Source code and simulation results: Poles and zeros of electromagnetic quantities in photonic systems

Summary

This publication supplements the article "Poles and zeros of electromagnetic quantities in photonic systems" with tabulated data and matlab code that allows to reproduce the results. The article elaborates how evaluating resonances based on contour integrals of scalar electromagnetic quantities extends to computing zeros. Furthermore, direct differentiation of underlying scattering problems is used to compute sensitivities with respect to design parameters.

Structure

The script 'main_text.m' can be used to reproduce the results provided in the paper. In tabulated form the results are contained in the directory tabulated. Furthermore, the script 'supplement.m'  can be used to reproduce results presented in the supplement. The directory RPExpand contains the software RPExpand v2, which is available on Zenodo with additional examples. 

Compute residues

The modal expansion of the Fourier transform is based on its residues at the dominant resonances. If the poles are simple, which often is the case, the residues can be obtained directly from the eigenvectors of the generalized eigenvalue problem used to obtain the poles or the zeros. Introducing the Vandermonde matrix

\(V = \begin{bmatrix} 1 & \dots & 1 \\ w_1 & \dots & w_M \\ \vdots & & \vdots \\ w_1^{M-1} &\dots & w_M^{M-1} \end{bmatrix}\),

the Hankel matrix \(H\) can be written as \(H = V A V^T\) with \(A\) being the diagonal matrix \(\mathrm{diag}(a_1,\dots,a_M)\) containing the residues \(a_m \). This decomposition is a consequence of the Cauchy's reisdue theorem if the poles are simple. Furthermore, we now that \(V^{-T}\) solves the generalized eigenproblem \(H^