1010 ISSN 0030-400X, Optics and Spectroscopy, 2015, Vol. 119, No. 6, pp. 1010–1014. © Pleiades Publishing, Ltd., 2015. Original Russian Text © T.M. Lysak, V.A. Trofimov, 2015, published in Optika i Spektroskopiya, 2015, Vol. 119, No. 6, pp. 1007–1011. “Wandering” Soliton in a Nonlinear Photonic Crystal 1 T. M. Lysak and V. A. Trofimov Lomonosov Moscow State University, Moscow, 119991 Russia e-mail: vatro@cs.msu.ru Received April 15, 2015 Abstract—On the basis of computer simulation, we demonstrate the possibility of a new type of “wandering” solitons implementation in nonlinear periodic layered structures. “Wandering” soliton moves across the lay- ers, repeatedly changing its direction of motion due to the reflection from the photonic crystal (PC) bound- aries with the ambient medium. The initial soliton is located inside a PC and occupies several of its layers. Its profile can be found as the solution of the corresponding nonlinear eigenvalue problem. “Wandering” soli- tons are formed as a result of a large perturbation of the wave vector, which leads to the soliton motion across photonic crystal layers. In the process of ref lection from the boundary with the ambient medium, the soliton partly penetrates into the ambient medium at a depth equal to the width of several PC layers. A slow return of light energy, which previously left the PC, can take place at this moment. DOI: 10.1134/S0030400X15120188 Among the different problems related to laser-mat- ter interaction, soliton formation and light localization inside photonic crystals (PC) attract great attention [1–15], partially due to the possibility of their wide application in information technologies. In our recent paper [16] the possibility of formation of a new type of oscillating soliton at the boundary of PC with the ambient medium was demonstrated on basis of com- puter simulation for sufficiently small intensities of laser radiation. Unlike the solitons, investigated in [10–15], the discovered soliton penetrates into the ambient medium at the depth of several PC layers and occupies several PC layers. Note that in the process of its formation, there was a single reflection of the soli- ton from the PC boundary with the ambient medium, opposite to the boundary of the surface soliton forma- tion. Continuing this research, we show the formation of the “wandering” soliton repeatedly reflecting from the PC boundary with the ambient medium. At the time of propagation direction changing, the soliton partly leaves the PC at a depth equal to the width of several PC layers. We consider propagation of a soliton occupying, e.g., more than ten layers, along the layers of PC at a perturbation of the propagation direction due to the presence of transverse component of wave vector. This means that the soliton is incident on the layered struc- ture at an angle. A detailed formulation of the problem can be found in [17, 18]. For simplicity, we describe the soliton propagation within the framework of the nonlinear Schrödinger equation that takes into account only propagation of the light pulse across the layers. In this case, assuming linear dependence of wave vector on frequency and in the absence of a preferred propagation direction (i.e., complex ampli- tude is a slowly varying amplitude just in time), the propagation of the soliton is described by the equation [19] (1) The following notations were used above: , , , (2) , , , . Here, is the time coordinate; is the spatial coordi- nate; and are the laser wavelength and the char- acteristic wavelength of the layered structure, respec- tively; is the third-order susceptibility; and are the widths of alternating layers with dielectric per- mittivities and , respectively; and and are, respectively, the dielectric permittivity and the 1 The article was translated by the authors. k ω (,) Azt 2 2 2 ε( ) β ε( ) α( ) 0, 0, 0 . z A A z iD i z z A A t z t z L ∂ ∂ ⎡ ⎤ + + + = ⎣ ⎦ ∂ ∂ > < < 0 str λ 1 ω , β π , 4π ω λ z D =− =−Ω Ω= = Ω (3) 2 0 α( ) 3πχ () z zE = str 0 ω 2π /λ c = 0 1 1 2 2 λ ε ε d d = + 0 λ z z → 0 λ c t t → 0 λ L L → t z λ 0 λ (3) χ 1 d 2 d 1 ε 2 ε ε( ) z α( ) z NONLINEAR AND QUANTUM OPTICS