Download - Cloaking and imaging effects in plasmonic checkerboards of negative and and dielectric photonic crystal checkerboards

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-

mailto:[email protected]

http://dx.doi.org/10.1016/j.photonics.2007.07.015

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.


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.)


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


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


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


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


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.


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