References and relation to this project

Primary motivation

1. George Shaker, Beñat Martinez de Aguirre Jokisch, Pengning Chao, and Steven G. Johnson, “Eigenvalue-accelerated LDOS optimization of high-Q optical resonances,” arXiv:2511.16643 (2025), https://arxiv.org/abs/2511.16643.

This supplies the central optimization idea: use ordinary LDOS topology optimization to discover a strong resonance, then recenter subsequent optimization with a shift-invert eigenvalue calculation so an increasingly narrow line does not make fixed-frequency optimization ill-conditioned. This repository now adapts that logic to a 3D defect whose x boundaries are frequency-dependent periodic-lead self-energies. It does not copy the paper's complex-frequency eigensolver: the moving pole and gradient come from retarded real-axis Loewner continuation, and the shifted objective includes both device and homogeneous-normalization frequency derivatives. Constrained MMA, detuning, fixed-atom, overlapping-mode rational, and PML gates are added around the local proposal.

1. Qimin Quan and Marko Loncar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Optics Express 19, 18529 (2011), arXiv:1108.2675, https://arxiv.org/abs/1108.2675.

This establishes the deterministic mirror/defect picture for ultra-high-Q nanobeams, including air-mode cavities and Gaussian-like envelope/Fourier reasoning. It motivates treating the periodic mirror as known physics and concentrating free-form optimization in the finite defect. A Gaussian envelope is a useful outcome/initialization principle, but this solver does not impose a Gaussian basis on the defect field.

Reduced slab and reference-cavity tools

1. Legume 1.0.0, “Guided mode expansion with Autograd,” https://legume.readthedocs.io/en/1.0.0/examples/06_Guided_mode_expansion_with_autograd.html.

Legume demonstrates fast differentiable guided-mode expansion for patterned slabs and radiative-Q evaluation. It remains a promising independent reduced-basis cross-check or future global initializer. The current solver instead uses a full-vector 3D Yee defect so that the large air strip, vertical radiation, and arbitrary free-form topology are represented without choosing guided/non-guided basis functions in advance.

1. MEEP documentation, “Resonant Modes and Transmission in a Waveguide Cavity,” https://meep.readthedocs.io/en/latest/Python_Tutorials/Resonant_Modes_and_Transmission_in_a_Waveguide_Cavity/.

This is a useful reference geometry for a periodic waveguide mirror and central defect. The repository uses its own wide/narrow rectangular nanobeam period, selected by a corrected-domain stop-band sweep, rather than importing the MEEP example verbatim.

Rational and reduced-order continuation

1. Yuji Nakatsukasa, Olivier Sète, and Lloyd N. Trefethen, “The AAA algorithm for rational approximation,” SIAM Journal on Scientific Computing 40, A1494–A1522 (2018), https://doi.org/10.1137/16M1106122.

AAA is used only as a real-axis pole validator because its greedy support selection is not differentiable.

1. Athanasios C. Antoulas, Sanda Lefteriu, and A. C. Ionita, Loewner-framework model reduction; see the foundational realization treatment by Mayo and Antoulas, “A framework for the solution of the generalized realization problem,” Linear Algebra and its Applications 425, 634–662 (2007), https://doi.org/10.1016/j.laa.2007.03.008.

The scalar and SIMO Loewner pencils provide a differentiable local pole surrogate and an independent shared-pole model-order check.

Surface Green functions

1. M. P. López Sancho, J. M. López Sancho, and J. Rubio, iterative surface Green-function matching/cyclic-reduction methods for semi-infinite periodic systems. The implementation uses the corresponding doubling recursion after exactly condensing internal slices of one Maxwell mirror period.

The derivations actually implemented in the code, including sign conventions and Schur blocks, are in DERIVATIONS.md; the references provide context, not a substitute for the tested discrete formulas.