Conservative finite-difference scheme for the problem of propagation of a femtosecond pulse in a nonlinear photonic crystal with nonreflecting boundary conditions

2015

Opt. Spectrosc.

Self Cite

By means of computer simulation, we demonstrate the possibility of formation of a new type of surface solitons in nonlinear layered structures. The initial soliton is located inside a photonic crystal and occupies several of its layers. Its profile can be found from the solution to the eigenvalue problem for a nonlinear Schrödinger equation with periodic linear and nonlinear coefficients. Surface solitons are formed as a result of perturbation of the wave vector, which leads to soliton motion across photonic crystal layers. After several reflections from the side boundaries of the crystal, under certain conditions, the soliton becomes attached to the side boundary. A characteristic feature of the discussed soliton is the fact that its penetration depth into the photonic crystal medium is equal to the thickness of several crystal layers.Among the different problems related to lasermatter interaction, the formation of solitons and localization of light inside photonic crystals (PC) attracts special attention [1][2][3][4][5][6][7][8][9][10][11][12][13], partially due to the possibility of their wide use in information technologies. Propagation of a soliton along PC layers is of special interest [4][5][6][7]. In the present work, based on computer modeling, we show the possibility of forming an oscillating soliton at the PC interface with the surrounding medium. It is important that only part of such a soliton is located inside the PC. The other part is located outside of the PC. Thus, one can speak about localization of energy of light in the near-surface layer at the PC boundary, i.e., about the surface soliton.We stress that the discussed solitons and those studied in [8-13] differ mainly in their transverse dimensions: the solitons under consideration penetrate into the surrounding medium by a distance equal to several PC layers and occupy several PC layers. It is important that these solitons can be created at relatively low laser pulse intensities.Let us analyze propagation of a soliton occupying, e.g., more than ten layers along a PC under the conditions of perturbation of its direction of propagation due to the wave vector having a transverse component; i.e., the soliton is incident on the layered structure at an angle. A detailed formulation of the problem can be found in [14,15].For simplicity, we describe propagation of the soliton within the framework of the nonlinear Schrödinger equation that takes into account only propagation across the layers. In this case, assuming linear dependence of wave vector k on frequency ω and in the absence of a preferred propagation direction (i.e., complex amplitude A(z, t) is slowly varying only in time), the propagation of the soliton is described by the equation [16] (1)The following notations were used above:(2)Here, is the time coordinate; is the spatial coordinate;and are the laser wavelength and the characteristic wavelength of the layered structure, respectively;is the third-order susceptibility; and are the widths of alternating layers with dielectric permittiviti...

mentioning

confidence: 99%

Formation of an oscillating soliton near a photonic crystal surface

Lysak

2015

Opt. Spectrosc.

Self Cite

“…Let us consider the 1D nonlinear Schrödinger equation which describe the light pulse propagation in 1D Photonic crystal (PC). In this case, the parameters D z , ϕ defined as: and relate to the initial complex amplitude distribution [ 3 ] corresponding to the wave propagation along the z-axis and BCs for the Eq (69) are written in the following way …”

Section: The Finite-difference Scheme Based On the Rosenbrock Methods mentioning

confidence: 99%

“…In the past decade, development of the artificial boundary conditions (ABCs) is the direction for increasing the computer simulation effectiveness. With respect to the Schrödinger equation the ABCs were proposed, for example, in [ 3 – 5 ] [ 32 – 39 ]. As a rule, these boundary conditions (BCs) were realized for the finite-difference schemes based on the split-step method in contrast to the CFDS used by us.…”

Section: Introductionmentioning

confidence: 99%

Implicit finite-difference schemes, based on the Rosenbrock method, for nonlinear Schrödinger equation with artificial boundary conditions

Trykin

2018

PLoS ONE

We investigate the effectiveness of using the Rosenbrock method for numerical solution of 1D nonlinear Schrödinger equation (or the set of equations) with artificial boundary conditions (ABCs). We compare the computer simulation results obtained during long time interval at using the finite-difference scheme based on the Rosenbrock method and at using the conservative finite-difference scheme. We show, that the finite-difference scheme based on the Rosenbrock method is conditionally conservative one. To combine the advantages of both numerical methods, we propose new implicit and conditionally conservative combined method based on using both the conservative finite-difference scheme and conditionally conservative Rosenbrock method and investigate its effectiveness. The combined method allows decreasing the computer simulation time in comparison with the corresponding computer simulation time at using the Rosenbrock method. In practice, the combined method is effective at computation during short time interval, which does not require an asymptotic stability property for the finite-difference scheme. We generalize also the combined method with ABCs for 2D case.

“…These conditions look like A| z=L1 = 0, A| z=L2 = 0, here L 1 , L 2 are coordinates of domain, which either coincide with the domain of photonic crystal or are located inside the photonic crystal if we want to find the solution that is located in some layer of PC. For problem (2.2), (2.4) two invariants (conservation laws) [31] take place:…”

Section: Statement Of Problem For Laser Pulse Propagation In Pcmentioning

confidence: 99%

“…It does not equal to zero for inclined incidence of laser beam with respect of direction along which the layers are placed. An investigation of a stability for the laser pulse propagation with respect to transverse perturbation of propagation direction is reasonability made by the usage of so-called nonreflecting (transparent) boundary conditions [30,31]:…”

Section: Statement Of Problem For Laser Pulse Propagation In Pcmentioning

confidence: 99%

Parameter Control of Optical Soliton in One‐dimensional Photonic Crystal

Mathematical Modelling and Analysis

Lysak

Matusevich

et al. 2010

Abstract. The paper deals with finding of soliton solution for Schrödinger equation with periodic linear and nonlinear properties of medium in 1D case. Such structure is named as photonic crystal. To find soliton solution the corresponding problem for finding of eigenfunctions and eigenvalues is formulated. Iterative process is proposed for solution of this problem. Using the technique of continuation on parameter we investigate a dependence of soliton location on its maximum intensity, on ratio between light frequency and frequency of structure, on ratio between dielectric permittivity of alternating linear and nonlinear layers and on position between centre of initial distribution of eigenfunction and center of considered photonic structure area. The results of this paper confirm the features of soliton self-formation investigated early in our papers [37,38,39,40,41,42], in which one considered a propagation of femtosecond laser pulse through nonlinear layered structure.