On band gaps in photonic crystal fibers

On band gaps in photonic crystal fibers

Abstract

We consider the Maxwell’s system for a periodic array of dielectric ‘fibers’ embedded into a ‘matrix’, with respective electric permittivities and , which serves as a model for cladding in photonic crystal fibers (PCF). The interest is in describing admissible and forbidden (gap) pairs of frequencies and propagation constants along the fibers, for a Bloch wave solution on the cross-section. We show that, for “pre-critical” values of i.e. those just below (where is the magnetic permeability assumed constant for simplicity), the coupling specific to the Maxwell’s systems leads to a particular partially degenerating PDE system for the axial components of the electromagnetic field. Its asymptotic analysis allows to derive the limit spectral problem where the fields are constrained in one of the phases by Cauchy-Riemann type relations. We prove related spectral convergence. We finally give some examples, in particular of small size “arrow” fibers () where the existence of the gaps near appropriate “micro-resonances” is demonstrated by a further asymptotic analysis.

1 Problem formulation and main result

We consider the Maxwell’s system

 ∇×^H =−ϵ∂^E∂t (1.1) ∇×^E =μ∂^H∂t,

where the electric permittivity and magnetic permeability adopt two different sets of constant values in the fibers along the direction which are positioned periodically in the cross-sectional -plane, and in the surrounding matrix.

More precisely, let be the reference periodic cell, be an open bounded subset of , , with sufficiently smooth boundary , and . Let denote the characteristic function of the -periodically extended , . Then

 ϵ(^y) =ϵ0χ0(y1,y2)+ϵ1χ1(y1,y2), μ(^y) =μ0χ0(y1,y2)+μ1χ1(y1,y2), ^y=(y1,y2,y3)∈R3,

where , , , are positive constants.

A Bloch wave type solution to (1.1) is sought in the form

 ^E=ei(ωt+ky3)E(y1,y2),    ^H=ei(ωt+ky3)H(y1,y2). (1.2)

The interest is in describing the pairs ( is the frequency and is the “propagation constant”) for which there exists a non-trivial solution of the form (1.2) with and quasi-periodic in .

Upon substituting (1.2) into (1.1), we find that , necessarily satisfy the following system of equations

 H3,2−ikH2 = −iωϵE1 (1.3) ikH1−H3,1 = −iωϵE2 (1.4) H2,1−H1,2 = −iωϵE3 (1.5)
 E3,2−ikE2 = iωμH1 (1.6) ikE1−E3,1 = iωμH2 (1.7) E2,1−E1,2 = iωμH3. (1.8)

Resolving then (1.4) and (1.6) for and , and (1.3) and (1.7) for and , gives the following representations of the cross-sectional components and in terms of the “axial” components and

 H1 = Missing or unrecognized delimiter for \big (1.9) H2 = Missing or unrecognized delimiter for \big (1.10) E1 = a−1(ikE3,1+iωμH3,2), (1.11) E2 = a−1(ikE3,2−iωμH3,1), (1.12)

assuming

 a(y):=ω2ϵ(y)μ(y)−k2 (1.13)

is not zero. Substituting then (1.9)–(1.12) into (1.5) and (1.8) reduces (1.3)–(1.8) to the following system for and only

 ∂1(ikaH3,2)−∂2(ikaH3,1)+∂1(iωϵaE3,1)+∂2(iωϵaE3,2) =−iωϵE3 (1.14) ∂1(ikaE3,2)−∂2(ikaE3,1)−∂1(iωμaH3,1)−∂2(iωμaH3,2) =iωμH3. (1.15)

Setting , multiplying (1.14) and (1.15) by smooth test functions and respectively, adding up and integrating over gives upon integration by parts the following equivalent weak formulation:

 β(u,ϕ)=ω2∫R2ρu⋅¯¯¯ϕ,∀ϕ=(ϕ1,ϕ2)∈[C∞0(R2)]2. (1.16)

Here the bilinear form is given by

 β(u,ϕ):=∫R2ω2a(ϵ∇u1⋅¯¯¯¯¯¯¯¯¯¯∇ϕ1+μ∇u2⋅¯¯¯¯¯¯¯¯¯¯∇ϕ2)+kωa({u1,¯¯¯¯¯ϕ2}−{u2,¯¯¯¯¯ϕ1}), (1.17)

where , as above,

 ϵ :=ϵ(y)=ϵ0χ0(y)+ϵ1χ1(y), μ :=μ(y)=μ0χ0(y)+μ1χ1(y), y∈R2, (1.18)

and

 ρ(y):=(ϵ(y)00μ(y))=χ0(y)(ϵ000μ0)+χ1(y)(ϵ100μ1).

Notice that the form (1.17) is symmetric, i.e. , as follows by direct inspection. Noticing further that, for any ,

 ∣∣{u1,¯¯¯¯¯u2}−{u2,¯¯¯¯¯u1}∣∣≤γ|∇u1|2+γ−1|∇u2|2,

it is readily seen that if satisfy

 k2<ω2min{ϵ(y)μ(y)}=min{ϵ0μ0,ϵ1μ1}, (1.19)

then is coercive on ; namely, there exists a constant such that,

 β(u,u)≥ν∥u∥2[H1(R2)]2,∀u∈[H1(R2)]2. (1.20)

Henceforth, we shall consider a ‘non-magnetic’ photonic crystal fiber, i.e. we set the magnetic permeability a constant, and we assume that in (1.18). We shall consider the set of pairs for which the problem (1.16) admits a non-trivial solution when approaches from below the critical line . (This line corresponds to the dispersion relation in the matrix - the wavenumber for a plane wave - which appears exactly where loses its coercivity.) More precisely, for a fixed small parameter , we can say that (1.16) admits a non-trivial solution for the pair if belongs to the spectrum of the self-adjoint operator generated by the bilinear form given by (1.17). This form can be conveniently represented as

 βε(u,v)=∫R2Aε∇u⋅¯¯¯¯¯¯¯∇v=∫R2Aεijpqup,q¯¯¯vi,j. (1.21)

Here is easily found by rearranging (1.17) upon substituting ; see (3.3) below. Then the domain and the range of the self-adjoint operator consist of all and respectively, such that ()

 βε(u,ϕ)=∫R2ρf⋅¯¯¯ϕ,∀ϕ∈[C∞0(R2)]2.

Notice that, for a fixed , the operator and hence its spectrum depend on the spectral parameter , cf. (1.17), (1.13), hence one generally deals with an operator pencil. We will see however that, as , it asymptotically bahaves as a conventional spectral problem (although for a “partially degenerating” operator).

So, for converging to zero, we aim to characterise the (Floquet-Bloch) spectrum . This brings us to our main result

Theorem 1.1.

Let be the spectrum of the operator generated by the bilinear form , see (1.21). Then the set converges to the set

 σ0=⋃θ∈[0,1)2σ(θ)

as in the sense of Hausdorff. Here is the set of , , such that there exists a non-zero solution to

 ∫Q∇u1⋅¯¯¯¯¯¯¯¯¯¯∇ϕ1+γ−1(divu⋅¯¯¯¯¯¯¯¯¯¯¯¯divϕ+divu⊥⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯divϕ⊥)=λk(θ)∫Qϵ1(u⋅¯¯¯ϕ+γχ0u1¯¯¯¯¯ϕ1),∀ϕ∈V(θ), (1.22)

where and the unitary mapping is defined by ;

 V(θ)={v∈[H1θ(Q)]2: divv=0 and divv⊥=0 in Q1}, (1.23)

where1 is the space of functions that are -quasiperiodic: for some .

Remark 1.1.

We say that sets converge to a set as in the sense of Hausdorff when the following two conditions hold:

1. For every there exists such that as .

2. Suppose such that converges to some as , then .

Symbolically, we represent this convergence as follows

 limε→0σε=σ0.
Remark 1.2.

For fixed , the eigenvalues are ordered, and repeated according to their multiplicity, as follows:

 0≤λ1(θ)≤λ2(θ)≤λ3(θ)≤…,λk(θ)→∞ as k→∞.

Moreover, for any fixed the eigenvalues are continuous with respect to (see Section 2, Lemma 2.4), and one has

 σ0=∞⋃i=1[minθλi(θ),maxθλi(θ)].
Corollary 1.1.

The set also admits the following equivalent representation

 σ0=⋃N∈N2σ(AN),

where is the set of , , such that there exists a non-zero solution to

 ∫NQ∇u1⋅¯¯¯¯¯¯¯¯¯¯∇ϕ1+γ−1(divu⋅¯¯¯¯¯¯¯¯¯¯¯¯divϕ+divu⊥⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯divϕ⊥)=λNk∫NQϵ1(u⋅¯¯¯ϕ+γχ0u1¯¯¯¯¯ϕ1),∀ϕ∈VN, (1.24)

where

 VN:={v∈[H1#(NQ)]2:  divv=0 and divv⊥ = 0 in F1∩NQ}.

Here for a given multi-index is a “multi-cell” of size , and is the -periodic extension of the set to the whole space .

Proof.

By Theorem 1.1, inclusions (3.34) and (3.35), one has

 ⋃θ∈[0,1)2σ(θ)=limε→0σε⊃¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯⋃N∈N2σ(AN)⊃⋃θ∈[0,1)2σ(θ).

The ‘limit spectrum’ has a gap if two adjacent bands do not overlap: i.e. if for some , one has

 maxθλi(θ)

An immediate consequence of Theorem 1.1 is the following result

Corollary 1.2.

Assume that is a gap of . Then, there exists a set such that for any pair equation (1.14)-(1.15) admits no non-trivial solution. Furthermore, one has .

Proof.

Suppose has a gap, i.e. there exists an such that . Let us fix such that .

We now argue that there exists a such that the set denoted by

 G=⋃ε0<ε<ε0{(ω2,ω2μ(ϵ1−ε2)) ∣∣ ω2∈[a,b]}

satisfies the statements of the corollary. For if not, there exists a sequence for which each set contains a pair such that (1.14)-(1.15) admits a non-trivial solution, i.e. belongs to . Since , it has a subsequence which converges to some . By Theorem 1.1, we conclude that which contradicts the fact is a gap in the spectrum . ∎

In Section 5 we prove the existence of gaps in the limit operator for several subclasses of two-dimensional photonic crystal.

2 Limit Bloch operators and some properties

In this section we shall study some properties of the family of self-adjoint operators , , associated with Theorem 1.1. In particular, we shall establish several important properties concerning the continuity of these operators, their domains and their spectra with respect to the quasi-periodicity parameter .

Suppose and let be defined by (1.23).

Remark 2.1.

is ‘equivalent’ to solving the conjugate Cauchy-Riemann equations in : for if we define a complex valued -quasiperiodic function then solves the Cauchy-Riemann equations in if, and only if, .

Note that is a closed subspace of and therefore a Hilbert space when equipped with the following equivalent norm

 ∥v∥2V(θ):=∫Q|v|2+∫Q|divv|2+∫Q|divv⊥|2. (2.1)
Remark 2.2.

The equivalence of the norm (2.1) follows from standard Poincaré type inequalities and the following equality, which is a simple consequence of integration by parts and the quasiperiodicity of the functions in question,

 ∫Q|∇v|2=∫Q|divv|2+∫Q|divv⊥|2,∀v∈[H1θ(Q)]2.

Denoting by the closure of in2 we introduce the non-negative quadratic form , defined by, cf. (1.22),

 αθ(u,v):=∫Q∇u1⋅¯¯¯¯¯¯¯¯¯∇v1+γ−1(divu⋅¯¯¯¯¯¯¯¯¯¯¯divv+divu⊥⋅¯¯¯¯¯¯¯¯¯¯¯¯¯¯divv⊥). (2.2)

It is clear that is closed in due to the norm (2.1). Therefore, generates a unique self-adjoint operator whose domain is a dense subset of and whose action associates to the unique solution of

 αθ(u,ϕ)=∫Qϵ1(f⋅¯¯¯ϕ+γχ0f1¯¯¯¯¯ϕ1)=:∫Qϵ1ρf⋅¯¯¯ϕ,∀ϕ∈V(θ). (2.3)

As is compactly embedded into we find that the resolvent of is compact and, in particular, the spectrum of is discrete: it consists of countably many isolated eigenvalues of finite multiplicity which converge to infinity. We order the eigenvalues, allowing for multiplicity, as follows:

 0≤λ1(θ)≤λ2(θ)≤λ3(θ)≤…,λk(θ)→∞ as k→∞.

An important distinct feature of the operators (as opposed to the usual Floquet-Bloch operators concerned with periodic elliptic PDEs in the whole space) is the non-trivial dependence on of the operator form domain . The statement of continuity of is therefore not a simple consequence of the Bloch-wave representation of functions belonging to , and relies on establishing that the underlying space is continuous with respect to , see Lemma 2.1. We shall show that the continuity of leads to the operators being continuous with respect to in the norm-resolvent sense, which in turn implies the continuity of , see Lemma 2.4.

In the rest of this section we shall introduce and prove the aforementioned results.

Lemma 2.1 (Continuity of V(θ)).

Let be such that for some . Then, for each there exists such that strongly in as

To prove this lemma we shall use the following characterisation of the spaces .

Lemma 2.2.

(i). Suppose , then if, and only if,

 ϕ=∇a+∇⊥b

for some with

 Δa=f1, Δb=f2, (2.4)

where with and contained in . Here, for a given scalar function , we denote by the vector .

(ii). Suppose . Then if, and only if,

 ϕ=c+∇a+∇⊥b

for some constant and with

 Δa=f1, Δb=f2, (2.5)

where , ,3 with and contained in .

Proof.

In both cases (i) and (ii) the necessity condition is easy to demonstrate. Let us now show the sufficiency condition. First, we shall consider case (i). Suppose, , and let us fix and set , . It is clear that and that their supports are contained in . Now, let us introduce the unique solutions of

 Δa=f1, Δb=f2.

We claim that . Indeed, this can be seen by noting that belongs to with , which implies .

For case (ii), fix ; then it is clear that for some such that . Now we repeat the above argument, that is let be the unique solutions of

 Δa=divϕ0,⟨a⟩=0, Δb=divϕ⊥0,⟨b⟩=0.

As above, we show that . ∎

Additionally, we shall use in the proof of Lemma 2.1 the following regularity results.

Lemma 2.3.
1. Suppose , and let be the unique solution of

 −Δu=f, (2.6)

for some . Then, and

 ∥u∥H2(Q)≤(1+1|θ|2)∥f∥L2(Q). (2.7)
2. Suppose and let , be the unique solution of

 Δu+4πiθ⋅∇u−4π2|θ|2u=f, (2.8)

for some such that . Then, and there exists a constant , independent of , such that

 ∥u∥H2(Q)≤C∥f∥L2(Q).
Proof.

Proof of (i): The eigenfunctions of the -quasiperiodic Laplacian form an orthonormal basis of . By decomposing and in terms of this basis, we have

 u(y)=∑z∈Z2aze2πi(θ+z)⋅y, f(z)=∑z∈Z2bze2πi(θ+z)⋅y,

Now (2.6) tells us , where . Since

 ∥u∥2H2(Q)=∑z∈Z2(1+4π2|θ+z|2)2|az|2=∑z∈Z2(1+4π2|θ+z|24π2|θ+z|2)2|bz|2.

For we see . Hence,

 ∥u∥H2(Q)≤(1+|θ|2|θ|2)∥f∥L2(Q).

Proof of (ii): The eigenfunctions of the periodic Laplacian form an orthonormal basis in . Decomposing and in terms of this basis we have

 u(y) =∑z∈Z2aze2πiz⋅y f(z) =∑z∈Z2bze2πiz⋅y,

By the assumptions and , we have and respectively. Now (2.8) tells us that for , . Since , one has

 ∥u∥2H2(Q) =∑z∈Z2z≠0(1+4π2|z|2)2|az|2=∑z∈Z2z≠0(1+4π2|z|24π2|θ+z|2)2|bz|2 ≤∑z∈Z2z≠0(1+4π2|z|24π2|z|2)2|bz|2≤4∑z∈Z2z≠0|bz|2,

i.e.

 ∥u∥2H2(Q)≤4∥f∥2L2(Q).

Proof of Lemma (2.1)..

Consider a sequence such that as . There are two separate cases to consider, the cases and .

Case 1: . Let us assume without loss of generality that . For fixed we know, by Lemma 2.2, that for some where

 Δa=f1, Δb=f2,

for some . We shall now construct the desired as follows: Set where solve

 Δaε=f1, Δbε=f2.

Notice, by Lemma 2.2, that . It remains to show strongly in . To this end, it is sufficient to show that and strongly in as . Let us show that as . By defining , one notices that and uniquely solves

 Δ˜aε=fε, (2.9)

where . In particular, with , and therefore by Lemma 2.3(i) one has

 ∥˜aε−a∥2H2(Q)≤C∥fε−f1∥2L2(Q). (2.10)

Furthermore, since , by an application of Lemma 2.3(i) one notices

 ∥aε∥2H2(Q)≤(1+1|θε|2)2∥f1∥2L2(Q)≤C∥f1∥2L2(Q),

where is independent of (here we used the fact that as ). Therefore, one has

 ∥fε−f1∥2L2(Q)≤∥(e2πi(θε−θ)⋅y−1)f1