Spectral topology and edge modes for one-dimensional non-Hermitian photonic crystals

Sp ectral top ology and edge mo des for one-dimensional non-Hermitian photonic crystals Junshan Lin ∗ and Hai Zhang † Abstract This w ork in v estigates edge mo des in non-Hermitian photonic crystals with brok en spec- tral recipro cit y . In suc h systems, the sp ectra of the underlying operators generally form closed lo ops ov er the complex plane with nontrivial sp ectral top ology , which gives rise to the so-called skin eect characterized by edge mo des lo calized at in terfaces. F or discrete lattice mo dels, the skin eect can be understo od through the sp ectral theory of T o eplitz matrices. Ho w ev er, this mathematical framework no longer applies to contin uous w a v e mo dels, where nite-dimensional appro ximations break down. In this work, we emplo y a transfer matrix approach to describe w a v e propagation in one-dimensional p erio dic media and in troduce a new sp ectral top ological in v arian t based on the eigenv alues of the transfer matrix. The new top ological in v arian t is equiv alen t to the winding num b er of the non-Hermitian sp ectrum and it enables the character- ization of edge mo des in one-dimensional non-Hermitian photonic crystals. The mathematical theory provides the theoretical foundation for the skin eect in contin uous wa ve mo dels. 1 In tro duction 1.1 Bac kground Non-Hermitian top ology has attracted signicant researc h in terest in recen t years, due to its ubiq- uit y in materials and unprecedented ph ysics phenomena induced by the band topology . W e refer to [3, 5, 8, 4, 12, 13, 14, 15, 17, 19, 20, 21] and references therein for the theoretical progress on understanding the top ological phases in non-Hermitian systems. Compared to Hermitian systems which attain sp ectrum on the real line, the spectrum of the non-Hermitian systems generally lies on the complex plane. As suc h, b esides top ology induced by eigenfunctions (eigenv ectors) of the op erator, the eigen v alues of the non-Hermitian op erator ma y attain top ological inv ariant not present in the Hermitian op erator, whic h could lead to new physics phenomena. One prominent disco v ery along this line of research is the so-called skin eect, whic h describ es a macroscopic accumulation of skin (edge) mo des lo calized on the b oundary of a bulk medium under op en b oundary condition [17, 21, 23]. The skin eect could b e attributed to the topologically non-trivial features of the spectrum for the non-Hermitian system [18, 22]. F or the tight-binding mo del, the relation b et ween the spectral top ology of the innite perio dic problem and the sp ectrum of the semi-innite bulk medium is w ell understoo d through the spectral theory of the T o eplitz matrices [18]. More precisely , the ∗ Departmen t of Mathematics and Statistics, A uburn Univ ersity , A uburn, AL 36849. jzl0097@auburn.edu. † Departmen t of Mathematics, HKUST, Clear W ater Bay , Ko wlo on, Hong K ong S.A.R., China. haizhang@ust.hk . H.Z w as partially supp orted by Hong Kong RGC grant GRF 16307024 and GRF 16301625. 1 innite perio dic operator is a Lauren t operator L p and the corresp onding semi-innite op erator is a T o eplitz op erator T . If the spectrum σ ( L p ) forms a closed curve on the complex plane with a nonzero winding n um b er, then the spectrum σ ( T ) = σ ( L p ) ∪ σ wind (cf. [6]), wherein σ wind := { λ ∈ C \ σ ( L p ) : w ind ( σ ( L p ) − λ )  = 0 } . (1) In the abov e, σ wind corresp onds to the sp ectrum of the skin mo des, whic h is the area enclosed b y the dispersion curv e σ ( L p ) . The spectral theory of T o eplitz op erators has been further dev elop ed to study the non-Hermitian skin eect for contin uous wa ve mo dels [1, 2]. Therein, the non-Hermitian system consists of an array of sub wa v elength resonators with an imaginary gauge potential. When the resonators are small, the dieren tial equation mo dels can be eectively describ ed b y the ca- pacitance matrices, whic h attain the T oeplitz structure. In this work, we study the non-Hermitian skin eect in contin uous wa v e mo dels, for which nite-dimensional appro ximations break down, and the sp ectral theory of T o eplitz op erators no longer applies. More specically , w e consider non-Hermitian photonic crystals with material loss and broken spectral reciprocity (see Denition 4). Numerical studies in [10, 24] indicate that the eigen v alues corresponding to the edge mo des of the nite-size photonic crystal accumulate within the region enclosed b y the sp ectrum σ ( L p ) of the innite p eriodic op erator, thereby manifesting the photonic skin eect. W e aim to develop a rigorous mathematical framework that elucidates the sp ectral top ology of non-Hermitian photonic crystals and the emergence of edge mo des in semi-innite photonic structures. Our main results show that, for one-dimensional photonic crystals, the sp ectrum of edge mo des is characterized by σ wind , which admits the representation in the form of ( 1). T o this end, we emplo y a transfer matrix formulation to describ e wa ve propagation in perio dic media. Beyond the winding n um b er of the sp ectral curv e asso ciated with the p erio dic non-Hermitian operator in the complex plane, w e in tro duce a new sp ectral top ological inv ariant dened via the eigenv alues of the transfer matrix. Remarkably , these t wo spectral top ological inv arian ts coincide. F urthermore, the newly in tro duced in v ariant precisely c haracterizes the magnitude of the transfer matrix eigenv alues, whic h allows for the characterization of the edge mo des in terms of the corresp onding eigen vectors. 1.2 Mathematical mo del for the one-dimensional photonic crystals W e consider a p erio dic lay ered medium with the p ermittivity and p ermeability tensors v arying along the z direction, whic h are given by ε ( z ) =   ε xx ( z ) ε xy ( z ) 0 ε xy ( z ) ε y y ( z ) 0 0 0 ε z z ( z )   , µ ( z ) =   µ xx ( z ) µ xy ( z ) 0 µ xy ( z ) µ y y ( z ) 0 0 0 µ z z ( z )   . (2) Eac h component of ε ( z ) and µ ( z ) is bounded and piecewisely contin uous with the p eriod p = 1 suc h that ε ( z + 1) = ε ( z ) and µ ( z + 1) = µ ( z ) . Consider a time-harmonic electromagnetic wa ve { E , H } propagating along the z direction with E = [ E x ( z ) , E y ( z ) , 0 ] T e − i ω t and H = [ H x ( z ) , H y ( z ) , 0 ] T e − i ω t . (3) Then the Maxw ell’s equations ∇ × E = i ω c µ H , ∇ × H = − i ω c ε E (4) 2 are reduced to the follo wing ODE system: Q d dz E ( z ) = i ω c µ ( z ) H ( z ) , Q d dz H ( z ) = − i ω c ε ( z ) E ( z ) , (5) where the orthogonal matrix Q =  0 − 1 1 0  . Here and henceforth, with the abuse of notations, we still use { E , H } to denote the transv erse eld comp onen ts of the electromagnetic eld b y ignoring the z comp onen t in (3). In what follo ws, for clarit y w e assume that the w av e sp eed c = 1 . By introducing the notations Φ ( z ) :=  E ( z ) H ( z )  , Q :=  Q 0 0 Q  , A ( z ) := i  0 µ ( z ) − ε ( z ) 0  , the ODE system (5) can b e written as d dz Φ ( z ) = ω Q − 1 A ( z ) Φ ( z ) , (6) wherein A ( z + 1) = A ( z ) . If the p ermittivity and p ermeabilit y functions are p ositiv e denite matrices satisfying ε ( z ) = ε ∗ ( z ) , µ ( z ) = µ ∗ ( z ) , where ∗ denotes the conjugate transp ose operation, the photonic crystal is said to b e Hermitian. Otherwise, we call the photonic crystal non-Hermitian, whic h are t ypically made of lossy materials. The fo cus of this w ork is on non-Hermitian photonic crystals with broken sp ectral recipro cit y . F or suc h congurations, the dispersion curv es ma y form closed loops o ver the complex plane and enclose non trivial regions. 1.3 Organization of the pap er The rest of the pap er is organized as follows. In Section 2, w e study the band structure of the pho- tonic crystals and relate the eigenv alues of the underlying dierential op erator with the eigenv alues of the corresp onding transfer matrix. The concepts of sp ectral recipro cit y and p oint gaps will be il- lustrated through detailed discussions. In Section 3, we inv estigate the edge mo des in semi-innite photonic crystals when disp ersion curves of the corresp onding innite p eriodic structures attain p oin t gaps. It is shown that the winding n um b er of each disp ersion curve is equiv alen t to a new top ological inv arian t dened via the eigenv alues of the underlying transfer matrix. F urthermore, w e prov e that the eigenfrequencies for the edge mo des can b e characterized b y the winding num b er of the disp ersion curves and the new top ological in v ariant. 2 Band structure of photonic crystals 2.1 General band theory and transfer matrix F or eac h k ∈ [ − π , π ] , w e consider the follo wing eigen v alue problem for z ∈ R : d dz Φ ( k ; z ) = ω Q − 1 A ( z ) Φ ( k ; z ) , (7a) Φ ( k ; z + 1) = e i k Φ ( k ; z ) . (7b) 3 In the abov e, the interv al B := [ − π, π ] is called the Brillouin zone, k is called the Blo ch w a v enum b er, ω is the eigenfrequency , and the corresponding eigenmo de Φ ( k ; z ) is called the Blo c h mode. The solution of the ODE system (7a) can be expressed via the transfer matrix. The 4 × 4 transfer matrix T ( ω ; z ) is dened in a wa y such that the solution at z = z 2 is related to the solution at z = z 1 b y the follo wing relation: Φ ( z 2 ) = T ( ω ; z 2 − z 1 ) Φ ( z 1 ) . (8) F or eac h xed ω ∈ C , the transfer matrix ma y b e expressed via the fundamental matrix of the ODE system (7a) that satises T ( ω ; 0) = I . It is clear that T ( ω , · ) is a contin uous matrix function that satises T ( ω ; − d ) = T ( ω ; d ) − 1 . Lemma 1. L et T ( ω ; z ) b e the tr ansfer matrix asso ciate d with the ODE system (7a) and T ( ω ; 0) = I . Then det ( T ( ω ; z )) = 1 for any z ∈ R . Pr o of. Using the Liouville’s formula (cf. [7]), w e ha ve det ( T ( ω ; z )) = det ( T ( ω ; 0))  exp  z 0 tr ( Q − 1 A ( s )) ds  = det ( T ( ω ; 0)) = 1 , where we hav e used the fact that tr ( Q − 1 A ( s )) = 0 . The eigen v alue of (7) can b e determined using the transfer matrix and the quasi-p erio dic b ound- ary condition (7b). More precisely , let M ( ω ) := T ( ω ; 1) , then for eac h k ∈ [ − π , π ] , ω is an eigenfrquency of (7) if and only if e i k is eigenv alue of the transfer matrix M ( ω ) suc h that M ( ω ) v = e i k v , wherein v ∈ C 4 is the corresponding eigen vector. As suc h the disp ersion relation ω ( k ) is determined b y the follo wing c haracteristic equation for M : det ( M ( ω ) − e i k I ) = 0 . (9) Setting λ = e i k , the ab o ve c haracteristic equation reads P ( ω ) := λ 4 + a 3 ( ω ) λ 3 + a 2 ( ω ) λ 2 + a 1 ( ω ) λ + 1 = 0 (10) for some co ecien ts a j ( ω ) ( j = 1 , 2 , 3) , whic h are all analytical in ω . Lemma 2. F or e ach c omplex-value d ω ∈ C , ther e exist four r o ots to the char acteristic e quation (10) denote d by λ i ( ω ) ( i = 1 , 2 , 3 , 4 ). Each function λ i ( ω ) is lo c al ly analytic al exc ept at br anch p oints. Mor e pr e cisely, for any ω 0 , λ i ( ω ) is analytic ne ar ω 0 if λ i ( ω 0 )  = λ j ( ω 0 ) , j  = i. Mor e over, λ 1 λ 2 λ 3 λ 4 = 1 . In view of the ab o ve discussions, we ha ve the following proposition that characterizes the eigen- v alues of (7) using the transfer matrix. Prop osition 3. F or e ach k ∈ B , the eigenvalues for (7) c onsist of al l ω ∈ C such that at le ast one of the eigenvalues for the tr ansfer matrix M ( ω ) satises λ i ( ω ) = e i k , i.e. ω ( k ) = { ω ∈ C : λ j ( ω ) = e i k for some 1 ≤ j ≤ 4 } . (11) 4 2.2 Band structure of the Hermitian photonic crystals 2.2.1 Flo quet-Bloch theory W e start from lay ered Hermitian media in whic h the permittivity and permeability functions are p ositiv e-denite matrices satisfying ε ( z ) = ε ∗ ( z ) and µ ( z ) = µ ∗ ( z ) . By eliminating the magnetic eld H in the Maxw ell’s equations (4), w e obtain the second-order Maxwell system ∇ × ( µ − 1 ∇ × E ) − ω 2 ε E = 0 . (12) F or eac h k ∈ B , let ν := ω 2 and consider the following eigenv alue problem that is equiv alent to (7): ∇ × ( µ − 1 ( z ) ∇ × E ( k ; z )) − ν ε ( z ) E ( k ; z ) = 0 , (13a) E ( k ; z + 1) = e i k E ( k ; z ) . (13b) It follo ws from the standard sp ectral theory for the self-adjoint second-order elliptic dieren tial op erator that (13) attains a sequence of real eigen v alues ordered in the follo wing w a y: ν 1 ( k ) ≤ ν 2 ( k ) ≤ · · · ≤ ν n ( k ) ≤ ν n +1 ( k ) ≤ · · · . W e focus on the non-negativ e eigenfrequencies ω n := √ ν n for (7), whic h is ordered as ω 1 ( k ) ≤ ω 2 ( k ) ≤ · · · ≤ ω n ( k ) ≤ ω n +1 ( k ) ≤ · · · . F or eac h n , the disp ersion relation ω n ( k ) is a contin uous function for k ∈ B . If max k ∈B ω n ( k ) < min k ∈B ω n +1 ( k ) , w e sa y there is a spectral gap betw een the band ω n ( k ) and ω n +1 ( k ) . The collection of the countable set of disp ersion relations { ω n ( k ) } ∞ n =1 for k ∈ B is called the band structure of the p eriodic medium. F rom the Flo quet-Bloch theory , the spectrum of the Maxwell operator L dened in (12) is giv en b y σ ( L ) = ∪ k ∈B ( ∪ ∞ n =1 ω n ( k )) . By virtue of prop osition 3, σ ( L ) can b e equiv alen tly describ ed b y the set Ω := { ω ∈ R : λ j ( ω ) ∈ T for some 1 ≤ j ≤ 4 } , wherein T = { e i k ; k ∈ ( − π , π ] } is the unit circle on the complex plane. The sp ectrum σ ( L ) is a collection of closed in terv als on the real line, whic h are separated by the spectral gap in terv als (max k ∈B ω n ( k ) , min k ∈B ω n +1 ( k )) for n = 1 , 2 , · · · . 2.2.2 Hermitian photonic crystals with sp ectral reciprocity Denition 4. A photonic crystal is said to attain sp e ctr al r e cipr o city if ω n ( k ) = ω n ( − k ) holds for any k ∈ B and n ∈ N + . Otherwise, the sp e ctrum of the photonic crystal is c al le d non-r e cipr o c al. W e rst consider the Hermitian photonic crystal for whic h the system (7) attains the sp ectral recipro cit y . In general, the sp ectral recipro cit y holds when the symmetry group of the p erio d medium attains a certain op eration g such that g Φ ( k, · ) = Φ ( − k , · ) . Example 1. (spatial inv ersion symmetry): ε ( z ) = ε ( − z ) , µ ( z ) = µ ( − z ) : Let k ∈ (0 , π ] and ( ω , Φ ( k ; z )) be an eigenpair of (7), then it can be v eried that Φ ( − k ; z ) := Φ ( k , − z ) satises ( 7a) with the boundary condition Φ ( − k ; 1) = e − i k Φ ( k ; 0) . Hence ( ω , Φ ( − k ; z )) is an eigenpair of (7) 5 with − k . Example 2. (time-reversal symmetry): ε ( z ) = ε ( z ) and µ ( z ) = µ ( z ) . Time-reversal symme- try says that if { E ( z , t ) , H ( z , t ) } is a solution of the time-dep enden t Maxw ell’s equations, so is { E ( z , − t ) , − H ( z , − t ) } . Note that for the time-harmonic electromagnetic eld that satises (7), using the quasi-p erio dic condition Φ ( k ; z + 1) = e i k Φ ( k ; z ) , the Bloch mode can be expressed as Φ ( k ; z + 1) = e i kz U ( k ; z ) , wherein  d dz + i k  U ( k ; z ) = ω Q − 1 A ( z ) U ( k ; z ) , (14a) U ( k ; 1) = U ( k ; 0) . (14b) Let ( ω , U ( k ; z )) b e an eigenpair of (14), wherein U ( k ; z ) = [ u 1 ( k ; z ) , u 2 ( k ; z )] T . Then the time- rev ersal symmetry implies that ( ω , V ( − k ; z )) is an eigenpair of (14) for − k , wherein V ( − k ; z ) = [ u 1 ( k ; z ) , − u 2 ( k ; z )] T . 2.2.3 Hermitian photonic crystals breaking the sp ectral recipro cit y F rom the discussion in the previous subsection, in order to break the sp ectral recipro cit y of the photonic crystal, one needs to break the spatial in version symmetry and time-rev ersal symmetry of the system. The former can be ac hieved b y using more than tw o lay ers with dieren t medium parameters, and the latter can b e ac hieved b y using magnetic photonic crystals [11]. T o this end, w e in tro duce the so-called A and F lay ers as follo ws. The A la yers are made of a non-magnetic dielectric material with anisotrop y o ver the xy plane: ε A ( δ, φ ) =   ε 0 + δ cos(2 φ ) δ sin(2 φ ) 0 δ sin(2 φ ) ε 0 − δ cos(2 φ ) 0 0 0 ε z z   , µ A = I . (15) The tensor ε A is real v alued, wherein the parameter δ describ es the magnitude of in-plane anisotrop y , while the angle φ denes the orientation of the principal axes of tensor ε A on the xy plane. The F la y ers are ferromagnetic with magnetization parallel to the z direction, for which the p ermittivit y and p ermeability tensors are giv en by ε F ( α ) =   ˜ ε 0 i α 0 − i α ˜ ε 0 0 0 0 ε z z   , µ F ( β ) =   1 i β 0 − i β 1 0 0 0 µ z z   . (16) The real parameters α and β are resp onsible for the magnetic F araday rotation. It is clear that ε F ( α )  = ε F ( α ) and µ F ( β )  = µ F ( β ) . The sp ectral asymmetry can b e achiev ed by stacking the A and F lay ers together. Let us consider a p erio dic medium for which eac h p erio dic cell consists of one F lay er sandwiched b y t w o A lay ers with dierent rotations φ 1 and φ 2 . The corresp onding medium parameters for the primitiv e cell are given by ε ( z ) =    ε A ( δ, φ 1 ) , 0 < z < L ; ε F ( α ) , L < z < 1 − L ; ε A ( δ, φ 2 ) , 1 − L < z < 1 . µ ( z ) =    µ A , 0 < z < L ; µ F ( β ) , L < z < 1 − L ; µ A , 1 − L < z < 1 . (17) 6 It the ab o ve, it is assumed that φ 1 − φ 2  = 0 , π 2 , and α, β  = 0 . The p eriodic medium do es not attain the spatial inv ersion symmetry by using three lay ers with dierent medium parameters. The time-rev ersal symmetry is also broken by using the F la y er. Hence, the corresp onding Maxw ell’s op erator may achiev e spectral non-recipro cit y ω n ( k )  = ω n ( − k ) . T o illustrate the sp ectral asymme- try , let us set ε 0 = 13 , ˜ ε 0 = 1 in (15) and (16), and consider the following example. Example 3. Consider a three-lay er p eriodic medium with the p ermittivit y and p ermeabilit y tensors in the form of (17), with the parameters δ = 6 , φ 1 = 0 , φ 2 = 0 . 8 , α = β = 0 . 5 . The corresp onding transfer matrix is expressed as a composition of the transfer matrices for the AF A la y er as M ( ω ) = T A ( ω , φ 1 ; L ) T F ( ω ; 1 − L ) T A ( ω , φ 2 ; L ) , where the explicit expression of the A la y er and F lay er are giv en in App endix B. Solving the c haracteristic equation (10) for each k ∈ B yields the band structure of the p erio dic medium as shown in Figure 2 (Left). - 0 0 0.2 0.4 0.6 0 0.2 0.4 0.6 -0.12 -0.08 -0.04 0 Figure 1: Band structures of the three-la y er p eriodic media which attain sp ectral asymmetry . The p ermittivit y and p ermeabilit y are giv en in the form of (17), with δ = 6 , φ 1 = 0 , φ 2 = 0 . 8 , α = β = 0 . 5 . Left: Hermitian photonic crystal with ε 0 = 13 , ˜ ε 0 = 1 ; Righ t: Non-Hermitian photonic crystal with ε 0 = 13 + 5i , ˜ ε 0 = 1 . . 2.3 Band structure of non-Hermitian photonic crystals When ε ( z )  = ε ∗ ( z ) or µ ( z )  = µ ∗ ( z ) , the eigenfrequency for (7) or (13) ma y b ecome complex-v alued. This occurs, for instance, when photonic crystals are made of lossy materials. F or eac h k ∈ B , w e order the eigenfrequencies in an increasing order by their real parts suc h that min k ∈B Re ω n ( k ) ≤ min k ∈B Re ω n +1 ( k ) and max k ∈B Re ω n ( k ) ≤ max k ∈B Re ω n +1 ( k ) . In addition, each sp ectral band ω n ( k ) is a contin uous function of k with ω n ( − π ) = ω n ( π ) . W e consider the photonic crystal ( ε ( z ) , µ ( z )) for which there is no ambiguit y to dene the disp ersion curv es { ω n ( k ) } ∞ n =1 with the ab o ve criteria. The denition of the sp ectral gap for Hermitian systems can not b e extended to non-Hermitian ones, as the eigen v alues are distributed ov er the complex plane. F or non-Hermitian systems, one ma y introduce tw o dierent types of complex-energy gaps, the so-called point gap and line gap [12]. A non-Hermitian system is said to attain a p oint gap if its disp ersion curv e forms a lo op that 7 encircles a reference p oint ω B ∈ C and the crossing the base point denes a gap closing transition. A line gap, on the other hand, is dened by a line ov er the complex plane that do es not in tersect with the sp ectral bands of the system. W e refer to Figure 2 for an illustration of these t w o t yp es of sp ectral gaps. Re #𝜔 Im #𝜔 Re #𝜔 Im #𝜔 𝜔 ! Figure 2: Poin t gap (left) and line gap (righ t) for the sp ectrum of non-Hermitian operators. . Similar to Hermitian photonic crystals, we distinguish tw o types of non-Hermitian photonic crystals: (i) The sp ectral reciprocity holds suc h that ω n ( k ) = ω n ( − k ) for all n . (ii) The sp ectral reciprocity is brok en with ω n ( k )  = ω n ( − k ) for some n . Again the sp ectral recipro cit y holds when the symmetry group of the p erio d medium attain an op eration g such that g Φ ( k , · ) = Φ ( − k , · ) . T o break sp ectral recipro cit y , one needs to exclude suc h op erations in the symmetry group of the perio dic media, such as the inv ersion symmetry and time-rev ersal symmetry . A point gap occurs for a non-Hermitian photonic crystal when the sp ectrum is non-reciprocal, since the sp ectral symmetry ω n ( k ) = ω n ( − k ) implies that the sp ectral band ω n ( k ) typically forms a trivial gap with an empty in terior region as illustrated in Figure 3 (left). The sp ec- tral non-recipro cit y allows the t w o segmen ts of the disp ersion curv e, { ω ( k ); k ∈ [ − π , 0] } and { ω ( k ); k ∈ [0 , π ] } , to not o verlap and form a closed lo op (cf. Figure 3, right). T o demonstrate the existence of p oint gaps for non-reciprocal and non-Hermitian photonic crystals, let us revisit the three-la y er p eriodic medium with the p ermittivit y and p ermeabilit y tensors in the form of (17). W e set all the parameters the same as in Example 3, except for p erturbing ε 0 from 13 to 13 + 5i for ε A suc h that the system become non-Hermitian. The corresponding band structure is sho wn Figure 2 (Righ t). It is seen that eac h of the rst v e bands forms a closed lo op o v er the complex plane enclosing a non-trivial interior region. As suc h, eac h dispersion curve encircles a reference base p oint ω B and the non-Hermitian photonic crystal attains point gaps. Remark 1 In general, the sp ectral non-recipro cit y for photonic crystals can only b e ac hiev ed when the medium parameter ε ( z ) or µ ( z ) is anisotropic with nonzero o-diagonal entries, no matter the system is Hermitian or non-Hermitian. This is b ecause the mo del for the electromagnetic w av e propagation in a one-dimensional isotropic medium is a scalar second-order ODE for the electric or magnetic eld. The corresp onding disp ersion relation is determined b y the nonlinear equation tr ( M ( ω )) = 2 cos( k ) (see, for instance, [16]), whic h implies that ω ( k ) = ω ( − k ) . 8 Re #𝜔 Im #𝜔 Re #𝜔 Im #𝜔 𝑘 𝑘 Figure 3: Spectrum of non-Hermitian op erators o ver the complex plane: a trivial gap when ω n ( k ) = ω n ( − k ) (left) and a non-trivial gap when the spectral symmetry is brok en (righ t). . 3 Sp ectral top ology and edge mo des in non-Hermitian photonic crystals In this section, we inv estigate the edge mo des in semi-innite photonic crystals for which the disp ersion curves for the quasi-p eriodic problem (7) attain p oint gaps. Based on the discussions in the previous section, w e consider non-Hermitian and non-reciprocal photonic crystals such that the sp ectrum lies on the complex plane and the dispersion curv es enclose non trivial in terior regions. W e sho w that the eigenfrequencies for the edge mo des are closely related to the winding directions of the disp ersion curv es. In addition, the winding n umber of each dispersion curve can b e c haracterized b y a new topological inv ariant dened in terms of the eigen v alues of the underlying transfer matrix. 3.1 T opological index for the sp ectral gap Consider a photonic crystal with the medium parameter ( ε ( z ) , µ ( z )) . Let T := { e i k ; k ∈ ( − π , π ] } b e the Brillouin zone ov er the complex plane. F or con v enience of notation, for each ξ ∈ T , w e rewrite the eigen v alue problem (7) as d dz Φ ( ξ ; z ) = ω ( ξ ) Q − 1 A ( z ) Φ ( ξ ; z ) , (18a) Φ ( ξ ; z + 1) = ξ Φ ( k ; z ) . (18b) Corresp ondingly , w e view the n -th spectral band as a function from T to C , which, with the abuse of notation, is still denoted by ω n . W e remark that ω n ( · ) can b e analytically contin ued lo cally . The corresp onding disp ersion curve is denoted by γ n = ω n ( T ) . W e express the union of dispersion curv es and their complemen t as Γ = ∞  n =1 γ n and Γ c = C \ Γ . F or an y ω ∈ C , let λ j ( ω ) ( j = 1 , 2 , 3 , , 4 ) be the eigen v alue branc hes of the transfer matrix M ( ω ) for the p eriodic medium that are dened in (34). W e in tro duce an index at ω as Ind ( ω ) = 1 2  # { i : | λ i ( ω ) | < 1 } − # { i : | λ i ( ω ) | > 1 }  . 9 The union of disp ersion curves Γ divides the complex plane in to connected comp onen ts. Let D b e one connected comp onen t in Γ c , we rst sho w that Ind ( ω ) is inv ariant for ω ∈ D . Lemma 5. L et D b e a c onne cte d c omp onent of Γ c . The value of Ind ( ω ) is indep endent of the choic e of ω ∈ D . Pro of W e only need to show that index Ind ( ω ) is lo cally inv ariant for ω ∈ D . Case 1 . ω 0 is not a branch point of the four eigenv alue functions λ j ( ω ) ( j = 1 , 2 , 3 , , 4 ). Then the v alues λ j ( ω 0 ) are pairwise distinct. In view of (11), eac h eigen v alue λ j ( ω ) has modulus either strictly less than 1 or strictly greater than 1 . By the standard perturbation theory , for ω suciently close to ω 0 , the four eigenv alue branc hes λ j ( ω ) can be c hosen con tinuously . Hence the quantities | λ j ( ω ) | v ary con tinuously in the small neighborho o d of ω . As suc h, the function | λ j ( ω ) | − 1 has constan t sign for all ω sucien tly close to ω 0 . Consequen tly , the counts # { i : | λ i ( ω ) | < 1 } and # { i : | λ i ( ω ) | > 1 } is lo cally constant near ω 0 , so is the index function Ind ( ω ) . Case 2 . If ω 0 is a branch point of the four eigen v alue functions λ j ( ω ) . Then the transfer matrix M ( ω ) has eigenv alues with algebraic m ultiplicity strictly greater than one at ω 0 . Although the individual eigen v alue branch need not admit con tin uous parameterizations near ω 0 , the m ultiset of eigen v alues (counting multiplicit y) depends con tinuously on ω in the optimal matching metric (cf. [9].) 1 . Consequen tly , the coun ts # { i : | λ i ( ω ) | < 1 } and # { i : | λ i ( ω ) | > 1 } are lo cally constant in ω , and hence so is the index Ind ( D , ω ; Γ) . In view of Lemma 5, for a giv en connected comp onen t D ⊂ Γ c , w e dene a comp onen t index asso ciated with D b y Ind ( D ) = Ind ( ω ) , ω ∈ D . It is clear that Ind ( D ) is w ell-dened. Denition 6. W e c al l that a c onne cte d c omp onent D of Γ c is a princip al gapp e d c omp onent if Ind ( D ) = 0 . 3.2 Homotop y in v ariance of Ind ( D ) for the principal gapp ed comp onen t W e view the non-Hermitian photonic crystal ( ε ( z ) , µ ( z )) as a con tinuous deformation from a Her- mitian photonic crystal ( ε H ( z ) , µ H ( z )) whose spectrum is real. More precisely , we introduce tw o con tin uous functions h ε , h µ : R × [0 , 1] → C 2 × 2 , suc h that for all z ∈ R , h ε ( z ; 0) = ε H ( z ) , h µ ( z ; 0) = µ H ( z ) , h ε ( z ; 1) = ε ( z ) , h µ ( z ; 1) = µ ( z ) . 1 Let X = { x 1 , x 2 , . . . , x n } and Y = { y 1 , y 2 , . . . , y n } b e tw o unordered n -tuples (multisets) of complex n umbers. The optimal matching distance b etw een X and Y , denoted as d ( X , Y ) , is dened as: d ( X, Y ) = min π ∈ S n max 1 ≤ i ≤ n | x i − y π ( i ) | where S n is the symmetric group of degree n , representing all p ossible p erm utations. 10 F or each t ∈ [0 , 1] , let ω n,t : T → C b e the n -th sp ectral band function and γ n,t = ω n,t ( T ) b e the corresp onding disp ersion curve. The t -dependent band diagram and its complemen t are giv en b y Γ t = ∞  n =1 γ n,t and Γ c t = C \ Γ t . (19) Let λ j ( ω ; t ) ( j = 1 , 2 , 3 , 4 ) b e the eigen v alue branc hes of the transfer matrix M ( ω ; t ) for the perio dic medium ( h ε ( z ; t ) , h µ ( z ; t )) that are dened in (34). F or each connected component D t ⊂ Γ c t , we denote the t-dep enden t index comp onen t b y Ind ( D t ; t ) := 1 2  # { i : | λ i ( ω ; t ) | < 1 } − # { i : | λ i ( ω ; t ) | > 1 }  , ω ∈ D t . W e rst sho w that Γ c 0 is a principal gapped component for the Hermitian medium ( ε H ( z ) , µ H ( z )) under the follo wing assumption. Assumption 7. (i) The sp e ctrum of the Hermitian phtononic crystal ( ε H ( z ) , µ H ( z )) c ontains a b andgap. A s a r esult, the set Γ c 0 is a c onne cte d and unb ounde d domain. (ii) Ther e exists ω 0 ∈ Γ 0 such that ther e ar e exactly two eigenvalues, say λ 1 and λ 2 , such that | λ 1 ( ω 0 ; 0) | = | λ 2 ( ω 0 ; 0) | = 1 and λ 1 ( ω 0 ; 0)  = λ 2 ( ω 0 ; 0) . Lemma 8. Under A ssumption 7, Γ c 0 is princip al gapp e d c omp onent, i.e., Ind (Γ c 0 ; 0) = 0 . Pro of Let ω ∗ ∈ Γ c 0 . Recall that det( M ( ω ∗ )) =  4 j =1 λ j ( ω ∗ ; 0) = 1 . Given that | λ 3 ( ω ∗ ; 0) |  = 1 and | λ 4 ( ω ∗ ; 0) |  = 1 , w e may assume without loss of generalit y that | λ 3 ( ω ∗ ; 0) | < 1 and | λ 4 ( ω ∗ ; 0) | > 1 . Since λ 3 ( ω ∗ ; 0) and λ 4 ( ω ∗ ; 0) do not coincide with other eigenv alues at ω ∗ , there exists a sucien tly small neighborho o d of ω ∗ , B ( ω ∗ , δ ) for some δ > 0 , suc h that λ 3 ( ω ; 0) and λ 4 ( ω ; 0) are analytic with | λ 3 ( ω ; 0) | < 1 and | λ 4 ( ω ; 0) | > 1 for all ω ∈ B ( ω ∗ , δ ) . Consider the pro duct of the remaining t wo eigen v alues: Q ( ω ) =: λ 1 ( ω ; 0) λ 2 ( ω ; 0) = 1 λ 3 ( ω ; 0) λ 4 ( ω ; 0) . The function Q ( ω ) is analytic on B ( ω ∗ , δ ) . By the open mapping theorem, the image Q ( B ( ω ∗ , δ )) is an op en set in C con taining the p oin t z 0 =: λ 0 ( ω ∗ ; 0) λ 2 ( ω ∗ ; 0) ∈ T . W e claim that there exists ˜ ω ∗ ∈ B ( ω ∗ , δ ) \ Γ 0 suc h that ( | λ 1 ( ˜ ω ∗ ; 0) | − 1) · ( | λ 2 ( ˜ ω ∗ ; 0) | − 1) < 0 . Then the conclusion of the lemma follo ws b y recalling that | λ 3 ( ˜ ω ∗ ; 0) | < 1 and | λ 4 ( ˜ ω ∗ ; 0) | > 1 . T o pro v e the claim, w e assume, to the contrary , that ( | λ 1 ( ω ; 0) | − 1) · ( | λ 2 ( ω ; 0) | − 1) ≥ 0 for all ω ∈ B ( ω ∗ , δ ) \ Γ 0 . Since | λ i ( ω ; 0) | − 1  = 0 for ω ∈ Γ c 0 and i = 1 , 2 , w e ha ve ( | λ 1 ( ω ; 0) | − 1) · ( | λ 2 ( ω ; 0) | − 1) > 0 for all ω ∈ B ( ω ∗ , δ ) \ Γ 0 . Hence, either | Q ( ω ) | = | λ 1 ( ω ; 0) λ 2 ( ω ; 0) | > 1 for all ω ∈ B ( ω ∗ , δ ) \ Γ 0 or | Q ( ω ) | = | λ 1 ( ω ; 0) λ 2 ( ω ; 0) | < 1 for all such ω . In other words, Q  B ( ω ∗ , δ ) \ Γ 0  lies either en tirely outside the unit circle T or en- tirely inside it. This, how ev er, con tradicts the previously established fact that Q  B ( ω ∗ , δ )  con tains an op en disc centered at a p oin t z 0 ∈ T . 11 Based on the ab ov e discussions, w e see that # { i : | λ i ( ˜ ω ∗ ; 0) | > 1 } = # { i : | λ i ( ˜ ω ∗ ; 0) | < 1 } = 2 . Therefore, Ind (Γ c 0 ; 0) = Ind ( ˜ ω ∗ ) = 0 . Theorem 9. Consider the p erio dic me dia  h ε ( · , t ) , h µ ( · , t )  with  h ε ( · , 0) , h µ ( · , 0)  = ( ε H ( z ) , µ H ( z )) and  h ε ( · , 1) , h µ ( · , 1)  = ( ε ( z ) , µ ( z )) . L et Γ t and Γ c t b e the union of the c orr esp onding disp ersion curves and their c omplement. Given a c onne cte d c omp onent D ⊂ Γ c 1 . If A ssumption 7 holds for  h ε ( · , 0) , h µ ( · , 0)  , and ther e exists a c ontinuous curve g : [0 , 1] → C such that (i) F or e ach 0 ≤ t ≤ 1 , g ( t ) ∈ Γ c t . (ii) g (1) ∈ D ⊂ Γ c 1 ; Then D is a princip al gapp e d c omp onent, i.e. I nd ( D ; 1) = 0 . Pro of : F or each 0 ≤ t ≤ 1 , we dene a comp onen t index function G ( t ) = 1 2 (# { i : | λ i ( g ( t ); t ) | < 1 } − # { i : | λ i ( g ( t ); t ) | > 1 } ) . By Lemma 8, we ha ve G (0) = 0 ; th us, to show that G v anishes iden tically , it suces to establish that it is lo cally constan t on [0 , 1] . T o this end, w e x t 0 ∈ [0 , 1] and sho w that G ( t ) is constan t for t sucien tly close to t 0 . Since g ( t ) ∈ Γ c t , in view of (11), eac h eigenv alue | λ j ( g ( t 0 ); t 0 ) | is either strictly less than 1 or strictly greater than 1 . Using the p erturbation theory for the transfer matrix M ( g ( t ); t ) , for t sucien tly close to t 0 , the eigenv alue set { λ j ( g ( t ); t ) : j = 1 , 2 , 3 , 4 } is con tinuously in t in the optimal matching metric. In the case when g ( t 0 ) is not a branch p oin t, the four eigenv alue branc hes λ j ( g ( t ); t ) are con tinuous for t near t 0 . The con tinuit y of the eigenv alue set in t implies that the counts # { i : | λ i ( g ( t ); t ) | < 1 } and # { i : | λ i ( g ( t ); t ) | > 1 } are lo cally constant in t , so is G ( t ) . This completes the pro of of the lemma. The pro of of the ab ov e theorem do es not require the op erator for the p eriodic medium ( ε ( z ) , µ ( z )) to b e non-Hermitian. Th us, the theorem holds for a generic photonic crystal ( ε ( z ) , µ ( z )) . If one applies the theorem to a Hermitian crystal ( ε ( z ) , µ ( z )) and chooses D = Γ c 1 and g ( t ) ≡ ω 0 ∈ C \ R , then it can b e concluded that Ind (Γ c 1 ; 1) = 0 . Namely , the complement of the sp ectrum for a Hermitian p erio dic medium is a principal gap component. Corollary 10. L et Γ b e the union of the disp ersion curves for the Hermitian photonic crystal ( ε ( z ) , µ ( z )) and Γ  = R , then Γ c is a c onne cte d and unb ounde d c omp onent, and ther e holds I nd (Γ c ) = 0 . 12 3.3 Jump of Ind ( D ) across the disp ersion curv e The disp ersion curv es Γ 1 for the p erio dic medium ( ε ( z ) , µ ( z )) divide the whole complex plane into connected components (cf. Figure 4). Let us consider the disp ersion curv e γ n , whic h is the image of the n -th band disp ersion function ω n : T → C . Due to the p erio dicit y of the sp ectral problem, it is clear that γ n is a closed curv e. The curv e γ n partitions the complex plane in to a set of disjoin t connected comp onen ts. Among them, there exists one unique un b ounded connected comp onent D e n and a nite n umber of bounded components D i n ( i = 1 , 2 , · · · , i 0 ). W e assume that each D i n is a Jordan domain with a simple closed b oundary γ i n that is smo oth except for nitely many corners. F or a given smo oth p oin t ζ ∈ γ i n , w e dene the out ward unit normal ν ( ζ ) as the one that p oin ts in to the un b ounded domain D e n . 𝛾 ! 𝛾 " 𝛾 # 𝛾 $ 𝛾 % Figure 4: A sc hematic plot of dispersion curv es γ 1 , γ 2 , γ 3 , · · · . . F or the simple closed curv e γ i n , we sa y that it is oriented coun terclo ckwise (or positively orien ted) if the mapping ω n preserv es the counterclockwise orientation of the unit circle T for ω n ( ξ ) ∈ γ i n . More precisely , let ξ 0 = (cos( k 0 ) , sin( k 0 )) ∈ T be such that ω n ( ξ 0 ) is a smooth p oin t of γ i n . Let τ = ( − sin( k 0 ) , cos( k 0 )) be the unit tangent vector to T at ξ 0 , and ω ′ n ( ξ 0 ) v be the tangen t v ector to the curv e γ i n at the point ζ 0 = ω n ( ξ 0 ) . The curv e γ i n is oriented coun terclo ckwise if the out w ard normal ν ( ζ 0 ) p oints in the direction of rotating the tangen t v ector ω ′ n ( ξ 0 ) v clo c kwise by π / 2 . Otherwise, the curve γ i n is said to b e orien ted clockwise (or negativ ely orien ted). Let D b e a connected comp onen t of Γ c 1 and assume that the assumptions in Theorem 9 hold suc h that D is a principal gapp ed comp onent with Ind ( D ) = 0 . F or instance, one may set D = ∞  n =1 D e n if the latter is connected. In this subsection, we study the topological index of a b ounded connected comp onen t D i n that is adjacent to the principal gapped comp onen t D . Our main result is that the top ological index attains a jump when crossing the dispersion curve γ n from D to D i n . W e rst giv e the winding num b er of the disp ersion curv e γ n . Theorem 11. L et D i n b e a b ounde d c omp onent p artitione d by γ n , with the b oundary given by the simple close d curve γ i n . If γ i n is p ositively oriente d, then the winding numb er W ( γ n ; ω B ) = 1 2 π i  ξ ∈ T 1 ω n ( ξ ) − ω B = 1 for ω B ∈ D i n . If γ i n is ne gatively oriente d, then the winding numb er W ( γ n , ω B ) = − 1 for ω B ∈ D i n . Pr o of. W e give the pro of when γ i n is p ositiv ely oriented, and the proof is similar when γ i n is neg- ativ ely orien ted. If ω n ( ξ ) is injectiv e for ξ ∈ T , then γ n is a simple closed curv e on the complex 13 plane and there holds γ n = γ 1 n . By the Jordan Curve Theorem, γ 1 n divides the complex plane in to a b ounded comp onen t D 1 n and an unbounded comp onen t C \ D 1 n . Since the mapping ω n ( z ) is orien- tation preserving, the winding num b er W ( γ n ; ω B ) = W ( γ 1 n ; ω B ) = 1 for any ω B ∈ D i n . If ω n ( ξ ) is not injective for ξ ∈ T , then for an y ω B in D i n , it is clear that the winding n um b er W ( γ i n ; ω B ) = 1 and W ( γ j n ; ω B ) = 0 for j  = i , since ω B ∈ D i n lies in the exterior of the domain enclosed b y γ j n . Therefore, W ( γ n , ω B ) =  i 0 j =1 W ( γ j n ; ω B ) = 1 . Next, we inv estigate the top ological index of the bounded comp onen t D i n . Theorem 12. L et D i n b e a b ounde d c omp onent p artitione d by γ n , with the b oundary given by the simple close d curve γ i n . A ssume that γ n ∩ γ m = ∅ for m  = n . F urther assume that (i) Ther e exists ξ 0 ∈ T such that ζ 0 = ω n ( ξ 0 ) ∈ γ i n is a smo oth p oint of γ n with a wel l-dene d outwar d unit normal ve ctor ν . A dditional ly, ζ 0 + tν ∈ D and ζ 0 − tν ∈ D i n for suciently smal l t > 0 . (ii) The function ω n ( ξ ) dene d via (18) is analytic at ξ 0 . If γ i n is p ositively oriente d, then Ind ( D i n ; 1) = W ( γ n ; ω B ) = 1 for ω B ∈ D i n . Otherwise, Ind ( D i n ; 1) = W ( γ n ; ω B ) = − 1 . Pr o of. W e assume that γ i n is p ositively orien ted. Step 1. The mapping ω n ( ξ ) ne ar ξ 0 . Since ω n ( ξ ) is analytic at ξ 0 , we may expand ω n ( ξ ) as ω n ( ξ ) = ω n ( ξ 0 ) + ∞  m =1 α n ( ξ − ξ 0 ) m . Note that ζ 0 = ω n ( ξ 0 ) is a smo oth point of γ n , w e deduce that ω ′ n ( ξ 0 )  = 0 . As such ω n ( ξ ) is a conformal mapping at ξ 0 and ω n ( ξ ) is lo cally injective at ξ 0 . Let τ = ( − sin( k 0 ) , cos( k 0 )) and ν = (cos( k 0 ) , sin( k 0 )) b e the unit tangent and normal v ectors to T at ξ 0 resp ectiv ely . Then at ζ 0 = ω n ( ξ 0 ) ∈ γ i n , the normal vector ω ′ n ( ξ 0 ) · ν to the curve γ n is obtained by rotating the tangent vector ω ′ n ( ξ 0 ) · τ clo c kwise b y π / 2 . Since γ i n is p ositively orien ted, or the mapping ω n ( ξ ) preserves the orien tation on T for z near ξ 0 , we deduce that the normal v ector ω ′ n ( ξ 0 ) · ν po in ts in to the un b ounded exterior domain C \ D i n . In view of the assumption (i), this implies that for sucien tly small t > 0 , ω n ( ξ 0 + tν ) ∈ D and ω n ( ξ 0 − tν ) ∈ D i n (20) resp ectiv ely . Step 2. Computing I nd ( D i n ; 1) . Recall that ω n ( z ) is a conformal mapping at ξ 0 . There exists an inv erse map near ζ 0 , which we denote b y f : B ( ζ 0 , δ 0 ) → B ( ξ 0 , η 0 ) . By virtue of (20), we hav e f  B ( ζ 0 , δ 0 ) ∩ D  ⊂ B ( ξ 0 , η 0 ) ∩ { ξ : | ξ | > 1 } (21) and f  B ( ζ 0 , δ 0 ) ∩ D i n  ⊂ B ( ξ 0 , η 0 ) ∩ { ξ : | ξ | < 1 } . (22) 14 Since Ind ( D ) = 0 implies that there are exactly t w o eigenv alues of the transfer matrix M ( ω ) , say λ 1 ( ω ) and λ 2 ( ω ) , that attain mo dulus strictly greater than 1 for ω ∈ D . F rom (21), w e see that the mapping f coincides with one of these tw o eigenv alue functions. Without loss of generality , let us assume that f ( ω ) = λ 1 ( ω ) for ω ∈ B ( ζ 0 , δ 0 ) . In view of (22), there holds | λ 1 ( ω ) | < 1 for ω ∈ B ( ζ 0 , δ 0 ) ∩ D i n . On the other hand, due to the uniqueness of the in verse map f in B ( ζ 0 , δ 0 ) , w e deduce that | λ 2 ( ω ) | > 1 , | λ 3 ( ω ) | < 1 , | λ 4 ( ω ) | < 1 (23) for ω ∈ B ( ζ 0 , δ 0 ) . W e conclude that I nd ( D i n ; 1) = 1 . F ollowing the parallel lines, it can b e shown that I nd ( D i n ; 1) = − 1 when γ i n is negatively orien ted. The abov e theorem gives the topological index of a bounded domain D i n enclosed b y the dis- p ersion curv e γ n , which do es not intersect with other dispersion curv es. The following prop osition considers the scenario when disp ersion curves ma y intersect. Prop osition 13. L et D i n b e a b ounde d c omp onent p artitione d by γ n with the simple close d curve γ i n as its b oundary. A ssume that γ n ∩ γ m  = ∅ for some m  = n . In addition, the assumptions (i)(ii) in The or em 12 hold. Then Ind   D i n \  m  = n D m ; 1   = W ( γ n ; ω B ) = 1 for ω B ∈ D i n when γ i n is p ositively oriente d, and Ind   D i n \  m  = n D m ; 1   = W ( γ n ; ω B ) = − 1 for ω B ∈ D i n when γ i n is ne gatively oriente d. In the ab ove, D m = i m 0  i =1 D i m is the set of al l b ounde d domains p artitione d by the disp ersive curve γ m . The pro of for the prop osition is similar to the pro of of Theorem 12. It is clear that (21) still holds, and (22) holds with D i n replaced b y D i n \  m  = n D m . As such, one can still concludes that only three eigenv alues attain mo dulus smaller than 1 in the domain D i n \  m  = n D m when γ i n is p ositiv ely orien ted. Remark 2 In the abov e discussions, for clarit y w e assume that eac h bounded comp onen t D i n is a Jordan domain with a simple closed b oundary γ i n . It should be pointed out that the argumen t ab o ve can be extended to the scenario when D i n is not a Jordan domain. F or instance, the positively orien ted disp ersion curve γ n sho wn in Figure 5 partitions the complex plane into D 1 n , D 2 n , and D e n . It can b e prov ed that Ind ( D 1 n ; 1) = W ( γ n ; ω B ) = 1 for ω B ∈ D 1 n , and Ind ( D 2 n ; 1) = W ( γ n ; ω B ) = 2 for ω B ∈ D 2 n . 15 𝛾 ! 𝐷 ! " 𝐷 ! # 𝐷 ! $ 𝐷 ! $ Figure 5: The domains enclosed b y γ n are not Jordan domains. . 3.4 Edge mo des in semi-innite photonic crystals with non-trivial sp ectral top ol- ogy W e consider a semi-innite photonic crystal sitting in the in terv al I − := ( −∞ , 0) or I + := (0 , ∞ ) . F or z ∈ I ± , the medium parameter is giv en by ( ε ( z ) , µ ( z )) , and the am bien t medium outside the photonic crystal is assumed to b e v acuum. The corresp onding sp ectral problem is form ulated as nding ( ω , Φ ) ∈ C × ( L 2 ( I ± )) 4 suc h that d dz Φ ( z ) = ω Q − 1 A ( z ) Φ ( z ) , z ∈ I ± ; (24a) Φ (0) =  E 0 (0) H 0 (0)  . (24b) In the abov e, { E 0 ( z ) , H 0 ( z ) } is the outgoing electromagnetic w av e in the am bient medium. More explicitly , { E 0 ( z ) , H 0 ( z ) } can b e expressed as  E 0 ( z ) H 0 ( z )  =  c 1 c 2  e i k 0 z and  E 0 ( z ) H 0 ( z )  =  c 1 c 2  e − i k 0 z for z > 0 and z < 0 respectively . In the ab o ve, k 0 := ω /c is the w av enum b er in the v acuum, and c 1 , c 2 ∈ C 2 are constant vectors that satisfy c 2 = Q c 1 and c 2 = − Q c 1 (25) when z > 0 and z < 0 respectively . The corresp onding innite perio dic problem is giv en b y (18), whic h attains the disp ersion curves { γ n = ω n ( T ) } n ∈ N + o v er the complex plane. W e consider ω ∈ C that is located in one of b ounded domains partitioned by the disp ersion curv e γ n . F or clarity of presentation, let us assume that γ n ∩ γ m = ∅ for m  = n , and consider ω ∈ D i n . Then the eigen v alues of the semi-innite problem (24) are describ ed in the following theorem. Note that the assumption on the non-in tersection of the disp ersion curv es is not essential as long as w e restrict ω to a smaller domain D i n \  m  = n D m ; w e refer to Prop osition 13 and Remark 5 for the discussions of this scenario. Theorem 14. L et D i n b e one of the simply c onne cte d b ounde d c omp onents p artitione d by γ n . (i) If Ind ( D i n ; 1) = W ( γ n ; ω B ) = 1 for ω B ∈ D i n , (26) 16 then e ach ω ∈ D i n is an eigenvalue for the semi-innite pr oblem (24) dene d over the interval I + , as long as ω is not a br anch p oint of λ j ( ω ) (j=1,2,3,4) for the tr ansfer matrix M ( ω ) . The c orr esp onding eigenfunction Φ ( z ) de c ays exp onential ly as z → ∞ . (ii) If Ind ( D i n ; 1) = W ( γ n ; ω B ) = − 1 for ω B ∈ D i n , (27) then e ach ω ∈ D i n is an eigenvalue for the semi-innite pr oblem ( 24) dene d over the interval I − , as long as ω is not a br anch p oint of λ j ( ω ) (j=1,2,3,4) for the tr ansfer matrix M ( ω ) . The c orr esp onding eigenfunction Φ ( z ) de c ays exp onential ly as z → −∞ . Remark 3 By virtue of Theorem 11, under suitable assumptions, (26) and (27) hold when γ i n , the b oundary of the D i n , is p ositiv ely and negativ ely oriented respectively . Hence, the ab ov e theorem relates the existence of the edge modes for the semi-innite problem (24) with the winding direction of γ i n . Pro of If Ind ( D i n ; 1) = 1 , there exist three eigenv alues of the transfer matrix M ( ω ) with mo dulus less than 1, and one eigenv alue with mo dulus larger than 1. F urthermore, ω is not a branch p oin ts of λ j ( ω ) (j=1,2,3,4). Therefore, without loss of generality , there hold | λ 1 ( ω ) | < | λ 2 ( ω ) | < | λ 3 ( ω ) | < 1 < | λ 4 ( ω ) | . Let Φ j := [ E j , H j ] T b e the corresponding eigen v ectors of M ( ω ) for j = 1 , 2 , 3 , 4 . W e choose the initial v alue of Φ ( z ) at z = 0 as the linear com bination: Φ (0) = α 1 Φ 1 + α 2 Φ 2 + α 3 Φ 3 . (28) Then the solution of the ODE system (24a) with the ab o v e initial v alue takes the following form at z = n : Φ ( z ) = α 1 λ n 1 Φ 1 + α 2 λ n 2 Φ 2 + α 3 λ n 3 Φ 3 , (29) whic h decays exp onen tially as n → ∞ . W e only need to show that there exists α 1 , α 2 , and α 3 suc h that the initial condition (24b) is satised. In view of (25), this b oils down to solving the following linear system of tw o equations: B α = 0 , where B = [ H 1 , H 2 , H 3 ] + Q [ E 1 , E 2 , E 3 ] ∈ C 2 × 3 . (30) The ab o ve linear system attains at least one solution α := [ α 1 , α 2 , α 3 ] T ∈ C 3 . This prov es that ω is an eigenv alue of the problem (24), with the corresp onding eigenfunction decaying in the form of (29). F ollo wing the parallel lines, it can b e sho wn that any ω ∈ Ω n is an eigen v alue of ( 24) ov er the interv al I − when W n = − 1 . The corresp onding eigenfunction Φ ( z ) deca ys exp onen tially as z → −∞ . Remark 4 Theorem 14 holds if other b ound boundary conditions are imp osed. F or instance, if the photonic crystal is terminated with a perfect electric or magnetic conducting w all at z = 0 , which giv es E (0) = 0 and H ( 0 ) = 0 , for the initial condition (24b), resp ectiv ely . Similarly , one can nd the initial v alue in the form of (28) that satises the ab o ve b oundary condition. 17 Remark 5 If γ n ∩ γ m  = ∅ for some m  = n , then Theorem 14 holds for ω ∈ D i n \  m  = n D m . More sp ecically , ω is an eigen v alue of (24) o ver the in terv al I + and I − when Ind ( D i n \  m  = n D m ; 1) = 1 and Ind ( D i n \  m  = n D m ; 1) = − 1 resp ectiv ely . Giv en ω ∈ D i n , if r ank ( B ) = 2 for the matrix B in (30), it follo ws that dim ( k er ( B )) = 1 . This giv es the bulk-edge corresp ondence for non-Hermitian photonic crystals. Prop osition 15. ( Bulk-e dge c orr esp ondenc e ) L et D i n b e one of the simply c onne cte d b ounde d c omp onents p artitione d by γ n with Ind ( D i n ; 1) = ± 1 . F or e ach ω ∈ D i n , ther e is only one e dge mo de satisfying (24) with || Φ || L 2 ( I ± ) = 1 if rank ( B ) = 2 . Similarly , if Ind ( D i n ; 1) = W ( γ n ; ω B ) = 2 , all four eigenv alues of the transfer matrix attain mo dulus smaller than 1, and one can expand Φ (0) = α 1 Φ 1 + α 2 Φ 2 + α 3 Φ 3 + α 4 Φ 4 . The matrix for the corresp onding linear system (30) becomes B = [ H 1 , H 2 , H 3 , H 4 ] + Q [ E 1 , E 2 , E 3 , E 4 ] ∈ C 2 × 4 . Again dim ( k er ( B )) = 2 if r ank ( B ) = 2 , as such the num b er of edge modes satisfying (24) with || Φ || L 2 ( I ± ) = 1 is 2. App endix A Ro ots of quartic functions Consider solving a quartic equation λ 4 + a 3 λ 3 + a 2 λ 2 + a 1 λ + a 0 = 0 . (31) F ollowing the F errari metho d and introducing λ = z − a 3 4 , the quartic equation (31) can b e reduced to the depressed quartic equation z 4 + b 2 z 2 + b 1 z + b 0 = 0 , (32) where b 2 = a 2 − 3 8 a 2 3 ; b 1 = a 1 − 1 2 a 2 a 3 + 1 8 a 2 3 ; b 0 = a 0 − 1 4 a 1 a 3 + 1 16 a 2 a 2 3 − 3 256 a 4 3 . It can b e shown that if w solv es the cubic p olynomial w 3 + 2 b 2 w 2 + ( b 2 2 − 4 b 0 ) w − b 2 1 = 0 , (33) then the depressed quartic function in (32) can b e decomp osed as the pro duct of tw o quadratic functions:  z 2 + b 2 2 + w 2 + √ w z − b 1 2 √ w  ·  z 2 + b 2 2 + w 2 − √ w z + b 1 2 √ w  = 0 . 18 Th us one can obtain the solutions of the quartic equation b y solving the tw o quadratic equations ab o ve: λ 1 , 0 = a 3 4 − √ w 2 + √ u + 2 , λ 2 , 0 = a 3 4 − √ w 2 − √ u + 2 , λ 3 , 0 = a 3 4 + √ w 2 + √ u − 2 , λ 4 , 0 = a 3 4 + √ w 2 − √ u − 2 , (34) where u ± := − 2 b 2 − 2 w ± 2 b 1 / √ w . In the ab o ve, w can be expressed explicitly using the cubic formula: w = − 1 3  2 b 1 + v + ∆ 0 v  , v = 1 3 √ 2 · 3  ∆ 1 +  ∆ 1 − 4∆ 3 0 , (35) where ∆ 0 = a 2 2 − 3 a 1 a 3 + 12 a 0 , ∆ 1 = 2 a 3 2 − 9 a 1 a 2 a 3 + 27 a 0 a 2 3 + 27 a 2 1 − 72 a 0 a 2 . In particular, w abov e is chosen to be a real ro ot of (33) when a j are real. The discriminant of the quartic function is dened as ∆ = − ∆ 1 − 4∆ 2 0 27 . App endix B T ransfer matrices for the constan t media It can b e computed that the transfer matrix for the A la y er tak es the following form ([11]): T A ( ω ; L ) =     t 2 ρ 1 + ˜ t 2 ρ 2 t ˜ t ( ρ 1 − ρ 2 ) i t ˜ t  ˜ ρ 2 n − 1 2 − ˜ ρ 1 n − 1 1  i t 2 ˜ ρ 1 n − 1 1 + i ˜ t 2 ˜ ρ 2 n − 1 2 t ˜ t ( ρ 1 − ρ 2 ) ˜ t 2 ρ 1 + t 2 ρ 2 −  i t 2 ˜ ρ 1 n − 1 1 + i ˜ t 2 ˜ ρ 2 n − 1 2  i t ˜ t  ˜ ρ 1 n − 1 1 − ˜ ρ 2 n − 1 2  i t ˜ t ( ˜ ρ 2 n 2 − ˜ ρ 1 n 1 ) − (i t 2 ˜ ρ 1 n 1 + i ˜ t 2 ˜ ρ 2 n 2 ) ˜ t 2 ρ 1 + t 2 ρ 2 t ˜ t ( ρ 2 − ρ 1 ) i t 2 ˜ ρ 1 n 1 + i ˜ t 2 ˜ ρ 2 n 2 i t ˜ t ( ρ 1 n 1 − ρ 2 n 2 ) t ˜ t ( ρ 2 − ρ 1 ) t 2 ρ 1 + ˜ t 2 ρ 2     , where n 1 =  ε 0 + δ , n 2 =  ε 0 − δ ; ρ 1 = cos( n 1 ω L ) , ρ 2 = cos( n 2 ω L ) , t = cos( φ ); ˜ ρ 1 = sin( n 1 ω L ) , ˜ ρ 2 = sin( n 2 ω L ) , ˜ t = sin( φ ) . The transfer matrix for the F lay er is T F ( ω ; L ) =     σ 1 + σ 2 i( σ 1 − σ 2 ) ˜ σ 1 ˜ m − 1 1 − ˜ σ 2 ˜ m − 1 2 i ˜ σ 1 ˜ m − 1 1 + i ˜ σ 2 ˜ m − 1 2 i( σ 2 − σ 1 ) σ 1 + σ 2 − i( ˜ σ 1 ˜ m − 1 1 + ˜ σ 2 ˜ m − 1 2 ) ˜ σ 1 ˜ m − 1 1 − ˜ σ 2 ˜ m − 1 2 ˜ m 2 σ 2 − ˜ m 1 σ 1 − i( ˜ m 1 σ 1 + ˜ m 2 σ 2 ) σ 1 + σ 2 i( σ 1 − σ 2 ) i( ˜ m 1 σ 1 + ˜ m 2 σ 2 ) ˜ m 2 σ 2 − ˜ m 1 σ 1 i( σ 2 − σ 1 ) σ 1 + σ 2     , where m 1 =  ( ˜ ε 0 + α )(1 + β ) , m 2 =  ( ˜ ε 0 − α )(1 − β ); ˜ m 1 =  ( ˜ ε 0 + α ) / (1 + β ) , ˜ m 2 =  ( ˜ ε 0 − α ) / (1 − β ); σ 1 = cos( m 1 ω L ) , σ 2 = cos( m 2 ω L ); ˜ σ 1 = sin( m 1 ω L ) , ˜ σ 2 = sin( m 2 ω L ) . 19 References [1] Habib Ammari et al. “Mathematical foundations of the non-Hermitian skin eect” . In: A r chive for R ational Me chanics and A nalysis 248.3 (2024), p. 33. [2] Habib Ammari et al. “Sp ectra and pseudo-sp ectra of tridiagonal k-T o eplitz matrices and the top ological origin of the non-Hermitian skin eect” . 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