General method for solving nonlinear optical scattering problems using fix point iterations

Abstract

In this paper we introduce a new fix point iteration scheme for solving nonlinear electromagnetic scattering problems. The method is based on a spectral formulation of Maxwell's equations called the Bidirectional Pulse Propagation Equations. The scheme can be applied to a wide array of slab-like geometries, and for arbitrary material responses. We derive the scheme and investigated how it performs with respect to convergence and accuracy by applying it to the case of light scattering from a simple slab whose nonlinear material response is a sum a very fast electronic vibrational response, and a much slower molecular vibrational response.

Figures (23)

• Figure 1: In this figure we display the geometry of the slab, the placement of the sources, and the material polarization response functions in each of the three regions defining the slab.
• Figure 2: The nonlinear scattering problem displayed using maps and spectral amplitudes defined in the text
• Figure 3: The amplitudes of Raman and electronic spectral response functions, $\hat{\chi}_R(\omega)$, and $\hat{\chi}_K(\omega)$, together with norm of spectral amplitude of the left source $A_+(a,\omega)$, scaled for convenience by factor of $0.4$. Because of translational invariance we assume, without loss of generality, that $a=0$.
• Figure 4: Logarithmic amplitude spectrum for the reflected field of a slab of length 50 wavelengths. The shape of the spectrum for iteration number 1, 15, and 30, for the fix point iterations of the map (\ref{['FixpointIteration']}), are included as solid blue, dashed yellow and dashed black curves. Zero padding $\omega_m=18$
• Figure 5: This is the spectrum from Figure \ref{['fig3']} on the restricted interval $0<\omega<1.8$. Zero padding is $\omega_m=18$