Self trapped modes in nonlocal nonlinear media

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(4) Instituto Tecnologico y de Estudios Superiores de Monterrey Campus Monterrey GRADUATE PROGRAMS IN INFORMATION TECHNOLOGIES AND ELECTRONICS The members of the thesis advisory board hereby recommend acceptance of the disser-. tation "Self-trapped modes in nonlocal nonlinear media," by Servando Lopez Aguayo, submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Information Technologies and Communications, Major in Optics.. Thesis Advisory Board. Dr./juflb Cesar Gutierrez Vega Tecnologico de Monterrey Advisor. Dr. Anton S. Desyatnikov Australian National University Reviewer. Prof. Yuri S. Kivshar Australian National University Reviewer. Dr. Carlos M. Hinojosa E. Tecnologico de Monterrey Reviewer. Dr. Rodolfo Rodríguez y M. Tecnológico de Monterrey Reviewer. Dr. Graciano Dieck Assad Director Of Graduate Program In Information Technologies and Electronics May 2007.

(5) Self-trapped modes in nonlocal nonlinear media by: Servando López Aguayo. TESIS A thesis submitted to the faculty of the Division of Information Technology and Electronics in partial fulfillment of the requirements for the degree of. DOCTOR of Philosophy. Monterrey, Mexico May 2007.

(6) Declaration I hereby declare that I composed this dissertation entirely myself and that it describes my own research.. Servando Lopez Aguayo Monterrey, N.L., Mexico May 2007.

(7) To my family..


(9) Acknowledgements I would like to start by thanking Dr. Julio Gutierrez, for his incredible and always present support at all moments, and also for his inspirational dedication to science. Massive thanks to Dr. Anton Destyanikov, for his invaluable guidance and advice in the soliton field, and for his always remembered words: "keep moving". Many thanks to Prof. Yuri Kivshar, because he showed me that there are people who can be incredibly brilliant in science, but also can be humble and always eager to help others. I would like to express also my gratitude to my thesis committee: Dr. Hugo Alarcon Opazo, Dr. Carlos Hinojosa, and Dr. Rodolfo Rodriguez, for their help in proof-reading my thesis. Many thanks to all my friends at Optics Center and Physics department at ITESM, thanks to Josue, Rodrigo, Polo, Raul, Ponce, Martha, and Carlos. Thank you "Pecera". I wish to thank also The amazing Nonlinear Physics Centre at ANU, all people there show me the beauty of cutting edge research work and I am indebt with all of you guys, specially thanks to Robert, Santiago and Andre. Very special thanks to the intellectual galotonic group: Josh, Pepe, Beto, Jorge, Alejandro, Luis Carlos, Javier, Oscar, Eduardo, and Ricardo, thanks guys for show me real friendship. Thanks also to all these beautiful girls: Jessica, Mina, Mrs Sthepy, Amanda, Rosa Elva, Miriam, Carolina, Rossy, Azael, and Yezabel. I want to thank all the people that I already feel like family: Armando, Gerardo, Luis, Jesus, Daniel, Monica, Carlitos, and Andy. Massive thanks to four very special persons: my precious sister Any, my inspirational girl Saray, my treasured neighbor Daniela, and the tremendous and always philosophical Eduardo Juarez. Finally, thank you to my entire family: my father for his wisdom and guidance, my mother for her unconditional love, and my brother, a future inspirational mathematician and bodybuilder.. in.


(11) Abstract The present thesis contains both, a theoretical and experimental study of selftrapped modes in nonlocal nonlinear media. First, we give a basic review about the soliton phenomena and the nonlocal nonlinear model. We proceed then to obtain the recently introduced azimuthons (azimuthally modulated self-trapped rotating singular optical beams) using both, a variational approach and numerical scheme of propagation to analyze their stability in general nonlocal nonlinear media. Remarkably, we found that nonlocal nonlinear media are the first known media in where azimuthons are stable. However, due at the present time the generation of azimuthally modulated beams (in the way required by azimuthons) can be difficult to achieve in experiments, we also have studied the light localization of a simple fundamental beam in Bessel photonic lattices, this as first step towards the generation of singular and azimuthally modulated beams. In fact, we have been successful to observe the first light localization in Bessel lattices in an anisotropic and nonlocal medium. As we have predicted stabilization of azimuthons with an enough degree of nonlocality, is natural to think that in the highly nonlocal limit, we will have also stable azimuthons. Thus, we have explored the highly nonlocal media and we have extended the concept of azimuthon to a more general one: we have found a novel kind of elliptically modulated self-trapped beams, the so-called ellipticons. Finally, we have shown that the highly nonlocal limit, indeed provides a powerful and useful tool to extrapolate some results to general nonlocal nonlinear media..

(12) VI.

(13) Contents 1 Introduction 1.1 Preliminary remarks 1.2 Optical solitons 1.2.1 A historical perspective 1.2.2 Existence of solitons in different physical systems 1.2.3 The nonlinear Schrodinger equation 1.3 Nonlocality in nonlinear optics 1.3.1 The nonlocal model 1.3.2 Physical effects of nonlocality 1.4 Thesis Outline. 1 1 2 2 2 3 9 9 13 16. 2 Azimuthons in nonlocal nonlinear media 2.1 Introduction 2.2 Rotating dipole solitons 2.2.1 Variational approach 2.2.2 Propagation dynamics and stability 2.3 Higher order azimuthons in nonlocal nonlinear media 2.3.1 Variational approach 2.3.2 Propagation dynamics and stability 2.4 Concluding remarks. 19 19 20 20 26 30 30 34 40. 3 Light localization in Bessel lattices 3.1 Introduction 3.2 Nonlinear waveguide arrays 3.3 Optical Lattices 3.4 Photorefractive crystals 3.4.1 Local model 3.4.2 Nonlocal anisotropic model 3.5 Experimental setup 3.5.1 Generation of optical Bessel lattices 3.5.2 Experimental results 3.6 Concluding remarks. 43 43 43 44 46 46 48 48 48 49 54. vii.

(14) 4 Strongly nonlocal nonlinear media 4.1 Introduction 4.2 Derivation of highly nonlocal nonlinear media equations 4.3 Helmholtz-Gauss Self-Trapped states 4.3.1 Derivation of Helmholtz-Gauss model 4.3.2 Oscillating modes 4.3.3 Quasi invariant HzG modes 4.4 Solitons in highly nonlocal nonlinear media 4.4.1 Derivation of soliton solutions 4.4.2 Laguerre and Hermite solitons 4.5 Ellipticons in HNN media 4.5.1 Soliton solutions in elliptical coordinates 4.5.2 Rotating elliptically modulated self-trapped beams 4.5.3 Explanation of stability and soliton revivals using ellipticons . . 4.6 Concluding remarks. 55 55 55 58 58 62 63 68 68 69 71 71 76 79 80. 5 Concluding remarks. 5.1 Related publications 5.2 Journal papers 5.3 Conference proceedings. 81 83 83 84. References. 85. List of Figures. 96. Vita. 97. viii.

(15) Chapter 1 Introduction. 1.1. Preliminary remarks. Nonlinear optics has been a subject of intense research over the last years. In fact, at the present time, each of the contemporary nonlinear optics subfields posses a huge amount of research literature: optical chaos [1], photorefractive crystals [2], organic polymers[3], quantum wells[4], optical fibers[5], and of course, optical solitons[6]. The future of the optical technologies could have solitons as a fundamental "bit" of information, and hence their importance in nonlinear optics: they could constitute the bases in the creation of all pure optical networks. In this chapter, we introduce the soliton phenomenon and then we give a review of the model that allows soliton propagation in optical materials: the Nonlinear Schrodinger equation [7]. Next, the general features of nonlocality are covered, and we present the nonlocal response function and its existence taking into account different degrees of nonlocality. We give a review about the new physics phenomena that nonlocality produces, being stabilization of self-localized structures the most important for the following chapters. Finally, we give an outline of all the chapters that compose this thesis..

(16) 1.2. Optical solitons. 1.2.1. A historical perspective. The term soliton was derived from the name solitary wave and it was introduced by Zabusky and Kruskal in 1965 as a name to solitary nonlinear waves that behaves as particles [8], i.e. solitary waves that after interaction between them, remain invariant in their amplitude and just suffer a change in their respective phase. The first reference to a solitary wave was reported by Scott Russell in 1834. In his studies, he saw a smooth, rounded, well-defined lump of water, that traveled without changing its shape and speed for many miles along the canal linking Edinburgh with Glasgow. His report, published in 1844, includes the following text [9]: "I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped -not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without, change of form or decreasing of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight of nine miles an hour, preserving its original figure some thirty feet long and a foot to foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with than singular and beautiful phenomenon which I have called the Wave of Translation" . This initial observation was followed by extensive theoretical and experimental research that established the existence of solitons as one of the most striking aspects of nonlinear wave phenomena. Solitons have been observed and studied in various fields ranging from optics and fluids [10]-[12]to soliton state and chemical systems [13], [14]. The stability of solitons is due to the balance between dispersive and nonlinear effects. Basically, the dispersion effect tends to spread the localization of wave while the nonlinear effect try to concentrate it.. 1.2.2. Existence of solitons in different physical systems. Solitons, as universal nonlinear phenomena, are studied in many fields of Physics. The following three examples of nonlinear partial differential equations have soliton solutions and are used in various physical systems. The Korteweg-de Vries equation: The Kdv equation is given by [15]: 5A. dA. where k is a constant. The Kdv equation arises in the derivation of a mathematical model that describe the lossless evolution of shallow water waves. It was derived by the mathematicians Korteweg and de Vries. The KdV also describes magnetohydrodynamic.

(17) waves in plasma, longitudinal dispersive waves in elastic rods, and thermally excited phonon packets in low-temperature nonlinear crystals. Other famous nonlinear equation is the sine-Gordon equation, that is given by[16]:. . = sm(0),. (1.2). this equation has been used to describe the Bloch wave motion in magnetic crystals, unitary theory for elementary particles, propagation of light through a crystal dislocation and lipid membrane, and it appears in differential geometry and relativistic field. Finally, as a third example of a nonlinear equation that present soliton solutions we show the Gross-Pitaevskii equation [17]: A,T.. fe2. '2\|/ _)- v(r)ty + f/o |^| 2 W. (1-3). equation that models the macroscopic dynamics of a Bose-Einstein condensate with a parabolic potential V(r) created by a magnetic trap, here f/o is a constant. In the last years the field of Bose-Einstein condensates have been a intensive field of research. In the following section, we will focus in another nonlinear differential partial equation that is broadly used in nonlinear optics, and it will constitute the core for all the research work done in this thesis.. 1.2.3. The nonlinear Schrodinger equation. To derive the master equation of propagation in optical nonlinear media, we start from the wave equation that is directly obtained from Maxwell's equations; this wave equation that describes the electric field in a medium associated with the optical wave propagation in such medium has the form [5]:. here e0 is the vacuum permittivity and c is the speed of light in the vacuum. This equation arises directly from the Maxwell equation. To continue with our derivation, we will do a series of special considerations that allows to simplify our model. As a first consideration, we assume that the induced polarization P is composed of two parts that can be separated in the form:. P(r,t) = P 1 (r,0 + PjvL(r,t),. (1-5). where PI refers to the linear induced polarization and F/VL refers to the much more smaller nonlinear part; because of the difference in magnitude between both quantities, the nonlinear part can be treated as a small perturbation of the linear part of the induced polarization, and both polarizations are given by:.

(18) P L (r,«) = eQX(l)(t OO 0 0. PL(r,t)=£0. - t') • E ( r , 0 <. (1.6). OO. x()(t-ti,t-t2,t-t3)-~E(r,ti)E(r,t2)E(r,t3)dtidt2dt3,. I. (1.7). —oo—oo—oo. where x ^ and x ^ are the first and third-order susceptibility tensors [18] respectively. In Eq.(1.6) and Eq.(1.7) we can see that there is a non instantaneous response of the medium. Usually the susceptibility tensors are modeled with delta functions, so at the end, we have just local terms (in time domain), and hence, we have assumed an immediate response from the medium. We would like to emphasize this issue, because in fact Eq.(1.6) and Eq.(1.7) were originally obtained assuming a local response of the medium (in the physical space), so a more detailed model of our linear and nonlinear polarization terms should include also the nonlocality in both, time and r. To the present derivation of our interest, we can just assume an instantaneous response as well as a pure local response in r. Following with the derivation of our master equation, we consider as another simplification, that the optical field maintain its polarization along its propagation, so a scalar approach can be used. As another consideration, the optical field is assumed to be quasi-monochromatic. We propose then a general solution of Eq.(1.4) in the following form: E(r, t) = - [A(r) exp(i/30z) exp(-iu>0t) + c.c] ej,. (1.8). where /30 is the propagation constant in terms of the optical wavelength A, and is given by A) = 2?rno/A, (being % the linear refractive index of the material) UJQ is the carrier frequency, e[ is the polarization unit vector and c.c. stands for complex conjugate. In this ansatz, E(r, t) is a slowly varying function of time relative to the optical period. The function A (X, Y, Z) describes the evolution of the beam envelope. The beam propagates along the spatial coordinate Z, and X and Y are the spatial coordinates associated with r. Inserting this ansatz in Eq.(1.4), plus using all the considerations before mentioned, and if besides we neglect the second derivative d?A/dz2 using the paraxial approximation, the beam envelope is found to satisfy the following equation: f)A. r)2 A. f)2A. = 0,. (1.9). here nni is the nonlinear part of the refractive index that depends on the beam intensity. Next we introduce the following scaled dimensionless variables as: X. Y. Z. w0. w0. Ld. x= — ,y= — ,z = —,u=. ^k0Ld\n2\A,. (1.10). where the Rayleigh range, Ld,(or diffraction length) is given by Ld = Po^l, and WQ is a transverse scaling parameter. Using these variables in Eq.(1.9) we obtain finally the.

(19) form of a general (2+l)-dimensional NLS equation: .du where F(I) is a function that depends of the intensity, (for example, in a Kerr medium, / = |u| 2 ). We will use this equation as starting point in all the other chapters of this thesis. After the derivation of the NLSE, we are going to analyze which are the contributions from the diffractive and nonlinear terms that constitutes our master equation. Then, focus first in the term that model diffraction, we neglect the nonlinear term from Eq.(l.ll), and we get that the obtained equation corresponds to the paraxial equation studied in the context of scalar diffraction theory [19]. du. 1 d2u. d2u\ 0. *22. 1J2. oz 2 \ox2 ay2 J To analyze Eq.(1.12) a basic mathematical tool used is the standard Fourier method, a tool that allows to obtain an expression from Eq.(1.12), in which every Fourier component of the initial localized wavepacket propagates with a certain characteristic velocity, and then, after a certain distance of propagation, our initial beam shape will have generally a very different form from its original form. This leads to the phenomenon of the diffraction in the context of the NLSE used for the description of spatial optical beams, whereas in other physical systems, for example, light pulses and Bose-Einstein condensates, the term preferred is dispersion, in spite of they mathematical structure is the same. To gain more insight about the action of the pure diffractive term of the NLSE, we will solve Eq.(1.12), and we will consider just one transverse spatial dimension (x) for simplifying purposes. Recalling that the definitions of our pairs of Fourier transforms are given by: 1 f°° _ u(z,x) = I u(z,kx)exp(—ikxx)dkx (1-13) V 2vr 7_oo u(z,kx) =. If00 — /. u(z, x)exp(ikxx)dx,. (1-14). V27T 7-00. we can use then the latter definitions to solve our differential equation Eq.(1.12) in the spatial frequency domain (kx). The Fourier equivalent of our pure diffractive equation then is: _ du _ z 2 _ ( xU. !Tz ~ " 2. '. '. and its solution is given by: k2?. (1.16). from this general solution, we can observe that the diffraction term just rearranges the phase relations among existing spatial frequency components, so we do not have any.

(20) new ones. We can gain even more insight taking an example with analytic solution, for example, a Gaussian beam. Taking u(Q, x) = exp(— x2/2), we have that its Fourier transform is given then by 5(0, kx) = exp(—k2/2), and after apply the inverse Fourier transform to Eq.(1.16), we obtain:. for long distances, we can observe that the intensity envelope is given by |u|2 a exp(— x2/(l + z2)), and hence the beam width grows as: x = x0Vl + x2,. (1.18). where x0 is in this case the initial minimum beam width. Thus diffraction modify the width and amplitude of the initial intensity envelope. We can also use a plane wave to explore the diffraction relation, so using &(x, y) = exp(i(kxx + kyy)), and substituting in Eq.(1.12), we get: k2 \ =- ,. (1.19). where k2 = k2 + k2; the velocity of group (Vg) if given by: V9 = § = k,. (1.20). thus our "dispersion" of diffraction term (D), is given in this case by:. and from this expression, we can see that in the case of spatial optical beams, we will have always a positive diffraction, contrary to the case of propagation of light pulses in fiber optics, where is possible to have either positive or negative dispersion [20]. Now we will focus on the action of the pure nonlinear term of the NLSE, so neglecting the diffraction effects in the NLSE, we get: r\. i-^ = ±F(I)u,. (1.22). several different models have been used for the functional F(7), being the most common functional the Kerr nonlinearity [6]; in this model, we have: F(/) = / = H 2 ,. (1.23). other example is the case of the saturation model that corresponds to a two-level atomic system (model that was introduced more than 25 year ago) and is used to characterize,.

(21) for example, photorefractive materials like LiNb03 [21], [22], in this case the functional F(I) takes the form: (1-24) u another example of the different kind of nonlinearities is the known as competing nonlinearity, which is used as example for PTS (p-toluene sulfonate) crystals that are working in the region close to 1600 nm, this nonlinearity has the following cubic-quintic form [23]: = n 2 |u|2 + n 3 H 4 , (1.25) where n-i and n$ have opposite signs. There are many others kinds of nonlinearities and in the followings sections we will introduce a Kerr nonlinearity but with the special characteristic to be nonlocal. It is important to remark that the NLSE equation in its general form does not admit analytical solutions, and hence, to study the NLSE, is of fundamental importance the use of some approximative methods such as variational or perturbation methods, as well as the use of numerical methods (such as the split step Fourier Method) to check the properties suggested by the approximative methods mentioned before. To study now the action of the pure nonlinear term, we have to solve: i ^ = ±F(I)u,. (1.26). in this case, the differential equation can be solve directly, thus, as a general solution we get: u(z,x) = tt(0,Oexp(=FiF(7», (1.27) from this general solution we can appreciate that a pure nonlinear term will just affect the phase of our beam, letting invariant the intensity. In other words, the action of a pure nonlinear term is to add new spatial frequency components. We can appreciate this using a Fourier transform to obtain the spatial frequency representation. Setting u(Q, x) = exp(— £ 2 /2), and besides using a Kerr medium, F(I) = I = |u| 2 , we have the following approximation for very short propagation distances (z <C 1)[24]: (1.28) here we can see that a new component in the frequency space is created, so the nonlinear term broads the spatial frequency components of our initial beam. We have analyzed diffraction and nonlinearity individually, and while the diffraction seems just to affect the amplitude of the beam, the nonlinearity seems to just affect the phase, and hence it is not yet clear how both phenomena can compensate each other, so there is still open the question: how is possible that exists a balance between diffraction and nonlinearity, allowing soliton formation? To answer this interrogant, it is of great help to consider another point of view: remembering how light is confined by optical.

(22) Diffraction. Self-focusing. Spatial soliton. Figure 1.1: Schematic illustration of the diffraction and self-focusing effects: (a) initial beam, (b) beam propagated where diffraction effects predominate, (c) beam propagated where self-focusing effect predominate, and (d) balance between diffraction and selffocusing: formation of spatial soliton. waveguides. In optical waveguides, the diffraction can be compensated by refraction, and to do this we just need that the refractive index of the material to be increased in the transverse area occupied by the beam (producing then a focusing phenomenon). It was discovered some time ago [25], that nonlinearity can also change the refractive index of the medium (phenomenon known as the optical Kerr effect [18]) in the same way: refractive index is larger in the area where there is larger intensity, so it is natural to think that under the correct scenario is possible to effectively cancel the spreading (produced by diffraction) and the self-focusing (produced by nonlinearity), allowing then the formation of soliton (see Fig.(1.1)). In nonlinear optics, is common to classify the solitons as either temporal (where the nonlinearity is the self-phase modulation, and this produces self-localization of pulses) or spatial (where the nonlinearity is the self-focusing -or self-defocusing- that produces self-trapping of beams). We would like to remark that in all this thesis have been deal just with spatial optical solitons. As an example, in the case of a Kerr nonlinearity, we can solve the NLSE and obtain an analytic soliton solution, given by: u(z, x) = sec h(x) exp(iz/2),. (1.29). we can observe from this soliton solution how the intensity envelope is independent from the propagation distance, and hence we have an invariant beam that only has phase changes in its propagation. Thus, diffraction and nonlinearity are in balance in this case and we have the formation of a spatial optical soliton beam. It is important to remark that rigorously speaking, the term soliton should be used.

(23) when a series of mathematical properties are fulfilled, such as integrability of the system and invariance after interaction with other localized wavepackets. However, in the nonlinear optics field is quite common to refer to all solitary waves as solitons, and in fact, we adopt this definition in all this thesis.. 1.3 1.3.1. Nonlocality in nonlinear optics The nonlocal model. Nonlocal response function Usually the refractive index of a nonlinear optical material is modeled by the interaction of the laser intensity with the nonlinear material in a localized way: more intensity in a certain region creates a stronger refractive index in that particular section. Nevertheless, in reality the refractive index in a particular point depends also of the beam's intensity in all the other material points. This dependence is known as the nonlocality of the material. Stronger dependence means stronger nonlocality. Nonlocality appears naturally in the thermal self-action of laser beams [26], and during several years the interest to the nonlocal propagation of light beams was renewed and several works were devoted to the study of one-dimensional [27], [28], [29] and two-dimensional [30] solitons. It have been shown that a diversity of media like photorefractive materials [31], nematic liquid crystals [32], media with thermal self-action [34], lead glasses [35], atomic vapors [36] and dipole-dipole interactions in Bose-Einstein condensates (BECs) can be correctly modeled as nonlocal nonlinear media [37]. We would like to remark that the nonlocality concept by itself is opposite to our logic and diary experience in our lives, so at first glance it could be a little hard to accept and completely understand it, but as happened with other controversial theories in their respective time (for example, relativity and quantum mechanics), nonlocality has proven until now to be really useful in the description and research work of many phenomena, and Indeed it has been already accepted as a valid model for the physical systems mentioned before. The study of solitons in nonlocality is still a rich field of research, and in the last years it has been showed that nonlocality can provide new physics to the soliton field. The nonlocal response function has been the model used to shown the nonlocal effects in the nonlinear optics. Mathematically speaking, it is a function that "averages" another function that depends of the intensity, all this using a mathematical convolution, given as a final result the refractive index change in the medium; in other words, our nonlocal response has the form: = sjR(\f-p\)F(I(p))dp,. (1.30). where R is the nonlocal response function and the parameter s gives the condition of focusing (+1) or defocusing (—1) medium. The integral is done usually just in the 9.

(24) transverse dimensions. It has been usual to normalized the nonlocal response function: R(p)dp=l,. (1.31). besides it has been usual to work with nonlocal response functions that are real ( i.e. no nonlinear loss or gain) , localized (like all physically reasonable response functions) and symmetric (e.g. excluding the asymmetric Raman response in the case of temporal solitons [38]). Next we present some examples of functions that have been used as nonlocal response: The first one, is the Gaussian kernel, so we have:. where a is a parameter that modeled the nonlocality degree. The Gaussian model have the advantage that from a mathematical point of view, is the easier nonlocal response to work with, due to it easier operational treatment; nevertheless, there is no known physical system which can be really described by a Gaussian response. A more physically nonlocal response is the one used in some thermal nonlinear models, such as the process of plasma heating on propagation of electromagnetic waves [39] , in this case, the nonlocal response function is given by: R(p) = 0(p),. '. (1.33). where i? represents a modified Bessel function of the second kind. Finally, another nonlocal response that corresponds to another physical system, is the exponential function. This exponential response occurs in materials with a nonlinear response determined by a transport mechanism and described by a generic diffusion equation, such as the presented in [40] , this response has the following representation: ).. (1.34). here a is the diffusion parameter. We show in Fig. (1.2) the three different nonlocal responses before mentioned. Observe that mathematically, the Gaussian kernel can be differentiable all the points, while the same is not possible with the others two at the point p = 0, even more, there is another striking difference with the nonlocal response in the form of the Bessel function here presented: the Bessel model has also a singularity in p = 0, a similar situation is presented in the model presented in [41]. All this mathematical points can change dramatically the stability domain of the self-trapped structures.. Degrees of nonlocality It is useful to study different scenarios of nonlocality, and it is usual to divide the different degrees of nonlocality in four cases. The first one is when the nonlocality is just localized; in this scenario as a response function we get: R(r) = 5(r), 10. (1.35).

(25) R(r). o5. -5. Figure 1.2: Schematic illustration of three different kinds of nonlocal responses: (a) Gaussian, (b) Exponential, and (c) Modified Bessel function of the second kind. if we substitute this response function in the nonlocal NLSE, we recover a standard pure local NLSE. Hence, for a depreciable degree of nonlocality, we can simplify our nonlocal nonlinear media to some local nonlinear media, such a as a Kerr medium. Then Eq.(1.30) is a general nonlocal model that also allows to recover the local model. In the case of pure localized media we have an extensive literature and research work done [6]. In this local limit, optical beams with a higher power than a threshold value, will collapse after certain propagation distance. Also, in this pure local media, is possible to have just solitons with circular symmetry, i.e. bright and vortex solitons with elliptical modulation will be always unstables. Furthermore, most of the circularly solitonic structures known as azimuthons, dipoles, etc, are also unstable in local media and just the fundamental scalar soliton and the vector dipole-mode solitons are known to be stable. If we slightly increase the degree of nonlocality of our pure local media, we arrive to the second scenario: weakly nonlocal media, in this case, due the width of the nonlocal response function is much smaller than the width of the intensity of the beams, is possible to do a series expansion of the intensity, and hence we obtain the following result for the refractive index N: - J + 7V2/, *R(r)dr.. (1.36) (1.37). here 7 is a parameter that describes the relative width of the nonlocal response. A nonlinearity in this particular case occurs in the theory of nonlinear effects in plasma [42]. In the limit of weak nonlocality, we deal with a perturbed NLSE that have a nonlinear-dispersion term [42], and its solutions can be found in an explicit and closed form[28]. 11.

(26) R(r). R(r). (d). Figure 1.3: Different degrees of nonlocality: (a) a pure local medium, (b) a weak nonlocal medium, (c) general nonlocal medium, and (d) strongly nonlocal limit. If we increase more the degree of nonlocality, we arrive then to our third scenario: general nonlocal media. In this case we must to use the nonlocal NLSE without any approximation, and besides usually, both, numerical and variational approaches are needed. Increasing even more the degree of nonlocality of our media, we arrive finally to our four scenario: strongly nonlocal nonlinear media. In this case, now is the nonlocal response function the term that can be expanded, so we have: (1.38) where = R(0),. (1.39). m=. (1.40). p=. (1.41). in this case, the NLSE is reduced suppressing to a very well know equation: the harmonic oscillator equation. Even more, in the strongly nonlocal limit, we have strictly a linear equation of propagation instead of a nonlinear. This remarkably concept was explored by Snyder and Mitchell in [43], where they called this kind of solitons: "accessible solitons". These kind of solitons allow solutions with different symmetries; in fact, we have explored in chapter 4, two different kind of self-trapped modes in the highly nonlocal limit. In Fig.(1.3) we can observe four different representations for the diverse degrees of nonlocality here explained. In this thesis, we have done research studies for the two last scenarios: general nonlocality and the strongly nonlocal limit. In the following sections we will present some of the peculiar characteristics that the nonlocality can introduce in the nonlinear phenomena. 12.

(27) 1.3.2. Physical effects of nonlocality. Changes to modulational instability Our first physical effect presented is the change introduced by nonlocality to modulational instability. Modulational instability means the exponential growth of a weak perturbation of the amplitude of the wave as it propagates [28]. It is known that modulational instability can act as a precursor for the formation of bright spatial solitons, while dark solitons require the absence of instability [44]. Many mechanisms can affect the modulational instability in nonlinear media; for example: saturation of nonlinearity [45] and coherence properties of the optical beams [46]. If we consider perturbed plane waves in the form V>(r, z) = [Jfr + oi(£, z) exp(iko-r - i(3z}} ,. (1.42). where po > 0 is the wave intensity and where ai(f. , z)=. I ai(k)exp(ik-(, + Xz)dk,. £ = r - koz,. (1.43). is the complex amplitude of the small perturbation in a coordinate frame moving. Is possible to show that A fulfills [28]: A2 = -k2po lak2 - sR(k}] ,. (1.44). here k denotes the spatial frequency, a = l/(4p 0 ), s is a parameter (either 1 or —1) that is selected according to if we are working with, respectively, focusing or defocusing nonlinearity, and R(k) is the Fourier transform of the particular nonlocal response function studied. It is the sign of A2 that determines how stable is the solution. For example, for A2 > 0, the initial perturbation grows during propagation with the value given by Re(A), and in this case we have modulational instability. For a detailed analysis of the possible scenarios of MI gain spectrum, see [29]. However, in summary, the importance of Eq.(1.44) is because it shows that the stability properties of the plane waves in a nonlocal medium are completely determined by the spectral properties of the nonlocal response function [28]. Hence, nonlocality can indeed change the modulational instability behavior. Therefore, as a main result, it can be shown that for response functions such as Gaussian and exponential, (functions that their Fourier spectrum is always positive definite), there are always a wavenumber band symmetrically centered about the origin, where A2 > 0 for enough small A; (this contrary to the case of the defocusing nonlinearity, where a plane wave solution remain always stable). This means that in the focusing case we will have long MI, regardless which response function was used (just fulfilling the requirements of Fourier spectrum positive definite), and then the use of nonlocality in self-focusing media keeps open always the possibility to obtain soliton solutions. 13.

(28) Collapse arrest Another feature of nonlocality is the capacity of collapse arrest [30], [47]. Collapse refers to the fact when the strong contraction of a wave in nonlinear media leads to a blowup of its amplitude. In spite of is possible to have quasicollapse dynamics in real physical systems, usually the collapse shows the limit of the applicability of the model in question. The collapse in nonlinear optics is a signal that the NLSE fails as envelope equation due it breaks the original scale on which it was derived using the multiscale asymptotic technique [48], [49]. It has been shown that collapse can be eliminated completely under a nonlocal response function. In fact, it is possible to show that there is a bound of the Hamiltonian of the nonlocal NLSE that is given by [30] : R*«x>,R(k')>0:H>. \\VU\\* -Ro^,. (1-45). where the definition of RQ and P are the same as in Eq.(1.39) and Eq.(1.41), respectively, and where \u\pdr>0,. \\U\\; =. (1.46). this inequality shows that the Hamiltonian is bounded from below by a conserved quantity given by .Ro^ &nd also is bound from above by H + RQ^-. These bounds represent a rigorous mathematical proof that a collapse cannot occur in general nonlocal media. It is important to remark that this inequality holds for all symmetric response functions with a positive definite Fourier spectrum with a finite value at the center. It can be formally shown that if the Fourier spectrum of the nonlocal response is positive definite, as do most physically reasonable response functions, then collapse will be always prevented. This issue of nonlocality can be easily understood, because when the beam starts to contract, there will be a point in the propagation where the radius of beam will be shorter than the radius response function, then eliminating the continuous contraction produced by the refraction index, because the convolution then will average the intensity concentration, "relaxing" the collapse. Interaction of solitons The interaction among solitons in nonlocal nonlinear media changes considerably from the traditional behavior in other nonlinear media [50]. For example, it is known that in pure local media, solitons with the same phase will experiment attraction because their amplitudes will interfere constructively, and this increasing in the intensity will give as a consequence an increasing in the refractive index, moving both soliton toward the central region of higher refractive index. In the other case, solitons with an opposite phase of TT radians will experiment repulsion in local media, this because their amplitudes will have destructive interference and then the refractive index will decrease in the central region, forcing the solitons to repeal each other. Any other phase difference will lead to a more kind of complicated dynamics accompanied by a strong energy exchange. In nonlocal nonlinear media, when the degree of nonlocality is strong enough, we just can have attraction behavior. We show in Fig(1.4) which is 14.

(29) (h). /=(). /=L/2. U U. Figure 1.4: Soliton interaction in nonlocal media: (a) solitons in phase, (b) relative phase difference of ?r/2, and (c) relative phase difference of TT. the dynamics of interaction between two bright solitons in nonlocal nonlinear media for cases with different phase difference. Nonlocality provides a potential in which self-trapped structures always attract, even in the case of dark solitons: it is known that dark solitons always suffer from a repulsive interaction if there is not any other external condition imposed, but, in the case of nonlocality, when a certain threshold is reached, then is possible to observe attraction behavior between dark solitons [51].. Stabilization Perhaps the most interesting -at least, from the engineering point of view- feature of nonlocality, is the stabilization of modulational [6] and azimuthal instabilities [54], [55]. It has been usual to assume that nonlocality tends to average the instabilities produced in all the self-trapped entities. Nonlocality successfully can stabilize solitary structures because the averaging characteristic of the convolution decreases the growth rate of the perturbations in the beam, providing a strong waveguide structure which acts as a confining potential. Then, nonlocality can prevent breakup of self-localized structures. In the chapter 2 we will present how nonlocality can stabilize solitonic structures such as dipoles and azimuthons. It is important to remark that usually stabilization of solitonic structures are reached with a strong degree of nonlocality. As a result of the stabilization characteristic of the nonlocality, the formation of multisoliton bound states with diverse two-dimension geometry [52] is expected. The stabilization produced by nonlocality is very different from the one discussed in the context of competing nonlinearities. In the last one, the stabilization happens 15.

(30) when the nonlinearity changes its character from focusing to defocusing. Stabilization from nonlocality is more related to the similar stabilization produced in vortex vector solitons [53], where one of the beams behaves as a confining potential. In a similar way, strong nonlocality tends to average perturbations of the beam intensity, producing then a reduced effect on the widely induced refractive index, helping to maintain more stable the spatial beam in question. The use of nonlocality for stabilization purposes has been already studied for the case of bright vortex solitons [54], [55]. Vortices are fundamental objects that can be found in many fields of physics [56], [61]. In optics, the vortices are characterized by finite-size beams with a phase singularities at the center. They have been produced experimentally in both, linear and nonlinear media [57]. The phase of these beams rotates around the singularity, and the number of rotations define the so-called topological charge of the vortex [58]. The bright vortex beams have the form of a bright right with an amplitude of zero in their center and they are important due to the characteristic to carry angular momentum. Unfortunately, these kind of beams are highly unstable in self-focusing nonlinear media, the latter due to the symmetry-breaking azimuthal instability [54]. In fact, after certain propagation distance they decay into several fundamental solitons [6]. In [54] and [55] is suggested that the symmetry-breaking azimuthal instability can be suppressed using a nonlocal nonlinear medium; the physical mechanism for the suppression of the azimuthal instability of the vortex soliton beams in nonlocal nonlinear media can be associated with effective diffusion processes introduced by the nonlocal response. As is was pointed out in [54], if a small azimuthal perturbation of the vortex beam alters its structure in certain point, the respective temperature distribution along the vortex ring becomes no uniform, producing then, that the intrinsic thermo diffusion processes smooth out the inhomogeneity, allowing hence stabilization of vortex beams. It has been found that single-charge optical vortices can be stabilized, whereas multi-charged vortices always decay into fundamental solitons, even for high levels of nonlocality.. 1.4. Thesis Outline. Nonlocality opens new physical phenomena in the soliton field, being the stabilization of self-trapped structures one of the most important. The mechanism behind the stabilization process is a generic feature found in many physical systems. In this chapter we have presented a basic review of the main properties introduced by nonlocality. This thesis is organized as follows: In this first chapter we have given a general introduction to nonlinear optics and the master equation of the light propagation in nonlinear media: the Nonlinear Schrodinger Equation. Then the spatial optical soliton is introduced. In this chapter we also have presented a description about the general features of nonlocal nonlinear media: difference in the interaction between solitons respect to local media, stabilization of self-localized structures, changes to the behavior of the modulation instability, and collapse arrest. Also we have introduced the nonlocal NLSE 16.

(31) and different nonlocal responses were presented. We also studied the four different scenarios with diverse nonlocality degree. In the second chapter, propagation of rotating dipoles is studied. It is shown that there is threshold value of nonlocality where stabilization is reached. This study is done using propagation of the rotating dipoles using a split step Fourier method. It is also shown that the kind of dipole studied is indeed the link between vortex solitons and dipoles without any azimuthal modulation. Then, we extend this previous analysis, and general azimuthons in nonlocal nonlinear media are covered. We present a basic review about what an azimuthon is, and then we use numerical and variational methods to find azimuthons in nonlocal nonlinear media. Again, we found that for enough degree of nonlocality, we achieve azimuthon stabilization. In the third chapter, light localization in Bessel Lattices is observed. This is the only chapter where the results are mainly experimental, but also a theoretical review of the experiment background is included: we give a summary of the principles of propagation in photorefractive crystals with highly anisotropy and nonlocal features. In the fourth chapter, we studied the highly nonlocal nonlinear media in its limit to find Helmholtz-Gauss (HzG) accessible soliton solutions. These solutions connect nondiffracting beams and the soliton field. First the reduction of the nonlocal NLSE to the harmonic equation is derived, and then using a ansatz that also is a solution of the Helmholtz equation, the HzG soliton solutions are founded. In the second part of this chapter, the term of ellipticon is introduced: solitons that have an elliptical modulation and whose natural coordinate system of study is the elliptical cylindrical coordinate system. This kind of solution has a closed and analytical form in highly nonlocal nonlinear media. Some basic features of these solutions are studied and their utility in general nonlocal nonlinear media is commented. Finally, in the fifth chapter, a general review, conclusions and future work to be done are covered.. 17.

(32) oo.

(33) Chapter 2 Azimuthons in nonlocal nonlinear media 2.1. Introduction. Recently, a novel kind of azimuthally modulated self-trapped rotating singular optical beams were introduced: the so-called azimuthons [59]. These novel beams are a link between the rotating soliton clusters and radially symmetric vortices. They are characterized for two independent integer indices: the number of intensity maxima or peaks, and the charge m of the phase dislocation trapped by the beam. Unfortunately, in the first two nonlinear media previously studied (Kerr and saturable medium) [59], it was found that azimuthons are unstable, and hence is not possible to produce them in laboratories. In the present chapter, we extend the study of azimuthons to nonlocal nonlinear media. In the first part of this chapter we introduce a self-trapped beam that is composed by two humps with any arbitrary azimuthal modulation, and due this characteristic, this kind of dipole soliton can be classified as the simplest case of azimuthon (single charge and two peaks). We study rotating dipoles in nonlocal nonlinear media and show that under enough level of nonlocality, we can achieve stabilization of them. In the second part of this chapter, we extend our previous study to include higher order azimuthons: self-trapped singular beams with azimuthal modulation. We study their physical characteristics in nonlocal nonlinear media. In both cases, dipole and general azimuthons, our research studies are done with a variational approach and a numerical propagation using a pseudo-spectral method. We found that, in both cases, when some enough value of nonlocality or power is reached, it is possible to have a stable propagation of rotating dipoles and general azimuthons. Hence, we found that nonlocality can help to achieve stabilization of the azimuthons, being then nonlocal media the first known media that can support stable azimuthons. 19.

(34) 2.2 2.2.1. Rotating dipole solitons Variational approach. As we pointed out in the past chapter, nonlocality is in reality an inherent characteristic of the general definition of susceptibility tensors [60] and therefore it could be considered always in electromagnetic models. Obviously, nonlocality can be neglected if the response of the medium is assumed to be instantaneous, approximation done frequently and that works satisfactorily in many situations. Nevertheless, in certain media, the nonlocality is strong enough that its effects must be included in the mathematical model. In this chapter we will focus on the property that an enough nonlocal degree can help to achieve stabilization of self-trapped structures. The stabilization with strong nonlocality became particularly important for vortex solitons, known to be azimuthally unstable in local media [61]. The stabilization of vortex solitons against symmetry breaking instability was demonstrated theoretically for the models with thermal nonlinearity [54] and with Gaussian kernel of nonlocality [55]. Stabilization of vortices in nonlocal media suggests the possibility to generate stable higher-order modes of spatial solitons, such as optical necklaces [62, 63, 64], soliton clusters [65], and azimuthons [59]. The simplest case of an azimuthon, found to be highly unstable in saturable medium [59], is a single-charged vortex azimuthally modulated that is formed with two peaks in its ring intensity profile. In terms of azimuthal indices, this structure has the topological charge m = I and the intensity index N = 2. The index m defines a phase twist, while the index N is just the number of the "intensity peaks" of the beam. In principle, such solution can be constructed as the coherent superposition of two fundamental bright solitons, tilted with respect to their common propagation direction. Stable fundamental solitons interact similar to classical particles while relative phase determines the character of interaction [6] as we pointed out in the previous chapter. Furthermore, the initial tilt of two beams results in a net angular momentum and, depending on the relative phase between solitons, the appearance of phase dislocations in their common wave-front. Such system is known as soliton spiraling [61]. However, in local media with Kerr-type nonlinearities, the bound state of two solitons is always unstable, so we can only appreciate soliton spiralling for a certain distance in their respective numerical simulations. In this chapter we extend the previous studies done with vortex solitons, and we study the simplest azimuthon, or rotating dipole soliton in nonlocal media. We assume a Gaussian response, the last one because with another nonlocal response kernel, the variational method in which our approach is based becomes then mathematically intractable. We demonstrate that the rotating two-soliton bound state can be stabilized, provided that the nonlocality (or equally in our model, the power of the beam) exceeds some value of nonlocality. Remarkably, this threshold is found to be lower for beams with higher angular momentum and higher rotation rate. We also find that the family of two peaks azimuthon extends from the non-rotating dipole soliton to the radially symmetric vortex soliton by continuous azimuthal transformation. Hence our 20.

(35) rotating dipole spatial soliton provides a link between optical vortices carrying phase dislocations and solitonic two-body system, producing stable rotating bound state of two particle-like beams. The key feature of stabilization of rotating dipoles can be explained if we take the analogy with composite solitons in local media [6]. Several mutually incoherent optical beams create common potential, e.g. bell-shaped soliton and dipole-type component. Because the common potential has the bell-like shape it supports linear azimuthal modes, e.g. dipole-mode. The rotating dipole-mode soliton is known to be stable in saturable media [61], and stabilization of similar scalar azimuthon in highly nonlocal media has the same nature. Thus, the azimuthal modes of spatial solitons in nonlocal media have several features in common with multi-component and thus partially spatially coherent beams. We begin our analysis for the rotating dipole solitons with the general Nonlinear Schrodinger Equation [6] , equation that as we pointed out in the past chapter, describes nonlinear propagation of paraxial optical beams:. iEz + A£ + n (/, r) E = 0,. (2.1). where A = V 2 is the transverse Laplacian, and the nonlinear term of the refractive index n depends on intensity / = \E\2. For a nonlocal response, we use the following refractive index: n(I,r) = j R(\r-p\)I(p)dp,. (2.2). where we consider a normalized Gaussian kernel of nonlocality, which is given by: R(t) = a-27r~l exp (-t2/a2). .. (2.3). Parameter a measures the transverse scale (strength or degree) of nonlocality. It is important to note that although there is still no known physical model with this kind of nonlocality, Krolikowski et. al. [28], [30] have demonstrated that models with symmetric and positive Fourier spectrum, such as the one with Gaussian response, can be adequate models reflecting generic properties of nonlocal media as we pointed out in the past chapter. Radially symmetric stationary solutions to NLSE can be obtained by following the standard procedure: we seek for solitary solutions in the form E(x, y, z) = V(r) exp(imip + iA.z). (2.4). and we substitute this ansatz into Eq.(2.1); after using the rescaled variables: f = r/a,p = p/a,V. = Trl/2(T-lU,. (2.5). we obtain: -kU + Urr + r~lUr - m2r-2U + UN(U2,r) = 0, 21. (2.6).

(36) 70. b). U(r) 0. Figure 2.1: Envelope of a fundamental soliton and corresponding refractive index profiles for strong, k=50 (a) and weak, k=l (b) nonlocality. here k = Aa 2 and we drop tilde f —> r for simplicity in the notation. The normalized convolution integral takes the form: re. 2. ~r ). Jo. pdpexp(-p2)I0(2rp)U*(p),. (2.7). where /o is the modified Bessel function of first kind. To solve this nonlinear integrodifferential equation, we use a shooting method within the iterative procedure Uj —» Uj+i with initial Gaussian envelope U0. At each iteration we compare the envelope f/j+i with the previous one and monitor the error parameter:. fUjdr. (2.8). The convergence is relatively fast, but the number of iterations varies with k value, because in our variables the larger k corresponds to larger nonlocality a. To obtain solutions with the accuracy e ~ 10~9 we need 10 or less iterations for k ~ 100, while 40 or more iterations are needed for k near 1. With the use of our numerical scheme, we show in Fig.(2.1) the fundamental soliton envelopes (m = 0) and corresponding induced refractive index changes for the cases of weak and strong nonlocality. Observe how in the case of weak nonlocality the width of refractive index induced and the intensity is the same, whereas in the case of strong nonlocality the width of the refractive index is larger than the width of the intensity beam. If we let the m value to take another integer value different from zero, it is also possible to obtain then radial profiles for the bright vortex soliton cases. Several examples of radial envelopes with different number of nodes n are shown in Fig.(2.2 ). Observe that a higher value for the n parameter implies that the maximum beam intensity is also larger. While the two indices m and n provide full classification of modes in linear media (the well known Laguerre-Gaussian beams), the existence of azimuthal higher-order modes of spatial solitons did not receive the full attention of researchers due to the complexity of the problem In fact, the so-called azimuthons were introduced for the local media in the case of a Kerr and saturable nonlinearity very recently at the time of this writing [59]. These new kind of self-trapped azimuthally modulated 22.

(37) Figure 2.2: Amplitude of the radial modes with n = l , 2, and 3 for (a) fundamental solitons (m=0), and (b) vortex soliton (m=l) solitons. The value is k=100 for all the radial modes. singular beams feature also two independent azimuthal indices, the topological charge ra and the number of peaks in intensity ring N; hence our azimuthally modulated dipole beam, is indeed the simplest non trivial azimuthon with parameters m = 1 and TV = 2. It is known that exact solutions can be found numerically by two-dimensional relaxation methods [52], however, this approach represents major difficulties due to the poor convergence for complex envelopes with singular phase profiles. On the other hand, variational solutions, can be obtained with much lesser effort and provide an acceptable approximation [66], but we need to have functions that can be mathematically useful (for example, they must be integrable) and also we need to check the solutions obtained with a numerical method. So with the Eq.(2.6) and Eq.(2.7) we can just obtain bright and vortex solutions. In order to propose an useful ansatz to obtain our dipole solutions, we would like to remember that the dipole-mode vector soliton was the first example of azimuthallymodulated spatial soliton[67], in fact these beams were observed in photorefractive media by Krolikowski et al [68]. This dipole-mode vector soliton is indeed a type of an optical vector soliton that has its origin from trapping of a dipole Hermite Gaussian 01-type mode by a fundamentalsoliton-induced waveguide created by another, incoherently coupled, co-propagating beam [61]. It can be shown that in the case where the vortex and dipole components are small - close to the so-called bifurcation line-, both solutions are linearly stable[53]. As long as we move away from the bifurcation line, the family of vortex solitons becomes azimuthally unstable, whereas the dipole-mode solitons remain stable in the complete domain of their existence. Thus, close to the bifurcation line this composite structure can be described as a linear mode [61]. This analogy with linear waveguide theory, (and besides remembering that stabilization in nonlocality is related to the stabilization produced in vector solitons) suggests the general ansatz for the azimuthally modulated component of a composite spatial soliton could be a structure in the form: (£) (cos(m^) + ipsm(mp)) exp (i±z) ,. (2.9). where parameter p determines the modulation depth of azimuthon intensity and varies 23.

(38) 0.0. 08. Figure 2.3: Schematic illustration of the dipole solutions: (a,b) low nonlocality scenario with k = l and (c,d) high nonlocality scenario with k=70. (a,c) Show the amplitude of the radial profiles, whereas their corresponding nonlinear potential is show in (b,d). All the solutions are show with 5 different modulation depths, from p=0 (m-pole case) to p = l (vortex case), intermediates values are p=0.25,0.5,and 0.75. from 0 to 1. We substitute this ansatz into corresponding averaged Lagrangian [66] and derive Eq. (2.6) for the radial dipole envelope U(r) with the nonlinear term given by /OO pdpexp(-p 2 )t/ 2 (p)x _ f I0(2rp) + 1 ( ^ 4 } I2m(2rP)} .. (2.10) (2.11). Two important limits are available with our ansatz. First, in the limit p —> 1, the intensity envelope is radially symmetric (\E\2 in Eq. (2.9)), and potential Eq. (2.10) is converging exactly to Eq. (2.7), i.e., Nv —» N at p —> 1. Thus our approach contains also exact vortex and bright solitons as the limiting case. From the other side, p —> 0, such solution represent the scalar multipole solitons, or optical necklaces, known to be radially unstable in local media. We focus here on the lowest order azimuthon with indices m = 1 and N = 2, also considering the basic radial mode n = 0 only. In this case, the approximation Eq. (2.9) is particularly close to azimuthon, even though it does not contain explicitly the rotational velocity as a parameter. Applying our numerical technique before described, we obtain profiles of stationary solutions with different p and k values; several examples are shown if Fig. (2.3). In all the cases studied, we found that the vortex solution (p = 1) has the lowest amplitude value, and conform we increase the p value, we observe also larger amplitude values (as well their corresponding refractive index induced) until we reach the maximum amplitude value, when p = 0, which is the m-pole case. The families of dipole solitons and their parameters are summarized in Fig.(2.4). We note that within our approximation the power Fig.(2.4) (a) does not depend on 24.

(39) 16C. 20. (b). (a) 10 Power 50. 3.0. Amplitude. 100 0 4. 50. 100. (d). (c). 1.5. Radius. Width " 5 0 ^ 1 0 0 0. 50. k. 1. Figure 2.4: Parameters of vortex soliton family (m=l) in function of the k value: (a) power P, (b) amplitude A=max—U—, (c) soliton ring width (distance between the half maximum intensity points of the single ring vortex soliton), and (d) the radius (distance of the maximum intensity to the vortex center) of the soliton ring. modulation, while the power of azimuthon is expected to vary with angular velocity and modulation depth [59]; this fact determines the limitation of the ansatz Eq. (2.9). Comparison of different envelopes in Fig.(2.5) shows that the difference between envelopes with different p is largely absorbed by the amplitude, see Fig.(2.4) (b), while the radius and the width of intensity ring is practically the same, see Fig.(2.4) (c,d). Nevertheless, the two-dimensional envelopes in Fig.(2.5) clearly demonstrate the big difference between dipole solitons with different modulation depth. It also shows the continuous transformation from the real dipole-type structure - through highly modulated vortex with 0 < p < I - to the radially symmetric vortex soliton. Thus the solutions we introduce here provide the link between soliton spiraling, or solitonic twobody problem, and optical vortices, the singular beams with phase dislocations. The NLSE equation is associated with an infinite number of conserved quantities, called the integrals of motion [69]. The first two integrals of motion, besides corresponds to physical variables: power (P) and angular momentum (M), and they are given by the relations: \U\2dr,. "-f M. Jo. (2.12) (2.13). = Im / issue E—dr, for our rotating dipole solitons, an important here is the value of angular momenJ-oo UV tum. For the ansatz Eq. (2.9) it is easy to calculate it [64], S = 2p/(l+p2),. (2.14). here S = M/P is the (orbital) angular momentum of the optical beam M, normalized 25.

(40) (c). Figure 2.5: Schematic illustration of dipole envelopes of the (top) intensity and (bottom) phase for three characteristic chases with k=0 and (a) p=0, (b) p=\/2, and (c) p = l . by the power obtained in base of its pure radial form given by: /•o. /. U2rdr.. (2.15). Jo. In terms of angular momentum per particle, the Eq.(2.14) gives a fractional value. We observe that for p — 0 we do not have angular momentum, while the angular momentum reaches its maximum in the vortex case (p ~ 1).. 2.2.2. Propagation dynamics and stability. Once the stationary solutions are obtained, there is still an open question about the stability of these soliton solutions. Because the non-integrability of our system, stability only can be tested with numerical methods. In fact, the stability of selftrapped beams against small perturbations is extremely important because just stable (or weakly unstable) self-trapped beams can be done in the laboratory [6]. In our studies we performed a numerical propagation using the split-step beam propagation method with fast Fourier transform, and we have evaluated the nonlinear term in the frequency space, so that the convolution is calculated as the inverse transform of the product of two Fourier images. We add typically 10 — 20% of initial noise to our envelopes and propagate for large values of z ~ 102 — 103 (~ 104 in some cases) normalized units. As a result of the initial noise added, at the first steps of propagation the optical beam radiates some energy, and then after a certain distance the beam remains more or less constant. Because we have broken the radial symmetry by adding noise, if the optical beam is unstable, the development of new structures is observed faster, such as splitting into two fundamental solitons in Fig.2.6(a) This scenario occurs when the nonlocality is weak; 26.

(41) Figure 2.6: Two different scenarios of dipole propagation: (a) Unstable with lower nonlocality at k = l and p=0.5 (b) Unstable with higher nonlocality at k=20 and p=0.75. (c) Stable rotating dipole soliton propagation with k=100 and p=0.25. then the dipole breaks into two mutually repelling beams, and this behavior is reminiscent of the azimuthal instability of vortices in local media [61]. Another scenario is the formation of a single soliton, showed in Fig.2.6(b). Here we have a subcritical nonlocality the azimuthon initially breaks up into two beams that are subsequently forced by the nonlocality-induced potential to collide and merge into a single fundamental soliton. If we increase enough the nonlocality degree, we can have stable propagation of the dipole, indeed stable spiraling is shown in Fig.2.6(c). This stable spiraling scenario corresponds to a balance of the effective nonlocality attractive force and centrifugal repulsive force. Doing simulations and finding what happens for enough large distance propagations, we have tested the solutions in the domain (k,p) and roughly determine the stability border, shown in Fig.2.7. The stability threshold decreases in the limit of the radially symmetric vortices. In the case of stable propagation, we obtain angular velocity by numerical averaging, and Fig.2.7 shows the dependence of the azimuthon angular velocity on the modulation parameter p. The velocity increases monotonically with p and depends only weakly on the power (two curves are for k = 60 and k — 80). For a vortex soliton, the rotation velocity is really a free parameter. We found that our dipoles with different modulation depths (different p values) have the minimal threshold k value required for stability. It has been observed that vortex solitons are much more robust than dipole solitons with the same k value (or the same power, see Fig.2.7). This feature can be associated with the rotation of soliton that helps to stabilize it. In addition, while the dipole soliton (at p = 0) experiences the mutual repulsion of its out-of phase peaks, the vortex soliton is radially balanced, and only stabilization needed is against the azimuthal modulational instability. Similar results hold for spatial solitons with higher azimuthal index m (note that the ansatz Eq.(2.9)can also describe higher order azimuthon, but with condition N = 27.

(42) Stable. Unstable. Figure 2.7: (a) Stability border of dipole solitons in nonlocal media with Gaussian response, (b) Numerically found relation between the angular velocity omega of a dipole soliton and its parameter p. 2m only). We show in Fig.2.8 two examples of stable propagation of (a) quadrupole and (b) hexapole-type azimuthons obtained with Eq.(2.9). In the propagation of the quadrupole, we see that the intensity of the beam rotates, while in the case of hexapole beam, it seems to be a stationary beam. Is there the possibility to have a quadrupole rotating or and hexapole stationary? this question is answered in the next section, in where instead of using the ansatz Eq.(2.9), we develop another approach to study the general azimuthon case. In another numerical experiment, we have tested the stability of rotating dipole solitons with respect to strong perturbations that lead to large-amplitude oscillations. We found that persistent oscillations of both the maximum intensity and the diameter D (the distance between two peaks) are observed, and both oscillations decay slowly during propagation, see Fig.2.9. Some surprising results we obtained testing the stability of double-ring radially symmetric (p — 1) solitons, including fundamental (m — 0) and vortex (m = 1) modes. We did not find any stable solutions, but observed, after modulational instability is fully developed, the soliton revivals, such as the one shown in Fig.2.10(a). This instability can be easily excited if we dephase the real and imaginary part of the initial condition by a small amount, (i.e if our initial optical soliton is given by t/(r)(cos(m#) + zsin(m# + <J)). Note that the structures which appear at intermediate stages of such dynamics, are very similar to the regular higher-order linear modes, such as Laguerre- and HermiteGaussian beams, see Fig.2.10. Thus the nonlocal media could be seen as a self-induced mode converter, where 28.

(43) Figure 2.8: Azimuthons with (a) m=2 and p=0.75, and (b) with m=3 and p=0.25. The value k=100 for both.. o o ; oo. Figure 2.9: (a) Evolution of a dipole soliton (k=60, p=0) excited by a strong initial perturbation and (b) Maximum intensity and diameter oscillations in the propagation.. 29.

(44) Figure 2.10: Examples of the soliton revivals observed in the instability-induced evolution of (a) double ring zero charge and (b) double ring single-charged vortex solitons. The value k=100 for both. conversion depends on the initial state as well as the properties of the medium, i.e. the nonlocality parameter. Using the strongly nonlocal limit approach, we can understand that this self-induced mode conversion is produced because there is an adjust of energy between the diverse modes that forms a particular family of self-trapped beams. Nevertheless, in general nonlocal media, because of strong dynamical reshaping, such beams radiate energy and after several cycles of revivals loose their regular structure. In the regime of high nonlocality (at k ~ 100), the induced potential does not allow the splinters of unstable solitons to fly away from initial ring, such as shown in Fig.2.10(a). Now, after the modulational instability develops, the most of the energy is still selftrapped and remains localized. At the same time its structure losses any order and we observe irregular dynamics. The study of the chaos and optical turbulence of these structures could be subject of future research works.. 2.3 2.3.1. Higher order azimuthons in nonlocal nonlinear media Variational approach. In the past section we study the propagation of rotating dipole in nonlocal nonlinear media, and we pointed out that this kind of beam can be classified as the simplest member of the azimuthon family, because in fact our rotating dipole soliton can be 30.

(45) classified as an azimuthon of single charge and two peaks. In this second part, we focus in azimuthons with more general parameters. Azimuthons are spatially localized self-trapped optical beams in nonlinear media, which provide an important missing link between the radially symmetric vortices and rotating soliton clusters. In [59], is reveled that these kind of beams have an angular momentum that has two different contributions. The first contribution comes from the internal energy flow of the beam, while the second one has a "particle" origin, and appears only when the beam is modulated. These two contributions can have opposite signs and then is possible to have a cancelation of their respective contribution of angular momentum, given as a result non rotating modulated singular beams, or stationary azimuthons. A striking difference from the vortex phase case, where it is given by 6 = m<p , is that in the case of the phase of the azimuthons, we have a nonlinear function of the polar angle ip, and hence, higher-order spatial solitons can be described in terms of azimuthal deformations of the vortex solitons. We have told that azimuthons can be described by two independent parameters: the topological charge m and their number of peaks N, but also we could consider an additional third dimensional parameter, the modulation depth n. This last parameter gives the strong of depth modulation, and varies from 0 to 1. Where 0 represents a null modulation and in the case of n = 1 we have a m-pole solution. In the case of soliton clusters [64], [70], [71], the number of humps of the azimuthons fulfils the relation: 'N > 4m, (2.16) however, rotating azimuthons that exist in saturable media can be found with the condition: N > 2m. (2.17) we show in this chapter that azimuthons in nonlocal media can exist in a broader region of conditions. We begin our analysis considering that the envelope of our beams varies really slower in comparison with the wavelength of the carrier, so we can use again the well know NLS equation for nonlinear paraxial beams (Eq.(2.1) ), and also, as we did it in the past section, we assume again a normalized Gaussian nonlocal response Eq.(2.2). This because as has been told previously, Eq.(2.2 is the one of the few known mathematical nonlocal responses that allows to obtain accessibly results using variational methods, and besides, the physics of nonlocal media are still in its early stages, so first we need to study the phenomenon in its simplest context possible, and then we can extrapolate and study nonlocal propagation using kernels which are more realistic and complicated. However, as has been shown by Krolikowsky [28],[30], nonsingular responses will give similar results if their spectrum is positive definite, at least qualitatively spoken. In the Eq.(2.2) the parameter a represents the degree of nonlocality and is important to remark that when a — 0 we can recover the case of propagation in Kerr media, while when a —> oo, we can talk about the concept of accessible solitons [43]: solitons that can be explained just by linear theory, simplifying enormously the mathematical tool-box necessary to understand their fundamental properties. This fact will be study in chapter 4, and indeed we will explore a more general azimuthon con31.

(46) cept but in the highly nonlocal limit. In the present approach we use now the following scaling: r = r'o,z = z'a2, (2.18) and omit primes. Than we look for stationary solutions in the rotating frame f = {r, 0}, where 9 = (p — u>z, E(x, y,z} =. - V(r, 9) exp(ikz),. (2.19). a k being the soliton constant in the rotating frame. Equation for the envelope V reads -W. - iuVe + V 2 F + V f dp exp ( - | f - p\2} \V(p)\2 = 0.. (2.20). Corresponding Lagrangian density takes now the form: L = k\V\2 + u, Im (VV?) + \VV\2 - i\V(f)\2 J dpexp (-\f- p\2) \V(p)\2.. (2.21). We propose to use then the following ansatz: V(r, 0) = # ( r ) y i - n sin2 (N6/2) exp (iif>(6)) ,. (2.22). ^(0) = — 0 + I - ( m - — ) tan" 1 f V T ^ tan ( —]] , (2.23) u;0 N \ u0J \ V2 / / where the real variational parameters are n, w0> and R(r). This anstaz was developed in [59] , and is indeed the azimuthon solution in the Kerr media. Substituting the ansatz into Lagrangian C = / L df we obtain - = k(2 - n) [R2] - u [R2] (flm^fT^l + — (l - ^f\~^nf\ +(2-n) 7T. \. + [R2r~2] (2m2Vl^. CJfl. + (~ + ^ ]. /. \R2} (2.24) L. -I. (l - v / T ^ 2 ). /-oo. - I [R2(r) e~r2] J. 2. PdpR. (p) e^. {(2 - n)2 I0(2rp) + n2 IN(2rp)/2}. (2.25). here dot indicates the derivative on r, [X] = f^° Xrdr, and Ij is a modified Bessel function of the first kind and order j . Following with our variational method, first we derive the variational equation for UQ and obtain: 0,0 = 2 [R2r~2] I [R2] . (2.26) Second we derive the equation for n and obtain: 2. - -k [R2] + [R2r~2] (1 - n)- 1 / 2 (2Qm - m2 + (N2/4 - Q2) (l (2.27) + TT [7?2(r) e~r2] f" pdpR2(p) e-»2 {(2 - n) I0(2rp) - n IN(2rp)/2} . 32. (2.28).