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

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


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][2][3][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][3][4][5][6][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][3][4][5][6][7][8][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][11][12][13][14][15][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 ω, a real propagation constant β, and a field confined around the core, but β is less than the largest propagation constant η 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 (±α max , β), where α max = (η 2 max − β 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][18][19][20][21][22][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 it 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 β and a real ω, and have a complex ω and a complex β, respectively.The families of resonant and leaky modes vary continuously with β and ω.Near a typical BIC with frequency ω * and propagation constant β * , a resonant mode has a complex frequency with Im(ω) ∼ |β − β * | 2 (quality factor Q ∼ 1/|β − β * | 2 ), and a leaky mode has a complex propagation constant with Im(β) ∼ |ω − ω * | 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 β * (if β * = 0) or the same ω * .If the amplitude of the perturbation is δ, then the resonant mode with the same β * has Im(ω) ∼ δ 2 , and the leaky mode with the same ω * has Im(β) ∼ δ 2 [29][30][31].However, we can still ask whether there is a BIC in the perturbed structure with a real pair (β, ω) near (β * , ω * ).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 (β, ω) near (β * , ω * ).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 β near β * and ω near ω * , respectively), we use a perturbation method to show that typically, the nearby resonant and leaky modes have Im(ω) ∼ (β − β * ) 4 and Im(β) ∼ (ω − ω * ) 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][40][41][42][43][44][45][46][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 δ, and show that the perturbed waveguide has no BIC for δ < 0 (or δ > 0) and two BICs for δ > 0 (or δ < 0).Since a pair of BICs split out of the non-generic BIC, δ = 0 is the bifurcation point of a saddle node bifurcation [48].On the other hand, as the positive (or negative δ) 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, and it is surrounded by resonant modes of ultrahigh Q factor, and thus it is also a super-BIC.Our theory reveals that a non-generic BIC is in fact a merging-BIC for δ → 0 and almost any perturbation profile F .
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 √ δ.To illustrate our theory, numerical examples are presented in Sections 3 and 4. The paper is concluded with some remarks in Section 5.

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., ε = ε(x, z).The dielectric function ε b of the layered background depends only on z.The waveguide core occupies a bounded domain in the xz plane.We further assume that ε is symmetric about x, i.e., ε(x, z) = ε(−x, z).As an example, we show a ridge waveguide in Fig. 1.A ridge waveguide with a rectangular core of width w and height hr.The background is a slab waveguide with a slab of thickness hs.The dielectric constants of the substrate (yellow region), the slab (light cyan region), the core (light blue region) and the cladding are ε1, ε2, ε3 and εt, respectively.
For a guided mode propagating along the y axis, the electric field can be written as Re[E(r)e −iωt ], where ω is the angular frequency, r = (x, z), E = Φ(r)e iβy , β is the propagation constant, and The frequency-domain Maxwell's equations give rise to the following equation for the complex amplitude Φ: where k = ω/c is the free space wavenumber and ŷ is the unit vector in the y direction.Since the field must decay as z → ±∞, the propagation constant satisfies For the same frequency, the background planar waveguide may support a few guided modes.We order the eigenmodes of the planar waveguide according to their propagation constants, denote the propagation constant of the j-th transverse electric (TE) mode by η te j and the corresponding vertical profile by u j (z), and those of the j-the transverse magnetic (TM) mode by η tm j and v j (z).Both u j (z) and v j (z) are real functions and they can be normalized so that where L is a characteristic length.Typically, the propagation constants satisfy Thus, the first TE mode has the largest propagation constant, i.e., η max = η te 1 .If β > η max , the guided mode is a classical one and it depends on β and ω continuously.A BIC is a special guided mode with β < η max , and it corresponds to an isolated point in the β-ω plane.In this paper, we focus on BICs with β satisfying η tm 1 < β < η te 1 .In that case, the BIC is compatible with the left-and right-going first TE mode u ± e i(βy−ωt) , where Since the BIC is a guided mode, it must decay as x → ±∞ and cannot couple with u + or u − .Clearly, we can formulate a scattering problem by sending right-going incident wave C + u + from x = −∞ and left-going incident wave C − u − from x = +∞, where C + and C − are given constants.The incident waves give rise to outgoing waves D − u − and D + u + for x → −∞ and x → +∞, 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 × 2 scattering matrix.Since β > k max{ √ ε 1 , √ ε 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 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 P and T be operators satisfying where f = f (x, z) is an arbitrary vector function and f x is the complex conjugate of f x .If the BIC {k, β, Φ} is non-degenerate, we have either PΦ = Φ or PΦ = −Φ, and it can be scaled such that T Φ = Φ.For the same k and β as the BIC, by choosing C − = ±C + , we can construct two scattering solutions satisfying PΨ = ±Ψ, where Ψ is the complex amplitude of the electric field.Moreover, the scattering solutions can be further scaled and shifted such that T Ψ = Ψ and εΨ, Φ = 0, where •, • is the inner product defined as We are concerned with non-generic BICs satisfying the following condition where Ψ is the one with the same parity symmetry (i.e.operation by P) as the BIC, and B is the operator satisfying for any differentiable vector function w(x, z).Condition ( 6) was identified in the robustness theory for BICs in waveguides with lateral leakage channels [38].It has been proved that if the BIC is generic, i.e., Eq. ( 6) is not satisfied, and η tm 1 < β < η te 1 , then it is robust with respect to any small perturbation that preserves the lateral mirror symmetry [38].
Given a particular BIC with frequency ω * and propagation constant β * , we can consider resonant and leaky modes for β near β * and ω near ω * , respectively.Both resonant and leaky modes satisfy outgoing radiation conditions as x → ±∞, and they are coupled with outgoing first TE mode u ± .In other words, the complex electric-field amplitude Φ of a resonant or leaky mode has the following asymptotic relation where c ± 1,te are nonzero coefficients.A resonant mode is defined for a real β.Since power is radiated out laterally to x = ±∞, the amplitude of the resonant mode must decay with time, thus, ω is complex and Im(ω) < 0. As a result, the TE and TM modes of the background planar waveguides are eigenmodes of 1D Helmholtz equations with a complex freespace wavenumber k.All propagation constants η te j and η tm j have a negative imaginary part.Therefore, Im(α te 1 ) < 0 and u ± diverges as x → ±∞.A leaky mode is defined for a real frequency ω.It also loses power laterally, and must decay as it propagates forward.This implies that β is complex and Im(β) > 0. Since the frequency is real, the propagation constants of the background planar waveguide are real, but since β is complex, we still have a complex α te 1 with a negative imaginary part, and u ± also diverges as x → ±∞.

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 ω * (freespace wavenumber k * = ω * /c), a real propagation constant β * , and a complex electric-field amplitude Φ * .Without loss of generality, we assume PΦ * = Φ * .The case for PΦ * = −Φ * is similar.We further scale and normalize the BIC such that T Φ * = Φ * and εΦ * , Φ * = 1.The scattering solution can be chosen to satisfy We are concerned with resonant and leaky modes for β near β * and ω near ω * , respectively.Our theory reveals a major distinction between the generic and non-generic BICs.Near a generic BIC, Im(ω) of the resonant modes is proportional to |β − β * | 2 , and Im(β) of the leaky modes is proportional to |ω − ω * | 2 .But near a non-generic BIC, the imaginary parts of ω and β of the resonant and leaky modes are much smaller, and they typically exhibit a fourth order dependence on |β − β * | and |ω − ω * |, respectively.

Resonant modes: perturbation with respect to β
We first analyze the resonant modes near a BIC.For any real β near β * , there is a resonant mode near the BIC.If δ = (β − β * )L is small, we can expand the freespace wavenumber k = ω/c and complex electric-field amplitude Φ of the resonant mode in power series of δ: Our objective is to determine the leading order for the imaginary part of k.We show that if the BIC is generic, then Im(k) ∼ δ 2 Im(k 2 ) and Im(k 2 ) < 0; if the BIC is non-generic, then Im(k 2 ) = 0 and typically Im(k) ∼ δ 4 Im(k 4 ) with a negative 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 L and it satisfies LT = T L and LP = PL.Collecting the O(δ j ) terms, we obtain where B is the operator defined in Eq. ( 7) with β replaced by β * .The right hand sides R j are listed in Appendix A. As shown in Refs.[38,41], a differential equation Lw = f is solvable if and only if Φ * , f = 0.If Pf = f and f → 0 as |r| → ∞, there exists a particular solution w that satisfies Pw = w and has asymptotic behavior w ∼ du ± * as x → ±∞, where u ± * is defined as in Eq. ( 4) with β replaced by β * , k replaced by k * , etc.Moreover, the coefficient d is a multiple of the integral Ψ * , f .If Ψ * , f = 0, we have d = 0 and then w → 0 as |r| → ∞.
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 Ψ * is chosen to satisfy the Eq. ( 9), we have Ψ * , R 1 = Ψ * , BΦ * /L.Therefore, if the BIC is generic, i.e., Ψ * , BΦ * = 0, we have On the other hand, if the BIC is non-generic, we have Ψ * , BΦ * = 0, thus d 1 = 0 and Im(k 2 ) = 0.Moreover, we must have Im(k 3 ) = 0, since otherwise Im(ω) will change sign as β passes through β * .This is not possible, because any resonant mode with radiation loss must have Im(ω) < 0, so that the field amplitude can decay with time.With k 1 , k 2 , and Φ 1 determined, as shown in Appendix A, Eq. ( 14) with j = 2 has a particular solution Φ 2 which satisfies PΦ 2 = Φ 2 and has the following asymptotic form where d 2 is the coefficient.Moreover, we show that the imaginary part of k 4 is proportional to for a non-generic BIC.Consequently, the resonant mode near a non-generic BIC has an ultra-high Q factor.

Leaky modes: perturbation with respect to ω
Next, we analyze the leaky modes near a BIC.For any real ω near ω * , the waveguide supports a leaky mode with a complex propagation constant β and complex electric-field amplitude Φ.If δ = (k − k * )L is small, we can expand the propagation constant β and complex electric-field amplitude Φ of the leaky mode in power series of δ: Substituting Eqs. ( 17)-( 18) into Eq.( 1) and collecting O(δ j ) terms, we obtain where L j are listed in Appendix A.

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 ε t = 1, ε 1 = 2.1025, and ε 2 = ε 3 = 11.0304.The height of the ridge and the thickness of the slab are h r = 0.03 µm and h s = 0.08 µm, respectively.We consider a non-degenerate BIC satisfying PΦ * = Φ * .By tuning the width of the ridge, a merging-BIC is obtained at w = w ♮ ≈ 0.3396 µm.The wavenumber k and propagation constant β 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 = Ψ * , BΦ * for different BICs.It is clear that for the merging-BIC at w = w ♮ , we have V c = 0. Therefore, the merging-BIC is indeed a non-generic BIC.
In Fig. 5, we show the Q factor of resonant modes for three different values of w.For w = 0.342 µm, 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 ∼ |β − β * | −2 .For w = w ♮ , there is only one non-generic BIC and the Q factor satisfies Q ∼ |β − β * | −4 .As shown in Fig. 5, for w = 0.338 µm < w ♮ , there is no BIC in the waveguide, and there are only resonant modes with a finite Q factor. In

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 ♮ and no BIC for w < w ♮ .Notice that a small change of w around w ♮ 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 where δ is a small real number and F (r) is a real function of x and z.We further assume that F is symmetric in x and has compact support.In the previous work on robustness [38], BICs in the perturbed waveguide are constructed through power series of δ by using the condition Ψ * , BΦ * = 0. Therefore, this robustness theory is not applicable to non-generic BICs satisfying Ψ * , BΦ * = 0.In the following, we assume the non-generic BIC in the unperturbed waveguide has a non-zero d 2 [defined in Eq. ( 16)], and introduce a characteristic function χ(F ) given by where A is proportional to d 2 and independent of F .It can be proved that χ(F ) is real, and it is clear that χ(−F ) = −χ(F ).Our main result is that for a sufficiently small δ, if χ(F ) > 0, then the perturbed waveguide has two BICs for δ > 0 and no BIC for δ < 0, and if χ(F ) < 0, then the perturbed waveguide has two BICs for δ < 0 and no BIC for δ > 0.
In the remainder of this section, we focus on the case δ > 0 and χ(F ) > 0, and show that there  indeed exist two BICs which are given by power series of √ δ: where k, β, and Φ are the freespace wavenumber, the propagation constant, and the complex electricfield amplitude of these two BICs, respectively.To justify the existence of these BICs, we need to show for each j ≥ 1, k j and β j can be solved and they are real, Φ j decays rapidly to zero as x → ±∞, and it can be chosen to satisfy In addition, there are two solutions for k j , β j and Φ j corresponding to the two BICs.
To prove the above results, we first substitute Eq. ( 23) into Eq.( 1), collect terms of different powers of δ j/2 , and obtain where the right hand sides B j are listed in Appendix B. For the equation of Φ j to have a solution that decays rapidly to zero as x → ±∞, the right hand side B j must satisfy the following two conditions Since the original BIC is non-generic and Ψ * is chosen to satisfy ε * Ψ * , Φ * = 0, we obtain Using this result, as shown in Appendix B, the condition Ψ * , B 2 = 0 gives rise to a real quadratic equation of β 1 : where A is mentioned earlier in this section.Since χ(F ) > 0, we obtain two real β 1 given by For each β 1 , we have a real k 1 and Eq. ( 25) has a particular solution Φ 1 that satisfies Eq. ( 24) and decays to zero as x → ±∞.For each β 1 given in Eq. ( 29) and j ≥ 2, the two conditions Φ * , B j = 0 and Ψ * , B j+1 = 0 give rise to a real linear system for k j and β j which is uniquely solvable and guarantees that Eq. ( 26) has a solution Φ j decaying at infinity and satisfying Eq. ( 24).Therefore, if χ(F ) > 0 and δ > 0, we have two BICs in the perturbed waveguide.On the contrary, if χ(F ) < 0, β 1 is complex, thus the perturbed waveguide (with δ > 0) does not have any BIC given as the power series (23).For perturbed waveguides with a negative δ, the results can be obtained by substituting δ and F with −δ and −F , respectively.
Notice that if δ regarded as a parameter, two BICs emerge at δ = 0 [for δ > 0 or δ < 0, depending on the sign of χ(F )].Therefore, δ = 0 (corresponding to the non-generic BIC) is a bifurcation point.Conversely, as δ 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 δ 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.
To verify our theory, we regard the silicon rib waveguide with w = w ♮ , 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 χ(F ) > 0. As shown in Fig. 7, for δ > 0, two BICs emerge from the non-generic BIC at δ = 0 with the local behavior For δ < 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 χ(F ) < 0. Therefore, two BICs exist in the perturbed waveguide with δ < 0 and no BIC exists for δ > 0.

CONCLUSION
In this paper, we built a theoretical framework for non-generic BICs in waveguides with lateral leakage channels.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 β * and a frequency ω * , we typically have Q ∼ |β − β * | −4 for the resonant mode with a real propagation constant β near β * .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.However, existing studies on super-BICs or merging-BICs are concerned with specific examples and specific parameters.We studied general perturbations to waveguides supporting non-generic BICs, and developed a bifurcation theory for BICs in the perturbed waveguide.Our work establishes interesting links among non-generic BICs, super-BICs and merging-BICs.Notice that non-generic BICs are defined for the unperturbed waveguide, while super-BICs or merging-BICs are related to perturbing or tuning of parameters.Therefore, the existence of a non-generic BIC is an intrinsic property of the waveguide.

APPENDIX A
In this appendix, we show that Im(k 2 ) defined in subsection 3.3 is proportional to −|d 1 | 2 .Moreover, if 1 = we can prove that Im(k 4 ) is proportional to −|d 2 | 2 .In addition, the physical significance of Im(k 2 ) is stated that it is associated with the leading-order radiation loss in the lateral direction.
Substituting Eqs. ( 10)-(11) into Eq.( 1) and collecting O(δ j ) terms, we have where Φ 0 = Φ * and k 0 = k * .To derive the imaginary part of k 2 , we first recall some fundamental formulas.We have the vector Green's theorem in 2D domain Ω: where n is the outer unit normal vector of ∂Ω, u(r) and v(r) are vector functions.Recall the vector identities Because Φ * decays to zero exponentially as |r| → 0, by using the vector Green's theorem and vector identities, we have Φ * , BΦ 1 = BΦ * , Φ 1 .The solvability of Eq. ( 14) with j = 2, i.e., Φ * , R 2 = 0, gives rise to According to the vector Green's theorem and vector identities, we can obtain where Ω H = (−H, H) × R and α 1 * is defined in Eq. ( 4) with corresponding quantities.Then we can get Im(k 2 ) = −α te 1 * |d 2 1 |/(k * L).Therefore, if the BIC is non-generic, we have d 1 = 0 and Im(k 2 ) = 0.In this case, Φ 1 → 0 and R 2 → 0 as |r| → ∞.It is clear that PR 2 = R 2 and then Eq. ( 14) with j = 2 has a particular solution Φ 2 which satisfies PΦ 2 = Φ 2 and has the following asymptotic form The coefficient d 2 is a multiple of Ψ * , R 2 .By using the same process as in the above, we can prove that Im(k 4 ) is proportional to −|d 2 | 2 .
The complex Poynting vector S for the resonant modes near BICs can be expanded as Equations ( 31)- (32) imply that the imaginary part of k 2 is associated with the leading-order radiation loss in the lateral direction.
Using the above results, we can rewrite Eq. ( 26) with j = 2 as As shown in Appendix A, the coefficient A is a multiple of d 2 since ε * Ψ * , Φ * = 0.In this paper, we assume that d 2 = 0.If χ(F ) > 0, we have two real β 1 = ± χ(F ).Accordingly, we can get a real K 1 .
According to previous results, for each β 1 given in Eq. ( 29) and j ≥ 2, Eq. ( 26) can be written as where G j = F j − 2β 1 β j−1 R2 .It is clear that G j is independent of the unknowns β j and K j .The condition Φ * , B j = 0 gives rise to a real relation between K j and β j : Although β j and K j are not obtained, we can reformulate Φ j as Φ j = β j Φ1 + w j , where Lw j = F j − Φ * , F j ε * Φ * .Since the solution w j is not unique, we can scale w j such that Φ j satisfies Eq. ( 24) if β j is real.Using above results, the condition Ψ * , B j+1 = 0 gives rise to a real linear equation of β j : Equations ( 34) and (35) determine real β j and K j corresponding to each β 1 given in Eq. ( 29).With

Figure 1 .
Figure1.A ridge waveguide with a rectangular core of width w and height hr.The background is a slab waveguide with a slab of thickness hs.The dielectric constants of the substrate (yellow region), the slab (light cyan region), the core (light blue region) and the cladding are ε1, ε2, ε3 and εt, respectively.

Figure 2 .
Figure 2. The wavenumber k and propagation constant β of BICs for different width w.

Figure 3 .
Figure 3.The imaginary parts of Ey and Hy for the non-generic BIC at w = w ♮ .

Figure 4 .
Figure 4.The quantity Vc for different BICs shown in Fig. 2.

Figure 5 .
Figure 5.The Q factor of resonant modes for three different values of width w.In the right panel, β * ≈ 32.8168[1/µm] for w = 0.342 µm.

Figure 7 .
Figure 7. β and k of BICs emerging from a non-generic BIC marked by a black hexagram.