1. Introduction

A bound state in the continuum (BIC) is a trapped or guided mode with a frequency in the frequency interval of radiation modes [1–10]. Mathematically, a BIC corresponds to an embedded eigenvalue in a related continuous spectrum [11–15]. In recent years, BICs have become a topic of significant research interest in the photonics community [7–10]. Photonic BICs have been found in many different structures including periodic structures sandwiched between two homogeneous media [15–21], waveguides with lateral leakage channels [22–24], layered anisotropic media [25–27], and rotationally symmetric periodic (RSP) waveguides such as fiber gratings and one-dimensional (1D) arrays of dielectric spheres [28–30]. A number of important applications like lasing [31], sensing [32,33], filtering [34] and switching [35], have been demonstrated experimentally [7–10].

In structures consisting of ordinary lossless dielectric material, BICs can only exist when the structure is unbounded in at least one spatial direction. An RSP waveguide is periodic along its axis (the $z$ axis in this paper) and rotationally symmetric about the same axis [28–30,36–38]. A guided mode of an RSP waveguide is associated with a Bloch wavenumber $\beta$ (for the $z$ direction) and an azimuthal index $m$, so that its electric field can be written as ${\boldsymbol{\Phi} }(\rho,z) e^{ i ( m \phi + \beta z)}$, where $(\rho, \phi, z)$ are the cylindrical coordinates and ${\boldsymbol{\Phi} }$ is periodic in $z$. A BIC is a special guided mode above the light line. Specifically, if the RSP waveguide is surrounded by air, then the free space wavenumber $k$ and Bloch wavenumber $\beta$ of a BIC must satisfy $k > |\beta |$. Corresponding to the cases $\beta =0$ and $\beta \neq 0$, the BIC is a standing wave and a propagating mode, respectively. For $m=0$, the BIC is either transverse-magnetic (TM) or transverse-electric (TE) polarized. If the RSP waveguide has a reflection symmetry in $z$, a typical standing wave with $m=0$ is symmetry-protected, but there also exist standing waves unprotected by symmetry [6,36]. Propagating BICs with $m=0$ have been found in many RSP waveguides [28,30,36]. For $m\neq 0$, the TE and TM components of any guided mode in an RSP waveguide are fully coupled. Standing waves with a nonzero $m$ have been found in a number of RSP waveguides such as 1D arrays of dielectric spheres or circular disks [28,36] and periodic nanowires [38]. It has been shown that propagating BICs with $m\ne 0$ exist in 1D arrays of dielectric spheres [29].

Existing studies have revealed that many BICs are robust, in the sense that they continue to exist when the structure is perturbed [39–47]. When the structure has a relevant symmetry, for example the in-plane inversion symmetry for periodic structures sandwiched between two homogeneous media, the symmetry-protected BICs are clearly robust with respect to small but arbitrary symmetry-preserving structural perturbations [5,11,12,19,22]. If the perturbation breaks the relevant symmetry, the BIC is usually (but not always) destroyed and becomes a resonant mode with a finite quality factor ($Q$ factor) [48,49]. Some BICs unprotected by symmetry can also be robust, but symmetry still plays a key role [39,40]. For example, in periodic structures with the in-plane inversion symmetry, some propagating BICs are robust with respect to small structural perturbations that preserve the inversion symmetry [39,40,45,47]. For some BICs, a topological charge can be defined using the far field polarization vector of the surrounding resonant modes, and the conservation of topological charge implies robustness [39,40]. On the other hand, mathematical theories on the robustness of BICs in periodic structures sandwiched between homogeneous media and in waveguides with lateral leakage channels have been established [45–47]. In these works, the existence of BICs in perturbed structures is proved constructively using power series in a small parameter $\delta$ related to the amplitude of the perturbation.

In this paper, we analyze the robustness of nondegenerate BICs in lossless dielectric RSP waveguides with a reflection symmetry in $z$. Specifically, we assume $k$ and $\beta$ of the BIC satisfy $|\beta | < k < 2\pi /L - |\beta |$, where $L$ is the period of the RSP waveguide. This condition ensures that the zeroth diffraction order is the only propagating order. It is shown that with respect to structural perturbations that preserve the reflection symmetry, generic BICs with $m=0$ (including both standing waves and propagating modes) are robust, and those BICs with $m\ne 0$ are not robust. The theory for BICs with $m=0$ can be relatively easily developed following the approach of Ref. [45]. A non-robust BIC with $m\ne 0$ is usually destroyed by a structural perturbation, even when the perturbation is symmetric in $z$, but the BIC can nevertheless persist under some special structural perturbations. We show that if the structural perturbation contains a tunable parameter, a generic non-robust BIC with $m \ne 0$ can continue to exist when the parameter is properly tuned. This implies that in any symmetric RSP waveguide with two real structural parameters, a generic non-robust BIC with $m\ne 0$ exists as a curve in the parameter plane.

The rest of this paper is organized as follows. In section 2, we recall some facts about BICs in RSP waveguides and introduce some useful scattering solutions. The theory is developed in section 3 mainly for the case of $m\ne 0$, but we also include a brief discussion for the case of $m=0$. To show the continual existence of generic non-robust BICs under small structural perturbations with a tunable parameter, we determine the BIC in the perturbed structure and the parameter value together, using power series in $\delta$, where $\delta$ is the amplitude of the perturbation. In section 4, we illustrate our theory by numerical examples including new propagating BICs with $m\ne 0$ in 1D arrays of circular disks and fiber gratings. The paper is concluded with a brief discussion in section 5.

2. Background

We consider an isotropic and lossless RSP waveguide that is periodic in $z$ with period $L$, has a reflection symmetry in $z$, and is surrounded by air. A particular example is a periodic array of circular disks shown in Fig. 1 below. The dielectric function $\varepsilon$ satisfies

 

Fig. 1. A periodic array of circular disks surrounded by air. The period of the array is $L$. The radius, thickness and dielectric constant of the disks are $R$, $d$ and $\varepsilon _1$, respectively.

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If $m=0$, the electromagnetic field can be decoupled into TE (${\textbf{H}}^e=H_\rho \hat {\rho }+H_z\hat {z},\;{\textbf{E}}^e=E_\phi \hat {\phi }$) and TM (${\textbf{E}}^h=E_\rho \hat {\rho }+E_z\hat {z},\;{\textbf{H}}^h=H_\phi \hat {\phi }$) polarized fields [36], where $\hat {\rho }$ and $\hat {\phi }$ are the unit vectors in the radial and azimuthal directions, respectively. If $m\neq 0$, the TE components

When $\rho >R$, the electric field of any Bloch mode satisfying an outgoing radiation condition can be expanded as [30]

A guided mode with a real $k$ has a mode profile satisfying $\sqrt {\rho }{\boldsymbol{\Phi} }\rightarrow 0$ as $\rho \rightarrow +\infty$. In general, guided modes exist below the light line ($k<|\beta |$) such that all $\kappa _p$ are pure imaginary. A BIC is a special guided mode above the light line (with a real $k$ and a real $\beta$ satisfying $k>|\beta |$). In this paper, we focus on BICs satisfying $|\beta | < k < 2\pi /L - |\beta |$. This condition implies that $\kappa _0>0$ and all other $\kappa _p$ ($p\ne 0$) are pure imaginary. Consequently, for such a BIC, the coefficients $c_0^e$ and $c_0^h$ must be zero.

In the rest of this section, we discuss some properties of the BICs and introduce related scattering solutions. The BICs are assumed to be nondegenerate. The scattering solutions are needed in the following section for analyzing the robustness of BICs. Since the RSP waveguide has a reflection symmetry in $z$ and no material loss, the BICs and the scattering solutions can be scaled to have a symmetry related to parity-inversion and time-reversal operations. To simplify the presentation, we define two operators ${\mathcal {P}}$ and ${\mathcal {T}}$ by

${\mathcal {T}}{\boldsymbol{\Phi} }={\boldsymbol{\Phi} }$, and ${\mathcal {P}}\mathcal {L}=\mathcal {L}{\mathcal {P}}$. This implies that ${\boldsymbol{\Phi}}$ satisfies either ${\mathcal {P}}{\boldsymbol{\Phi} }=-{\boldsymbol{\Phi} }$ or ${\mathcal {P}}{\boldsymbol{\Phi} }={\boldsymbol{\Phi} }$. More specifically, $\Phi _z$ is either even or odd in $z$ and the other components have the opposite parity in $z$.

A resonant mode is associated with a complex $k$ and a real $\beta$, and satisfies an outgoing radiation condition. A BIC can be regarded as a special resonant mode with a real $k$, a real $\beta$, and an infinite $Q$ factor, where $Q=-0.5\mbox {Re}(k)/\mbox {Im}(k)$ [51]. When $\beta =0$, $\Phi _z$ of a nondegenerate resonant mode is either odd or even in $z$. Therefore, either $c_0^h$ or $c_0^e$ must be zero. In other words, a resonant mode with $\beta =0$ radiates out power in a single polarized radiation channel.

Next, we construct scattering solutions with the same symmetry properties as the BICs. We start with a scattering solution excited by the following TE cylindrical incident wave

For $\beta =0$, since $\mathcal {P}{\textbf{e}}_0^e={\textbf{e}}_0^e$ and $\mathcal {P}{\textbf{e}}_0^h=-{\textbf{e}}_0^h$, the incident wave in Eq. (14) can be reduced to

In summary, if $\beta \ne 0$, we can scale the BIC such that ${\mathcal {P}\mathcal {T}}{\boldsymbol{\Phi} }={\boldsymbol{\Phi} }$, and construct scattering solutions satisfying ${\mathcal {P}}{\mathcal {T}}\hat {\boldsymbol{\Psi} }^l=\hat {\boldsymbol{\Psi} }^l$, $l\in \{e,h\}$. If $\beta =0$, we can scale the BIC such that $\mathcal {T}{\boldsymbol{\Phi} }={\boldsymbol{\Phi} }$ and $\mathcal {P}{\boldsymbol{\Phi} }={\boldsymbol{\Phi} }$ or $\mathcal {P}{\boldsymbol{\Phi} }=-{\boldsymbol{\Phi} }$. We can also construct diffraction solutions satisfying ${\mathcal {P}}\hat {\boldsymbol{\Psi} }^e=\hat {\boldsymbol{\Psi} }^e$, ${\mathcal {P}}\hat {\boldsymbol{\Psi} }^h=-\hat {\boldsymbol{\Psi} }^h$ and $\mathcal {T}\hat {\boldsymbol{\Psi} }^l=\hat {\boldsymbol{\Psi} }^{l}$, $l\in \{e,h\}$.

3. Theory

In this section, we analyze the robustness of BICs in RSP waveguides. It is assumed that an RSP waveguide with dielectric function $\varepsilon _*({\rho,z})$ supports a nondegenerate $\mathrm {BIC}$ with free space wavenumber $k_*$, Bloch wavenumber $\beta _*$, and electric field ${\textbf{E}}_*({\textbf{r}})={\boldsymbol{\Phi} }_*({\rho,z})\exp {[i (m\phi +\beta _* z)]}$. The RSP waveguide is periodic in $z$ with period $L$, is symmetric in $z$ and surrounded by air. The dielectric function $\varepsilon _*({\rho,z})$ satisfies Eq. (1). As in section 2, we assume $k_*$ and $\beta _*$ of the BIC satisfy

We are concerned with the robustness of generic nondegenerate BICs with respect to small but arbitrary structural perturbations that preserve the periodicity and reflection symmetry in $z$. Existing numerical results suggest that the BICs with $m=0$ are robust with respect to such perturbations [30,36]. This is indeed true and can be proved rigorously following the approach of Ref. [45]. This means that for any real function $F(\rho,z)$ satisfying Eq. (1) and $F(\rho,z) = 0$ for $\rho > R$, and for any small real number $\delta$, the perturbed RSP waveguide with the dielectric function $\varepsilon (\rho,z) = \varepsilon _* (\rho,z) + \delta F(\rho, z)$, has a BIC near the original one in the unperturbed structure. On the other hand, as the numerical examples have suggested [36], the BICs with $m\ne 0$ are nno-robust with respect to symmetric and periodic perturbations. For arbitrarily chosen $F$ and $\delta$, there is usually no BIC (near the original BIC) in the perturbed structure. However, for specially chosen perturbations, there could still be a BIC in the perturbed structure. Following the approach first developed in Ref. [52], we consider a perturbed RSP waveguide with the dielectric function

In order to show the existence of a BIC in the perturbed RSP waveguide [with the dielectric function given in Eq. (21)] for a properly chosen $\gamma$, we construct the BIC and find $\gamma$ together through power series in $\delta$. Let ${\boldsymbol{\Phi}}$, $k$, $\beta$ be the electric field profile, free space wavenumber and Bloch wavenumber of the desired BIC, we expand these quantities and $\gamma$ as

Inserting Eqs. (21)–(25) into Eq. (3) and comparing the different orders of $\delta$, we obtain

Equations (29) and (30) give rise to a linear system

We call a BIC satisfying the above condition a generic BIC. For the perturbation profile $G$, we require that $(a_{23}, a_{33})$ is not a multiple of $(a_{22}, a_{32})$, namely,

Consequently, if the BIC is generic and the perturbation profile $G$ is generic, we can solve $k_j$, $\beta _j$ and $\gamma _j$ from Eq. (31). This implies that conditions (29) and (30) are satisfied, thus Eq. (28) has a solution ${\boldsymbol{\Phi} }_j$ satisfying condition (26). In particular, the fact that ${\cal PT} {\textbf{f}}_j = {\textbf{f}}_j$ implies ${\cal PT} {\boldsymbol{\Phi} }_j = {\boldsymbol{\Phi} }_j$. Moreover, since for any constant $C$, ${\boldsymbol{\Phi} }_j+C{\boldsymbol{\Phi} }_*$ is also a solution, we can define a unique solution satisfying the orthogonality condition:

The theory above is developed for BICs with $m\ne 0$ and $\beta _* \ne 0$, and it remains valid for standing waves with $m \ne 0$ and $\beta _*=0$. As discussed earlier, we can assume the BIC with $\beta _* = 0$ and the related scattering solutions satisfy ${\cal P} {\boldsymbol{\Phi} }_* = \pm {\boldsymbol{\Phi} }_*$, ${\cal T} {\boldsymbol{\Phi} }_* = {\boldsymbol{\Phi} }_*$, ${\cal P} {\boldsymbol{\Psi} }^e_* = {\boldsymbol{\Psi} }^e_*$, ${\cal P} {\boldsymbol{\Psi} }^h_* = -{\boldsymbol{\Psi} }^h_*$, ${\cal T} {\boldsymbol{\Psi} }^l_* = {\boldsymbol{\Psi} }^l_*$ for $l \in \{e, h\}$, and Eq. (20). For the case when the BIC satisfies ${\cal P} {\boldsymbol{\Phi} }_* = {\boldsymbol{\Phi} }_*$, we have $a_{21}=a_{31}=0$ as before and also $a_{33}=d_{3j}=0$. For this case, a generic BIC with $\beta _*=0$ is defined as one satisfying $a_{32} \ne 0$, i.e.,

The case when the BIC satisfies ${\cal P} {\boldsymbol{\Phi} }_* = - {\boldsymbol{\Phi} }_*$ is similar. Therefore, generic standing waves with $m \ne 0$ are not robust with respect to symmetric structural perturbations, but they can continue to exist if a structural parameter is properly tuned. Notice that such standing waves are not symmetry protected. If ${\boldsymbol{\Phi} }_*$ satisfies ${\cal P} {\boldsymbol{\Phi} }_* = {\boldsymbol{\Phi} }_*$, the $z$ components of the BIC, $E_z$ and $H_z$, are even and odd in $z$, respectively. A generic symmetric perturbation will turn this BIC to a resonant mode with $c_0^h \ne 0$ and $c_0^e=0$. In some sense, such a standing wave is partially symmetry protected.

For the case of $m=0$, any nondegenerate BIC has a definite polarization. If we proceed as above, the second or third rows of both the coefficient matrix and the right hand side of system (31) are zero. This implies that $\gamma _j$ can be chosen arbitrarily. In other words, $\gamma$ is a free parameter. As the existence of the BIC can be proved for $\gamma =0$, that is, for dielectric function given by $\varepsilon (\rho,z) = \varepsilon _* (\rho,z) + \delta F(\rho, z)$, the BIC is robust with respect to symmetric structural perturbations. Assuming the BIC and the related scattering solutions satisfy the same scaling and normalization conditions used above, we only need the additional assumption

In previous studies [45,53], the robustness of BICs in periodic structures sandwiched between two homogeneous media have been thoroughly analyzed. For 2D structures that are invariant in $x$, periodic in $y$, bounded in $z$, and symmetric in both $y$ and $z$, any guided mode has a wave vector $(\alpha, \beta )$ with $\alpha$ and $\beta$ associated with $x$ and $y$, respectively. It is known that in such a 2D structure with 1D periodicity, whether $\alpha$ is zero or not, generic nondegenerate BICs (with a frequency in the range of a single propagation diffraction order) are always robust [45,53]. A symmetric RSP waveguide is analogous to such a 2D structure, with the cylindrical coordinates $\rho$, $\phi$, $z$ correspond to $z$, $x$ and $y$, respectively, $m$ corresponds to $\alpha$, and $\beta$ remains fixed. However, as shown in this section, BICs in RSP waveguides exhibit different robustness for $m=0$ and $m\ne 0$, respectively. This is related to the fact that $m$ is a fixed integer that cannot be varied when the structure is perturbed, while $\alpha$ (for BICs in 2D structures with 1D periodicity) is a solution parameter that can be varied.

4. Numerical examples

In this section, we present numerical examples to illustrate our theory. Since there are many numerical examples for BICs with $m=0$ or $m\ne 0$ but $\beta =0$ [28–30,36–38], we concentrate on the case $m \ne 0$ and $\beta \ne 0$. According to the theory developed in section 3, such a BIC in an RSP waveguide is non-robust, but can often be found by tuning a single structural parameter. The numerical examples in this section are calculated using a numerical modal method based on piecewise polynomials of $z$ [54,55]. The method is similar to the widely used Fourier modal method, also called rigorous coupled-wave analysis (RCWA), and it has been validated by reproducing the examples in Refs. [36] and [30].

First, we consider a periodic array of circular dielectric disks shown schematically in Fig. 1. The disks are surrounded by air. Their radius, thickness, and dielectric constant are $R$, $d$ and $\varepsilon _1$, respectively. For fixed $d=L/2$ and $\varepsilon _1 = 3$, we look for a propagating BIC with $m=1$ by varying the radius $R$. For $R\approx 1.8335L$, we obtain a BIC with $k_* \approx 0.4936 (2\pi /L)$ and $\beta _* \approx 0.3678 (2\pi /L)$. In Fig. 2, we show the six field components of the BIC in the $xz$ plane for three periods in $z$. Due to a scaling, the electromagnetic field is dimensionless, but consistency among all six components is maintained. From Fig. 2, it is clear that the amplitudes of field components decay rapidly away from the core. This BIC with $m=1$ is non-robust with respect to structural perturbations that preserve the reflection symmetry in $z$. For fixed $d=L/2$ and $\varepsilon _1=3$, when $R$ is moved away from $1.8335L$, the BIC is destroyed, so that all resonant modes for $\beta$ near $\beta _* = 0.3678 (2\pi /L)$ have a finite $Q$ factor. In Figs. 3(a) and (b), we show the imaginary part of the normalized frequency $\omega L/(2\pi c) = kL/(2\pi )$, and the $Q$ factor [$Q = -0.5 \mbox {Re}(\omega )/\mbox {Im}(\omega )$] of the resonant modes for $R$ near $1.8335L$ and $\beta$ near $\beta _*$. Similarly, if $d=L/2$, $R$ is fixed at $1.8335L$, and $\varepsilon _1$ is moved away from $3$, the BIC is destroyed. The corresponding $\mbox {Im}(k)$ and $Q$ factor of the resonant modes are shown in Figs. 3(c) and (d).

 

Fig. 2. Scaled field patterns on the $xz$ plane of a propagating BIC with $m=1$ in a periodic array of circular disks. $H_x$, $H_y$ and $H_z$ are scaled magnetic field components multiplied by the free space impedance.

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Fig. 3. Imaginary part of free space wavenumber $k$ and $Q$ factor of resonant modes near a BIC in a periodic array of circular disks. The BIC is obtained for dielectric constant $\varepsilon _1 = 3$ and radius $R = 1.8335L$. Results for $R$ near $1.8335L$ are shown in panels (a) and (b). Those for $\varepsilon _1$ near $3$ are shown in panels (c) and (d).

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Next, we consider a fiber grating consisting of three dielectric media with dielectric constants $\varepsilon _1$, $\varepsilon _2$ and $\varepsilon _3$, respectively. One period of the structure contains one segment of $\varepsilon _1$ medium with length $2d_1$, two segments of $\varepsilon _2$ medium with length $d_2$, and one segment of $\varepsilon _3$ medium with length $d_3$. The radius $R$ of the fiber grating is independent of $z$. If we choose the center of the segment with dielectric constant $\varepsilon _3$ as the origin, the dielectric function $\varepsilon (\rho,z)$ has a reflection symmetry in $z$ and is periodic in $z$ with period $L=2d_1 + 2d_2 + d_3$. The fiber grating is surrounded by air, thus $\varepsilon (\rho,z) = 1$ if $\rho > R$. In Fig. 4, we show one period of the fiber grating for $-L/2 < z < L/2$. Instead of a single segment of length $2d_1$, this unit cell contains two segments of $\varepsilon _1$ medium with length $d_1$. For fixed $R=L$, $d_1=d_2=L/8$, $d_3 = L/2$, $\varepsilon _2 = 3$ and $\varepsilon _3 = 3.4$, we search for a BIC with $m=1$ by varying $\varepsilon _1$. A BIC with $k_* \approx 0.6466 (2\pi /L)$ and $\beta _* \approx 0.1282 (2\pi /L)$ is obtained at $\varepsilon _1 \approx 3.5565$. The field components of this BIC are shown in Fig. 5. The non-robustness of the BIC is verified in Fig. 6, where the $Q$ factor of the resonant mode with fixed $\beta = \beta _*$ is shown as a function $\varepsilon _1$. Clearly, as $\varepsilon _1$ moves away from 3.5565, the BIC is destroyed and becomes a resonant mode with a finite $Q$ factor.

 

Fig. 4. One period of a fiber grating consisting of segments with dielectric constants $\varepsilon _1$, $\varepsilon _2$ and $\varepsilon _3$.

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Fig. 5. Scaled field patterns on the $xz$ plane of a propagating BIC with $m=1$ in a fiber grating. $H_x$, $H_y$ and $H_z$ are scaled magnetic field components multiplied by the free space impedance.

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Fig. 6. Quality factor of resonant modes in a fiber grating for fixed Bloch wavenumber $\beta = \beta _* = 0.1282 (2\pi /L)$ and dielectric constant $\varepsilon _1$ near 3.5565. A BIC with $m=1$ and Bloch wavenumber $\beta _*$ is obtained at $\varepsilon _1 = 3.5565$.

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To verify the theory developed in section 3, we introduce two real parameters $\delta$ and $\gamma$ by

 

Fig. 7. Parametric dependence of a BIC in a fiber grating. Panel (a) shows a curve in the parameter plane on which the BIC exists continuously. Panel (b) shows $k$ and $\beta$ of the BIC as functions of parameter $\delta$.

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5. Conclusion

Existing works on BICs in RSP waveguides are mostly concerned with specific structures, such as fiber gratings and 1D arrays of spheres or circular disks. There are also numerical studies about the dependence of BICs on specific structural parameters, such as the radius of the spheres and the dielectric constant of the disks. The theoretical analysis of this paper is developed for BICs in a general RSP waveguide with a reflection symmetry in $z$. The concept of robustness is defined with respect to small but arbitrary perturbations that preserve the periodicity and reflection symmetry in $z$. It is shown that generic BICs with azimuthal index $m=0$ are robust, and those with $m\ne 0$ are non-robust, but can nevertheless continue to exist if a parameter in the perturbation is properly tuned. The theory is developed by proving the existence of a BIC in the perturbed structure mathematically. It is applicable to BICs with free space and Bloch wavenumbers satisfying condition (19). In addition, the BICs are assumed to be generic, as defined by condition (36) in general and condition (38) or (41) for special cases. For the case of non-robust BICs with $m\ne 0$, the continual existence is only proved when the perturbation profile $G$ associated with the tunable parameter $\gamma$ is generic, as defined by condition (37) or (39).

The robustness theory developed in this paper can be compared with the theory for BICs in symmetric 2D structures (with 1D periodicity) given by a dielectric function $\varepsilon (y,z)$ that is independent of $x$, periodic in $y$, bounded in $z$, and symmetric in both $y$ and $z$. The cylindrical coordinates $\rho$, $\phi$ and $z$ for describing RSP waveguides correspond to $z$, $x$ and $y$, respectively. In such a 2D structure, a generic BIC with wave vector $(\alpha, \beta )$ and a frequency in the band for single propagating diffraction order, is robust with respect to all symmetry-preserving perturbations. This is true for both $\alpha = 0$ and $\alpha \ne 0$. Notice that $\alpha$ corresponds to $m$ for BICs in RSP waveguides, but unlike the fixed integer $m$, $\alpha$ is a changeable solution parameter for BICs in 2D structures. Consequently, BICs in 2D structure with $\alpha \ne 0$ and those in RSP waveguides with $m\ne 0$ exhibit distinct robustness properties.