## Abstract

The band spectrum and associated Floquet–Bloch eigensolutions, arising in acoustic and electromagnetic waveguides, which have periodic structure along the guide while remaining of finite width, are found. Homogeneous Dirichlet or Neumann conditions along the guide walls, or an alternation of them, are taken. Importantly, in some cases, a total stop band at zero frequency is identified providing space for low-frequency localized modes; geometric defects in the structured waveguide also create these modes. Numerical and asymptotic techniques identify dispersion curves and trapped modes. Some cases demonstrate maxima and minima of the spectral edges within the Brillouin zone and also allow for ultraslow light or sound. Imaging applications using anomalous dispersion to generate subwavelength resolution are possible and are demonstrated.

## 1. Introduction

There is considerable interest in the transport properties of waves propagating in periodic structures. A recent survey (Dowling *et al*. 2008) gives several thousand research articles on bandgaps for electromagnetic waves (photonic bandgaps), but far less, only a hundred or so, for bandgaps for elastic waves. Nevertheless, technological applications in photonic and phononic bandgap elastic materials look equally exciting in opening new applications in light and audio filters, nanoscopic photonic and phononic lasers, perfect reflectors, light-emitting diodes and photonic or phononic integrated circuits. One of the most exciting potential applications is the possibility to control light propagation through opto-elastic mechanisms, by making use of elastic stop bands in photonic crystal fibre preforms (Guenneau *et al*. 2003; Russell *et al*. 2003). This application relies upon the appearance of localized eigenmodes, within large enough supercells, created by defects of the periodic structures. These eigenmodes are sought as Floquet–Bloch waves and correspond to nearly dispersionless curves (flat bands) on band diagrams: their group velocity vanishes when the size of the supercell goes to infinity, which corresponds to the physical situation of stationary waves.

There have been suggestions that anomalous dispersion occurring in bandgap structures might lead to so-called negative refraction (Pendry 2000), when light or sound is refracted according to inverted Snell–Descartes laws (Gralak *et al*. 2000; Guenneau *et al*. 2007). Anomalous wave propagation also includes new features such as waves slowing down around inflection points of band diagrams (Figotin & Vitebskiy 2006), and even possibly freezing and then propagating in the opposite direction. The model structures we investigate herein demonstrate some of these features in a surprisingly simple setting.

The solutions in the periodic guide are characterized by the frequency, *ω*, and the product of the Bloch wavenumber, *k*_{0}, and the periodic cell length *d*. The spectral problem, for some range of frequencies, does not permit propagating solutions and thus stop bands (Movchan *et al*. 2002) occur. Trapped modes, which are non-trivial eigenfunctions of the system that have exponential decay in at least one of the media, can also be found. They correspond to modes that are evanescent, i.e. cut-off, in some part of the guide and are localized. Simultaneously, some dispersion curves correspond to waves that have zero group velocity, and thus are standing waves that do not transmit energy through the guide. Some curves also have very slight curvature in the *ω*–*k*_{0} plane and allow for very slow propagation of energy. All three of these features—trapped modes, standing or stationary waves and slow waves—are of considerable practical and current interest (Figotin & Vitebskiy 2006; Mok *et al*. 2006). We investigate whether these features, and subwavelength imaging, can occur for acoustic (out-of-plane elastic) and electromagnetic propagations in the periodic planar waveguide illustrated in figure 1; a sequel (Adams *et al*. in preparation) considers the more complicated case of in-plane elastic periodic guides.

The plan of the article is as follows: before embarking upon a numerical or asymptotic study, it is illuminating to draw parallels between the Navier equations of elasticity and those of Maxwell for electromagnetism and deduce the fundamental cases that we consider herein (§§2 and 3). The periodic system has an exact dispersion relation, provided the guide walls are straight and the boundary conditions are the same all along the guide wall. We complement this exact solution with numerical techniques that then translate across to less simple systems such as alternating boundary conditions (transfer matrix methods), and to non-straight-walled geometries (finite elements); these numerical approaches are briefly summarized in §3. Numerical results are presented for the fundamental cases of interest for a straight guide and also for a deformed periodic guide; asymptotic approaches are also highlighted in §4, which connect with recent work on trapped modes in infinite structures (Postnova & Craster 2006). A further generalization is considered wherein the boundary conditions are alternated on the horizontal guide edges, and this provides a simple example where the stop bands achieve minimum thickness within the Brillouin zone rather than at the edges, thereby providing a concrete example of this phenomenon and complementing results in Harrison *et al*. (2007). We describe physical implications of the anomalous dispersion and trapping phenomena and illustrate them by describing an imaging device. This leads to the design of an endoscope whereby an acoustic line source is imaged through the system via anomalous dispersion. We describe various situations arising in acoustics (and their counterparts in electromagnetism), showing filtering and imaging phononic bandgap structures of a new type. We emphasize that our structure is not covered by the usual analysis of one- or two-dimensional photonic crystals. It lies in-between, being quasi-one dimensional, and so displays its own specific novel features. This imaging device is described and some concluding remarks are given in §6.

## 2. Elastic and electromagnetic spectral problems

Harmonic time dependence, exp(−i*ωt*), for the physical variables with *ω* as the wave frequency is assumed understood, and henceforth suppressed. An elastodynamic wave represented by the displacement field * u*(

*x*,

*y*,

*z*) within a heterogeneous isotropic elastic medium of spatially varying density

*ρ*and Lamé parameters and is described by the vector Navier equation as in Ben-Menahem (1987); in the weak sense in (2.1)We consider the spectrum of the periodic waveguide structure of figure 1 assuming that the stresses and displacements are continuous across interfaces and that Bloch conditions connect the ends of the elementary cell; either stress-free or clamped conditions hold on the guide walls. Elasticity and electromagnetism are identical, for piecewise continuous media, in the out-of-plane elastic and transverse electromagnetic cases. The out-of-plane elastic case is physically identical to acoustics and, compared with electromagnetism, allows for greater variety in the physically allowable boundary conditions.

Electromagnetic waves within heterogeneous isotropic magneto-dielectric media are characterized by the permittivity, *ϵ*(*x*, *y*, *z*), and permeability, *μ*(*x*, *y*, *z*), with and *c*_{L} is the speed of light. The electric and magnetic fields * E*(

*x*,

*y*,

*z*) and

*(*

**H***x*,

*y*,

*z*), respectively, are solutions of Maxwell's equations that can be manipulated to(2.2)

Out-of-plane elastic (acoustic) and transverse electromagnetic problems are identical modulo changes of parameter labelling, and so can be considered together. Equation (2.1) has fully coupled shear and compressional wave solutions. We consider two-dimensional guides, in *x*, *y*, with spatially varying elastic parameters independent of the third Cartesian coordinate, *z*. Thus, the out-of-plane shear waves decouple from in-plane coupled shear and compressional waves. Taking the parameters to be piecewise continuous, we invoke Helmholtz's decomposition theorem introducing Lamé potentials *Φ*(*x*, *y*, *z*) and * Ψ*(

*x*,

*y*,

*z*) to give (Sternberg 1960)(2.3)Taking the vector Lamé potential to be transverse, i.e. , so

*Φ*vanishes, corresponds to an anti-plane shear displacement . Here

*Ψ*

_{t}and the displacement

*u*

_{3}satisfy(2.4)For transverse electromagnetic waves, the magnetic and electric fields

**H**_{t}and

**E**_{t}, respectively, and the transverse vector Lamé potential

**Ψ**_{t}are equivalent, provided(2.5)where

*c*

_{T}stands for the shear wave-speed.

Thus out-of-plane shear and transverse electromagnetic cases are treated simultaneously; the vector in-plane elastic case, with coupled compressional and shear waves, inherently brings in more mathematical technicalities (Adams *et al*. in preparation), although much of the physics is similar. An important point is that completeness of the spectrum can be proved and thus that no exotic or corner singularities arise (see appendix A).

In acoustics/elastic SH waves, Dirichlet [D] and Neumann [N] conditions on the guide walls correspond to clamped and freely vibrating walls, respectively; [D] and [N] are used hereafter to distinguish these cases. The two fundamental solutions of equation (2.4) of interest have either *u*_{3}=0, [D], or , [N], on *y*=±*h*. Both have continuity of *u*_{3} and across material interfaces and have electromagnetic counterparts. The transverse electric (TE) and transverse magnetic (TM) cases are equivalent to the [N]/[D] acoustic problems, respectively (see appendix A). Notably, taking, say, TE and then alternating Dirichlet and Neumann conditions, as we shall do later, is not physically meaningful in electromagnetism whereas in acoustics it is.

### (a) Modal analysis of the periodic spectral problem

We non-dimensionalize and fix upon notation using elastic/acoustic variables henceforth; all variables and parameters decorated with tilde are non-dimensional. We use (*j*) in superscript, and subscript, to denote a variable, and parameter, defined only in material *j*, for *j*=1, 2. Non-dimensional variables and parameters are introducedThe non-dimensional Helmholtz equations are now(2.6)and the equations are valid within the domains occupied by homogeneous material *j*. Further expressions for stress in terms of displacement emerge as(2.7)Stress-free, or clamped, boundary conditions, , [N], or , [D], respectively, are applied all along the guide walls . Interface conditions ensuring the continuity of normal stress and displacement at material junctions are imposed(2.8)for . The change of phase between =−*b* and *a* is fixed using Floquet–Bloch conditions on the normal stress and displacementand to ensure propagating solutions we impose that is real. We pose a single-mode solution in each of the materials, of the form(2.9)valid for *k*_{n(j)}≠0, in which *A*_{1}, *A*_{2}, *B*_{1} and *B*_{2} are constants, and is the *n*th modal solution to the homogeneous problem(2.10)(2.11)where the upper (lower) terms in curly brackets refer to the [N] ([D]) cases. The key difference between the [D] and [N] cases is the exclusion of *n*=0 in the clamped [D] case. In the stress-free [N] case, *n*=0 leads to a constant solution for , whereas for [D] this corresponds to the trivial solution and is hence omitted. Using (2.9) with conditions (2.8) leads to a dispersion relation linking and (2.12)(2.13)It is initially surprising that the dispersion relation is the same for both [D] and [N] cases; the only difference being the exclusion of wavenumbers *k*_{j(0)} in the clamped [D] case. This relation is the waveguide analogue of that derived in Movchan *et al*. (2002) for a one-dimensional problem using layers of infinite height.

In cases for which and , we get *k*_{n(1)}=0 and (2.9) is degenerate for *j*=1. In this situation, we pose the solution in the form , which leads to the following dispersion relation replacing (2.12):(2.14)An analogous result holds when and .

For the clamped homogeneous guide [D] of material *j*, there exists a non-zero cut-off frequency below which no waves can propagate, whereas in the free case [N] is always permitted. By expanding the dispersion relation (2.12) about , for *j*=1, 2, and invoking the mean value theorem, one can show that there always exists a mode for which min, even for very weak periodic and piecewise material contrasts. This mode is evanescent in one of the materials, while propagating in the other, and is henceforth referred to as a trapped mode. This phenomenon is not confined to material contrast as even periodic widenings of the guide can have a similar effect.

## 3. Discretization of the Floquet–Bloch spectral problems

Given the exact dispersion relation, it seems unnecessary to develop numerical schemes; however, later we generalize to deformed guides, or guides with alternating boundary conditions, and these require accurate numerical schemes.

One efficient approach is to use matrix-based techniques as in Pagneux & Maurel (2002). Within a homogeneous section of guide, the wavefield is written as(3.1)Here, the notation , is introduced, and , ; their amplitudes represent the contribution by the *n*th modal solution in the right- and left-going directions, respectively. The hatted variables represent modal solutions introduced in equation (2.11). The angle made in the complex plane by *k*_{n} and the positive real axis, *θ*, satisfies so that the wave is physically meaningful.

We introduce vectors * a*,

*defined by (likewise*

**b***), and note that the Floquet–Bloch conditions (2.9) are equivalent to , . Invoking continuity of normal stress and displacement across a junction at , one can equate these sums to yield(3.2)Here a superscript ± denotes field variables on the ± side of interface. We introduce inner product 〈.|.〉 that is defined by(3.3)The problem exhibits an orthogonality relationship whenever this inner product is taken within homogeneous material*

**b***j*: , where , for

*n*=1, 2, 3, … and where

*δ*

_{nm}=1 if

*n*=

*m*and 0 otherwise. Further, the result forces the matrices , and to be diagonal. This corresponds physically to no-mode conversion at an interface: an incoming mode is only transmitted and reflected into the same mode. This observation is attributed to the fact that the vertical structure of the

*n*th mode is independent of material parameters and so a separation of variables is possible allowing us to consider the problem mode by mode. In §4 consideration is given to a similar problem where the entire guide is a single homogeneous material, but has periodically alternating boundary conditions. Using the matrix approach is still valid, the orthogonality relationship within each section of guide is preserved; however, the aforementioned no-mode conversion property at the interface no longer holds: a single incoming wave is transmitted and reflected into multiple modes.

Taking the inner product of the sum (3.2) with a single mode *m* yields the following results (Pagneux & Maurel 2002) that we shall require later:(3.4)(3.5)where and . We denote the material corresponding to +, − in the above formulation by a superscript and subscript, respectively. For example , the matrix used to compute * b*(0

^{+}) in terms of

*(0*

**b**^{−}), denotes matrix with materials 1 and 2 corresponding to − and +, respectively. Combining this with results for a homogeneous material, we arrive at an eigenvalue problem that is solved numerically using the QZ algorithm (Moler & Stewart 1973)(3.6)(3.7)The first equality is the result of the Floquet–Bloch conditions on

*(*

**a***x*),

*(*

**b***x*), and where , are matrices with entries , along the leading diagonal, respectively. The system is energy conserving, and hence the determinant of the transfer matrix is identically 1. The transfer matrix is rearranged using row and column swaps to form 2×2 blocks on the leading diagonal corresponding to the transfer matrices for single-mode problems, and we denote the block corresponding to mode

*n*by . The determinant of , being zero, where is the identity matrix of appropriate dimensions, implies that the product of determinants of each is zero. The condition implies first that the dispersion relation (2.12) is satisfied, and second that , where tr(.) is used to denote the matrix trace. The condition that is real is equivalent to imposing that and thus allows one to conclude, without even solving the problem, if a bandgap is present at the frequency prescribed.

We also use finite-element techniques for problems with geometric variations. The weak formulation of (2.4) is(3.8)and this is ideally suited to finite-element analysis; the main novelty in this application is to incorporate the Floquet–Bloch conditions. To enforce the Floquet–Bloch conditions of the weak solution, *u*_{3}, in (3.8), we link values of test functions *v* on opposite sides of the basic cell *Ω* and assume a phase shift between them as described in Nicolet *et al*. (2004). The upshot is that an eigenvalue problem emerges, which is solved efficiently using the Lanczos algorithm (Cullum & Willoughby 2002).

## 4. Results

The finite-element and modal approach is used, together with cross-verification using a spectral collocation method with Chebyshev basis functions, described in Skelton *et al*. (2007), to generate band diagrams and the associated eigenfields. We begin by discussing results from the two cases described, namely [D]/TM and [N]/TE in acoustics/electromagnetism, respectively, and then go on to address problems featuring geometric perturbations and periodic mixed boundary conditions.

Typical results are shown in figure 2 that show the variation of frequency with of the propagating waves that pass through the layered structure, for different material parameters, for both the stress-free [N] and clamped [D] cases. We reiterate that, as mentioned in §5, the dispersion relations (2.12) give curves in the clamped case, which are necessarily also solutions to the stress-free case. The solid curves correspond to solutions of both the [D] and [N] cases, and the dashed curves are solutions of the free [N] case only with *n*=0 (hence excluded from [D]). In each case, the guide has *a*=3, *b*=3 and material 1 is aluminium (*c*_{T}=3130 m s^{−1} and *ρ*=2700 kg m^{−3}). Material 2 takes parameters linearly interpolated between those of aluminium and those of tin: , for 0<*σ*<1, and likewise *ρ*_{2}; where a subscript t is used to denote material properties of tin (*c*_{T}=1670 m s^{−1} and *ρ*=7300 kg m^{−3}). Using the correspondence between clamped/TM and free/TE cases, this figure also corresponds to an electromagnetic problem where the contrast between permittivity and permeability is close to 2.7, which can be achieved by alternating, say, homogeneous layers of silica and a magnetic material such as core Ferrite U 60 that displays a permeability that can be as large as 1×10^{−5} H m^{−1} (corresponding to a relative permeability of 8). Another possibility is to alternate layers of silica and magneto-optical material, whose permeability can be tuned through the Faraday rotation effect induced by an ambient magnetic field. The latter design would inherently lead to some anisotropic features, including non-reciprocity effects (Postava *et al*. 2005). The induced asymmetry in the waves propagating forwards and backwards could be exploited to make delay lines in virtually lossless optical media, as proposed in Ania-Castanon *et al*. (2006). Our numerical algorithms can be adapted to such periodic waveguides consisting of anisotropic layers, but scattering matrix-based methods might also prove handy for this purpose (Neviere 1994; Botten *et al*. 2004). The Bloch parameter represents a phase shift across the Brillouin zone, and hence the graph is 2*π*/*d* periodic and even; therefore only values are shown.

Figure 2*b–d* shows the presence of bandgaps where no propagation is possible, and the size and number of these bandgaps changes as the material disparity increases from figure 2*b* to figure 2*d*; the nature of this relationship is shown in figure 3. The width of the stop bands is determined by the dispersion relations at the edge of the Brillouin zone as is typically the case (Harrison *et al*. 2007), although we later generate counterexamples. At the two frequencies bounding the bandgap, by continuity; physically, this derivative corresponds to energy flux through the cell and thus solutions here are standing waves. As the material contrast increases (as *σ* increases), most curves have ; however, there is some disparity in this value between modes. Hence as *σ* increases some modes move towards faster than others and gaps in which no propagation is possible appear. In a homogeneous aluminium guide (*σ*=0), the lowest mode in the [N] case passes through the origin and meets the second mode at and . This is found analytically by exploiting the fact that the wavelength there must be 2*d* and finding the corresponding wavenumbers and frequencies using (2.11). As *σ* increases, the lowest mode has a value of lower than the second mode and thus approaches the axis faster, leading to the formation of a bandgap. The most important difference between the clamped and free cases is the appearance of a total stop band at in the clamped case; there exist non-zero cut-off frequencies in each material and below both of these no propagation is possible.

It is also worth noting that in the clamped [D] case wave trapping, available only when material contrast is non-zero, is more apparent than in the stress-free [N] case: at excitation frequencies , the only propagating solutions available are trapped in the sense outlined in §1 (and at least one such solution can always be found), whereas in the [N] problem, propagating and non-trapped solutions exist down to .

We now proceed to investigate the nature of the trapping of the lowest mode in the clamped [D] case, and figure 4*a–d* shows the evolution of this mode with changing material contrasts: *σ*=0, 1/20, 1/10 and 1/5, respectively; shaded areas correspond to the presence of stop bands; and the dashed lines are drawn at cut-off frequencies in the two materials. When the mode lies between these lines, it is trapped and amplitude localization occurs; the mode is always trapped sufficiently close to . If the mode passes through a cut-off frequency, as occurs in figure 4*b*,*c*, we obtain a mode with shape . Defining , from a rearrangement of (2.14), as this occurs only if has a real solution in . When the material contrast is sufficiently large, the mode is trapped for all values of , or equivalently the upper cut-off frequency ( in this instance) enters the stop band, as shown in figure 4*d*. As the material contrast further increases, other modes are trapped below the greater cut-off frequency and these trapped modes become increasingly flatter. Thus, the flux of energy through the cell (given by ) decreases: at *σ*=1, the variation between minimum and maximum frequencies obtained by the lowest mode is 3.87×10^{−3}, and three modes exist completely below cut-off frequency . Such anomalous dispersion suggests the possibility of slow sound, in a way similar to that achieved for light (Figotin & Vitebskiy 2006; Mok *et al*. 2006).

Figure 3*a* shows the variation of the size of bandgaps with *σ* for the stress-free [N] case. For the homogeneous (*σ*=0) problem, every gives rise to at least one real (propagating) solution for *k*_{n} and hence no bandgaps are present. As the materials differ, bandgaps form and their size with changing *σ* is smooth with the exception of some sharp kinks. One example of such a kink is shown by the middle circle on figure 3*a*, and the physical mechanism responsible for these kinks is demonstrated in figure 3*b–d*. Figure 3*b* shows a portion of the plane zoomed to show the modes that bound the bandgap at *σ*=0.4, corresponding to the leftmost circle in figure 3*a*, and shows that the flatter of the two modes below the bandgap forms the lower bound. However, this flat mode approaches faster than the mode below it as *σ* increases. Thus, the distance between these two modes decreases until some critical value at which both modes form the lower bound of the bandgap in question, as shown in figure 3*c*. It is at this point that the kink in curve in figure 3*a* occurs, shown by the middle circle at *σ*=0.427. In figure 3*d*, the flatter of the two modes below the bandgap has passed the other mode, which continues to form the lower bound until *σ*=1. The size of this bandgap decreases to the right of this kink indicating that the mode bounding it from above is approaching faster than that bounding it from below. This phenomenon happens to some modes, but not all, for example bandgaps formed between the first and second modes and between the second and third modes, as shown in figure 2, correspond to the two curves that begin at *σ*=0 in figure 3 and are free from kinks. This indicates that within 0<*σ*<1, no modes overtake the lowest three as they approach the axis.

The interactions between modes as *σ* changes, leading to the formation of bandgaps, are more readily observed by plotting for each mode the maximum and minimum attained, and this is shown in figure 5. A bandgap forms when there is a difference between the maximum achieved by one mode and the minimum of another and these regions are shaded in the figure. The bandgap shaded nearest to the axis shows that formed by the maximum of mode 1 and minimum of mode 2, as shown in figure 2*b–d*. The nature of the crossings of the maxima and minima allows for bandgaps to appear and disappear with increasing *σ*, and those bandgaps that disappear are shown in figure 3*a* by curves reaching a bandgap size of 0. For example, figure 5 shows a bandgap forming at *σ*=0 bounded by the fourth and fifth modes, but disappears at *σ*=0.317 when the maxima of the lower bound exceed the minima of the upper bound. This is shown in the previous figure as a curve growing with *σ* initially, reaching a maximum bandgap size of 0.0144 at *σ*=0.146 and then falling until it disappears at *σ*=0.317. The inset in the figure shows the same kink circled in figure 3*a*. The maxima of two modes cross one another at a point circled on the inset of figure 5, corresponding to figure 3*c*, and the bandgap continues to be bounded by the maximum of these two curves.

There is a dispersion curve, shown in figure 4*a*, near the cut-off frequency for the first mode of a homogeneous guide in the [D] case, namely . With the addition of weak contrast, such that , this dispersion curve drops below *π*/2 for . Figure 6*a* shows the localization of the mode to one of the media when the guide is very long and this is then a long wave in the sense of Postnova & Craster (2006). The localization is similar although caused by material rather than geometric perturbations; the dispersion curve is then very flat and this mode is both localized and virtually stationary. As the domain becomes shorter, the dispersion curve then varies more dramatically until eventually it is no longer trapped.

The effect of periodic geometric perturbations is considered as these will induce similar behaviour to that of the periodic material contrast. We take a single homogeneous material, but alter the boundary. For illustration, a small asymmetric notch profile, as shown in figure 7, is taken for which the overall area of the basic cell is preserved. Interestingly, the dashed curve reaches its global maximum for a Bloch parameter that does not occur at the edges =0 and *π* of the reduced Brillouin zone, as one would expect for a one-dimensional periodic structure (Harrison *et al*. 2007). Since the next curve reaches its global minimum around the same value of , it is fair to say that the width of the associated stop band shrinks deep into the Brillouin zone and is not fixed by values at the edges. This quasi-one-dimensional structure then contradicts the usual one-dimensional assertion that the Bloch spectrum associated with the one-dimensional Schrödinger operator with a periodic potential is included within the spectrum computed at the edges of the Brillouin zone, where the split in eigenvalue degeneracy is predicted, leading to gaps (Reed & Simon 1978). We numerically checked that our very unlikely event actually occurs for every dispersion curve above the dotted one (at least for the next 100 or so dispersion curves). Indeed the minimum thickness of the bandgap at the top of figure 7*a* is directly in the middle of the Brillouin zone.

This remarkable result is interpreted as follows: the split in degeneracy of the curves away from the =0 and *π* vertical axes is associated with a side-wall effect associated with the presence of upper and lower boundaries that are subjected to homogeneous Neumann data. It is indeed apparent on figure 7*b* that the abrupt change in the profile of the upper and lower boundaries of the guide leads to a trapping effect, which suggests the possibility of vanishing group velocity deep within the Brillouin zone, associated with the presence of inflexion points, sometimes referred to as slow sound–light phenomenon (Figotin & Vitebskiy 2006).

Noting that the trapping occurs when a mode is locally cut on in one material and cut off in the other, it is natural to consider a perturbation about this cut-off frequency and develop an asymptotic theory. We treat a gently thickening guide, the thickness variation repeating periodically along the guide. The methodology is similar to, say, Postnova & Craster (2006) and references therein, where trapping is sought in infinite structures or in finite structures such as rings (Gridin *et al*. 2004). Let us take the thickness variation within a cell to be symmetric about the centreline and the upper boundary to be(4.1)where *ϵ* is a small parameter and the cell is and *g* is a smooth function; *α*>0 (<0) corresponds to a thickening (thinning) guide. Using a change of variables and together with an expansion and yields, to leading order, with *λ*_{0}=*nπ*/2 for *n*=1, 2, … in the Dirichlet [D] case; the leading-order solution being just that for an infinite guide at cut-off. Proceeding to the next term in the expansion leads to a consistency condition that identifies *f*_{0} as the solution of a differential-eigenvalue problem(4.2)subject to Bloch conditions at *ξ*=−*B*, *A*: , where , and likewise for *f*_{0ξ}. Solutions are readily obtained numerically and the eigenfrequencies are(4.3)So for, say, *n*=1 the lowest Dirichlet mode then, if a mode lies beneath *π*/2 and is trapped. If *α*>0 then Sturm–Liouville theory suggests the possibility of a trapped mode and indeed this is observed numerically; in figure 8, we compare the asymptotics with computations from finite-element simulations.

## 5. Periodic mixed boundary conditions

Unlike in electromagnetics, it is perfectly legitimate to consider mixed Dirichlet and Neumann boundary conditions on the transverse walls of an acoustic guide. For instance, it can be clamped from above and freely vibrating below. This possibility opens new vistas in the design of stop band acoustic guides that are homogeneous, but with straight and parallel boundaries, mixing [D] and [N] homogeneous data. Figure 9 shows dispersion diagrams from problems with various arrangements of mixed boundary conditions, and we proceed to highlight the important features of each. In each case, we denote the conditions imposed on the cell by a series of pairs of letters, each pair representing one section and the two letters in the pair corresponding to the conditions on the upper and lower guide walls, respectively. For example, the configuration ND-ND, shown in figure 9*a*, corresponds to Neumann conditions on top and Dirichlet conditions on bottom in the first and second layers. Here we can see a zero-frequency stop band, but no other gap at higher frequencies. In figure 9*b*, a NN-DD configuration, we note the presence of further stop bands and two nearly completely flat bands associated with modes trapped in the layers with Dirichlet data above and below. The inset in figure 9*b* shows a dotted line that corresponds to an aluminium–tin guide with the same boundary conditions. Interestingly, the stop band for the homogeneous material (solid line) sits within the aluminium–tin stop band suggesting that one could create trapped modes for a finite homogeneous section within an otherwise periodic aluminium–tin guide.

In figure 9*c*, showing an ND-DN guide, the nature of the stop bands is very different, as the gaps are larger around the axis and smaller around the axis, unlike for figure 9*b*. Also, there is no flat band, as each layer has mixed Neumann and Dirichlet boundary conditions, which is less conducive for trapping than only Dirichlet data. Finally, in figure 9*d*, for a macrocell with conditions NN-DD-ND-DD-NN, we display the dispersion curves for mixed Dirichlet and Neumann data so as to mimic a defect in the lower boundary of the basic cell, in view of the choice of [N], [D], [D], [D] and [N] homogeneous data. Figure 9*c*,*d* has stop bands whose width is set by minima and maxima of dispersion curves well within the Brillouin zone.

## 6. Concluding remarks

We summarize our main results and suggest extensions for future work. This quasi-one-dimensional periodic structure has previously been unexplored and the appearance of stop bands in a layered waveguide of finite thickness created from a periodic cell subject to Floquet–Bloch conditions on its sides and homogeneous Dirichlet and/or Neumann conditions on its upper and lower boundaries are analysed. Such spectral problems model TE and TM microwaves propagating within layered waveguides with infinite conducting upper and lower walls. They can equally well model anti-plane shear waves propagating within elastic waveguides with freely vibrating and/or clamped upper and lower walls.

The spectrum of the waveguide very much depends upon the conditions on the upper and lower walls; for mixed Dirichlet and Neumann data, the stop bands occur even for homogeneous guides. We find many situations where the stop bands are thinner for values of the Bloch parameter strictly within the first Brillouin zone. This suggests that analogous cases should occur in doubly periodic structures, for which the accurate location of stop bands then requires the plotting of dispersion surfaces, not only dispersion curves; this supports a claim in Harrison *et al*. (2007). It is also related to the phenomenon of slow light/sound (Figotin & Vitebskiy 2006), and periodic guides with boundary perturbations exhibit this behaviour (cf. figure 10) and hence can be simply constructed, unlike their doubly periodic counterparts which must fulfil drastic criteria for the material properties (such as magneto-optic photonic crystals). Potential applications lie in improved optical and elastic delay lines and have therefore attracted a great deal of attention (Mok *et al*. 2006). For instance, scientists explored ways to slow down the propagation of light or sound by introducing some disorder in layered structures. This phenomenon known as Anderson localization was analysed using the transfer matrix methods in Barnes & Pendry (1991) for electromagnetic waves and Godin (2006) for elastic waves.

The existence of zero-frequency stop bands for both the TM and the clamped acoustic case is shown (figure 2). This suggests the possibility of low-frequency resonant structures opening new vistas in subwavelength imaging through negative refraction as proposed by Pendry (for instance a flat lens or an endoscope). This physical phenomenon relies upon a special feature associated with a non-vanishing eigenvalue , when goes to zero, interpreted either as a singular perturbation (Poulton *et al*. 2001) or via non-commuting limits (Nicorovici *et al*. 1995). However, this would also require the acoustic band to display negative curvature in the [D] case. Direct analysis of the dispersion relation shows that this cannot occur: assuming without loss of generality that , it follows that the trapped *n*=1 mode has and . Writing the dispersion relation as , which is real and differentiable in the range , and then differentiating *g* at , using the fact that , yields: ; inspection shows that the numerator is negative for all . Further, for all *ϵ* sufficiently small, , and thus the smallest root of must occur with regardless of material parameters. The variation of the dispersion relation in the high-contrast case is shown in figure 11, and the inset shows that the dispersion relation initially increases with . It follows that for the smallest root and the stationary point here is a minimum, thereby discounting the possibility of negative curvature of the lowest mode.

Our original idea was to create a defect mode within the zero-frequency stop band and subsequently meet the criteria for existence of a low-frequency phason/plasmon enhancing propagation of evanescent waves (Pendry *et al*. 1996). An alternative approach would be to increase the material contrast between the layers to create a locally resonant structure similar to that proposed by O'Brien & Pendry (2002) to create artificial magnetism in high-contrast dielectric photonic crystals. That may well lead to ultrasonic waveguides displaying a negative shear modulus similar to that experimentally achieved for an elastic waveguide with embedded Helmholtz resonators (Fang *et al*. 2006). Unfortunately, as exemplified in figure 11, the variation of the dispersion relation in a high-contrast case (*α*=1/10, *β*=10) does not meet the prerequisite criteria. Indeed, the dispersion relation has positive slope within a neighbourhood of , regardless of material contrast (cf. the first crossing of the *x*-axis by the dispersion relation, circled in the inset, which corresponds to the lowest mode root). Thus, the lowest mode cannot exhibit negative curvature near . Recently, to overcome this theoretical obstacle to acoustic subwavelength imaging via negative refraction, we have introduced effective boundary conditions on the upper and lower walls of the waveguide. Preliminary numerical tests involving certain types of Robin boundary conditions look promising, and such adequate effective boundary conditions could be obtained via high-contrast homogenization or by considering corrugated waveguides with boundaries undergoing fast oscillations. The work required to quantify this is underway, and so despite our proof that negative refraction cannot exist for the simple boundary conditions used in this article, progress on this should not be discounted.

Another route for the design of a high-resolution, subwavelength, endoscope is to make use of anomalous dispersion in the Bragg regime of layered waveguides. One can achieve so-called all-angle-negative refraction in photonic crystals (Gralak *et al*. 2000), which then displays some of the hallmarks of negative refraction: one can make a convergent flat lens, yet not a subwavelength one (Luo *et al*. 2002). In figure 12, a lensing effect of an elastic source through an aluminium–tin endoscope is shown. As we increase the number of layers, the resolution of the image increases until for 24 layers we achieve subwavelength resolution. This is demonstrated by the comparison of the profile of the displacement field in the source and image planes in figure 12*b*. This uses anomalous dispersion; that is, we choose a range of working frequencies corresponding to the neighbourhood of the intersection of the first dispersion curve (or elastic band) of figure 2*a* with the second dispersion curve (or optical band) of figure 2*d*. The subwavelength imaging is therefore achieved through a combination of both negative group velocity and strong anisotropy induced by the walls of the guide. These are frequencies where the homogeneous material has positive group velocity and the endoscope has negative group velocity. Such high-resolution endoscopes may have potential applications in medical imaging for ultrasonic waves and would work in many ways as a metamaterial displaying very high anisotropy along the imaging direction. Swiss rolls, which consist of concentric cylinders of insulated metal, form one such device designed for magnetic resonance imaging (typical working frequency of 24 MHz that lies within the radio frequency range). An array of Swiss rolls is used as a flux ducting mechanism and was shown to behave as a fibre bundle (Wiltshire *et al*. 2003), thus enabling deep subwavelength near-field imaging: it is, to date, the champion of subwavelength imaging with a resolution of up to 64th of a wavelength. The multi-layered endoscope we propose could also have applications in ultrasonic non-destructive near-field imaging.

We conclude with some remarks on the modal method. Using matrix methods for periodic waveguides enables one to gain some analytical insight (Lekner 1994; Felbacq *et al*. 1998), but it can, under some circumstances, become rather problematic. When dealing with a straight-walled guide with either stress-free or clamped conditions on both faces, one benefits from the single mode into single-mode property mentioned previously. As a result there are only a finite number of modes to consider and only a finite number of evanescent modes are available, which do not cause numerical problems. However, attempting to apply a matrix approach to a problem in which mixed boundary conditions occur, one finds that orthogonality between modes under the integral relationship (3.3) no longer holds, and a single incident mode is transmitted and reflected into an infinite sum of modes. In such cases, the large determinants of the matrices involved present potential numerical difficulties, and these pitfalls are readily observed when examining the straight-walled guide using equations of in-plane elasticity (Adams *et al*. in preparation). Papers analysing elastic stop band properties of layered waveguides are fairly sparse in the existing literature (Chen & Wang 2007), but they suggest a richer underlying physics than in layered optical waveguides. Similar problems also occur in the analysis of elastic stop band properties of doubly periodic structures: for instance, the celebrated Rayleigh method requires a suitable normalization of an infinite linear algebraic system: it can then be truncated with good accuracy to certain multipole order (dipole order in case of dilute composites; Poulton *et al*. 2000).

## Acknowledgments

We thank Duncan Williams (DSTL) and Mike Lowe (NDT Group, Imperial College) for their useful discussions and acknowledge support from the EPSRC and a CASE MoD DSTL studentship awarded to S.D.M.A. One of us, R.V.C., thanks the Mathematics Departments at the Universities of British Columbia and Alberta for their hospitality, the Royal Society for an International Visitor grant and the EPSRC for support under grant EP/E046029/1. Part of this work was performed while S.G. was working as a *Chargé de Recherche* at the Fresnel Institute (UMR CNRS 6133, University of Marseille).

## Footnotes

- Received February 13, 2008.
- Accepted April 22, 2008.

- © 2008 The Royal Society