Structural perturbation theory for bound states in the continuum via bifurcation of zeros and extrema of the dispersion relation
Abstract
In a lossless periodic structure, a bound state in the continuum (BIC) corresponds to a real zero and a local maximum of the imaginary part of a complex dispersion relation k = k(β), where β is the Bloch wave number. A perturbation of the structure deforms the dispersion curve and may destroy, move or split the BIC, as demonstrated in existing studies involving lossless symmetry-preserving perturbations, symmetry-breaking perturbations and dissipative perturbations. We present a comprehensive perturbation theory, emphasizing the evolution of the real zeros and extreme points of Im[k(β)] under various structural perturbations. In particular, our theory reveals the existence of lasing threshold modes (LTMs), which are also zeros of Im[k(β)] when the perturbation involves gain with or without balanced loss. Using local Taylor expansions and Puiseux series, we determine the number, locations, and leading-order scaling of real zeros and extreme points for various types of BICs under different types of structural perturbations. The theory recovers known results and predicts new behavior for super-BICs under 𝒫𝒯-symmetric perturbations. Specifically, a propagating super-BIC corresponding to a fourth-order zero of Im[k(β)] splits into two real zeros representing either two BICs or two LTMs, and a symmetric standing wave splits into two BIC-LTM pairs. Our theory provides a general framework for studying BICs and nearby resonant modes in both lossless and non-Hermitian periodic structures.
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BibTeX
@misc{yuan2026structural,
title = {{Structural perturbation theory for bound states in the continuum via bifurcation of zeros and extrema of the dispersion relation}},
author = {Lijun Yuan and Ya Yan Lu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28321},
eprint = {2608.28321},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28321},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Structural perturbation theory for bound states in the continuum via bifurcation of zeros and extrema of the dispersion relation
AU - Lijun Yuan
AU - Ya Yan Lu
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28321
UR - https://arxiv.org/abs/2608.28321
AB - In a lossless periodic structure, a bound state in the continuum (BIC) corresponds to a real zero and a local maximum of the imaginary part of a complex dispersion relation k = k(β), where β is the Bloch wave number. A perturbation of the structure deforms the dispersion curve and may destroy, move or split the BIC, as demonstrated in existing studies involving lossless symmetry-preserving perturbations, symmetry-breaking perturbations and dissipative perturbations. We present a comprehensive perturbation theory, emphasizing the evolution of the real zeros and extreme points of Im[k(β)] under various structural perturbations. In particular, our theory reveals the existence of lasing threshold modes (LTMs), which are also zeros of Im[k(β)] when the perturbation involves gain with or without balanced loss. Using local Taylor expansions and Puiseux series, we determine the number, locations, and leading-order scaling of real zeros and extreme points for various types of BICs under different types of structural perturbations. The theory recovers known results and predicts new behavior for super-BICs under 𝒫𝒯-symmetric perturbations. Specifically, a propagating super-BIC corresponding to a fourth-order zero of Im[k(β)] splits into two real zeros representing either two BICs or two LTMs, and a symmetric standing wave splits into two BIC-LTM pairs. Our theory provides a general framework for studying BICs and nearby resonant modes in both lossless and non-Hermitian periodic structures.
ER -