DOI: 10.5281/zenodo.15880360
title: Source code and simulation results: Uncovering hidden resonances in non-Hermitian systems with scattering thresholds

description: This publication offers the necessary data and scripts to replicate the findings of the article titled "Hidden resonances in non-Hermitian systems with scattering thresholds". Additionally convergence studies are provided. The article aims to offer a new perspective on resonances in the vicinity of scattering thresholds and provide access to hidden modes on different Riemann sheets. 

Usage

All Matlab files can be run without solving scattering problems as the required data is stored in .mat files in the data directory. In order to run the simulations with JCMsuite you must delete the data directory and replace corresponding place holders with a path to your installation of JCMsuite. Free trial licenses are available, please refer to the homepage of JCMwave. 

Requirements



JCMsuite (tested with version 6.4.1)

MATLAB (tested with version R2023b)


FEM convergence

We acquire the snapshots with the finite element method (FEM) solver JCMsuite. To estimate the error, the specular reflection has been collected at 24 equidistantly sampled points within the range of interest and at two additional sampling points on either side of the branch points (for further details we refer to the file convergence.m). The error is defined as \(\mathrm{min}\,\mathrm{abs}\left( R_0^n(\omega)-R_0^8(\omega)\right)\), where the superscript denotes the polynomial order of the FEM basis functions. Furthermore, the energy conservation (incoming energy minus reflection plus absorption) has been investigated. 

All the data for the paper have been generated using \(n=5\). The error at the data points can therefore be expected to be below \(3\times10^{-7}\). 

AAA convergence

The AAA algorithm adaptivly increases the degree \(m\) of the rational approximation until the error of the model with respect to all sample points falls below a given threshold \(t\) as long as \(m\) is smaller than half the number of sample points \(N\). We use \(t = 10^{-6}\) and \(t = 5\times 10^{-7}\) to make sure that it is larger than the error introduced through the FEM discretization. In the file AAAconvergence.m, error and model size are compared for different values of \(t\) and different numbers of support points. It can be observed that the error with respect to more than 500 reference points is smaller by orders of magnitude, while at the same time the size of the model is reduced and saturates quickly if the transformed variable \(\tilde{k}\) is used instead of \(k\). Here, 80 support points suffice for errors below \(10^{-6}\) for a spectrum containing three branch points and more than eight resonances (if hidden resonances are included).

Sampling scheme

We adopt a sampling scheme with additional samples in the vicinity of the branch points. This is achieved with equidistant samplings in the transformed space. For details we refer to the matlab scripts. 
description: FB, FB, and SB acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project 390685689) and by the German Federal Ministry of Education and Research (BMBF Forschungscampus MODAL, project 05M20ZBM). JDF and CR acknowledge financial support by the Helmholtz Association in the framework of the innovation platform “Solar TAP”. The work of NF and AFK is part of the Dutch Research Council (NWO) and was performed at the research institutes AMOLF and ARCNL. ARCNL (Advanced Research Center for Nanolithography) is a public-private partnership between the University of Amsterdam, Vrije Universiteit Amsterdam, University of Groningen, NWO, and the semiconductor equipment manufacturer ASML.