Different structural changes give sharply different outcomes around an optical bound state in the continuum, or BIC. In the model, a lossless symmetry-preserving perturbation retains a nearby BIC. The reported lossless symmetry-breaking and dissipative cases instead leave the nearby local maximum of the dispersion relation's imaginary part below the real axis, with no nearby real zero remaining. Under gain, two real zeros are classified as LTMs; under PT symmetry, one is a BIC and the other an LTM.
The preprint follows two features of the complex dispersion relation: real points where its imaginary part is zero, and nearby highs or lows, called extreme points. It asks how many such features appear, where they are located, and how their leading-order locations scale with the perturbation amplitude. Those are the local markers used to compare the modeled BIC cases.
Following the dispersion relation
The setting is a two-dimensional, lossless periodic structure that supports a BIC. A structural perturbation is represented by adding a profile to the unperturbed dielectric function and multiplying that profile by an amplitude. The unperturbed structure supplies the reference, while the analysis compares lossless symmetry-preserving and symmetry-breaking cases with dissipative, gain and PT-symmetric profiles.
To make its predictions, the authors use local Taylor expansions and Puiseux series. These calculations determine the number and real locations of zeros and extreme points, together with their leading-order scaling as the perturbation amplitude changes. Numerical dispersion-curve checks in cylinder-array examples reproduce the predicted BIC and LTM patterns.
A real zero is not automatically a BIC
Classification is a separate step. The paper distinguishes a BIC from an LTM by examining whether the associated wave-field coefficients decay at infinity. If a non-decaying component is present, the zero is assigned to the LTM category. That is why two real zeros in the PT-symmetric case can have different identities: one is a BIC and the other an LTM.
The perturbation classes diverge
For a generic propagating BIC, a nearby local maximum persists across the perturbation classes considered. The lossless symmetry-preserving case retains a BIC. In the reported lossless symmetry-breaking and dissipative cases, the maximum lies below the real axis and no nearby real zero remains. The local maximum therefore survives in the calculations even when the associated BIC zero does not.
The gain case has a different pattern in the same generic setting. It has two simple real zeros, and the paper classifies both as LTMs. PT symmetry also has two zeros, but one is classified as a BIC and the other as an LTM. The conjugate LTMs are described as CPA states. The contrast is therefore not the number of zeros, but the classification attached to their associated fields.
A numerical check
A related result appears for generic antisymmetric standing waves. Reciprocity fixes the local maximum at zero Bloch wave number. With a dissipative or gain profile that is even in the coordinate y, the analysis calls this point a cBIC. Here the maximum stays at a fixed value, unlike the nearby maximum tracked in the generic propagating case.
The numerical check used a periodic cylinder-array example. The generic BIC was at approximately for the Bloch wave number and for the free-space wave number. Here, the first starred quantity is the Bloch wave number and the second is the free-space wave number; L is the structure's period, and π is the circle constant. The factor in parentheses sets the scale. At a gain perturbation amplitude of 0.0001, the two zero locations were approximately 0.2333 and 0.2488 in the same units, and both were classified as LTMs. The PT example showed a BIC and an LTM, with complete absorption at the corresponding CPA frequency.
Taken together, the calculations draw a line between two questions: where a real zero appears and how its associated field coefficients behave at infinity for BIC or LTM classification. The analytical map is checked against numerical dispersion patterns in the modeled structures, within the cases examined. The calculations organize the outcome around perturbation type, reciprocity and field decay.
Paper data and sources
Original title: Structural perturbation theory for bound states in the continuum via bifurcation of zeros and extrema of the dispersion relation
Authors: Lijun Yuan, Ya Yan Lu
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text