Source code and simulation results: Efficient rational approximation of optical response functions with the AAA algorithm

This publication provides data published in the article "Efficient rational approximation of optical response functions with the AAA algorithm" [1] in tabulated form along with the Matlab scripts that have been used to produce them. These scripts interface the finite element method solver JCMsuite [2,3]. The article presents rational approximations of optical response functions based on an extended version of the AAA algorithm [4] that allows to efficiently reconstruct sensitivty spectra and gives access to sensitivities of poles, residues, and zeros. Furthermore, the rational approximation of a scalar observalbe is used to construct solutions of the source free Maxwell's equation, i.e., a nonlinear eigenvalue problem. 

The physical Structure

The example is based on the chiral metasurface introduced in [5]. For the sake of simplicity we added infinite layers of SiO\(_2\) to the top and the bottom of the structure. The original structure has a SiO\(_2\) substrate and a layer of PMMA polymethyl methacrylate (PMMA)  deposited on top. PMMA can be modelled with the same refractive index of 1.45 as SiO\(_2\). Furthermore, our simulations include the 13 nm indium tin oxide (ITO) coating which drastically reduces the Q-factor as it is slightly absorbing. The accuracy of the discrete model is verified by assessing reflection, transmission, and absorption at 241 evenly spaced points within the specified range. Energy conservation requires that the discrepancy between their sum and the energy entering the system is zero. The numerical discretization is chosen such that the maximum relative error is less than \(3\times10^{−5}\).

Dispersion

Tabulated data for ITO has been taken from the refractiveindex.info database (T. A. F. König et al., 2014, https://doi.org/10.1021/nn501601e) and the data for TiO2 was kindly provided the authors of [5]. The permittivity \(\varepsilon = (n+ik)^2\) is locally approximated as a rational function, i.e., only data in a vicinity of the frequency range of interest is considered. As we aim for a function with the symmetry \(f^\ast(\omega) = f(-\omega^\ast)\) we add the complex conjugated data at negative frequencies and enforce the symmetry in a second step. The partial fraction decomposition of the required function is of the form: \(\varepsilon(\omega) = \varepsilon_\infty + \sum_{j=1}^{4}a_j/(\omega-\omega_j) - a_j^\ast/(\omega+\omega_j^\ast)\) with the residues \(a_j\) and the poles \(\omega_j\). We expect 4 pairs of poles to sufficiently approximate the data within the range of interest (4 with positive and 4 with negative real parts).

Requirements

• JCMsuite (at least 6.2.0)
• MATLAB (tested with version R2023b)

In order to run the simulations with JCMsuite you must replace corresponding place holders with a path to your installation of JCMsuite. Free trial licenses are available, please refer to the homepage of JCMwave.

Usage

With the content of 'spectra.zip' you can reproduce results presented in the paper. Running the script 'plots.m' will not start any expensive simulation but use the provided data. With 'dispersion.m' the fits to the material data can be reproduced. Additionally, tabulated data is contained in 'data/ascii'. The archive 'eigenmodes.zip' must be extracted in the same directory as 'spectra.zip'.

References

[1] Fridtjof Betz, Martin Hammerschmidt, Lin Zschiedrich, Sven Burger, Felix Binkowski: Efficient rational approximation of optical response functions
with the AAA algorithm, https://doi.org/10.48550/arXiv.2403.19404.

[2] Jan Pomplun, Sven Burger, Lin Zschiedrich, Frank Schmidt, Adaptive finite element method for simulation of optical nano structures, Physica Status Solidi B 244, 3419 (2007), http://dx.doi.org/10.1002/pssb.200743192.

[3] Fridtjof Betz, Felix Binkowski, Sven Burger, RPExpand: Software for Riesz projection expansion of resonance phenomena, SoftwareX 15, 100763 (2021), https://doi.org/10.1016/j.softx.2021.100763.

[4] Y. Nakatsukasa, O. Sète, and L. N. Trefethen, The AAA Algorithm for Rational Approximation, SIAM Journal on Scientific Computing 40, A1494 (2018), http://dx.doi.org/10.1137/16M1106122.

[5]  X. Zhang, Y. Liu, J. Han, Y. Kivshar, and Q. Song, Chiral emission from resonant metasurfaces, Science 377, 1215 (2022), http://dx.doi.org/%2010.1126/science.abq7870.