Gap localization of TE-Modes by arbitrarily weak defects

This paper considers the propagation of TE-modes in photonic crystal waveguides. The waveguide is created by introducing a linear defect into a periodic background medium. Both the periodic background problem and the perturbed problem are modelled by a divergence type equation. A feature of our analysis is that we allow discontinuities in the coefficients of the operator, which is required to model many photonic crystals. It is shown that arbitrarily weak perturbations introduce spectrum into the spectral gaps of the background operator.


Introduction
Electromagnetic waves in periodically structured media, such as photonic crystals and metamaterials, are a subject of ongoing interest. Typically, the propagation of waves in such media exhibit band-gaps (see [20,23]), i.e. intervals on the frequency or energy axis where propagation is forbidden. Mathematically, these correspond to gaps in the spectrum of the operator describing a problem with periodic background medium. The existence of these gaps for certain choices of material coefficients was proved in [9,14,18] and in [15] for the full Maxwell case. Using layer potential techniques this question has been studied in [3,4,5].
In this paper, we consider the propagation of TE-polarized waves in photonic crystals. TEpolarization (transverse electric) here means that the direction of the electric field is confined to a plane perpendicular to the direction of the magnetic field. When the periodicity is perturbed by point or line defects, localization may take place in band gaps, analogous to the situation in solid-state physics and semiconductor devices. The use of line defects in photonic crystals has been proposed in the context of wave guide applications. The gap localization gives rise to guided modes which decay exponentially into the bulk structure and propagate along the direction of the line defect. For this reason, it is of great importance to know whether a given line defect produces gap modes.
Is it possible to give rigorous sufficient conditions which imply localization in gaps? In particular, does localization also occur when arbitrarily weak defects are introduced? Here, "weakness" either means a perturbation of small magnitude in the material coefficients or a perturbation of finite magnitude, but small lateral extent. The latter are particularly interesting for optical applications, since defects are usually created by inserting materials with differing dielectric constant ε into the photonic crystal structure.
Weak localization results are quite different from results for sufficiently strong defects (like for example [1,12,13,24,25,27]) and there are surprisingly few of them in the literature. The first rigorous results on weak gap localization for periodic Schrödinger operators were given by Parzygnat et al. in [29]. Brown et al. showed weak gap localization in [7] for a periodic Helmholtz-type operator corresponding to TM-mode polarization. We also refer to the paper of Parzygnat et al. [29] for a thorough discussion of the literature on strong and weak localization for Schrödinger operators. In the slightly different setting of the coupling of two waveguides through narrow windows, weak localisation results for the Helmholtz equation were obtained in [26,30].
For TE-polarized waves in periodic media, the problem is very challenging and we present, for the first time, conditions ensuring weak gap localization. The method we present here relies on [7], but the proofs are considerably more difficult due to the different structure of the operator. The chief difficulty is the fact that here the perturbation is of the same order as the principal part of the differential operator. Moreover, we will be working with operators (1) with non-smooth coefficients ε(x), requiring a sophisticated functional analytic setting.

Problem statement
We consider the propagation of electromagnetic waves in a non-magnetic, inhomogeneous medium described by a varying dielectric function ε(X) with X = (x, y, z). Assuming that the magnetic field H has the form H = H(x, y)ẑ, whereẑ denotes the unit vector in the z-direction, we look for time-harmonic solutions to Maxwell's equations. This leads to the equation for the z-component H of the magnetic field. Note that in the context of polarized waves, we assume that all fields and constitutive functions depend only on x = (x, y). The periodic background medium is characterised by ε 0 (x), where for simplicity we assume that the unit square [0, 1] 2 is a cell of periodicity.
2.1. Line defects. Letx = (1, 0) andŷ = (0, 1). We now introduce a line defect, which we assume to be aligned in thex-direction and to preserve the periodicity in this direction. In addition, the defect is assumed to be localized in theŷ-direction. The new system is therefore described by a dielectric function ε 1 (x), periodic inx-direction, i.e. (2) and there exists some R > 0 such that ε 1 (·, y) may differ from ε 0 (·, y), if |y| < R and equals ε 0 (·, y) if |y| > R.
Since the system is still periodic in thex-direction, we can apply Bloch's theorem [28,22] to reduce our problem to a problem on the strip Ω := (0, 1) × R. Thus, the generalized eigenfunctions of (1) have the form e ikxx ψ (kx) (x), where k x ∈ [−π, π], ψ (kx) is periodic in thê x-direction and satisfies Equivalently, we may look for functions u (kx) satisfying k x -quasiperiodic boundary conditions and solving the equation For our purposes, it is slightly more convenient to use (5), since unlike in (3), the differential operator is not changed. Note that the boundary condition (4) now depends on k x . Suppose now (Λ 0 , Λ 1 ) is a band gap of the unperturbed system (1) with ε = ε 0 . We will give conditions which ensure that localized modes appear in the interval below Λ 1 under arbitrarily weak perturbation. The unperturbed system is periodic with respect to two directions, and the application of Bloch's theorem leads to the usual Bloch functions ψ s (x, k x , k y ) and corresponding band functions λ s (k x , k y ) with s ∈ N, x ∈ [0, 1] 2 and (k x , k y ) ∈ [−π, π] 2 , see, e.g. [7] for more details. Let M ∈ N be such that Λ 1 is the minimum of the M -th band function and let k 0 = (k 0 x , k 0 y ) be a value of the quasi momentum at which Λ 1 is attained, i.e.
We note that the minimum is attained; for more details see [7,Proposition 3.2]. For simplicity, we assume that λ M (k 0 x , k y ) = Λ 1 for all k y different from k 0 y . We intend to deal with the more general case in forthcoming work. We note that due to analyticity of the function k y → λ M (k 0 x , k y ) in a complex neighbourhood of the interval [−π, π] (see e.g. [21, Theorem VII.3.9]), we have close to k 0 y , for some α > 0. (This also holds if k 0 y = ±π, due to the periodic boundary behaviour of λ M (k 0 x , ·)). One of the main features of this paper is that we do not require the functions ε i to be continuous. The smoothness we require of the ε i is merely that ε i ∈ L ∞ . This is motivated by physical applications, where, to produce the typical band-gap spectrum, ε 0 is usually piecewise constant. See, for instance, [9,14,15]. Moreover, we make the following assumptions on the perturbation: (i) ε i ≥ c 0 > 0 for some constant c 0 and i = 0, 1. (ii) The perturbation is nonnegative, i.e.
We are now in a position to state our main result.

Remark 2.2.
(1) In fact, we have a slightly weaker condition for localization. (9) can be replaced by where G 1 is the Green's operator introduced by (19). From the proof of Lemma 4.3, we have , thus (11) follows from (9).
(2) Condition (9) is satisfied for sufficiently weak perturbations, so arbitrarily weak perturbations induce spectrum into the gap.
x ) ∈ L 2 (Ω) precisely expresses the type of localization we expect in the context of line defects, i.e. the eigensolutions u (k 0 x ) decay in the direction perpendicular to the line defect, whereas they are k 0 x -quasi periodic in thex-direction. This is different from the localization by point defects: these induce defect eigenfunctions that are square integrable over the whole space.

The periodic Green's function
In this section, we recall the mathematical formalism needed and introduce the operators to be studied first in the L 2 -setting. Later on, we shall introduce realizations of the same operators in negative Sobolev spaces, required to apply the perturbation theory to nonsmooth coefficients.
The unperturbed operator L 0 is defined in a standard way using the representation theorem (see [21]) from the sesquilinear form where u, v are H 1 -functions on Ω satisfying k x -quasiperiodic boundary conditions inx-direction. The pertubed operator L 1 is defined in a similar way, replacing ε 0 by ε 1 . As our technique is based on exploiting the Bloch representation of the Green's functions (or, equivalently, the resolvent operators), combined with a variational approach, we first review the definition and properties of the Green's function.
Central to our analysis is the Green's function G 0 (x, x ′ ) (see e.g. [10]) satisfying We note that the Green's function in (12) is then also subject to k 0 x -quasiperiodic boundary conditions: for all integers m. It is very useful to have a representation of G 0 (x, x ′ ) in terms of eigenfunctions, i.e. Bloch waves. We shall now derive such a representation. The set of Bloch waves ψ s (x, k) is known to form a complete system in the space of squareintegrable functions defined on the whole space. Likewise, the Bloch functions ψ s (x, k 0 x , k y ) with the x-component of the quasimomentum fixed, form a complete system in the space of square-integrable functions on the strip Ω. This means that any such function f can be expanded in terms of Bloch waves: (14) f where U denotes the Floquet transform in theŷ-direction and the series converges in the L 2sense.
Here, U f (·, k y ), ψ s (·, k 0 x , k y ) is the L 2 -inner product over the unit square [0, 1] 2 . Note that in (14), we only integrate over the k y -component of the quasi-momentum. To simplify notation, we will also write λ s (k y ) := λ s (k 0 x , k y ) and ψ s (x, k y ) = ψ s (x, k 0 x , k y ) in the following. As in [10, Chapter 1], (14) immediately implies the following representation: Formula (15) is extremely powerful. As we shall show, it allows us to analyze rigorously the interaction of the defect with the Bloch waves of the unperturbed system. It is convenient to Then G 0 is a symmetric and positive operator in L 2 (Ω). We also need to introduce the analogous Green's operator for L 1 (subject to k 0 x -quasiperiodic boundary conditions in thê x-direction). Since the spectrum of the differential operator L 1 is contained in the positive half-axis, G 1 := (L 1 + 1) −1 exists and is a symmetric positive operator in L 2 (Ω). Note that the coefficients of L 1 describe the perturbed system and have no periodicity in theŷ-direction. As a consequence, G 1 cannot be expressed in terms of Bloch waves, as in (15).
We will show that the essential spectra of L 0 and L 1 coincide, so any new spectrum introduced in the gap can only consist of eigenvalues. The key idea of our approach is then to transform the eigenvalue problem (10) into an eigenvalue problem for the Green's operator G 1 . In fact, suppose that u = u (k 0 x ) = 0 solves (10) together with the boundary condition (4), i.e.

The operator theoretic formulation
In this paper we shall only assume ε 0 , ε 1 ∈ L ∞ with a positive lower bound. In order to deal with the lack of smoothness in the coefficients, we will work in negative Sobolev spaces. In particular, rather than study the operators G 0 and G 1 directly, we will consider their realisations in the space H −1 qp (Ω), introduced below, denoted by G 0 and G 1 , respectively. Recall that we work with quasimomentum k 0 x fixed. To construct the H −1 -realisations, we introduce the space of quasi-periodic H 1 -functions on Ω As ε 0 is bounded and bounded away from zero, we can introduce a new scalar product on which is equivalent to the usual scalar product in H 1 (Ω). When there is no danger of confusion, we denote the associated norm · H 1 .
where the ·, · -notation indicates the dual pairing, i.e. w, ϕ is the action of the linear functional w on the function ϕ. We shall also use w[ϕ] to denote the dual pairing.
φ is an isometric isomorphism, and hence the scalar product on H −1 qp (Ω) given by induces a norm which coincides with the usual operator sup-norm on H −1 qp (Ω). We next introduce the realisations of L 0 and G 0 in H −1 qp (Ω).
the last line follows by (18). Thus L 0 + 1 is symmetric.
Since φ is bijective it follows that L 0 + 1 is bijective, thus (L 0 + 1) −1 : is defined on the whole space and is also symmetric. Therefore, G 0 = (L 0 + 1) −1 is self-adjoint. Hence L 0 + 1, and so L 0 itself is self-adjoint. (1) The map φ corresponds to the operator L 0 + 1 and φ −1 : From the definitions of φ and L 0 follows the useful identity (4) We note that just as in [6,Section 5], the L 2 -and H −1 -spectra coincide: . Then by the previous remark, 1/µ ∈ ρ(L 0 + 1), so for a given f ∈ H −1 qp (Ω). We now introduce the solution operator for the perturbed problem. Let G 1 be the operator We see that G 1 is well-defined, since it can be constructed via a form in the same way as G 0 , noting that the norms in the H −1 qp -spaces constructed from both sesquilinear forms B 0 and B 1 are equivalent. Note that G 0 | L 2 = G 0 and G 1 | L 2 = G 1 , which are both symmetric operators in L 2 . Moreover, again, as in [6,Section 5], the L 2 -and H −1 -spectra coincide: σ(G 1 ) = σ(G 1 ). We also denote L 1 = G 1 −1 − 1. We conclude the section with the proof of some more simple properties of G 1 and G 0 which will be useful later. Recall that, by assumption, ε 1 ≥ ε 0 .
Proof. Let f ∈ H −1 qp (Ω) and u = G 1 f . Choose ϕ = u in (19). It then follows that is bounded by assumption. Proof. Choose a sequence (w n ) ∈ (L 2 (Ω)) N such that w n → w in H −1 qp (Ω). By continuity of G i : since G i ≥ 0 as operators in L 2 .

Birman-Schwinger-type reformulation
An essential feature of our approach is to first perform a Birman-Schwinger-type reformulation of the problem. In this way, we bring the unperturbed Green's operator into play. We will show below (see Lemma 5.9) that G 1 − G 0 is compact as an operator in H −1 qp (Ω). Hence the spectra of G 0 and G 1 can only differ by eigenvalues.
qp (Ω) and choose sequences (v n ) and (u m ) in L 2 (Ω) such that v n → v and u m → u in H −1 qp (Ω). We first note that Now, using the convergence in H −1 qp (Ω) and the symmetry of the By a similar calculation to (22), this equals Kv, u H −1 = u, Kv H −1 , proving the result.
Moreover, (L 0 + 1)G 0 u = u ∈ H −1 qp (Ω) and Combining these three equalities, we get The first term is non-negative by Lemma 4.4. Also, . Proof. This is obvious, since L 0 is a self-adjoint operator in H −1 qp (Ω). In order to proceed, it is most convenient to modify the equation (21) by suitably projecting out the null space of K. Set K = ran K and let P : H −1 qp (Ω) → K be the orthogonal projection. On K, we introduce a new inner product given by (23) f, g K := Kf, g H −1 .
We first show the definiteness of this inner product.
Lemma 5.5. We have the estimates Therefore,

proving (i) and (ii).
Since K ≥ 0, Thus we have (iii). Finally, as G 1 − G 0 = G 0 K, using (ii) we have and rearranging gives the desired inequality.
Note in particular that this means that for small perturbations, the only dependence of the bound for Ku H −1 on the perturbation ε 1 is through the term 1 ε 0 − 1 ε 1 ∞ . We now introduce the operator we wish to study. Let (26) A µ := P (I − µG 0 −1 ) −1 K : K → K.
Lemma 5.6. Equation (20) has a non-trivial solution u iff −1 is an eigenvalue of A µ .
Proof. Let v = P u, where u is a solution of (20). Applying P to (21) shows that v solves Conversely, one easily checks that a solution v = 0 of (27)   Proof. Let u, v ∈ K. Then where we have used Lemma 5.1 and Lemma 5.3.
Observe that the second integral in the second line of the calculation is just ( It is therefore clear that f vanishes on all functions ϕ supported outside the support of 1 We are now in a position to establish that the spectrum of A µ consists only of eigenvalues. We first show that G 1 − G 0 enjoys the same property.
Lemma 5.9. The essential spectra of G 0 and G 1 coincide.

Existence of spectrum for weak perturbations
We next estimate the eigenvalues of A µ using variational methods. Lemma 5.6 enables us to study our spectral problem by applying variational methods to the equation (27). As a mathematical subtlety, note that K is in general not complete with ·, · K as an inner product. However, this does not affect our arguments, since the spectral theory of symmetric compact operators is applicable on Pre-Hilbert spaces (see [17]).
We next seek both lower and upper bounds on (29).
6.1. Lower bound. Lemma 6.2. For all u ∈ K and µ ∈ (Λ 1 + 1) −1 , (Λ 0 + 1) −1 we have Proof. Let (v n ) ∈ (L 2 (Ω)) N such that v n → Ku ∈ H −1 qp (Ω). Then, as in the proof of Lemma 5.7, Using the expansions in terms of Bloch functions (14) and (15), we have Next, let M be the index introduced in (6). Then, as all terms in the series with s < M are non-negative, we have < 0, we can now add the missing bands back in to get as required.
Inequality (30) also shows that for a fixed µ in the spectral gap, the size of the perturbation has to reach a threshold before it is possible for µ to lie in the spectrum of G 1 .

6.2.
Upper bound. Now we show that the minimum of the Rayleigh quotient of A µ diverges to −∞ as µ approaches 1 Λ 1 +1 from above. To do this, we have to construct a suitable test function; in order to bring the interaction with the gap edge into play, we use the edge Bloch wave ψ M . Here, M is as introduced in (6). We recall our assumption that there exists a ball D such that ε 1 − ε 0 > 0 on D.
Proof. As G 1 : . Reversing all the steps with u replaced byũ, we get From now on, u will always denote the test function in K given in Lemma 6.6. In considering the Rayleigh quotient for our test function, expressions involving Ku = (L 0 −L 1 )G 1 u will arise. To be able to make use of the resolvent representation via Bloch waves in L 2 (Ω), we need to regularize Ku. First, define and extend T u quasi-periodically from Ω to R 2 . Next, we introduce a mollifier (χ n ) n≥0 with support in [0, 1] 2 and set u n = T u * χ n and v n = − div(T u * χ n ). Similarly, In particular, in both cases, the sum is finite.
We now show that this gives us the desired smooth approximation of Ku.
where the boundary term in the integration by parts vanishes, as all functions satisfy quasiperiodic boundary conditions in thex-direction. On the other hand, Hence, As T u − T u * χ n → 0 in L 2 (Ω), we see that v n → Ku in H −1 qp (Ω).
Before finally considering the Rayleigh quotient, we need two more auxilliary results.
Lemma 6.9. There exist c > 0, δ > 0 and N ∈ N such that for all |k − k 0 y | < δ and n > N .
Proof. Integrating by parts, we have where, by Lemma 6.8 the convergence is uniform in k. Now, in view of the location of the support of the functions, and using the quasi-periodicity of ψ M , (0,1) 2 U T u(·, k) · ∇ψ M (·, k) = 1 2π By Lemma 6.6, this is non-zero at k = k 0 y . Consider the map k → [(L 0 − L 1 )ψ M (·, k)] [G 1 u]. This is continuous in k and so there exists δ > 0 such that it is non-zero for all |k − k 0 y | < δ. Uniformity of the convergence then proves the result for all n > N for some N ∈ N. Lemma 6.10. For the test function u given in Lemma 6.6 we have A µ u, u → −∞ as where we have again used Lemma 6.7 and the last estimate follows by Lemma 5.5 (iii). We are left with the contribution from the M -band which we divide up into integration over two disjoint regions: Let δ be as in Lemma 6.9 and B δ (k 0 y ) denote the ball of radius δ around k 0 y . Then and using that for |k − k 0 y | < δ we have 1 λ M (k) + 1 ≥ c 1 > 0 (by choosing a smaller δ, if necessary) and | U v n , ψ M L 2 | 2 ≥ c by Lemma 6.9, we have . Now observe that from (7) we have .
Combining the results of Lemma 6.1, Lemma 6.2 and Lemma 6.10, we obtain our main result, Theorem 2.1, from the Intermediate Value Theorem. In particular, any arbitrarily weak perturbation induces spectrum into the gap.

Concluding Remarks and open problems
We provided a sufficient rigorous criterion for localization in gaps by arbitrarily weak line defects, for the case of TE-polarized electromagnetic waves. We arrive at our results by comparing the Green's operators of the perturbed and unperturbed systems. While Green's functions techniques have been a part of the theoretical physics literature for a long time (see e.g. [10]), our method combines Green's functions and variational methods. For example, we do not use series expansion of the difference of the operators G 0 and G 1 to get approximations, and the variational approach avoids in an elegant way the need to control the remainder terms.
The method presented here is, in principle, also applicable to both the case when the band edge under consideration is degenerate and to the full Maxwell equations, at the expense of greater technical complexity. We plan to deal with these in forthcoming work.
Another open problem is the following. We know now sufficient conditions to create gap modes which are localized in theŷ-direction centering on the line defect. If the modes were additionally localized in thex-direction, we would have a bound state of the operator −∇·ε −1 1 ∇ on the whole of R 2 . This would go against physical intuition, since then light would stand still in the defect. It would be desirable to show that there are no modes that are localized in thex-direction, i.e. the perturbation creates truly guided modes. This would equivalently mean, that there is no flat band created in the gap (for a discussion, see [24]). The absence of bound states for periodic Helmholtz operators with line defects has been proven in [19]. However, to show absence of bound states for periodic divergence type operators seems to be extremely difficult. For periodic operators with sufficiently smooth coefficients, this question is investigated addressed in [16]. Let f j = f (1) j be a bounded sequence in H 1 qp (Ω, e γ|y| ). Let Ω p := (0, 1) × (−p, p) for any p ∈ N. Since H 1 qp (Ω p ) embeds compactly into L 2 (Ω p ), we may extract from (f (1) j ) a subsequence (f (2) j ) converging in L 2 (Ω 1 ) and from (f (2) j ) a subsequence converging in L 2 (Ω 2 ) and so forth. We claim that the diagonal sequence (f (p) p ) is Cauchy in L 2 (Ω). This is seen as follows: given any ε > 0, determine first a p 0 so large that for all p ≥ p 0 , using (34). Now determine a p 1 ≥ p 0 so large that f