General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics

Abstract

The auxiliary differential equation finite-difference time-domain method for modeling electromagnetic wave propagation in dispersive nonlinear materials is applied to problems where the electric field is not constrained to a single vector component. A full-vector Maxwell's equations solution incorporating multiple-pole linear Lorentz, nonlinear Kerr, and nonlinear Raman polarizations is presented. The application is illustrated by modeling a spatial soliton having two orthogonal electric field components. To the best of our knowledge, the numerical technique presented here is the first to model electromagnetic wave propagation with two or three orthogonal vector components in dispersive nonlinear materials. This technique offers the possibility of modeling sub-wavelength interactions of vector spatial solitons.

FDTD Computational Study of Ultra-Narrow TM Non-Paraxial Spatial Soliton Interactions

“…Some limitations, however, are that it makes a scalar approximation, relies on paraxiality, and also depends on slowly-varying envelope conditions for validity without proper modifications Recently, a new FDTD algorithm was described, which can accommodate more than one electric-field component in media possessing both instantaneous and dispersive nonlinearities, as well as linear material dispersion. Known as the general vector auxiliary differential equation (GVADE) method [15], it has been applied to the study of soliton interactions with nanoscale air gaps embedded in glass [16]. Ultra-narrow solitons involve significant interactions between both longitudinal and transverse electric-field components [14] and the GVADE method accounts for this physics.…”

Dynamic Characteristics of Linear and Nonlinear Wideband Photonic Crystal Filters

“…Particularly, in case of non-saturable Kerr nonlinearity polarization current density is given in following form [4]:…”

“…Applying the FDTD technique expanded with auxiliary differential equation for the nonlinear medium [4] with polarization current given in form of (5) and assuming perfectly-matched layer [1] at the boundary of computation region, we can compute time-dependent electromagnetic field distribution in nonlinear saturable media.…”

Scattering of Spatial Optical Solitons by Subwavelength Air Holes

“…Reformulated ADE FDTD methods have been reported which eliminate the need to solve a system of equations at each time-step, where is the number of poles of the chromatic dispersion [10], [11]. In [9], we reported in detail the GVADE FDTD method, which extends this technique to nonlinear optics problems where the electric field has two or three orthogonal vector components and validated the GVADE FDTD method for the case of temporal soliton propagation. This was an advance over previous FDTD Maxwell's equations models of optical solitons [12], which included only the nonlinear Kerr polarization, and described only a single electric field vector component.…”

“…Furthermore, there are no restrictions on sharp interfaces and the interaction of beams with arbitrary geometries can be modeled. We recently reported the general vector auxiliary differential equation (GVADE) formulation of FDTD [9], a general method which we believe to be the first technique to model electromagnetic wave propagation with two or three orthogonal electric field vector components in dispersive nonlinear materials, that can be applied to any geometry.…”