A high-order numerical method for the nonlinear Helmholtz equation in multidimensional layered media

Abstract

a b s t r a c tWe present a novel computational methodology for solving the scalar nonlinear Helmholtz equation (NLH) that governs the propagation of laser light in Kerr dielectrics.The methodology addresses two well-known challenges in nonlinear optics: Singular behavior of solutions when the scattering in the medium is assumed predominantly forward (paraxial regime), and the presence of discontinuities in the optical properties of the medium. Specifically, we consider a slab of nonlinear material which may be grated in the direction of propagation and which is immersed in a linear medium as a whole. The key components of the methodology are a semi-compact high-order finite-difference scheme that maintains accuracy across the discontinuities and enables sub-wavelength resolution on large domains at a tolerable cost, a nonlocal two-way artificial boundary condition (ABC) that simultaneously facilitates the reflectionless propagation of the outgoing waves and forward propagation of the given incoming waves, and a nonlinear solver based on Newton's method.The proposed methodology combines and substantially extends the capabilities of our previous techniques built for 1D and for multi-D. It facilitates a direct numerical study of nonparaxial propagation and goes well beyond the approaches in the literature based on the ''augmented" paraxial models. In particular, it provides the first ever evidence that the singularity of the solution indeed disappears in the scalar NLH model that includes the nonparaxial effects. It also enables simulation of the wavelength-width spatial solitons, as well as of the counter-propagating solitons.

Efficient numerical method for analyzing optical bistability in photonic crystal microcavities

“…For two-dimensional (2D) structures, the nonlinear interactions between light and Kerr media can be modeled by a nonlinear Helmholtz equation. In general, the nonlinear Helmholtz equation can be discretized by a standard numerical method such as the finite difference or finite element method, and solved by an iterative method, such as Newton's method or the nested iteration scheme [14,15]. However, this approach is not very popular, since the iterative methods often converge very slowly or fail to converge for interesting problems with strong nonlinear effects and unstable solutions.…”

Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium

“…In nonlinear optics, one may often focus on the propagation of monochromatic waves, such as continuous high intensity laser beams. In this case, some reasonable assumptions (see [3]) simplify the Maxwell's models to a nonlinear Helmholtz (NLH) equation [5,13]…”

“…When the Kerr medium is inhomogeneous which means the whole Kerr medium will be cut into pieces by some other mediums, the linear medium for example. In this case, the discontinuities of the coefficients v and ε in equation 2will be more complex, we refer [2,3] for the detail.…”

“…To solve (3) in the interval [0, Z max ], some boundary conditions are needed. It is well-known that, at the interfaces z = 0 and z = Z max , the electronic field E and its first derivative are continuous [3],…”

“…According to this fact, the so-called two-way boundary conditions which are also used in [3,14,15] are as follows…”