Circularly polarized states and propagating bound states in the continuum in a periodic array of cylinders
aa r X i v : . [ phy s i c s . op ti c s ] F e b Circularly polarized states and propagating bound states in the continuumin a periodic array of cylinders
Amgad Abdrabou ∗ and Ya Yan Lu Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong, China (Dated: February 26, 2021)Bound states in the continuum (BICs) in a periodic structure sandwiched between two homoge-neous media have interesting properties and useful applications in photonics. The topological natureof BICs was previously revealed based on a topological charge related to the far-field polarizationvector of the surrounding resonant states. Recently, it was established that when a symmetry-protected BIC (with a nonzero topological charge) is destroyed by a generic symmetry-breakingperturbation, a pair of circularly polarized resonant states (CPSs) emerge and the net topologi-cal charge is conserved. A periodic structure can also support propagating BICs with a nonzerowavevector. These BICs are not protected by symmetry in the sense of symmetry mismatch, butthey need symmetry for their robust existence. Based on a highly accurate computational methodfor a periodic array of slightly noncircular cylinders, we show that a propagating BIC is typicallydestroyed by a structural perturbation that breaks only the in-plane inversion symmetry, and whenthis happens, a pair of CPSs of opposite handedness emerge so that the net topological charge isconserved. We also study the generation and annihilation of CPSs when a structural parameter isvaried. It is shown that two CPSs with opposite topological charge and same handedness, connectedto two BICs or in a continuous branch from one BIC, may collapse and become a CPS with a zerocharge. Our study clarifies the important connection between symmetry and topological chargeconservation.
I. INTRODUCTION
Bound states in the continuum (BICs) are trapped orguided modes with their frequencies in the radiation con-tinua [1, 2]. They exist in a variety of photonic struc-tures including periodic structures sandwiched betweentwo homogeneous media [3–13], waveguides with localdefects [14, 15], waveguides with lateral leakage chan-nels [16–20], etc. In structures that are invariant or pe-riodic in one or two spatial directions, a BIC is a spe-cial point with an infinite quality factor ( Q factor) ina band of resonant states [11, 21–23], and it becomes ahigh- Q resonance if the structure is perturbed generically[24, 25]. High- Q resonances lead to strong local field en-hancement [26–28] and abrupt features in reflection andtransmission spectra [29], and are essential for sensing,lasing, switching and nonlinear optics applications.For theoretical interest and practical applications, itis important to understand how a BIC is affected by aperturbation of the structure. If the BIC is protectedby a symmetry [3–8, 14, 15], i.e., there is a symmetrymismatch between the BIC and the compatible radia-tion modes, it naturally continues its existence if theperturbation preserves the symmetry [3, 14, 30]. There-fore, a symmetry-protected BIC is robust with respectto symmetry-preserving perturbations. If a perturbationbreaks the symmetry, a symmetry-protected BIC typ-ically, but not always, becomes a resonant state witha finite Q factor [24, 25, 31]. In periodic structuressandwiched between two homogeneous media, there arealso BICs with a nonzero Bloch wavevector, and they ∗ Corresponding author: [email protected] propagate in the periodic directions [9–12, 29, 32, 33].Such a propagating BIC is not protected by symmetry inthe usual sense, but can also be robust with respect tosymmetry-preserving perturbations [34–36]. More pre-cisely, in a periodic structure with an up-down mirrorsymmetry and an in-plane inversion symmetry, a genericlow-frequency propagating BIC (with only one radiationchannel) continues its existence if the structure is per-turbed by a perturbation preserving these two symme-tries [35, 36].The BICs in periodic structures exhibit interestingtopological properties. Zhen el al. [34] realized that aBIC in a biperiodic structure is a polarization singu-larity in momentum space (the plane of two wavevec-tor components), and defined a topological charge basedon the winding number of the far-field polarization vec-tor. Since the far field of a resonant state is typically el-liptically polarized, Bulgakov and Makismov refined thedefinition using the major polarization vector [32]. Thetopological charge can be used to classify the BICs andillustrate their generation, interaction and annihilationprocesses when structural parameters are tuned [32, 34].Importantly, the topological charge is a conserved quan-tity that cannot be changed by small structural perturba-tions. However, this does not imply that the BICs (with anonzero topological charge) are robust with respect to ar-bitrary structural perturbations, because a resonant statecan be circularly polarized and also has a nonzero topo-logical charge. In a recent work, Liu el al. [37] showedthat symmetry-protected BICs in a photonic crystal slab,protected by the in-plane inversion symmetry, turn topairs of circularly polarized resonant states (CPSs) whenthe structure is perturbed breaking the symmetry.Most propagating BICs are found in periodic struc-tures with both the up-down mirror symmetry and thein-plane inversion symmetry. It is known that if one ofthese two symmetries is broken, a propagating BIC isusually, but not always, destroyed [31, 33]. It has beenshown that CPSs can exist in structures without the up-down mirror symmetry and in-plane inversion symme-try [39]. In this paper, we show that CPSs emerge whenpropagating BICs are destroyed by generic and arbitrar-ily small perturbations that break only the in-plane inver-sion symmetry. We consider vectorial BICs in a periodicarray of of dielectric cylinders, introduce a small defor-mation of the cylinder boundary, and show that pairsof CPSs emerge when propagating BICs with topologicalcharge ± II. EIGENMODES IN PERIODIC STRUCTURES
We consider a 2D structure that is invariant in x , pe-riodic in y with period L , bounded in z by | z | < D forsome D >
0, and surrounded by air (for | z | > D ), where { x, y, z } is a Cartesian coordinate system. Let ε = ε ( y, z )be the relative permittivity of this structure and its sur-rounding medium, then ε ( r ) = ε ( y + L, z ) for all r = ( y, z )and ε ( r ) = 1 for | z | > D . We study time-harmonic elec-tromagnetic waves that depend on time t and variable x as exp[ i ( αx − ωt )], where ω is the angular frequencyand α is a real wavenumber in the x direction. Fromthe frequency-domain Maxwell’s equations, it is easy toobtain the following system ∇ · (cid:18) εη ∇ E x (cid:19) + ∇ · (cid:18) αkη P · ∇ e H x (cid:19) + εE x = 0 , (1) ∇ · (cid:18) η ∇ e H x (cid:19) − ∇ · (cid:18) αkη P · ∇ E x (cid:19) + e H x = 0 , (2)where E x is the x component of the electric field, e H x is the x component of a scaled magnetic field (magneticfield multiplied by free space impedance), k = ω/c is thefree space wavenumber, c is the speed of light in vacuum,and η = k ε ( r ) − α , ∇ = " ∂ y ∂ z , P = " − . (3) The other four field components can be obtained fromthe following equations: " E y E z = iη (cid:16) α ∇ E x + k P · ∇ e H x (cid:17) , (4) " e H y e H z = iη (cid:16) α ∇ e H x − k εP · ∇ E x (cid:17) . (5)If α = 0, the equations for E x and e H z are decoupled, wesay the corresponding wave is scalar. The case α = 0 isreferred to as vectorial.Due to the periodicity in y , any eigenmode of the struc-ture is a Bloch mode with an electric field given by E ( r ) = F ( r ) e iβy , (6)where β is a real Bloch wavenumber satisfying | β | ≤ π/L ,and F is a periodic in y with period L . For | z | > D , thefield can be expanded in plane waves as E ( r ) = ∞ X m = −∞ c ± m e i ( β m y ± γ m z ) , ± z > D, (7)where β = β , and β m = β + 2 πmL , γ m = p k − α − β m . (8)A guided mode satisfies the condition E → as z →±∞ . For a positive k , guided modes usually exist when k < p α + β , so that all γ m are pure imaginary and allplane waves in the right hand side of Eq. (7) are evanes-cent in z . A BIC is also a guided mode, but it satisfiesthe condition k > p α + β , thus γ and probably a fewother γ m are real positive. We study BICs with a real k ,a real α , and a real β such that p α + β < k < s α + (cid:18) πL − | β | (cid:19) . (9)The above condition implies that γ is positive and allother γ m are pure imaginary. Since the plane wavesexp[ i ( β y ± γ z )] can propagate to infinity, the coeffi-cients c ± of the BIC must vanish.A resonant state is also an eigenmode, but it satisfiesan outgoing radiation condition as z → ±∞ . Because ofthe radiation loss, the amplitude of a resonant state de-cays in time, thus the frequency ω or k must be complexwith a negative imaginary part. This implies that thereal and imaginary parts of γ are positive and negative,respectively, and exp[ i ( β y + γ z )] is an amplifying out-going plane wave as z → + ∞ . We are concerned withresonant states with only one radiating plane wave foreither z > D or z < − D . Therefore, it is assumed thatIm( γ m ) > m = 0. Resonant states form bandswhere each band corresponds to k being a complex-valuedfunction of α and β . The Q factor of a resonant mode isgiven by Q = − . k ) / Im( k ). A BIC is a special point(with a real k ) in a band of resonant states.We assume the periodic structure has an up-down mir-ror symmetry, i.e., ε ( r ) = ε ( y, − z ) for all r , then it issufficient to study the field of any eigenmode in the up-per half space ( z > z or odd in z . For a BIC with a frequencyand wavevector satisfying Eq. (9), it is possible to definea topological charge based on the far field polarizationvector of the surrounding resonant states [32, 34]. Let C be a closed contour in the α - β plane. Each point on C corresponds to a resonant state in a band that containsthe BIC. The resonant state contains a far-field outgoingplane wave (for z → + ∞ ) with a vector amplitude c +0 .Its projection on the x - y plane is b E = c +0 x c +0 y e i ( αx + βy + γ z ) , (10)where c +0 x and c +0 y are the x and y components of c +0 .For any fixed z > D , the real projected electric fieldRe( b E e − iωt ) is typically elliptically polarized, and the ma-jor polarization vector (along the major axis of the po-larization ellipse) forms an angle θ with the x axis. If itis possible to define θ continuously as ( α, β ) traversesalong C in counterclockwise direction from a startingpoint back to the same point (the ending point), and θ | end − θ | start = 2 πq for some q , where θ | start and θ | end are the values of θ at the starting point and the endingpoint respectively, then q is the winding number (of theprojected major polarization vector) on C . Alternatively, q can be evaluated by the integral formula q = 12 π I C d α · ∇ α θ ( α ) , (11)where α = ( α, β ) and ∇ α = ( ∂ α , ∂ β ). A BIC is a po-larization singularity, since it does not have a far field( c ± = 0). The topological charge of a BIC is defined asthe winding number q , if C is sufficiently close to the BICand encloses the BIC in the α - β plane [32, 34].It is important to note that the major polarization vec-tor or the angle θ is undefined, if the projected far-fieldplane wave is circularly polarized. This implies that a cir-cularly polarized state (CPS), i.e., a resonant state with acircularly polarized far field, is also a polarization singu-larity. If C contains the wavevector of a CPS, the windingnumber on C is undefined. The definition of topologicalcharge requires the underlying assumption that no CPSsexist in a small neighborhood (in the α - β plane) of theBIC.Finally, we recall that the polarization state of theplane wave given by Eq. (10) can be characterized bythe Stokes parameters [40] S = | c +0 x | + | c +0 y | , S = | c +0 y | − | c +0 x | , (12) S = 2 | c +0 x c +0 y | cos( ϕ y − ϕ x ) , (13) S = 2 | c +0 x c +0 y | sin( ϕ y − ϕ x ) , (14) where ϕ x = arg( c +0 x ), ϕ y = arg( c +0 y ) are the phases of c +0 x and c +0 y , respectively. Since a BIC has no radiation,it corresponds to the point ( S , S , S , S = 0). A CPSsatisfies the condition ( S , S , S / S ) = (0 , , ± S / S = +1 for the left CPS and − S = 0. The principle value of angle θ is given by θ =arg( S + i S ) / III. CIRCULAR CYLINDERS
A periodic array of circular dielectric cylinders sur-rounded by air, as shown in Fig. 1, is a simple structure x yz O yz a L Ω FIG. 1. A periodic array of circular dielectric cylinders sur-rounded by air. The array is periodic in y with period L . Thecylinders are parallel to the x axis. The radius and dielectricconstant of the cylinders are a and ε c , respectively. supporting many different BICs [7, 12, 29, 30, 32]. In gen-eral, a BIC may be a standing wave ( α = β = 0), maypropagate along the y axis ( α = 0 and β = 0), along the x axis ( α = 0 and β = 0), or in both x and y directions( α = 0 and β = 0). Those BICs with α = 0 are scalarones with either the H or E polarization. The BICs with α = 0 are vectorial ones with nonzero E x and e H x . InTable I, we list four BICs for periodic arrays with a fixed TABLE I. A few BICs in periodic arrays of circular cylinderswith radius a and dielectric constant ε c = 15. a/L αL βL kL q BIC . . − . . . .
45 0 0 . . . . . . dielectric constant ε c = 15. The topological charges ofthe BICs are listed in the last column. More BICs in thisperiodic array can be found in Ref. [32].Normally, the resonant states and the BICs are com-puted by solving an eigenvalue problem for the Maxwell’sequations. To take advantage of the special geometryof the circular cylinders, we use a semi-analytic methodbased on cylindrical wave expansions. Since the struc-ture is invariant in x and periodic in y , it is sufficientto solve the eigenmodes in one period of the structure,i.e., a 2D domain Ω ∞ given by | y | < L/ | z | < ∞ .Inside Ω ∞ , there is a square Ω given by | y | < L/ | z | < L/
2. We assume one cylinder is located at the cen-ter of Ω. For given α and ω , the electromagnetic fieldin Ω can be expanded in vectorial cylindrical waves withunknown coefficients [41]. If β is also specified, we can ex-pand the electromagnetic field in plane waves (also withunknown coefficients) for | z | > L/
2. Relating the fieldat y = ± L/ y , and imposing continuity con-ditions at z = L/
2, we can obtain an operator A , suchthat A ( k, α, β ) u | z = L/ = , (15)where u is a column vector for E x and e H x , and u | z = L/ denotes u at z = L/ | y | < L/
2. Notice that A isan operator that acts on a vector of two single-variablefunctions. Since A depends on k , Eq. (15) is a nonlineareigenvalue problem. In practice, y ∈ ( − L/ , L/
2) is sam-pled by N points, u | z = L/ becomes a vector of length 2 N ,and A is approximated by a (2 N ) × (2 N ) matrix. Themethod can be extended to the case where the bound-ary of the cylinders are slightly and smoothly deformed.Details are given in Appendix.For computing resonant states, α and β are given, welook for a complex k such that Eq. (15) has a nontrivialsolution. One possible approach is to solve k from λ ( A ) = 0 , (16)where λ is the eigenvalue of A with the smallest magni-tude. A BIC is a special resonant state with Im( k ) = 0.For propagating BICs, it is more efficient to treat α and/or β also as unknowns. We can solve Eq. (16) for areal k , a real α , and/or a real β . IV. SLIGHTLY NONCIRCULAR CYLINDERS
In this section, we consider a periodic array of slightlynoncircular cylinders, study the emergence of CPSs whenpropagating BICs are destroyed, and also the annihila-tion and generation of CPSs. We assume the boundaryof the cylinder centered at the origin is given by y = − a sin( τ ) + δ cos( gτ ) , z = a cos( τ ) , (17)for 0 ≤ τ < π and g = 2 or 4, where δ > ε c = 15. The deformed cylinders for δ = 0 . a are shown in Figs. 2(a) and 2(b) for g = 2 and g = 4, respectively. The small boundary deformationis a perturbation that breaks the reflection symmetry in y (the in-plane inversion symmetry), but preserves thereflection symmetry in z (the up-down mirror symmetry).To calculate vectorial resonant states, we extend themethod described in the previous section. Importantly,a general electromagnetic field in Ω (the square domain a O z yz y (b) a (a) Ω Ω ε = ε c ε = ε = ε c ε = Ω Ω FIG. 2. Cross-sections Ω of two deformed cylinders with aboundary given by Eq. (17) for δ = 0 . a and the cases (a) g =2 and (b) g = 4. Ω is the exterior domain outside the cylinderand inside the square Ω = ( − L/ , L/ × ( − L/ , L/ containing one cylinder centered at the origin) can stillbe expanded in cylindrical waves, although the expan-sions are more complicated due to the deformation ofthe cylinder boundary. As shown in Appendix, the eigen-value problem for resonant states is reduced to Eq. (15),where A is a (2 N ) × (2 N ) matrix depending on k , α and β , and N is the number of sampling points for aninterval of length L . It is highly desirable to compute aCPS without calculating all nearby resonant states. Tofind a left or right CPS, we solve a real α , a real β and acomplex k , from Eq. (16) and S = 0 , S / S = ± , (18)where S , S and S are the Stokes parameters.First, we show that when the deformation parameter δ is increased from zero, all four BICs listed in Table Iare destroyed and pairs of CPSs emerge. If the topolog-ical charge of the BIC is 1 (or − / − /
2) and different handed-ness emerge. The net topological charge is conserved. InFig. 3, we show the emergence of CPSs from BIC for a -0.5 0 0.50.30.350.40.450.5 FIG. 3. A pair of CPSs with topological charge 1 / for a periodic array of slightly noncircular cylinderswith g = 4. periodic array with a = 0 . L and g = 4. The purplearrows indicate the direction of increasing δ . The curveshows the wavevector ( α, β ) for a pair of left and rightCPSs, and it is shown in color to indicate the value of δ .These two CPSs exhibit a symmetry with respect to the β axis. For each left CPS with wavevector ( α, β ), there isalso a right CPS with wavevector ( − α, β ). The complexfrequencies of these two CPSs are exactly the same. InFigs. 4(a) and 4(b), we show CPSs emerged from BIC (a) (b) FIG. 4. Pairs of CPSs emerged from BIC for periodic arraysof slightly noncircular cylinders with (a) g = 2 and (b) g = 4. for periodic arrays with g = 2 and g = 4, respectively.The radius of the original cylinders is a = 0 . L . There isno apparent symmetry between the two CPSs emergedfrom the BIC, but there is still a symmetry with respectto the β axis. Since the structure is invariant in x , theresonant states (or BICs) for ( α, β ) and ( − α, β ) are re-flections of each other. The CPSs with wavevectors ( α, β )and ( − α, β ) have the opposite handedness.Next, we show that as δ is increased, the CPSs emergedfrom BICs of opposite topological charge may collapse toa CPS with a zero charge. BIC and BIC in Table Iare found in the same periodic array of circular cylinderswith radius a = 0 . L , but their topological chargesare − δ is increased from zero,both BIC and BIC are destroyed. In Figs. 5(a) and5(b), we show the emergence of CPS pairs from BIC -1 -0.5 0 0.5 1-20-15-10-505 10 -3 -1 0 1-505101520 10 -3 (a) (b) FIG. 5. Emergence of CPSs from BIC and BIC , and an-nihilation of CPSs of opposite charges for periodic arrays ofslightly noncircular cylinders with (a) g = 2 and (b) g = 4. and BIC for periodic arrays of deformed cylinders with g = 2 and g = 4, respectively. The CPSs emerged fromBIC and BIC carry topological charges − / / δ reaches a critical value δ AP ( δ AP ≈ . × − L for g = 2 and δ AP ≈ . × − L for g = 4), two CPSs with the same handedness, one fromBIC and the other from BIC , collapse to a CPS witha zero topological charge. Since these two CPSs cease toexist for δ > δ AP , we call δ AP an annihilation point (AP).Finally, we show that CPSs connected to a singleBIC may encounter self-generation and self-annihilationpoints as δ is increased. For a periodic array with a = 0 . L and g = 2, a continuous branch of CPSsemerging from BIC is shown in Fig. 6(a). The curve -3 (a)(c) (d)(b) FIG. 6. (a) A continuous right CPS emerged from BIC fora periodic array with g = 2, with δ increased from 0 to δ AP ,decreased to δ GP , and increased again. The topological chargeof the CPS has changed from 1 / − / / Q factor, (c) α and (d) β of the rightCPS as multivalued functions of δ . The blue and red curvescorrespond to topological charge 1 / − /
2, respectively. in Fig. 6(a) depicts the wavevector ( α, β ) of a CPS when δ is first increased from zero to δ AP ≈ . L , then de-creased to δ GP ≈ . L , finally increased again. Thetopological charge of the CPS has changed from 1 / − /
2, and back to 1 / δ , the critical value δ AP is annihilationpoint where two CPSs of opposite topological charge andsame handedness collapse to a CPS with a zero charge,but these two CPSs are not connected to different BICs.In fact, the CPS with topological charge − / / δ GP . We call δ AP aself-annihilation point and δ GP a self-generation point.However, annihilation and generation are relative termsdepending on how the structure is tuned. For example, δ GP can also be regarded as a self-annihilation point, ifthe structure is tuned by a decreasing δ . It is clear thatthe continuous branch of CPSs emerged from BIC , asshown in Fig. 6(a), exhibits a multi-valued dependenceon δ . In Figs. 6(b), (c) and (d), we respectively showthe Q factor, and wavevector components α and β asmultivalued functions of δ . Despite the generation andannihilation of CPSs as δ is varied, the net topologicalcharge is conserved and remains at 1 / V. CONCLUSION
Many applications of BICs are realized in periodicstructures sandwiched between two homogeneous media.It is known that the existence and robustness of BICs,including the propagating BICs in periodic structures,depend crucially on symmetry [11, 34]. More precisely, ithas been proved that some propagating BICs are robustwith respect to structural perturbations that preserve thein-plane inversion symmetry and the up-down reflectionsymmetry, even though these BICs do not have a symme-try mismatch with compatible radiating waves [35, 36].If the perturbation breaks one of these two symmetries,the propagating BICs are typically destroyed and becomeresonant states with a finite Q factor [33]. The topologi-cal charge of a BIC, defined using the polarization vectorof the resonant states surrounding the BIC in momentumspace, is an interesting concept and is useful for under-standing the evolution of BICs as structural parametersare varied [32, 34]. The definition of topological chargedoes not require the in-plane inversion symmetry. Whenstructural parameters are varied, the topological charge isalways conserved, but this does not imply that the BICsare robust with respect to arbitrary structural perturba-tions, because a CPS is also a polarization singularityin momentum space and it can have a nonzero topolog-ical charge. Therefore, the conservation of topologicalcharge is only valid when both BICs and CPSs are in-cluded. For propagating BICs, the connection betweensymmetry-breaking perturbations and the emergence ofCPSs has not been clearly established in existing litera-ture. Using a periodic array of slightly noncircular cylin-ders as an example, we show that pairs of CPSs emergewhen propagating BICs are destroyed by arbitrarily smallperturbations that break only the in-plane inversion sym-metry. We also study the generation and annihilation ofCPSs when structural parameters are varied. It is shownthat pairs of CPSs of opposite topological charge can col-lapse at special CPSs with a zero topological charge, butthe net topological charge is still conserved. ACKNOWLEDGMENTS
The authors acknowledge support from the ResearchGrants Council of Hong Kong Special Administrative Re-gion, China (Grant No. CityU 11305518).
Appendix: Construction of matrix A To obtain a matrix A satisfying Eq. (16) for the case ofa periodic array of slightly noncircular cylinders, we needto find cylindrical wave solutions that are valid inside andoutside a single cylinder, determine a matrix C mapping u to the normal derivative of u on ∂ Ω (the boundaryof Ω), and finally construct the matrix A . As shown inFig. 2, Ω is a subdomain of Ω corresponding to the cross section of the cylinder, Ω = Ω \ Ω , ε ( r ) = ε or ε for r ∈ Ω or Ω , respectively. The boundary of Ω is Γ.For given k and α and an integer p , we construct avectorial cylindrical wave solution that depends on twoarbitrary coefficients c p and r p . The x components ofthis solution are assumed to be E ( p ) x = X a pq J q ( ρ r ) J q ( ρ a ) e iqθ , r ∈ Ω c p J p ( ρ r ) J p ( ρ a ) e ipθ + X b pq Y q ( ρ r ) Y q ( ρ a ) e iqθ , r ∈ Ω e H ( p ) x = X s pq J q ( ρ r ) J q ( ρ a ) e iqθ , r ∈ Ω r p J p ( ρ r ) J p ( ρ a ) e ipθ + X t pq Y q ( ρ r ) Y q ( ρ a ) e iqθ , r ∈ Ω where r and θ are the polar coordinates of r , ρ j = ( k ε j − α ) / , a is the radius of the circle to which Γ is close, J q and Y q are the first and second kinds of Bessel functionsof order p , the sums are for q from −∞ to + ∞ , and a pq , b pq , s pq , t pq (for all q ) are unknown coefficients.Let ν = ( ν y , ν z ) be the outward unit vector normal toΓ, and τ = ( − ν z , ν y ) be a unit vector tangential to Γ,then the tangential field components of this cylindricalwave solution are e H ( p ) τ = iη " ε ∂E ( p ) x ∂ν + γ k e H ( p ) x ∂τ , (A.1) E ( p ) τ = iη " ∂ e H ( p ) x ∂ν − γ k ∂E ( p ) x ∂τ . (A.2)We discretize Γ using M points (assuming M is odd) andtruncate the sums by | q | ≤ ( M − /
2. The continuityconditions of E x , e H x , E τ and e H τ at those M points onΓ give rise to the following linear system: A A A A A A A A A A A A a p b p s p t p = c m g g + r m h h , where A jk for 1 ≤ j, k ≤ M × M matrices, g j and h j for 1 ≤ j ≤ M , a p , b p , s p and t p are also column vectors of length M , a p is thevector for all a pq , etc. Solving the above linear system,we obtain column vectors f ( p ) jk for 1 ≤ j, k ≤
2, such that b p = c p f ( p )11 + r p f ( p )12 , t p = c p f ( p )21 + r p f ( p )22 . (A.3)Therefore, the cylindrical wave solution can be written as u p = c p u (1) p + r p u (2) p , where u (1) p and u (2) p are completelydetermined.The general field in Ω can be expanded in the abovecylindrical waves as u = ∞ X p = −∞ u p = ∞ X p = −∞ ( c p u (1) p + r p u (2) p ) . (A.4)If ∂ Ω is sampled by 4 N points, we can truncate the index p in the sum above to − N ≤ p ≤ N −
1, and evaluate u and the normal derivative of u at the 4 N points on ∂ Ω.This leads to (8 N ) × (8 N ) matrices D and D such that u | ∂ Ω = D " cr , ∂ u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ω = D " cr , (A.5)where u | ∂ Ω and ∂ ν u | ∂ Ω are column vectors of length 8 N , c and r are column vectors of length 4 N with entries c p and r p , respectively. For simplicity, ∂ ν is simply takento be ∂ z or ∂ y on the horizontal and vertical sides of Ω.Therefore, we have matrix C = D D − , such that ∂ u ∂ν (cid:12)(cid:12)(cid:12)(cid:12) ∂ Ω = C u | ∂ Ω . (A.6)If β is given, u and ∂ y u satisfies the following quasi- periodic condition: u | y = L/ = e iβL u | y = − L/ , (A.7) ∂ u ∂y (cid:12)(cid:12)(cid:12)(cid:12) y = L/ = e iβL ∂ u ∂y (cid:12)(cid:12)(cid:12)(cid:12) y = − L/ . (A.8)Since the structure has the up-down mirror symmetry,we have either u ( y, z ) = u ( y, − z ) or u ( y, z ) = − u ( y, − z ).Substituting these conditions into Eq. (A.6), we obtain amatrix B such that ∂ u ∂z (cid:12)(cid:12)(cid:12)(cid:12) z = L/ = B u | z = L/ . (A.9)For z > L/
2, the field can be expanded in plane wavesas in Eq. (7). Using the plane wave expansion for u ,evaluating u and ∂ z u at z = L/
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