Bound states in the continuum on periodic structures surrounded by strong resonances
aa r X i v : . [ phy s i c s . op ti c s ] M a r Bound states in the continuum on periodic structures surrounded by strongresonances
Lijun Yuan ∗ College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, China
Ya Yan Lu
Department of Mathematics, City University of Hong Kong, Hong Kong
Bound states in the continuum (BICs) are trapped or guided modes with their frequencies inthe frequency intervals of the radiation modes. On periodic structures, a BIC is surrounded bya family of resonant modes with their quality factors approaching infinity. Typically the qualityfactors are proportional to 1 / | β − β ∗ | , where β and β ∗ are the Bloch wavevectors of the resonantmodes and the BIC, respectively. But for some special BICs, the quality factors are proportional to1 / | β − β ∗ | . In this paper, a general condition is derived for such special BICs on two-dimensionalperiodic structures. As a numerical example, we use the general condition to calculate special BICs,which are antisymmetric standing waves, on a periodic array of circular cylinders, and show theirdependence on parameters. The special BICs are important for practical applications, because theyproduce resonances with large quality factors for a very large range of β . I. INTRODUCTION
Bound states in the continuum (BICs), first studied byVon Neumann and Wigner for quantum systems [1], aretrapped or guided modes with their frequencies in thefrequency intervals of radiation modes that carry powerto or from infinity [2]. For light waves, BICs have beenanalyzed and observed for many different structures, in-cluding waveguides with local distortions [3–7], waveg-uides with lateral leaky structures [8–11], and periodicstructures sandwiched between or surrounded by homo-geneous media [12–39]. The BICs on periodic structuresare particularly interesting, because they are surroundedby families of resonant modes (depending on the wavevec-tor) with quality factors ( Q -factors) tending to infinity,and they give rise to collapsing Fano resonances corre-sponding to discontinuities in the transmission and re-flection spectra [40, 41]. The high- Q resonances and therelated strong local fields [42, 43] can be used to enhancelight-matter interactions for applications in lasing [44],nonlinear optics [45], etc. Collapsing Fano resonancescan be exploited in filtering, sensing, and switching ap-plications [46, 47].If the structures are symmetric, the BICs and theradiation modes may have incompatible symmetry sothat they are automatically decoupled. These so-calledsymmetry-protected BICs are well known [12–18]. Theirexistence can be rigorously proved [5, 12, 16, 18], andthey are robust against small structural perturbationsthat preserve the required symmetry. On periodic struc-tures, there are also BICs that do not have a symmetrymismatch with the radiation modes [19–39]. These BICsare often considered as unprotected by symmetry, but forsome important cases, they appear to depend crucially on ∗ Corresponding author: [email protected] the symmetry, and they continue to exist when geomet-ric and material parameters are varied with the relevantsymmetries kept intact [25, 36–39]. In fact, it has beenshown that under the right conditions, these BICs arerobust against any structural changes that preserve therelevant symmetries [37].On periodic structures, a BIC is a guided mode, butit belongs to a family of resonant modes, and can beregarded as a special resonant mode with an infinite Q -factor. Let β ∗ be the Bloch wavevector of a BIC, thenthere is a related family of resonant modes depending onwavevector β . The Q -factors of the resonant modes typ-ically satisfy Q ∼ / | β − β ∗ | . Clearly, a resonant modewith an arbitrarily large Q -factor can be obtained if β ischosen to be sufficiently close to β ∗ , and an arbitrarilylarge local field can be induced by an incident wave withthe wavevector β . However, practical applications of thestrong field enhancement can be limited by the difficultyof controlling β to high precision, in addition to otherpractical issues such as fabrication errors [48], materialdissipation [43], variations in different periods, finite sizes[49–51], etc. In [45], we showed that for symmetric stand-ing waves, which are BICs with β ∗ = and are unpro-tected by symmetry in the usual sense, the Q -factors ofthe associated resonant modes satisfy an inverse fourthpower asymptotic relation, i.e., Q ∼ / | β − β ∗ | . Inthat case, resonances with large Q -factors and stronglocal fields can be realized with a much more relaxedcondition on β . This property has been used to showthat optical bistability can be induced by a very weakincident wave [45], and it should be useful in other ap-plications that require a significant field enhancement.In general, resonant modes near antisymmetric stand-ing waves (ASWs), which are symmetry-protected BICswith β ∗ = , only satisfy the inverse quadratic asymp-totic relation, but Bulgakov and Maksimov found a fewexamples for which the inverse fourth power relation issatisfied [49].In this paper, we derive a general condition for thosespecial BICs with the Q -factors of the associated resonantmodes satisfying the inverse fourth power relation. Forsimplicity, the theory is developed for two-dimensional(2D) periodic structures. We use a perturbation methodassuming | β − β ∗ | is small. The condition is given inintegrals involving the BIC itself and related diffractionsolutions for incident waves with the same frequency andsame wavevector. With this general condition, it be-comes feasible to systematically search the parameter val-ues of the periodic structure supporting the special BICs.As numerical examples, we calculate special BICs on aperiodic array of circular dielectric cylinders, and showtheir dependence on the parameters. II. BICS AND RESONANT MODES
We consider 2D dielectric structures which are invari-ant in z , periodic in y with period L , and bounded inthe x direction by | x | < D for some constant D , where { x, y, z } is a Cartesian coordinate system. The surround-ing medium for | x | > D is assumed to be vacuum. There-fore, the dielectric function ǫ satisfies ǫ ( x, y + L ) = ǫ ( x, y )for all ( x, y ), and ǫ ( x, y ) = 1 for | x | > D . For the E po-larization, the z -component of the electric field, denotedas u , satisfies the Helmholtz equation ∂ u∂x + ∂ u∂y + k ǫu = 0 , (1)where k = ω/c is the free space wavenumber, ω is theangular frequency, c is the speed of light in vacuum, andthe time dependence is assumed to be e − iωt .A Bloch mode on the periodic structure is a solutionof Eq. (1) given as u = φ ( x, y ) e iβy , (2)where φ is periodic in y with period L and β is the realBloch wavenumber. Due to the periodicity of φ , β can berestricted to the interval [ − π/L, π/L ]. If φ → | x | →∞ , then u given in Eq. (2) is a guided mode. Typically,guided modes that depend on β and ω continuously canonly be found below the light line, i.e., for k < | β | . ABIC is a special guided mode above the light line, i.e., β and k satisfy the condition k > | β | . For a given structure,BICs can only exist at isolated points in the βω plane.In the homogeneous media given by | x | > D , we canexpand a Bloch mode in plane waves, that is u ( x, y ) = ∞ X j = −∞ c ± j e i ( β j y ± α j x ) (3)where the “+” and “ − ” signs are chosen for x > D and x < − D respectively, and β j = β + 2 πjL , α j = q k − β j . (4) If k < | β j | , then α j = i q β j − k is pure imaginary,and the corresponding plane wave is evanescent. For aBIC, one or more α j are real, then the correspondingcoefficients c ± j must vanish, since the BIC must decay tozero as | x | → ∞ .Above the light line, if the frequency ω is allowed tobe complex, there are Bloch mode solutions that dependon a real wavenumber β continuously. These solutionsare the resonant modes, and they satisfy outgoing ra-diation conditions as x → ±∞ . Due to the time de-pendence e − iωt , the imaginary part of the complex fre-quency of a resonant mode must be negative, so thatits amplitude decays with time. The Q -factor is givenby Q = − . ω ) / Im( ω ). The expansion (3) is stillvalid, but the complex square root for α j must be de-fined to maintain continuity as Im( ω ) →
0. This can beachieved by using a square root with a branch cut alongthe negative imaginary axis (instead of the negative realaxis), that is, if ξ = | ξ | e iθ for − π/ < θ ≤ π/
2, then √ ξ = p | ξ | e iθ/ . Notice that α and probably a few other α j have negative real parts. Therefore, a resonant modeblows up as | x | → ∞ . As β is continuously varied fol-lowing a family of resonant modes, Im( ω ) may becomezero at some special values of β , and they correspond tothe BICs. Therefore, although a BIC is a guided mode,it belongs to a family of resonant modes, and it can beregarded as a special resonant mode with an infinite Q -factor. III. PERTURBATION ANALYSIS
Given a BIC on a periodic structure with frequency ω ∗ and Bloch wavenumber β ∗ , we are interested in the com-plex frequency ω and the Q -factor of the nearby resonantmode for wavenumber β close to β ∗ . For simplicity, wescale the BIC such that1 L Z Ω | φ ∗ | d r = 1 , (5)where Ω is one period of the structure given by − L/ 1, and L = ∂ x + ∂ y + 2 iβ ∗ ∂ y + k ∗ ǫ − β ∗ . (13)In addition, φ j must satisfy proper outgoing radiationconditions as | x | → ∞ .Equation (10) is simply the governing Helmholtz equa-tion of the BIC. The first order term φ satisfies the inho-mogeneous Eq. (11) which is singular and has no solution,unless the right hand side is orthogonal to φ ∗ . Multiply-ing φ ∗ (the complex conjugate of φ ∗ ) to both sides ofEq. (11) and integrating on domain Ω, we obtain k = ω c = β ∗ R Ω | φ ∗ | d r − i R Ω φ ∗ ∂ y φ ∗ d r k ∗ R Ω ǫ | φ ∗ | d r . (14)It is easy to show that k is real. Therefore, in generalIm( ω ) is proportional to ( β − β ∗ ) .For k given above, Eq. (11) has a solution. Similarto the plane wave expansion (3), φ can be written downexplicitly for | x | > D . Importantly, φ contains only asingle outgoing plane wave as x → ±∞ , that is φ ∼ d ± e ± iα ∗ x , x → ±∞ , (15)where d ± are unknown coefficients and α ∗ = p k ∗ − β ∗ .A formula for k can be derived from the solvability con-dition of Eq. (12). In particular, the imaginary part of k has the following simple formulaIm( k ) = Im( ω ) c = − Lα ∗ (cid:0) | d +0 | + | d − | (cid:1) k ∗ R Ω ǫ | φ ∗ | d r . (16)A special case of Eq. (16) was previously derived in [45].Additional details on the derivation of Eqs. (14) and (16)are given in Appendix.Notice that if φ radiates power to x = ±∞ , d +0 and d − are nonzero, then Im( ω ) = 0. In that case, the imag-inary part of the complex frequency satisfiesIm( ω ) ∼ − cLα ∗ (cid:0) | d +0 | + | d − | (cid:1) k ∗ R Ω ǫ | φ ∗ | d r ( β − β ∗ ) , (17)and the Q -factor satisfies Q ∼ k ∗ R Ω ǫ | φ ∗ | d r Lα ∗ (cid:0) | d +0 | + | d − | (cid:1) ( β − β ∗ ) − . (18) On the other hand, if φ does not radiate power to in-finity, then d ± = 0, Im( ω ) = 0, and Eqs. (17) and(18) are no longer valid. In that case, Im( ω ) must alsobe zero, since otherwise, Im( ω ) changes signs when β passes through β ∗ . This is not possible, since Im( ω )of a resonant mode is always negative. Therefore, if φ is non-radiative, we expect Im( ω ) ∼ ( β − β ∗ ) and Q ∼ ( β − β ∗ ) − . IV. STRONG RESONANCES On periodic structures, the Q -factors of resonantmodes around certain special BICs satisfy an inversefourth power asymptotic relation Q ∼ ( β − β ∗ ) − . Thishappens when the first order perturbation φ does notradiate power to infinity, i.e., d ± = 0. However, to checkthis condition, it is necessary to solve φ from Eq. (11).This is not very convenient. Ideally, one would like tohave a condition that involves the BIC φ ∗ only. Thisdoes not seem to be possible. In the following, we derivea condition that involves the BIC φ ∗ and related diffrac-tion solutions for incident waves with the same ω ∗ and β ∗ as the BIC.For Eq. (1) with k replaced by k ∗ , we consider twodiffraction problems with incident waves e i ( β ∗ y + α ∗ x ) and e i ( β ∗ y − α ∗ x ) given in the left and right homogeneous me-dia, respectively. The solutions of these two diffractionproblems are denoted as u l and u r , respectively, and theysatisfy u j = ϕ j ( x, y ) e iβ ∗ y , j ∈ { l, r } , (19)where ϕ l and ϕ r are periodic in y with period L . It shouldbe pointed out that the existence of a BIC implies thatthe corresponding diffraction problems have no unique-ness [12, 16], but the diffraction solutions are uniquelydefined in the far field as | x | → ∞ . In fact, ϕ l and ϕ r have the following asymptotic formulae ϕ l ∼ ( e iα ∗ x + R l e − iα ∗ x , x → −∞ T l e iα ∗ x , x → + ∞ ,ϕ r ∼ ( T r e − iα ∗ x , x → −∞ e − iα ∗ x + R r e iα ∗ x , x → + ∞ , (20)where R l , T l , R r and T r are the reflection and transmis-sion amplitudes associated with the left and right inci-dent waves, respectively. It is well known that the scat-tering matrix S = (cid:20) R l T r T l R r (cid:21) is unitary. Notice that u l and u r are easier to solve than φ , since they satisfy ahomogeneous Helmholtz equation with a zero right handside.Equation (11) for φ can be written as L φ = 2 G ,where G = − i∂ y φ ∗ + ( β ∗ − k ∗ k ǫ ) φ ∗ . (21)Since G → x → ±∞ , the followingintegrals F j = Z Ω ϕ j Gd r , j ∈ { l, r } (22)are well defined. On the other hand, ϕ j and φ (in gen-eral) do not decay to zero as | x | → ∞ , it is not immedi-ately clear whether ϕ j L φ is integrable on the unboundeddomain Ω. However, for any h ≥ D , we can define arectangular domain Ω h given by | y | < L/ | x | < h ,and evaluate the integral on Ω h , then take the limit as h → ∞ . Clearly, the limit must exist andlim h →∞ Z Ω h ϕ j L φ d r = 2 F j , j ∈ { l, r } . In Appendix, we show thatlim h →∞ Z Ω h ϕ l L φ d r = 2 iLα ∗ (cid:0) d +0 T l + d − R l (cid:1) , (23)lim h →∞ Z Ω h ϕ r L φ d r = 2 iLα ∗ (cid:0) d +0 R r + d − T r (cid:1) . (24)Therefore, F l = iLα ∗ (cid:0) d +0 T l + d − R l (cid:1) ,F r = iLα ∗ (cid:0) d +0 R r + d − T r (cid:1) . Using the unitarity of the scattering matrix S , it is easyto show that | d +0 | + | d − | = | F l | + | F r | ( Lα ∗ ) . (25)If ( F l , F r ) = (0 , ω ) ∼ − c (cid:0) | F l | + | F r | (cid:1) Lα ∗ k ∗ R Ω ǫ | φ ∗ | d r ( β − β ∗ ) . (26)Clearly, the condition d +0 = d − = 0 is equivalent to F l = F r = 0 . (27)BICs are most easily found on structures with suitablesymmetries. If the structure has a reflection symmetry inthe y direction, it is often possible to find ASWs whichare symmetry-protected BICs with β ∗ = 0. Assumingthe origin is chosen so that ǫ ( x, y ) = ǫ ( x, − y ), then theASWs are odd functions of y . From Eq. (14), it is easyto see that k = 0, thus G = − i∂ y φ ∗ and F j = − i Z Ω ϕ j ∂φ ∗ ∂y d r , j ∈ { l, r } . (28)Notice that symmetric standing waves (which are evenfunctions of y ) may also exist on periodic structures witha reflection symmetry in y . In [45], it is shown that d ± =0 is always true for the symmetric standing waves. Thisis so, because k = 0 and G = − i∂ y φ ∗ are still valid, thus G is odd in y . Meanwhile, ϕ l and ϕ r are even in y .Therefore, F l = F r = 0.Propagating BICs (with β ∗ = 0) are often found onstructures with an additional reflection symmetry in the x direction. With a properly chosen origin, the dielectricfunction satisfies ǫ ( x, y ) = ǫ ( x, − y ) = ǫ ( − x, y ) (29)for all ( x, y ). In that case, we can reduce the condition F l = F r = 0 to a single real condition. In [37], it is shownthat if the BIC u ∗ = φ ∗ e iβ ∗ y is a single mode, then it iseither even in x or odd in x , and it can be scaled to satisfythe PT -symmetric condition u ∗ ( x, y ) = u ∗ ( x, − y ) . (30)In particular, the ASWs should be scaled as pure imagi-nary functions.It is also shown in [37] that there is a complex number C with unit magnitude, such that u e = C ( u l + u r ) and u o = C ( u l − u r ) are even and odd in x , respectively, andare also PT -symmetric. As in Eq. (19), we associate twoperiodic functions ϕ e and ϕ o with u e and u o , respectively.It is easy to see that φ ∗ , ϕ e , ϕ o and G given in Eq. (21)are all PT -symmetric. Furthermore, let F e and F o bedefined as in Eq. (22) for j ∈ { e, o } , then F e = C ( F l + F r )and F o = C ( F l − F r ). This leads to | F e | + | F o | = 2( | F l | + | F r | ) . (31)If ( F e , F o ) = (0 , ω ) ∼ − c (cid:0) | F e | + | F o | (cid:1) Lα ∗ k ∗ R Ω ǫ | φ ∗ | d r ( β − β ∗ ) . (32)Clearly, the condition F l = F r = 0 is equivalent to F e = F o = 0 . (33)If a function satisfies the PT -symmetric condition (30),its real part is even in y and its imaginary part is oddin y . Therefore, F e and F o are always real. If the BICis even in x , then F o is always zero, and it is only nec-essary to check one real condition F e = 0. Similarly, ifthe BIC is odd in x , the only condition is F o = 0. ForASWs on periodic structures with the double reflectionsymmetry (29), G = − i∂ y φ ∗ is real and even in y , andthe corresponding diffraction solutions u e and u o are alsoreal even functions of y . V. NUMERICAL EXAMPLES In this section, some numerical examples are presentedto validate and illustrate the theoretical results developedin the previous sections. As shown in Fig. 1(a), we con-sider a single periodic array of dielectric circular cylinderssurrounded by air. The radius and dielectric constant ofthe cylinders are a and ǫ , respectively. BICs on such xy (a) L a ǫ a/L F e -0.4-0.200.20.4 (b) FIG. 1: (a): A periodic array of circular cylinderssurrounded by air. (b) F e of the first ASW as afunction of radius a for ǫ = 8 . a = 0 . L , ǫ = 8 . E polarization, the array supports five ASWs and one prop-agating BIC. The frequencies and Bloch wavenumbers ofthese BICs are listed in the first and second columns ofTable I, respectively. ω ∗ L/ (2 πc ) β ∗ L/ (2 π ) a : Eq. (35) a : approximation0 . . . . . . . . . . . . . . . . . . . TABLE I: Frequencies and Bloch wavenumbers of sixBICs on a periodic array of circular cylinders withradius a = 0 . L and dielectric constant ǫ = 8 . 2, andtheir exact and approximate coefficients a .First, we check the formula for Im( ω ) for ordinary BICswhere φ radiates power to infinity. The periodic ar-ray has reflection symmetries in both x and y directions,thus, Eq. (32) is applicable. In terms of the normalizedfrequency and normalized wavenumber, Eq. (32) can bewritten as Im( ω ) L πc ∼ − a (cid:20) ( β − β ∗ ) L π (cid:21) , (34)where a is a dimensionless coefficient given by a = π (cid:0) | F e | + | F o | (cid:1) L k ∗ α ∗ R ǫ | φ ∗ | d r . (35)Recall that F e and F o are real, and one of them is alwayszero. For each BIC listed in Table I, we calculate a by Eq. (35), and also find an approximation of a bya quadratic polynomial fitting the numerical values ofIm( ω ) for β = β ∗ and β ∗ ± . π/L . As shown in the thirdand fourth columns of Table I, the exact and approximate values of a agree very well. This confirms that Eqs. (34)and (35) are correct.We are interested in the special BICs surrounded bystrong resonances with Q ∼ / ( β − β ∗ ) . It is known thatthe symmetric standing waves (even in y ) are examplesof such special BICs [45], and they exist when a and ǫ lieon a curve in the aǫ plane [33]. Bulgakov and Maksimov[49] found a number of ASWs which also have this spe-cial property. Using the perturbation theory developed inprevious sections, we can find these special BICs system-atically by searching the parameters a and ǫ , such that F e = 0 for an x -even BIC or F o = 0 for an x -odd BIC.The first ASW listed in Table I, with ω ∗ L/ (2 πc ) = 0 . a = 0 . L and ǫ = 8 . 2, is even in x . We calculate F e for this BIC as a function of a with a fixed ǫ = 8 . 2. Theresult is shown in Fig. 1(b). Since F e is real and changessigns, it must have a zero. It turns out that F e = 0 for a = 0 . L . The frequency of the corresponding ASWis ω ∗ L/ (2 πc ) = 0 . (a) x/L -1.5 -1 -0.5 0 0.5 1 1.5 y / L -0.500.5 (b) x/L -1.5 -1 -0.5 0 0.5 1 1.5 y / L -0.500.5 FIG. 2: Wave field patterns of two special x -even ASWson periodic arrays of circular cylinders with ǫ = 8 . a = 0 . L . (b) The fifth ASWfor a = 0 . L .For other values of ǫ > F e of the first ASW canstill reach zero for a properly chosen a . Those values of a and ǫ such that F e = 0 for the first ASW give riseto a curve in the aǫ plane, shown as the red solid linein Fig. 3(a). The corresponding frequency ω ∗ is shownwith a as the solid red curve in Fig. 3(b). It appearsthat as ǫ is increased, the related a increases and ap-proaches a constant as infinity. It should be pointed thatthe first ASW exists for all ǫ > < a ≤ . L [18].The curve represents those parameter values such thatthe ASW becomes a special BIC surrounded by strongresonances.For the other BICs listed in Table I, we also attemptto find parameters a and ǫ such that F e = F o = 0. Itseems that only the fifth ASW, with ω ∗ L/ (2 πc ) = 0 . a = 0 . L and ǫ = 8 . 2, can be tuned to a special BIC.For ǫ = 8 . 2, the fifth ASW, which is also even in x , gives a/L ǫ (a) a/L ω L / ( π c ) (b) FIG. 3: (a): Parameters of the periodic array for twospecial ASWs: the first ASW (red solid curve) and thefifth ASW (blue dashed curve). (b) The correspondingfrequencies of the two special ASWs. Points A and B correspond to Figs. 2(a) and 2(b), respectively. F e = 0 for a = 0 . L . Its frequency is ω ∗ L/ (2 πc ) =0 . ǫ , we also found the corresponding valuesof a such that F e = 0 for the fifth ASW. The results aregiven as a curve in the aǫ plane, i.e., the blue dashed linein Fig. 3(a). The corresponding frequency ω ∗ is shownas the blue dashed line in Fig. 3(b). Notice that ǫ has alower bound around 4 . 67, and it is achieved as a → . L .In addition, a as a function of ǫ , has a minimum around ǫ = 5 . 35, and it seems to approach a constant as ǫ tends to infinity.In order to evaluate F e for an x -even BIC, we need tocalculate the x -even diffraction solution u e = C ( u l + u r ),where C is chosen so that u e is PT -symmetric and C = e − iτ for a real constant τ . As shown in [37], this leads to ϕ e ∼ α ∗ x ± τ ) , x → ±∞ . (36)In Fig. 4, we show the diffraction solutions correspondingto the two special ASWs shown in Fig. 2.In Sec. III, we argued that if Im( ω ) = 0, then Im( ω )should also be zero, and Im( ω ) should be proportional to( β − β ∗ ) in general. For the two ASWs shown in Fig. 2,we check this result by computing the complex frequen-cies of some nearby resonant modes directly. In Fig. 5,we show Im( ω ) as functions of β in a logarithmic scalefor some resonant modes near these two ASWs. The nu-merical results confirm the fourth order relation betweenIm( ω ) and β . VI. CONCLUSION BICs on periodic structures are surrounded by reso-nant modes with Q -factors approaching infinity. High- Q resonances and the resulting strong local field en-hancement have important applications in lasing, non-linear optics, etc. On 2D periodic structures, the Q - (a) x/L -1.5 -1 -0.5 0 0.5 1 1.5 y / L -0.500.5 -1012 (b) x/L -1.5 -1 -0.5 0 0.5 1 1.5 y / L -0.500.5 -202 FIG. 4: (a) and (b): Diffraction solutions ϕ e corresponding to the special ASWs shown in Figs. 2(a)and 2(b), respectively. β L/ (2 π ) -2 -1 - I m ( ω ) L / ( π c ) -8 -6 -4 FIG. 5: Imaginary parts of the complex frequencies vs. β for resonant modes near ASWs shown in Fig. 2(a) ( ◦ )and Fig. 2(b) ( ∗ ), respectively.factors of the resonant modes near a BIC usually sat-isfy Q ∼ / ( β − β ∗ ) , where β and β ∗ are the Blochwavenumbers of the resonant mode and the BIC, respec-tively. In this paper, we derived a general condition forspecial BICs so that their nearby resonant modes have Q ∼ / ( β − β ∗ ) . These special BICs produce high- Q res-onances for a very large range of β , and they are usefulbecause precise control of β may be difficult in practice.The conditions for the special BICs are given in integralsinvolving the BIC and related diffraction solutions, andthey imply that the first order perturbation φ does notradiate power to infinity. Numerical examples are givenfor two families of ASWs on a periodic array of circularcylinders.In practical applications, the BICs always dissolve intoresonant modes with finite Q -factors, because the struc-tures are always finite and fabrication errors will breakthe required symmetries and periodicity. We expect thespecial BICs have advantages over the ordinary BICs inpractical structures with fabrication errors and in finitestructures, but a rigorous analysis is still under devel-opment. In addition, the results of this paper are re-stricted to 2D structures. Clearly, it is worthwhile toderive similar conditions for special BICs on bi-periodicthree-dimensional (3D) structures and rotationally sym-metric 3D structures. ACKNOWLEDGMENTS The authors acknowledge support from the Basicand Advanced Research Project of CQ CSTC (GrantNo. cstc2016jcyjA0491), the Scientific and Technologi-cal Research Program of Chongqing Municipal Educa-tion Commission (Grant No. KJ1706155), the Programfor University Innovation Team of Chongqing (Grant No.CXTDX201601026), and the Research Grants Council ofHong Kong Special Administrative Region, China (GrantNo. CityU 11304117). APPENDIX For operator L given in Eq. (13), it is easy to verifythat φ ∗ L φ − φ L φ ∗ = ∇ · (cid:0) φ ∗ ∇ φ − φ ∇ φ ∗ (cid:1) + 2 iβ ∗ ∂ y ( φ φ ∗ ) , where ∇ is the 2D gradient operator. The integral on Ωof the right hand side can be reduced to an integral on ∂ Ω (the boundary of Ω) by the divergence theorem. It iszero, since φ ∗ and φ are periodic in y and φ ∗ → | x | → ∞ . Meanwhile, φ ∗ satisfies Eq. (10),thus R Ω φ ∗ L φ d r = 0. From Eq. (11) for φ , it is clearthat Z Ω φ ∗ [ − i∂ y φ ∗ + 2( β ∗ − k ∗ k ǫ ) φ ∗ ] d r = 0 . This leads to Eq. (14). Meanwhile, R φ ∗ ∂ y φ ∗ d r is pureimaginary, since Z Ω φ ∗ ∂ y φ ∗ d r = Z Ω ∂ y | φ ∗ | d r − Z Ω φ ∗ ∂ y φ ∗ d r = − Z Ω φ ∗ ∂ y φ ∗ d r . Therefore, k is real.Similarly, we have R Ω φ ∗ L φ d r = 0. Multiplying bothsides of Eq. (12) and integrating on Ω, we obtain k = R Ω (1 − k ǫ ) | φ ∗ | d r + R Ω R d r k ∗ R Ω ǫ | φ ∗ | d r , where R = φ ∗ [ − i∂ y φ + 2( β ∗ − k ∗ k ǫ ) φ ]. Therefore,Im( k ) = Im (cid:2)R Ω R d r (cid:3) k ∗ R Ω ǫ | φ ∗ | d r . From the complex conjugate of Eq. (11), we obtain Z Ω h φ L φ d r = Z Ω h R d r , where Ω h is the rectangular domain defined in Sec. IV.The right hand side above requires an integration byparts that switches the integral from φ ∂ y φ ∗ to − φ ∗ ∂ y φ .Meanwhile, it is easy to verify that Z Ω h φ L φ d r = Z ∂ Ω h φ ∂φ ∂ν ds + Z Ω h (cid:20) ( k ∗ ǫ − β ∗ ) | φ | − |∇ φ | − iβ ∗ φ ∂φ ∂y (cid:21) d r , where ∂ Ω h is the boundary of Ω h and ν is its unit outwardnormal vector. The second term in the right hand sideabove is real. Therefore,Im (cid:20)Z Ω h R d r (cid:21) = Im (cid:20)Z ∂ Ω h φ ∂φ ∂ν ds (cid:21) . Since φ is periodic in y , the line integrals at y = ± L/ Z ∂ Ω h φ ∂φ ∂ν ds = Z L/ − L/ (cid:20) φ ∂φ ∂x (cid:21) x = hx = − h dy, where P ( x, y ) | x = hx = − h denotes P ( h, y ) − P ( − h, y ).For | x | > D , the equation for φ is quite simple. It isnot difficult to see that φ = d ± e ± iα ∗ x + X j =0 d ± j ( x ) e i πjy/L e ± γ ∗ j x for x > D and x < − D respectively, where γ ∗ j = − iα ∗ j ispositive, d ± are unknown coefficients, and d ± j ( x ) ( j = 0)are unknown linear polynomials of x . The above giveslim h → + ∞ Z L/ − L/ (cid:20) φ ∂φ ∂x (cid:21) x = hx = − h dy = − iLα ∗ ( | d +0 | + | d − | ) , and Im (cid:2)R Ω R d r (cid:3) = − Lα ∗ ( | d +0 | + | d − | ), and finallyEq. (16).To show Eq. (23), we notice that ϕ l L φ − φ L ϕ l = ∇ · [ ϕ l ∇ φ − φ ∇ ϕ l ] + 2 iβ ∗ ∂ y ( φ ϕ l ) . Since ϕ l satisfies the Helmholtz equation and both φ and ϕ l are periodic in y , we have Z Ω h ϕ l L φ d r = Z ∂ Ω h (cid:20) ϕ l ∂φ ∂ν − φ ∂ϕ l ∂ν (cid:21) ds. In the right hand side above, the integrals on the twoedges at y = ± L/ Z Ω h ϕ l L φ d r = Z L/ − L/ (cid:20) ϕ l ∂φ ∂x − φ ∂ϕ l ∂x (cid:21) x = hx = − h dy. Based on the asymptotic formula (20), it is easy to showthat as h → + ∞ , the right hand side above tend to2 iLα ∗ ( d +0 T l + d − R l ). This leads to Eq. (23). The prooffor Eq. (24) is similar. [1] J. von Neumann and E. Wigner, “ ¨Uber merkw¨urdigediskrete eigenwerte,” Z. Physik , 291-293 (1929).[2] C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos,and M. Soljaˇci´c, “Bound states in the continuum,” Nat.Rev. Mater. , 16048 (2016).[3] D. V. Evans and C. M. Linton, “Trapped modes in openchannels,” J. Fluid Mech. , 153-175 (1991).[4] J. Goldstone and R. L. Jaffe, “Bound states in twistingtubes,” Phys. Rev. B , 14100-14107 (1992).[5] D. V. Evans, M. Levitin and D. Vassiliev, “Existencetheorems for trapped modes,” J. Fluid Mech. , 21-31(1994).[6] D. V. Evans and R. Porter, “Trapped modes embeddedin the continuous spectrum,” Q. J. Mech. Appl. Math. (2), 263-274 (1998).[7] E. N. Bulgakov and A. F. Sadreev, “Bound states in thecontinuum in photonic waveguides inspired by defects,”Phys. Rev. B , 075105 (2008).[8] Y. Plotnik, O. Peleg, F. Dreisow, M. Heinrich, S. Nolte,A. Szameit, and M. Segev, “Experimental observation ofoptical bound states in the continuum,” Phys. Rev. Lett. , 183901 (2011).[9] M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar,“Surface bound states in the continuum,” Phys. Rev.Lett. , 070401 (2012).[10] S. Weimann, Y. Xu, R. Keil, A. E. Miroshnichenko, A.T¨unnermann, S. Nolte, A. A. Sukhorukov, A. Szameit,and Y. S. Kivshar, “Compact surface Fano states embed-ded in the continuum of the waveguide arrays,” Phys.Rev. Lett. , 240403 (2013).[11] C. L. Zou, J.-M. Cui, F.-W. Sun, X. Xiong, X.-B. Zou, Z.-F. Han, and G.-C. Guo, “Guiding light through opticalbound states in the continuum for ultrahigh-Q microres-onantors,” Laser & Photonics Rev. , 114-119 (2015).[12] A.-S. Bonnet-Bendhia and F. Starling, “Guided waves byelectromagnetic gratings and nonuniqueness examples forthe diffraction problem,” Math. Methods Appl. Sci. ,305-338 (1994).[13] P. Paddon and J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,”Phys. Rev. B , 2090-2101 (2000).[14] S. G. Tikhodeev, A. L. Yablonskii, E. A Muljarov, N.A. Gippius, and T. Ishihara, “Quasi-guided modes andoptical properties of photonic crystal slabs,” Phys. Rev.B , 045102 (2002).[15] S. P. Shipman and S. Venakides, “Resonance and boundstates in photonic crystal slabs,” SIAM J. Appl. Math. , 322-342 (2003).[16] S. Shipman and D. Volkov, “Guided modes in periodicslabs: existence and nonexistence,” SIAM J. Appl. Math. , 687–713 (2007).[17] J. Lee, B. Zhen, S. L. Chua, W. Qiu, J. D. Joannopoulos,M. Soljaˇci´c, and O. Shapira, “Observation and differenti-ation of unique high-Q optical resonances near zero wavevector in macroscopic photonic crystal slabs,” Phys. Rev.Lett. , 067401 (2012).[18] Z. Hu and Y. Y. Lu, “Standing waves on two-dimensionalperiodic dielectric waveguides,” Journal of Optics ,065601 (2015).[19] R. Porter and D. Evans, “Embedded Rayleigh-Bloch sur-face waves along periodic rectangular arrays,” Wave Mo- tion , 29-50 (2005).[20] D. C. Marinica, A. G. Borisov, and S. V. Shabanov,“Bound states in the continuum in photonics,” Phys.Rev. Lett. , 183902 (2008).[21] R. F. Ngangali and S. V. Shabanov, “Electromagneticbound states in the radiation continuum for periodicdouble arrays of subwavelength dielectric cylinders,” J.Math. Phys. , 102901 (2010).[22] C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J.D. Joannopoulos, and M. Soljaˇci´c, “Bloch surface eigen-states within the radiation continuum,” Light Sci. Appl. , e84 (2013).[23] C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. John-son, J. D. Joannopoulos, and M. Soljaˇci´c, “Observationof trapped light within the radiation continuum,” Nature , 188–191 (2013).[24] Y. Yang, C. Peng, Y. Liang, Z. Li, and S. Noda, “An-alytical perspective for bound states in the continuumin photonic crystal slabs,” Phys. Rev. Lett. , 037401(2014).[25] B. Zhen, C. W. Hsu, L. Lu, A. D. Stone, and M. Soljaˇciˇc,“Topological nature of optical bound states in the con-tinuum,” Phys. Rev. Lett. , 257401 (2014).[26] E. N. Bulgakov and A. F. Sadreev, “Bloch bound statesin the radiation continuum in a periodic array of dielec-tric rods,” Phys. Rev. A , 053801 (2014).[27] E. N. Bulgakov and A. F. Sadreev, “Light trappingabove the light cone in one-dimensional array of dielectricspheres,” Phys. Rev. A , 023816 (2015).[28] E. N. Bulgakov and D. N. Maksimov, “Light guidingabove the light line in arrays of dielectric nanospheres,”Opt. Lett. , 3888 (2016).[29] R. Gansch, S. Kalchmair, P. Genevet, T. Zederbauer,H. Detz, A. M. Andrews, W. Schrenk, F. Capasso, M.Lonˇcar, and G. Strasser, “Measurement of bound statesin the continuum by a detector embedded in a photoniccrystal,” Light: Science & Applications , e16147 (2016).[30] L. Li and H. Yin, “Bound States in the Continuum indouble layer structures,” Sci. Rep. , 31908 (2016).[32] L. Ni, Z. Wang, C. Peng, and Z. Li, “Tunable opticalbound states in the continuum beyond in-plane symme-try protection,” Phys. Rev. B , 245148 (2016).[33] L. Yuan and Y. Y. Lu, “Propagating Bloch modes abovethe lightline on a periodic array of cylinders,” J. Phys.B: Atomic, Mol. and Opt. Phys. , 05LT01 (2017).[34] E. N. Bulgakov and A. F. Sadreev, “Bound states in thecontinuum with high orbital angular momentum in a di-electric rod with periodically modulated permittivity,”Phys. Rev. A , 013841 (2017).[35] Z. Hu and Y. Y. Lu, “Propagating bound states in thecontinuum at the surface of a photonic crystal,” J. Opt.Soc. Am. B , 1878-1883 (2017).[36] E. N. Bulgakov and D. N. Maksimov, “Topological boundstates in the continuum in arrays of dielectric spheres,”Phys. Rev. Lett. , 267401 (2017).[37] L. Yuan and Y. Y. Lu, “Bound states in the continuumon periodic structures: perturbation theory and robust- ness,” Opt. Lett. (21) 4490-4493 (2017).[38] E. N. Bulgakov and D. N. Maksimov, “Bound statesin the continuum and polarization singularities in peri-odic arrays of dielectric rods,” Phys. Rev. A , 063833(2017).[39] Z. Hu and Y. Y. Lu, “Resonances and bound states inthe continuum on periodic arrays of slightly noncircularcylinders,” J. Phys. B: At. Mol. Opt. Phys. , 035402(2018).[40] S. P. Shipman and S. Venakides, “Resonant transmissionnear nonrobust periodic slab modes,” Phys. Rev. E ,026611 (2005).[41] S. Shipman and H. Tu, “Total resonant transmission andreflection by periodic structures,” SIAM J. Appl. Math. , 216-239 (2012).[42] V. Mocella and S. Romano, “Giant field enhancement inphotonic lattices,” Phys. Rev. B , 155117 (2015).[43] J. W. Yoon, S. H. Song, and R. Magnusson, “Criticalfield enhancement of asymptotic optical bound states inthe continuum,” Sci. Rep. , 18301 (2015).[44] A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman,and B. Kant´e, “Lasing action from photonic bound statesin continuum,” Nature , 196-199 (2017).[45] L. Yuan and Y. Y. Lu, “Strong resonances on periodic arrays of cylinders and optical bistability with weak in-cident waves,” Phys. Rev. A , 023834 (2017)[46] J. M. Foley, S. M. Young, and J. D. Phillips, “Symmetry-protected mode coupling near normal incidence fornarrow-band transmission filtering in a dielectric grat-ing,” Phys. Rev. B , 165111 (2014).[47] X. Cui, H. Tian, Y. Du, G. Shi, and Z. Zhou, “Normal in-cidence filter using symmetry-protected modes in dielec-tric subwavelength gratings,” Sci. Rep. , 36066 (2016)[48] Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samu-sev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bog-danov, and A. V. Lavrinenko, “Transition from opticalbound states in the continuum to leaky resonances: Roleof substrate and roughness,” ACS Photonics , 723-727(2017).[49] E. N. Bulgakov and D. N. Maksimov, “Light enhance-ment by quasi-bound states in the continuum in dielectricarrays,” Opt. Express (13), 14134-14147 (2017)[50] A. Taghizadeh and I.-S. Chung, “Quasi bound states inthe continuum with few unit cells for photonic crystalslab,” Appl. Phys. Lett.111