arXiv subjects

Lijun Yuan

Publications and source records attributed to Lijun Yuan.

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

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↗

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↗

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↗

arXiv2021

Resonant field enhancement in lossy periodic structures supporting complex bound states in the continuum

Resonant modes in a lossy periodic structure sandwiched between two lossless homogeneous media form bands that depend on the Bloch wavevector continuously and have a complex frequency due to radiation and absorption losses. A complex bound state in the continuum (cBIC) is a special state with a zero radiation loss in such a band. Plane waves incident upon the periodic structure induce local fields that are resonantly enhanced. In this paper, we derive a rigorous formula for field enhancement, and analyze its dependence on the frequency, wavevector and amplitude of the incident wave. For resonances with multiple radiation channels, we determine the incident wave that maximizes the field enhancement, and find conditions under which the field enhancement can be related to the radiation and dissipation quality factors. We also show that with respect to the Bloch wavevector, the largest field enhancement is obtained approximately when the radiation and dissipation quality factors are equal. Our study clarifies the various factors related to field enhancement, and provides a useful guideline for applications where a strong local field is important.

physics.optics↗

arXiv2020

On the robustness of bound states in the continuum in waveguides with lateral leakage channels

Bound states in the continuum (BICs) are trapped or guided modes with frequencies in radiation continua. They are associated with high-quality-factor resonances that give rise to strong local field enhancement and rapid variations in scattering spectra, and have found many valuable applications. A guided mode of an optical waveguide can also be a BIC, if there is a lateral structure supporting compatible waves propagating in the lateral direction, i.e., there is a channel for lateral leakage. A BIC is typically destroyed (becomes a resonant or a leaky mode) if the structure is slightly perturbed, but some BICs are robust with respect to a large family of perturbations. In this paper, we show (analytically and numerically) that a typical BIC in optical waveguides with a left-right mirror symmetry and a single lateral leakage channel is robust with respect to any structural perturbation that preserves the left-right mirror symmetry. Our study improves the theoretical understanding on BICs and can be useful when applications of BICs in optical waveguides are explored.

physics.optics↗

arXiv2020

Conditional robustness of propagating bound states in the continuum on biperiodic structures

For a periodic structure sandwiched between two homogeneous media, a bound state in the continuum (BIC) is a guided Bloch mode with a frequency in the radiation continuum. Optical BICs have found many applications, mainly because they give rise to resonances with ultra-high quality factors. If the periodic structure has a relevant symmetry, a BIC may have a symmetry mismatch with incoming and outgoing propagating waves of the same frequency and compatible wavevectors, and is considered as protected by symmetry. Propagating BICs with nonzero Bloch wavevectors have been found on many highly symmetric periodic structures. They are not protected by symmetry in the usual sense (i.e., there is no symmetry mismatch), but some of them seem to depend on symmetry for their existence and robustness. In this paper, we show that the low-frequency propagating BICs (with only one radiation channel) on biperiodic structures with an inversion symmetry in the plane of periodicity and a reflection symmetry in the perpendicular direction are robust to symmetry-preserving structural perturbations. In other words, a propagating BIC continues its existence with a slightly different frequency and a slightly different Bloch wavevector, when the biperiodic structure is perturbed slightly preserving the inversion and reflection symmetries. Our study enhances theoretical understanding for BICs on periodic structures and provides useful guidelines for their applications.

physics.optics↗

arXiv2020

Parametric Dependence of Bound States in the Continuum on Periodic Structures

Bound states in the continuum (BICs) have some unusual properties and important applications in photonics. A periodic structure sandwiched between two homogeneous media is the most popular platform for observing BICs and realizing their applications. Existing studies on BICs assume the periodic structure has a $C_2$ rotational symmetry about the axis perpendicular to the periodic layer. It is known that all BICs turn to resonant states with finite quality factors if the periodic structure is perturbed by a generic perturbation breaking the $C_2$ symmetry, and a typical BIC continues to exist if the perturbation keeps the $C_2$ symmetry. We study how typical BICs depend on generic structural parameters. For a class of BICs with one opening radiation channel, we show that in the plane of two generic parameters, the BICs exist continuously as a curve. Consequently, BICs can exist on periodic structures without the $C_2$ symmetry, and they can be found by tuning a single structural parameter. The result is established analytically by a perturbation theory with two independent perturbations and validated by numerical examples. Our study reveals a much larger family for BICs on periodic structures, and provides new opportunities for future applications.

physics.optics↗

arXiv2019

Resonant field enhancement near bound states in the continuum on periodic structures

On periodic structures sandwiched between two homogeneous media, a bound state in the continuum (BIC) is a guided Bloch mode with a frequency within the radiation continuum. BICs are useful, since they give rise to high quality-factor ($Q$-factor) resonances that enhance local fields for diffraction problems with given incident waves. For any BIC on a periodic structure, there is always a surrounding family of resonant modes with $Q$-factors approaching infinity. We analyze field enhancement around BICs using analytic and numerical methods. Based on a perturbation method, we show that field enhancement is proportional to the square-root of the $Q$-factor, and it depends on the adjoint resonant mode and its coupling efficiency with incident waves. Numerical results are presented to show different asymptotic relations between the field enhancement and the Bloch wavevector for different BICs. Our study provides a useful guideline for applications relying on resonant enhancement of local fields.

physics.optics↗

arXiv2019

Perturbation theories for symmetry-protected bound states in the continuum on two-dimensional periodic structures

On dielectric periodic structures with a reflection symmetry in a periodic direction, there can be antisymmetric standing waves (ASWs) that are symmetry-protected bound states in the continuum (BICs). The BICs have found many applications, mainly because they give rise to resonant modes of extremely large quality-factors ($Q$-factors). The ASWs are robust to symmetric perturbations of the structure, but they become resonant modes if the perturbation is non-symmetric. The $Q$-factor of a resonant mode on a perturbed structure is typically $O(1/δ^2)$ where $δ$ is the amplitude of the perturbation, but special perturbations can produce resonant modes with larger $Q$-factors. For two-dimensional (2D) periodic structures with a 1D periodicity, we derive conditions on the perturbation profile such that the $Q$-factors are $O(1/δ^4)$ or $O(1/δ^6)$. For the unperturbed structure, an ASW is surrounded by resonant modes with a nonzero Bloch wave vector. For 2D periodic structures, the $Q$-factors of nearby resonant modes are typically $O(1/β^2)$, where $β$ is the Bloch wavenumber. We show that the $Q$-factors can be $O(1/β^6)$ if the ASW satisfies a simple condition.

physics.optics↗

arXiv2019

Bound states with complex frequencies near the continuum on lossy periodic structures

On a lossless periodic dielectric structure sandwiched between two homogeneous media, bound states in the continuum (BICs) with real frequencies and real Bloch wavevectors may exist, and they decay exponentially in the surrounding homogeneous media and do not couple with propagating plane waves with the same frequencies and wavevectors. The BICs are of significant current interest, because they give rise to high-$Q$ resonances when the structure or the Bloch wavevector is slightly perturbed. In this paper, the effect of a small material loss on the BICs is analyzed by a perturbation method and illustrated by numerical results. It is shown that bound states with complex frequencies near the continuum appear, but they behave differently depending on whether the BIC is symmetry-protected or not. The Bloch wavevector of a bound state with a complex frequency can be real if the original BIC is symmetry-protected, and it is usually complex if the original BIC is not symmetry-protected. Our study improves the theoretical understanding on BICs and provides useful insight for their practical applications.

physics.optics↗

arXiv2019

Unidirectional Reflectionless Transmission for Two-Dimensional $\mathcal{PT}$-symmetric Periodic Structures

Unidirectional reflectionless propagation (or transmission) is an interesting wave phenomenon observed in many $\mathcal{PT}$-symmetric optical structures. Theoretical studies on unidirectional reflectionless transmission often use simple coupled-mode models. The coupled-mode theory can reveal the most important physical mechanism for this wave phenomenon, but it is only an approximate theory, and it does not provide accurate quantitative predictions with respect to geometric and material parameters of the structure. In this paper, we rigorously study unidirectional reflectionless transmission for two-dimensional (2D) $\mathcal{PT}$-symmetric periodic structures sandwiched between two homogeneous media. Using a scattering matrix formalism and a perturbation method, we show that real zero-reflection frequencies are robust under $\mathcal{PT}$-symmetric perturbations, and unidirectional reflectionless transmission is guaranteed to occur if the perturbation (of the dielectric function) satisfies a simple condition. Numerical examples are presented to validate the analytical results, and to demonstrate unidirectional invisibility by tuning the amplitude of balanced gain and loss.

physics.optics↗

arXiv2019

Excitation of bound states in the continuum via second harmonic generations

A bound state in the continuum (BIC) on a periodic structure sandwiched between two homogeneous media is a guided mode with a frequency and a wavenumber such that propagating plane waves with the same frequency and wavenumber exist in the homogeneous media. Optical BICs are of significant current interest, since they have applications in lasing, sensing, filtering, switching, and many light emission processes, but they cannot be excited by incident plane waves when the structure consists of linear materials. In this paper, we study the diffraction of a plane wave by a periodic structure with a second order nonlinearity, assuming the structure has a BIC and the frequency and wavenumber of the incident wave are one half of those of the BIC. Based on a scaling analysis and a perturbation theory, we show that the incident wave may induce a very strong second harmonic wave dominated by the BIC, and also a fourth harmonic wave that cannot be ignored. The perturbation theory reveals that the amplitude of the BIC is inversely proportional to a small parameter depending on the amplitude of the incident wave and the nonlinear coefficient. In addition, a system of four nonlinearly coupled Helmholtz equations (the four-wave model) is proposed to model the nonlinear process. Numerical solutions of the four-wave model are presented for a periodic array of circular cylinders and used to validate the perturbation results.

nlin.PS↗

arXiv2018

Bound states in the continuum on periodic structures surrounded by strong resonances

Bound states in the continuum (BICs) are trapped or guided modes with their frequencies in the frequency intervals of the radiation modes. On periodic structures, a BIC is surrounded by a family of resonant modes with their quality factors approaching infinity. Typically the quality factors are proportional to $1/|β- β_*|^2$, where $β$ and $β_*$ are the Bloch wavevectors of the resonant modes and the BIC, respectively. But for some special BICs, the quality factors are proportional to $1/|β-β_*|^4$. In this paper, a general condition is derived for such special BICs on two-dimensional periodic structures. As a numerical example, we use the general condition to calculate special BICs, which are antisymmetric standing waves, on a periodic array of circular cylinders, and show their dependence on parameters. The special BICs are important for practical applications, because they produce resonances with large quality factors for a very large range of $β$.

physics.optics↗

arXiv2017

Bound States in the Continuum on Periodic Structures: Perturbation Theory and Robustness

On periodic structures, a bound state in the continuum (BIC) is a standing or propagating Bloch wave with a frequency in the radiation continuum. Some BICs (e.g., antisymmetric standing waves) are symmetry-protected, since they have incompatible symmetry with outgoing waves in the radiation channels. The propagating BICs do not have this symmetry mismatch, but they still depend crucially on the symmetry of the structure. In this Letter, a perturbation theory is developed for propagating BICs on two-dimensional periodic structures. The study shows that these BICs are robust against structural perturbations that preserve the symmetry, indicating that these BICs are in fact implicitly protected by symmetry.

physics.optics↗