BLOCH WAVES IN AN ARBITRARY TWO-DIMENSIONAL LATTICE … · 2018. 1. 11. · 1 BLOCH WAVES IN AN ARBITRARY TWO-DIMENSIONAL 2 LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS 3 ORY SCHNITZER

BLOCH WAVES IN AN ARBITRARY TWO-DIMENSIONAL1

LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS2

ORY SCHNITZER∗ AND RICHARD V. CRASTER†3

Abstract. We study waves governed by the planar Helmholtz equation, propagating in an infi-4nite lattice of subwavelength Dirichlet scatterers, the periodicity being comparable to the wavelength.5Applying the method of matched asymptotic expansions, the scatterers are effectively replaced by6asymptotic point constraints. The resulting coarse-grained Bloch-wave dispersion problem is solved7by a generalised Fourier series, whose singular asymptotics in the vicinities of scatterers yield the8dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing9in the absence of the scatterers; there are also empty-lattice waves that are only weakly perturbed.10Characterising the latter is useful in interpreting and potentially designing the dispersion diagrams11of such lattices. The method presented, that simplifies and expands on Krynkin & McIver [Waves12Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs13entailing resonant subwavelength elements distributed over a lattice with periodicity on the order of14the operating wavelength.15

Key words. Bloch waves, Periodic media, Singular perturbations16

AMS subject classifications. 34D15, 35P20, 35B27, 78M3517

1. Introduction. There is immense current interest in wave phenomena in ar-18

tificial periodic media. In subwavelength metamaterials (SWM), that are constructed19

using tiny resonant elements, the periodicity is small compared with the operating20

wavelength [35]. SWM mimic natural materials whose macroscopic properties are21

endowed by an underlying atomic structure. Ingenious designs — introduced in elec-22

tromagnetics but since adapted to acoustics, elasticity and seismology — have enabled23

unusual effective properties and capabilities unfamiliar in nature, including negative24

refractive index and cloaking. Photonic (similarly platonic and phononic) crystals25

constitute a separate class of artificial materials, in which operating wavelengths are26

typically on the order of the periodicity of the microstructure [32]. Here, wave manip-27

ulation is enabled by the surprisingly coherent outcome of multiple scattering events,28

mimicking the way electron waves are sculptured in solid-state crystals. Studies of29

photonic crystals were initially focused on the existence of complete photonic band30

gaps [42]. Nowadays, however, artificial crystals broadly interpreted are in the spot-31

light as a lossless alternative to SWM, with the plethora of phenomena demonstrated32

including slow light [3], dynamic anisotropy [2], defect and interface modes [14], cloak-33

ing [9], topologically protected edge states [20], and unidirectional propagation [29]34

amongst others.35

Artificial “metacrystals”, made out of subwavelength particles, inclusions, or mi-36

crostructured resonators periodically distributed with spacing on the order of the37

operating wavelength, are an amalgam of SWM and photonic crystals. The small-38

ness of the scattering elements in this setup suggests a relatively weak modulation of39

waves propagating through the crystal. When this is indeed the case, the effect of the40

lattice can be captured by a perturbation approach [28] in the spirit of the “empty-41

lattice approximation” popularised in solid-state physics [16]. There are scenarios,42

however, where the modulation cannot be reasonably regarded as small, regardless of43

the scatterer dimensions. An elementary example is the propagation of flexural waves44

in a periodically pinned plate, where the time-harmonic displacement field is governed45

∗Department of Mathematics, Imperial College London, London SW7 2AZ, UK

1

This manuscript is for review purposes only.

2 O. SCHNITZER AND R. V. CRASTER

by the two-dimensional Biharmonic reduced-wave equation and a zero-displacement46

“Dirichlet” condition on pinned boundaries. Since the Green function of the latter47

equation is regular, a valid leading-order approximation is obtained by simply ignor-48

ing the finite size of the scatterers, treating these as Dirichlet point constraints. The49

resulting problem offers a convenient framework for detailed mathematical studies50

of Bloch-wave dispersion surfaces and scattering from finite and semi-infinite arrays51

[10, 36, 12], and was also utilised for demonstrating the method of “High-Frequency52

Homogenisation” [1, 23].53

A related scenario is the propagation of planar electromagnetic waves through a54

doubly periodic array of perfectly conducting cylinders in transverse-electric polari-55

sation, where the time-harmonic electric-field component along the cylindrical axes56

satisfies the Helmholtz equation and vanishes on the cylindrical boundaries [31]. The57

acoustic analogue of the latter problem entails soft cylinders on which the velocity58

potential vanishes. Mathematically, the singular nature of the Helmholtz Green func-59

tion implies that, contrary to the Biharmonic case, one cannot prescribe the value of60

the wave field at a point. Accordingly, the small-scatterer limit needs to be handled61

with care. One approach is to reduce an exact multipole-expansion solution in the62

latter limit [27]. A more general and systematic approach is to employ the method of63

matched asymptotic expansions [13], where “inner” expansions valid in the vicinities64

of the scatterers are matched with an “outer” expansion valid away from the scat-65

terers. The pertinent essentials of the approach are outlined in [7] and [24] in the66

context of low-frequency scattering from isolated subwavelength Dirichlet scatterers,67

and in the context of singularly perturbed eigenvalues of the Laplace operator in [41].68

A matched asymptotics solution of the Bloch-wave dispersion problem for a dou-69

bly periodic array of small cylindrical Dirichlet scatterers was given by Krynkin and70

McIver [17]. In that work, the analysis of the inner region and the matching procedure71

follow closely the approach in [7], generalised to a cylinder of arbitrary cross-sectional72

shape. The outer solution is sought from the start as a quasi-periodic Green function,73

namely an infinite double sum of Hankel functions whose convergence is accelerated74

using the addition theorem and highly developed techniques for manipulation of lat-75

tice sums. The main result is an asymptotic dispersion relation, for which several76

accurate and semi-accurate methods of solution are suggested. A key finding is that77

deviations from empty-lattice eigenvalues, identified there as lattice-sum poles, can78

be either appreciable or negligible, i.e., logarithmically or algebraically small with79

respect to the inclusion size.80

In this paper, we revisit the doubly periodic Helmholtz–Dirichlet problem. We81

aim to streamline the approach of Krynkin and McIver [17] thereby allowing the82

methodology to be more readily applied, and also to generalise to lattices with ele-83

mentary unit cells occupied by multiple scatterers (e.g. the honeycomb lattice that is84

of much current interest due to its connection to graphene and isolated Dirac points).85

In our approach, those empty-lattice Bloch waves that are weakly perturbed are sys-86

tematically identified a priori by considering the existence (and dimension of the87

space) of empty-lattice waves that vanish at the positions of all scatterers. In addi-88

tion to simplifying the analysis, this approach provides insight when interpreting the89

resulting dispersion surfaces, and potentially in the design of tailored media. We also90

take a different matching route from that of Krynkin and McIver [17]. They adopt an91

infinite asymptotic expansion in inverse logarithmic powers, and following [7], iden-92

tify, in the course of the analysis, a Poincare perturbation parameter, corresponding93

to a summation of the infinite logarithmic series. In contrast, we note that the inner94

and outer problems are linear and similar at all logarithmic orders and hence it is95


LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS 3

more efficient to group all logarithmic and O(1) terms together and apply van Dyke’s96

matching rule [38, 8, 15].97

Another key difference is that we bypass the machinery of addition theorems98

and lattice sums. Rather, in the case of strongly perturbed modes, we represent the99

outer-region solution in terms of a generalised Fourier series. The dispersion relation100

is then derived by matching the singular asymptotics of the latter in the vicinities101

of the scatterers with the respective inner solutions. The present analysis thereby102

corrects the Fourier-series example of Refs. [1] and [23] for the singular Helmholtz103

case; the Fourier series solutions there were prescribed to vanish at the scatterers’104

positions, whereas those series actually diverge there.105

In §§2.1 we formulate the Bloch-wave dispersion problem for an arbitrary two-106

dimensional Bravais lattice of small Dirichlet scatterers. In §§2.2 we outline the107

asymptotic procedure for replacing the finite scatterers with point-singularity con-108

straints, leading to the coarse-grained eigenvalue problem of §§2.3. Weakly and109

strongly perturbed modes are analysed in §§2.4 and §§2.5, respectively, and dispersion110

diagrams for square and hexagonal lattices are shown and discussed in §§2.6. The ap-111

proach is generalised in §3 to lattices with multiply occupied elementary cells, and the112

dispersion diagram for a honeycomb lattice is shown and interpreted. In §4 we briefly113

consider the applicability of the asymptotic procedure to solving scattering problems114

involving a finite collection of scatterers, and employ the resulting scheme towards115

demonstrating the strong dynamic anisotropy suggested by the infinite-lattice disper-116

sion surfaces. Lastly, in §5 we give concluding remarks and discuss further research117

directions.118

2. Small Dirichlet scatterers at Bravais lattice points.119

2.1. Problem formulation. Consider the dimensionless Helmholtz equation in120

two dimensions,121

(1) ∇2u+ ω2u = 0.122

Here u(x) is an arbitrarily normalised field, x being the position vector normalised123

by a characteristic length scale L, and ω is the frequency scaled by the wave speed124

divided by L. An infinite number of identical Dirichlet scatterers, of characteristic125

dimension εL and arbitrary cross-sectional shape, are distributed with their centroids,126

say, at the vertices of a two-dimensional Bravais lattice,127

(2) R = na1 +ma2,128

where a1 and a2 are lattice base vectors and n and m are arbitrary integers. Accord-129

ing to Bloch’s theorem [16], waves propagating through the crystal satisfy a Bloch130

condition in the form131

(3) u(x) = U(x)eik·x, U(x + R) = U(x),132

where U possesses the periodicity of the lattice and k is the Bloch wave vector. Given133

(3), we only need to consider an elementary unit cell encapsulating a single scatterer134

centred at the origin x = 0, to which we attach polar coordinates (r, θ). The Dirichlet135

constraint is then written as136

(4) u = 0 on r = εκ(θ),137



where κ(θ) is the shape function associated with the scatterer. Our goal here is to138

calculate the dispersion surfaces ω = Ω(k). The latter inherit the periodicity of the139

reciprocal lattice, i.e. Ω(k + G) = Ω(k), where140

(5) G = nb1 +mb2,141

n and m being arbitrary integers, and b1 and b2 the reciprocal lattice base vectors142

defined through ai · bj = 2πδij . Additional symmetries further limit the k vectors143

required to be considered to the “irreducible Brillouin zone” [14], which we shall define144

on a case-by-case basis.145

2.2. Matched asymptotics. Henceforth we consider the asymptotic limit ε→146

0, with ω and κ(θ) = O(1). To this end, we conceptually decompose the elementary147

cell into an “outer” domain where r = O(1) and an “inner” domain where ρ =148

r/ε = O(1). As prompted in the introduction, logarithmic terms are treated on par149

with O(1) terms. Since Bloch eigenmodes are determined only up to a multiplicative150

constant, we may assume without loss of generality that u = O(1) in the outer domain.151

The inner region u is, at most, of comparable magnitude.152

Accordingly, we assume an inner expansion of the form153

(6) u ∼ Φ(ρ, θ; ε) + a.e.(ε), ρ,Φ = O(1),154

where “a.e.” denotes algebraic error. Substituting (6) into (1) and (4), we find that155

Φ satisfies Laplace’s equation156

(7) ∇2ρΦ = 0, ∇2

ρ =1

ρ

∂

∂ρ

(ρ∂

∂ρ

)+

1

ρ2∂2

∂θ2,157

subject to the Dirichlet condition holding on the surface of the scatterer158

(8) Φ = 0 on ρ = κ(θ),159

and matching with the outer region as ρ → ∞. Regardless of the shape function,160

κ(θ), at large distances [4]161

(9) Φ ∼ A ln ρ+B +O(1/ρ) as ρ→∞,162

where terms algebraically growing with ρ are disallowed as they become algebraically163

large in ε in the outer region. In (9), A is a constant that, once known, determines164

through the solution of the inner problem the constant B and similarly smaller terms165

in the far-field expansion. For example, for a circular cylinder, κ = 1, the solution166

is simply Φ = A ln ρ. Since this inner problem is harmonic, solutions for other cross-167

sectional shapes are obtained by conformal mapping of the solution for a circular168

cylinder. Specifically, let z = ρ exp(iθ) and w = f(z) be a conformal mapping from169

ρ > κ(θ) to |w| > 1, where f(z) ∼ Sz + O(1) as |z| 1 and S > 0 is a parameter170

determined by the shape of the inclusion. Then Φ/A = Re[lnw] = Re[ln f(z)] and171

hence Φ/A ∼ ln |z|+ lnS +O(1/|z|) as |z| → ∞. It follows that, for a general shape,172

(10) Φ ∼ A ln(Sρ) +O(1/ρ) as ρ→∞,173

where A is a constant to be determined from matching with the outer region. To174

leading algebraic error, only the large ρ expansion (10) of the inner solution is relevant.175

Hence shape dependence can be effectively captured by assuming a circular cylinder176



with effective radius (ε/S)L; the effective radius is a familiar concept in potential177

theory and is provided in the literature for various cross-sectional shapes [13, 40, 17].178

Without loss of generality we set S = 1 and henceforth interpret ε as an effective179

radius.180

In the outer region we assume the expansion181

(11) u ∼ φ(x; ε) + a.e.(ε), r, φ = O(1)182

where φ satisfies (1), the Bloch condition (3), and matching with the inner region as183

r → 0. The solution is necessarily of the form184

(12) φ = ψ(x) + aH(1)0 (ωr),185

where ψ(x) is a solution of (1) that is regular at r = 0, a is an undetermined constant,186

and H(1)0 is the zeroth-order Hankel function of the first kind. The latter is the187

fundamental outward-radiating solution of (1), which is logarithmically singular as188

r → 0. Higher-order singularities, i.e. products of constant tensors and gradients of189

H(1)0 are disallowed at this order since such solutions are algebraically singular as190

r → 0 and hence become algebraically large in the inner region.191

We may now match the inner and outer expansions using Van Dyke’s matching192

rule (specifically, see pg. 220 in [38] and [8]), treating logarithmic terms, such as ln ε,193

on par with O(1) terms. Thus, on the one hand, the inner expansion to O(1), written194

in terms of the outer variable, and then expanded to O(1), is195

(13) A ln(r/ε).196

On the other hand — noting the regularity of ψ and that197

(14) H(1)0 (ωr) ∼ 2i

π[ln(rω) + γ − ln 2] + 1 as r → 0,198

γ ≈ 0.5772 being the Euler constant — the outer expansion to O(1), written in terms199

of the inner variable, and then expanded to O(1), is200

(15) ψ(0) + a

2i

π[ln(rω) + γ − ln 2] + 1

,201

upon rewriting in terms of r. Comparing expansions (13) and (15), one finds202

(16) a = −πi2

ψ(0)

ln 2ωε − γ + πi

2

203

and A = (2i/π)a. Hence, the presence of a scatterer in general causes a perturbation204

which is O(1/ ln 1ε ), formally small, but appropriately regarded as being O(1) for all205

practical purposes as well as when using the matching rule.206

2.3. Effective eigenvalue problem. The preceding analysis gives rise to an207

effective eigenvalue problem, which can be formulated in two equivalent ways. Thus,208

we may seek a regular eigenfunction ψ that satisfies an inhomogeneous Bloch condition209

owing to the aperiodicity of aH(1)0 (ωr), a being proportional to ψ(0) through (16).210

An alternative formulation is obtained by noting from (12) that φ satisfies the forced211

Helmholtz equation212

(17) ∇2φ+ ω2φ = 4aiδ(x),213



where δ(x) denotes the Dirac delta function. In addition, φ satisfies the Bloch condi-214

tion (3) and the asymptotic constraint215

(18) limr→0

[φ(x, ω,k) + a

2i

πlnε

r

]= 0,216

that follows from the matching condition (16) in conjunction with (12).217

2.4. Empty-lattice Bloch waves and weakly perturbed modes. Prior to218

solving the effective Bloch eigenvalue problem formulated above, we note that in219

the absence of scatterers there are plane-wave solutions of (1) that satisfy the Bloch220

condition (3). Indeed, substituting the generalised Fourier series221

(19) u =∑G

UG exp [i(k + G) · x]222

into (1), where UG are constants, we find the empty-lattice dispersion relation223

(20) ω2 = |k + G|2.224

It is evident from (5) that for any given k there are infinitely many “eigenpairs”225

(ω,k). A distinction can be made between those pairs that are simple and those that226

are degenerate. Generally, we write the empty-lattice eigenmode space as227

(21) u =

D∑j=1

Uj exp [i(k + Gj) · x] ,228

where D denotes the level of degeneracy, and GjDj=1 those reciprocal lattice vectors229

satisfying (20) for the eigenpair (ω,k). Since (21) vanishes at the origin if230

(22)

D∑j=1

Uj = 0,231

for degenerate eigenpairs we may form a subspace, of dimension D−1, of empty-lattice232

plane-wave solutions that vanish at x = 0. Empty-lattice waves in the latter space233

also satisfy the true eigenvalue problem, with a = 0 (and hence φ = ψ), and hence234

are perturbed only weakly — algebraically in ε — from the empty-lattice dispersion235

relation (20). In appendix A it is verified that there are no φ eigensolutions satisfying236

(20) for which a 6= 0.237

2.5. Strongly perturbed modes. Consider next the possibility of eigenpairs238

of the effective eigenvalue problem that do not satisfy the empty-lattice dispersion239

relation (20). Since eigensolutions with a = 0 necessarily satisfy (20), we may assume240

here that a 6= 0.241

For the outer field φ we write the generalised Fourier-series solution242

(23) φ(x)/a =∑G

φ′G exp [i (k + G) · x] ,243

which satisfies the Bloch condition (3). The coefficients φ′G are determined by substi-244

tuting (23) into (17) and applying orthogonality relations. Denoting the area of the245

unit cell by A, we find246

(24) φ(x)/a =4i

A∑G

exp [i (k + G) · x]

ω2 − |k + G|2.247



The series solution (24) is conditionally convergent for |x| 6= 0 and diverges for x = 0,248

as would be expected from the asymptotic constraint (18). In appendix B, we derive249

the singular asymptotics of (24),250

(25) φ(x)/(ai) ∼ 2

π

[lnr

2+ γ]

+ σ(ω,k) + o(1) as r → 0,251

where the limit252

(26) σ(ω,k) =4

AlimR→∞

∑|G|<R

1

ω2 − |k + G|2+A2π

lnR

253

converges and is easy to compute. Substituting (25) into (18) furnishes the asymptotic254

dispersion relation governing the strongly perturbed modes as255

(27) σ(ω,k) +2

π

(lnε

2+ γ)

= 0.256

2.6. Square and hexagonal lattices. We now demonstrate the preceding re-257

sults by calculating the dispersion surfaces for square and hexagonal (or triangular)258

lattices. Consider first a square lattice,259

(28) a1 = 2ex, a2 = 2ey, b1 = πex, b2 = πey, A = 4.260

Following common practice, we plot in Fig. 1 (left panel) the dispersion surfaces for261

ε = 0.05 along the edges of the irreducible Brillouin zone, which in the present case262

is bounded by straight lines in reciprocal space connecting the symmetry points263

(29) Γ = 0, X = b1/2, M = (b1 + b2)/2.264

Red solid lines depict strongly perturbed eigenvalues, which are solutions of the dis-265

persion relation (27). Blue solid lines depict weakly perturbed modes. As shown266

in §§2.4, the eigenpairs of the latter are given to algebraic order by the degenerate267

empty-lattice eigenpairs; they are readily identified from (20) in conjunction with268

(28). As shown in §§2.4, if the degeneracy in the absence of a scatterer was D, the269

degeneracy is reduced to D − 1. The black dashed lines depict simple empty-lattice270

eigenpairs that are no longer part of the dispersion surfaces.271

Let us interpret the dispersion diagram in light of the distinction between weakly272

and strongly perturbed empty-lattice waves. To begin with, the notable zero- fre-273

quency band gap is not specific to a square lattice, but in fact exists for any lattice274

of the class considered in this study. This is because the zero-frequency light line,275

obtained by substituting G = 0 in (20), is always simple, hence according to the276

discussion in §§2.4 the perturbation from it must be appreciable. Next we note that277

in the empty-lattice case the first X point, (ω,k) = (π/2,X), is doubly degenerate.278

In the present case that eigenpair is therefore simple, which explains the partial gap279

opening above X, with the originally degenerate first band from X to M splitting280

into a weakly perturbed blue curve and a strongly perturbed red curve that joins the281

second strongly perturbed red curve from Γ to X. Also of interest are the high sym-282

metry crossing points at k = Γ,X, and M, where the original four-degeneracy in the283

empty-lattice case is reduced to three. Dispersion surfaces in the vicinity of degener-284

ate symmetry points are either conical or paraboloidal, while those in the vicinity of285

simple symmetry points are necessarily paraboloidal [16, 6]. Thus, by noting which286



Γ X M Γ

ω

0

0.5

1

1.5

2

2.5

3

3.5

Γ M K Γ0

1

2

3

4

M

K

Fig. 4

cellX

M

ΓΓ

Fig. 3

Brillouin

Fig. 1. Dispersion curves for square (left) and hexagonal (right) lattices of Dirichlet scatterersof effective radius ε = 0.05. Blue and red lines respectively depict weakly and strongly perturbedeigenpairs, the latter coinciding with degenerate empty-lattice eigenpairs. Dotted-black lines depictsimple empty-lattice eigenpairs. Inset show unit cells and first Brillouin zone (square — scale 3:10,hexagonal — 2:10).

surviving empty-lattice curves have zero or conversely nonzero slopes at symmetry287

points, it becomes possible to qualitatively sketch dispersion diagrams without any288

computation.289

Clearly, the above qualitative features of the dispersion surfaces are independent290

of the effective radius ε, at least as long as ε 1. Indeed, it is clear from (27)291

that ε merely affects the magnitude of the strongly perturbed modes. The appar-292

ent dominance of the blue weakly perturbed curves in Fig. 1 is misleading; within293

the irreducible Brillouin zone, rather than along its edges, degenerate empty-lattice294

eigenpairs are rare. This is clarified by the isofrequency contour shown in Fig. 3 for295

the frequency marked by the horizontal dashed green line in Fig. 1.296

Consider next the triangular (or hexagonal) lattice, with lattice base vectors297

(30) a1 = ex −√

3ey, a2 = ex +√

3ey,298

and reciprocal-lattice base vectors299

(31) b1 = π

(ex −

1√3ey

), b2 = π

(ex +

1√3ey

).300

The cell area is A = 2√

3. The irreducible Brillouin zone is formed by straight lines301

in reciprocal space connecting the symmetry points302

(32) Γ = 0, M = b1/2, K =|b1|√

3ex.303

Fig. 1 (right panel) shows the dispersion curves for ε = 0.05. Note that the dimension304

of the eigenmode space at the second Γ point is now 5, having been reduced by one305

from the empty-lattice degeneracy there, D = 6. Note also the crossing at the first K306

point, where the level of degeneracy has been reduced from D = 3 to 2. The latter307

implies that any additional constraint further reducing the degeneracy there will open308

an omnidirectional gap.309

3. Unit cells occupied by multiple scatterers.310



3.1. Effective eigenvalue problem. We here generalise to the case where in311

each cell there are P > 1 particles at x = xj , j = 1 . . . P . It is assumed that312

|xm−xn| ε for m 6= n; otherwise, the neighbouring scatterers share a common inner313

region, in which case we are back to a Bravais lattice with the particle multiplicity314

captured by an effective ε. The analysis of the present case closely follows that of §2.315

Thus, the outer potential (12) generalises to316

(33) φ = ψ(x) +

P∑p=1

apH(1)0 (ω|x− xp|),317

where ψ(x) is a regular function, and φ satisfies the Bloch condition (3). It follows318

from (33) that φ satisfies the governing equation319

(34) ∇2φ+ ω2φ = −4

i

P∑p=1

apδ(x− xp),320

where matching with the inner region of each scatterer provides P conditions,321

(35)

[2i

π

(ln

2

ωε− γ)− 1

]aj = ψ(xj) +

P∑p 6=j

apH(1)0 (ω|xp − xj |), j = 1 . . . P.322

From (33), the right hand side of (35) is the limit323

(36) limx→xj

[φ(x)− ajH(1)

0 (ω|x− xj |)],324

whereby, using (14), conditions (35) simplify to325

(37) limx→xj

[φ(x)− aj

2i

πln

ε

|x− xj |

]= 0, j = 1 . . . P.326

In the case of multiply occupied unit cells, empty-lattice waves survive as weakly327

perturbed modes mainly at high symmetry points, though in some incidental cases328

also along edges of the irreducible Brillouin zone. We next focus on the strongly329

perturbed modes, identifying and analysing in §§3.3 the weakly perturbed empty-330

lattice waves in the context of an important example.331

3.2. Dispersion relation. Generalising the Fourier-series solution (24) to sat-332

isfy (34) gives333

(38)A4iφ =

P∑p=1

ap exp [ik · (x− xp)]∑G

exp [iG · (x− xp)]

ω2 − |k + G|2.334

The singular asymptotics of the double sum in (38) is obtained by generalising the335

derivation in appendix §B, which gives [cf. (25)]336

337

(39) φ/i ∼[

2

π

(ln|x− xj |

2+ γ

)+ σ(ω,k)

]aj338

+4

A

P∑p 6=j

ap∑G

exp [i(G + k) · (xj − xp)]

ω2 − |k + G|2as x→ xj , j = 1 . . . P.339

340



Substituting (39) into (37), we find a set of P equations for the scattering coefficients341

ajPj=1:342

343

(40)

[2

π

(lnε

2+ γ)

+ σ(ω,k)

]aj344

+4

A

P∑p 6=j

ap∑G

exp [i(G + k) · (xj − xp)]

ω2 − |k + G|2= 0, j = 1 . . . P.345

346

The dispersion relation is obtained by requiring the determinant of the coefficient347

matrix to vanish. Clearly, for P = 1 we retrieve (27). For P = 2, the dispersion348

relation can be written as349

(41)

[2

π

(lnε

2+ γ)

+ σ(ω,k)

]2=

16

A2

∣∣∣∣∣∑G

exp [iG · (x1 − x2)]

ω2 − |k + G|2

∣∣∣∣∣2

.350

The sum on the right hand side is conditionally convergent, yet at a fast algebraic351

rate. The convergence deteriorates as the two scatterers approach; as already noted,352

however, the extreme case where the centroid-to-centroid separation is O(ε) is actually353

covered by the analysis of §2.354

3.3. Honeycomb lattice. A honeycomb lattice is equivalently a triangular lat-355

tice with two scatterers in each elementary cell, positioned so that closest neighbours356

are equidistant. Choosing the elementary cell as the parallelogram generated by the357

hexagonal base vectors a1 and a2 [cf. (30)], the two scatterers are positioned at358

(42) x1 =2

3a1 +

1

3a2, x2 =

1

3a1 +

2

3a2.359

The dispersion curves are plotted in Fig. 2 along the edges of the irreducible Brillouin360

zone as defined for the underlying triangular lattice in §§2.6. (The inset shows an361

alternative choice of the elementary cell.) Red curves depict strongly perturbed modes362

calculated from the asymptotic dispersion relation (41). Blue lines depict empty-363

lattice eigenpairs that remain as weakly perturbed empty-lattice waves; in contrast,364

the dotted and dash-dotted lines depict, respectively, simple and doubly degenerate365

empty-lattice eigenpairs that do not satisfy the asymptotic Bloch problem.366

Following the discussion in §§2.4, one might expect that for doubly occupied unit367

cells the degeneracy D of empty-lattice eigenpairs reduces to D− 2. Namely, that no368

simple or doubly degenerate pairs remain, triply degenerate pairs survive as simple369

eigenpairs, etc. This is consistent with the narrow gap opening up above the first370

band, the level of degeneracy at the second Γ point being reduced to 4, and the fact371

that most of the blue curves in the right panel of Fig. 1 are replaced by red ones.372

However, the above rule of thumb does not universally hold, as demonstrated by the373

remaining blue curves in Fig. 2, and by the third M point, which according to the374

above rule of thumb should have detached from the crossing of the light lines but375

remains there nevertheless.376

These and other features of the dispersion diagram are understood by examining377

the existence of empty lattice plane waves that vanish at both x1 and x2. Alluding378

to the general form (21) of empty-lattice wave solutions, the conditions for this are379

(43)

D∑j=1

Uj exp (iGj · x1) = 0,

D∑j=1

Uj exp (iGj · x2) = 0,380



Γ M K Γ

ω

0

1

2

3

4

5

cell

Brillouin

Γ K

M

Fig. 2. Same as Fig. 1 but for a honeycomb lattice (see §§3.3). Note that now there aredegenerate empty-lattice eigenpairs that do not remain as weakly perturbed modes; the latter aredepicted by black dash-dot lines.

where D and Gj = njb1+mjb2 are determined from (20) for any (ω,k) pair satisfying381

the empty-lattice dispersion relation. For D = 1, there are no nontrivial solutions of382

(43). Consistently with the rule of thumb suggested above, for D > 1, and if the two383

equations (43) are independent, the dimension of the space of solutions reduces to384

D − 2. If, however, they happen to be dependent, its dimension is D − 1.385

As an example, consider for example k traversing from Γ to M. Setting k = tb1/2,386

0 ≤ t ≤ 1, the empty-lattice dispersion relation (20) becomes387

(44)3ω2

π2= 4m2 + 2m(2n+ t) + (2n+ t)2.388

From (44) we see that the second Γ point is degenerate with D = 6, and (nj ,mj)6j=1389

= (0,±1), (±1, 0), (1,−1), (−1, 1). For 0 < t ≤ 1, the third light line is degenerate390

with D = 2 and (nj ,mj)2j=1 = (0,−1), (−1, 1). In the former case, (43) read391

U1ei23π + U2e−i

23π + U3ei

43π + U4e−i

43π + U5ei

23π + U6e−i

23π = 0,(45)392

U1ei43π + U2e−i

43π + U3ei

23π + U4e−i

23π + U5e−i

23π + U6ei

23π = 0,(46)393394

which are independent. Thus, as predicted by the rule of thumb, the second Γ point395

is four-degenerate. In contrast, in the latter case, (43) read396

U1e−i23π + U2e−i

23π = 0,(47)397

U1e−i43π + U2ei

23π = 0,(48)398399

which are clearly dependent, explaining the lower blue curve in Fig. 2.400

4. Scattering by a finite collection of scatterers. There is a close connection401

between the Bloch dispersion problem, which is defined on a unit cell of an infinite402

lattice, and the scattering properties of a truncated finite variant of the same lattice.403

In order to demonstrate this in the context of the dispersion surfaces calculated in the404



-20 0 20-30

-20

-10

0

10

20

30

-0.2

-0.1

0

0.1

0.2

-π/2 π/2

-π/2

π/2

Γ X

M

Fig. 3. Dynamic anisotropy. Scattering at ω = 1.82 from a finite square lattice of Dirichletinclusions of radius ε = 0.05, subjected to a point-source, acting as the incident field, at the origin

ui = H(1)0 (ωr). Left panel: Isofrequency contour of the corresponding infinite lattice (Blue points

— weakly perturbed eigenpairs; Dashed lines — empty-lattice eigenpairs). Right panel: Re[u].

preceding sections, we shall employ a scattering formulation known as Foldy’s method405

[11, 19], which here readily follows from the inner-outer asymptotic procedure of §2.2.406

To be specific, we consider scattering from N Dirichlet scatterers of effective radius407

ε, positioned at x = xlNl=1, and subjected to an incident field ui(x). The field u408

satisfies the Helmholtz equation (1), Dirichlet condition u = 0 on the boundary of409

each scatterer, and a radiation condition at large distances on the scattering field410

u − ui. In the limit ε 1, the “outer” solution of the scattering problem is readily411

seen to be [cf. (12) and (16)]412

(49) u ∼ ui(x) +

N∑l=1

alH(1)0 (ωrl) + a.e.(ε),413

where rl = |x−xl| and the coefficients al are determined from N matching conditions,414

(50) al +πi

2

1

ln 2ωε − γ + πi

2

ui(xl) +∑p 6=l

apH(1)0 (ω|xl − xp|)

= 0, l = 1 . . . N.415

Numerically solving the above linear system is fairly straightforward, even for a large416

number of scatterers. Once the coefficients alNl=1 have been determined, the field417

is obtained from (49). The above approximate scheme was derived by Foldy, who418

conjectured isotropic scattering and obtained the “scattering coefficient” — essen-419

tially the ε-dependent prefactor in (50) — in a semi-heuristic manner. The validity420

of Foldy’s assumption and scattering coefficient was later confirmed by reducing, in421

the long-wavelength limit, the exact solution for an isolated scatterer [25], and also422

by asymptotic matching [24]. Foldy’s approach is applicable, with appropriate mod-423

ifications, not only to the Helmholtz–Dirichlet problem, but whenever the material424

properties, and the type of waves considered, are such that the reaction of each scat-425

terer is dominantly isotropic. For the Biharmonic–Dirichlet problem mentioned in the426

introduction Foldy’s method is particularly intuitive [10], since the equations analo-427

gous to (50) are obtained by applying regular Dirichlet point constraints.428



Fig. 4. Same as in Fig. 3 but for a hexagonal lattice at ω = 1.88.

Here we employ Foldy’s method to demonstrate the strong dynamic anisotropy429

implied by the dispersion relations calculated in §2.6. To this end, we solved (50) for430

finite square and hexagonal lattices, with a point source replacing a centrally located431

scatterer. In Figs. 3 and 4, the right panels depict the total fields Re[u], calculated by432

solving (50) with a point source at the origin, ui = H(1)0 (ωr), the frequencies being433

ω = 1.82 and ω = 1.88 in the square (Fig. 3) hexagonal cases (Fig. 4). The left panels434

show in reciprocal lattice space the corresponding isofrequency contours, calculated435

from the infinite-lattice dispersion relation (27). The latter complement the dispersion436

diagrams in Fig. 1, and are particularly convenient for interpreting the behaviour of437

finite lattices. At the selected frequencies, the dispersion contours are approximately438

straight lines connecting the crossing points of the empty-lattice light circles. The439

group velocities, which give the permissible directions for energy propagation [32], are440

normal to the isofrequency contours, hence the highly directional response.441

5. Discussion. In this paper we studied wave propagation in lattices of subwave-442

length Dirichlet scatterers. To be precise, the asymptotic approach we use assumes443

that ε 1 with ω = O(1), namely it is implicit that wavelengths are both com-444

parable to the periodicity and yet large compared to scatterer dimensions. Hence,445

whilst the analysis is robust at lower frequencies, it inevitably breaks down in the446

high frequency regime, ω = O(1/ε), when the wavelength is comparable to scatterer447

size. Provided we are away from that regime the scheme we present is very versatile448

and, to summarise, our method entails: (i) effectively replacing the finite scatterers449

by singular point constraints by applying the method of matched asymptotic expan-450

sions; (ii) identifying the space of empty-lattice waves that are weakly perturbed;451

and (iii) deriving a dispersion relation for the strongly perturbed modes by extract-452

ing the singular “inner” asymptotics of a generalised Fourier-series solution of the453

“outer” eigenvalue problem. Using this method, we generated dispersion curves for454

the fundamental square, hexagonal, and honeycomb lattices, pointing out the essen-455

tial role played by degeneracy and the existence of weakly perturbed modes. We456

also demonstrated using Foldy’s method the strong dynamical anisotropy implied by457

those dispersion curves. Our method can be readily employed to study more involved458

lattices of small Dirichlet scatterers. In particular, the topological properties of lat-459

tices breaking mirror symmetry are currently under intensive investigation [21]. In460



the present formulation this can be easily achieved by asymmetrically positioning two461

or more scatterers in a unit cell, or, alternatively, assigning symmetrically positioned462

scatterers different effective radii.463

The two dimensional Dirichlet–Helmholtz problem considered herein, which has464

realisations in electromagnetics and acoustics, serves to demonstrate that small scat-465

terers are not necessarily weak scatterers (though they are e.g. in the 3D variant of466

this problem and for Neumann scatterers in both 2D and 3D). We already mentioned467

in the introduction the Biharmonic pinned-plate problem, and other important ex-468

amples include wire media [34], Faraday cages [25, 5], lattices of high-contrast rods469

[33], and plasmonic nanoparticle waveguides and metasurfaces [22]. In addition, ongo-470

ing research into photonic and mechanical analogies of topological insulators and the471

integer-quantum-hall effect has stirred interest in media breaking time-reversal sym-472

metry. This has led to novel metamaterial designs incorporating small-scale resonant473

mechanical components [37] and high-contrast opto-magnetic rods [39]. We expect474

that the type of analysis carried out herein can be adopted to study all of these ex-475

amples, and in general media consisting of wavelength-scale lattices built out of small476

strongly scattering elements. For any given example this would entail repeating the477

inner-outer asymptotics, which enable replacing the small scale elements by point478

constraints; this could be more involved than in the present case [18], especially if the479

strong scattering results from a subwavelength resonance. Asymptotic coarse-grained480

descriptions of such media would not only be technically advantageous, but may also481

offer new insight by highlighting the existence of both weakly and strongly perturbed482

modes.483

It is worth noting that the Foldy methodology, often utilised in studies of random484

scattering [26], appears somewhat under-employed in the periodic photonic, phononic485

and platonic literature. As demonstrated here it provides an ideal setting, although486

limited to small scatterers, for investigating many of the phenomena of interest (such487

as for the dynamic anisotropy shown in Figs. 3 & 4) and their dependence upon488

lattice geometry in an algebraic setting; the solutions are accessed far more rapidly489

than, say, the more usual finite element approaches [30], popularised by COMSOL490

and other commercial packages, commonly used that require refined meshes for very491

small scatterers.492

Appendix A. Solvability condition. Consider the effective eigenvalue prob-493

lem formulated in §§2.3 for the outer field φ, in the case where (ω,k) satisfy the494

empty-lattice dispersion relation (20). The corresponding space of empty-lattice plane495

waves is given by (21), and we choose an arbitrary plane wave u from that space. The496

complex conjugate of the latter, u∗, satisfies497

(51) ∇2u∗ + ω2u∗ = 0498

and the Bloch condition (3) at frequency ω and Bloch wave vector −k. Subtracting499

(51) multiplied by φ from (17) multiplied by u∗, followed by an integration over the500

unit cell, yields501

(52)

∫∫ (u∗∇2φ− φ∇2u∗

)dA = 4iau∗(0).502

Using Green’s theorem, and the facts that φ exp(−ik · x) and u∗ exp(ik · x) both503

possess the periodicity of the lattice, the left-hand side can be shown to vanish. We504



therefore find, upon substituting (21),505

(53) a

D∑j=1

Uj = 0.506

Since we may choose the Uj ’s arbitrarily, it must be the case that a = 0.507

Hence, to conclude, when (ω,k) satisfy the empty-lattice dispersion relation (20),508

only φ eigensolutions with a = 0 — necessarily empty-lattice plane waves — are509

permitted. Furthermore, it then follows from (18) that only the empty-lattice plane510

waves that vanish at the position of the scatterer, as characterised in §§2.4, remain511

unperturbed to algebraic order in ε.512

Appendix B. Double-sum asymptotics. We here derive the asymptotics513

of the outer Fourier-series solution (24) in the limit where r = |x| → 0. As already514

noted, substituting x = 0 yields a diverging sum and hence this limit is singular. To515

overcome this, we separately sum terms corresponding to reciprocal lattice vectors G516

whose magnitudes are smaller and larger than an arbitrary large radius R satisfying517

1 R 1/r. This gives, after obvious approximations,518

(54)A

4iaφ(x) ∼

∑|G|<R

1

ω2 − |k + G|2−∑|G|>R

exp (iG · x)

|G|2+ o(1) as r → 0.519

Both sums in (54) diverge logarithmically as R→∞ but the singularity must cancel520

out. Writing G · x = Gr cos θ, the second sum in (54) becomes521

(55)∑|G|>R

exp (iG · x)

|G|2=∑G>R

exp (iGr cos θ)

G2,522

which, in the considered limit, can be approximated by an integral,523

(56)∑|G|>R

exp (iG · x)

|G|2∼ 1

AG

∫ ∞R

∫ 2π

0

exp (iGr cos θ)

GdθdG+ o(1),524

where AG = 4π2/A denotes the area of a unit cell in reciprocal space. Integrating525

with respect to θ gives after a change of variables ξ = G/R,526

(57) ∼ 2π

AG

∫ ∞1

ξ−1J0(rRξ) dξ + o(1).527

Standard asymptotic evaluation of the latter integral for rR 1 gives528

(58)∑|G|>R

exp (iG · x)

|G|2∼ 2π

AG

(ln

2

rR− γ)

+ o(1).529

Substituting into (54), and defining the limit (26), we find the asymptotic result (25)530

stated in the text. The derivation in the case of multiply occupied cells follows along531

the same lines.532

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