Scattering modeling by a nonlinear slab: exact solution of the full vector problem

TL;DR

This work develops a fully vectorial Maxwell framework for nonlinear, anisotropic slab scattering under conical incidence, incorporating nonlinear susceptibility tensors $\chi^{(n)}$ without pump-depletion or scalar-field simplifications. A tensorial finite-element method with Picard iterations solves the coupled harmonic equations, exploiting $y$-periodicity to reduce the problem to 1D and using a virtual-antenna excitation with exact outgoing boundaries. The method is demonstrated on KTP and LiNbO3 configurations, revealing polarization- and rotation-dependent second- and third-harmonic generation, energy transfer among harmonics, and phase-mismatch–driven oscillations, all validated by energy-conservation checks and convergence studies. The results provide a general, practical tool for designing and interpreting nonlinear optical experiments and devices, capable of handling arbitrary incidence, polarization, and crystal orientation beyond conventional approximations.

Abstract

We investigate the scattering of light by a nonlinear, anisotropic slab under conical incidence and arbitrary polarization, within the framework of Maxwell's equations, where the nonlinearities are described by nonlinear susceptibility tensors. We develop a fully tensorial numerical method, free from standard simplifications such as the undepleted pump approximation or scalar field assumptions, based on an iterative scheme where each step is solved via the finite element method. The two-dimensional problem is reduced to one dimension by exploiting symmetry arguments. Energy considerations are also addressed. Several numerical experiments involving a potassium titanyl phosphate (KTP) slab and a lithium niobate (LiNbO3) photonic crystal are presented, including cases with incident TE and TM waves, as well as a rotation-based study highlighting the anisotropic capabilities of our numerical model. This work provides a practical and general tool to help the optics research community overcome the limitations of existing models. It may facilitate the design of more advanced experiments to test nonlinear optical theories or improve nonlinear devices.

Figures (7)

• Figure 1: 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 2: Nonlinear scattering from a KTP slab under TE- and TM-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 $A_0 = 1,5\times10^{10} V/m$ and an incidence angle $\theta=\pi/4 \ rad$. The two solid black vertical bars indicate the slab interfaces.
• Figure 3: Reflection ($R_1$, $R_2$) and transmission ($T_1$, $T_2$) coefficients of the slab at $\omega_I$ and $2\omega_I$, as well as normalized losses $(Q)$, are shown as a function of the crystal rotation angle $\phi$ around the $x$-axis (a) and of the polarization angle $\gamma$ around the incident wave vector $\mathbf{k}_{inc}$ (b). The simulations were carried out for a plane wave with an amplitude $A_0 = 10^{10} V/m$ and an incidence angle $\theta=\pi/4$, using a KTP slab of thickness $2 \, \mathrm{\mu m}$. Energy conservation is verified within $6 \times10^{-7}\%$ for all the studied angles.
• Figure 4: Nonlinear scattering from a LiNbO3 slab under TM-polarized illumination. The real parts of the fundamental field $\mathbf{E}_1$, the second-harmonic $\mathbf{E}_2$ and the third harmonic $\mathbf{E}_3$ are plotted along the $x$-axis ($y$$=$$0$), for an incident plane wave from the left with an amplitude $A_0 = 2\times10^{9} V/m$ and an incidence angle $\theta=\pi/4 \ rad$.
• Figure 5: Transmission coefficients $T_2$ and $T_3$ as a function of the photonic crystal length. The experiment was carried out for three different configurations: TE polarization including only 2HG and; TE and TM polarization including second- and third-order nonlinear effects. The dotted vertical bars indicate the positions of the layer interfaces.