# Revisiting the boundary conditions for second-harmonic generation at metal-dielectric interfaces

## TL;DR

This paper revisits the boundary conditions for second-harmonic generation at metal-dielectric interfaces, clarifying the discontinuity of electric fields, and provides a correction method for numerical simulations.

## Contribution

It introduces a spectral decomposition method that simplifies boundary condition considerations and offers a correction recipe for commercial simulation tools.

## Key findings

- Confirmed the discontinuity of the tangential electric field component at the interface.
- Reaffirmed the standard convention for surface dipole radiation.
- Provided a correction method for COMSOL Multiphysics to accurately model SHG.

## Abstract

We study second-harmonic generation (SHG) arising from surface nonlinearity at a metal-dielectric interface using a spectral decomposition method. Since our method avoids the need to consider the generalized boundary condition across the metal-dielectric interface in the presence of a perpendicular surface source, we retrieve the known discontinuity of the tangential component of the electric field ($E_{\parallel}^ {2\omega}$) for a general geometry, based on a purely mathematical argument. Further, we reaffirm the standard convention of the implementation of this condition, namely, that the surface dipole source radiates as if placed outside the metal surface for arbitrary geometries. We also study and explain the spectral dependence of the discontinuity of the tangential component of the electric field at second harmonic. Finally, we note that the default settings of the commercial numerical package COMSOL Multiphysics fail to account for the $E_{\parallel}^ {2\omega}$-discontinuity. We provide a simple recipe that corrects the boundary condition within these existing settings.