Bache, Morten

Bang, Ole

Esbensen, B. K.

Krolikowski, W.

Wlotzka, A.

Physical Review A

B. K. Esbensen

A. Wlotzka

M. Bache

O. Bang

W. Krolikowski

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Modulational instability and solitons in nonlocal media with competing nonlinearities

We investigate propagation and spatial localization of light in nonlocal media with competing nonlinearities. We show that the competing focusing and defocusing nonlinearities enable coexistence of dark or bright spatial solitons in the same medium by varying the intensity of the beam

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K.; Wlotzka, A.; Bache, Morten; Bang, Ole; Krolikowski, W.Published in:Advanced Photonics CongressPublication date:2012Document VersionPublisher's PDF, also known as Version of recordLink back to DTU OrbitCitation (APA):Esbensen, B. K., Wlotzka, A., Bache, M., Bang, O., & Krolikowski, W. (2012). Modulational instability andsolitons in nonlocal media with competing nonlinearities. In Advanced Photonics Congress (pp. NW3D.4).Optical Society of America.NW3D.4.pdf Advanced Photonics Congress © 2012 OSAModulational instability and solitons in nonlocalmedia with competing nonlinearitiesB.K. Esbensen1, A. Wlotzka, M. Bache1, O. Bang1, W. Krolikowski21DTU Fotonik, Department of Photonics Engineering, Technical University of Denmark, Lyngby, Denmark2Department of Physics, University of Karlsruhe, Karlsruhe, Germany3Laser Physics Centre, Australian National University, Canberra, Australian Capital Territory 0200, Australiawzk111@gmail.comAbstract: We investigate theoretically propagation and spatial localization of light innonlocal media with competing nonlinearities. We show that the competing focusing anddefocusing nonlinearities enable coexistence of dark or bright spatial solitons in the samemedium by varying the intensity of the beam.© 2012 Optical Society of AmericaOCIS codes: 190.4420, 190.6135.Recently there has been growing interest in the nonlocal nonlinear media where the nonlinear response of themedium in a particular location depends on the light intensity in a certain neighborhood of this location [1,2]. Nonlocalnonlinearities have been identified in a variety of physical systems where the nonlocal response is caused by eithertransport processes such as heat or ballistic atomic transport or nematic liquid crystals and Bose Einstein condensateswith long range interaction. The concept of nonlocal nonlinearity has been recently extended to media with the so-called synthetic nonlinearities where the nonlinear response is a result of two or more competing processes or effects.Such competing nonlinear response occurs naturally in Bose Einstein condensates with dipolar interaction. It has beenshown, e.g. that simultaneous presence of nonlocal nonlinearities of opposite sign leads to stabilization of complexsoliton structures which are otherwise unstable in medium with one type of nonlocal nonlinearity [3]. On the otherhand, it turns out that competing nonlinearities may also destabilize dark soliton states and lead to repulsion of in-phasebright solitons [4].In this work we study analytically and numerically the modulational instability of plane waves and soliton formationin nonlocal media with competing nonlinearities. In particular, we will explore the interplay between nonlocality andthe nonlinearity and its effect on stability of bright and dark solitons. We will also discuss the regime in which thenonlocal medium simultaneously support bright and dark solitons.We will consider propagation of optical beam with the slowly varying amplitude u(x;z) and corresponding in-tensity I = ju(x;z)j2 in nonlinear media with the intensity-dependent refractive index change Dn(x; I) = a1RR1(x x0)I(x0;z)dx0+a2RR2(x x0)I(x0;z)dx0, where x and z represent transverse and longitudinal coordinates, respectively.a1 and a2, represent the strength and sign of the two nonlinear contributions, respectively. Here a1 will be consid-ered positive, corresponding to a self-focusing nonlinearity, while a2 negative, corresponding to a self-defocusingnonlinearity. The nonlocal response functions R1;2(x) define the nonlocal character of the nonlinearity. Without lossof generality we will use Gaussian nonlocal response Ri(x) = 1=ppsi exp  jxj2=s2i . Then the evolution of the 1-dimensional beams in a phenomenological model of a nonlinear medium is represented by the following nonlocalnonlinear Schro¨dinger equationi¶uz+¶ 2uxx+Dnu= 0: (1)Modulational instability (MI) is one of the most fundamental effects associated with wave propagation in nonlinearmedia. It signifies the exponential growth of a small perturbation of the plane wave (PW) amplitude during propagation.Following the standard procedure we find that the perturbation evolves with propagation as a(x;z) µ exp(l z) wherethe growth rate l is determined by the following relationl 2 = k2A20 k2=A20 2R˜(k): (2)where R˜(k) = R˜1(k)+ R˜2(k) is the Fourier transform of the nonlocal response function. Therefore plane wave solutionsare stable if perturbations at all wave numbers k do not grow with propagation. This is the case as long as l isNW3D.4.pdf Advanced Photonics Congress © 2012 OSAFig. 1. (a) Plot of the functions 2R˜(k) and k2=A20 for A0 = 1 and A0 = 2. (b) MI gain curves forA0 = 2, A0 = 4 and A0 = 6. a1 = 2, a2 =  1, s1 = 1 and s2 = 3. (c) Plot of the functions 2R˜(k)and k2=A20 when ja2j> a1, for A0 = 0:5 and A0 = 2; (d) The MI gain curves for A0 = 2, A0 = 4 andA0 = 6. In (c-d) a1 = 1, a2 = 2, s1 = 1 and s2 = 3.purely imaginary. Physically, modulational stability means that small-amplitude waves can propagate along with thebackground intense PW, although their propagation parameter l depends on the PW intensity A20. In terms of MI, theregime ja2j>a1 is particularly interesting, because the stability properties become intensity-dependent. If s2 >s1 lowintensity PWs are modulationally stable, whereas high intensity PWs are unstable. So we may have MI even thoughthe strength of the defocusing nonlinearity exceeds that of the focusing, ja2j > a1. These unique stability propertiesare a pure consequence of the competition between nonlocal nonlinearities and has not been observed before. In mediawith just one self-defocusing (self-focusing) nonlocal nonlinearity, PWs would be stable (unstable), independently ofthe intensity. Since each of the individual response functions, R j, is normalized, R˜(0) = a1+a2. If a1 > ja2j we willtherefore always have 2R˜(k)> k2=A20, and thus l 2 > 0, in a certain band symmetrically centered about the origin, fork sufficiently small. So independent of the details in the behavior of the symmetric response function, we always havelong wave MI in this regime [Fig.1(a,b)]. When ja2j> a1, the stability properties depend on the exact behavior of theresponse function. If R˜(k)  0 for all k 2 R, PW solutions are modulationally stable. On the other hand, if R˜(k) > 0for k 2 I  R, PWs will be unstable if the intensity A20 is sufficiently high [see Fig.1(c,d)].Fig. 2. (a,b) Evolution of the ”dark beams” from the initial amplitude distribution u(x;z)=A tanh(x x0) tanh(x+x0). (a) Unstable background (A0 = 0:85), breakup of the initial structure and formationof bright solitons. (b) Stable background (A0 = 0:75) and formation of two dark solitons. (c-d)Coexistence of bright and dark solitons for a1 = 1, a2 =  2, s1 = 1 and s2 = 3. (c) dynamics ofsoliton propagation (d) stable intensity profile. The inset depicts details of dark solitons.As the stability properties of the plane wave depends crucially on the intensity one can envisage a situation whenboth bright and dark solitons may coexist on the same background as long as its intensity is below the MI threshold.Indeed such situation is illustrated in Fig.2(d) where we show the evolution of two dark and one bright solitons.References1. A. Snyder and J. Mitchell, Science 276, 1538 (1997).2. W. Krolikowski, O. Bang, N. I. Nikolov, D. Neshev, J. Wyller, J. J. Rasmussen, and D. Edmundson, J. Opt. B:Quantum Semiclass. Opt. 6, S288–S294 (2004).3. D. Mihalache, D. Mazilu, F. Lederer, L.C. Crasovan, Y.V. Kartashov, L. Torner, B.A. Malomed, Phys. Rev. E74, 066614 (2006); J. Cuevas, B.A. Malomed, P.G. Kevrekidis, D.J. Frantzeskakis, Phys. Rev. A 79, 053608(2009).4. Y. Du, Z. Zhou, H. Tian and D. Liu, J. Opt. 13, 015201 (2010).

We investigate analytically and numerically propagation and spatial localization of light in nonlocal media with competing nonlinearities. In particular, we discuss conditions for the modulational instability of plane waves and formation of spatial solitons. We show that the competing focusing and defocusing nonlinearities enable coexistence of dark or bright spatial solitons in the same medium by varying the intensity of the beam

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Modulational instability and solitons in nonlocal media with competing nonlinearities

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