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Transcript of Cloaking and imaging effects in plasmonic checkerboards of negative and and dielectric photonic...
Invited Paper
Cloaking and imaging effects in plasmonic checkerboards
of negative e and m and dielectric photonic
crystal checkerboards
S. Guenneau a,*, S. Anantha Ramakrishna b, S. Enoch a,Sangeeta Chakrabarti b, G. Tayeb a, B. Gralak a
a Institut Fresnel, UMR CNRS 6133, University of Aix-Marseille, 13390 Marseille Cedex 20, Franceb Department of Physics, Indian Institute of Technology, Kanpur 208016, India
Received 5 February 2007; received in revised form 11 July 2007; accepted 14 July 2007
Available online 11 August 2007
www.elsevier.com/locate/photonics
Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–72
Abstract
Negative refractive index materials are known to be able to support a host of surface plasmon states for both polarizations of
light. This makes possible unique effects such as a perfect lens. Checkerboards consisting of alternating cells of positive and
negative refractive index represent a very singular situation in which the density of modes diverges at the corners. This raises the
question as to whether such effects will still be observed in a real dissipative system of finite size. We have considered several
aspects of such structures including these and symmetry aspects (rectangular against triangular checkerboards). We have also
studied silver checkerboards whose transverse extent is finite. Negative refractive index checkerboards bring new electromagnetic
paradigms both through the intriguing possibilities they appear to offer, and the challenges they present to our understanding of the
diffraction process. Most intriguing of all is the possibility of a triangular checkerboard lens whose resolution is limited not by
wavelength, but only by the losses in the constituent materials, while a ray picture suggests it behaves as a perfect mirror. The
resolution of this lens increases without limit as the losses tend to zero as shown by the generalized lens theorem. We finally show
that light confinement can be achieved to certain extent using dielectric triangular photonic crystal (PC) checkerboards displaying
the all-angle-negative-refraction (AANR) within the Bragg regime in p polarization. Effectively even a single rectangular or
triangular PC can act as an open resonator that confines light in its neighborhood. This cloaking effect has been previously observed
in PC slabs displaying the AANR effect. We show that the cloaking is enhanced for three triangular PC wedges sharing a vertex and
further improved for 12 triangular PC wedges arranged in a checkerboard fashion.
# 2007 Elsevier B.V. All rights reserved.
Keywords: Negative refraction; Perfect lenses; Plasmons; Photonic crystals
1. Prologue: perfect lenses and checkerboards
Negatively refracting media have gained popularity
over the past few years due to their non-intuitive physics
as well as their scope for application [1,2]. A material is
considered to have a negative refractive index only if the
* Corresponding author.
E-mail address: [email protected] (S. Guenneau).
1569-4410/$ – see front matter # 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.photonics.2007.07.015
real parts of both e and m are simultaneously negative at a
given frequency v. This was first pointed out by Veselago
in 1968 [3]. This will occur only in a very narrow range of
frequencies and in real life, NRMs will be necessarily
dissipative and dispersive. One such device is the ‘perfect
lens’ or superlens first proposed by Pendry [4]. In the
simplest case, this is just a slab of NRM (with e ¼ �1,
m ¼ �1) which involves both the far-field as well as
the near-field modes in the imaging process. Conse-
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–7264
quently, the resolution of the image is not restricted by
the conventional diffraction limit for an ideal, lossless
NRM. According to the conventional diffraction limit,
the maximum image resolution cannot exceed
D ¼ ð2pcÞ=v ¼ l. This is true no matter how perfect
the lens and however large its aperture. However, in the
perfect lens, evanescent modes (with imaginary wave-
vector kz) undergo amplification via the resonant surface
plasmon modes on the surface of the NRM while the
propagating modes (with real wavevector kz) undergo
phase reversal. Thus, all waves are collected and ‘perfect’
focussing results. This in turn leads to a generalized lens
theorem [5] for more complex NRM satisfying certain
symmetry such as checkerboards [6].
We first meet the paradigm of a perfect checkerboard
lens which should not let any ray go through. After
solving this paradox thanks to the generalized lens
theorem [6], we investigate numerically light confine-
ment in chess boards thanks to the finite element method.
We then build up quickly upon the finite element algo-
rithm towork out some of the counter-intuitive plasmonic
properties of NRM checkerboards: we meet puzzling
physics such that infinite degenerated modes at corners
leading to an infinite local density of states (LDOS). This
only applies in the limit when the material within the
NRM cells satisfies lim d! 0eðvþ idÞ ¼ lim d! 0mðvþidÞ ¼ �1 which is maybe the less unlikely event.
Presence of a finite amount of dissipation, however
small, could be expected to wipe out all of the enhanced
plasmonic properties of the checkerboards. Never-
theless, we numerically show using the transfer matrix
method that plasmonic properties of our checkerboards
are preserved to certain extent when dissipation is
present and also when they are of finite extent, although
the LDOS becomes now finite. We further show that
light confinement increases with the size of the finite
dissipative checkerboards.
We finally propose to use photonic crystal checker-
boards of some dielectric constant at hand which
display some negative refraction effect in p polarization
and enable light confinement to certain extent [7]. Such
checkerboards, built by putting next to each other
perfect lenses (also known as Alice’s mirrors [8]) in a
periodic fashion, may open new vistas for light
localization and enhanced Raman scattering.
2. A generalized lens theorem for
complementary media
The original ‘perfect lens’ presupposed a slab of
material with e ¼ �1 and m ¼ �1. However, focussing
will occur under more general conditions [5,6]. Any
system for which
e1 ¼ þeðx; yÞ; m1 ¼ þmðx; yÞ; �d< z< 0;
e2 ¼ �eðx; yÞ; m2 ¼ �mðx; yÞ; 0< z< d;(1)
will show identical focussing. Focussing will always
occur irrespective of the medium in which the lens is
embedded. This is true for any medium which is mirror
antisymmetric about a plane.
Thus, in general, a negatively refracting medium is
complementary to an equal thickness of vacuum and
optically ‘cancels’ its presence. The compensating
action extends to both the evanescent as well as the
propagating modes. Due to this, there is perfect
transmission and the phase change of the transmitted
wave is zero. In such systems, it is possible to excite
surface plasmon modes at all spatial frequencies.
A corner made of a NRM shares the perfect property
of other negatively refracting lenses. This has been first
shown in [5] using the technique of coordinate
transformation. A pair of negatively refracting corners
is capable of bending light in a loop and forming a series
of images such that as n! � 1, light circulates within
the loop forever. In the electrostatic limit, all the surface
plasmon modes are degenerate at vp=ffiffiffi
2p
. At this
frequency, the density of states diverges and e ¼ �1, if
we assume a simple Drude form for the plasma,
e ¼ 1� v2p=vðvþ ihÞ, where h is a small positive
parameter.
2.1. Triangular and square corner lenses
In Reverend Abbott’s Flatland, a romance of many
dimensions, the social class of characters transpires from
their shape. The higher their symmetry, the upper their
social background e.g. a square is higher up than a
triangle and so on. It turns out that similar rules may apply
for perfect lenses in terms of enhancement of imaging
and cloaking effects: a square of NRM may be better than
a slab lens of NRM in this respect, but an equilateral
triangle of NRM is the noble one! We illustrate this aspect
with finite element computations reported on Fig. 1. We
can see that two squares of NRM can be put together in a
checkerboard fashion so that they form a so-called
perfect corner reflector. The 2D plot of Fig. 1 compares
indeed very well with results reported in [9] for the
analogous case of two intersecting planes separating four
regions of space alternating complementary heteroge-
neous anisotropic dissipative media. Spatial oscillations
of plasmons at interfaces were shown to depend in a
logarithmic way on the absorption [9]. We also see that
three triangles of NRM can form an open resonator [10].
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–72 65
Fig. 1. Resonant modes of square and triangular non dissipative NRM checkerboards (m ¼ e ¼ �1) with cells of side length d ¼ 0:1. Upper left:
l ¼ 4:732þ i0:816 for two squares of NRM; Upper right: l ¼ 4:844þ i0:795 for three equilateral triangles of NRM; Bottom left: l ¼ 4:798þi0:007 for 32 squares of NRM; Bottom right: l ¼ 4:973þ i0:006 for 27 triangles of NRM.
2.2. Building up checkerboards
Such ‘black corners’ possess quite counter-intuitive
properties such that an infinite local density of state
(LDOS) at the corner in the limit of no dissipation when
the structure covers the whole of the space [9]. One way
to do this is either to repeat the structure periodically
[6], and by doing so paving the plane in accordance with
the crystallographic rules as is illustrated in Fig. 1, or to
let the structure grow infinitely. In the latter case, it was
shown in [9] that the LDOS at the corner becomes
infinite in the limit of zero absorption, due to infinite
degeneracy of the resonance mode. In the former case,
the imaginary part of the resonant wavelength decreases
when the number of cells increases, so that the LDOS at
every corner will become infinite in the limit of infinite
non dissipative checkerboards. We also remark that in
[9], the corner mode was a bound state whenever some
dissipation was introduced in the system. For finite
checkerboards, this is certainly not true: only leaky
modes are present in the system, being it with or without
dissipation. Photon resonances may change from leaky
state to periodic Bloch modes only in the limit of infinite
dissipative checkerboards.
In the next section, we adopt the first point of view
(periodic checkerboards).
2.3. Plasmonic resonances in infinite
checkerboards
Some intriguing physics is at work in infinite non-
absorptive checkerboard structures, for which the
imaginary part of the resonances should vanish. It
has been shown in [6] that a source placed in one cell of
a rectangular checkerboard produces an image in every
other cell. This straightforwardly applies to triangular
checkerboards as well, since they meet the property of
mirror antisymmetry of the generalized lens theorem.
These checkerboard structures retain their image
transfer properties irrespective of whether they consist
of homogeneous isotropic media or inhomogeneous
anisotropic media, as long as they exhibit mirror
antisymmetry and adjacent cells are complementary to
each other.
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–7266
Fig. 2. A pair of complementary checkerboard layers with e ¼ �1 and m ¼ �1. Positive and negative refractive index are schematically depicted by
white and colored regions. Left: a ray can be transmitted with no change in direction or retro-reflected for a rectangular checkerboard; Right: all rays
incident on the checkerboard are reflected for a triangular checkerboard.
All modes are degenerate at a given frequency
vp=ffiffiffi
2p
and the density of modes is infinite. These
systems are extremely singular and contain a very large
number of corners between positive and negative cells
where the density of surface plasmon states diverges.
Dissipation affects sub-wavelength imaging badly, and
the divergence in the local density of states can only
worsen the situation. It is due to this fact that the effect
of dissipation on sub-wavelength resolution becomes an
important issue, which we shall investigate after the
following electromagnetic paradox.
2.4. Contradictions between the ray and the wave
pictures
Let us consider a finite checkerboard of NRM
(n ¼ �1) with rectangular cells. From Fig. 2, it appears
that a ray can either be transmitted or retro-reflected. in
Fig. 3. Left panel: p-polarized plane wave of wavelength l ¼ 0:5 ¼ 5d (fiv
lens of dissipative NRM (e ¼ �1þ i4� 10�2, m ¼ 1); Right panel: line sour
subwavelength image located half a cell underneath the checkerboard. Note th
in the limit of zero absorption for a line source located on the upper boun
a rectangular checkerboard. For the triangular checker-
board shown in Fig. 2, the ray picture indicates the
possibility of reflection. However, in both situations,
perfect transmission is found to occur. These contra-
dictions with the ray analysis arise due to the localized
fields at the corners which the rays cannot describe [1].
This extraordinary transmission occurs via the excita-
tion of the surface plasmon modes of the system.
Nevertheless, the type of plasmonic guidance involved
here via the interfaces between positive and negative
index media differs substantially from the extraordin-
ary transmission experimentally demonstrated in
[11,12]. As illustrated by the paradox of the ray picture
showing no transmission and the complementary
theorem showing perfect lensing (the optical path
cancels), it is imperative to investigate numerically
finite structures of NRM which are of the checkerboard
type. This is why we also numerically checked the
e times as large as the cells) incident from above on the checkerboard
ce of same wavelength located half a cell above the checkerboard with a
at the generalized lens theorem states that a perfect image would occur
dary of the checkerboard (with the image on its lower boundary.)
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–72 67
statement of the generalized lens theorem through
finite element computations, as reported on Fig. 3.
We numerically show in Fig. 3, the response of two
small, slightly dissipative checkerboard systems
consisting of cells alternating air and NRM
(e ¼ �1þ i0:04 and m ¼ þ1), for a line source placed
one half of a cell above the checkerboard for the p-
polarization. We note the flip of the sign for the field
when it crosses the checkerboard, which we attribute to
m being non-negative. On left panel of Fig. 3, the
checkerboard is shown to act along the lines of the
generalized lens theorem, even though it is of finite
horizontal extent. Noteworthily, the dissipation in
NRM is almost comparable to that of silver for optical
frequencies. Hence, the field enhancement is weakened
on interfaces (which is why PML need not be taken
far away), but subwavelength imaging is displayed.
We numerically checked that the finite element
algorithm converges even in the limit of non-
dissipative NRM checkerboards for which the Maxwell
operator loses ellipticity, which can be rigorously
established [13].
3. Transfer matrix versus finite elements
Our next move consists in getting a deeper insight
into the electromagnetic response of NRM checker-
boards, thanks to on one hand the transfer matrix
method (which is particularly well suited for the
analysis of rectangular NRM checkerboards) and on the
other hand the finite element method (which is more
versatile but less analytical) (Fig. 4).
Fig. 4. Ray analysis. Upper left panel: one source gives rise to two images th
along closed trajectories around the corner in the triangular perfect corner, bu
checkerboard are trapped (closed trajectories around corners).
3.1. Transfer matrix analysis of finite
checkerboards
We have studied 2-D checkerboards oriented on the
x–z plane as shown in Fig. 2. We have assumed
invariance along the y-axis. Numerical solutions were
carried out using the PHOTON code based on the
transfer matrix method [14]. We used a free-space
wavelength l ¼ 4:53 mm, periodic boundary condi-
tions along x and a typical layer thickness of
0.45 mm = l=10. The period along x is irrelevant to
the effects discussed.
It should be noted that the numerical results are
sensitive to the degree of discretization used [15]. On
Fig. 5, we show the transmission and reflection
coefficients obtained for a dissipative checkerboard
whose NRM cells are defined by e ¼ �1þ i4� 10�2
and m ¼ 1. To ensure convergence of the numerical
scheme, we took a discretization of 262 points along x.
There is a resonance around kx� 1:1k0. Away from this
resonance, the transmittivity is nearly unity and the
reflectivity is asymptotically zero.
A detailed discussion of the genuine numerical
problems occurring in this analysis and of the
robustness of transmission properties of such checker-
board lenses against dissipation can be found in [15].
3.2. Finite element analysis of dissipative
triangular and rectangular checkerboard systems
We now examine the response of a NRM checker-
boards which are finite in both x and y directions, when
rough a triangular perfect lens; Lower left panel: some rays are trapped
t some rays diverge; Right panel: all rays emitted by a source within the
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–7268
Fig. 5. The transmission and reflection coefficients for a pair of
dissipative checkerboard layers with e ¼ þ1 and e ¼ �1þ i4�10�2 and m ¼ þ1 for p-polarized waves. For subwavelength
kx� 1:1k0, a resonant excitation of surface plasmon modes occurs.
their cells exhibit either a four-fold (squares) or a six-
fold (regular triangles) geometry. Fort this, we solve the
Maxwell system using finite edge elements (also known
as Whitney forms) which naturally fulfill transmission
conditions for the tangential components of the
electromagnetic field at interfaces between positive
and negative index media (hence exhibiting two anti-
parallel wavevectors at both sides of such interfaces).
Also, outgoing wave conditions ensuring well-posed-
ness of the problem (existence and uniqueness of the
solution) are enforced through implementation of
Berenger’s perfectly matched layers [16]. These ones
provide a reflectionless interface between the region of
interest containing the checkerboard and the PML at all
incident angles.
We both look at the spectral problem (looking for
resonant frequencies of the checkerboard) and the
scattering problem (when a harmonic line source
radiates some field in the neighborhood of the
checkerboard). On left panel of Fig. 3, it is obvious
that the electromagnetic response of the checkerboard
contradicts the ray picture (the plane wave goes through
the checkerboard without reflection, with an attenuation
due to dissipation in NRM).
We observed in these calculations that the eigen-
functions are large in magnitude, always located at the
intersecting corners and very small in the bulk of the
material. The working wavelength is typically larger
than the cell’s size, so that these corner modes are
exponentially decaying away from the corners. We
observe some surface plasmon excitations running
along the interfaces separating complementary media.
Since there is a large degeneracy of the modes, the leaky
modes could be localized at any corner. From Fig. 6, we
can see that the ratio of the real part to the imaginary
part of some representative eigenfrequencies increases
with increasing size of the checkerboard (i.e. the
leakage reduces). The rate of increase is faster for the
case of triangular structure. From Fig. 1, we see that the
amplitude of the cornermode increases with the size of
the system (actually it goes up faster for triangular than
for square lattices). Also, it is worthwhile noting that the
corner mode for the triangular perfect corner exhibit a
C6v symmetry (structure invariant through a rotation
2p=6) while its counterpart for the square perfect corner
is of C4v symmetry. We preserve the symmetry of the
structure for larger checkerboards but degeneracy
means symmetry of modes is broken, which is a
common feature in waveguide theory (see [17] for mode
classes and degeneracies).
4. Dielectric photonic crystal chess boards
Last, we explore light confinement through negative
refraction in finite checkerboards when we replace the
NRM by a photonic crystal [18]. Notomi was the first
physicist to suggest light trapping using two square
Photonic Crystals displaying the AANR which shared
the same vertex [25]. Here, we consider a triangular
array of cylindrical fibers of circular cross-section and
high-refractive index (n ¼ 4) embedded within an air
matrix. We meet the criteria for all-angle negative
refraction in p polarization (the magnetic field is
parallel to the axis of cylinders) for a pitch array a and
fibers’ radii r ¼ 0:45a at l ¼ 3:66a (see Fig. 7). The
essential condition for the all angle negative refraction
(AANR) effect is that the equifrequency surfaces (EFS)
should become convex everywhere about some point in
the reciprocal space, and the size of this EFS should
shrink with increasing frequency. Further the EFS
should be larger than the free space dispersion surface
and the frequency should be within the first Bragg zone
[19–21]. Such a theoretical prediction of AANR was
experimentally confirmed around 13.5 GHz and
enabled to image a line source at that frequency
through a PC slab lens [22].
4.1. Cloaking light with one PC triangle
The influence of resonance poles on the image process
in the checkerboard is strongest for Bloch modes of the
dispersion diagram with the smallest group velocity. For
the corresponding finite PC checkerboard, the phenom-
enon attenuates due to leakage (complex poles) and
impedance mismatch with that of air. The difficulty to
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–72 69
Fig. 6. Field radiated by a line source of wavelength l ¼ 0:05 ¼ d=2 in the presence of triangles of NRM with e ¼ m ¼ �1þ i4� 10�4 (perfect
lens in upper left corner), three triangles of NRM sharing a vertex (perfect corner in upper right corner) and 12 triangles of NRM (checkerboards in
lower left and lower right corner). The magnitude of the field increases with the number of triangles. Note that in the upper left corner, we depict the
modulus of the field to make the source and images more visible.
find out adequate resonance poles (minimizing the
leakage of the mode) lies in the continuous nature of the
spectrum: there is an infinity of admissible complex
frequencies, but very few good candidates for the corner
mode of the PC checkerboard structure. The resonance
poles were found thanks to an algorithm discussed in
Fig. 7. Dispersion diagram for a triangular array of circular rods r ¼0:45a where a if the period. Horizontal axis: the Bloch vector
describes the first Brillouin zone GJX. Vertical axis: normalized
frequencies a=l, where a is the wavelength. Inset: isofrequency
contour at the intersection of the light line with the optical band
showing negative group velocity at l ¼ 3:66a.
[23](note that AANR occurs only around a given
frequency for infinite PC).
We start by looking at resonances of a single PC
triangle, since this avoids the degeneracy of the modes
induced by the crystalline symmetry of the checker-
board, on top of the existing degeneracy of the modes
induced by the symmetry of the lattice along which rods
are arranged in each single cylinder. We find that the
ratio between the real and imaginary part of the
eigenfrequency decreases when the size of the number
of rods increases, as shown on Fig. 8. This suggests that
light confinement improves with the size of the PC
triangle. On Fig. 8, one can also see two images building
up in a symmetric fashion when a line source radiates in
the neighborhood of the PC triangle. We numerically
checked that the focussing effect is enhanced with the
size of the PC triangle.
4.2. Improving light confinement with 3 and 12 PC
triangles
For a line source operating on resonance close to the
intersection of surface band with light cone, surface states
at interfaces between PC and air dominate: a large field
amplitude can be seen on Fig. 9. The finite reflectivity
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–7270
Fig. 8. Upper panel: scattering by a line source located such that it preserves the symmetry of the regular triangle (left) and region of the complex
plane with two resonances associated with modes of the middle panel (PC triangles with 20 rods on their side). Middle panel: resonance modes for
one PC triangle with 20 rods on its side arranged along a triangular array of pitch a at wavelengths l ¼ 3:7052aþ i0:1049a (left) and l ¼3:7016aþ i0:0075a (right); Lower panel: same for resonance modes for a PC triangle with 30 rods on its side at wavelengths l ¼ 3:6903aþi0:0602a (left) and l ¼ 3:6802aþ i0:0025a (right).
observed at the PC terminations is due to impedance
mismatch between PC and air, since the effective
refractive index of PC only approaches �1 asymptoti-
cally for large enough number of rods. This effect may be
reduced by changing the rod’s shapes on the interfaces
between PC and air (as proposed in [10]) or otherwise by
increasing the size of the structure. The amplitude of the
field in the middle corner becomes larger when a line
source excites checkerboard resonances on Fig. 9. The
eigenfield looks clearly more confined when an addi-
tional row of triangles is added.
5. Epilogue
In conclusion, we have shown that, within the
accuracy of the numerical calculations presented here,
cloaking and focussing effects in checkerboards of
NRM are reasonably robust against dissipation. This is
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–72 71
Fig. 9. Upper panel: resonance modes for three PC triangles of pitch array a at wavelengths l ¼ 3:6797aþ i0:21139a (left) and field radiated by a
harmonic line source located at point (0, 5.78) with wavelength l ¼ 3:6798a (right); Lower panel: resonance modes for 12 triangles at wavelengths
l ¼ 3:68176aþ i0:007727a (left) and field radiated by harmonic line source located at point ð0; 4Þ with wavelength l ¼ 3:66a (right).
surprising in the face of the expectation that dissipation
in such singularly degenerate systems can wipe out all
such effects. We have also shown some numerical
evidence of interesting focussing effects in weakly
dissipative checkerboard systems of small extent, using
both transfer matrix and finite element methods, we
checked with the FEM package that a plane wave
merely suffers a eip phase shift when it moves through
the NRM checkerboard when taking negative cells with
e ¼ �1þ i0:01 and m ¼ 1 while a line source is neatly
imaged on the other side (with same phase shift). But
there is even more: anomalous localized resonances
studied in great mathematical details for a cylindrical
perfect lens in quasi-static limit in [24] are present in
our checkerboards. We observed numerically with the
full-wave FEM package that a line source lying in close
neighborhood of the NRM checkerboard is cloaked by
the system so that its radiations in free space are
minimized! We then used a multiple algorithm
discussed in [23] to check that such a cloaking effect
can be achieved to certain extent in checkerboard
photonic crystals displaying the negative refraction
effect. We checked that light confinement can be
achieved even using a single PC triangle with an
effective refractive index close to �1. Here, we noticed
an interplay between the imaging process and light
confinement: the triangular lens acts as an open
resonator [26] when the line source is located so that
its two images preserve the three-fold symmetry of this
optical system. But the confinement is improved when
some PC triangles are added in a checkerboard fashion.
Acknowledgments
SAR acknowledges partial support from the Depart-
ment of Science and Technology, India under grant no.
SR/S2/CMP-54/2003. SG, SE, GT and BG acknowl-
edge funding from the EC funded project PHORE-
MOST under grant FP6/2003/IST/2-511616. The
authors thank A.S. Bonnet-Ben Dhia and F. Zolla for
enlightening discussions on Lax-Milgram’s lemma and
Fredholm’s alternative for wave transmission between
media with opposite sign dielectric constant [13], and
the referee for constructive critical comments.
S. Guenneau et al. / Photonics and Nanostructures – Fundamentals and Applications 5 (2007) 63–7272
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