A generalized perturbative approach for the computation of nonlinear scattering problems

TL;DR

This work introduces a generalized perturbative framework for nonlinear optical scattering that avoids the computational burden of fully iterative, coupled nonlinear solves. By expanding the fields in a perturbative series across harmonics and exploiting a cascading, triangular system, it decouples the problem into sequential linear solves while maintaining accuracy through higher-order terms. The method is demonstrated on KTP and LiNbO3 structures, focusing on second-harmonic generation and coupled second- and third-order effects, with quantitative benchmarks showing substantial reductions in computation time and good agreement with rigorous solutions. The approach promises significant efficiency gains for high-dimensional nonlinear scattering problems and optimization tasks, while also enabling analytical insights into nonlinear interactions in complex media.

Abstract

We present a perturbative technique for modeling the scattering of light by a nonlinear material. This approach eliminates the need for an iterative algorithm to solve the fully coupled nonlinear problem. We demonstrate its effectiveness in the cases of a nonlinear anisotropic slab and a nonlinear periodic crystal, both illuminated by a plane wave under conical incidence and arbitrary polarization. Quantitative comparisons of the accuracy and computational time with a previously published rigorous model are provided.

Figures (8)

• Figure 1: Illustration of the cascading triangular system. The indices $j$ and $p$ denote the perturbative order and the harmonic number, respectively. Each pointed field $\mathbf{E}_p^{(j)}$ depends only on the connected elements.
• Figure 2: Schematic view of a nonlinear slab illuminated by a plane wave with its wave vector $\mathbf{k}_{inc}$ on its left side. $\mathbf{k}_{inc}$ is within the $(x,y)$ plane.
• Figure 3: Nonlinear scattering from a KTP slab under TE-polarized illumination. The real parts of the fundamental field $\mathbf{E}_1$ and the second-harmonic $\mathbf{E}_2$ are plotted along the $x$-axis ($y=0$), for an incident plane wave from the left with an amplitude $A_0=6\times10^{9}V/m$ and an incidence angle $\theta=30^{\circ}$. The three curves are associated to the rigorous method, the second-order perturbative method, and the fourth-order perturbative method, respectively.
• Figure 4: Transmission coefficient $T_2$ of the second harmonic as a function of the photonic crystal length, computed using both the rigorous and perturbative methods. The simulations were performed for a TE-polarized plane wave at normal incidence. The dotted vertical bars indicate the positions of the layer interfaces.
• Figure 5: Relative errors between the complex field amplitudes obtained with the rigorous and perturbative approaches as a function of the incident-wave amplitude, for various expansion orders, plotted on a logarithmic scale.

Theorems & Definitions (3)

• Proposition 1
• Proposition 2
• Proposition 3