1. Introduction

A bound state in the continuum (BIC) is a trapped or guided mode that decays to zero in a spatial direction along which radiation modes with the same frequency and wavevector (when appropriate) can propagate to or from infinity [1–4]. In photonics, BICs have been found on various structures including waveguides with a local distortion [5], periodic structures sandwiched between two homogeneous media [6–9], waveguides with lateral leakage channels [10–20], rotationally symmetric periodic structures surrounded by a homogeneous medium [21,22], and anisotropic multilayer structures [23]. Applications of the BICs are mostly related to resonances with arbitrarily large quality factors ($Q$ factors) that appear when the structure or the wavevector are properly perturbed [24–28]. High-$Q$ resonances give rise to strong local field enhancement and abrupt variations in transmission/reflection spectra that are useful for lasing [29], sensing [30,31], imaging [32,33], switching [34], and nonlinear optics [35].

An important theoretical question is about the robustness (i.e., the continual existence) of the BICs with respect to small structural perturbations. If a BIC is protected by a symmetry; i.e., there is a symmetry mismatch between the BIC and the radiation modes, it is typically robust with respect to symmetry-preserving perturbations. For periodic structures with the in-plane inversion symmetry and the up-down mirror symmetry, some BICs unprotected by symmetry are also robust with respect to perturbations preserving these two symmetries [36–40]. An optical waveguide may have a lateral leakage channel, if the structure away from the waveguide core has guided modes that can propagate in the lateral direction. A BIC on such a structure may or may not be symmetry-protected. In a recent work [19], Bykov et al. studied a waveguide with a rectangular core in a background with a slab, showed that even the BICs unprotected by symmetry are robust with respect to variations in the structural parameters.

In this paper, we analyze the robustness of BICs in optical waveguides with respect to arbitrary structural perturbations. Based on an all-order perturbation method first developed in [38], we show analytically that a generic BIC in a waveguide with a lateral mirror symmetry and a single radiation channel is robust with respect to any perturbation that preserves the lateral mirror symmetry. The rest of this paper is organized as follows. In Sect. 2, we present the necessary background concerning the waveguide structure and the BICs. In Sect. 3, we formulate a related scattering problem and present some useful properties of the scattering solutions. Section 4 contains the main result about robustness, including the precise conditions and the proof. In Sect. 5, we present numerical examples to illustrate the continual existence of BICs when structural parameters are varied. The paper is concluded with a brief discussion in Sect. 6.

2. Guided modes and BICs

We consider a three-dimensional (3D) $z$-invariant lossless dielectric waveguide with a core embedded in a layered background. The dielectric function $\varepsilon$ of the waveguide depends only on the two transverse variables $x$ and $y$, that is, $\varepsilon =\varepsilon (\textbf {r})$ for $\textbf {r}=(x,y)$. We assume the structure is symmetric in $x$, the core is contained in the region given by $|x| < W/2$ for some $W>0$, and the background is a layered medium with a dielectric function $\varepsilon _s$ that depends only on $y$. Thus

In addition, we assume the background itself is a 2D waveguide, typically consisting of a slab, a substrate and a cladding. We denote the dielectric constants of the cladding and the substrate by $\varepsilon _0$ and $\varepsilon _2$, respectively, and assume $\varepsilon _2 \ge \varepsilon _0 \ge 1$. As an example, we show a ridge waveguide in Fig. 1, where the ridge is an isosceles trapezoid.

 

Fig. 1. A waveguide with an isosceles trapezoidal ridge of height $h_r$, base $W$ and base angle $\pi /2-\theta$. The thickness of the slab is $h_s$. The dielectric constants of the slab and the ridge, the substrate, and the cladding are $\varepsilon _1$, $\varepsilon _2$, and $\varepsilon _0$ satisfying $\varepsilon _1 > \varepsilon _2 \ge \varepsilon _0\ge 1$.

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For such a 3D $z$-invariant waveguide, a guided mode decays to zero as $r=\sqrt {x^2 + y^2} \to \infty$, and depends on $z$ as $\exp ( i \gamma z)$, where $\gamma$ is a real constant (the propagating constant). The electric field of the guided mode can be written as $\mbox {Re} [ \textbf {E}\, e^{i (\gamma z - \omega t)} ]$, where $\omega$ is the angular frequency, and $\textbf {E}$ is a vector function that depends only on $\textbf{r}$. From the frequency-domain Maxwell’s equations, it is easy to show that $\textbf {E}$ satisfies

Since $\varepsilon$ is real, it is easy to verify that the vector field

Since the structure has a reflection symmetry in $x$, the field components of a guided mode is either even in $x$ or odd in $x$. More precisely, let $\textbf {E}(\textbf {r})$ be the electric field of a guided mode and $\hat{\textbf {E}}(\textbf {r})$ be the vector field given by

The layered background given by the dielectric function $\varepsilon _s(y)$ is a 2D waveguide with transverse-electric (TE) and transverse-magnetic (TM) modes. To distinguish these modes from the eigenmodes of the original 3D waveguide, we call them slab modes. A TE slab mode is characterized by a zero $y$ component of the electric field and a scalar function $u(y)$ satisfying the eigenvalue problem:

Notice that if $\gamma < k \eta ^\textrm {te}$, then $\alpha ^\textrm {te} > 0$ and the field given in Eq. (12) is outgoing as $x \to +\infty$. If $\gamma > k \eta ^\textrm {te}$, then $\alpha ^\textrm {te}$ is pure imaginary with a positive imaginary part, and the field decays to zero as $x \to +\infty$. For $x < -W/2$, the TE slab mode with the same $z$ dependence has an electric field given by

The case for the TM slab mode is similar. It has a zero $y$ component in the magnetic field and a scalar function $v(y)$ satisfying the eigenvalue problem

Assuming the same dependence on $z$, the electric field of the TM slab mode can be written down as

At a fixed frequency, the layered background has a finite number of slab modes. We order the TE and TM slab modes separately according to their effective indices, and denote them as $\left \{ u_j, \eta _j^\textrm {te} \right \}$ for $j = 0$, 1, $\ldots$, $N-1$, and $\left \{ v_j, \eta _j^\textrm {tm} \right \}$ for $j = 0$, 1, $\ldots$, $M-1$, where $N$ and $M$ are the numbers of TE and TM modes, respectively. It is known that the TE and TM modes are interlaced, namely, their effective indices satisfy

In the slab region given by $|x| > W/2$, a general time-harmonic field is a superposition of the finite number of slab modes and a continuum of radiation modes [41]. If the field depends on $z$ as $e^{i \gamma z}$ and is outgoing as $x \to \pm \infty$, then the electric field can be written as

In the above, $\alpha _j^\textrm {te}$, $\alpha _j^\textrm {tm}$, $\displaystyle \textbf {f}_{+, j}^\textrm {te}$, $\textbf {f}_{-, j}^\textrm {te}$, $\textbf {f}_{+, j}^\textrm {tm}$, $\textbf {f}_{-, j}^\textrm {tm}$ are defined for the $j$th TE or TM mode following Eqs. (13), (20), (14), (16), (21) and (22); $a_j^\pm$ and $b_j^\pm$ are the coefficients of the slab modes; and the terms with the subscript “$\diamond$” are radiation modes that depend continuously on a real wavenumber $\beta$ for the $y$ direction. The wavenumber for the $x$ direction is

If $\textbf {E}$ is the electric field of a guided mode (of the 3D waveguide) with a frequency $\omega$ and a propagation constant $\gamma$, then it must satisfy Eq. (23) for properly chosen coefficients $\{ a_j^\pm , a_\diamond ^\pm (\beta ), b_j^\pm , b_\diamond ^\pm (\beta ) \}$. A regular guided mode satisfies the condition $k \eta _0^\textrm {te} < \gamma$, thus all $\alpha _j^\textrm {te}$, $\alpha _j^\textrm {tm}$ and $\alpha _{\diamond }(\beta )$ are pure imaginary with positive imaginary parts, and $\textbf {E} \to 0$ as $x \to \pm \infty$. A BIC is a special guided mode with a positive propagation constant satisfying

This means that $\alpha _0^\textrm {te}$ and probably other $\alpha _j^\textrm {tm}$ or $\alpha _j^\textrm {te}$ are positive, thus the layered background has at least one slab mode that can radiate power to $x = \pm \infty$ and propagate in the $z$ direction as the BIC. Since the BIC must decay to zero as $x \to \pm \infty$, it is clear that $a_0^\pm$ and probably the coefficients of other slab modes (that radiate out power in the $x$ direction) must be zero. We are particularly interested in BICs satisfying

In that case, $\alpha ^\textrm {te}_0$ is real, all other $\alpha ^\textrm {te}_j$, $\alpha ^\textrm {tm}_j$, and $\alpha _{\diamond }(\beta )$ are pure imaginary. Therefore, only the fundamental TE slab mode can radiate power to $x = \pm \infty$, and the coefficient $a_0^{\pm }$ of the BIC must be zero. Due to the reflection symmetry of the structure in the $x$ direction, we have either $a_0^+= a_0^-$ or $a_0^+= - a_0^-$, if the BIC satisfies Eqs. (9) or (10), respectively.

3. Scattering solutions

For $\omega$ and $\gamma$ satisfying Eq. (30), the fundamental TE slab mode can propagate in the slab region ($|x|> W/2$) with the $(x,z)$ dependence given by $\exp [ i (\gamma z \pm \alpha _0^\textrm {te} x)]$, where $\alpha _0^\textrm {te} = [ (k\eta _0^\textrm {te})^2 - \gamma ^2]^{1/2} > 0$. This implies that we can consider scattering problems using the fundamental TE slab mode as incident waves. The solutions of the scattering problems are needed in Sect. 4 for proving the robustness of BICs. If

To construct a scattering solution satisfying the symmetry Eqs. (9) or (10), we need to specify incident waves in both left and right slab regions. For Eq. (9), we let the incident waves be $C \textbf {u}_+$ for $x < -W/2$ and $C \textbf {u}_{-}$ for $x > W/2$, where $C$ is a nonzero constant, then any solution of this scattering problem satisfies

Although the far field given in the right hand side above already satisfies Eq. (9), the total field $\textbf {E}(\textbf {r})$ may not, because the solution is not unique if a BIC exists at the same $\omega$ and $\gamma$. However, we can define a vector field $\hat{\textbf {E}}(\textbf {r})$ as in Eq. (8), then $\hat{\textbf {E}}$ and $(\textbf {E} + \hat{\textbf {E}})/2$ solve the same scattering problem. We can replace our original solution by $(\textbf {E} + \hat{\textbf {E}})/2$, then the new solution, still denoted as $\textbf {E}(\textbf {r})$, satisfies Eq. (9).

By choosing a proper constant $C$, we can ensure that Eq. (7) is also satisfied. Since $|R+T|=1$, we have $R+T = e^{ i \varphi }$ for a real phase $\varphi$. If we let $C = e^{-i\varphi /2}$, then Eq. (32) becomes

Due to the possible non-uniqueness, the above solution may not satisfy Eq. (7). However, we can define $\tilde{\textbf {E}}$ as in Eq. (6), then $\tilde{\textbf {E}}$ and $(\textbf {E}+\tilde{\textbf {E}})/2$ solve the same scattering problem. If we replace the original solution by $(\textbf {E}+\tilde{\textbf {E}})/2$, then the new $\textbf {E}(\textbf {r})$ satisfies Eq. (7).

Since we are concerned with the case where a BIC exists at the same $\omega$ and $\gamma$, we denote the electric fields of the BIC and the scattering solution by $\textbf {E}_0$ and $\textbf {E}^{(s)}$, respectively. Assuming both $\textbf {E}_0$ and $\textbf {E}^{(s)}$ satisfies Eqs. (7) and (9), we can replace $\textbf {E}^{(s)}$ by $\textbf {E}^{(s)} + C_1 \textbf {E}_0$ for a real constant $C_1$ (if necessary), such that the following orthogonality condition is satisfied:

In summary, we have constructed a scattering solution $\textbf {E}^{(s)}$ that satisfies Eqs. (7), (9) and (34). Similarly, we can construct another scattering solution that satisfies Eqs. (7), (10) and (34).

4. Robustness of BICs

In this section, we study the robustness of BICs in optical waveguides based on an all-order perturbation method developed in our previous works [38,40]. The robustness refers to the continual existence of a BIC under small structural perturbations. The original unperturbed waveguide is given by a dielectric function $\varepsilon _0(\textbf {r})$ satisfying Eqs. (1) and (2). It is assumed that the unperturbed waveguide has a non-degenerate BIC with an electric field $\textbf {E}_0(\textbf {r})$, a frequency $\omega _0$ and a propagation constant $\gamma _0$. Moreover, $\omega _0$ and $\gamma _0$ must satisfy Eq. (30), where $k$ should be replaced by $k_0=\omega _0/c$, and $\eta _0^\textrm {te}$ and $\eta _0^\textrm {tm}$ are effective indices of the fundamental TE and TM slab modes at frequency $\omega _0$. Without loss of generality, we assume $\textbf{E}_0(\textbf {r})$ satisfies Eq. (7) and (9). The dielectric function of the perturbed waveguide is given by

Following the procedure developed in [38,40], we construct the BIC in the perturbed waveguide by expanding $\textbf{E}(\textbf {r})$, $\gamma$ and $k = \omega /c$ in power series of $\delta$:

Note that for any real $\gamma$ near $\gamma _0$, the governing Eqs. (3) and (4) have a solution satisfying an outgoing radiation condition as $x \to \pm \infty$, but the frequency $\omega$ is complex in general. Similarly, for any real $\omega$ near $\omega _0$, there is an outgoing solution with a complex $\gamma$. These outgoing solutions with a complex $\omega$ or a complex $\gamma$ are the resonant and leaky modes, respectively. Our objective is to determine a BIC that decays to zero as $r \to \infty$, and it only exists for a particular pair $(\omega , \gamma )$ near $(\omega _0, \gamma _0)$. Therefore, both $\omega$ and $\gamma$ must be determined together with the field.

Inserting expansions Eqs. (36)–(38) into Eqs. (3) and (4), and comparing the coefficients of $\delta ^j$ for $j \geq 1$, we obtain the following equation for $\textbf{E}_j(\textbf {r})$:

If such an $\textbf {E}_j$ exists, we can show (see Appendix A) that

The above can be written as

Since $\mathscr {B} \textbf {E}_0$ satisfies Eq. (7), $a_{11}$ and $a_{21}$ are real. In addition, $a_{12} > 0$, and $a_{22}=0$ due to the orthogonality Eq. (34). Therefore, all entries of $\textbf {A}$ are real. Meanwhile, since $\textbf {F}_j$ depends on $\gamma _m$, $k_m$ and $\textbf {E}_m$ for $0\le m < j$, $\gamma _m$ and $k_m$ are real, and $\textbf {E}_m$ satisfies Eq. (7), $\textbf{F}_j$ also satisfies Eq. (7), and thus $b_{1j}$ and $b_{2j}$ are real. Therefore, when $\textbf {A}$ is invertible, $\gamma _j$ and $k_j$ can be solved, and they are real.

Although Eqs. (41) and (42) are derived assuming Eq. (39) has a solution, the linear system (43) does not depend on $\textbf {E}_j$. In the following, we show that if $\gamma _j$ and $k_j$ satisfy Eq. (43) and $\textbf {A}$ is invertible, then Eq. (39) indeed has a solution $\textbf {E}_j$ that decays to zero exponentially as $r \to \infty$ and satisfies Eqs. (9) and (7).

Since $\textbf {F}_j$ depends on $\textbf {E}_m$ for $0\le m < j$ and $\textbf {E}_m \to 0$ exponentially as $r \to \infty$, the right hand side of Eq. (39) tends to zero as $r \to \infty$ and can be regarded as a source distributed around the waveguide core. Therefore, we require the solutions of Eq. (39), if they exist, to satisfy an outgoing radiation condition. Since only the fundamental TE slab mode can radiate out power (in the $x$ direction), the outgoing radiation condition gives rise to

The second condition, Eq. (42), can be used to show that $d_j^\pm = 0$. For any positive $h$, let $\Omega _h$ be the domain given by $|x| < h$ and $|y| < \infty$, then Eqs. (39) and (42) clearly imply that

In Appendix B, we show that

It can be easily verified that the right hand side of Eq. (39) satisfies Eq. (7). Thus, for any solution $\textbf {E}_j$, we can construct a vector field $\tilde{\textbf {E}}_j$ as in Eq. (6), then $\tilde{\textbf {E}}_j$ and $(\textbf {E}_j + \tilde{\textbf {E}}_j)/2$ also satisfy Eq. (39) and they decay to zero exponentially as $r \to \infty$. Therefore, we can replace $\textbf {E}_j$ by $(\textbf {E}_j + \tilde{\textbf {E}}_j)/2$ (if necessary), and assume $\textbf {E}_j$ satisfies Eq. (7).

The above proof is for a BIC satisfying the symmetry Eq. (9). If the BIC satisfies Eq. (10), we need to use the scattering solution that also satisfies Eq. (10), then $d_j^+= -d_j^-$, and $d_j^+-d_j^-$ should appear in the right hand side of Eq. (49). If $s(\textbf {r})$ does not satisfy Eq. (1), the above proof fails, because we no longer have $d_j^+ = \pm d_j^-$, then $d_j^\pm \ne 0$ in general, and $\textbf {E}_j$ does not decay to zero as $r \to \infty$. To ensure that matrix $\textbf {A}$ is invertible, the BIC of the unperturbed waveguide is required to satisfy

5. Numerical examples

In this section, we present some numerical examples to demonstrate the robustness of BICs in optical waveguides. We start with a silicon ridge waveguide with a rectangular ridge, a SiO$_2$ substrate, and an air cladding. The dielectric constants of the cladding, the ridge and the slab, and the substrate are $\varepsilon _0=1$, $\varepsilon _1 = 12.25$, and $\varepsilon _2 = 2.1025$, respectively. The geometric parameters, as shown in Fig. 1, are $W = 0.68 \mu m$, $h_r = 0.07 \mu m$, $h_s = 0.15 \mu m$, and $\theta = 0$. This waveguide has a BIC with freespace wavenumber $k_0 \approx 0.7266 (2\pi /\mu m)$ (i.e. freespace wavelength $\lambda _0 \approx 1.3762 \mu m$) and propagation constant $\gamma _0 \approx 1.4794 (2\pi / \mu m)$. In Fig. 2(a), we show the dispersion curves of the fundamental TE and TM slab modes (solid blue curves) and the dispersion curves of some resonant modes (solid red lines, for real part of $k=\omega /c$ only). The BIC, marked as “$\ast$” in Fig. 2(a), satisfies Eq. (30) and is a special point on a band of resonant modes. In Fig. 2(b), we show the $Q$ factor of this particular band for $\gamma$ near $\gamma _0$. The BIC has been scaled to satisfy Eqs. (7) and (5) with $L=1 \mu m$. The $z$-components of the electric field and a scaled magnetic field of the BIC are shown in Figs. 2(c) and 2(d), respectively. The scaled magnetic field is obtained by multiplying the free space impedance to the original magnetic field. Due to the normalization, Eq. (5), all field components are dimensionless. From Fig. 2(c), it is clear that the BIC satisfies symmetry Eq. (10). Following Sect. 3, we calculate a scattering solution satisfying Eqs. (7), (10) and (34). The $z$ components of the scattering solution; i.e., $E^{(s)}_z$ and $H^{(s)}_z$, are shown in Figs. 2(e) and 2(f).

 

Fig. 2. A rectangular ridge waveguide with a BIC. (a) Dispersion curves of the fundamental TE and TM slab modes (solid blue curves) and a few resonant modes (solid red curves), and a BIC with $(\gamma _0, k_0) \approx (1.4794, 0.7266) (2 \pi / \mu m)$. (b) $Q$ factor of the resonant modes in a band with the BIC. (c) and (d) the imaginary parts of $E_z$ and $H_z$ of the BIC, respectively. (e) and (f) the imaginary parts of $E^{(s)}_z$ and $H^{(s)}_z$ of The scattering solution.

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It is easy to verify that the BIC satisfies Eq. (50). Therefore, according to the theory developed in Sect. 4, the BIC should be robust with respect to any perturbation that preserves Eq. (1); i.e., the left-right mirror symmetry. To validate the theory, we consider a perturbation profile

 

Fig. 3. BIC in a ridge waveguide with a varying dielectric function. (a) $Q$ factor of resonant modes on a band with a BIC, for $\delta = 0.2$. (b) Dispersion curve of the same band. The BIC is located at $(\gamma , k) \approx (1.4947, 0.7288) (2 \pi / \mu m)$ and shown as the “$\ast$”. (c) and (d) Values of $\gamma$ and $k$ of the BIC for different $\delta$ (solid blue curves) and their first order approximations (red dashed lines).

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As a second test for the theory, we consider a waveguide with an isosceles trapezoidal ridge. For the same set of parameters, $\varepsilon _0$, $\varepsilon _1$, $\varepsilon _2$, $W$, $h_r$, and $h_s$, we study the waveguide allowing $\theta$, shown in Fig. 1, to increase from $0^\circ$ to $10.2^\circ$. The numerical results reveal a BIC that depends on $\theta$ continuously. The propagation constant $\gamma$ and freespace wavenumber $k$ of the BIC are shown as functions of $\theta$ in Figs. 4(c) and 3(d), respectively. From Fig. 4(d), it is clear that $k$ and the wavelength of the BIC vary more rapidly at larger values of $\theta$. For $\theta = 5^{\circ }$, the BIC, as usual, is a special point on a band of resonant modes. The $Q$ factor and the dispersion curve of this band are shown in Figs. 4(a) and 4(b), respectively. The BIC is obtained with $(\gamma , k) \approx (1.6082, 0.7544) (2\pi /\mu m)$ and shown as the “$\ast$” in Fig. 4(b). Additional numerical results indicate that the BIC ceases to exist for $\theta > 10.2^{\circ }$. It appears that the BIC in the waveguide with $\theta \approx 10.2^\circ$ fails to satisfy Eq. (50); i.e., $a_{21}=0$ and matrix $\textbf{A}$ is not invertible. In Fig. 4(e), we show $a_{21}$ as a function of $\theta$. It indicates clearly that $a_{12}$ can become zero when $\theta$ is further increased.

 

Fig. 4. BIC in a trapezoidal ridge waveguide. (a) $Q$ factor of resonant mode on a band with a BIC, for $\theta =5^\circ$. (b) Dispersion curve of the same band. The BIC is located as $(\gamma , k) \approx (1.6082, 0.7544) (2 \pi / \mu m)$ and shown as the “$\ast$”. (c) and (d) $\gamma$ and $k$ of the BIC for different values of $\theta$. (e) The $(2,1)$ entry of matrix $\textbf {A}$ as a function of $\theta$.

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6. Conclusion

A 3D $z$-invariant optical waveguide may have a lateral leakage channel if the waveguide core is embedded in a background which itself is a 2D waveguide (typically a slab waveguide). A guided mode of the 3D waveguide is a BIC if its propagation constant $\gamma$ is smaller than the propagation constant of the fundamental TE slab mode. Such a BIC is useful for low-refractive-index light guiding [4,13,18,20]. Resonant effects due to high-$Q$ resonances associated with the BIC can also be realized if one specifies an incident TE slab mode in the layered background. In this paper, we studied BICs satisfying Eq. (30), so that only the fundamental TE slab mode can have the same $z$ dependence as the BIC and still propagate in the lateral direction. For waveguides with the left-right mirror symmetry; i.e., Eq. (1), we showed that any BIC satisfying Eqs. (30) and (50) is robust with respect to structural perturbations that preserve Eq. (1). The left-right mirror symmetry is imposed so that the solutions satisfy either Eq. (9) or Eq. (10), and the radiation channels to the left and right (via the fundamental TE slab mode) are essential the same. Equation (50) is given assuming the BIC and the scattering solution are properly scaled, have the same lateral symmetry, and are orthogonal to each other; i.e., they are required to satisfy Eqs. (7), (34), and (9) or (10).

It is straightforward to extend our result to waveguides without the left-right mirror symmetry, but with leakage channels in only one lateral direction, for example, the positive $x$ direction. A guided mode of such a waveguide is a BIC, if its propagation constant $\gamma$ is smaller than the propagation constant of the right fundamental TE slab mode (of the layered structure for $x>W/2$). If $\gamma$ is larger than the propagation constants of all other right slab modes and all left slab modes (of the layered structure for $x< -W/2$, if they exist), then the BIC exists in a continuum with only one radiation channel. Such a BIC, if it is non-degenerate and satisfies the generic Eq. (50), is robust with respect to any structure perturbation. The proof is slightly simpler than the one given in Sect. 3, because we no longer need to worry about the symmetry Eqs. (9) and (10). It is only necessary to ensure the BIC and the scattering solution satisfy Eqs. (7) and (34).

The BICs studied in this paper are quite different from those in periodic structures sandwiched between two homogeneous media. They have a different number of direction along which the field is confined, and they exist in radiation continua provided by different waves. The left-right mirror symmetry assumed in this paper and the up-down mirror symmetry for biperiodic structures with propagating BICs [40], serve the same purpose, that is, to lock together the radiation channels in the opposite directions. For biperiodic structures, besides the up-down mirror symmetry, the in-plane inversion symmetry is also needed to ensure the robustness of propagating BICs [40]. In contrast, for waveguides with lateral leakage channels, robustness of BICs does not require any additional symmetry. This difference is related to our assumption that the waveguide is $z$-invariant and the perturbation is $z$-independent.