## Abstract

The paper presents analytical and numerical models describing localized electromagnetic defect modes in a doubly periodic structure involving closely located inclusions of elliptical and circular shapes. Two types of localized modes are considered: (i) an axi-symmetric mode for the case of transverse electric polarization with an array of metallic inclusions; (ii) a dipole type localized mode that occurs in problems of waveguide modes confined in a defect region of an array of cylindrical fibres, and propagating perpendicular to the plane of the array. A thin bridge asymptotic analysis is used for case (i) to establish double-sided bounds for the frequencies of localized modes in macro-cells with thin bridges. For the case (ii), the electric and magnetic fields independently satisfy Helmholtz equations, but are coupled through the boundary conditions. We show that the model problem associated with localized vibration modes is the Dirichlet problem for the Helmholtz operator. We characterize defect modes by introducing a parameter called the ‘effective diameter’. We show that for circular inclusions in silica matrix, the effective diameter is accurately represented by a linear function of the inclusion radius.

## 1. Introduction

This paper deals with analysis of long wavelength vibrations of two-dimensional and finite structures containing two different materials. The materials are distributed in such a way that there exist thin bridges of one phase placed in between two regions occupied by the other phase. We derive asymptotic estimates for frequencies of localized vibration modes that include two-sided bounds for the case of thin bridges of variable thickness. These modes will be classified accordingly to whether they satisfy Neumann (type I) or Dirichlet (type II) boundary conditions. For periodic structures containing macro-cells with thin bridges, we analyse filtering properties and construct dispersion diagrams that include stop bands for certain intervals of frequencies. We pay particular attention to dispersion curves of low group velocity, which correspond to localized modes (standing waves).

Thin bridge structures have been of interest to many authors. For example, Batchelor & O'Brien (1977) and Nunan & Keller (1984) considered static problems and analysed the contributions to the effective transport properties of the flux through the thin bridges. The paper by McPhedran *et al*. (1988) used functional equation techniques for source densities to derive a highly accurate multipole series representation of the solution to an electrostatic problem. This work was generalized to problems of elasticity by McPhedran & Movchan (1994) and Movchan *et al*. (1997).

Dynamical thin bridge problems have been extensively studied in the book by Kozlov *et al*. (1999) and the paper by Maz'ya & Movchan (2003) both for the cases of the Helmholtz operator and for elasticity. These problems have attracted renewed interest in the important fields of photonic crystals (Joannopoulos *et al*. 1995) and photonic crystal fibres (Zolla *et al*. 2005). In the latter case, the thin bridge structure is used as a means of confining light within the central core or defect, and may even be used to trap light within an air region surrounded by silica incorporating air holes with thin bridges of silica between them (Russell 2003). The first studies able to accurately calculate the leakage of electromagnetic flux through large systems incorporating multiple thin bridges were made by White *et al*. (2002) and Kuhlmey *et al*. (2002) using the multipole method. Defect modes in photonic crystals have also been analysed by Centeno & Felbacq (1999) who studied a finite size two-dimensional photonic crystal containing a micro-cavity and the corresponding defect modes for different values of the angle of conical incidence.

Further work has included the analysis of photo-elastic coupling (also known as Brillouin scattering) within the central core of such thin-walled micro-structures (Russell *et al*. 2003; Guenneau *et al*. 2004). The former reference contains comparisons between experimental and numerical results, whereas the latter contains asymptotic models derived using the methods of Movchan & Movchan (1995) and Movchan *et al*. (2002) for electro-mechanical coupling within thin-bridges leading to one-dimensional differential equations of the visco-elastic type.

The structure of the paper is as follows. In §2, we address the asymptotic derivation of lower-dimension models for thin bridges of variable thickness. Section 3 includes an asymptotic estimate of the fundamental frequency for a model spectral problem and derivation of two-sided bands for fundamental frequencies in structures involving a finite size core connected to thin ligaments (type I modes). Section 4 contains a comparison of the asymptotic estimates for a particular thin-bridge structure with accurate numerical results for a doubly periodic array with a macro unit cell, which duplicates the given structure. The particular geometry we investigate is composed of radially arranged elliptical inclusions, for which we present dispersion diagrams displaying stop bands. Type II localization (Dirichlet modes) is considered in §5. In particular, we address the case of guiding obliquely incident electromagnetic waves by a set of cylindrical fibres of smooth cross-section in the low-frequency limit. We show that the confined electric and magnetic fields decouple to leading order and generate localized eigenmodes of the Dirichlet type. Section 6 includes numerical results, which are used to investigate the accuracy of a simple model in which a single disc with Dirichlet boundary conditions has its radius chosen to match the frequency of the fundamental modes in the physical structure.

## 2. Basic settings and main results

We consider the Maxwell system in the doubly periodic array of cylindrical fibres of elliptical or circular cross-sections. Taking a time-harmonic dependence e^{−iωt}, we look for eigenvalues *ω* (angular frequency) and associated electric eigenfield ** E**(2.1)or magnetic eigenfield

**(2.2)Here,**

*H**c*denotes the wave speed in vacuum and

*n*the refractive index. We note that if we take the divergence of both sides of (2.1) and (2.2), and assume materials to be non-magnetic, we obtain(2.3)

The macro-cell of the periodic structure has the cross-section shown in figure 1. This is a honeycomb structure with an elementary cell containing six inclusions. Such an elementary cell will be referred to as the ‘defective’ cell, and compared to the perfect hexagonal structure.

Assuming that the *x*_{3}-axis is directed along the fibres, we formulate the problems for the *E*_{3}, *H*_{3} components of the electric and magnetic fields in the following cases: (i) transverse electric (TE) polarization; (ii) conical propagation when the electric and magnetic fields are given by(2.4)with *β* being the conical parameter.

We solve the spectral problem for the class of Floquet–Bloch eigenfunctions *E*_{β}, *H*_{β} where each field satisfies the Floquet–Bloch condition(2.5)Here ** k**=(

*k*

_{1},

*k*

_{2}) is the Bloch vector, and

*l*^{(1)},

*l*^{(2)}are the basis vectors of the lattice, with

*m*

_{1}and

*m*

_{2}being integers.

The spectral problem may possess localized vibration modes that will be called the ‘defect modes’. By localized modes, we mean eigenfunctions whose associated eigenfrequencies depend very little on ** k**. Strictly speaking, these modes should be characterized by horizontal flat curves on band diagrams, but we will adopt a looser definition since most localized modes are never perfectly confined and tend to radiate some energy away from the region of localization (in our case, the centre of the macro-cell

*Ω*). In practice, the better the localization the flatter the associated curves are on the band diagram.

In the first case (i), the magnetic field is given by ** H**=

*H*

_{3}(

*x*

_{1},

*x*

_{2})

*e*_{3}. If we consider perfectly conducting metallic inclusions, then

*H*

_{3}satisfies the Neumann problem for the Helmholtz operator together with Floquet–Bloch conditions (2.5). Indeed, using the identity(2.6)equation (2.2) becomes(2.7)since

**is divergence free.**

*H*It is then enough to assume that the reciprocal of the dielectric constant *n*^{−2}(*x*_{1}, *x*_{2}) is constant outside the inclusions and vanishes inside the inclusions to deduce that the field satisfies the Helmholtz equation in the matrix and homogeneous Neumann conditions on the boundary of each inclusion.

The thin bridge asymptotic analysis of §3 will be used in §4 to establish double sided bounds for the frequencies of localized modes in macro-cells with thin bridges. These formulae are tested numerically in §5.

In the conical propagation case (ii), the fields *E*_{β} and *H*_{β} each independently satisfy Helmholtz equations. However they are coupled through the boundary conditions. It is shown in §6 that the model problem associated with localized vibration modes is the Dirichlet problem for the Helmholtz operator. We characterize defect modes by introducing a parameter called the ‘effective diameter’. We show that for circular inclusions in silica matrix, the effective diameter is accurately represented by a linear function of the inclusion radius.

## 3. Thin bridge asymptotics

Let be a thin domain defined for *j*=1, …, *K* by(3.1)where *ϵ* is a small positive non-dimensional parameter, and are smooth non-negative functions defining the profile of the *j*-th thin bridge. Let the amplitude *u*_{ϵ}(*x*_{1}, *x*_{2}) of a time-harmonic function be a solution of the Helmholtz equation(3.2)Assume that *u*_{ϵ} satisfies the homogeneous Neumann boundary conditions on the upper and lower parts *Γ*_{ϵ,±} of (3.3)where .

The appropriate boundary conditions at *x*_{1}=±*l*^{(j)}/2 will be specified later. Note that the choice of homogeneous Neumann conditions on the sides of the bridge is appropriate for a traction free boundary problem of anti-plane shear in elasticity, and will also be shown in §6 to be appropriate for a certain class of modes in micro-structured fibres in electromagnetism.

The aim of this section is to outline an asymptotic approximation of *u*_{ϵ} in *Π*_{ϵ} (to simplify the notations, we drop the superscript *j* from now on). Accordingly, we introduce the scaled variable(3.4)so that in the scaled domain *Π*_{1} the Helmholtz equation becomes(3.5)where . Also, *u*_{ϵ} satisfies the Neumann boundary condition(3.6)on the scaled boundary . In (3.6), ** n** denotes the unit outward normal, with denoting the first derivative,(3.7)Using (3.7), the boundary condition (3.6) can be written in the form(3.8)

We seek the asymptotic approximation of *u*_{ϵ} outside neighbourhoods of the ends *x*_{1}=±*l*/2 of *Π*_{1} in the form(3.9)Direct substitution leads to the following model problems for the leading order term *U*_{0} on the scaled cross-section *Π*_{1} of the bridge(3.10)It follows from (3.10) that *U*_{0} is independent of *ξ*, that is, *U*_{0}=*U*_{0}(*x*_{1}). Similarly, collecting terms of order *ϵ*^{2}, we obtain the boundary value problem for *U*_{1}(3.11)Sinceandthe solvability condition for the problem (3.11) is(3.12)where (*x*_{1})=*h*_{+}(*x*_{1})+*h*_{−}(*x*_{1}).

Assuming that equation (3.12) is supplied with appropriate boundary conditions at *x*_{1}=±*l*/2, and the boundary value problem for *U*_{0} is solved, we can then determine the second term *ϵ*^{2}*U*_{1} in (3.9) as a solution of (3.11).

## 4. Asymptotic estimates of frequencies of axi-symmetric localized modes for a macro-cell with thin bridges

We consider a doubly periodic array where a macro-cell *Ω* has the form shown in figure 1*a*. The exterior boundary of the hexagonal macro-cell is denoted by *Γ*. We will relate the localized vibrational modes for the Neumann problem in the array to the solutions of the spectral problem in a finite structure shown in figure 1*b*.

### (a) Floquet–Bloch waves in a doubly periodic array with honeycomb structure of defects

Assuming that we deal with the *H*_{3} polarization of electromagnetic waves propagating in the (*x*_{1}, *x*_{2}) plane, we can formulate the following problem for *H*_{3}:(4.1)(4.2)(4.3)where *l*^{(1)}=*d*(1, 0), , ** k**=(

*k*

_{1},

*k*

_{2}) is the Bloch vector,

*m*

_{1}and

*m*

_{2}are integer numbers, and

*d*is the period of the structure. The dispersion diagram in figure 2 gives the normalized frequency

*ωd*/

*c*as a function of the Bloch vector, as it traverses the boundary of the irreducible Brillouin zone (the triangular contour

*ΓXM*) shown in figure 1

*c*. Waves corresponding to points on the boundary of the irreducible Brillouin zone have zero group velocity (these are standing waves). In figure 2

*b*, we display the standing wave eigensolution of the eigenfrequency

*ω*

_{*}(see dispersion diagram in figure 2

*a*; for this case, the solution is ‘localized’ in the core region of the macro-cell and the maximum field variation occurs within the thin bridges. The frequency of this ‘localized mode’ on the dispersion diagram is close to the partial stop band, and we would like to derive an asymptotic estimate for it.

### (b) Multi-structure

In order to estimate analytically the frequency of the eigenstate in figure 2*b*, we use an asymptotic method. For this analysis, it is adequate to assume periodic boundary conditions on the sides of the elementary cell within the periodic structure (see figure 1). The domain in figure 1*b* will be referred to as ‘the multi-structure’.

Let us consider the multi-structure *Ω*_{ϵ} which incorporates thin ligaments of the normalized thicknesses *ϵh*^{(j)} connected to a two-dimensional domain *Σ* with Lipschitz boundary ∂*Σ* at one end and to a non-zero potential wall at the other end (see figure 1*b*)(4.4)here are thin bridges as defined in (3.1). It is also noted that the eigenstate in question has zero average over the elementary cell(4.5)Outside the thin bridge region, in the part *Σ* of the multi-structure *H*_{3} is approximated by a constant, *Χ* say, and within the complementary area of the macro-cell *Ω*\*Ω*_{ϵ} excluding the thin ligaments by a constant *C*. We note that, for low frequencies,(4.6)where _{Σ}, denote the areas of the ‘core region’ *Σ* and of the complementary part *Ω*\*Ω*_{ϵ}, respectively. The right-hand side of (4.6) is small since, for low frequencies, the inertia of thin bridges is neglected. We conclude that the boundary condition on *Γ*_{D} should be chosen as follows:(4.7)where *Χ*=const. is the amplitude of vibration of the central core region.

In the paper by Movchan & Guenneau (2004), we assumed homogeneous Dirichlet data on *Γ*_{D} but still obtained a good estimate, because *Ω*\*Ω*_{ϵ} was much larger than *Σ*. In the present situation (porous multi-connected medium), this assumption is no longer valid.

### (c) Asymptotic approximation of the frequency of the localized mode

The boundary conditions are chosen to be of Neumann type on the lateral surface *Γ*_{N}=*Γ*_{+}∪*Γ*_{−} as in §3, and of the Dirichlet type at the outer end regions *Γ*_{D} of the thin ligaments. We aim to estimate the fundamental frequency of a certain spectral problem in *Ω*_{ϵ} and to relate this frequency to the one of the localized mode identified in §5. The spectral problem of interest is(4.8)

(4.9)

(4.10)

We consider the case when the normalized fundamental frequency is small, and hence the term *ω*^{2}/*c*^{2}(*x*_{1})*U*_{0}(*x*_{1}) in equation (3.12) describing the field within the thin ligament may be neglected.

The boundary layers occur at the end regions of thin ligaments. These boundary layers are characterized by exponential decay when the boundary conditions for the functions (the leading term in the asymptotic expansion of the thin bridge solutions) are chosen in the form(4.11)We note that the functions are written in local coordinates associated with thin bridges as outlined in §3.

Integrating (4.8) over *Ω*_{ϵ} and applying the divergence theorem we obtain that, to order ,(4.12)where are local longitudinal coordinates along the ligaments , and *Σ*_{Σ} is the area of the central core region *Σ*. The functions are solutions of the differential equations(4.13)Since are smooth functions, we deduce that(4.14)The first condition in (4.11) leads to(4.15)whereas the second condition gives(4.16)Combining (4.12), (4.14)–(4.16), we obtain that(4.17)and(4.18)

From (4.17) and (4.18) it then follows that the asymptotic approximation for the frequency of the fundamental mode of the finite size structure *Ω*_{ϵ} with thin ligaments has the form(4.19)whereSimilar asymptotic formulae for three-dimensional structures with ligaments of constant thickness were presented in Kozlov *et al*. (1999).

### (d) Two-sided bounds

Let 〈_{j}〉 and be defined by the formulae(4.20)Here, _{j} is a continuous positive function.

We also assume that the total area of scaled ligaments of variable thickness remains constant,(4.21)

The Cauchy–Schwarz inequality yields(4.22)or equivalently,(4.23)Hence, we have the upper bound(4.24)

Using the Cauchy–Schwarz inequality for sums and the inequality (4.23) we obtain(4.25)Thus, the right-hand side of (4.19) allows for the following estimate:(4.26)

Finally, we arrive at the double-sided bound(4.27)

### (e) Thin ligaments of constant curvature: the asymptotic optimization

Let us consider a particular case when all thin ligaments are bounded by curves of constant curvature, that is, let us assume thatwhere *a*_{j} and *h*^{(j)} are some constants characterizing the curvature and the minimal width of the *j*-th ligament. From (4.14) it then follows thatwhereand the estimate (4.19) becomes(4.28)

Under the constraint that the total area occupied by thin ligaments is fixed, that is,(4.29)where is a given constant (see equation (4.21)), we can develop an asymptotic optimization similar to Maz'ya & Movchan (2003), assuming in addition that , *j*=1, …, *K*. In this case the asymptotic approximation (4.28) simplifies to(4.30)Using the Cauchy–Schwarz inequality for sums we obtainand thus,(4.31)This lower bound can be achieved when *l*^{(j)}=*l* and *a*^{(j)}=*a* (all ligaments have the same length and the same curvature). In this case(4.32)

## 5. Comparison with results of numerical simulations

The numerical simulations in this section were set up with the Finite Element software Getdp (Dular *et al*. 1998).

### (a) Macro-cell with infinitely conducting circular inclusions

For the geometry shown on figure 2*b*, we numerically solve the spectral problem (4.1)–(4.3) for TE polarization. In figure 2*a* we draw the band diagram for which the second dispersion curve flattens at the edges *Γ*, *M* and *K* of the first Brillouin zone. The first dispersion curve, known as the acoustic curve, is linear and symmetric in the neighbourhood of *Γ*=(0, 0). Its slope is associated with effective properties in the long-wave limit approximation, and this is well described in the literature. The second dispersion curve, known as the optical band, represents the eigenfrequencies associated with the mode shown in figure 2*b*. This computation suggests that the gradient field is ‘localized’ within thin bridges. The approximate normalized eigenfrequency (see formula (4.28)) is *ω*_{asymp}=*ωd*/*c*=2.58; the finite element computation gives a slightly smaller value of *ω*_{FEM}=*ω*^{*}*d*/*c*=2.3825 at the boundary of the Brillouin zone1. The estimates for the eigenfrequencies for different radii of the circular inclusions are collected in table 1.

### (b) Macro-cell with infinitely conducting elliptical inclusions

We now consider the case of thin bridges of varying thickness. Our numerical comparison of asymptotic estimates and finite element results is confined to the case of ellipses of the same area but different eccentricities. Table 2 includes the asymptotic estimates of eigenfrequencies (see formula (4.28)) together with the corresponding values obtained from the finite element computations. One can see that the asymptotic formula (4.28) gives a sufficiently accurate prediction for the case when the ellipses are aligned with the sides of the hexagonal macro-cell. On the contrary, the asymptotic estimate is less accurate for ellipses oriented in the radial directions: in this case the width of the bridges between the inclusions (compared to their length) increases.

Comparison of dispersion diagrams in figures 3 and 4 suggests that the spectral problem for the structure involving elliptical inclusions oriented along the sides of the macro-cell has an eigensolution localized within the central core region of the macro-cell. For both types of structures there is a full band gap near the frequency *ω*^{*} estimated by the formula (4.28).

## 6. The dipole type localization: Dirichlet modes

In this section we consider a different type of localization, which may occur in inhomogeneous photonic crystal structures in the case of oblique incidence. The elementary cell is shown in figures 5–7, and the refractive index *n*_{1} of the material of the inclusions is chosen to be smaller than the refractive index *n*_{2} of the surrounding matrix. In this case, for a certain angle of incidence, there exist eigenmodes of the dipole type (see figures 5–7), which are similar to those which occur for the case of Dirichlet boundary conditions imposed on the inclusion boundaries.

### (a) Oblique incidence of an electromagnetic signal

Now consider the case of oblique incidence of an electromagnetic wave on an array of cylindrical fibres aligned with the *x*_{3}-axis, and assume the cross-sectional geometry to be the same as in figure 2*b*. In this case, the electric and magnetic fields are defined by(6.1)We note that the knowledge of *E*_{3}, *H*_{3} is sufficient to determine *E*_{t}, *H*_{t} through the Maxwell's equations. With reference to the paper by White *et al*. (2002), and denoting *β*=*n*_{eff}*ω*/*c*, we write the Helmholtz equations for *H*_{3}, *E*_{3}(6.2)and the boundary conditions on the contours ∂*Ω*\*Γ* of interior inclusions which represent the continuity of the tangential components of ** E** and

**,(6.3)and(6.4)where [.] denotes the jump across ∂**

*H**Ω*\

*Γ*and

**is the unit tangential vector on the boundary. Let**

*τ**n*

_{1}and

*n*

_{2}be the refractive indices associated with the materials of circular cylindrical inclusions and of the matrix

*Ω*, respectively. The conical parameter

*β*is chosen in such a way that(6.5)so that the waves inside the inclusions become evanescent.

Consider a thin bridge region and apply the scaling outlined in §3, so that the Helmholtz equation within the bridge *Π*_{1} takes the form(6.6)where we used local coordinates (*y*_{1}, *y*_{2}) and the scaled coordinate *ξ*_{2}=*ϵ*^{−1}*y*_{2}. This leads to the conclusion that the leading order approximations of *E*_{3} and *H*_{3} are linear in the transverse variable *ξ*_{2} within the thin bridge. Let , be associated with interior of inclusions, whereas , correspond to a thin bridge. We use the representation(6.7)In this case(6.8)where *k*=0, 1 and hence within the thin bridge(6.9)In addition, it follows from the boundary conditions (6.3) and (6.4) that(6.10)The latter together with (6.9) yields that , are independent of *ξ*_{2}, that is,(6.11)Hence, we deduce that and similarly , i.e. the functions *E*_{3} and *H*_{3} are continuous across the thin bridge, to leading order approximation. Proceeding to the next order and engaging the second group of transmission conditions written in terms of derivatives, we deduce(6.12)where the left-hand side is independent of *ξ*_{2} due to (6.9). Hence,(6.13)Similarly, for the component we write(6.14)Taking into account that is linear in *ξ*_{2} and that , we deduce that(6.15)Hence, we have proved that the fields and and their transverse derivatives are continuous across the thin bridge (ideal contact conditions). Since these fields are evanescent in the cylindrical inclusions, they are also negligibly small inside thin bridges. The latter implies that problems of Dirichlet type formulated in the core of the macro-cell can be used for estimating frequencies of the localized eigenmodes.

The results of White *et al*. (2002) and Kuhlmey *et al*. (2002) show that the first localized mode within the micro-structured optical fibre (MOF) *Ω* is the dipole mode in the central core region.

### (b) Frequency estimates for a MOF with circular inclusions

We would like to make an analytical estimate of the frequency of the first localized mode. Compared to §5, the thin bridges play little part in the asymptotic evaluation of the frequency of the fundamental localized mode, partly because all the Dirichlet data on ∂*Ω*\*Γ* is homogeneous. In this case, the eigenfield is similar to that of the fundamental mode in a disc with the homogeneous Dirichlet boundary conditions. Since *E*_{3} and *H*_{3} are solutions of the same governing equations, we can represent the common problem as(6.16)In polar coordinates (*r*, *θ*), the dipole solutions of this problem have the form(6.17)where *Χ*_{1m} is the *m*-th zero of the Bessel function *J*_{1}. Note that *E*_{3} and *H*_{3} would typically correspond to real and imaginary parts of this solution. From (6.16) and (6.17) we obtain(6.18)For a MOF consisting of six circular air holes of radius *a* and constant centre-to-centre spacing *Λ*=6.75 μm surrounded by an infinite matrix of silica, we determine numerically the complex frequency *ω*^{*} of the localized mode using the multipole method (CUDOS MOF facilities; Kuhlmey, 2005).

In this case, the effective radius *R*_{eff} of the disc is found in the form(6.19)

In table 3, we show the variation of *R*_{eff} obtained from equation (6.19) with the hole radius *a* for the micro-structured fibre geometry considered by Kuhlmey *et al*. (2002) and White *et al*. (2002). The centres of cylinders lie on the circle of radius *Λ*=6.75 μm, which is equal to the distance between the centres of neighbouring cylinders. The effective refractive index *n*_{eff} for this finite system is in fact complex, with the imaginary part determining flux leakage in the transverse plane as the mode propagates along the *x*_{3}-axis. The values of *n*_{eff} given in table 3 were obtained using the CUDOS MOF utilities (Kuhlmey 2005). Naturally, the real part of *n*_{eff} was used to generate the values of *R*_{eff} in table 3.

Note that as *a* increases, the real part of *n*_{eff} decreases, the imaginary part of *n*_{eff} increases and *R*_{eff} decreases. The first of these trends can easily be understood on the basis of an effective index model, while the second and third are less intuitively obvious. In table 3 we show that *R*_{eff} is a linear function of *a*, which is represented accurately by the following empirical expression:(6.20)Note from table 3 that *R*_{eff} corresponds to a radius always located within the thin bridge region.

### (c) Numerical simulations for a macro-cell with elliptical inclusions

The previous computations are extended to the case of a doubly periodic array of macro-cells containing elliptical cylinders arranged to preserve the axial (sixfold) symmetry, as shown in figures 6 and 7. Note that all modes are now of real frequency (there is no dissipation in the periodic structure). In this case, we model the problem using edge finite elements (vector case) together with Floquet–Bloch boundary conditions on the sides of the macro-cell. For the computations, we used the Getdp software (Dular *et al*. 1998).

In contrast with the transverse case, there is no acoustic curve containing the origin *Γ*=(0, 0). The parabolic shape of the first curve in the centre of the diagram is approximated in closed form in Guenneau *et al*. (2003) for the case of high-contrast cylindrical inclusions of circular cross-section.

We note from figures 6 and 7 that the field is mostly confined within the centre of the macro-cell. Figures 6 and 7 suggest that the fundamental eigenmode (which is a dipole mode) is highly localized in the central core region (the disc of radius *R*_{eff}) of the macro-cell when elliptical inclusions are elongated in the tangential direction within the macro-cell.

In table 4, we report the dependence of the effective radius *R*_{eff} on the eccentricity and orientation (radial or tangential) of the elliptical holes. We note that *R*_{eff} reduces when ellipses are elongated in the radial direction and it increases for ellipses elongated in the tangential direction. Assuming that the area of the ellipses is fixed, we can approximate *R*_{eff} by a linear function of *a* for the ellipses oriented in the radial direction,(6.21)Comparing figures 5–7 we note that the first photonic band gap is larger for the case of ellipses elongated in the tangential direction within the hexagonal macrocell.

## 7. Concluding remarks and discussion

Our work was motivated by the papers by Kuhlmey *et al*. (2002) and White *et al*. (2002), which deal with a finite structure. Here, we have analysed solutions of an eigenvalue problem for Bloch–Floquet waves in a doubly periodic medium. Asymptotic approximations were presented for the cases when the inclusions within the macro-cell are close to touching. This approach clearly identified the way to estimate the frequency of localized eigenmodes, together with the boundaries of low frequency stop bands on the dispersion diagrams. For the case of a TE polarization, with Neumann data on the inclusion boundaries, we have obtained explicit asymptotic formulae for the frequencies of the localized (locally axi-symmetric) modes for different shapes of thin bridges between the inclusions. For the case of the oblique incidence, we have studied localized eigenmodes of dipole type. In this case, we have computed the ‘effective radius’ of the region of localization and have shown that, for certain geometries, it could be approximated by a linear function of the diameter of the inclusion.

Further generalization of our work is to the case of multi-scaled structural composites. An example of such a structure is shown in figure 8, where the macro-cell (studied earlier) has now additional split-ring resonators inside every circular hole. Some of the eigenmodes are highly localized within the split-ring resonators, and there is a low frequency band gap whose boundary is close to the resonant frequency of split-ring resonators. This frequency allows for explicit analytical estimate, as outlined in the main text of the paper,(7.1)Here, *n* is the refractive index of the dielectric material within the split-ring resonator, *ϵh* and *l* are the constant thickness and the length of the thin bridge in the resonator, is the area of the core of the ring, and is the area of the hexagonal cell excluding the resonators. The corresponding eigenmode is shown in figure 8*c*. The results of numerical calculations presented in figure 8 correspond to *ϵh*=0.012, =0.159^{2}*π*, *l*=0.1 and , , and . The asymptotic approximation agrees well with the finite element computation, which gives *ω*_{FEM}=0.35308. Other frequencies and shown in figure 8*a* correspond to high-order localized modes as shown in figure 8(*d*,*e*).

## Acknowledgments

Ross McPhedran acknowledges support of the London Mathematical Society under grant 4424 and the Australian Research Council through its Centre of Excellence Program. Sébastien Guenneau acknowledges support of the EC funded project PHOREMOST under grant FP6/2003/IST/2-511616. Sébastien Guenneau, Alexander Movchan and Natasha Movchan are grateful for support of the Franco-British Partnership Program ALLIANCE (project PN 05.026).

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2006.1800 or via http://www.journals.royalsoc.ac.uk.

↵Here,

*d*is the characteristic size of the honeycomb cell, and*c*is the speed of light in a vacuum.- Received August 8, 2006.
- Accepted November 28, 2006.

- © 2007 The Royal Society