Modal expansion techniques have long been used as an efﬁcient way to calculate radiation of sources in closed cavities. With one set of cavity modes, calculated once and for all, the solution for any arbitrary conﬁgu-ration of sources can be generated almost instantaneously, providing clear physical insight into the spatial variation of Greens function and thus the local density of states. Nanophotonics research has recently generated an explo-sion of interest in generalizing modal expansion methods to open systems, for example using quasinormal mode / resonant state expansion . Yet one major practical obstacle remains: numerical generation of resonator modes is slow and unreliable, often requiring considerable skill and hand guiding.
Here, we present a practical numerical method for generating suitable modes, possessing the trifecta of traits: speed, accuracy, and reliability. Our method is capable of handling arbitrarily-shaped lossy resonators in open systems. It extends existing methods that expand modes of the target struc-ture using modes of a simpler analytically solvable geometry as a basis . This process is guaranteed to succeed due to completeness, but is ordinarily inefﬁcient because optical structures are usually piecewise uniform, so the resulting ﬁeld discontinuities cripple convergence rates. Our key innovation is use of a new minimal set of basis modes that are inherently discontinuous, yet remarkably simple. We choose to implement our method for the General-ized Normal Mode Expansion (GENOME)  which unlike its alternatives , is valid for any source conﬁguration, including the important case of sources exterior to the scatterer. We achieve rapid exponential convergence, with 4 accurate digits after only 16 basis modes, far more than is necessary. This also means lightning-speed simulation results, faster by 2-3 orders of mag-nitude compared to mode generation using COMSOL. Finally, our method is extremely reliable, as it culminates in a small dense linear eigensystem. No modes go missing, nor are there spurious modes that need to be manually discarded, which is critical to the success of modal expansion methods.
 M. B. Doost et al., Phys. Rev. A 90, 013834 (2014), C. Sauvan et al.,
Phys. Rev. Lett., 110 237401, (2013)
 P. Chen, D. Bergman and Y. Sivan, Phys. Rev. Appl., accepted (2018).
Parry Y. Chen, Egor A. Muljarov, and Yonatan Sivan, "Lightning-fast solution of scattering problems in nanophotonics: an effortless modal approach (Conference Presentation)," Proc. SPIE 11345, Nanophotonics VIII, 113450T (Presented at SPIE Photonics Europe: April 08, 2020; Published: 1 April 2020); https://doi.org/10.1117/12.2569671.6146310453001.
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