BLOCH WAVES IN AN ARBITRARY TWO-DIMENSIONAL 1 LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS 2 ORY SCHNITZER * AND RICHARD V. CRASTER † 3 Abstract. We study waves governed by the planar Helmholtz equation, propagating in an infi- 4 nite lattice of subwavelength Dirichlet scatterers, the periodicity being comparable to the wavelength. 5 Applying the method of matched asymptotic expansions, the scatterers are effectively replaced by 6 asymptotic point constraints. The resulting coarse-grained Bloch-wave dispersion problem is solved 7 by a generalised Fourier series, whose singular asymptotics in the vicinities of scatterers yield the 8 dispersion relation governing modes that are strongly perturbed from plane-wave solutions existing 9 in the absence of the scatterers; there are also empty-lattice waves that are only weakly perturbed. 10 Characterising the latter is useful in interpreting and potentially designing the dispersion diagrams 11 of such lattices. The method presented, that simplifies and expands on Krynkin & McIver [Waves 12 Random Complex, 19 347 2009], could be applied in the future to study more sophisticated designs 13 entailing resonant subwavelength elements distributed over a lattice with periodicity on the order of 14 the operating wavelength. 15 Key words. Bloch waves, Periodic media, Singular perturbations 16 AMS subject classifications. 34D15, 35P20, 35B27, 78M35 17 1. Introduction. There is immense current interest in wave phenomena in ar- 18 tificial periodic media. In subwavelength metamaterials (SWM), that are constructed 19 using tiny resonant elements, the periodicity is small compared with the operating 20 wavelength [35]. SWM mimic natural materials whose macroscopic properties are 21 endowed by an underlying atomic structure. Ingenious designs — introduced in elec- 22 tromagnetics but since adapted to acoustics, elasticity and seismology — have enabled 23 unusual effective properties and capabilities unfamiliar in nature, including negative 24 refractive index and cloaking. Photonic (similarly platonic and phononic) crystals 25 constitute a separate class of artificial materials, in which operating wavelengths are 26 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 of 29 photonic crystals were initially focused on the existence of complete photonic band 30 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 demonstrated 32 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 the 37 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 of 39 waves propagating through the crystal. When this is indeed the case, the effect of the 40 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 of 43 the scatterer dimensions. An elementary example is the propagation of flexural waves 44 in a periodically pinned plate, where the time-harmonic displacement field is governed 45 * Department of Mathematics, Imperial College London, London SW7 2AZ, UK 1 This manuscript is for review purposes only.
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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
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
This manuscript is for review purposes only.
8 O. SCHNITZER AND R. V. CRASTER
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
This manuscript is for review purposes only.
LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS 9
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
This manuscript is for review purposes only.
10 O. SCHNITZER AND R. V. CRASTER
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
This manuscript is for review purposes only.
LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS 11
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
This manuscript is for review purposes only.
12 O. SCHNITZER AND R. V. CRASTER
-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
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
This manuscript is for review purposes only.
LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS 13
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
This manuscript is for review purposes only.
14 O. SCHNITZER AND R. V. CRASTER
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
This manuscript is for review purposes only.
LATTICE OF SUBWAVELENGTH DIRICHLET SCATTERERS 15
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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