due to electronic resonances in the UV and near UV visible part of the spectrum. The standard phase matching techniques involve a polarization change between one of the fundamental inputs and the second harmonic.

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Figure 1 Phase-matching conditions for (a) Type I SHG and (b) Type II SHG in birefringent media where e and o identify the e- and o-polarized beams.

Type I SHG: Here there is only a single fundamental input beam so that ∆φ = ½ [k3(2 ω) 2k1( ω )]L and the fundamental and harmonic are orthogonally polarized. Writing k3= 2 ωn 3/c and k1 = ω n 2/c, n1 =n3 is required. This can be achieved in a birefringent crystal, for example as shown in Figure 1a in which no > ne, i.e. the “ordinary” polarized refractive index is larger than the “extraordinary” polarized index. Type II SHG: Here there are two orthogonally polarized fundamental beams with different refractive indices, for example n0 = n1 > ne = n 2. One photon from each polarization is used to form a harmonic photon. Therefore now ∆φ = ω[n 3 -½(n 1 +n2)]L/c so that n 3 = ½(n1+n2) is required for phase-matching. This case is shown schematically in Figure 1b. Very recently a new approach has been developed for bulk crystals called Quasi-PhaseMatching (QPM) SHG.7 It has proven possible to use periodic electric fields in ferroelectric crystals like lithium niobate and lithium tantalate to periodically reverse the ferroelectric domains and hence the nonlinear coefficient d(2)333. For a domain period Λ = 2 π/ κ, the phase mismatch becomes ∆φ = ½ [k3 (2 ω) - 2k1 (ω) ± κ ]L. Therefore, choosing the period to produce wavevector conservation gives ∆φ = 0. SHG Phase-Matching in Waveguides The best conversion efficiency occurs when a minimum beam cross-sectional area can be maintained over the full length of the sample. This can be achieved in fiber or channel waveguides whose cross-sectional areas can be of order a few λ2 .8 The usual geometry used is a channel waveguide by which we mean a waveguide whose shape does not have circular symmetry but is more rectangular in nature. There is a “core” guiding region surrounded by media of lower refractive index. Typically a discrete number of modes with propagation wavevectors β m, n (along z) are allowed for a given index difference between the core guiding region and the dimensions and a given shape of the guiding region where m and n are integers starting from 0. The corresponding field distributions fm n(x,y) are oscillatory across the two channel dimensions with m zeros across the x-dimension and n along y. There are two sets of orthogonal modes, each of which contains electric fields along all three axes, including the

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propagation direction. However, for each set, usually one field component dominates, TEmn (TMmn) with the dominant electric field parallel (perpendicular) to the long channel crosssectional dimension (y) which is also usually parallel to the air-material interface. An example of the x-dependence of the TEmn field distributions, identified as TEm is shown in Figure 2.

Figure 2 Representative field distributions for some lowest order slab waveguide modes TE m .

Wavevector matching in waveguides requires finding conditions for which ∆φ = ½ 0. (Phase-matching in waveguides usually involves only one [β mn (2 ω ) - 2 β m’n’( ω )]L fundamental input.). Because there are in principle mn values for the propagation wavevectors β at each frequency, one for each mode, there are more degrees of freedom in achieving phasematching than there are in bulk media. The phase matching condition can be expressed more conveniently in terms of the waveguide mode effective indices, Neff = β /kvac as ∆φ = kvac( ω )[N mn(2ω ) - N m’n’ (ω)]L 0. As will be discussed later, there is a price to be paid for this flexibility in terms of efficiency. In addition to the Type I and Type II SHG phase-matching approaches, there are others which are unique to waveguides and they will now be discussed: (i) Modal Dispersion Phase Matching (MDPM) Here phase matching is implemented from a lower order fundamental mode to a higher order second harmonic mode, typically with both modes having the same polarization, although the mixed mode case is of course also possible. The modal dispersion curves in Figure 3 show by a circle the MDPM case for the simpler example of a slab waveguide. (The channel case would require a three dimensional plot with two axes, one for each transverse dimension.) First note that at a given frequency, the effective indices undergo dispersion with increasing waveguide dimension, from cut-off where Neff = ns, the substrate index for the slab case (air above), to Neff = nf, the guiding film index. Furthermore, there is dispersion with frequency in the refractive indices of both the substrate and film which in the visible and near infrared requires phase-matching from a lower to higher order mode. For the channel case, the MDPM condition needed is in a three-dimensional plot. The possible multiplicity of such MDPM conditions in multi-mode waveguides shows the flexibility of using waveguides versus bulk media. The loss in efficiency inherent to using modes of different orders is clear from the expression for SHG in a slab waveguide: (14) where the integral term is called the “overlap integral”. It effectively projects the second harmonic polarization onto the second harmonic field at every point in the waveguide and sums the product. For the case shown in Figure 3, the corresponding fields are shown in Figure 4a and it is clear that the overlap integral is small. However, a number of solutions to this problem have been successfully demonstrated. For example, in Figure 4b about one half 7

of the guiding film is made from a material with d (2)eff =0 so that no interference occurs between the two halves of the waveguide. 9 Another solution has been to reverse the sign of the nonlinearity at the position that the harmonic field reverses sign so that the overlap integral is again maximized, Figure 4c. 10 Both schemes have recently been implemented in poled polymer channel waveguides. 10,11

Figure 3 Dispersion in effective indices in a slab waveguide and the phase matching conditions for MDPM (circle), QPM (solid vertical arrow) and Cerenkov (range given by horizontal arrow).

Figure 4 (a) Field overlap condition for overlap integral for a TE0 ( ω) and a TE1(2 ω). Waveguide with (b) one half of the core linear and one half nonlinear and (c) with nonlinearity reversed at the harmonic field reversal plane.

(ii)

Quasi-Phase-Matching (QPM) This technique is very powerful because it allows phase-matching between the lowest order, co-polarized fundamental and harmonic modes.12 Only a brief description is given here because a subsequent chapter will deal with this concept in greater detail. A grating is introduced with periodicity Λ = 2 π/κ so that ∆φ = ½ [β 00 (2ω) - 2 β 00 ( ω) - κ]L 0. In terms of effective indices, ∆φ = k vac (ω )[N00 (2 ω) - N00 ( ω) - κ/2k vac ( ω)]L 0. As shown in Figure 3, this corresponds to a vertical translation between the dispersion curves and Λ can be chosen to minimize the effects of fabrication parameters on phase-matching. This modulation can be produced by either a linear grating, for example by modulating the waveguide dimensions, or by a nonlinear grating achieved by periodically reversing the nonlinearity d(2) with distance

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down the waveguide. Certainly the second approach is much more effective in optimizing the SHG efficiency. It has been implemented with various techniques in ferroelectric media such as LiNbO3 and LiTaO3 . 12,13 There has also been some work on polymer waveguides, but the efficiencies have been disappointing.14 (iii)

Cerenkov Phase Matching (CPM) This is an “automatic” wavevector-matching condition (parallel to the interfaces) made possible by the dispersion of material refractive index with wavelength in the visible and near infrared. l5 In Figure 3, note that there is a region of waveguide dimension “h” in which ns(2 ω ) > Neff( ω ). Because in the waveguide the projection of the wavevectors of the interacting electromagnetic modes must be conserved, then radiation fields are generated into the bulk at an angle θ from the surface given by ns(2 ω)cosθ = N eff( ω). The above discussion has been facilitated by considering only a slab waveguide, but clearly the concepts can be extended easily to the two dimensional confinement case. Other d(2) Parametric Processes SHG is only one of many possible parametric d(2) processes. Clearly sum and difference frequency generation also require phase-matching and they have been implemented in bulk and waveguide media. In fact, the application of difference frequency mixing to the shifting of the wavelength of communications signals has been investigated. An example of shifting from the 1310 nm communications band to the 1550 nm is to mix the 1310 signal with a laser at 708 nm to produce a signal 1540 nm.16 Alternatively, a shift in the signal frequency within the erbium amplifier band can be achieved.17

Figure 5 The range of a single OPO as compared to the range of numerous common narrow bandwidth sources.

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One of the most powerful applications of second order processes is to optical parametric generators (OPG), amplifiers (OPA) and oscillators (OPO).2 This is essentially a down-conversion process in which a pump beam at frequency ωp breaks up into two beams at frequencies ω i and ω s so that ωp = ω s + ω i, i.e a pump photon breaks up into a signal and an idler photon. The choice of the signal and idler frequencies is determined by the wavevector matching condition kp = ks + ki. The wavevector matching condition can be tuned either by changing the temperature or the direction of the wavevectors. When either the signal or the idler is used to seed the interaction, this is called an OPA. When noise is used to supply the seed, this is an OPG, and when the crystal is placed in a cavity (in order to enhance the output) the device is called an OPO. An example of the tuning range of an OPO based on the crystal BBO is shown in Figure 5 for comparison with the output of a number of discrete laser systems. 18 Tunability from 300 to 2400 nm is achievable with a pump beam derived from the fourth harmonic of a Nd:YAG laser. There have been reports of many other impressive OPO systems, with pulse widths varying from cw right down to tens of femtoseconds.19 Second Order Materials Materials are the key to successful applications of nonlinear optics. By its very nature, nonlinear optics requires high intensities for efficient implementation. However, the larger the nonlinearity, the smaller the intensity required. This is true for both the second and third order nonlinearities. Many of the most impressive recent advances in nonlinear optics have involved the development of both new second order materials and new techniques for phase-matching them as well as existing materials. Until about ten years ago, the only source of second order materials was single crystals which could be grown in non-centrosymmetric lattices. At a molecular level, the potential well in which the electrons sit must be anharmonic, i.e. the potential well should be of the form Ax2 + Bx3. For example, this rules out symmetric molecules. As a result, for a given level of excitation, the electron displacement is larger in one direction than the other so that the induced dipole has a second harmonic component. Such molecules can be packed in a lattice in which these induced polarizations cancel out, leading to a macroscopically centrosymmetric crystal. Alternatively, other lattice configurations can be non-centrosymmetric and therefore show macroscopic second order nonlinearities.

Figure 6 Electric field poling of polymers.

10

Recently it has proven possible to take non-centrosymmetric molecules and artificially arrange them into non-centrosymmetric configurations. One example is Langmuir-Blogett films which are deposited one monolayer at a time.20 More promising seem to be poled polymers.21 The poling process is shown in Figure 6. The molecules (chromophores) require a permanent dipole moment in addition to a large molecular second order nonlinearity to be incorporated into a polymer. When the polymer is heated to its glass transition temperature, it becomes soft and the chromophores can be oriented towards the direction of a strong applied electric field. This partial alignment leads to a macroscopic d(2) when the polymer is cooled down to room temperature and the external field is removed. Such materials have been used for both frequency conversion and electro-optical effects. 21,22 Three classes of materials are usually considered for their d(2) activity. 23 Of the inorganic crystals, ferroelectrics have shown nonlinearities in the 10s of pm/V range. Semiconductors, for example in the GaAs-based family, have typical nonlinearities in the 150 pm/V range. Organic media have nonlinearities ranging from 20 pm/V (LB films, poled polymers, crystals) to 100 pm/V (poled polymers), to 600 pm/V (the crystal DAST). As discussed, candidate materials must also have a good transparency range and be phasematchable.

Table 1 Representative list of d (2) active materials, their largest diagonal and off-diagonal nonlinear coefficients and their transparency window.20,23 Material

d ij (pm/V)

BaB 2 O 4

d ii (pm/V)

Transparency (nm)

1.6

190 → 2500

LiB 3 O5

1.1

0.06

180 → >2800

LiNbO 3 *

5.8

30

350 → >3000

KTiOPO4*

4.4

18.5

350 → 3000

-18

-27

400 → >2500

125

0

900 →

84

30

480 → 1800

90

30

450 → 1700?

30

600

750 → 1700

100

500 → 1800

KNbO 3

*

GaAs * NPP DMNP

*

DAST †

DANS *

‡BAMSAB*

21 polymer ‡ - Langmuir Boldgett film N-(4-nitrophenyl)-L-prolinol NPP: DMNP: 3,5-dimethyl-1-(4-nitrophenyl)pyrazole Dimethyl-amino-4-N-methylstilbazolium tosylate DAST: DANS: 4-dimethylamino-4| nitrostilbene BAMSAB: 4-(dimethylamino)-4|-(methylsulfonyl)azobenzene

450 → 1700?

* - waveguides † - poled

A representative list of materials is given above, along with their transparency ranges. 23 The trends are quite clear. Materials for operation into the UV region of the spectrum, for example the borates, have in general quite modest nonlinearities, of order 1 pm/V. The ferroelectrics have larger nonlinearities, 10s of pm/V, but their transparency window extends 11

from the just below the visible to around 3000-4000 nm. Organic materials have nonlinearities in the 100s of pm/V range, but their transparency window requires operation above 500 nm, and below 1800 nm (set by vibrational overtones). The large nonlinearity of semiconductors limits their utility to wavelengths beyond the near infrared, although it is noteworthy that they are transparent well into the infrared. Unfortunately, the known trends indicate that the larger the nonlinearity, the more limited the transparency window.

Waveguide SHG Results The progress in producing efficient SHG in channel waveguides has been spectacular. It is summarized in Table 2. It is important to note that this table only contains the figure of merit and in some cases the absolute conversion efficiency is limited by high waveguide losses, for example in the case of the semiconductor and polymer work quoted where the losses were 10's of dB/cm.10,17 There are two noteworthy developments in the application of semiconductors to SHG. Asymmetric quantum wells (AQW) lead to asymmetric wave functions for the electrons (see Figure 7a for an example), and hence a non-centrosymmetric response for a single quantum well. By stacking quantum wells, a d (2) xzx = 10 pm/V has been obtained.24 An alternative scheme uses the d ( 2 ) xyz ~ 150 pm/V component of GaAlAs semiconductors which normally is very difficult to implement in a phase-matched geometry. Using a combination of wafer bonding, selective etching and organometallic vapor deposition, it has proven possible to grow the QPM structure shown in Figure 7b.17 An effective nonlinearity of ~ 90 pm/V was obtained when the fundamental was propagated along the [1,1,0] direction. However, although the losses were high, 33 dB/cm at the harmonic frequency, improvements in growth technology could reduce these losses significantly.

Table 2 FOM for waveguide SHG in terms of the figure of merit for cases in which both the fundamental and harmonic are guided and for η ' = -4 for the Cerenkov case. Note η ∝ λ . Phase-Match

Waveguide SiO2/Ta 2 O5 /KTP

25

η %/[W-cm2 ]

λ (nm)

960

830

QPM

KTP

26

800

830

QPM

LiTaO 327

1500

830

QPM

LiNbO328

44

1550

MDPM

DR1 10

44

1550

76

1550

Birefringent

QPM

AlGaAs

17

η ' %/[W-cm]

CPM CPM

12

DMNP core fiber

29

40

830

SiO2/Ta2 O5 /KTP

25

130

830

Figure 7 (a) An example of the quantum well structure and the electron wave functions for an 17 AQW. 24 (b) QPM AlGaAs structure fabricated for SHG.

Electro-optic Effect Although not usually discussed in such terms, the electro-optic effect is also a χ (2) phenomenon. 2 The nonlinear polarization can be written in the form: (15)

where ω>>Ω , i.e. E(Ω ) is a DC or slowly varying field. Furthermore, ri j k is the electro-optic coefficient = χ(2)i j k /n 4 . For the simplest case this leads to ∆ n i = -½ni nj 2 r ijk Ek ( Ω ). Such an electric field induced index change can be used in, for example, a Mach-Zehnder switch of the type shown below in Figure 8. The applied voltage leads to changes in the effective indices of the channel waveguides which in turn leads to the phase difference ∆φ ∝ n3 r eff V/d L in the light π , the throughput is switched from maximum propagated through the channels. When ∆φ to zero etc. Such applications require a variety of different material properties. Some of the important ones, and their values for three different materials options are listed in Table 3. Of these, two are key: the space-bandwidth product which gives a modulator’s bandwidth for a 1 cm long device; and the voltage-length product which predicts the voltage required to switch a 1 cm long device. The very large bandwidth of electro-optic polymer devices has recently been verified (110 Ghz).22 Photorefractive Effect This phenomenon leads to optically induced changes in the refractive index and is classified as second order because the electro-optic effect plays a key role. The material requirements are as follows:30

13

Figure 8 Geometry (upper) for an electro-optic modulator in which the index change is different in the two channels, as shown in the lower figure.

Table 3 Figures of merit for electro-optic materials Figure of Merit

GaAs

LiNbO3

EO Polymer

EO coefficient reff (pm/V)

1.5

31

10-60

Dielectric Constant ∈

10

28

4

Refractive Index n

3.5

2.2

1.6

n3 r

(pm/V)

64

248

123

∈

6.4

8.7

31

Loss (dB/cm @ λ = 1300 nm)

2

0.2

0.5-2.0

Space-Bandwidth product (GHz-cm)

10

10

120

0.3-1.0

5

5-10

3 r/

n

Voltage-Length product (V-cm)

Electrons are promoted from trap states into a “conduction band” by the absorption 1. of light. Therefore some absorption at a suitable wavelength is required. 2. Conduction of electrons in response to Coulomb forces, diffusion and/or local or external electric fields.

14

Refractive index change via the electro-optic effect. 3. 4. Subsequent optical beams in the medium are affected by the refractive index changes. Therefore the best candidate materials are electro-optic materials with trap states due to the incorporation of selected impurities, for example KNbO3 , LiNbO3 , SBN etc.30 The unique properties of the interaction of light with photorefractive materials is illustrated in Figure 9. Consider the interference between two optical beams (9a). Electrons are promoted to the conduction band in the bright regions of the interference pattern. Due to Coulomb repulsion, a charge separation is created by electron conduction as indicated in Figure 9c. This leads to a space charge field as shown in Figure 9d, and in turn to an index change via the electro-optic effect, Figure 9e. Note that the resulting index modulation is shifted by π /2 relative to the intensity modulation introduced by the interference pattern. This has a large effect on any optical beams traveling in the medium. For example, power can be interchanged between the two beams. This and other effects will be discussed in more detail in another chapter in this book.

Figure 9 Generation of phase-shifted gratings via the photorefractive effect. Details described in 30 text.

Third Order Nonlinear Phenomena A large number of effects derive from the χ (3) susceptibility tensor. Starting from the they can for convenience be classed nonlinear polarization into four categories: 2 1. Incoherent All-Optical Effects This leads to self and cross phase modulation, an intensity-dependent refractive index ∆ n(I), and applications such as all-optical switching, logic, bistability, and temporal and spatial solitons 2. Coherent Frequency Degenerate Effects Degenerate Four-wave Mixing phase conjugation

15

3.

Nonlinear Spectroscopy This leads to the generation of new frequencies and spectroscopies such as third harmonic generation, coherent Stokes (Anti-stokes) Raman spectroscopy, and Raman gain spectroscopy. 4. Stimulated Scattering Stimulated Raman, Brillouin and Rayleigh Scattering Here examples of these categories will be discussed, in the reverse order to that given above. Example of Nonlinear Spectroscopy: Coherent Anti-Stokes Raman Spectroscopy (CARS) This technique can be used to investigate the normal modes of materials, for example vibrations, rotation etc.2 This CARS example deals specifically with vibrational modes. Consider two waves of frequency ω 1 , wavevector k1 and frequency ω 2 , wavevector k2 with ω 1 > ω 2 incident onto a material with a molecular vibration at the frequency ω q ω1ω 2 . If the mixing field E1E2* modulates the molecular vibration, it coherently ( excites the vibration in space with the spatial modulation exp[i(k2 - k1). r ]. Now the ω 1 ( ω 2 ) beam is scattered by the excited vibration producing an output CARS field at the frequency ω 3 = 2 ω1ω 2 (2ω 2 - ω 1 , coherent Stokes Raman Scattering) with wavevector k 3. Therefore this process involves the mixing of three incident fields and hence is a χ(3) process. Under appropriate wavevector matching conditions, i.e. k3 + k2 - 2k 1 = 0, a strong CARS signal is measured. Formally, this process can written in terms of a nonlinear polarization as (16) where it has been assumed for convenience that all the beams are y-polarized so that χ (3) χ yyyy (- ω 3 ; -ω 2,ω 1 ,ω 1). This leads to the coupled mode equation at ω 3 :

(3) eff

=

(17)

Because of the usual dispersion of the refractive index in the visible, |k2 | + |k 3 | > 2|k 1 | so that wavevector matching requires that the beams be non-collinear. Usually the angle between k 2 and k1 is only a few degrees. When the difference frequency ω 2 - ω 1 ωq , the nonlinear susceptibility has the form

(18)

where the background susceptibility χ b(3) contains all of the other contributions and χq(3) describes the interaction with the q’th vibrational mode. Because of the resonance at ω 2 - ω 1 ωq , the signal is enhanced and the location of ω q can be identified by tuning ω 2 - ω 1. In fact, monolayers have been investigated in waveguide geometries.31 Example of a Stimulated Process: Stimulated Raman Scattering (SRS) This was one of the first effects observed in the early days of nonlinear optics.2 A strong pump beam (Ep ) at frequency ωp leads via a stimulated process to the generation of a Stokes beam (Es) at ω s shifted from the pump by a vibrational frequency ωq = ω p - ω s . 16

Although it is initiated by Raman scattering from vibrations excited by thermal fluctuations at some unknown distance into a material, the analysis is the easiest to perform if one assumes both the incident and scattered signals are propagating in the medium. Just as in the CARS case, the mixing of the pump and Stokes beams excites vibrations with amplitude Q 0 in the molecules. The distortion in the “electron cloud” leads to a modulation of the molecular polarizability so that α = α0 + [∂α/∂ q]q where The detailed calculation of the vibrational amplitude Q 0 is tedious, can be found in textbooks and here only the result is given for ωq = ω p - ω s : 2 (19)

where µ red is the reduced mass of the vibration. The total polarization induced in a medium with a density of molecules N is given by

(20) The nonlinear polarization terms describing the evolution of the pump and Stokes beams are:

(21)

Applying coupled mode theory gives: (22)

As long as the pump is undepleted, these equations clearly lead to exponential growth (stimulated emission) of the Stokes beam. The gain coefficient for the Stokes power is (23)

Note that it depends on the pump power. Taking the combination dI/dz ∝ E * dE/dz + E dE* /dz leads to the Manley Rowe relations for the energy flow between the two beams, i.e. (24)

This means that as the Stokes grows by one photon, the pump beam depletes by one photon. Since the energy difference goes into exciting the vibrations which are in turn 17

dissipated to the lattice. The other well-known stimulated process is Stimulated Brillouin Scattering. 2 In this case, the stimulated Stokes beam travels in the opposite direction to the pump beam so that acoustic phonons of wavevector kp + k p are required. The corresponding frequency shift is ω p - ωB = Ω s = 2kp /v s where Ω s is the phonon frequency and vs is the sound wave velocity. Degenerate Four Wave Mixing 2 As indicated below, in this interaction there are two counter-propagating pump beams and an incident signal beam which collectively generate a fourth, conjugate beam which propagates backwards to the signal beam. If the pump beams are misaligned, then the conjugate no longer retraces along the path of the signal but the process etc. still occurs with perhaps a reduced efficiency due to the reduced overlap volume of the beams.

Figure 10 Degenerate four wave mixing geometry.

The input field in this case is:

(25) (3)

EEE. In fact it is the behavior and clearly there are a great number of terms in the product χ of the signal and conjugate beams that is of prime interest, usually in the regime of negligible depletion of the pump beams. The key nonlinear polarization terms are: (26a)

(26b) It is a straight-forward process to evaluate the coupled mode equations for the signal and conjugate beams: (27)

18

Solving gives:

(28)

There are two regimes of interest. The first involves significant gain for both beams. One photon from each of the pump beams provides gain for the signal and the conjugate. Furthermore, the conjugate beam is actually the complex conjugate of the signal beam at the input, hence the nomenclature phase conjugation. The second frequent application is to the evaluation of the third order susceptibility χ(3) . Note that for L I 2 . Assume for simplicity a Kerr-law nonlinearity, i.e. When the nonlinearly induced phase difference is π, destructive interference occurs and the output signal is zero. The important point is that the nonlinear phase shift dz is the key parameter, and for this device a minimum value of π is needed. This minimum value depends on the particular device of interest. Probably the most useful alloptical device reported to date is a nonlinear directional coupler for which a φ N L >2 π is required of the nonlinear material.32 Based on the discussion above, it proves useful to compare materials via nonlinear phase shift based material Figures Of Merit (FOM). 32 Consider a nonlinear material described by a intensity dependent refractive index change ∆ n = n 2 I so that the nonlinear phase shift φ NL = k v a c n 2 IL e f f where the effective length L e f f can be limited by material absorption (or scattering) losses. Here we assume that one, two or three photon absorption can occur, i.e. α = α 1 + α 2 I + α 3 I 2 and limit Le f f to approximately α-1 . 19

Limit #1: One Photon absorption dominates α1 >> α 2 I, α3 I

2

Limit #2: Two Photon absorption dominates α 2 I >> α1 , α 3 I 2 Limit #3: Three Photon absorption dominates α 3 I 2 >> α 2 I, α1 These three FOM assess the potential of a material to produce a 2 π phase shift in the presence of various sources of loss. Note that in the presence of both one and two photon absorption, the appropriate combined FOM is W/[1 + WT] > 1. The problem of finding suitable materials is now one of characterizing promising materials over broad spectral ranges. A subsequent chapter in this volume will deal with this problem. Material Systems and Nonlinearities A change in the refractive index of a material due to the propagation of a high power beam through it can occur due to a large variety of physical mechanisms. The most interesting involve resonant interactions between an electromagnetic field and some state or excitation over a limited frequency range. In this limit, the nonlinear index change can usually be “turned on” very quickly if enough energy is put into the material, and usually decays with some characteristic relaxation time for the “excitations” created. Detuned from these resonances, virtual transitions occur, and the response time ∆ t is typically limited by ∆ω∆ t ~ 1 where ∆ω is the frequency detuning from the resonance. In this limit, most nonlinear mechanisms resemble an ideal Kerr nonlinearity, namely that ∆ n = n 2 I. Furthermore, now a χ (3) formalism is usually a useful description. It was found that there are usually trade-offs between the nonlinearity n 2 due to a specific mechanism, the absorptive loss α associated with it, and the characteristic relaxation time τ. Namely n 2 /ατ ~ a constant to within a couple of orders of magnitude over a wide range of materials and mechanisms. 33 Although it is not always a valid description of a nonlinear index change for a specific mechanism, it is useful to define an effective n2 in order to compare the magnitudes of various nonlinearities. A table of such comparisons is given below. For any material, an index change can be induced onto one beam by itself or by another beam of a different frequency and/or polarization. The general relationship between the different nonlinear refractive indices can be given in terms of χ (3) tensor elements whenever such a formalism is valid. In general (29) The first term gives the usual refractive index n 0 = [1 + χ ( 1 ) ] ½.The second term can be written in terms of intensity-dependent refractive indices, namely (30) where ω can be either ω 2 or ω1 (orthogonally polarized case), I1 is the intensity of beam at ω1 for which the index is changed and I 2 that of the intensity of a second beam at ω2 or ω1 ( ⊥ polarization).

20

NONLINEAR MECHANISMS 1.

Electronic (∑ atoms)

2.

Electrostriction

3.

Electronic (conjugation)

4.

Charge transfer molecules

|n 2 (cm 2 /W)| non-resonant

2x10- 1 4 →10 -16 3x10 -14 →10 -16

non-resonant

10 -11 → 10 -13

resonant

10- 5

non-resonant

2x10-13→ 10 -14

resonant

10- 8

5.

Cascading ( χ ( 2 )χ( 2 ))

non-resonant

10- 9 → 10 -14

6.

Electronic, excited states

resonant

10 -10 → ??

7.

Carrier generation (semiconductors) resonant

10 -6

8.

Semiconductor Kerr

in-gap

10 -11

below ½ gap

10 -13 →0

9.

Exciton bleaching (semiconductors) resonant

10 - 5

10.

Liquid Crystals (reorientation)

2) states, either simultaneously, or sequentially. Within the scope of this review, it is not feasible to discuss all the contributions to the nonlinearities and neglected are transitions between vibronic sub-levels which give a fine structure to the transitions between electronic levels, electric quadrapole and magnetic dipole contributions etc. The scenarios which govern the gross nonlinear response are shown schematically in Figure 12a and 12b for the interaction of two photons with an oversimplified molecular system. These processes are normally associated with third order nonlinear optics. One photon changes the optical properties of the medium which can then be experienced by the second (or subsequent) photon(s). The first case is for electric dipole allowed transitions from the ground state to a one photon allowed state (opposite parity to ground state), Figure 12a. This occurs for molecules with well-defined parity states, or with mixed parity states. For high enough incident intensities, the excited state population becomes large enough so that the probability of the absorption of subsequent photons is reduced, i.e. α is reduced from it’s low intensity value α 1 to α = α 1 + α 2I with α 2 < 0. For intensities approaching the saturation intensity, the next higher order term α 3 I 2 will be positive etc. (α goes to zero when the ground and excited state populations are equal.) This is called “saturable absorption”. Simultaneously the dispersion in the refractive index is “bleached out”, as also shown in Figure 12a. This can be written as n = n 1 + n 2 I where n 2 >0 for ω > ω ij and n 2 0 and α 3 < 0. Although this case is relatively rare, it has been observed recently in the polydiacetylene PTS. 4 The corresponding variation in n with intensity, n = n 0 + n2 I + n 3 I 2 , is also quite different from the one photon case. For example, for frequencies less than the resonance frequency, n2 > 0 and n3 < 0 where-as, for ω > ω ij, n 2 < 0 and n 3 > 0 . Near saturation the response of n 3 and α3 is not instantaneous over an exciting pulse and hence they are fluence dependent. Note that the signs for α2 are different for the one and two photon cases. This implies that at certain wavelengths they could cancel leading to an effective α 2 → 0. Standard nomenclature is that the nonlinear response within a few linewidths of the absorption maximum is called the "resonan" response. "Near-resonant" refers to the spectral region where there is a strong spectral dependence to the nonlinearity n2 and "non-resonant" is defined by the long wavelength limit where n2 becomes essentially independent of wavelength. Multiphoton processes involving the simultaneous interaction of three, four etc. photons with a molecular system can also occur. In general they require very high intensities to interfere with the one and two photon contributions. The three photon case leads to three 3 2 photon absorption and dispersion, i.e. to terms with ∆ n ∝ n3 I + n4I etc. and ∆α ∝ α 3 I 2 + I3 α 4 3 etc. where terms in I and beyond are due to saturation of the three photon process. Intensity dependent changes in the refractive index and absorption are linked by the Kramers-Kronig relationship, even if the exact physical origin of the nonlinearity is unknown.

23

(36)

where P.V. is the principal value of the integral. For example, in a pump probe experiment to evaluate ∆ n( ω,Ω ) where the probe is at frequency ω and the pump (strong beam) is at Ω, it is necessary to measure ∆α with the same strong beam ( Ω ), and a tunable probe (frequency over a broad spectral range. Although it is preferable to make measurements of the absorption change over as wide a frequency range as possible, the most important spectral ranges are those containing resonances ω ij. Semiconductor Nonlinearities Semiconductors have many physical mechanisms in which an intensity-dependent refractive index can be induced, i.e. ∆ n = f(intensity, carrier density). Note that the index change is also a function of the carrier density in the conduction band. The dominant mechanisms are due to the absorption or emission of photons moving electrons across the band gap or creating excitonic states. (Some of the energy involved is dissipated in the lattice, leading to strong thermo-optic effects for long pulses or cw excitation.) It proves convenient to classify the nonlinearities as passive (conduction band initially populated in thermal equilibrium with the valence band) and active nonlinearities in which the population of the conduction band is initially out of thermal equilibrium due to electrical or optical pumping. Passive Nonlinearities: 34,35 Bandfilling 1. - associated with photon absorption → carrier generation - all parameters depend on detuning from band gap - 0 < |n2 | < 10 - 6 cm2 /W | ∆n sat | < 0.1 τ ~ 10 ns 2. Exciton bleaching - associated with photon absorption - parameters depend on detuning from exciton line - 0 < |n 2 | < 10 -5 cm 2 /W |∆ n sat | 10 and the FOM T < 0.2. Furthermore, for propagation along [1,-1,0], the self and cross phase-matching coefficients and the nonlinearities for both polarizations are essentially equal. In fact, this material resembles very closely a Kerr-law medium and has been used to demonstrate many all-optical switching devices, and spatial soliton phenomena. Active interband (between valence and conduction band) nonlinearities have proven very useful for various communications functions such as demultiplexing, clock recovery, wavelength shifting and others. 3 6 Electrons are “pumped” into the conduction band either by direct electrical injection (via electrodes) or by absorption of radiation. When in the gain regime, an incident photon stimulates an electron transition from the conduction to the valence band producing device gain. Furthermore, the change in population in the two bands induces an index change which has the opposite sign to that in Figure 14. A 3 picojoule input pulse energy is sufficient for creating a π phase shift! However, the recovery time is in the nsec

25

range. It has been shortened by maintaining strong pumping into the conduction band so that the switching process only utilizes a small fraction of the total available phase shift, reducing the recovery time down to the 10 psec range.

Figure 14 (a) FOM W for semiconductors versus normalized (to the exciton linewidth) detuning from the band gap. (b) FOM T for bulk semiconductors for the two photon nonlinearity.

There are also multiple ultrafast mechanisms which occur near and at the transparency point, i.e. the point at which stimulated emission and absorption are just balanced. 36,37 (Note that is not the lossless case!) As indicated schematically in Fig 15, an incident photon flux “burns” a hole via stimulated emission into the equilibrium electron distribution in the conduction band. This occurs on a sub-100 fsec time scale. Over a time period of 100s of femtoseconds the electron distribution is redistributed to a new equilibrium distribution by “carrier heating”. This second mechanism produces a relatively large index change, i.e. effective n 2 .

26

Figure 15 The time sequence of the electron distribution, “hole burning” and “carrier heating”, in the conduction band due to stimulated emission.

Table 4 The properties of the ultrafast mechanisms are summarized in the table below. Mechanism Ultrafast Kerr (40 fsec) Carrier Heating (carriers thermalize in band, 100s fsec)

λ µm

n2 cm 2/W

α cm-1

W

T

1.5

-3x10 -12

40

0.5

4

1.5

-12

40

0.75

3

Spectral Hole Burning (photons burn "hole" in conduction band, 10m) have made them awkward to work with, leading to the development of new, more nonlinear glasses as indicated in the table below. Table 5 Nonlinear glasses Glass

Absorption Edge (nm)

λ (nm)

α

n 2 cm 2/W

SiO 2

300

1550

0), the selffocusing helps to collimate the beam increasing the transmittance of the aperture. Scanning the sample farther toward the detector returns the normalized transmittance to unity. Thus, 56

the valley followed by peak signal shown by the solid line in Fig. 12 is indicative of positive NLR, while a peak followed by valley shows self-defocusing. The EZ-scan can be described in nearly identical terms except we monitor the complementary information of what light leaks past the obscuration disk, or eclipsing disk. Since in the far field, the largest fractional changes in irradiance occur in the wings of a Gaussian beam (see Fig. 10), the EZ-scan can be more than an order-of-magnitude more sensitive than the Z-scan. Figure 13 demonstrates the sensitivity of this method by comparing a Z-scan and an EZ-scan on neat toluene with nanosecond 532 nm pulses under identical experimental parameters (only replacing an aperture by a disk). Note that the vertical scale for the Z-scan is expanded by a factor of 10, and the signal is inverted for the EZ-scan since what is transmitted by an aperture is blocked by a disk. Using this method we have observed a peak optical path length change of as small as λ/2200 with a signal-tonoise ratio greater than 5 ( ∆Φ0=2 π/2200, where ∆Φ0 is defined as the integrated peak-onaxis phase shift).30

Figure 13. A Z-scan and EZ-scan on toluene. From Ref. 30.

It is an extremely useful feature of the Z-scan (or EZ-scan) method that the sign of the nonlinear index is immediately obvious from the data. In addition the methods are sensitive and simple single beam techniques. We can define an easily measurable quantity ∆ T pv as the difference between the normalized peak and valley transmittance: Tp - T v . The variation of ∆ T pv is found to be linearly dependent on the temporally averaged induced phase distortion, defined here as ∆Φ0 (for a bound-electronic n2, ∆Φ 0 involves a temporal integral of Eq. 11). For example, in a Z-scan using an aperture with a transmittance of ≅ 40%; (38) With experimental apparatus and data acquisition systems capable of resolving transmission changes ∆ T pv ≅1%, Z-scan is sensitive to less than λ/225 wavefront distortion (i.e., ∆ Φ0=2 π/225). The Z-scan has a demonstrated sensitivity to a nonlinearly induced optical path length change of nearly λ/103 while the EZ-scan has shown a sensitivity of λ/10 4.

57

In the above picture we assumed a purely refractive nonlinearity with no absorptive nonlinearities (such as multiphoton or saturation of absorption). Qualitatively, multiphoton absorption suppresses the peak and enhances the valley, while saturation produces the opposite effect. If NLA and NLR are simultaneously present, a numerical fit to the data can extract both the nonlinear refractive and absorptive coefficients. The NLA leads to a symmetric response about Z=0, while the NLR leads to an asymmetric response (if ∆Tpv is not too large), so that the fitting is unambiguous. In addition, noting that the sensitivity to NLR in a Z-scan is entirely due to the aperture, removal of the aperture completely eliminates the effect. In this case, the Z-scan is only sensitive to NLA. Nonlinear absorption coefficients can be extracted from such “open aperture” experiments. A further division of the apertured Z-scan (referred to as “closed aperture” Z-scan) data by the open aperture Z-scan data gives a curve that for small nonlinearities is purely refractive in nature. In this way we can have separate measurements of the absorptive and refractive nonlinearities without the need of computer fits with the Z-scan. Figure 14 shows such a set of Z- scans for ZnSe. Separation of these effects without numerical fitting for the EZscan is more complicated. The single beam Z-scan can be modified to give nondegenerate nonlinearities by focusing two collinear beams of different frequencies into the material and monitoring only one of the frequencies (different polarizations can be used for degenerate frequencies). 31 This “2-color Z-scan” can separately time resolve NLR and NLA by introducing a temporal delay in the path of one of the input beams. This method is particularly useful to separate the competing effects of ultrafast and cumulative nonlinearities.

Figure 14. Z-scans of ZnSe showing closed aperture (upper left), open aperture (lower left) and closed divided by open aperture data (upper right). The solid lines are theoretical fits. Adapted from Ref. 7.

58

Pump-Probe Z-scan Pump-probe (or excite-probe) techniques in nonlinear optics have been commonly employed in the past to deduce information that is not accessible with a single beam geometry. The most significant application of such techniques concerns the ultrafast dynamics of the nonlinear optical phenomena. There has been a number of investigations that have used Z-scan in pump-probe scheme. The general geometry is shown in Fig. 15 where collinearly propagating excitation and probe beams are used. After propagation through the sample, the probe beam is then separated and analyzed through the far-field aperture. Due to collinear propagation of the pump (excitation) and probe beams, we are able to separate them only if they differ in wavelength or polarization. The time-resolved studies can be performed in two fashions. In one scheme, Z-scans are performed at various fixed delays between excitation and probe pulses. In the second scheme, the sample position is fixed (e.g. at the peak or the valley positions) while the transmittance of the probe is measured as the delay between the two pulses is varied. The analysis of the 2-color Z-scan is naturally more involved than that of a single beam Z-scan. The measured signal, in addition to being dependent on the parameters discussed for the single beam geometry, will also depend on parameters such as the excite-probe beam waist ratio, pulsewidth ratio and the possible focal separation due to chromatic aberration of the lens. However, these can easily be handled theoretically. Figure 16 shows a temporally-resolved, 2-color Z-scan for ZnSe using 30 ps, 532 nm pulses as the excitation source and 40 ps, 1.064 µm pulses as the temporally delayed probe.32

Figure 15. The experimental setup for performing a time-resolved, 2-color Z-scan.

Figure 16. The results of a time-resolved, 2-color Z-scan performed on ZnSe using a 532 nm pump and 32 1.06 µm probe showing nonlinear refraction vs. time (left) and nonlinear absorption vs. time (right).

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Degenerate Four-Wave Mixing Another commonly used method to determine the dynamics of the nonlinear response of a material is time-resolved degenerate four-wave mixing (DFWM).33 O n e implementation of this technique is shown in Fig. 17 where the interference of the temporally and spatially coincident forward pump (irradiance If ) and probe (Ip) sets up a nonlinearity that is examined by the backward pump (Ib) as a function of its temporal delay. One interpretation is that If and Ip set up a grating whose dynamics is investigated by Ib scattering off this grating into the detector shown in Fig. 17 (often referred to as the “conjugate” direction).34 While this does not adequately describe the signal within the pulsewidth, it gives a reasonable picture of the longer time response of this method. Figure 18 shows the response in this experiment performed on ZnSe using 30 ps, 532 nm pulses. 35 Clearly there is a fast response following the pulse shape and a slower response with a decay time of 100’s of picoseconds (dominated by carrier diffusion washing out the grating). While this technique gives information about the dynamics of the nonlinear response, absorptive and refractive nonlinearities both contribute to the signal and their effects are difficult to separate. That is, in the grating picture, both absorptive and refractive gratings scatter the backward pump into the detector.

Figure 17. Experimental setup for time-resolved degenerate four-wave mixing (DFWM). From Ref. 35.

Figure 18. DFWM experiment performed on a sample of ZnSe at 532 nm. From Ref. 35.

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Streak Camera Imaging The dynamics of the nonlinear lensing effects can be dramatically demonstrated by monitoring the spatial beam profile in real time with the use of a streak camera. Figure 19 shows the spatial profile at low and high input energies for 30 ps, 532 nm pulses incident on ZnSe after propagation in the relatively near field.36 At high inputs the 2PA creates carriers which are long lived with respect to the pulsewidth used and build up in time with the integrated energy. Thus, the defocusing from these carriers is observed in Fig. 19 to get stronger with increasing time in the pulse.

Figure 19. Spatial scans at different times (separated by 7.4 ps) in the pulse after transmission through ZnSe and propagation to the near field as measured using a streak camera/vidicon combination. The left side shows low energy (i.e. the input shape) and the right side shows the defocusing at high input. From Ref. 36.

EXCITED-STATE

NONLINEARITIES

Both Figs. 16 and 18 show a fast response mimicking the input pulse (i.e. a response time less than the input pulsewidth) along with another more slowly responding nonlinearity. The rapid response is either a combination of degenerate 2PA, β(2ω) and n2 (2 ω) effects for the DFWM experiment (see Fig. 18), or the separate effects of nondegenerate 2PA, β( ω;2 ω), for the 2-color Z-scan and nondegenerate n2 (ω ;2ω), see Fig. 16. Here ω correspond to a wavelength of 1.064 µm. The longer time response in Figs. 16 and 18 (and 19) is due to the nonlinear absorption and refraction induced by 2PA generated carriers. The generation rate for these carriers of density N is given by (39) The absorption from these carriers is referred to as free-carrier absorption, FCA, and the refraction as free carrier refraction, FCR and both effects are linear in the carrier density. While the FCA and FCR depend on the carrier density independent of their generation mechanism, when the carriers are generated via 2PA these effects appear as fifth order

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nonlinearities or, as an effective, pulsewidth dependent χ (5). 37 However, they are best described in terms of absorptive (σ a) and refractive (σr) cross sections as;

(40)

Sometimes k is included in the phase equation changing the units of σr to length cubed. The sign of σ a is intrinsically positive while, in principle σr can have either sign. In fact, however, for below gap excitation, σr is always negative leading to self-defocusing. Once excited, these carriers can undergo a variety of processes including several types of recombination and diffusion which have not been included in Eq. 40. For short pulse excitation (e.g. ps) with pulsewidths less than recombination and diffusion times Eq. 39 is rd th adequate to describe the response within the duration of the pulse. Combining the 3 and 5 order responses gives, (41) The dynamics of these carriers is seen in the time-resolved Z-scan of Fig. 16, the DFWM data of Fig. 18 and the streak camera imaging of Fig. 19. In the Z-scan data carrier recombination dominates the decay while in the DFWM experiment carrier diffusion between peaks and valleys of the grating dominates. These decays would need to be included in Eq. 39 to describe these dynamics. The order of the response is seen in the DFWM data as the inset of Fig. 18 where the signal at two time delays is plotted as a function of the input irradiance I (all three input irradiances varied simultaneously). At zero delay the slope of the signal versus I is three (third-order, χ(3) response) while at a 200 ps delay (well past the overlap of the pulses), the slope is five indicating the fifth-order response.

Figure 20. A plot of the index change, ∆n, divided by the irradiance, I, as a function of I. From Ref. 37.

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This higher-order response for nonlinear refraction is also observed in Z-scans performed at different irradiance inputs of 532 nm picosecond pulses. At low inputs the third-order bound-electronic response dominates while at higher inputs the fifth-order selfdefocusing from the 2PA generated free carriers becomes important. Figure 20 shows the index change divided by the peak-on-axis input irradiance, I0, in ZnSe at 532 nm, as a function of I0 . The index change is calculated from the measured ∆Φ 0 . For a purely thirdorder response, ∆ n=n 2I0, and this figure would show a horizontal line. The slope of the line shown in Fig. 20 shows a fifth-order response while the intercept gives n2, the negative bound-electronic defocusing at 532 nm. This helps explain some of the discrepancies of measured values of 2PA coefficients in GaAs (see Fig. 9). These higher order nonlinearities seen in semiconductors give some indication of the importance of the careful characterization needed to interpret the measured nonlinear loss and phase. It would seem reasonable that σ r and σ a would be related by causality through Kramers-Kronig relations. However, after excitation there can be rapid redistribution of the carriers within the bands due to various mechanisms. This redistribution leads to socalled band-filling nonlinearities, and for the time scales of picoseconds used in the experiments shown, this prohibits the use of Kramers-Kronig relations for the cross sections (note that after excitation and redistribution the absorption and refraction due to these carriers are related by Kramers-Kronig relations). EXCITED-STATE NONLINEARITIES VIA ONE-PHOTON ABSORPTION As discussed in the previous section, excited carriers can lead to NLA and NLR. In other materials such as molecular systems, the creation of excited states can lead to analogous nonlinearities described by identical equations (Eqs. 40) where N is interpreted as the density of excited states. Again, how they are generated is unimportant. If the carriers or excited states are created by 2PA the resulting nonlinearities are fifth order, i.e., an effective χ(5). Depending on the absorption spectra, these states can also be created by linear absorption where, neglecting decay within the pulse, (42) Concentrating on molecular nonlinearities we refer to these nonlinearities as excited-state nonlinearities, ESA and ESR in analogy to FCA and FCR respectively. Assuming that depletion of the ground state can be ignored (i.e., no saturation), (43) By temporal integration of Eqs. 43 with 42 we find; (44) where F is the fluence (i.e., energy per unit area). This equation is exactly analogous to Eq. 10 which describes 2PA, except that the irradiance is replaced by the fluence and the 2PA coefficient, β is replaced by ασa/2 ω . Thus, experiments such as Z-scan will monitor a third-order nonlinear response that could easily be mistaken for 2PA. However, there must be some linear absorption present, however small, for ESA to take place. Two-photon-

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absorption does not require linear loss. Unfortunately this is not enough to differentiate the processes as there can be linear absorption present in 2PA materials unrelated to the NLA process, e.g. from impurities or other absorbing levels. A temporally resolved measurement, such as DFWM or time-resolved Z-scan, would also show the excited-state lifetime assuming the pulsewidth was short compared to this lifetime. Another way to determine the mechanism is to measure the nonlinear response for different input pulsewidths, again assuming the pulses can be made shorter than the excited state lifetime. Figure 21 shows this measurement performed on a solution containing chloro-aluminum phthalocyanine (CAP).38 While the energy in the pulses was held fixed while the irradiance was changed by a factor of two by changing the pulsewidth, the nonlinear transmittance remained the same in the open aperture Z-scans. This clearly indicates that the NLA is fluence rather than irradiance dependent and, therefore, must be described by a real state population, i.e., ESA. In CAP, at 532 nm, the ESA cross section σa is considerably larger than the ground-state cross section. This type of absorber is referred to as a reverse-saturable absorber since the absorption increases with increasing input. Such effects are useful in optical limiting.39 For large inputs the ground state can become depleted reducing the overall NLA.

Figure 21. Z-scans performed on a sample of CAP at 532 nm. The left shows open aperture Z-scans for pulsewidths of 29 ps (squares) and 61 ps (triangles) and the right shows closed aperture Z-scans (after absorption is divided out) for the same pulsewidths. Adapted from Ref. 38.

Associated with ESA is ESR as given by the second term in Eq. 41, which is simply due to the redistribution of population from ground to excited state. This is analogous to the index change in a laser from gain saturation which leads to frequency pulling of the cavity modes.2 Ground state absorbers are being removed and excited state absorbers are being added. Depending on the spectral position of the input with respect to the peak linear and peak excited-state absorption, the NLR can be of either sign. For reverse saturable absorbing materials the NLR is most likely controlled by the addition of excited-state absorbers, and their spectrum since the cross section is larger. Thus N is determined by Eq. 42. Figure 21 also shows the NLR in CAP for two different pulsewidths demonstrating that it is also fluence dependent and, thus, dependent on real state populations. TWO-BEAM INTERACTIONS Here we give examples of “nondegenerate” nonlinearities, where here the breaking of degeneracy is not just frequency, but propagation direction, e.g. 2-beam coupling. There

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are many phenomena that can occur when two beams are coincident in space and time on a nonlinear sample. Among these are cross-phase modulation where a pump beam modulates the phase of a probe through the optical Kerr effect, 2PA induced on the probe, excited species created by the pump affecting the probe or temperature changes induced by the pump changing the index seen by the probe. In these cases, if the probing beam, p, is much weaker than the pump (exciting beam e), the changes in the weak probe beam can be twice as large as the changes in the strong beam (the strong beam is unaffected by the weak probe). This factor of two comes from the cross term or grating term in the nonlinear interaction and is sometimes referred to as weak-wave retardation. 12 From Eq. 4 with two input fields and keeping only the third-order self-action term (ignoring their vector nature other than keeping track of the wave-vector dependence) gives;

(45)

We plug this equation into the reduced wave equation given in the slowly varying amplitude and phase approximation by Eq. 6. Looking just at the terms with k vectors in the pump , and probe, exp exp , directions separately, we have;

(46)

where it has been assumed that Ep 1. Technically, fundamental solitons are of great interest because they do not change their shape as they propagate i.e. they remain sech-shaped. Higher-order solitons N = 2, N = 3,… change shape [they breathe but, fortunately, so does the chirp!] as they propagate but keep returning to Nsech(t), periodically. The period is known as the soliton period and is a useful length scale of the system. Apart from that, higher-order solitons are not expected to be useful in a switching device or communication system.

Figure 16. A Square input pulse gives birth to an envelope soliton . The transverse direction is time (pulse) or distance (beam).

This is because a higher-order, bright, envelope soliton is a bound state of N =1, sechshaped, fundamental solitons, with zero binding energy, so it is extremely vulnerable to perturbations. Hence an N = 2 soliton, for example, could degenerate quickly to two N = 1 solitons. This means that N is the soliton content or soliton number for a given input pulse. Figure 17 contains a diagramatic explanation of what happens when either a Bsech(t) input pulse is used for a nonlinear system described by the nonlinear Schrödinger equation, or a rectangular input pulse B rect(t). The soliton "content" is shown as a function of B. For example, if a pulse Bsech(t) is entered into a dispersive material then, provided 0.5 < B < 1.5, a fundamental soliton is created. On the other hand, if B rect(t) is entered then a fundamental envelope soliton pulse is created only if 0.5 π < B < 1.5 π . These conclusions come from an exact mathematical treatment of the nonlinear Schrödinger equation, using inverse-scattering theory [15] [IST] that is to be discussed later on. The conclusions are easily confirmed out by numerical simulations.

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Figure 17(a). How solitons are created from Bsech(t) input pulses or Brect(t) input pulses.

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Figure 18 Simple idea of diffraction.

4. DIFFRACTION, SELF-FOCUSING AND BRIGHT SPATIAL SOLITON FORMATION We are well aware that a beam of light, even if it is emerging from the now familiar laser pointer used by lecturers, will spread out and change shape as it propagates (see

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figure 18). In air, this is all that will happen. But what about light propagation in nonlinear media? In this case light always heads for the region of highest refractive index (think of how a lens works), and a nonlinear material, such as a quantum well AlGaAs structure, forces the index up, precisely, and only, where the beam is travelling. This is just what is needed to stop light rays from inside the beam from bending outwards and causing a change in beam shape. When beam spreading is exactly stopped by the nonlinearity, a spatial soliton [16] is formed and the beam can now be used as a stable optical guide, just like the familiar optical fibre. The spatial soliton is, however, better that the fixed, ‘hard’ waveguide such a fibre constitutes. The reason is that it is a ‘soft’ guide that is easily tunable by changing the beam size. Many applications can be imagined for these novel waveguides, such as using them for optical wiring, to steer one beam with another in addressable arrays, and to act as switching devices and as logic units for a whole range of information processing and computing. Spatial solitons [16,17,18-30] are a significant outcome of modern soliton and materials science because of the ease with which they can be manipulated. Although there are still worries about how fast many materials can react to light, or changes of beam direction, work on spatial solitons should herald an era of new, all-optical processing devices that are cheap and easy to implement. Russell’s observation did not change physics or technology in his own century nor most of our own. But with the real possibility of optical soliton communication systems and all-optical processors now on the horizon, it may well transform them next century. It is many years now since the basic physics, concerning the self-trapping [30-32], of a powerful beam of light in a nonlinear medium was thoroughly discussed. Some of the imagery evoked in those early, pioneering, discussions will be produced here, through the consideration of a beam of light with a rectangular distribution of energy, a radius r and a wavelength λ , propagating through a dielectric, non-magnetic, medium that has a linear refractive index n 0 . First, as stated above, the beam will want to spread out, due to diffraction. Indeed, at a given radius r the light (laser) beam can be imagined to act as its own aperture and rays of light will diffract through it, with an approximate diffraction angle θ d = λ/2nr. If the material becomes nonlinear, in a Kerr-like manner, [i.e. third-order only, with nonlinearity proportional to intensity], with a nonlinear coefficient α , then the refractive index of the linear medium will change, by the small amount α|E|2 , where E is the electric field carried by the beam. If the medium has a positive α then the refractive index increases to n + (α|E| 2) and, as shown in figures 19 and 20, it is possible to balance the tendency of the beam to self-diffract with a nonlinear tendency to self-focus. In these figures can be seen a simplistic, use of diffraction through a circular aperture of radius r, but the example is designed to illustrate what nonlinearity can do [30,31] to achieve diffraction-free beams. In figure 19, rays setting out from the beam axis will either escape, or be totally internally reflected. This approximate model uses an aperture to create a beam of radius r. Since diffraction does not permit a geometrical transmission, rays making an angle θ c with the axis are totally internally reflected at the beam boundary but other rays can escape the geometrically defined boundary. In principle, therefore, beam broadening occurs. Because the electric field E of the beam causes a nonlinear change in refractive index, the critical angle depends upon E and, hence, upon power. As figure 20 shows, most of the beam power is enclosed [30-32] within ray angles making an

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Figure 19 Showing that the critical angle depends upon electric field or power.

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Figure 20 Illustration of circular hole/single slit diffraction plus introduction of the concept of critical power.

angle

to the beam axis. Figure 19 is not completely true (it lacks rigour) [33]

but we ought to be able to learn quite a lot from it! The main idea being introduced is that if the aperture is circular then 2θd is the direction of the first minimum in the (far field) Fraunhofer diffraction pattern. This means that the bulk of the energy in a diffracting beam is, more or less, associated with rays making an angle of θd, or less, to the horizontal axis. For a single slit the first minimum in the pattern should be at θd but, of course, the intensity will not be zero on the scale of figure 20 because the distance between the slit and the screen is not infinite, as it should be for (Fraunhöfer diffraction: far field), when the source is at infinity. For figure 20, the observation plane is rather close to the circular hole, or slit, so the actual geometrical shadow if it would also be wider than that which is shown. The greater the distance between hole/slit and screen the more the Fresnel (: near field) pattern goes over to a Fraunhöfer pattern. These points need not worry us in this kind of what we call a ‘hand waving’ or ‘ball park’ exercise, so we will set

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(approximately) and use it to get an estimate of some distance [30] LD at which a beam can double in width, due to diffraction. Since this is the case, then θc = θ d is the critical condition for the beam to have its diffraction killed by the nonlinearity. This leads to a critical power Pc , which, rather importantly, does not depend upon r. Another way to look at this problem is to introducc the concept of nonlinear length LNL, as demonstrated in figure 21. At balance, the diffraction balances the self-focusing and diffraction-free beams are the result. Such diffraction-free beams are called self-trapped and are spatial solitons. Troublesome questions now arise about Pc. Can it be controlled to produce stable spatial solitons?

Figure 21. Introducing diffraction length (LD ) and nonlinear length (LN ) .

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The important feature is that, if P > P cr , rays are forced towards the axis of the beam, in a self-focusing action. If P < Pc r, then diffraction takes over. These properties of the medium occur because Pcr is fixed, i.e. ε0, n, c, α and λ are fixed parameters of the system and it is necessary for any launched input power P to be precisely equal to P cr for stability to occur [21,23,30,32]. Inevitably, in any real experiment the power launched into a waveguide will be either P < Pcr or P > P cr, even if the difference is only infinitesimal. In either case, such a fluctuation will lead to instability taking the form of a 'runaway' to diffraction or to self-focusing. This, then, is the problem with self-trapped beams in which diffraction and self-focusing is finely balanced in a bulk (infinite with no boundaries) medium. Mathematically, it means that such beams are, by their nature, unstable in a multi-dimensional system and this has been established, for a long time, in the literature. The situation described above, although interesting, in principle, at first sight appear to have very little experimental promise because of its instability. Some forms of stabilisation have been proposed, however, that are proving to be rather successful. The first stabilisation method involves controlling the perturbations that can lead to instability by a method based upon a modulation due to interference fringes [34]. This is, relatively, complicated so it will not be discussed here because the aim of this chapter is to study straightforward spatial solitons, which can be created within [16] a nonlinear planar waveguide. This procedure reduces the propagation to a one-dimensional diffraction. Given the propagation distance as the other dimension this is then called (1+1) propagation. The instability, referred to above, that arises in a three-dimensional medium has a P c r that is fixed, in the sense that self-adjustment of the beam such as its radius, while focusing, cannot change it. On the other hand, if a beam is launched into a planar waveguide that has interfaces parallel to the x-z plane, then guiding confinement will occur in the y-direction. Suppose then that the guide is weakly nonlinear, so that the modal field intensity is unaffected by nonlinearity, to first order. In this case, the power needed for diffraction and self-focusing to balance becomes [21]

where 2d is the thickness of a guiding layer with linear refractive index n0 . r is now the largest beam dimension and πrd is, approximately, the area of a beam with most of its energy confined within an elliptic cross-section, and within the guide. This is shown in Figure 22, which also contains a schematic explanation of the stability. This new value of P cr does not now arise from the cancellation of r2 but has, left within it, a factor

A s

r changes, due to attempts to self-diffract or self-focus, the system is always forced back to stability. This is the elegant feature of a waveguide being able to induce (1+1) propagation and created some excitement in the field [16] when it was announced. Naturally, since open waveguides, made of transparent dielectric materials, are used, real beams carry some part of their energy outside the waveguide. This is because tangential field components are continuous at

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Figure 22 Stable beam propagation in a planar waveguide .Launched powers P < Pcr or P > Pc r return towards P = P c r. Diffraction/self-focusing balance in the x-direction; linear guiding in y-direction; propagation in z-direction.

the boundaries but, as shown in figure 23, most of the energy is carried, inside the waveguide, as sketched in figure 22. Hence the 'balancing' arguments are correct, even for realistic cases.

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Figure 23 Contour plot of the intensity distribution of a stabilised beam in a planar waveguide. Note that the contours are further apart as the field weakens into the cladding regions.

In figures 24 and 25 an outline of what self-focusing means, physically, is given. Basically, if an unbounded crystal is used and beam of power P is entered into it with P > P c , or P >> Pc then phenomenon of self-focusing occurs. This means that because the power of the beam is more than is needed to achieve the balance power (Pc ) needed to kill diffraction, the beam will continue to focus. In figure 24 pure self-focusing is conveniently illustrated in terms of an equivalent focusing [30,32] lens of focal length LN . In fact, while L N is (equivalently) only associated with a focusing lens, diffraction is associated with a defocusing lens of focal length LD . The combined effect is obtained in this, geometrical, ray picture by obtaining a combined focal length, in the usual way. Of course, self-trapping (spatial soliton creation) is always going to be possible so the lens model is only useful upto a point. According to figure 24 as P → Pc , LND → ∞, yet a self-trapped filament can be formed. A thorough investigation of focusing region is needed, therefore, but this is far beyond the scope of this article. It has been addressed, however, and some answers can be found in the literature that abounds on this topic [32]. Figure 25, sketches out the case when beams go supercritical and hints that several filaments (self-trapped) beams can be created, each with P = Pc. The figure finishes with a final sketch of a spatial soliton labelled as a self-trapped beam.

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Figure 24 An illustration of self-focusing. Only the equivalent positive (focusing) lens is shown, to demonstrate the dramatic consequences of self-focusing. The focusing region is not accounted for by this simple theory. Self-trapped filaments can form, for example, rather than the beam becoming smaller and smaller.

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Figure 25 Paraxial geometrical optics will not accurately describe the focusing region. Many filaments (spatial solitons) could be created, for example.

5. GENERATING THE NONLINEAR SCHRÖDINGER EQUATION [NLS] The nonlinear Schrödinger equation [NLS] is a familiar sight in the literature and one of the preferred generic forms is (5.1) [the literature often shows ½ multiplying the second term but a simple re-scaling of Z places the 2, as in (5. l)], where U is an envelope function that is slowly varying, with respect to Z. This envelope function has been seen pictorially in the previous sections and is literally the “envelope” that “holds”, or “encloses” the rapidly oscillating wave. In the form (5.1) the equation has been stripped of dimensions and any particular physical application. As shown in figure 25, it is entirely generic in character.

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Figure 26 pulse/beam solitons are solutions of the same equation.

Even though it is generic in form, equation (5.1) has preferred forms, as outlined in figure 27, so that applications to beams of pulses can be emphasised.

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Figure 27. Preffered form of the NLS.

Having stated what the NLS looks like, how does the equation arise? There are several answers to this question to be found in the literature, all leading to the generic equation. At one extreme, a complicated first principles calculation can be undertaken. At the other, a simple, straightforward, examination of the dispersive nature of the system also reveals the nonlinear Schrödinger equation. It is this latter approach that will be adopted here. Suppose that the dispersion equation is written in terms of kz , the z-component of the wavenumber in the manner shown in figure 28.

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Figure 28 Summary of expansion of dispersion equation about an operating point ( ω 0 , k0 ). This procedure is like Taylor expanding the characteristics of an electronic device about an operating (bias) point.

Dispersion in a system simply means that the wavenumber (k) has a frequency (ω ) dependence that is not a straight line. This means that phase velocity vp = ω/k and group velocity

varies, according to which point on the dispersion curve is selected.

Temporal pulses and spatial beams are modulations of carrier waves [9,14]. If only temporal pulses are being considered, then it is clear that the act of producing an envelope introduces a frequency spread ∆ ω about the carrier frequency ω 0. Provided that we consider only a small bandwidth around ω0 , all frequency components in a linear system lie fairly close to the centre (carrier) frequency ω0 . If, in addition, there is a small spread of wavenumbers about k0 then the wavenumber of the complete beam-pulse-like disturbance can be Taylor expanded [9,15,35] around (ω0,k0) and the series can be safely truncated at the second-order or third-order terms. The number of terms to be kept will depend upon the application we have in mind. Figure 28 shows propagation down the z-axis of such a

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pulse/beam envelope, associated with some envelope function u(x,z,t) that varies much more slowly than the rapidly varying exp i[ ω0 t-k 0 z], as the propagation proceeds. Note also that y is ‘frozen’ so no variation with y is allowed. Because of the slowly varying assumption only

will eventually appear in the NLS, amounting to a neglect of

In the expansion given in figure 28, since the system is confined in the y-direction, i.e. guiding along z occurs and the x- and z-directions are each allowed to reach ± ∞ . The guiding confinement means that the wavenumber being used here is already the guided wavenumber. The expansion looks at the deviation, from the guided wavenumber k z0, because of the introduction of wavenumber and frequency bandwidths [ ∆ kx , ∆ k z ] and ∆ω, respectively. Since the whole point of the exercise is to look at beam and pulse propagation in a nonlinear medium, the deviation, away from k z0 , originating from the power is also added in, intuitively, as a term

|u|² term, where u is the

(complex) envelope amplitude. Because both spatial and temporal effects are included, the final envelope equation accounts, simultaneously, for diffraction and dispersion. Some details will now be given that may help with the derivation. These are the relationships •

k x , ω, |u| ² are to be treated as independent variables (5.2a)

• kz is a dependent variable; k depends upon ω and |u|² (5.2b)

(5.2c) [k x , ω are independent variables in this theory] (5.2d)

(5.2e)

[k x |u|² are independent variables] (5.2f)

At the operating point k z = k z0 = k, k x = 0, ω = ω0 so that

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(5.3a) (5.3b)

Figures 29, 30 and 31 show further developments that lead to the familiar generic equation. Figure 29 contains the fundamental step that ∆ ω , ∆ k x and ∆ k z are in the space-time domain,

and

exist as operators on u(x,z,t). In other words, the NLS is

the application of the inversion to the time-space domain of the operator

to u(x,z,t).

Figure 29(a) Using the operator derived in figure 27 upon u to get the NLS. The result of leaving out diffraction is given.

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Figure 29(b) NLS appropriate to diffraction cases.

Figure 30 Illustration of length scales in dispersion and diffraction limited propagation in a nonlinear medium.

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Figure 31 How to reach the generic form of the NLS

Bright (B) or dark (D) solitons can be generated from the temporal, or spatial, equations by considering the sign of γ and/or the sign of β 2. This is the application of the famous Lighthill criterion, which tests whether β2γ < 0, or β 2γ > 0. Diffraction always has the same sign and so is always there, even if the material is not; hence a laser beam in a vacuum will always spread out because only material-induced self-focusing can stop it. Nevertheless, in a self-defocusing material dark (D) soliton beams are possible. The situation is summarised in figure 32. Dark solitons will not be discussed any more here, except to point out that they are, strictly speaking, ‘holes’, or ‘absence of light’ in a continuous infinitely extending background. In practice, finite backgrounds can be used such as a ps hole ‘dug out’ from an ns pulse. These ‘holes’ propagate like the bright pulses or beams. A dark soliton has a tanh cross-section. For spatial solitons the preferred notation is (5.4) and the general (fundamental) solution for this equation is displayed and demonstrated in figure 33.

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Figure 32 Summary of bright/dark conditions.

Figure 33(a) First steps in checking the general, fundmental solution of the NLS. 216

Check

Figure 33(b) Final check on the fundamental (first-order) solution of the NLS, formulated for a spatial soliton beam.

In this beam case, ξ is interpreted as the angle at which the beam centre propagates, relative to the z-axis, and θ 0 is just an arbitrary fixed phase that can be set to zero, or any other value, without any loss of generality. There are other options, and interpretations, depending on the application in mind and figure 33 contains an appropriate form for beams. For pulses, it is common to use the following notation and form (5. 5) in which , even though it is dimensionless, t is regarded as the local time, the ½ is in its ‘traditional position’. The general solution to equation (5.5) can be written down like the general solution to (5.4) except for a different interpretation. The first point to make is that equation (5.5) describes the motion of a pulse, relative to a frame of reference moving with a speed equal to the group velocity v g [see figure 31 for the relevant transformation]. The question of whether a laboratory frame, or a moving frame, is used is really relevant to the temporal (pulse) case. Figure 34 illustrates the frames of reference and emphasises that in the moving frame the measurement of time is local to the pulse i.e. it is not absolute, or global, time. Hence, we tend to put the pulse centre at

and let T range from

217

negative to positive values as the pulse is transversed in time. In the vast literature on this topic, the symbol t is usually used, as shown in equation (5.5), but it is really the local T, as emphasised in figure 34. The general solution is also shown in figure 34 and relates immediately to that shown in figure 33. If x0 = 0 in figure 33, then A Ω z of figure 34 is to be compared to ξ z in figure 33. For spatial solitons ξ is the angle the beam axis makes to the z-axis. For temporal solitons Ω is a frequency. A little consideration will show, noting the change in the position of the ½ and the 2 in equations (5.5) and (5.4), that the solutions given in figures 33 and 34 are identical.

Figure 34 Frames of reference.

218

In dimensionless units [36], writing u = Asech

where q = AΩZ

,

where A is the amplitude and now Ω is a shift in reciprocal dimensionless velocity. The phase displacement is

is, in

fact, a frequency and the soliton velocity, relative to the already moving frame is So when Ω is zero, the soliton is at rest in the frame moving at the group velocity. This ½, versus 2, position accounts for the z being measured as z → 2 z in the spatial beam formula and leads to [note the plus sign etc] as opposed to [note the minus sign etc]. To recap, the interpretation is (1) ξ is the angle of beam propagation, where AΩ is the relative soliton pulse displacement (2) Ω is a frequency. In this formulation, for pulses, 2A is the mean soliton energy, Ω is a mean frequency and the mean time is q/A. Hence, [A, Ω , q, φ] characterise the temporal soliton at any point during its propagation. It cannot be overemphasised that, since we are dealing with the general formula it is important we are already in the moving frame of the group velocity, so quantities like Ω (a frequency displacement) are in addition to this and describe a soliton moving relative to this frame. In the spatial soliton case, ξ ≠ 0 means that the beam is pointing in a certain direction and that is easy to understand. For pulses, however, usually something has to be happening to the pulse environment for pulses to drift relative to group velocity frame. Such events may be the presence of amplifiers, causing perturbations, or self-frequency shifts acquired when pulses get very short.

6. INVERSE SCATTERING TRANSFORM (IST) METHOD The generic equation for pulse and beam envelope soliton propagation is (6.1) Although Korteweg and de Vries managed to solve their particular nonlinear equation, attempts to formalise exact solutions immediately encountered the problem that the usual Fourier transform method does not work on such an equation. This is because of the presence of the nonlinear term 2|U|2 U. For most of this century this mathematical difficulty remained unresolved. Thirty years ago, however, Gardner, Greene, Kruskal and Muira [37] invented what is known as the inverse scattering method (IST) to extract soliton solutions from the Korteweg-de Vries shallow water problem. A beautiful extension by Lax to a wide class of nonlinear problems soon followed. The honours, however, go to Zakharov and Shabat for their discovery of how to deal with (6.1), so it is widely known as the Zakharov and Shabat problem [38]. At first sight, the IST method appears to be very abstract, especially to physicists. It even seems mysterious but, once mastered, the technique is very appealing and really does have a lot of practical uses. The idea at the heart of the method is to break up the nonlinear equation (6.1) into two ordinary linear differential equations that look like a scattering problem so dear to the hearts of physicists. This move takes us into familiar quantum mechanics territory, in so far as we have become accustomed to generating both eigenvalues and eigenfunctions from some kind of potential function that is acting as a scattering agent. Furthermore, if the number of eigenvalues reveals just how many solitons are present in an evolving beam, or pulse, this would be a very strong bonus, indeed. If 219

these same eigenvalues told us the amplitude and velocity of the solitons present, then this would be even better. Finally, if the scattering potential function is actually the initial value, U(0,S) then this would be an excellent outcome! It turns out that this is precisely the case. To achieve such a remarkable mathematical coup was always going to be a difficult task and, indeed, when the first case applied to the Korteweg-de Vries (KdV) equation, was presented, it was regarded as possibly just luck! Zakharov and Shabat changed that view and now the IST is applicable to quite a broad class of evolution equations. Possibly more than 100 equations. The main problem [2,9,39] is to find linear operators that permit the creation of an auxiliary eigenvalue problem to replace the nonlinear partial differential equation. There is still a strong measure of intuition needed here, even today. The auxiliary problem uses the pulse or beam shape U(Z = 0,S) as a potential. For this choice of linear operator the eigenvalue must remain fixed as U(Z = 0,S) evolves to U(Z,S), provided that U(Z,S) satisfies (6.1). In fact, given U(Z = 0,S), the eigenvalue problem becomes an ordinary (direct) scattering problem so the evolution of the eigenfunctions, with Z, becomes trivial. Given the eigenfunctions at Z ≠ 0 the function U(Z,S) can be found by inverting the problem; just like taking an inverse Fourier transform, only more complicated. It is interesting that Gelfand, Levitan and Marchenko [2,9,39] had provided a method for doing such as inverse, quite a few years before it was needed for soliton problems. As will be seen below, we will not need to go all the way to the inverse because even at the eigenvalue equation stage, very powerful quantitative statements about the nonlinear system can be extracted. Having stated in words what the idea is, let us now be more specific. Let us be clear, first of all, that Fourier analysis does not apply to a nonlinear problem but it is important to appreciate how beautifully simple Fourier analysis is in the linear case. Suppose that the nonlinear term 2|U| 2 U is absent, then (6.1) becomes (6.2) where k is wavenumber and ω is angular frequency, the For a plane wave dispersion relation is k = ω 2 and, from the Fourier transform, (6.3) (6.4) where (0,ω) is the Fourier transform of the INITIAL (input) function at Z = 0. Here then is a major clue to generalising the well-known Fourier method to nonlinear problems, because the evolution of U(0,S) to U(Z,S) is trivially asserted by equation (6.3). In other words, given U(0,S) (0, ω) can easily be found and once (0, ω ) is well-known (6.3) evolves the information to yield U(Z,S). In practice, (6.3) is, of course, the familiar INVERSE TRANSFORM i.e. in mathematical language, the inverse mapping of (0, ω) onto U(Z,S). This little example is very helpful, however, in enabling the task to be seen. If only we could use just U(0,S), coupled to a simple evolution like (6.3) then even if the inverse is more complicated than (6.4) the nonlinear task would be complete. How is it possible then to proceed? The first point is that using a traditional Fourier decomposition is out, but if the nonlinear differential equation could be reduced to linear calculations, using only

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U(Z = 0,S) then the calculation could proceed. The result of such n o n l i n e a r decomposition will be, as stated earlier, to make the problem like a traditional, linear, scattering problem of the type that appears in undergraduate quantum mechanics texts. Indeed, it is well known that if we could collect enough scattering data, such as reflection and transmission coefficients for any wave scattering off a potential well or bump then a reconstruction of what is actually doing the scattering is possible, in principle. This is working backwards [INVERSELY] from the scattered field to a knowledge of the scattering object. It is like hearing the sounds from a distant drum and using these sounds to reconstruct the shape of the drum. In our case the “drum”, i.e. the potential, is the initial value U(0,S). From U(0,S) we ought to be able to construct U(Z,S) by some form of inversion. We will not perform the inversion, however, because it turns out that the initial break down into linear equations yields a pair of equations that permits a very powerful and straightforward assessment of soliton presence. The problem, then, is even easier, in principle, than it looked at first sight. Unfortunately, breaking down equations like (6.1), in the first place, is not easy and required a burst of inspiration from Zakharov and Shabat, through a consideration of the following. Zakharov and Shabat recast equation (6.1) by introducing operators and , with the properties (6.5) where

and

are linear operators. Zakharov and Shabat showed that

(6. 6)

where 0 < p < 1. At this stage, things do not look too bad! Furthermore (6.7) where ψ 1, ψ2 are eigenfunctions and ξ is an eigenvalue. In other words (6.7) is solved just like a scattering problem in quantum mechanics but note that ξ remains constant, with respect to Z, i.e. if the eigenvalue can be found for U(Z = 0,S) it remains the same for all U(Z,S). Equations (6.7) yield the linear coupled equations [37] (6.8a) (6.8b) where Z i is the initial value Z = Zi , and the eigenvalue is, in general, complex. The beautiful thing about ξ, however, is that (for pulses) its real part ξ r gives the velocity of the soliton, relative to the frame of reference moving at the group velocity and ξ i gives the amplitude i.e. [40]

221

(6.9)

The interpretation of (6.8a, 6.8b) is as follows. U(Z,S) is the starting (initial ) value of the nonlinear solution of (6.1) before any evolution to further points along Z is achieved. We usually set out at Z = 0, so that (6.10a) (6.10b) An example, which will serve to illustrate the power of equations (6.10) will now be given. It shows how to determine the ‘soliton content’ of an input pulse or beam that has a rectangular shape i.e.

(6.11)

According to equations (6.10), and writing Λ = -iξ , the solutions ψ 1, ψ 2 are (6.12)

(6.13)

(6.14) where k2 = N2 - Λ 2 and g is some constant. Note that U(0,S) = 0 for S < - ½ and S > ½ and that equations (6.10) reveal that either ψ 1 or ψ 2 is zero in these regions. In other words, to get functions that do not diverge [‘blow up’], ψ1 ≠ 0, ψ 2 = 0 in the region S < - ½ and ψ 1 = 0, ψ 2 ≠ 0 in the region S > ½. Because the coefficient of ψ 1 in the region S < - ½ is normalised to unity, the coefficient of ψ 2 is equal to some constant value, which we call g. In the region - ½ < S < ½, ψ 1 ≠ 0, ψ 2 ≠ 0 but ψ 1 = 0, precisely, at S = ½ and ψ 2 = 0, precisely, at S = ½, to avoid divergences.

222

Figure 35 Dispersion relation for scattering off the rectangular distribution (6.11), using the coupled linear equations (6.8).

The boundary conditions at S ± ½ give

(6.15a) (6.15b) (6.15c) From equations (6.15a) and (6.15c)

(6.16)

223

and substitution of equation (6.16) into (6.15b) gives the dispersion equation (6.17)

and Equation (6.17) can be solved graphically as shown in figure 35. The ψ 1 , ψ 2 solutions for the three eigenstates, predicted by assuming that figure 36.

Figure 36

224

Eigenstates for N = 2π (1.7)

are shown in

Figure 37 Deducing the soliton content of Nrect(S)

The crossing points on the Λ = 0 axis of ( Λ ,k) plots are found from (6.18) and these occur at (6.19) On the Λ = 0 axis, however, k = N so equation (6.18) tells us the critical values of N needed to get n = 1, n = 2, n = 3,.… crossing points of the curves i.e. it reveals the values of N needed to generate 1,2,3,… solitons from a given Nrect(S) input condition. Figure 37 illustrates this answer for two crossing points and sets out the soliton existence conditions. The procedure described above permits the determination of whether a given input will lead to solitons, as the input evolves, during propagation. It also determines how many solitons lie “buried” in the input condition. The eigenvalue equations (6.10), on working backwards yield only the unmodified nonlinear Schrödinger equation yet this can still be used to determine soliton content [40,41] when extra terms, needed to account for

225

damping for instance need to be added to modify the original nonlinear Schrödinger equation. All we need to do is to use the exact numerically determined solution of the modified NLS at any point Z = Z 1 and use that in place of the function U(0,S) in equations (6.10). The value of the eigenvalues ξ will then reveal, given the starting function U(Z 1,S), a certain soliton content at that point of the evolution. What this means is that if any point during the evolution is used as an input to an unmodified nonlinear Schrödinger equation then the ξ = ξ r + ξ i values show how many solitons could emerge and what they will be like in terms of amplitude and velocity. In other words, at every point along the propagation direction the soliton content of a pulse, or beam, can easily be found. Indeed, it is rather easy to program (6.10) with modern mathcad software. Plots of ξ, as a function of Z, show just when the system is capable of supporting solitons and a precise way of finding out when the pulse or beam finally becomes devoid of soliton content. The power to determine this comes from the direct scattering equations (6.10). There is no need to worry about taking the inverse. A great deal can be learned about the system at this crucial, eigenvalue, stage. The algorithm is illustrated in figure 38.

Figure 38 How to deduce if an evolving pulse has any soliton content.

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REFERENCES 1. 2. 3. 4. 5.

6. 7.

8. 9. 10. 11. 12.

13. 14. 15. 16.

17. 18.

19. 20.

21. 22. 23.

J.S. Russell, Report on waves, Proc. Roy. Soc. Edinburgh, 319-320, (1844). P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge University Press, Cambridge (1983). G.B. Airy, Tides and Waves, Encyc. Metrop., Fellowes, London (1845). G.G. Stokes, On the theory of oscillatory waves, Camb. Trans. 8, 441-473, (1847). J. Boussinesq, Théorie de l’intumescence liquid appelée onde solitaire ou de translation, ce propageant dans un canal rectangulaire, Comptes Rendus Acad Sci (Paris), 72, 755-778, (1871). Lord Rayleigh, On waves, Phil. Mag. 1, 257-279, (1876). D.J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Mag. 39, 422-443, (1895). N. Zubusky, and M. Kruskal, Interaction of solitons in a collisionless plasma and the recurrence of initial states, Phys Rev Lett 15, 240-243, (1965). M. Remoissenet, Waves Called Solitons, Springer-Verlag, Berlin (1995). D.L. Lee, Electromagnetic Properties of Integrated Optics, John Wiley & Sons, New York (1986). A.C. Scott, F.Y.F. Chu and D.W. McLoughlin, The soliton: a new concept in applied science, Proc. IEEE 61, 1443-1483, (1973). (a) B.M. Oliver, Bell Telephone Laboratories Technical Memorandum MM-51-15010, Case 33089, March 8, (1951). (b) S.C. Bloch, Introduction to chirp concepts with a cheap chirp radar, Am. J. Phys. 41, 857-864, (1973). R. Dawkins, The Blind Watchmaker, Penguin, London (1986). G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, San Diego (1995). A.K. Zvezdin and A.F. Popkov, Contribution to the nonlinear theory of magnetostatic spin waves, Sov. Phys. JETP 57, 350-355, (1983). J. S. Aitchison, Y. Silberberg,, A.M. Weiner, D.E. Leaird, M.K. Oliver, J.L. Jackel, E.M. Vogel and P.W.E. Smith, Spatial optical solitons in planar glass waveguides, J. Opt. Soc. Am. B. Opt. Phys., 8(6), 1290-1297 (1991). J.S. Aitchison, K. Al-Hemyari, C.N. Ironside, R.S. Grant and W. Sibbett, Observation of spatial solitons in AlGaAs waveguides, Electron Lett., 28, 1879-1880, (1992). (a) Y. Silberberg, Spatial optical solitons in Optical Solitons, Ed. J Satsuma, SpringerVerlag, Berlin (1992). (b) P.V. Mamyshev, A. Villeneuve, G.I. Stegeman and J.S. Aitchison, Steerable optical waveguides formed by bright spatial solitons, Electronics Letters 30, 726-727, (1994). A. Villeneuve, J.S. Aitchison, J.U. Kang, P.G. Wigley and G.I. Stegeman, Optics Letters 19, 761-763, (1994). G.I. Stegeman, A. Villeneuve, J.S. Aitchison and C.N. Ironside, Nonlinear integrated optics and all-optical waveguide switching in semiconductors, Fabrication, Properties and Applications of Low-Dimensional Semiconductors, Ed M Balkanski and I Yanchev, Kluwer Academic Publishers, Netherlands (1995). A.D. Boardman and K. Xie, Theory of spatial solitons, Radio Science, 28, 891-899, (1993). A.D. Boardman, K. Kie and A.A. Zharov, Polarisation interaction of spatial solitons in optical planar waveguides, Phys. Rev. A., 51, 692-705, (1995). A.D. Boardman and K. Xie, Dynamics of spatial soliton coupling. Studies in Classical and Quantum Nonlinear Optics. Ed Ole Keller, 2-30, Nova Press, New York (1995).

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24. A.D. Boardman, K. Xie and A. Sangarpaul, Stability of scalar spatial solitons in cascadable nonlinear media, Phys. Rev. A 52, 4099-4106, (1995). 25. A.D. Boardman and K. Xie, Magnetic control of optical spatial solitons, Phys. Rev. Letters, 75, 4591-4594, (1995). 26. A.D. Boardman and K. Xie, Waveguide-based devices: linear and nonlinear coupling, Low-Dimensional Semiconductor Devices, Ed. M. Balkanski, Kluwer Publishers, Amsterdam, (1996). 27. J. Boyle, S.A. Nikitov, A.D. Boardman, J.G. Booth and K.M. Booth, Nonlinear selfchannelling and beam shaping of magnetostatic waves in ferromagnetic films, Phys. Rev. B, 53, 1-9, (1996). 28. A.D. Boardman and K. Xie, Spatial solitons in χ (2) and χ(3) dielectrics and control by magneto-optic materials. Proceedings of Minnesota International Mathematics Workshop, Springer-Verlag (1997). 29. Boardman, A.D. and Xie, K. Soliton-based switches, logic gates and transmission systems. Ed. M. Balkanski, Kluwer Publishers, Amsterdam (1997). 30. S.A. Akhmanov, A.P. Sukhorukov and R.V. Khokhlov, Self-focusing and diffraction of light in a nonlinear medium, Sov. Phys. Usp., Engl. Transl., 93, 609-636, (1968). 31. M. S. Sodha, A.K. Ghatak and V.K. Tripath, Self-focusing of laser beams, Tata McGraw-Hill, New Delhi (1974). 32. O. Svelto, Self-focusing, self-trapping and self-phase modulation of laser beams, Progress in Optics, 11, 1-51, (1974). 33. F.A. Jenkins and H.E. White, Fundamentals of Optics, McGraw-Hill, New York (1950). 34. A. Barthelemy, S. Maneuf and F. Froehly, Propagation et autoconfinement de faisceaux laser par non-linearite de Kerr, Opt. Comm. 55, 201-206, (1985). 35. A.D. Boardman, S.A. Nikitov, K. Xie and H.M. Mehta, Bright magnetostatic spinwave envelope solitons in ferromagnetic films, JMMM, 145, 357-378, (1995). 36. J.P. Gordon and H.A. Haus, Random walk of coherently amplified solitons in optical fiber transmission, Optics Letters 11, 665-667, (1986). 37. C.S. Gardner, J.M. Green, M.D. Kruskal and R.M. Miura, Method for solving the Korteweg de Vries equation, Phys. Rev. Lett. 19, 1095-1097, (1967). 38. V.E. Zakharov and A.B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, Soviet Physics JETP 34, 62-69,(1972). 39. G.L. Lamb, Elements of Soliton Theory, John Wiley & Sons, New York (1980). 40. A.N. Satsuma and G.M. Dudko, Initial value problems of one-dimensional selfmodulation of nonlinear waves in dispersive media, Prog. Theor. Phys. Suppl. 55, 284306, (1974). 41. V.V. Afanasjev, J.S. Aitchison and Y.S. Kivshar, Splitting of high-order spatial solitons under the action of two-photon absorption, Optics Comm, 116, 331-338, (1995).

228

SPATIAL SOLITONS IN QUADRATIC NONLINEAR MEDIA

Lluís Torner Laboratory of Photonics Department of Signal Theory and Communications Universitat Politècnica de Catalunya Gran Capità UPC-D3, Barcelona, ES 08034, Spain

1

Introduction

Self-focusing and self-trapping of light have been investigated since the early days of nonlinear optics. 1,2 Interest in this field has been maintained by the fascinating range of new phenomena encountered and their potential applications, such as soliton propagation, all-optical switching, and logic for ultrafast signal processing devices. For many years such effects have been pursued using the optical Kerr effect in cubic nonlinear media, 3 – 1 0 and since more recently using the photorefractive effect. 11 However, selfinduced trapping of light also occurs in quadratic nonlinear media (hereafter referred to as χ ( 2 ) media). 12 In this case, both spatial and temporal solitons form through the mutual focusing and trapping of the waves parametrically interacting in the nonlinear medium. In these lectures we focus on spatial solitons that form with cw light signals in planar waveguides and in bulk crystals, but most of the mathematical results hold also in the case of temporal solitons. Also, following the usual convention, throughout these lectures we make no distinction between the solitary waves that we study and mathematical solitons, referring to both of them as solitons. Self-focusing effects in quadratic nonlinear processes were long known to be possible under specific conditions, namely when the parametric interaction is very weak and yields an effective third-order effect for the pump wave. 1 3 , 1 4 By and large, however, the full extent of the self-focusing and its implications were not fully appreciated until recently. 1 5 The remarkable exception is the theoretical work of Karamzin and Sukhorukov more than twenty years ago, 12 who investigated the mutual focusing of beams in parametric processes and identified its implications for the formation of solitons. In the last few years, spatial soliton propagation has been observed experimentally in secondharmonic generation settings by Torruellas and co-workers in bulk potassium titanyl

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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phosphate (KTP), 16 and by Schiek et. al., in planar waveguides made of lithium niobate (LiNbO 3 ). 17 Generation of strings of solitons through the modulational instability of wide intense beams has been observed experimentally by Fuerst and co-workers.18 Solitons in quadratic nonlinear media form in planar waveguides and in bulk media, and the lowest-order bright solitons are stable on propagation in both cases. Stable light bullets are also theoretically possible. Those solitons might have important applications to different optical devices, including switching and routing devices and optical cavities containing χ ( 2 ) crystals. The aim of these lectures is to present the background of the topic in an unified point of view. We shall only discuss the basic properties of the solitons in the simplest configuration, namely second-harmonic generation, and we shall concentrate in Type I phase-matching. Naturally, the families of solitons are richer in the case of Type II phase-matching geometries because of the additional degree of freedom that this phase-matching offers, and that is important regarding potential applications of the solitons, but Type I geometries are simpler and capture the main features of the solitons. Hence, except for the last part of the lectures where we discuss a few applications of the solitons, we shall focus on Type I phase-matching. Also, we shall only study bright solitons. The lectures are organized as follows. In Section 2 we shall describe the physical setting considered and the evolution equations used to model the light propagation in quadratic nonlinear crystals. Section 3 is devoted to the basic properties, including stability, robustness and excitation, of the simplest families of bright solitons that exist in the absence of Poynting vector walk-off. In Section 4 we shall meet the solitons that exist in the presence of Poynting vector walk-off and discuss briefly their salient properties. Section 5 is devoted to the dynamics on evolution of beams with topological phase dislocations, or optical vortices. In this part we shall meet the modulational instabilities existing in quadratic nonlinear media and we will discuss how different combinations of topological charges of the input light signals produce certain patterns of bright solitons. To end the lectures, in Section 6 we shall discuss briefly discrete solitons and in Section 7 we shall summarize our main conclusions.

2 2.1

Physical Setting Evolution Equations

We consider cw light beams travelling in a medium with a large, non-resonant χ (2) nonlinearity under conditions of second-harmonic generation. We focus on spatial solitons in Type I phase-matching settings. The electric field of each of the waves is written in the form E (r, t) = A ( r) exp[ikz – iω t], and we consider experimental conditions for which both the scalar and the slowly varying envelope approximations for the fields hold. These conditions are expected to be fulfilled under most relevant experimental situations, but they might fail for narrow beams, very high input powers, and in highly anisotropic crystals.

230

We shall study the formation of solitons in planar waveguides and in bulk media. Most of the generic expressions shall be written for the bulk case, but it is assumed that they hold in both cases. Throughout the lectures we use the subscripts 1 and 2 for quantities associated with the fundamental and second-harmonic waves, respectively. The appropriate χ ( 2 ) nonlinear coefficients involved in the two-wave mixing process are included in the normalization of the fields. Under such conditions, the beam evolution can be described by the reduced equations 19

(1) where a1 and a2 are the normalized envelopes of the fundamental and second-harmonic waves, ξ is the normalized propagation coordinate, r = –1, and α = – k 1 / k 2 . Here k 1,2 are the linear wavenumbers at both frequencies. In all realistic situations α – 0.5. The parameter β is a measure of the linear phase mismatch between the fundamental and second-harmonic waves and is given by β = k 1 η2 ∆ k, where ∆ k = 2 k 1 – k 2 , is the wave vector mismatch and η is a characteristic beam width. The parameter δ and the unit vector characterize the magnitude and direction of the Poynting vector walk-off due to the fact that the energy and wave fronts generally propagate in different directions in a birefringent medium. One has, δ = k 1 ηρ w , with ρw being the walk-off angle, and we note that walk-off is absent for propagation along the principal optical axes of the crystal. Finally, the transverse coordinate is given in units of η , and the propagation coordinate is normalized in such a way that z/l d = 2 ξ, with l d = k 1 η2 /2, being the diffraction length of the fundamental beam. In the one-dimensional case, equations (1) also hold for pulsed light. Then, diffraction is replaced by dispersion, Poynting vector walk-off is replaced by group velocity mismatch, and r and α are given by the group velocity dispersions. 2.2

Limit of Large Wave vector Mismatch

The self-focusing nature of the wave-mixing process at the regime of large phase-mismatch between the waves (β >> 1) and small conversion to the second-harmonic, is simply exposed by noticing that in such conditions the governing equations approximately reduce to the nonlinear Schrödinger equation (NLSE) 2 0 – 2 2 (2) which in one-dimensional geometries is known to have stable soliton solutions. The properties of the χ (2) solitons are impacted by this fact. 19 However, the NLSE does not allow stable solitons for bulk geometries and a question might arise about the implications of this fact to χ (2) trapping . Actually, the beam evolution in the χ ( 2 ) medium quickly violates the approximations required to derive the NLSE, and stable solutions do exist in bulk quadratic nonlinear media.

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However, an important point must be emphasized. Most experimentally relevant solitons in quadratic media form for small and negative wave vector mismatches, at exact phase-matching, and in general under other conditions where (2) does not hold. Therefore, such solitons exhibit new properties, dynamics, and features. Thus, they ought to be treated accordingly. 2.3

Experimental

Values

We shall shortly be giving some values of β and δ that are relevant to experimental conditions, but the impact of these parameters on the formation of solitons is best elucidated by recalling their relation to the linear lengths that characterize the lowpower beam evolution in the χ (2) medium, namely the diffraction length l d , the coherence length lc = π / |∆ k|, and the walk-off length l w = η /ρ . One has (3) Non-critical or temperature-tuned phase-matching configurations that yield a negligible 0; by contrast, δ ≠ 0 occurs when angle-tuning phasewalk-off correspond to δ matching is used and walk-off is present. When the diffraction and walk-off lengths are comparable one has δ ~ 1. Most crystals with large χ(2) nonlinear coefficients are also highly birefringent, as is the case for most organic materials, and for that case walk-off would dominate diffraction and large values of δ would result for angle-tuned configurations. Under such conditions, equations (1) may not be valid. The role of the parameter β is exposed by recalling that for focused beams the value of ∆ k gives only partial information about whether or not it corresponds to near phase-matching. This is because focused beams contain a broad spectrum of transverse wave vectors, and each spectral component of the beam experiences a different wave vector mismatch. The effective phase mismatch, as measured by the second-harmonic generation efficiency, depends on the diffraction properties of the beam, and for the case we are studying here it is given by the value β . Outside exact phase-matching, which 2 –1/2 corresponds to β = 0, it is useful to write β = ( η/η0 ) , with η 0 = (k 1  ∆k ) being a characteristic width. Thus, for a given input beam width η , a wave vector mismatch such that η ≈ η0 , effectively corresponds to near phase-matching, whereas a larger ∆ k , with |β| >> 1, corresponds to a large phase mismatch. The mutual trapping of beams in the χ (2) crystal is governed by the interplay between the linear lengths contained in β and δ, in addition to the nonlinear length determined by the light intensity and by the input conditions. Regarding the linear lengths, favourable conditions for self-trapping occur when all lengths are comparable, so that β ~ ±3, and δ ~ ±1. Typical experimental conditions that yield these values, and which are representative of the actual parameters involved in the experimental observation of solitons in KTP cut for Type II phase-matching, 16 are η ~ 15 µm, and ρ w ~ 0.1° – 0.5°. For such parameters, the value of δ falls in the range 0.3 – 1.5, and one needs a coherence length of some l c ~ 2.5 cm, to have β ~ ±3. For typical materials and wavelengths, say

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–4

λ ~ 1 µm, this coherence length corresponds to a refractive index difference ∆ n ~ 10 between the fundamental and second-harmonic waves. The above parameters yield a diffraction length of l d ~ 1 mm, so that ξ in the range 0 – 20, corresponds to a few cm. Large effective wave vector mismatches, in the sense mentioned above, correspond to β ~ 30, whereas exact phase-matching corresponds to β = 0. Large values of δ, say δ ~ 5, which for the above parameters is obtained in configurations with a walk-off angle of some ρ w ~ 1.5°, correspond to a moderate Poynting walk-off. They are representative of the experimental conditions encountered, e.g., in organic materials with very large quadratic nonlinearities, at appropriate wavelengths close to those where non-critical 23,24 phase-matching occurs. To grasp the order of magnitude of the optical intensities associated to the normalized quantities used along the lectures and that are required to form solitons in quadratic nonlinear media, notice that for a beam width η ~ 20 µ m and the typical nonlinear coefficients of KTP, a normalized power (defined below) of some I ~ 50, leads to an actual peak intensity of the order of ~ 10 GW/cm2 . Existing inorganic materials (e.g., potassium niobate), semiconductors (e.g., gallium arsenide) and organic materials (e.g., N-4-nitrophenil-(L)-prolinol, NPP, or dimethyl amino stilbazolium tosylate, DAST), with larger quadratic nonlinear coefficients would require lower power requirements. 2 3 , 2 4 No need to say that, this is so provided that the appropriate nonlinear coefficients are accessible at suitable wavelengths and phase-matching geometries with a reasonable walk-off and low absorption, and provided that long enough, mechanically stable, high quality samples can be made. Quasi-phase-matching of the largest nonlinear coefficients of suitable materials, e.g., LiNbO 3 , also leads to reduced power requirements and to suitable operating conditions, so that it holds promise for future use. 2 5 – 2 7

2.4

Conserved Quantities

Central to the beam evolution described by equations (1) is the fact that they constitute an infinite-dimensional, Hamiltonian dynamical system.1 0 , 28 To expose this fact, it is convenient to rewrite the equations in terms of the new fields A 1 = a1 , and A2 = a2 exp(–i βξ ) . T h e n , the conserved Hamiltonian writes

(4) The governing equations can now be written in the canonical form (5) where the symbol δ F indicates a Fréchet or variational derivative and Ã2 = We shall also make use of two additional conserved quantities: the total power or energy flow given by the Manley-Rowe relation (6)

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and the total transverse beam momentum (7) In Section 5 we shall consider light beams without azimuthal symmetry, and in such cases it is useful to monitor the evolution of the longitudinal component of the total angular momentum of the light beams given by (8) with being the transverse beam momentum density in expression (7). In the absence of Poynting vector walk-off, L is also a conserved quantity of the beam evolution. When walk-off is present, one readily finds that (9) For the right-hand-side of this equation to vanish in the presence of walk-off, the transverse momentum of the second-harmonic beam has to be parallel to Otherwise, the total angular momentum is not conserved in the presence of Poynting vector walk-off. It is also important to examine the rate of energy exchange between the fundamental and second-harmonic waves. Writing the fields in the form a 1,2 = R 1,2 exp( i φ 1,2 ), where R 1,2 and φ 1 , 2 are real quantities, one arrives at (10) Therefore, to cancel the energy exchange between the fundamental and second-harmonic beams, their wave fronts, including the nonlinear wave vector shifts induced by the wave interaction, ought to verify φ 2 ( ξ , r ⊥ ) = 2φ1 ( ξ, r ⊥ ) + β ξ. This is what happens when a soliton is formed: the transverse complex shapes of the two interacting beams induce the appropriate wave front distortions to cancel diffraction and Poynting vector walk-off, while the energy exchange between the waves is also cancelled. It is worth emphasizing an obvious but important fact: such process occurs dynamically, and it takes an infinite distance to form a true stationary soliton out of input conditions that do not coincide with such a soliton, as it is always the case in practice. 3 3.1

The Simplest Bright Solitons Families of Stationary Solitons

We first consider stationary soliton solutions in the absence of Poynting vector walk-off (i.e., δ = 0). The simplest solutions have the form (11) with κ v being the wave number shifts induced by the nonlinear wave interact ion. For the solutions to be stationary one needs κ 2 = 2 κ1 + β. According to (10) this relation

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Figure 1: Typical shape of (1+1) solitons for different values of the wave vector mis32,33 match. The solitons correspond to I = 30. ensures that there is no power exchange between the waves. The coupled equations obeyed by U1 and U2 are given by

(12) By and large, these equations have a rich variety of solutions, including solutions with different dark shapes, solutions with multiple peaks, and solutions with exotic shapes. In these lectures we are only interested in the lowest-order solutions. In the case of (2+1) solitons in bulk media (i.e., solitons with two transverse coordinates plus one longitudinal coordinate), those correspond to nodeless, radially symmetric solutions with no azimuthal angular dependence. In equations (12), r, α and β contain material and linear wave parameters, while the nonlinear wave number shift κ1 parametrizes the families of solutions. In all plots presented in these lectures r = –1, and α = –0.5. In the case of (1+1) solitons, one analytical solution with a bright shape is known. Namely, 12,19,29–31 (13) where s is the transverse coordinate. This solution occurs at β = –2 ( α – 2r), with the special value κ1 = –2 r. The whole family of solutions for different values of κ 1 and at other values of β can be found by solving equations (12) numerically using a shooting or a relaxation algorithm. Figure 1 shows the typical shape of a few representative solitons. In the case of (2+1) solitons no analytical solutions are known, but whole families of solitons are readily obtained by solving the equations numerically. 32–35 3.2 Similarity Rules Even though the families of stationary soliton solutions are only known numerically,

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important information about their properties can be obtained analytically by examining the scaling properties of eqs. (12). One readily finds that such equations are invariant under the similarity transformations (14) with µ ≠ 0 being an arbitrary parameter. Thus, the soliton solutions can be transformed into each other following such rules. This self-similarity has important consequences. For example, when doing the numerical calculations it can be used to solve equations (12) in an efficient way. More important, it has direct physical consequences. In particular, it shows that the general features of the solutions at either side of phase-matching are similar. Similarly, it implies that at phase-matching ( β = 0), all the solitons in the absence of Poynting vector walk-off are self-similar. Therefore, at phase-matching the relation between the amplitude and the width of the family of solitons is amplitude × (width)² = constant.

(15)

Outside phase-matching, the stationary solutions exhibit different amplitude-width relations. We shall examine them below. 3.3

Stability

One crucial property of the families of solitons is their stability under propagation. Here we refer to the stability of the solitons when propagated in the system with the same number of “transverse dimensions” than the system where they have been found. In other words, modulational instabilities against higher-dimensional perturbations are not considered. Otherwise, solitons in quadratic nonlinear media are known to be modulationally unstable in both (1+1) and (2+1) geometries if the corresponding perturbations are allowed to grow in the experimental setting considered.36,37 We shall discuss this issue in Section 5. The stability of the families of stationary solutions can be elucidated by using different methods.38—42 In these lectures we shall only discuss a geometrical approach. 42 Such geometrical derivation of the stability criterion is useful by itself, because in many cases it gives direct insight into the stability of the solitons, and also because it shows the universality of the stability criterion for similar systems. One first finds that the stationary solutions with the form (11) occur at the extrema of the Hamiltonian for a given energy flow, i.e., they correspond to (16) Now, the stability of the stationary solutions can be elucidated by noticing that the global minimum of H gives stable solutions, whereas local maxima yield unstable solutions. This is a consequence of Lyapunov theorem about dynamical systems applied to this case.10,28 Therefore, to elucidate the stability of the families of solitons one has to plot the curve H = H (I) and identify its lower and upper branches. The condition of

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Figure 2: Hamiltonian and wave number shift versus energy for the families of (1+1) solitons. (a) and (c): phase-matching and positive β; (b) and (d): negative β. Solid lines: stable solitons; dashed lines: unstable solutions. 42 marginal stability, that separates stable from unstable solutions, is given by the point that separates the lower from the upper branches of the curve H = H (I ). There are different ways to determine that point. For example, examination of the corresponding curves, shown in Figure 2 for the case of (1+1) solitons, leads immediately to the so-called Vakhitov-Kolokolov criterion, 43 given by (17) The mathematical proof is given by Whitney’s theorem about two-dimensional maps applied to this case. 44 In the case of spatial solitons in quadratic nonlinear media, the lower branches of the curves in Figure 2 are found to correspond to the global minimum of H , and the upper branches to local extrema. Therefore, criterion (17) holds. Physically, the main conclusion to be raised from Figure 2 in the case of (1+1) solitons, and from the similar plot but for (2+1) solitons,34,35 is that at positive phasemismatch and at phase-matching, all the lowest-order stationary solutions in the absence of walk-off are stable. Very narrow regions of solutions that would be unstable exist at negative phase-mismatch near the cut-off condition for the soliton existence, but those have a very limited physical relevance to the experimental formation of solitons. This is so because of several reasons. First, because solutions near cut-off correspond to increasingly broader beams; second, because in reality the input beams never match exactly the shape of the unstable stationary solutions; and third, because above the threshold light intensity for the existence of solitons there is always a stable soliton. Therefore, the excitation of the stable solitons that exist reasonably far from cut-off is what dominates the dynamics of the beam evolution.

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3.4

Two Useful Properties

The families of stationary soliton solutions contain important information. In these lectures, we shall only discuss two of them, that have direct experimental implications. One important property of the families of solitons is their amplitude-width relations. At phase-matching the amplitude-width relation is given by (15), and otherwise it has to be calculated numerically. Figures 3(a)-(b) show the outcome in the case of (2+1) solitons at β = ±3. 35 One important consequence of the existence of soliton families with a given amplitude-width relation is that under the influence of any small perturbations which lead to adiabatic changes of the energy flow along the propagation direction, like small material or radiative losses or gain, the shape of the solitons tends to adiabatically evolve following the corresponding relation. Figures 3(a)-(b) show that in the case of solitons in quadratic nonlinear media, at reasonably large amplitudes the width increases rather slowly as the amplitude of the solitons decreases. Therefore, such solitons retain their shape a great deal while they propagate under the influence of the non-conservative perturbations, provided that they are small. It is worth noticing that the behavior shown in Figs. 3(a)-(b) is consistent with the behavior of the soliton width for different soliton energies that were observed experimentally.16 Another important feature of the family of solitons is the fraction of the total energy that is carried by each of the waves, fundamental and second-harmonic, that form the soliton. Figure 3(c) shows how this fraction depends on the wave vector mismatch for two representative values of the total energy flow of the soliton. 33 This particular plot corresponds to (1+1) solitons, and it shows that at positive phase-mismatch most of the energy is carried by the fundamental wave, whereas at phase-matching the energies are comparable and at negative β the largest amount of energy is carried by the secondharmonic. Naturally, all this has important implications when it comes to the excitation of solitons with different input light conditions.

Figure 3: (a) and (b): width of the (2+1) solitons as a function of their amplitude. 35 (c): fraction of energy carried by each wave forming the (1+1) solitons as a function of the wave vector mismatch. Solid line: I = 18; dashed line: I = 30. 33 238

Finally, notice that expression (16) provides the starting point of a variational approach to obtain approximate analytical expressions of the families of solitons. One needs to calculate I and H for given beam shapes and then optimize their parameters to minimize H. See ref. 45. 3.5

Excitation, or “Oscillating Solitons”

In practice, the input light conditions do not coincide with the shape of the stationary solitons. Therefore, the excitation of solitons with arbitrary input light beams is a central issue, that has to be investigated in detail. This is particularly true in the case of solitons that form in the presence of walk-off, because walk-off poses severe restrictions to the formation of solitons in realistic experimental conditions. Also, notice that at present, the longest χ (2) crystals available are a few centimeters long, which corresponds to a few tens of linear diffraction lengths, so that in single pass experiments only the soliton evolution during a relatively short distance is relevant. In general, one has to elucidate both the soliton content of the input conditions considered, and the dynamics of the evolution of such input conditions. In the case of mathematical solitons, namely soliton solutions of so-called completely integrable evolution equations, a great deal of this information can be obtained using the tools provided by the Inverse Scattering Transform; in particular, the soliton content of the input conditions can be determined a priori. However, this is not so for solitons of nonintegrable evolution equations, as it is the case of the system (1). Therefore, the study of the dynamics of the soliton excitation relies heavily on numerical experiments. Those are performed by solving the evolution equations (1), typically with a split-step Fourier or a finite-difference standard scheme, for given arbitrary input conditions. Motivated by its experimental relevance, one might consider sech-like, or Gaussian input beams with the form (18) with A and B being the amplitudes of the fundamental and second-harmonic beams, respectively. The excitation of bright solitons has been examined for a wide variety of input conditions, both in the case of (1+1) and (2+1) geometries. The numerical experiments show that solitons emerge from the input beams in a wide variety of input conditions, in terms of wave vector mismatches and input light shapes and intensities, not necessarily close to those given by the stationary soliton solutions. Solitons also emerge with inputs which fall very far from those solutions indeed, and in particular when only the fundamental beam, or only the second-harmonic plus a small fundamental seed, is supplied at the input face of the χ (2) crystal. In such a case, the input beams reshape, exchange power and adjust themselves through radiation of dispersive waves, and then they mutually trap. See refs. 46 and 47 for details. Here we shall only recall one point that is found: the excitation of solitons with arbitrary inputs, that hence contain one stationary soliton plus some amount of radiation,

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Figure 4: Excitation of solitons with the input conditions (18). Left: Peak amplitude of the fundamental beam, scaled to the input value, as a function of propagation distance. The plot is for (2+1) propagation. Right: Detail of the beam evolution in the case A = 20, B = 0.47 produces “oscillating states”. The amplitude of the oscillations decreases as the beams shed dispersive waves away, but after the initial reshaping process the leak is extremelly small. Figure 4 shows the typical beam evolution during the first tens of diffraction lengths. The plot corresponds to propagation in bulk media. The oscillations might have several experimental consequences. For example, when only fundamental beams are initially launched with energy well above the threshold for the existence of stationary solitons, the numerical simulations predict fast oscillations that produce sharp intermediate stages. 47 Under those conditions the peak power at the centre of the beams reaches very large values. Such values might be high enough to damage the crystal. Also, at such values equations (1) might not be valid, and the actual beam evolution inside the crystal might uncover new important features that ought to be investigated. 3.6

Solitons

in

Quasi-Phase-Matched

Samples

By and large, quasi-phase-matching (QPM) offers an attractive approach to produce highly-efficient parametric wave-mixing interactions in quadratic nonlinear optical media (for a comprehensive review, see refs. 25-27). Conventional phase-matching techniques used to compensate for the wave vector mismatch between the waves of difference frequencies that interact in the nonlinear medium are based on the birefringence and thermal properties of the materials involved. As a consequence, the overall efficiency of the interaction can be limited by a variety of effects, mainly Poynting vector spatial beam walk-off due to the different propagation directions of energy and phase fronts in anisotropic media, non-convenient operation temperatures or crystalline optical axes orientations, and the value of the quadratic nonlinear coefficients accessible through the polarization of the waves involved in the interaction.

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With QPM, the coupling between the fundamental and second-harmonic waves can be chosen so that non-critical phase-matching can be achieved at a suitable temperature, without Poynting vector spatial walk-off between the interacting beams, by using the largest second-order coefficients of the material employed, and with an optimized overlap between the guided modes involved in waveguide devices. Such advantages hold very promise for many applications, in both waveguide and bulk settings, and in particular to the formation of solitons. QPM can be potentially used for a variety of materials and operating frequencies. Typical experimental conditions give coherence lengths l c = π /|∆ k| in the range 2 – 20 µ m, with the actual value depending on the material involved and the wavelength used. In the normalized units of equations (1), one has β ~ 500 – 1000. For the typical values of the other various involved parameters, given in Section 2, β ~ 500 corresponds to a coherence length of some 7 µm. These are the conditions encountered, e.g., in the QPM of the diagonal d 33 nonlinear coefficient of LiNbO3 at λ ~ 1.5 µm. QPM relies on the periodic inversion of sign of the nonlinear χ( 2 ) coefficient at given multiples of the coherence length l c , and the so-called m -th order QPM corresponds to a periodic domain inversion with period m = 2ml c (see refs. 25-27 and references therein, and Marty Fejer’s lectures). In general, the biggest nonlinearity is obtained for 1-st order QPM. In such a case, the corresponding evolution equations at leading-order, after averaging over the periodicity of the QPM structure, are analogous to equations (1), but with effective nonlinear coefficients 2/ π smaller than the actual material coefficients and a global phase shift between the fundamental and second-harmonic waves. Hence, 48 so are the solitons. Higher-order corrections have been considered in ref. 49. Here we only wish to discuss the robustness of the soliton formation and evolution against random deviations of the domain length. Such random deviations occur as a consequence of the fabrication tolerances of the QPM domains. The physical nature and statistical properties of the random deviations from the nominal lengths depend a great deal on the specific fabrication technique used to implement the QPM sample. Here we only discuss the so-called duty-cycle errors, 25 that occur when the periodicity of the domains is very well controlled, but the positions where the domain walls actually form differ from the nominal ones. In the resulting structure, the ending wall of the n -th where domain is located at the position is the nominal length of each domain and Rn is the random shift. Such errors are short-range correlated along the longitudinal coordinate. Therefore, they have a small, adiabatic impact on the solitons, because when the correlation length is much shorter than the characteristic soliton length the stochastic effects can be averaged out to a large extent over every characteristic soliton length. On the contrary, long-range correlations might be far from being averaged out over a soliton period, hence they impact more strongly the soliton evolution. This is the case of the so-called random-walk errors that occur when the ending wall of the n -th domain is located at Figure 5 shows the typical outcome of the excitation of a soliton in a QPM sample with duty-cycle domain length errors. The plots show the fundamental beams propa-

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Figure 5: Excitation of a soliton in QPM samples with duty-cycle domain-length errors. Dashed lines: case of an ideal structure; solid lines: case of structures with different random domain errors, with typical deviation: in (a) 10 % ; in (b) 20 %. 48 gated some 20 diffraction lengths. The dashed lines show the soliton in the case of an ideal structure and the solid lines correspond to structures with random errors with a Gaussian distribution around the nominal domain length. On average the duty-cycle random errors induce small radiative losses in the solitons. Because losses are small, once the soliton is formed after the input beams reshape slightly, they evolve adiabatically following the amplitude-width relation of the family of solitons. 48,50

4 4.1

Walking Solitons Motivation

By their very nature, solitons in quadratic nonlinear media are made out of the mutual trapping of several waves. Here we consider the formation of spatial solitons under conditions for second-harmonic generation, therefore the solitons exist due to the mutual trapping of the fundamental and second-harmonic beams. In general, except under suitable conditions, in the low-power regime the beams propagate along different directions due to the Poynting vector walk-off present in anisotropic media and this fact has important experimental implications when it comes to the choice of suitable materials, input light wavelength and general conditions suitable to the formation of solitons. However, when a soliton is formed the interacting waves mutually trap and in the presence of Poynting vector walk-off the beams drag each other and propagate stuck, or locked together. Such a beam locking opens the possibility to specific applications of the solitons, 51–56 and it also poses new challenges to the understanding of the soliton formation. This is so because the “walking” solitons existing in the presence of walkoff exhibit new features in comparison to the non-walking solitons. Investigation of these new features is important regarding their potential applications, but also from a fundamental viewpoint because the approach and outcome have implications to walking solitons existing in other analogous but different physical settings.

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4.2 Families of Solitons Next, we examine stationary solutions of equations (1), with δ ≠ 0, describing mutually trapped beams walking off the ξ axis. Those have the form (19) where U and φ are real functions, η = s – v ξ is the transverse coordinate moving with the soliton, and φ v (ξ,s) = κ v ξ + ƒv (η). Here v is the soliton velocity, and ƒv ( η) are the wave fronts of the solitons. According to (10), to avoid energy exchange between the waves one needs κ 2 = 2 κ1 + β , as for non-walking solitons. However, one also needs that the wave fronts verify ƒ 2(η) = 2ƒ 1 ( η ) everywhere, or alternatively U v ( η ) and ƒ v ( η) ought to be symmetric and anti-symmetric functions of η, respectively. The solitons that exist in the absence of walk-off fulfil the former condition, whereas with this exception all the walking solitons fulfil the latter. Substitution of (19) into (1) yields the system of coupled nonlinear ordinary differential equations fulfilled by the functions U v ( η ) and ƒ v ( η). They can be solved numerically to obtain the families of walking solitons.57–60 Recall that κ 1 and v parametrize the families of walking solitons. Experimentally, such parameters correspond to the light intensity and to the angular deviation of the solitons from the longitudinal propagation axis. Walking soliton solutions exist for values of κ 1 and v such that solitons are not in resonance with linear dispersive waves. The resonance condition can be readily 57 calculated to obtain (20) Figure 6 shows the typical amplitude and wave front shape of a walking soliton. As shown in the plot, the walking solitons have curved wave fronts. Such curvature depends on all the parameters involved, including the wave vector mismatch, the walk-off measure, the soliton energy and the soliton velocity. 57–60 As it is the case of the non-walking solitons studied previously, important information about the families of walking solitons can be obtained from the conserved quantities of the wave evolution, as follows. One first finds that the stationary walking solitons with the form given by (19) occur at the extrema of the Hamiltonian for a given energy flow and a given transverse momentum, i.e., they occur at (21) This is an important result that has important implications to the soliton stability. One can also use it to find approximate analytical expressions of the shape of the walking solitons using a variational approach. The walking solitons have curved wave fronts and therefore their transverse momentum is not simply proportional to their velocity. The actual relation between the velocity and the momentum for the walking solitons can be elucidated by examining the evolution of the energy centroid of the bound state constituted by the fundamental and second-harmonic beams propagating stuck together. One finds 51

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Figure 6: Typical shape of a (1+1) walking soliton. The plots is for β = –3, δ = 1, and I = 30. In (a)-(b): v = –0.5; in (c)-(d): v = 0.5. 57

(22) In contrast to this result, the travelling-wave solutions that occur in the absence of Poynting vector walk-off have a flat wave front and a tilt given by the soliton velocity. In such conditions, one finds the particle-like results (23) 2

where H v = 0 is the Hamiltonian of the zero-velocity solitons and the term Iv /2 represents the kinetic part of the Hamiltonian of the solitons walking with velocity v. However, such is not the case of the solitons that exist in the presence of Poynting vector walk-off. 4.3 Stability The stability of the families of walking solitons can be elucidated by different ways. Here we shall use the geometrical approach discussed previously for the case of non-walking solitons. Once again, our starting point is the variational expression (21). Because the families of stationary walking solitons realize the extrema of H for given I and J, one concludes that solutions that realize the global minimum of H are stable, whereas those that realize a local maxima are unstable on propagation. Therefore, we ought to examine the surface H = H(I, J) and identify its lower and upper sheets (Fig. 7). In the case of “smooth” H = H (I, J) surfaces, the curve that separates the lower and upper sheets of the surface can be determined by noticing that over the curve the vector normal to the surface is contained on the horizontal plane. A similar procedure holds in the case of sharp surface foldings. Sheet-crossings ought to be treated separately. The vector normal to the two-parametric surface is given by the expression (24)

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Figure 7: Sketch of the procedure to determine geometrically the condition of marginal stability of the walking solitons. is contained where the symbol ∂ ( F, G ) / ∂(a,b) stands for a Jacobian matrix. The vector on the horizontal plane when the last term in the RHS of (24) vanishes, so that (25) This is the condition of marginal stability for the families of walking solitons that is derived using a linear stability approach.59 To identify the stable and the linearly unstable walking solitons one has to evaluate the condition (25) for the families of walking solitons. Actually, only the families that exhibit multi-valued surfaces H = H (I, J) need to be studied in detail. See ref. 60 for the details. The main conclusion is that, similarly to the families of solitons that exist in the absence of walk-off, narrow regions of unstable solutions exist near cut-off, but with such exceptions all the walking solitons are stable on propagation. 4.4 Excitation The excitation of walking solitons with different input beams is governed to a large extent by Eq. (22). For our present purposes it is better to write it as (26) This expression has to be used with caution because it holds for the families of stationary walking solitons, but not for the input light conditions. The difference is that unless the input conditions exactly match the shape of the walking soliton solutions, the beam dynamics and reshaping towards the formation of a walking soliton always produces some radiation that takes energy and momentum away. However, when the radiation produced is small Eq. (26) provides a direct estimation of the velocity of the walking soliton that eventually gets excited. The ratio I 2 /I depends strongly on the linear wave vector mismatch between the waves and also on the total energy flow. In particular, at positive β the solitons have small I 2 /I, and they walk slowly. At phase-matching and at negative β the solitons have larger I 2 /I, thus they walk faster.

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5 5.1

Vortices Motivation

Vortices, or topological wave front dislocations, are ubiquitous entities that appear in many branches of physics. 61 Regardless the physical setting considered, the vortices display fascinating properties and a strikingly rich dynamical behavior. Optical vortices are not an exception. They are spiral, or screw dislocations of the wave front that has a helical phase-ramp around a phase singularity. They appear spontaneously in speckle-fields, in the doughnut laser modes, and in optical cavities, and otherwise they can be generated with appropriate phase masks or by the transformation of laser modes with optical components. Optical vortices have been investigated in linear media, cubic nonlinear media, and photorefractive crystals. 62–70 In this section we shall examine the phenomena generated by intense beams containing optical vortices in bulk quadratic nonlinear media under conditions for secondharmonic generation. With moderate input powers and wide beams, when only the fundamental pump beam is initially launched light undergoes frequency doubling together with the generation of a phase dislocation nested in the second-harmonic beam. However, with intense, tightly focused beams soliton effects become crucial and a whole new range of phenomena appears. In particular, the composite states of mutually trapped beams containing the phase dislocations self-break inside the quadratic crystal into separate beams, that then form sets of spatial solitons. 71–73 Such a behavior defines the principle of operation of a new class of devices that can process information by mixing topological wave front charges and producing certain patterns of spatial solitons. The number of output solitons can be controlled by the value of the “array” of topological charges of the input light signals. Next, we shall present the basic ingredients needed to understand the device behavior. Namely, the existence of solitary-wave vortices and their instability, and the dynamics generated by the mixing of beams with different topological charges. 5.2 Bright Vortex Solitary Waves We examine families of stationary solutions of equations (1), with δ = 0, that correspond to solitary-wave vortices. Specifically, we consider topological phase dislocations nested in the centre of beams with a bright, Gaussian-like shape. Such solitary waves appear as higher-order solutions of the governing equations. They have the generic form (27) where ρ is the radial cylindrical coordinate, and ϕ is the azimuthal angle. In the case of solutions with a phase dislocation nested in the centre of the beams, m v are the topological charges of the dislocations and sgn(mv ) their chirality. The transverse profiles U v are assumed to be real, radially symmetric functions. To avoid power exchange between the fundamental and second-harmonic waves one needs κ 2 = 2κ1 + β, and m 2 = 2m1.

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Figure 8: Typical shape of bright vortex solitary waves. Solid lines: fundamental beam; dashed lines: second-harmonic. Conditions: β = 3, κ1 = 3. 72 The coupled equations obeyed by U1 and U 2 are

(28) These equations have a rich variety of solutions. In particular, there are families of bright, higher-order solutions of two types. Namely, higher-order solutions without vorticity, that are characterized by the number of nodes, or field zeroes of the beams, and solutions without nodes, but with phase dislocations of increasingly higher charge. Here we only consider the latter. Figure 8 shows the typical shape of a solitary-wave vortex with charge m1 = 1 and with charge m1 = 2. The whole families of solutions in the case of Type I geometries can be found in ref. 72. Type II geometries produce analogous results. 73 However, in the case of Type II the problem is richer, because it involves three waves (see appendix A). In particular, the families of solitary-wave vortices occur for the combinations of topological charges that verify m 2ω = m o ω + m e ω ,

(29)

where moω and meω correspond to the ordinary-polarized and the extraordinary-polarized fundamental beams. The important point for our present purposes is that the bright vortex solitary waves are unstable. In general, such instability produces the self-breaking of the beams along the azymuthal direction. Such self-breaking is due to the modulational instability of the top of the ring-shaped beams, somehow analogous to other spatial and temporal modulational instabilities arising in χ (2) media,36,37,109–111 and similar to azimuthal modulational instabilities that occur in χ ( 3 ) media.6 3 , 6 5 , 6 7 , 7 4

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Figure 9: (a) and (b): Growth rate of perturbations with different azimuthal indices for vortices with m1 = 1. (c): Typical decay of the solitary-wave vortex in Fig. 8(a). 72 5.3 Azimuthal Modulational Instability (AMI) To examine the stability of the solitary-wave vortices against azimuthal perturbations one can seek solutions of the form

(30) Inserting (30) into (1), and linearizing around the perturbations, one obtains the set of four coupled linear partial differential equations obeyed by ƒv ( ρ,ξ) and gv ( ρ,ξ). Such equations have many possible types of solutions, but now we are only interested in those that display exponential growth along ξ. To obtain them one can use the method described in refs. 74 and 75. Figure 9 shows the typical outcome of such an stability analysis. See refs. 72 and 73 for the details. The main result predicted by the plots is that the larger the parameter κ1 , hence the light intensity, the stronger the instability. For vortices with charge m1 = 1 , t h e perturbations with n = 3, together with and n = 2 exhibit the largest growth rate. For m1 = 2, the perturbations that tend to dominate are n = 5, together with n = 4. Therefore, under ideal conditions when all the perturbations are excited with equal strength, the vortex solitary waves tend to split into the corresponding number of beams. This is so at the initial states of the evolution, where (30) is justified. By and large, numerical simulations confirm such predictions. Figure 9 (c) shows the result of a representative simulation.71 5.4

From Topological Charge Information to Sets of Solitons

By now we have introduced the ingredients needed to describe the principle of operation of a class of devices that mix wave front topological charge dislocations nested in focused light beams and produce certain patterns of bright spatial solitons. The devices can operate in different regimes, as follows. Let us consider Type II geometries.

248

Figure 10: Sets of solitons obtained with different combinations of the topological charges of the input light. In a first regime only the fundamental beams are input in the crystal, with a high enough intensity to form spatial solitons. Then, a second-harmonic is generated with the topological charge m 2ω = m oω + me ω , and the beams self-split into a certain number of solitons because of the AMI of the solitary-wave vortices. Ideally, the number of output solitons is given by the index of the azimuthal perturbations with the highest growth rate, or by the interplay between different perturbations when there are some with similar growth rates. In practice, the several existing asymmetries in the experimental set-up, including Poynting vector walk-off, can seed a dominant perturbation. In a second regime, a coherent second-harmonic seed is present at the input together with the two fundamental beams. When m 2ω = mo ω + me ω the beam evolution is similar to that of the regime mentioned above. A totally new situation is encountered when the topological charges of the input light are chosen to verify m 2ω ≠ m o ω + me ω . Then, the number of output solitons is dictated by the different dynamics experienced by the different azimuthal portions of the beams.76 Down-conversion schemes, where an intense second-harmonic light beam is input in the crystal together with noise at the fundamental frequency, are another possibility. All such processes are driven by the azimuthal dependence of (10), that in the case of Type II phase-matching writes (31) The central result is that the “information” coded in the value of the input array [moω , me ω , m2ω ] is transformed into a certain number of output soliton spots. Figure 10 shows the outcome of typical numerical experiments with different combinations of topological charges and energies of the input light beams. 249

6

Discrete Solitons

6.1 Motivation Discrete solitons on nonlinear lattices are a subject of intense research and continuously renewed interest, as they appear in many models of energy transport in a variety of physical, chemical and biological scenarios (see, e.g., refs. 77-82, and references therein). Discrete solitons form as a consequence of the balance between linear coupling that tends to spread the excitation across the lattice, and nonlinearity. Different types of nonlinearities, and among them cubic nonlinearities that yield evolution equations belonging to the family of the NLSE, have been investigated for many years. However, discrete solitons also exist in quadratic lattices. Besides its potential application to arrays of optical waveguides, discrete solitons on quadratic lattices might have applications to mathematically analogous, but physically different systems. As a matter of fact, parametric interaction of modes in general is a universal phenomenon, hence similar discrete solitons might also exist in other branches of nonlinear science. Naturally, the same is true for the continuous solitons. 6.2 A Quadratic Lattice Consider the differential-difference normalized evolution equations

(32) which come from the standard discretization of equations (1). In general, these equations might hold as a model for the parametric interaction of two generic modes in different scenarios, not only in Optics. In any case, β is the phase-mismatch between the modes, and n the position on the lattice. The parameters α 1 and α 2 give the strength of the linear spreading effects. In the numerics we set α 1 = 0.5 and α 2 = 0.25, to allow comparison with the spatial solitons discussed previously in the continuous case. For later use it is convenient to introduce the quantities Q n = An , Pn = B n exp(–iβ ξ) . We shall make use of two conserved quantities of the corresponding evolution equations, namely the norm (33) and the Hamiltonian

(34) Those are the discrete versions of the continuous quantities (4)-(6). However, notice that one crucial difference between the continuous and the discrete systems is that no analog to the transverse momentum (7) is known in the discrete case.

250

Figure 11: Typical shape of two discrete solitons ( β = 0). (a): κ 1 = 1; (b): κ 1 = 5. 6.3

The Simplest Discrete Solitons

Let us consider solutions of (32) with the form A n = q n exp( iκ 1 ξ), B n = p n exp( i κ2 ξ) , where for simplicity here the numbers q n , pn are assumed to be real. As always, A n and B n ought to be phase-locked, so that κ 2 = 2 κ 1 + β. One readily finds that such stationary solutions realize the extrema of the Hamiltonian for a given norm, with κ 1 being the corresponding Lagrange multiplier. The difference equations obeyed by the series of amplitudes {qn} and {p n } are

(35) Similarly to other discrete systems, eqs. (35) have several different types of localized solutions. Here we only consider the simplest ones, namely those that have a bright shape and a constant phase across the lattice (i.e., “unstaggered”), and that have a single maximum that is located in one lattice site (i.e., “on-site” solitons). To elucidate the existence of such solutions one might solve numerically (35) searching for converging series {q n }, { pn }, with the symmetry conditions q o > q n for n > 0, and q n = q – n , and similarly for {pn }. Other families of solutions, including “inter-site” (i.e., qn = q– n + 1 ) and “staggered” solitons, also exist. Figure 11 shows the typical shapes of two discrete solitons. The most interesting is the strongly localized soliton in Fig. 11(b). By and large, the properties and features of discrete solitons can be drastically different from those of their continuous counterparts. Solitons of the continuous and the discrete NLSE provide a paradigmatic example. Discreteness modifies the shape and confinement features of the solitons, and modifies the existence conditions and properties of walking solitons that move across the lattice. In particular, this includes the amplitude-width relations of the families of solitons, and in the case of solitons supported by quadratic nonlinearities, the fraction of energy (or norm) in each mode forming the soliton, that we discussed in 3.4 for the continuous solitons. As an illustrative example, let us examine the ratio of the peak amplitudes of the two modes forming the discrete soliton, i.e. q 0 /p 0 . For the continuous families of solitons

251

such ratio is given by 32,33 (36) According to the similarity rules (14), at phase-matching ( β = 0) all the continuous solitons are self-similar. Thus, the ratio q(0)/p (0) is identical for all members of the 1.2069. The discreteness of (35) breaks such soliton family. Numerically, q (0)/ p(0) similariry rules, so that in particular q0 / p0 is no longer constant at β = 0. For example, in the case of the weakly-localized soliton of Fig. 11(a) one has q0 / p 0 1.20, whereas for the strongly localized soliton of Fig. 11(b) one finds q 0 / p0 1.33. Discreteness introduces a variety of other differences. Other discretizations of (1) different from (32), and the corresponding discrete solitons, might be also of interest.

7 Concluding Remarks In these lectures we have only examined the basics of solitons in quadratic nonlinear media. Specifically, we only considered the simplest bright solitons, mostly in Type I phase-matching geometries. However, much more is known.8 3 Important investigations have been reported by many authors about Type II geometries, 8 4 – 8 6 dark-like and symbiotic solitons, different types of temporal solitons,1 9 – 2 1 , 2 9 – 3 2 , 8 7 – 9 1 including Thirring and gap solitons, 9 2 – 9 5 higher-order soliton solutions, 9 6 – 9 9 soliton interactions and collisions, 1 0 0 – 1 0 3 the effects of higher-order nonlinearities that may be present in addition to the quadratic nonlinearity,1 0 4 – 1 0 9 magneto-optic effects, 1 1 0 modulational instabilities in various geometries and dimensionalities,3 6 , 3 7 , 7 1 – 7 3 , 1 1 1 – 1 1 3 some of which have been observed experimentally,1 8 and so forth.1 1 4 – 1 1 6 A few applications of the solitons, mainly those potentially relevant to all-optical switching schemes but also other applications, have been examined and some experimentally observed. 52,53,117 The field is growing vigorously, theoretically and experimentally, and new areas where spatial, temporal and spatio-temporal solitons in quadratic nonlinear media might have important applications are emerging already. Optical cavities containing quadratic nonlinear crystals and quantum optical devices are fascinating and promising examples. In principle, new and existing materials with very high quadratic nonlinearities hold very promise for the future to reduce the power requirements to form solitons and hence render them closer to practical applications. From a broader viewpoint, it is worth stressing that solitons in quadratic nonlinear media can potentially have important implications not only to various parts of nonlinear optics, but also to other branches of nonlinear science. That is so because parametric wave mixing is a universal phenomenon. 118 Therefore, the formation of stable multidimensional soliton entities in mathematically similar, but physically different settings is a potentially important possibility that has to be explored.

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Acknowledgements This work has been supported by the Spanish Government under grant PB95-0768. My own work in the topic addressed in these lectures has been done in close collaboration with many colleagues, including George I. Stegeman, Curtis R. Menyuk, Dumitru Mihalache, Dumitru Mazilu, Juan P. Torres, Nail N. Akhmediev, William E. Torruellas, Ewan M. Wright, Dmitrii V. Petrov, José-Maria Soto-Crespo and Maria C. Santos. I am most grateful to all of them. The numerical work has been carried out at the C 4Centre de Computació i Comunicacions de Catalunya. Support by the European Union through the Human Capital and Mobility Programme is gratefully acknowledged.

A

Type II

The normalized evolution equations for the slowly-varying field envelopes in Type II phase-matching geometries for second-harmonic generation can be written as

(37)

where a 1, a 2 and a3 are the normalized envelopes of the ordinary polarized fundamental beam, the extraordinary polarized fundamental beam and the second-harmonic beam, respectively. In the case of spatial solitons α 1 = –1, α 2 –1, and α 3 –0.5. The equations for Type I are obtained from these equations by setting a 2 = a 1 = a ω , α1 = α 2 = r, α 3 = α, δ 2 = 0, δ 3 = δ, and a 3 = a 2 ω . Equations (37) have several conserved quantities, including the corresponding Hamiltonian. For our present purposes, we only need the energy flow given by the Manley-Rowe relation (38) the unbalancing between the energies of the two fundamental waves Iu = I 1 – I 2 , and the transverse beam momentum

(39) Non-walking solitons of (37) are a two-parameter family, whereas walking solitons constitute a three-parameter family. Physically, such parameters are related to the total energy flow, to the unbalancing energy and to the soliton velocity. The extra degree of freedom relative to Type I geometries, namely the unbalancing energy, has important implications. 84 In particular, it can be used to control the velocity of the walking

253

solitons. The analogous of Eq. (26) but for Type II geometries writes (40) Changing the polarization of the input light at the fundamental frequency modifies the ratios I 2 /I and I 3 /I, hence it changes the velocity of the excited walking soliton.5 3 , 5 5 The properties of the families of solitons in Type II geometries can be found in refs. 84 and 85. Walking solitons have been also studied. 86

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PHOTOTOREFRACTIVE SPATIAL SOLITONS

Mordechai Segev¹ , Bruno Crosignani² , Paolo. Di Porto², Ming-feng Shih¹ , Zhigang Chen¹ , Matthew Mitchell¹ and Greg Salamo¹ ¹Electrical Engineering Department, Princeton University, Princeton, New Jersey 08544 ²Dipartimento di Fisica, Universita' dell'Aquila, 67010 L'Aquila, Italy and Fondazione Ugo Bordoni, Roma, Italy ³Physics Department, University of Arkansas, Fayetteville, Arkansas 72701

INTRODUCTION The advent of Nonlinear Optics, mainly due to the work of Bloembergen and coworkers at the beginning of the '60, has opened the way to a number of fundamental discoveries and applications [1]. No matter how sophisticated the microscopic or phenomenological approach adopted to deal with this topic, one of the central problems to the complete understanding of nonlinear processes is always the solution of the associated wave propagation equation, which is, of course, intrinsically nonlinear. This circumstance can be considered in many cases as an obstacle to a full comprehension of the problem, due to the well-known mathematical difficulties usually encountered when trying to solve this type of equations. However, it is precisely the nonlinear nature of the wave equation which gives rise to a wealth of solutions, and thus of possible behaviors of the propagating field, and makes nonlinear optics much more interesting that its linear counterpart. A typical situation is the one associated with the so-called optical Kerr effect. In this case, the third-order polarizability takes a particularly simple form and the nonlinearity is characterized by a contribution to the refractive index n proportional to the intensity I of the propagating field, that is n=n1 +n2 I. Inserting this expression into the wave propagation equation allows one to deduce the equation of evolution of the field amplitude in the form of a spatio-temporal partial differential equation, usually referred to as nonlinear Schrödinger equation (NLSE), whose solutions have been investigated in great detail starting with the pioneering work of Chiao et al. [2] and of Zakharov and Shabat [3]. In

Beam Shaping and Control with Nonlinear Optics Edited by Kajzar and Reinisch, Kluwer Academic Publishers, New York, 2002

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particular, this analysis has shown the existence of a peculiar kind of solutions (solitons), able to propagate without distortion (or periodically recovering their initial shape), and has given rise to an entirely new field of applied research. [4] [5]. It is now well established that solitons are a general phenomenon in nature: they appear in many fields in which waves propagate and can exhibit both dispersion and nonlinearities, including surface waves in fluids (where they were initially discovered), volume fluid waves (deep sea waves), charge density waves in plasma and in solid state, etc. [6]. In Optics, two distinct types of solitons are known: temporal solitons [4] and spatial solitons [5]. Temporal solitons are pulses of very short duration that maintain their temporal shape while propagating over long distances. They are now routinely generated [7] and are the backbone of future highspeed telecommunication links. Conversely, it is much more difficult to generate and observe spatial solitons, in particular those that resulted from the Kerr effect (so-called Kerr-type solitons), which are optical beams that propagate without diffraction (stationary solutions of the NLSE), for several reasons. First, because the nonlinear change in the refractive index scales with the optical intensity and since the values assumed by n 2 a r e -16 cm² /Watt), characteristically very small (e.g., in silica glasses n2 is of the order of 10 very large optical power densities are required [8]. Second, Kerr-type spatial solitons are stable only in two-dimensional systems, i.e., one longitudinal dimension along which the beam propagates and one transverse dimension in which the beam diffracts or self-traps (this configuration is often referred to as a (1+1) D system). Full 3D optical beams undergo catastrophic self-focusing [9], and (1+1) D beams become transversely unstable [10] when they propagate in self-focusing Kerr-like nonlinear media. This means that Kerr-type spatial optical solitons can be observed in single-mode waveguides, as demonstrated in [8]. In this respect, there is an apparent asymmetry between temporal and spatial solitons, notwithstanding the intrinsic spatio-temporal symmetry of the wave equation, due to the additional dimensionality in the latter case. Optical spatial solitons have undergone a sharp and dramatic conceptual change during the last few years. It has been driven by the discovery of three distinct nonlinear mechanisms that have been shown to support three dimensional [i.e., (2+1) D] solitons [11], that is, beams that are self-trapped in both transverse dimensions: the photorefractive nonlinearity [12,13], the quadratic nonlinearity [14], and the vicinity of an electronic resonance in atoms (or molecules) [15]. This article is dedicated to solitons in photorefractive media. The photorefractive nonlinearity occurs in high-quality lightly-doped electro-optic crystals, such as BaTiO 3 (barium titanate), LiNbO3 (lithium niobate), SBN (strontium barium niobate). The photorefractive effect [16] was originally interpreted as an "optical damage" of the crystal provoked by the optical beam. It exhibits a reversible variation of the refractive index induced by the spatial variation of the optical intensity. This mechanism, which is, by comparison to standard nonlinear optical effects, rather slow but is effective at extremely small optical powers, possesses the capability of recording in real time the information encoded in the spatial modulation of the beam and, because of this capability, has been widely used in the frame of real-time holography and optical phaseconjugation. [17], [18],[19]. These properties suggested to look for propagation of spatial solitons in photorefractive materials, hoping that, in the nonlinear regime where the induced refractive index variation affects the very beam which has produced it, the beam diffraction could be compensated by its self-focusing and self-trapped (non-diffracting) propagation would result [12]. In fact, this hope has been more than justified by the great deal of theoretical and experimental results which have been obtained in the last five years and which have shown the existence of one and two-dimensional photorefractive spatial solitons, bright and dark, of vortex solitons, of photovoltaic solitons, together with a

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number of interesting applications (like, e.g., soliton-induced waveguides in bulk photorefractive media) and of self-trapping of incoherent light beams. In this review, our scope is to introduce the general topic of nonlinear propagation in photorefractive media and focus more specifically on photorefractive solitons. We will start with the basic band transport model of the photorefractive effect [20,21], which suffices for treating most of the situations we will be dealing with. As a first application, this will allow us to consider the concept of photorefractive gratings and to introduce the formalism of two-wave mixing as the simplest example of nonlinear propagation. We will then consider the propagation of a generic beam in a photorefractive crystal and introduce the formalism for describing this kind of propagation. Following that we will examine the conditions under which it should be possible to find self-trapped solutions, that is, particular transverse beam profiles which propagate without distortion (photorefractive solitons). After reviewing the main theoretical results which have been established up to now, we will present some of the impressive experimental evidence which confirms the great possibilities of application of the PR effect and discuss interactions between (or among) solitons. THE PHOTOREFRACTIVE EFFECT The photorefractive nonlinearity gives rise to light-induced changes in the refractive index of certain types of (typically non-centrosymmetric) crystals. More specifically, the spatial ditribution of the intensity of the optical beam (or beams) gives rise to an inhomogeneous excitation of charge carriers, which in turn produces, by redistributing themselves through diffusion and drift, a space charge separation whose associated electric field modifies the refractive index of the crystal via the Pockels effect. This modification, the photorefractive effect, can be qualitatively described by means of the simple band transport model introduced in [21]. More precisely, let us refer to Fig. 3.1 of Ref. [18], in which the energy diagram of a typical photorefractive material is shown. A large concentration (1018 -10 19 cm - 3 ) of identical, uniformly distributed, donor impurities, whose energy state lies somewhere in the middle of the bandgap, can be ionized by absorbing photons. Correspondingly, the generated electrons are excited in the conduction band where they are free to diffuse or to drift under the combined influence of self-generated and external field (if any). During this process, some of the electrons are captured by ionized donors neutralizing them, the successive ionization rate being proportional to the local illumination. In this way, the ensemble of the non-mobile donors acquires an inhomogeneous charge distribution which tends to be positive in the illuminated regions and to vanish in the dark regions. The combined presence of this charge distribution, of the electrons in the conduction band and of a number of ionized acceptors present in the crystal [22], gives rise to a low-frequency electric field (space-charge field, E). It is this field that, through the standard mechanism of Pockels' effect (δn α r E, where r is some electro-optic coefficient of the crystal), produces the refractive index variation responsible for the photorefractive (PR) effect. In order to translate in quantitative terms the above considerations, we need to write down the equations necessary to determine the space charge field E. The first one is the rate equation describing the donor ionization rate as a result of the competition between thermal and light induced ionization and recombination with free electron charges, that is

(1)

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where I is the optical intensity, ND the donor density , the ionized donor density, Ne the electron density, β the thermal generation rate, s the photoionization cross-section and γ the recombination rate coefficient. The second equation is the Gauss Law , where

(2)

is the low-frequency dielectric constant and the charge density ρ is given by ,

(3)

-q and N A representing the electron charge and the acceptor density, respectively. The last relevant equation is the continuity equation (4) where the current density (5) is the sum of the drift and diffusion contributions, µ representing the electron mobility. The set of Eqs. (1)-(5) has to be supplemented by the appropriate boundary condition: ,

(6)

where is the distance between the two crystal faces to which the external bias static potential V is applied. We will first consider the set of Eqs.(1)-(6) as if the optical light intensity, associated with the field Eop propagating at optical frequencies were a prescribed function. The main task of determining Eop will then be faced by solving in a self-consistent way the pertinent wave equation, written in the presence of the induced tensorial refractive index contribution associated with the linear electrooptic (Pockels) effect . More precisely, by choosing x,y,z along the principal dielectric axes of the crystal, for the typical values assumed by the space-charge field, it is possible to write [17] (7) where rijk is the linear electrooptic tensor (of rank three). THE SPACE-CHARGE FIELD We start dealing with the steady-state situation in which the time derivatives in Eqs.(1) and (4) can be assumed to vanish (actually, as we shall see in Sect.4, this corresponds to consider time larger than the so-called dielectric relaxation time). In this asymptotic

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regime, it is possible to show the possibility of achieving self-trapped beam propagation (screening solitons). The derivation presented in the next Sections closely follows that worked out in [23]. Let us first eliminate between Eqs.(1) and (2). With the help of Eq.(3), we obtain, in terms of suitable normalized quantities and assuming Ne