Photonic band structure by direct second-kind free-space boundary integral equations
Abstract
We present a direct boundary integral equation (BIE) formulation of the Bloch eigenvalue problem for two-dimensional photonic crystals that uses only the free-space Helmholtz Green function. The unknowns are the Cauchy data of the Bloch mode on the inclusion boundary and on two cell walls. On the inclusion boundary, the transmission conditions are imposed via Müller's formulation. The wall equations, in turn, come from a stack formulation, in which Green's formula is applied on stacks of three cells, so that the wall self-interactions and the interactions between arcs sharing endpoints cancel exactly. The resulting system is of the second kind, Fredholm of index zero, and free of corner singularities, and the walls may be curved to keep the cell boundary away from the inclusions. Its operators are assembled once per frequency and reused for all Bloch vectors. We prove that every Bloch mode yields a nonzero null vector, and that a null vector comes from a Bloch mode exactly when its cell representation vanishes outside the cell, which gives an inexpensive test for spurious null vectors. The band frequencies are computed with a high-order Nyström discretization and the search of Bruno, Santana and Trefethen, which locates them as poles of adaptive Antoulas--Anderson (AAA) rational approximants to a randomly scalarized resolvent sampled on the real frequency axis. Numerical examples validate the method against exact and published band structures and include a challenging triangular lattice of air pores in a dielectric, whose cell requires curved walls.
Related
- Existence and multiplicity results for third-order ordinary differential equations with derivative dependence subject to the Riemann–Stieltjes integral boundary conditions
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- A Coupled System of p-Laplacian Langevin Equations with ψ-Hilfer Fractional Derivatives and Lebesgue–Stieltjes Integral Boundary Conditions in Weighted Banach Spaces
BibTeX
@misc{perezarancibia2026photonic,
title = {{Photonic band structure by direct second-kind free-space boundary integral equations}},
author = {Carlos Pérez-Arancibia and Tao Yin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28494},
eprint = {2610.28494},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28494},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Photonic band structure by direct second-kind free-space boundary integral equations
AU - Carlos Pérez-Arancibia
AU - Tao Yin
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28494
UR - https://arxiv.org/abs/2610.28494
AB - We present a direct boundary integral equation (BIE) formulation of the Bloch eigenvalue problem for two-dimensional photonic crystals that uses only the free-space Helmholtz Green function. The unknowns are the Cauchy data of the Bloch mode on the inclusion boundary and on two cell walls. On the inclusion boundary, the transmission conditions are imposed via Müller's formulation. The wall equations, in turn, come from a stack formulation, in which Green's formula is applied on stacks of three cells, so that the wall self-interactions and the interactions between arcs sharing endpoints cancel exactly. The resulting system is of the second kind, Fredholm of index zero, and free of corner singularities, and the walls may be curved to keep the cell boundary away from the inclusions. Its operators are assembled once per frequency and reused for all Bloch vectors. We prove that every Bloch mode yields a nonzero null vector, and that a null vector comes from a Bloch mode exactly when its cell representation vanishes outside the cell, which gives an inexpensive test for spurious null vectors. The band frequencies are computed with a high-order Nyström discretization and the search of Bruno, Santana and Trefethen, which locates them as poles of adaptive Antoulas--Anderson (AAA) rational approximants to a randomly scalarized resolvent sampled on the real frequency axis. Numerical examples validate the method against exact and published band structures and include a challenging triangular lattice of air pores in a dielectric, whose cell requires curved walls.
ER -