arXiv subjects

Ya Yan Lu

Publications and source records attributed to Ya Yan Lu.

At least 19 recordsLinked to original sources

arXiv2026

Structural perturbation theory for bound states in the continuum via bifurcation of zeros and extrema of the dispersion relation

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 $k = k(\beta)$, where $\beta$ 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 $\mathrm{Im}[k(\beta)]$ under various structural perturbations. In particular, our theory reveals the existence of lasing threshold modes (LTMs), which are also zeros of $\mathrm{Im}[k(\beta)]$ 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 $\mathcal{PT}$-symmetric perturbations. Specifically, a propagating super-BIC corresponding to a fourth-order zero of $\mathrm{Im}[k(\beta)]$ 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.

physics.optics↗

arXiv2026

On the Robustness of Propagating Bound States in the Continuum

Bound states in the continuum (BICs) are localized eigenmodes with their frequencies in the radiation continuum of scattering states. The existence of a BIC implies the loss of uniqueness for scattering problems with given incident waves. Perturbed wave systems close to the ideal ones with a BIC exhibit strong resonance effects that are essential to numerous practical applications. A question of fundamental importance is whether a BIC is robust, i.e., whether it can continue its existence when the structure is slightly perturbed. In an earlier work [Yuan and Lu, Optics Letters, Vol.~42, pp.~4490-4493, 2017], for a class of BICs governed by the two-dimensional (2D) Helmholtz equation, which are not trivially protected by symmetry, we uncovered the conditions that ensure robustness and formally constructed the BIC in perturbed systems using a perturbation method. In this paper, we present a rigorous theory on the robustness of BICs in 2D dielectric structures with a single periodic direction. Specifically, we analyze the solvability and provide estimates for each order in the perturbation series, and prove the convergence of the series.

math.AP↗

arXiv2026

Local Robustness of Bound States in the Continuum through Scattering-Matrix Eigenvector Continuation

We consider the diffraction of time-harmonic plane waves by a periodic structure, governed by the Helmholtz equation. Bound states in the continuum (BICs) are quasi-periodic fields that remain $L^{2}$-bounded over one period and occur at frequencies embedded in the continuous spectrum. Perturbations that break a BIC can lead to ultra-strong resonances, enabling various applications in photonics. Employing the implicit function theorem, we demonstrate how a simple BIC continuously deforms into a propagating field as system parameters vary in a neighborhood, with the frequency adjusting accordingly. In this setting, the incident coefficients of the field persist as an eigenvector of the scattering matrix with a fixed eigenvalue. By introducing a mapping $\mathcal{P}$ from the parameters to these coefficients, the zeros of $\mathcal{P}$ correspond precisely to BICs. When such a zero is isolated and the dimensions of the domain and codomain coincide, the BIC can be related to the mapping degree of $\mathcal{P}$ in a small neighborhood. This perspective clarifies the phase singularity associated with BICs and provides a general topological interpretation of their local robustness with respect to the given parameters. Moreover, it yields a practical numerical criterion for detecting and verifying BICs via computation of the mapping degree of $\mathcal{P}$.

math-ph↗

arXiv2026

An efficient numerical method for simulating two-dimensional non-periodic metasurfaces

Metasurfaces are extremely useful for controlling and manipulating electromagnetic waves. Full-wave numerical simulation is highly desired for their design and optimization, but it is notoriously difficult, even for two-dimensional metasurfaces, when they comprise a huge number of subwavelength elements. This paper focuses on two-dimensional non-periodic metasurfaces that contain only a relatively small number of distinct subwavelength elements. We develop an efficient numerical method based on Neumann-to-Dirichlet operators, the finite element method and local function expansions. Our method drastically reduces the total number of unknowns and is capable of simulating two-dimensional metasurfaces with $10^{5}$ subwavelength elements on a personal computer. Numerical examples demonstrate that the method maintains high accuracy while offering significant advantages in both computational time and memory usage compared to the classical full-domain finite element method, making it particularly suited for the analysis of large metasurfaces.

physics.optics↗

arXiv2025

Asymptotically circularly polarized bound states in the continuum

We study a class of bound states in the continuum (BICs) in all-dielectric periodic structures, near which resonant states approach ideal circularly polarized states (CPSs). We term these BICs {\em asymptotically circularly polarized BICs} ({\em acp}-BICs) and identify two types: single-angle and all-angle. Single-angle {\em acp}-BICs permit convergence to left- or right-handed CPSs only along a single momentum-space direction, whereas all-angle {\em acp}-BICs exhibit convergence to CPSs of a single handedness throughout the entire momentum space, rendering them exceptionally promising for chiral optical applications. We reveal that the existence of {\em acp}-BICs is underpinned by total reflection of circularly polarized waves. Moreover, all-angle {\em acp}-BICs qualify as super-BICs, with uniform nearby polarization being an intrinsic property. In addition, a bifurcation theory is developed to analyze the emergence of genuine CPSs from {\em acp}-BICs under $C_{2}$-symmetric structural perturbations. Our results suggest {\em acp}-BICs as a platform for singular and chiral optical responses in all-dielectric systems.

physics.optics↗

arXiv2025

Parametric dependence of unidirectional guided resonances in periodic structures

Unidirectional guided resonances (UGRs) in periodic structures are special resonant modes that exhibit strict one-sided radiation, even though radiation in both sides is allowed, offering significant advantages for various applications. Under a structural perturbation, a UGR typically turns to a regular resonant mode that radiates to both sides. Existing numerical results indicate that to find UGRs in any periodic structure, it is necessary to tune at least one parameter. In this work, we develop a rigorous theory on the parametric dependence of UGRs. We show that in the presence of a single radiation channel, a UGR can exist continuously with respect to a structural parameter, provided that another parameter (associated with a generic perturbation) is properly tuned. Moreover, from a periodic structure with a generic bound state in the continuum (BIC), it is always possible to obtain a continuous family of UGRs by tuning one parameter. This implies that UGRs with arbitrarily large quality factor can be easily obtained. Our work provides a theoretical basis for designing useful photonic devices based on UGRs.

physics.optics↗

arXiv2025

Existence of Friedrich-Wintgen Bound States in the Continuum: Cavity with a Thin Waveguide Opening

Bound states in the continuum (BICs) are localized states embedded within a continuum of propagating waves. Perturbations that disrupt BICs typically induce ultra-strong resonances, a phenomenon enabling diverse applications in photonics. This work investigates the existence of BICs in two-dimensional electromagnetic cavities coupled to thin waveguides for H-polarized waves. Our focus is on Friedrich-Wintgen BICs (FW-BICs), which arise from destructive interference between two resonant modes and were identified numerically in rectangular cavities with waveguide openings by Lyapina et al. [J. Fluid Mech., 780 (2015), pp. 370--387]. Here, we rigorously establish the existence of FW-BICs in a broader class of cavity geometries by introducing perturbations to the refractive index under regularity constraints. We show that BICs correspond to intersections of two curves derived implicitly from the governing equations constructed via the mode-matching method. Crucially, we prove that such intersections are guaranteed for sufficiently small waveguide widths, provided that two eigenvalues of the cavity cross and the associated eigenfunctions exhibit non-vanishing coupling to the radiation channel at the cavity-waveguide interface. Furthermore, our approach remains applicable for studying the emergence of FW-BICs under parameter-dependent boundary perturbations to the cavity.

math-ph↗

arXiv2025

Super Bound States in the Continuum: Analytic Framework, Parametric Dependence, and Fast Direct Computation

In periodic structures such as photonic crystal (PhC) slabs, a bound state in the continuum (BIC) is always surrounded by resonant states with their $Q$-factor following $Q\sim 1/|{\bm β}-{\bm β}_*|^{2p}$, where ${\bm β}$ and ${\bm β}_*$ are the Bloch wavevectors of the resonant state and the BIC, respectively. Typically $p=1$, but special BICs, known as the super-BICs, have $p\geq 2$. Super-BICs can significantly enhance the $Q$-factor of nearby resonant states and reduce scattering losses due to fabrication imperfections, making them highly advantageous in practical applications. However, super-BICs, requiring the tuning of structural parameters for their realization, are generally not robust. In this work, we develop a theory to classify super-BICs, determine the minimal number $n$ of tunable structural parameters needed, and show that super-BICs form a manifold of dimension $(m-n)$ in an $m$-dimensional parameter space. We also propose a direct method for computing super-BICs in structures with different symmetry. Numerical examples demonstrate that our method is far more efficient than existing methods when $n > 1$. In addition, we study the effect of structural perturbations, focusing on the transition from super-BICs to generic BICs. Finally, we analyze a class of degenerate BICs that can be regarded as Dirac points, and show that they are the intersections of super-BICs in a relevant parameter space. Our work advances the theoretical understanding on super-BICs, and has both direct and potential applications in optical design and light-matter interactions.

physics.optics↗

arXiv2025

Existence of Friedrich-Wintgen bound states in the continuum: system of Schrödinger equations

A bound state in the continuum (BIC) is an eigenmode with the corresponding eigenvalue embedded in the continuous spectrum. There is currently a significant research interest on BICs in the photonics community, because they can be used to induce strong resonances that are useful for lasing, sensing, harmonic generation, etc. The existence of BICs in classical or quantum wave systems has only been established for some relatively simple cases such as BICs protected by symmetry. In 1985, Friedrich and Wintgen (Physical Review A, Vol. 32, pp. 3232-3242, 1985) suggested that BICs may appear from the destructive interference of two resonances coupled to a single radiation channel. They used a system of three one-dimensional Schrödinger equations to illustrate this process. Many BICs in classical wave systems seem to follow this mechanism and are now called Friedrich-Wintgen BICs. However, Friedrich and Wintgen did not show the existence of BICs in their system of three Schrödinger equations. Instead, they approximated the original system by a model with one Schrödinger equation and two algebraic equations, and only analyzed BICs in the approximate model. In this paper, we give a rigorous justification for the existence of BICs in the original system of three 1D Schrödinger equations.

physics.optics↗

arXiv2025

Relationship between total reflection and Fabry-Perot bound states in the continuum

Bound states in the continuum (BICs) have interesting properties and important applications in photonics. A particular class of BICs are found in Fabry-Perot (FP) cavities formed by two parallel periodic dielectric layers separated by a distance $h$. A periodic dielectric layer can totally reflect a plane incident wave with a particular frequency and a particular wavenumber. Existing FP-BICs are found when $h$ is close to the values deduced from a phase-matching condition related to the reflection coefficient, but they are obtained in FP-cavities where the periodic layers have a reflection symmetry in the periodic direction. In this paper, we further clarify the connection between total reflections and FP-BICs. Our numerical results indicate that if the wavenumber is zero or the periodic layers have a reflection symmetry in the periodic direction, FP-BICs can indeed be found near the parameters of total reflections. However, if the wavenumber is nonzero and the periodic layer is asymmetric (in the periodic direction), we are unable to find a FP-BIC (with a frequency and a wavenumber near those of a total reflection) by tuning $h$ or other structural parameters. Consequently, a total reflection does not always lead to a FP-BIC even when the parameters of the FP-cavity are tuned.

physics.optics↗

arXiv2024

Perturbation theory for resonant states near a bound state in the continuum

In this work, we develop a perturbation theory to analyze resonant states near a bound state in the continuum (BIC) in photonic crystal slabs. The theory allows us to rigorously determine the asymptotic behavior of $Q$-factor and the far-field polarization. We show that the resonant states close to a BIC can be nearly circularly polarized if the scattering matrix satisfies a certain condition. Moreover, our theory offers a novel perspective on super-BICs and provides a clear and precise condition to efficiently identify them in both symmetric and asymmetric structures without requiring a merging process. For practical applications, we find a super-BIC in a square lattice of rods on a dielectric substrate. Our theory addresses the non-Hermitian nature of the system, can be generalized to treat other structures that support BICs, and has potential applications in resonance and chiral optics.

physics.optics↗

arXiv2024

An efficient method for calculating resonant modes in biperiodic photonic structures

Many photonic devices, such as photonic crystal slabs, cross gratings, and periodic metasurfaces, are biperiodic structures with two independent periodic directions, and are sandwiched between two homogeneous media. Many applications of these devices are closely related to resonance phenomena. Therefore, efficient computation of resonant modes is crucial in device design and structure analysis. Since resonant modes satisfy outgoing radiation conditions, perfectly matched layers (PMLs) are usually used to truncate the unbounded spatial variable perpendicular to the periodic directions. In this paper, we develop an efficient method without using PMLs to calculate resonant modes in biperiodic structures. We reduce the original eigenvalue problem to a small matrix nonlinear eigenvalue problem which is solved by the contour integral method. Numerical examples show that our method is efficient with respect to memory usage and CPU time, free of spurious solutions, and determines degenerate resonant modes without any difficulty.

physics.comp-ph↗

arXiv2023

Non-generic bound states in the continuum in waveguides with lateral leakage channels

For optical waveguides with a layered background which itself is a slab waveguide, a guided mode is a bound state in the continuum (BIC), if it coexists with slab modes propagating outwards in the lateral direction; i.e., there are lateral leakage channels. It is known that generic BICs in optical waveguides with lateral leakage channels are robust in the sense that they still exist if the waveguide is perturbed arbitrarily. However, the theory is not applicable to non-generic BICs which can be defined precisely. Near a BIC, the waveguide supports resonant and leaky modes with a complex frequency and a complex propagation constant, respectively. In this paper, we develop a perturbation theory to show that the resonant and leaky modes near a non-generic BIC have an ultra-high $Q$ factor and ultra-low leakage loss, respectively. We also show that a merging-BIC obtained by tuning structural parameters is always a non-generic BIC. Existing studies on merging-BICs are concerned with specific examples and specific parameters. We analyze an arbitrary structural perturbation (to a waveguide supporting a non-generic BIC) given by $δF({\bf r})$, where $F({\bf r})$ is the perturbation profile and $δ$ is the amplitude, and show that the perturbed waveguide has two BICs for $δ>0$ (or $δ<0$) and no BIC for $δ<0$ (or $δ>0$). This implies that a non-generic BIC is a merging-BIC (for any perturbation profile $F$) when $δ$ is regarded as a parameter. Our study indicates that non-generic BICs have interesting special properties that are useful in applications.

physics.optics↗

arXiv2022

Frequency perturbation theory of bound states in the continuum in a periodic waveguide

In a lossless periodic structure, a bound state in the continuum (BIC) is characterized by a real frequency and a real Bloch wavevector for which there exist waves propagating to or from infinity in the surrounding media. For applications, it is important to analyze the high-$Q$ resonances that either exist naturally for wavevectors near that of the BIC or appear when the structure is perturbed. Existing theories provide quantitative results for the complex frequency (and the $Q$-factor) of resonant modes that appear/exist due to structural perturbations or wavevector variations. When a periodic structure is regarded as a periodic waveguide, eigenmodes are often analyzed for a given real frequency. In this paper, we consider periodic waveguides with a BIC, and study the eigenmodes for given real frequencies near the frequency of the BIC. It turns out that such eigenmodes near the BIC always have a complex Bloch wavenumber, but they may or may not be leaky modes that radiate out power laterally to infinity. These eigenmodes can also be complex modes that decay exponentially in the lateral direction. Our study is relevant for applications of BICs in optical waveguides, and it is also helpful for analyzing photonic devices operating near the frequency of a BIC.

physics.optics↗

arXiv2022

Computing diffraction anomalies as nonlinear eigenvalue problems

When a plane electromagnetic wave impinges upon a diffraction grating or other periodic structures, reflected and transmitted waves propagate away from the structure in different radiation channels. A diffraction anomaly occurs when the outgoing waves in one or more radiation channels vanish. Zero reflection, zero transmission and perfect absorption are important examples of diffraction anomalies, and they are useful for manipulating electromagnetic waves and light. Since diffraction anomalies appear only at specific frequencies and/or wavevectors, and may require the tuning of structural or material parameters, they are relatively difficult to find by standard numerical methods. Iterative methods may be used, but good initial guesses are required. To determine all diffraction anomalies in a given frequency interval, it is necessary to repeatedly solve the diffraction problem for many frequencies. In this paper, an efficient numerical method is developed for computing diffraction anomalies. The method relies on nonlinear eigenvalue formulations for scattering anomalies and solves the nonlinear eigenvalue problems by a contour-integral method. Numerical examples involving periodic arrays of cylinders are presented to illustrate the new method.

physics.comp-ph↗

arXiv2022

Parametric dependence of bound states in the continuum: a general theory

Photonic structures with high-$Q$ resonances are essential for many practical applications, and they can be relatively easily realized by modifying ideal structures with bound states in the continuum (BICs). When an ideal photonic structure with a BIC is perturbed, the BIC may be destroyed (becomes a resonant state) or may continue to exist with a slightly different frequency and a slightly different wavevector (if appropriate). Some BICs are robust against certain structural perturbations, but most BICs are nonrobust. Recent studies suggest that a nonnegative integer $n$ can be defined for any generic nondegenerate BIC with respect to a properly defined set of structural perturbations. The integer $n$ is the minimum number of tunable parameters needed to preserve the BIC for perturbations arbitrarily chosen from the set. Robust and nonrobust BICs have $n=0$ and $n\ge 1$, respectively. A larger $n$ implies that the BIC is more difficult to find. If a structure is given by $m$ real parameters, the integer $n$ is the codimension of a geometric object formed by the parameter values at which the BIC exists in the $m$-dimensional parameter space. In this paper, we suggest a formula for $n$, give some justification for the general case, calculate $n$ for different types of BICs in two-dimensional structures with a single periodic direction, and illustrate the results by numerical examples. Our study improves the theoretical understanding on BICs and provides useful guidance to their practical applications.

physics.optics↗

arXiv2022

Real transmission and reflection zeros of periodic structures with a bound state in the continuum

For lossless periodic structures with a proper symmetry, the transmission and reflection spectra often have peaks and dips that are truly $100\%$ and $0\%$, respectively. The full peaks and zero dips typically appear near resonant frequencies, and they are robust with respect to structural perturbations that preserve the required symmetry. However, current theories on the existence of full peaks and zero dips are incomplete and difficult to use. For periodic structures with a bound state in the continuum (BIC), we present a new theory on the existence of real transmission and reflection zeros that correspond to the zero dips in the transmission and reflection spectra. Our theory is relatively simple, complete, and easy to use. Numerical examples are presented to validate the new theory.

physics.optics↗

arXiv2022

Approximating transmission and reflection spectra near isolated nondegenerate resonances

A linear scattering problem for which incoming and outgoing waves are restricted to a finite number of radiation channels can be precisely described by a frequency-dependent scattering matrix. The entries of the scattering matrix, as functions of the frequency, give rise to the transmission and reflection spectra. To find the scattering matrix rigorously, it is necessary to solve numerically the partial differential equations governing the relevant waves. In this paper, we consider resonant structures with an isolated nondegenerate resonant mode of complex frequency $ω_\star$, and show that for real frequencies near $ω_0 = \mbox{Re}(ω_\star)$, the transmission and reflection spectra can be approximated using only the scattering matrix at $ω_0$ and information about the resonant mode. We also present a revised temporal coupled-mode theory that produces the same approximate formulas for the transmission and reflection spectra. Numerical examples for diffraction of plane waves by periodic structures are presented to validate our theory.

physics.optics↗