1. Introduction

Some optical waveguides, such as the strip or ridge waveguides, consist of a core in a layered background which itself is a planar waveguide (usually, a slab waveguide) [1–4]. Such a waveguide may have only leaky modes for which power is lost laterally by coupling with outgoing propagating modes of the background planar waveguide [2–7]. It has been observed long time ago that by tuning the structural parameters, the leakage loss of a leaky mode in such a waveguide can be sharply reduced [2–9]. In fact, the leakage loss can be completely inhibited, and in that case, the leaky mode becomes a bound state in the continuum (BIC) [10–16]. More precisely, a BIC in such a waveguide with lateral leakage channels (assuming there is no material loss) is a true guided mode with a real angular frequency $\omega$, a real propagation constant $\beta$, and a field confined around the core, but $\beta$ is less than the largest propagation constant $\eta _{\rm max}$ of all propagating modes of the background planar waveguide. Notice that the BIC coexists with a propagating mode of the background planar waveguide having the in-plane wavevector $(\pm \alpha _{\rm max}, \beta )$, where $\alpha _{\rm max} = ( \eta ^2_{\rm max} - \beta ^2)^{1/2}>0$. A scattering problem can be formulated with the above propagating mode serving as incoming and outgoing waves. The existence of a BIC implies that the scattering problem does not have a unique solution.

Photonic BICs exist in many different structures [17–23], and have found useful applications in lasing, sensing [24,25], switching [26], nonlinear optics [27,28], etc. For lossless structures with a single invariant or periodic direction, a BIC is associated with a real frequency and a real propagation constant (or Bloch wavenumber), and is often regarded as a special member in a continuous family of resonant or leaky modes. Both resonant and leaky modes are eigenmodes satisfying outgoing radiation conditions. They are defined for a real $\beta$ and a real $\omega$, and have a complex $\omega$ and a complex $\beta$, respectively. The families of resonant and leaky modes vary continuously with $\beta$ and $\omega$. Near a typical BIC with frequency $\omega _*$ and propagation constant $\beta _*$, a resonant mode has a complex frequency with $\mbox {Im}(\omega ) \sim |\beta -\beta _*|^2$ (quality factor $Q \sim 1/|\beta -\beta _*|^2$), and a leaky mode has a complex propagation constant with $\mbox {Im}(\beta ) \sim |\omega - \omega _*|^2$.

For practical applications, it is important to understand how small perturbations of the structure affect the BICs. In the perturbed structure, there is usually no BIC with the same $\beta _*$ (if $\beta _* \ne 0$) or the same $\omega _*$. If the amplitude of the perturbation is $\delta$, then the resonant mode with the same $\beta _*$ has $\mbox {Im}(\omega ) \sim \delta ^2$, and the leaky mode with the same $\omega _*$ has $\mbox {Im}(\beta ) \sim \delta ^2$ [29–31]. However, we can still ask whether there is a BIC in the perturbed structure with a real pair $(\beta, \omega )$ near $(\beta _*, \omega _*)$. A BIC is called robust with respect to a set of perturbations, if for any sufficiently small perturbation in that set, there is a BIC in the perturbed structure with $(\beta, \omega )$ near $(\beta _*, \omega _*)$. Symmetry protected BICs are clearly robust with respect to symmetry-preserving perturbations, but BICs unprotected by symmetry can also be robust [32,33]. In addition, if some tunable parameters are introduced in the perturbation, even a non-robust BIC can continue its existence in the perturbed structure if the tunable parameters are properly chosen [34,35]. In fact, the minimum number of tunable parameters needed is a unique integer for the BIC and it is independent of the specific perturbations [36].

It is known that some BICs in optical waveguides with lateral leakage channels are robust [11,37,38]. More precisely, if the following three conditions are satisfied: (1) the waveguide has a lateral mirror symmetry; (2) only one propagating mode of the background planar waveguide has a propagation constant larger than that of the BIC; (3) the BIC is generic, then the BIC is robust with respect to any sufficiently small perturbation that preserves the lateral mirror symmetry [38]. The first two conditions above ensure that there is only one independent radiation channel. The third condition is given precisely in Section 2 and it involves an integral related to the BIC and a corresponding scattering solution.

In this paper, we study non-generic BICs in optical waveguides with lateral leakage channels. It is assumed that conditions (1) and (2) above are still satisfied, but the BIC is non-generic, namely, the integral mentioned above is zero. Since the BIC is surrounded by resonant and leaky modes (for $\beta$ near $\beta _*$ and $\omega$ near $\omega _*$, respectively), we use a perturbation method to show that typically, the nearby resonant and leaky modes have $\mbox {Im}(\omega ) \sim (\beta -\beta _*)^4$ and $\mbox {Im}(\beta ) \sim (\omega -\omega _*)^4$, respectively. This implies that a resonant mode near a non-generic BIC has an ultra-high quality factor ($Q$ factor), and a leaky mode near this BIC has ultra-low leakage loss. It should be mentioned that BICs surrounded by resonant modes with an ultra-high $Q$ factor have been found in many studies [39–47], and they are referred to as super-BICs by some authors [45,47]. Moreover, a BIC surrounded by leaky modes with ultra-low leakage loss has been observed in an early work [9]. Our theory reveals that a non-generic BIC is always a super-BIC.

The other purpose of this work is to find out whether BICs can persist under structural perturbations. The existing theory on robustness covers only generic BICs [38]. Our study indicates that non-generic BICs are indeed not robust, and the perturbed waveguide may or may not have BICs. We consider a general perturbation to the dielectric function given by an arbitrary profile $F$ (that preserves the lateral mirror symmetry) multiplied by an amplitude $\delta$, and show that the perturbed waveguide has no BIC for $\delta < 0$ (or $\delta >0$) and two BICs for $\delta > 0$ (or $\delta < 0$). Since a pair of BICs split out of the non-generic BIC, $\delta =0$ is the bifurcation point of a saddle node bifurcation [48]. On the other hand, as the positive (or negative $\delta$) tends to 0, the two BICs merge to the non-generic BIC, therefore, we can say that the non-generic BIC is a merging-BIC [42–44,46,49]. In existing works on merging-BICs, one studies how two or more BICs on a dispersion surface (or curve) of resonant modes merge together as a structural parameter tends to a particular value. The resulting BIC in the structure with that particular parameter value is called a merging-BIC. It is known that resonant modes near a merging-BIC have an ultra-high $Q$ factor, and thus a merging-BIC is always a super-BIC. Our theory reveals that a non-generic BIC is in fact a merging-BIC for almost any perturbation profile $F$ as $\delta \to 0$.

The rest of this paper is organized as follows. In Section 2, we recall some facts about resonant modes, leaky modes, and BICs in waveguides with lateral leakage channels, and introduce generic and non-generic BICs. In Section 3, we analyze resonant and leaky modes near a BIC, in a fixed waveguide, using a perturbation method. In Section 4, a bifurcation theory for BICs in a perturbed waveguide is developed based on power series in $\sqrt {\delta }$. To illustrate our theory, numerical examples are presented in Sections 3 and 4. The paper is concluded with some remarks in Section 5.

2. Basic definitions

We consider a three-dimensional (3D) $y$-invariant lossless open dielectric waveguide consisting of a waveguide core and a layered background which itself is a planar waveguide parallel to the $xy$ plane, where $y$ is the waveguide axis and $x$ is the lateral variable of the 3D waveguide. The dielectric function of the structure depends only on two transverse variables $x$ and $z$, i.e., $\varepsilon =\varepsilon (x,z)$. The dielectric function $\varepsilon _b$ of the layered background depends only on $z$. The waveguide core occupies a bounded domain in the $xz$ plane. We further assume that $\varepsilon$ is symmetric about $x$, i.e., $\varepsilon (x,z)=\varepsilon (-x,z)$. As an example, we show a ridge waveguide in Fig. 1.

 

Fig. 1. A ridge waveguide with a rectangular core of width $w$ and height $h_r$. The background is a slab waveguide with a slab of thickness $h_s$. The dielectric constants of the substrate (yellow region), the slab (light cyan region), the core (light blue region) and the cladding are $\varepsilon _1$, $\varepsilon _2$, $\varepsilon _3$ and $\varepsilon _t$, respectively.

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For a guided mode propagating along the $y$ axis, the electric field can be written as $\mbox {Re} [ {{\bf E}}({{\bf r}}) e^{-i\omega t}]$, where $\omega$ is the angular frequency, ${{\bf r}}=(x,z)$, ${{\bf E}}={{\bf \Phi }}({{\bf r}})e^{i{\beta }y}$, $\beta$ is the propagation constant, and ${{\bf \Phi }}({{\bf r}}) \to {\bf 0}$ as $|{{\bf r}}|=\sqrt {x^2+z^2} \to \infty$. The frequency-domain Maxwell’s equations give rise to the following equation for the complex amplitude ${{\bf \Phi }}$:

If $\beta >\eta _{\rm max}$, the guided mode is a classical one and it depends on $\beta$ and $\omega$ continuously. A BIC is a special guided mode with $\beta <\eta _{\rm max}$, and it corresponds to an isolated point in the $\beta$-$\omega$ plane. In this paper, we focus on BICs with $\beta$ satisfying $\eta _1^{\mathrm {tm}} < {\beta }< \eta _1^{\mathrm {te}}$. In that case, the BIC is compatible with the left- and right-going first TE mode ${\bf u}^\pm e^{ i ({\beta }y - \omega t)}$, where

Since the BIC is a guided mode, it must decay as $x \to \pm \infty$ and cannot couple with ${\bf u}^+$ or ${\bf u}^-$. Clearly, we can formulate a scattering problem by sending right-going incident wave $C^+ {\bf u}^+$ from $x=-\infty$ and left-going incident wave $C^- {\bf u}^-$ from $x=+\infty$, where $C^+$ and $C^-$ are given constants. The incident waves give rise to outgoing waves $D^- {\bf u}^-$ and $D^+ {\bf u}^+$ for $x \to -\infty$ and $x \to +\infty$, respectively. Because of the BIC, the solution of the scattering problem is not unique, but the amplitudes of the outgoing waves $D^+$ and $D^-$ are well-defined and related to $C^+$ and $C^-$ by a $2 \times 2$ scattering matrix. Since ${\beta }> k \max \{ \sqrt { \varepsilon _1}, \sqrt { \varepsilon _t} \}$, the incident waves will not induce outgoing waves in the substrate and the cladding. Therefore, power is balanced, the scattering matrix is unitary, and $|C^+|^2 + |C^-|^2 = |D^+|^2 + |D^-|^2$.

Since the structure is lossless and symmetric in $x$, the BIC and the corresponding scattering solutions can be scaled to have some useful symmetry. Let ${\mathcal {P}}$ and ${\mathcal {T}}$ be operators satisfying

We are concerned with non-generic BICs satisfying the following condition

Given a particular BIC with frequency $\omega _*$ and propagation constant $\beta _*$, we can consider resonant and leaky modes for $\beta$ near $\beta _*$ and $\omega$ near $\omega _*$, respectively. Both resonant and leaky modes satisfy outgoing radiation conditions as $x \to \pm \infty$, and they are coupled with outgoing first TE mode ${\bf u}^\pm$. In other words, the complex electric-field amplitude ${{\bf \Phi }}$ of a resonant or leaky mode has the following asymptotic relation

3. Resonant and leaky modes near BICs

In this section, we use a perturbation method to analyze the resonant and leaky modes near a BIC in an optical waveguide described in the beginning of Section 2. We consider a BIC with a real frequency $\omega _*$ (freespace wavenumber $k_* = \omega _*/c$), a real propagation constant $\beta _*$, and a complex electric-field amplitude ${{\bf \Phi }}_*$. Without loss of generality, we assume ${\mathcal {P}}{{\bf \Phi }}_*={{\bf \Phi }}_*$. The case for ${\mathcal {P}}{{\bf \Phi }}_*=-{{\bf \Phi }}_*$ is similar. We further scale and normalize the BIC such that ${\mathcal {T}}{{\bf \Phi }}_*={{\bf \Phi }}_*$ and $\left\langle\varepsilon {{\bf \Phi }}_*,{{\bf \Phi }}_*\right\rangle =1$. The scattering solution can be chosen to satisfy

We are concerned with resonant and leaky modes for $\beta$ near $\beta _*$ and $\omega$ near $\omega _*$, respectively. Our theory reveals a major distinction between the generic and non-generic BICs. Near a generic BIC, $\mbox {Im}(\omega )$ of the resonant modes is proportional to $|\beta -\beta _*|^2$, and $\mbox {Im}(\beta )$ of the leaky modes is proportional to $|\omega -\omega _*|^2$. But near a non-generic BIC, the imaginary parts of $\omega$ and $\beta$ of the resonant and leaky modes are much smaller, and they typically exhibit a fourth order dependence on $|\beta -\beta _*|$ and $|\omega - \omega _*|$, respectively.

3.1 Resonant modes: perturbation with respect to $\beta$

We first analyze the resonant modes near a BIC. For any real $\beta$ near $\beta _*$, there is a resonant mode near the BIC. If $\delta =(\beta -\beta _*)L$ is small, we can expand the freespace wavenumber $k=\omega /c$ and complex electric-field amplitude ${{\bf \Phi }}$ of the resonant mode in power series of $\delta$:

Our objective is to determine the leading order for the imaginary part of $k$. We show that if the BIC is generic, then $\mbox {Im}(k)\sim \delta ^2\mbox {Im}(k_2)$ and $\mbox {Im}(k_2)< 0$; if the BIC is non-generic, then $\mbox {Im}(k_2)= 0$ and typically $\mbox {Im}(k)\sim \delta ^4\mbox {Im}(k_4)$ with a negative $\mbox {Im}(k_4)$.

To obtain the above results, we substitute Eqs. (10)–(11) into Eq. (1), collect the $O(1)$ terms, and obtain the following equation satisfied by the BIC:

The above equation defines an operator ${\mathcal {L}}$ and it satisfies ${\mathcal {L}}{\mathcal {T}}={\mathcal {T}}{\mathcal {L}}$ and ${\mathcal {L}}{\mathcal {P}}={\mathcal {P}}{\mathcal {L}}$. Collecting the $O(\delta ^j)$ terms, we obtain

The solvability condition of Eq. (13), i.e., $\left\langle{{\bf \Phi }}_*,{\bf R}_1\right\rangle =0$, leads to $2k_*k_1=-\left\langle{{\bf \Phi }}_*,{\mathcal {B}}{{\bf \Phi }}_*\right\rangle /L$. Moreover, since ${\mathcal {T}}{\mathcal {B}}={\mathcal {B}}{\mathcal {T}}$, we can show that $k_1$ is real. With $k_1$ determined, we have ${\mathcal {P}}{\bf R}_1={\bf R}_1$ and ${\bf R}_1\rightarrow 0$ as $|{{\bf r}}|\rightarrow \infty$. Eq. (13) has a particular solution ${{\bf \Phi }}_1$ that satisfies ${\mathcal {P}}{{\bf \Phi }}_1 ={{\bf \Phi }}_1$ and has the following asymptotic form

A formula for $k_2$ can be deduced from the solvability condition of Eq. (14) with $j=2$. As shown in Appendix A, this condition implies that the imaginary part of $k_2$ is proportional to $-|d_1|^2$. Since the amplitude ${{\bf \Psi }}_*$ is chosen to satisfy the Eq. (9), we have $\left\langle{{\bf \Psi }_*}, {\bf R}_1 \right\rangle =\left\langle{{\bf \Psi }_*}, {\mathcal {B}} {{\bf \Phi }}_* \right\rangle /L$. Therefore, if the BIC is generic, i.e., $\left\langle{{\bf \Psi }_*}, {\mathcal {B}} {{\bf \Phi }}_* \right\rangle \neq 0$, we have $d_1\ne 0$, $\mbox {Im}(k_2) < 0$, $\mbox {Im}(\omega ) \sim |\beta -\beta _*|^2$, and $Q\sim |\beta -\beta _*|^{-2}$.

On the other hand, if the BIC is non-generic, we have $\left \langle{{\bf \Psi }_*}, {\mathcal {B}} {{\bf \Phi }}_* \right\rangle = 0$, thus $d_1=0$ and $\mbox {Im} (k_2)=0$. Moreover, we must have $\mbox {Im}( k_3)=0$, since otherwise $\mbox {Im}(\omega )$ will change sign as $\beta$ passes through $\beta _*$. This is not possible, because any resonant mode with radiation loss must have $\mbox {Im}(\omega ) < 0$, so that the field amplitude can decay with time. With $k_1$, $k_2$, and ${{\bf \Phi }}_1$ determined, as shown in Appendix A, Eq. (14) with $j=2$ has a particular solution ${{\bf \Phi }}_2$ which satisfies ${\mathcal {P}}{{\bf \Phi }}_2 ={{\bf \Phi }}_2$ and has the following asymptotic form

3.2 Leaky modes: perturbation with respect to $\omega$

Next, we analyze the leaky modes near a BIC. For any real $\omega$ near $\omega _*$, the waveguide supports a leaky mode with a complex propagation constant $\beta$ and complex electric-field amplitude ${{\bf \Phi }}$. If $\delta =(k-k_*)L$ is small, we can expand the propagation constant $\beta$ and complex electric-field amplitude ${{\bf \Phi }}$ of the leaky mode in power series of $\delta$:

Substituting Eqs. (17)–(18) into Eq. (1) and collecting $O(\delta ^j)$ terms, we obtain

The solvability condition of Eq. (19) leads to a real $\beta _1L=-2k_*/\left\langle{{\bf \Phi }}_*,{\mathcal {B}}{{\bf \Phi }}_*\right\rangle $. The integral $\left\langle{{\bf \Phi }}_*,{\mathcal {B}}{{\bf \Phi }}_*\right\rangle $ is typically non-zero. With $\beta _1$ determined, following the same process as in the previous subsection, we can show that if the BIC is generic, then $\mbox {Im}(\beta _2)> 0$ and $\mbox {Im}(\beta )\sim |\omega -\omega _*|^2$. On the other hand, if the BIC is non-generic, then $\mbox {Im}(\beta _2) = \mbox {Im}(\beta _3)=0$, and typically $\mbox {Im}(\beta _4)>0$. In that case, the leaky mode near a non-generic BIC has $\mbox {Im}(\beta )\sim |\omega -\omega _*|^4$. Consequently, a leaky mode near a non-generic BIC has ultra-low leakage loss.

3.3 Numerical examples

To validate our theory, we consider a silicon rib waveguide with silica substrate and air cladding, as shown in Fig. 1. The dielectric constants are $\varepsilon _t=1$, $\varepsilon _1=2.1025$, and $\varepsilon _2=\varepsilon _3=11.0304$. The height of the ridge and the thickness of the slab are $h_r=$ 0.03 ${\mu \mbox {m}}$ and $h_s=$ 0.08 ${\mu \mbox {m}}$, respectively. We consider a non-degenerate BIC satisfying ${\mathcal {P}}{{\bf \Phi }}_*={{\bf \Phi }}_*$. By tuning the width of the ridge, a merging-BIC is obtained at $w=w_\natural \approx 0.3396$ ${\mu \mbox {m}}$. The wavenumber $k$ and propagation constant $\beta$ of the BICs for different width $w$ are shown in Fig. 2. The merging-BIC is marked by a black hexagon. The imaginary part of electromagnetic field components $E_y$ and $H_y$ of the merging-BIC are shown in Fig. 3. In Fig. 4, we show the quantity $V_c=\left\langle{{\bf \Psi }_*}, {\mathcal {B}} {{\bf \Phi }}_* \right\rangle $ for different BICs. It is clear that for the merging-BIC at $w=w_\natural$, we have $V_c=0$. Therefore, the merging-BIC is indeed a non-generic BIC.

 

Fig. 2. The wavenumber $k$ and propagation constant $\beta$ of BICs for different width $w$.

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Fig. 3. The imaginary parts of $E_y$ and $H_y$ for the non-generic BIC at $w=w_\natural$.

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Fig. 4. The quantity $V_c$ for different BICs shown in Fig. 2.

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In Fig. 5, we show the $Q$ factor of resonant modes for three different values of $w$. For $w=0.342\;{\mu \mbox {m}}$, the waveguide has two BICs corresponding to the red and green squares in Figs. 2, 4, and 5. As shown in Fig. 5, the $Q$ factor of the resonant modes near these two BICs satisfies $Q\sim |\beta -\beta _*|^{-2}$. For $w=w_\natural$, there is only one non-generic BIC and the $Q$ factor satisfies $Q\sim |\beta -\beta _*|^{-4}$. As shown in Fig. 5, for $w=0.338\;{\mu \mbox {m}}<w_\natural$, there is no BICs in the waveguide, and there are only resonant modes with a finite $Q$ factor.

 

Fig. 5. The $Q$ factor of resonant modes for three different values of width $w$. In the right panel, $\beta _*\approx 32.8168[1/{\mu \mbox {m}}]$ for $w=0.342\;{\mu \mbox {m}}$.

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In Fig. 6, we show the imaginary part of $\beta$ of leaky modes for three different values of $w$. For $w=0.342\;{\mu \mbox {m}}$, it is clear that $\mbox {Im}(\beta )$ of the leaky modes near the two BICs satisfies $\mbox {Im}(\beta )\sim |\omega -\omega _*|^{2}$. For $w=w_\natural$, $\mbox {Im}(\beta )$ satisfies $\mbox {Im}(\beta )\sim |\omega -\omega _*|^{4}$. For $w=0.338\;{\mu \mbox {m}}<w_\natural$, the waveguide can only support leaky modes.

 

Fig. 6. $\mbox {Im}(\beta )$ of leaky modes for three different values of width $w$. In the right panel, $k_*\approx 12.8403[1/{\mu \mbox {m}}]$ for $w=0.342\;{\mu \mbox {m}}$.

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4. Bifurcation theory for non-generic BICs

In the previous section, we found a merging-BIC by tuning the ridge width, and showed that the merging-BIC is in fact a non-generic BIC. We also showed that the waveguide has two BICs for $w>w_\natural$ and no BIC for $w<w_\natural$. Notice that a small change of $w$ around $w_\natural$ can be regarded as a perturbation of the waveguide. In this section, we consider a general perturbation to waveguides with a non-generic BIC, and analyze the existence of BICs in the perturbed waveguide.

Using the same notations for the unperturbed waveguide and the non-generic BIC, we consider a perturbed waveguide with a dielectric function given by

In the remainder of this section, we focus on the case $\delta >0$ and $\chi (F)>0$, and show that there indeed exist two BICs which are given by power series of $\sqrt {\delta }$:

To prove the above results, we first substitute Eq. (23) into Eq. (1), collect terms of different powers of $\delta ^{j/2}$, and obtain

Since the original BIC is non-generic and ${{\bf \Psi }}_*$ is chosen to satisfy $\left\langle\varepsilon _*{{\bf \Psi }}_*,{{\bf \Phi }}_*\right\rangle =0$, we obtain $\left\langle{{\bf \Psi }_*},{\bf B}_1\right\rangle = 0$. The condition $\left\langle{{{\bf \Phi }}}_*,{\bf B}_1\right\rangle = 0$ leads to a real linear relation $2k_*k_1=-\beta _1\left\langle{{\bf \Phi }}_*,{\mathcal {B}}{{\bf \Phi }}_*\right\rangle $. Using this result, as shown in Appendix B, the condition $\left\langle{{\bf \Psi }_*},{\bf B}_2\right\rangle =0$ gives rise to a real quadratic equation of $\beta _1$:

For each $\beta _1$, we have a real $k_1$ and Eq. (25) has a particular solution ${{\bf \Phi }}_1$ that satisfies Eq. (24) and decays to zero as $x\rightarrow \pm \infty$. For each $\beta _1$ given in Eq. (29) and $j\geq 2$, the two conditions $\left\langle{{{\bf \Phi }}}_*,{\bf B}_j\right\rangle = 0$ and $\left\langle{{{\bf \Psi }}_*},{\bf B}_{j+1}\right\rangle = 0$ give rise to a real linear system for $k_j$ and $\beta _j$ which is uniquely solvable and guarantees that Eq. (26) has a solution ${{\bf \Phi }}_j$ decaying at infinity and satisfying Eq. (24). Therefore, if $\chi (F)>0$ and $\delta >0$, we have two BICs in the perturbed waveguide.

On the contrary, if $\chi (F)<0$, $\beta _1$ is complex, thus the perturbed waveguide (with $\delta >0$) does not have any BIC given as the power series (23). For perturbed waveguides with a negative $\delta$, the results can be obtained by substituting $\delta$ and $F$ with $-\delta$ and $-F$, respectively.

Notice that if $\delta$ is regarded as a parameter, two BICs emerge at $\delta =0$ [for $\delta >0$ or $\delta <0$, depending on the sign of $\chi (F)$]. Therefore, $\delta =0$ (corresponding to the non-generic BIC) is a bifurcation point. Conversely, as $\delta$ tends to 0, these two BICs merge to the non-generic BIC. This implies that the non-generic BIC is actually a merging-BIC when $\delta$ is the tuning parameter. Existing studies on merging-BICs are concerned with specific examples and specific parameters [9,37,43,44,46,49]. Our study reveals that a non-generic BIC is a merging-BIC with respect to any general perturbation. Moreover, our theory is applicable to the usual parameter-tuning process for searching merging-BICs. If the dielectric function $\varepsilon$ depends on a parameter $s$ and there is a merging-BIC at $s=s_*$, then $\varepsilon ({\bf r}, s) = \varepsilon ({\bf r}, s_*) + (s-s_*) \partial _s \varepsilon ({\bf r}, s_*) + \dots \approx \varepsilon _* + \delta F$, where $\varepsilon _* = \varepsilon ({\bf r}, s_*)$, $\delta = s - s_*$, and $F = \partial _s \varepsilon ({\bf r}, s_*)$. Therefore, by checking the sign of $\chi (F)$, we can tell whether the merging process occurs for $s$ decreasing or increasing to $s_*$ and the two BICs for $s$ near $s_*$ exhibit a dependence on $|s-s_*|^{1/2}$ as in Eq. (23).

To verify our theory, we regard the silicon rib waveguide with $w=w_\natural$, studied in subsection 3.3, as the unperturbed waveguide. In the following, we change the dielectric constant of the ridge and show the bifurcation phenomenon near the non-generic BIC. More specially, we let the perturbation profile $F$ satisfy $F= -1$ and $F= 0$ in and outside the ridge, respectively. For such a profile $F$, we can verify that $\chi (F)>0$. As shown in Fig. 7, for $\delta >0$, two BICs emerge from the non-generic BIC at $\delta =0$ with the local behavior $\beta -\beta _*=\mathcal {O}(\sqrt {\delta })$ and $k-k_*=\mathcal {O}(\sqrt {\delta })$. For $\delta <0$, there is no BIC. On the other hand, if we assume that $F= 1$ and $F= 0$ in and outside the ridge respectively, we have $\chi (F)<0$. Therefore, two BICs exist in the perturbed waveguide with $\delta <0$ and no BIC exists for $\delta >0$.

 

Fig. 7. $\beta$ and $k$ of BICs emerging from a non-generic BIC marked by a black hexagram.

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5. Conclusion

In this paper, we built a theoretical framework for non-generic BICs in waveguides with lateral leakage channels, and revealed their relation with merging-BICs and super-BICs. The definition of non-generic BICs is associated with the robustness theory developed in Ref. [38]. The generic and non-generic BICs are defined by a special integral which is non-zero and zero, respectively. We developed a perturbation theory for resonant and leaky modes near generic and non-generic BICs. It is shown that for a non-generic BIC with a real propagation constant $\beta _*$ and a frequency $\omega _*$, we typically have $Q\sim |\beta -\beta _*|^{-4}$ for the resonant mode with a real propagation constant $\beta$ near $\beta _*$. BICs surrounded by resonant modes with an ultra-high $Q$ factor have been found in many works and they are referred to as super-BICs by some authors [45,47]. Such a special BIC is usually obtained by merging a few BICs in a single dispersion surface/curve through tuning a structural parameter, and it is also referred to as a merging-BIC. We also studied general structural perturbations to waveguides supporting a non-generic BIC, and developed a bifurcation theory for BICs in the perturbed waveguide. The theory implies that a non-generic BIC can be regarded as a merging-BIC for almost any perturbation profile, and gives useful information to the parameter-tuning process for searching merging-BICs. In essence, non-generic BICs, super-BICs and merging-BICs are different names for the same class of special BICs. Since the definition of a non-generic BIC involves only the BIC and the associated scattering solution, the existence of a non-generic BIC is an intrinsic property of the waveguide.