Self-Similar Hermite Gaussian Spatial Solitons in Two-Dimensional Nonlocal Nonlinear Media

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1 Commun. Theor. Phys. (Beijing, China 53 (010 pp c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 5, May 15, 010 Self-Similar Hermite Gaussian Spatial Solitons in Two-Dimensional Nonlocal Nonlinear Media YANG Bin ( Ê, 1 ZHONG Wei-Ping (, and Milivoj R. Belić 3 1 Department of Computer Technology, Shunde Polytechnic, Shunde 58300, China Department of Electronic Engineering, Shunde Polytechnic, Shunde 58300, China 3 Texas A&M Univsersity at Qatar, P.O. Box 3874 Doha, Qatar (Received July 3, 009 Abstract We study analytically and numerically the propagation of spatial solitons in a two-dimensional strongly nonlocal nonlinear medium. Exact analytical solutions in the form of self-similar spatial solitons are obtained involving higher-order Hermite Gaussian functions. Our theoretical predictions provide new insights into the low-energy spatial soliton transmission with high fidelity. PACS numbers: 4.65.Tg Key words: nonlinear optics, spatial soliton, self-similar wave 1 Introduction Important classes of nonlinear material systems are those exhibiting nonlocal nonlinearities. [1] Such nonlinearities arise in several branches of physics, ranging from Bose Einstein condensates to plasmas. In nonlinear (NL optics, nonlocality can be encountered, for example, in liquid crystals, [] thermal nonlinear media, [3] photorefractive materials, [4] etc. The interest in properties of spatial solitons in nonlocal nonlinear (NN media, called the nonlocal spatial optical solitons, has greatly grown during recent years, both theoretically [5 9] and experimentally. [10] The nonlocal spatial solitons are modeled by the NN Schrödinger equation (NNSE. [5 9] In the general NNSE, the NL term has the nonlocal form associated with a symmetric and real-valued response kernel. Moreover, the general NNSE also describes several other physical situations. [11 1] Snyder and Mitchell [13] have simplified the NNSE to a linear model in the strongly nonlocal case. So far, many properties of nonlocal spatial solitons have adequately been described by the model and various related phenomena have theoretically been clarified. The experimental observations can also be interpreted in the framework of nonlocal nonlinearity. [5 9] It has been demonstrated that the localized wave packets in cubic NL materials with a symmetric NN response of arbitrary shape and degree of nonlocality can be described by a general NNLSE, and the nonlocality of the nonlinearity prevents collapse in Kerr media in all physical dimensions, resulting in stable soliton waves under the proper conditions. [11 1] In this paper, starting from the Snyder Mitchell model, we construct higher-order spatial solitons in NN media. We find and display these solitons in the strongly NN media, propagating in a self-similar manner. The paper is organized as follows. Section presents the NNSE model and Sec. 3 introduces the method for finding self-similar solutions of NNSE in the highly nonlocal limit. Section 4 discusses and displays various solitary solutions to NNSE. Section 5 brings conclusions. NNLSE Model The starting point in the analysis is NNSE, describing evolution of complex amplitude u(r, z of a paraxial beam propagating along the z axis: [1,11] i u z + 1 u + N(Iu = 0, (1 where stands for the transverse Laplacian, vector r = (x, y represents the transverse coordinates, and I = u is the beam intensity. All coordinates are made dimensionless by a suitable choice of scaling quantities. The nonlinearity N(I is assumed in the general NN form: N(I(r = + R(r r I(r d r, ( where the response function R(r is determined by the specific physical process responsible for the medium nonlocality. On general physical grounds it is assumed that R(r is real, symmetric, positive definite, and normalized, R(rd r = 1. The NNSE naturally arises in NL optics, where it describes the propagation of a slowly-varying field envelope in the paraxial approximation, in a wave-guiding potential. The nonlocal index change, caused by the propagating beam, may involve some transport process in the medium. Supported by the Science Research Foundation of Shunde Polytechnic under Grant No. 008-KJ06. Work at the Texas A&M University at Qatar is Supported by the NPRP Project with the Qatar National Research Foundation

2 938 YANG Bin, ZHONG Wei-Ping, and Milivoj R. Belić Vol. 53 In the limit when the response function R(r is much broader than the intensity distribution, the nonlinear term becomes proportional to the response function, N(I = PR, where P = I(rd r is the beam power. Assuming the intensity distribution is peaked at the origin, one can expand the response function about the origin, to find N(I = P(R 0 + R r. [1] In this case the highly nonlocal NLSE becomes the linear Schrödinger equation with harmonic potential. A general treatment without external potential [13] lead to a model in which the change in the nonlinearity is proportional to a function of the power, N(I = α (Pr. Although linear in u, the model still describes a highly NL phenomenon of solitons through the NL dependence of the coefficient α on the beam power P. For this reason the model is referred to as the highly nonlocal NLSE, and the solutions are known as the accessible solitons. [13] It has been used in Ref. [14], for example, to explain the observation of optical spatial solitons in nematic liquid crystals. Here, we will be concerned with this limit of the general NNSE. We should note that, while the stability and collapse of solutions to the multidimensional NNSE is still an open problem, [1,11 1] no such difficulty arises for the highly nonlocal NLSE, because this equation is linear. 3 Self-Similar Soliton Solutions of NNLSE In the Cartesian coordinate system and in the limit of a strong nonlocal nonlinearty, the evolution of the wave equation in two-dimensional (D NN media is described by the general NNSE [5 9,13] i u z + 1 ( u x + u y s(x + y u = 0, (3 where s(> 0 is the coefficient proportional to α (P, containing the influence of beam power. Note that P is a constant of motion, equal to the total input power P 0. The second term in Eq. (3 represents the diffraction and the last term accounts for the optical nonlinearity. Apparently, this equation is equivalent to the D quantum harmonic oscillator (QHO. Many solutions to QHO are known, ranging from 1D to 4D. However, all of those solutions are for the time (or z independent case. We provide here z-dependent solutions to Eq. (3, using the selfsimilar method of solution. The separation of variables u(z, x, y = X(z, xy (z, y in Eq. (3 leads to the following coupled equations: i X z + 1 X x sx X µx = 0, (4a i Y t + 1 Y z sy Y + µy = 0, (4b where µ is a constant. We solve Eq. (4a using the selfsimilar method. Following Refs. [4 7], we define the complex field as X(z, x = A(z, xexp[iφ(z, x], where A(z, x and φ(z, x are real functions. Substituting X(z, x into Eq. (4a, and requiring that the real and imaginary parts of each term be separately equal to zero, we obtain the following coupled equations A φ z + 1 [ A ( φ ] x A Asx µa = 0, (5a x A z + φ A x x + A φ x = 0. (5b To search for a self-similar solution of Eqs. (5a and (5b, we introduce a set of transformations A(z, x = (k 1 / w(zf(θ and φ(z, x = a 1 (z + b 1 (zx + c 1 (zx, where k 1 is the normalization constant. Other variables are introduced as follows: w(z is the beam width, F(θ is the real function to be determined, θ(z, x is the selfsimilar variable, a 1 (z is the phase offset, b 1 (z is the frequency shift, and c 1 (z represents the wave front curvature, or the beam chirp. Inserting these transformations into (5b, and making the coefficient of each power of x equal to zero, we obtain θ(z, x = x/w, b 1 (z = 0, and c 1 (z = (1/w(dw/dz. Thus, by means of these transformations, an ordinary differential equation for F(θ is readily obtained from Eq. (5a: F θθ F w( da 1 dz + µ Next, we introduce another transformation, F(θ = e θ / f(θ, and after some treatment, we find: w ( w d w dz + sw θ = 0. (6 d w dz = sw + 1 w 3, (7a da 1 dz = n + 1 w µ. (7b The combination of these equations with Eq. (6 results in: d f df θ + nf = 0, (8 dθ dθ where n (= 0, 1,,... is a non-negative integer. Equation (8 is the well-known Hermite differential equation, and its solutions H n (θ are known as the Hermite polynomials, namely f(θ = H n (θ. Now, we can solve the beam width equation. From Eq. (7a, one arrives at: 1 ( dρ dz + sw 4 0 (ρ 1(ρ λ ρ = 0, (9 where ρ = w/w 0, λ = 1/sw0 4, and the subscript 0 denotes the initial value of the corresponding parameter at z = 0. Taking w(z z=0 = w 0 and dw(z/dz z=0 = 0, and integrating the differential equation (9 yields: w = w 0[1 + (λ 1sin ( sw 0z]. (10 As seen, the beam width is determined by s and w 0. The beam diffraction initially overcomes the beam-induced refraction when λ > 1, and the beam initially expands, with w /w 0 oscillating between a maximum λ and a minimum 1. On the other hand, when λ < 1, the reverse happens;

3 No. 5 Self-Similar Hermite Gaussian Spatial Solitons in Two-Dimensional Nonlocal Nonlinear Media 939 the beam initially contracts, with w /w0 oscillating between a maximum 1 and a minimum λ. When λ = 1, the diffraction is exactly balanced by the nonlinearity, and the beam preserves its width as it travels in a straight path along z axis. Therefore, the phase-front curvature and the phase offset of the beam are given by: sw c 1 (z = 0 (λ 1sin(4 sw0 z 1 + λ (λ 1cos(4 sw0 (11a z, a 1 (z = a 10 (n + 1arctan[ λtan( sw0 z] µz.(11b sλw0 4 In the end, we obtain the self-similar solution of Eq. (4a: X(z, x = k 1 exp[ia 1 (z] exp [ x w w + ic 1(zx ] ( x H n, w where k 1 = (1/ n n! π 1/ ; w, c 1, and a 1 are determined by Eqs. (10 and (11. Using the same method, we obtain the Hermite Gaussian (HG self-similar solution of Eq. (4b: Y (z, y = k exp[ia (z] w where ( 1 1/ k = m m!, π exp [ y w + ic (zy ] ( y H m, w a (z = a 0 (m + 1arctan[ λ tan( sw0z] + µz, sλw0 4 sw c (z = 0 (λ 1sin(4 sw0 z 1 + λ (λ 1cos(4 sw0 z. Put together, these solutions provide the full exact selfsimilar solution of Eq. (3: u nm (z, x, y = k exp[i(a 1 + a ] w exp [ x + y ] w + i[c 1 (zx + c (zy ] ( x ( y H n H m, (1 w w are known as accessible solitons, [13] they are solutions to a linear differential equation, and can be represented as superpositions of HG functions. Hence, by constructing the appropriate system parameters and initial values, the higher order HG beam transmits in the same way as the lower one in the medium, regardless of any other conditions. The solutions in Eq. (1 are the exact solutions to Eq. (3, but only approximate solutions to Eq. (1. In order to check the behavior of solutions in Eq. (1 under propagation, we use the following input beam u 11 (0, x, y = (xy/ πexp[ (x + y /], assume that the material response in Eq. ( is Gaussian, i.e. R(r r = 1 [ πσ exp (x x + (y y ] σ, and propagate it according to Eq. (1. We take σ = 100, so that we are in the highly nonlocal regime. Figure 1 shows the comparison of the analytical solution with where k = k 1 k. As one can see, the solution is expressed in terms of D HG functions, as one would normally expect of a QHO problem, but the z-dependence is explicitly present in the parameters w, a, and c. This dependence causes the beams to oscillate, repeating a breathing cycle. 4 Some Properties of HG nm Soliton Solutions We find that the beam is determined by the degree (n, m of Hermite polynomials, the coefficient s, and the beam width w. The power of the light beam P = u(z, x, y dxdy can be calculated according to the orthogonality relations of H n (θ. It is a constant of motion, which we set equal to 1. It can be seen that the power in general is only half of the sech soliton power, and has nothing to do with the degree of Hermite polynomials. Since Eq. (3 is linear equation, the NL effect of wave collapse cannot occur. Even though such solutions Fig. 1 Comparison of analytical solution with numerical simulation for the initial Hermite Gaussian (u 11 width with different λ. (a Initial intensity profile. (b Analytical solution of Eq. (11 for λ = 0.6 (top row and 1.6 (bottom row. (c Numerical simulation of Eq. (1 for λ = 0.6 (top row and 1.6 (bottom row row. Propagation distances are: sw 0z = π/6, π/4, π/3, π/ from left to right, completing 1/ of the breathing cycle.

4 940 YANG Bin, ZHONG Wei-Ping, and Milivoj R. Belić Vol. 53 numerical simulation for the initial beam with different λ parameters (w 0 = 1, a 10 = 0, a 0 = 0. Figure 1(a displays the self-similar evolution of an initial beam propagating in this medium. The numerical solution of Eq. (1 is performed to ascertain the stability of solutions in the highly nonlocal region, and to compare them with the analytical solutions. As expected, no collapse is seen, and the numerical solution is stable and in good agreement with the analytical solution. As seen in Fig. 1, the beam changes from four spots at the initial position into four bigger spots at sw0 z = π/; this distance marks 1/4 of the full breathing cycle. The distances between the four wave packets contract as they propagate, when λ < 1. On the contrary, the intervals expand and the sizes of packets decrease with the propagation, when λ > 1. The wave repeats periodically this compression/expansion process, and the interval between packets oscillates periodically as the wave propagates. Hence, for λ 1 these wave packets represent an example of a soliton breather. Furthermore, we find that when λ = 1, the beam width does not change with the propagation distance. It is clearly seen that the beam becomes an accessible soliton. [1] The beam width, the wave front curvature of the soliton, and the phase offset can be easily found, and they are given by w = w 0, c = 0, and a = a 1 + a = a 0 [(n + m + 1z]/w0. Thus, the exact self-similar soliton solution of Eq. (3 can be written as: u sol nm (z, x, y = k ( x ( y { [ (n + m + 1z ]} H n H m exp i a 0 w 0 w 0 w 0 w0 exp ( x + y w0. (13 A spatial soliton of this kind exists only in the strongly nonlocal media and is often called the strongly nonlocal optical spatial soliton. It is, obviously, identical in form with the standard HG modes of D QHO. It is interesting that a strongly nonlocal spatial soliton of any width can propagate stably in the media as long as sw0 4 equals exactly 1 (namely λ = 1. It should be noted that s is a parameter determined by the material properties, which also fixes the initial beam width. We see that the spatial solitons in Eq. (13 are determined by two parameters (n, m. For a fixed number n and different m, the soliton HG nm forms a family that exhibits some common characteristics. We now turn to discuss the distributions of the amplitude, intensity, and the positions of the zero points (η nm = 0 and the extreme points (d η nm /dxdy = 0, where η nm = H n (x/w 0 H m (y/w 0 exp[ (x + y /w0 ]. Clearly, these distributions are determined by the form of the HG functions. First, we present the n = 0 case. In Fig. analytical solutions of several low-order solitons HG 0m are depicted along the y-axis direction, for n = 0 and different m. Figure (a depicts the distribution of zeros and extreme points in the y direction. It can be seen that HG 00 soliton has no zero points and one extreme point at (x, y = (0, 0. The soliton forms a ball that represents the fundamental Gaussian soliton. The solution with m = 1 has one zero and two extreme points. As m increases, so does the number of zeros and extreme points. Fig. HG 0m soliton groups, for different m, when n = 0. Here the parameter m has the values m = 0, 1, from left to right. (a Distribution of zero points and extreme points along the y axis direction. (b Analytical solution of Eq. (13.

5 No. 5 Self-Similar Hermite Gaussian Spatial Solitons in Two-Dimensional Nonlocal Nonlinear Media 941 Next, we show the cases with arbitrary n and m. In Fig. 3(a we depict some properties of the arbitrary n and m solutions. The extreme points are arranged in the form of a rectangular matrix. There exist (n + 1 (m + 1 extreme spots. The farther the position of the spot from the centre of the transverse axes, the greater the optical intensity. Specially, optical intensity is zero at the beam centre when n (or m is even. On the other hand, optical intensity is the smallest extremum at the centre when n is odd. When n = m (6= 0, the soliton forms a square matrix of spots. Figure 4 displays shapes of a few soliton square distributions. Fig. 3 The optical field distributions of analytical solution Eq. (13, for different paraments. The parameters are: (a n = 1, m is odd, for example, m = 1, 3, 5; (b n = 1, m is even, for example, m =, 4, 6 from left to right, respectively. Fig. 4 Typical square matrix solitons. The parameters are: n = m = ; n = m = 3; n = m = 4 from left to right, respectively. 5 Conclusions In summary, the self-similar wave of Hermite Gaussian spatial solitons in strongly nonlocal nonlinear media is studied, both analytically and numerically. An exact analytical solution in the form of self-similar wave is obtained and solutions in the form of HGnm functions are found. HGnm spatial solitons display many interesting properties, which include unchanging as well as breathing beam widths. Therefore, they may find wide applications as stable higher order solutions in optical numerical simulations, laser devices emitting ultra-short pulses and all-optical networks.

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