F. S. S. Rosa, D. A. R. Dalvit and P. W. Milonni- Casimir Interactions for Anisotropic...

8/3/2019 F. S. S. Rosa, D. A. R. Dalvit and P. W. Milonni- Casimir Interactions for Anisotropic Magnetodielectric Metamaterials

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arXiv:0807.3572v1

[quant-ph]22Jul2008

Casimir Interactions for Anisotropic Magnetodielectric Metamaterials

F. S. S. Rosa,1 D. A. R. Dalvit,1 and P. W. Milonni1

1Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA(Dated: July 22, 2008)

We extend our previous work [Phys. Rev. Lett. 100, 183602 (2008)] on the generalization of theCasimir-Lifshitz theory to treat anisotropic magnetodielectric media, focusing on the forces betweenmetals and magnetodielectric metamaterials and on the possibility of inferring magnetic effects bymeasurements of these forces. We present results for metamaterials including structures with uni-axial and biaxial magnetodielectric anisotropies, as well as for structures with isolated metallic ordielectric properties that we describe in terms of filling factors and a Maxwell Garnett approxima-tion. The elimination or reduction of Casimir stiction by appropriate engineering of metallic-basedmetamaterials, or the indirect detection of magnetic contributions, appear from the examples con-sidered to be very challenging, as small background Drude contributions to the permittivity act toenhance attraction over repulsion, as does magnetic dissipation. In dielectric-based metamaterialsthe magnetic properties of polaritonic crystals, for instance, appear to be too weak for repulsionto overcome attraction. We also discuss Casimir-Polder experiments, that might provide anotherpossibility for the detection of magnetic effects.

PACS numbers: 12.20.-m, 78.20.Ci, 81.05. Zx

I. INTRODUCTION

The last decade has witnessed a huge activity in thedevelopment of metamaterials (MMs) [1], boosted by thepossibility that such engineered media may give rise tonovel optical properties at selected frequency ranges, in-cluding negative refraction [2], perfect lensing [3], andcloaking [4], among others. Such striking phenomena,inaccessible with natural materials, are all possible dueto the significant magnetic activity built into metamate-rials, starting at microwave frequencies and going all theway up to the optical range. Generally speaking, meta-materials are made of micro- and nanostructures care-

fully designed to collectively endow them with a partic-ular electromagnetic property. It is generally desirablethat these structures should be smaller than the wave-length of the incident radiation, so that they are seen bythe incoming waves as artificial atoms. This fact of-ten allows the use of an effective medium approximationto describe metamaterials in terms of an effective elec-tric permittivitiy tensor () and an effective magneticpermeability tensor (), which incorporate the typicaloptical anisotropy of metamaterials.

Recent years have also witnessed an increased interestin Casimir physics [5, 6] thanks to improved precisionmeasurements [7] of the force between material objectsseparated by micron and sub-micron gaps. Quantum vac-uum fluctuations are modified by the presence of mate-rial boundaries, and this typically results in an attrac-tive Casimir force that depends sensitively on the shapeand the optical properties of the boundaries. While theCasimir force offers new possibilities for nanotechnology,such as actuation mediated by the quantum vacuum, italso presents some challenges, as micro- and nanoelec-tromechanical systems (MEMS and NEMS) may sticktogether and cease to work due to the attractive na-ture of van der Waals and Casimir forces. A strongly

suppressed Casimir attraction, or even repulsive Casimirforces, would provide an anti-stiction effect. RepulsiveCasimir forces between two objects 1 and 2, immersedin a background medium 3, may come in a variety ofways. One possibility involves non-magnetic media only,for which repulsion happens when the electric permittiv-ities evaluated at imaginary frequencies satisfy the re-lation 1(i) < 3(i) < 2(i) [8]. Another possibil-ity, first predicted by Boyer [9], involves magnetodielec-tric media: there is a repulsive force when a perfectlyconducting plate is placed near a perfectly permeableone with vacuum in between. Some years later it wasshown that Casimir repulsion can also occur betweenreal (i.e., non-ideal) magnetodielectric media, as long one

medium is mainly electric and the other one is mainlymagnetic [10]. However, this possibility has been con-sidered unphysical [11], as naturally occurring materials,even strong magnets at low frequencies [12], do not showsignificant magnetic response at near-infrared and opti-cal frequencies, which has been assumed as a prerequisitefor repulsion between Casimir plates separated by typi-cal experimentally relevant distances of d = 0.1 1m.On the other hand, recent developments in nanofabri-cation have resulted in metamaterials with magnetic re-sponse in the visible range of the electromagnetic spec-trum [13, 14, 15], fueling the hope for Casimir repulsion[16, 17, 18, 19]. The expectation is that, by tuning this

magnetic response to the right frequency range and mak-ing it strong enough, one could produce an experimen-tally measurable Casimir repulsion between, say, a MMslab and a thin metallic plate, or at least a significantlyreduced attraction.

Unfortunately, this is easier said than done. The majorissue is that the Casimir force between real dispersive ma-terials is a broadband frequency phenomenon, as shownby the Lifshitz formula expressing the force between twosemispaces as an integral over all (imaginary) frequencies
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with an exponential cut-off c/d [20]. For Casimir repul-sion purposes, this requires a magnetic response strongenough to dominate the electric response of the materialin a broad range of frequencies, which typically is notthe case for the magnetic resonances present in metama-terials. In addition, several metamaterials have metallicinclusions that produce a low-frequency Drude-like be-havior in (), whose contribution to the Liftshitz for-

mula dominates over any possible magnetic response thatthe metamaterial may have, making attractive a Casimirforce that would otherwise be predicted to be repulsive.

We have recently addressed many of these issues inthe context of the Casimir-Lishitz theory and metamate-rials [21]. The purpose of the present work is to furtherinvestigate the physics of Casimir interactions betweenmetamaterials, focusing on effects not previously consid-ered in depth in the Casimir literature, such as opticalanisotropy in magnetodielectrics and the feasibility of thecrossover from attractive to repulsive Casimir forces withrealistic metallic-based and dielectric-based metamateri-als.

II. CASIMIR-LIFSHITZ FORCE BETWEENANISOTROPIC MAGNETODIELECTRIC

MATERIALS

A. The scattering approach

Techniques for the evaluation of the Casimir force haveevolved very quickly in the last few years, paving theway for precise analytical [22] and numerical [23] cal-culations in non-trivial geometries. A particularly ap-pealing method is the so-called scattering approach, pi-oneered in Casimir physics by Balian and Duplantierto compute the free energy of the electromagnetic fieldin regions bounded by material boundaries of arbitrarysmooth shape [24]. The free energy is expressed as aconvergent multiple scattering expansion of ray trajecto-ries propagating between the material boundaries. Thismethod, first used for perfect conductors, was extendedto real materials in recent works [25, 26], allowing in prin-ciple the computation of the Casimir interaction betweenarbitrarily shaped material scatterers.

Since a thorough discussion of the scattering approachwould take us too far afield, we simply present the for-mula for the zero-temperature Casimir energy per unitarea A between two parallel plates separated by a vac-

uum gap of width d:

E(d)

A=

0

d

2logdet

1 R1eKdR2eKd

, (1)

where Rj = Rj(k,k, p , p, = i) is the reflectionoperator associated with reflection on the j-th plate(j = 1, 2). Here k and k

are the transverse wave vec-

tors (i.e, projected onto the planar interfaces) for inci-dent and reflected waves, respectively, and p and p are

their respective polarizations (transverse electric (TE) ortransverse magnetic (TM)). The operator exp(Kd) rep-resents one-way propagation between the two plates, andhas matrix elements

k|eKd|k = exp(d

k2 + 2/c2) (2)(k k). (2)

When both plates present homogeneity in the plane ofthe interface, only specular reflection takes place, and

Rj is also diagonal in the transverse momentum basis.This means that

k|Rj |k = Rj (2)(k k), (3)where Rj is the 2 2 reflection matrix on the j-th plate.Note that the reflection matrices here are evaluated atimaginary frequencies = i, and this requires the well-known analytic properties of the permittivities and per-meabilities in the complex frequency plane. For generalanisotropic media these reflection matrices are defined as

Rj =

rTE,TEj (i, k) r

TE,TMj (i, k)

rTM,TEj (i, k) rTM,TMj (i, k)

, (4)

where rp,p

j is the ratio of the amplitudes of a reflectedfield with p-polarization and an incoming field with p-polarization.

Using Eqs. (3) and (2) in Eq. (1), we get after somemanipulations

E(d)

A=

0

d

2

d2k(2)2

logdet

1 R1 R2e2K3d

,

(5)and the expression for the force per unit area follows:

F(d)

A= 2

0

d

2

d2k

(2)2K3 Tr

R1 R2 e2K3d1 R1 R2 e2K3d ,

(6)where K3 =

k2 +

2/c2. A positive (negative) value of

the force corresponds to attraction (repulsion). Despitethe fact that we have assumed homogeneity on each ofthe planar interfaces (which is a reasonable assumptionwhen describing metamaterials with an effective mediumapproach), Eq. (6) is still fairly general: it may be ap-plied to dispersive, dissipative and anisotropic media; allthat is needed are the appropriate reflection matrices.

Let us consider the setup depicted in Fig. 1, where wehave a metallic semi-space occupying the region z < dfacing a magnetodielectric semi-space z > 0. The reflec-tion matrix R1 characterizing the metal-vacuum inter-

face is given by the standard Fresnel coefficients [27]

rTE,TE1 (i, k) =K3

k2 + 1(i)

2/c2

K3 +

k2 + 1(i)2/c2

,

rTM,TM1 (i, k) =1(i)K3

k2 + 1(i)

2/c2

j(i)K3 +

k2 + 1(i)2/c2

,

rTE,TM1 (i, k) = rTM,TE1 (i, k) = 0, (7)


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FIG. 1: Typical setup used throughout this paper to computethe Casimir-Lifshitz force between a metal and a metamate-rial.

where 1() is the permittivity of the metal. The ele-ments ofR2 are only given by Fresnel-like formulas whenthe MM is isotropic, in which case

rTE,TE2iso (i, k) =2(i)K3

k2 + 2(i)2(i)

2/c2

2(i)K3 +

k2 + 2(i)2(i)2/c2

,

rTM,TM2iso (i, k) =2(i)K3

k2 + 2(i)2(i)

2/c2

2(i)K3 +

k2 + 2(i)2(i)2/c2

,

rTE,TM2iso (i, k) = rTM,TE2iso

(i, k) = 0, (8)

where 2, 2 are respectively the permittivity and the per-meability of the metamaterial.

However, as magnetodielectric MMs can generally beoptically anisotropic, the reflection matrix R2 for theMM-vacuum interface is in general not given by the usualFresnel formulas (8). In their most general form meta-materials can be bi-anisotropic, meaning that the consti-tutive relations have the form [28]

D = E+ H, (9)B = E+ H. (10)

Here and are the magneto-optical permittivities, andthey describe magnetic-electric cross-coupling. There areindeed some metamaterials in which the magneto-opticaltensors and are not negligible [29], but since theseproperties can be almost entirely suppressed by using asufficiently symmetric unit cell [30], we assume hence-forth that D = E and B = H. We also assumeagain that the material tensors and are functions offrequency only, neglecting any possible spatial dispersion.

Even without bi-anisotropy the physics of(uni)anisotropic materials is still very rich [31, 32].

It is very common to describe them according to theirdegree of symmetry; in crystallographic theory this leadsto Bravais lattices and their associated point groups[33]. This classification is also very useful for the studyof metamaterials, since they may usually be described interms of unit cells (split-ring-resonators [34], nanopillars[13], nanorods [14], nanospheres [35, 36], etc.) arrangedin a periodic lattice. The most extreme anisotropic

situation is when the only symmetry of the unit cell isinversion with respect to the origin. In this case, knownas the triclinic system, both the permittivity and thepermeability tensors have nine non-zero components [32]in a given orthogonal coordinate system, making theformulation very cumbersome. Although it is certainlypossible to diagonalize at least one of the tensors bychoosing a suitable basis, the angles formed by theeigenvectors depend upon frequency in the triclinicsystem [27, 37]. Since the force (6) is an integral over allfrequencies, this frequency-dependent diagonalization isof little help for purposes of calculating Casimir forces.Fortunately, it is still possible to investigate anisotropiceffects in the Casimir force without going into such aninvolved case, so we restrict ourselves in the next twosubsections to basically two types of anisotropy.

B. Reflection matrices for uniaxial (out-of-plane)planar metamaterials

In this subsection we calculate the reflection matrixR2 for the case of a planar interface between vacuumand a uniaxial magnetodielectric medium that is isotropicon the interface plane, i.e., whose electric and magneticanisotropic directions coincide and are perpendicular to

the interface. In optical terminology, this is an exampleof a uniaxial medium [27] with the optic axis coincidingwith the anisotropic direction. It is known that for uniax-ial lattices, that is, the ones belonging either to the trig-onal, tetragonal and hexagonal crystallographic systems[33], the electromagnetic tensors are diagonal in the co-ordinate system defined by any two orthogonal directionsin the symmetry plane and the optic axis [37]. Therefore,choosing the interface as the xy plane and the anisotropicmedium to be the half-space defined by z > 0 (see Fig.2), the permittivity and permeability tensors are givenby

ij = xx 0 00 xx 0

0 0 zz

; ij = xx 0 00 xx 00 0 zz

,(11)where we have used yy = xx and yy = xx, whose fre-quency dependence is implicit. Although the calculationof the reflection matrix for a metamaterial with a singleout-of-plane anisotropic direction is relatively simple andakin to the isotropic case, for the sake of completenesswe briefly review this calculation, which is relevant toseveral metamaterials having such anisotropy [38].


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Metamaterial

k

kref

in

x

y

FIG. 2: An incident plane wave impinging on a uniaxial meta-material with its optic axis perpendicular to the z = 0 plane.

Let us assume that a plane wave with wave vectork and polarization p impinges upon the interface fromz < 0 (region 3, vacuum) towards the metamaterial (re-gion 2). Given the rotational symmetry about z, withoutloss of generality we can choose our coordinate system sothat the plane of incidence (defined by k and z) coincideswith the xz plane (see Fig. 2). In order to solve thereflection-refraction problem we have to know how wavespropagate in the anisotropic medium. In this particularcase of uniaxial anisotropy orthogonal to the interface itmay be shown by direct substitution that TE waves

ETE = E0yei(kxx+kzz)eit, (12)

are solutions to Maxwells equations provided that

k22,xzz

+k22,zxx

=2

c2xx [in the MM] (13)

k21,x + k21,z =

2

c2[in vacuum]. (14)

In a similar fashion, TM waves

HTM = H0yei(kxx+kzz)eit, (15)

are solutions to Maxwells equations provided that

k22,xzz

+ k22,z

xx=

2

c2xx [in the MM] (16)

k21,x + k21,z =

2

c2[in vacuum]. (17)

Therefore there is no polarization-mixing, and conse-quently the off-diagonal elements of the reflection matrixvanish: rTE,TM2 (i, k) = r

TM,TE2 (i, k) = 0. This al-

lows one to consider separately the reflection of TE andTM waves.

Let Ein = E0yei(k1,xx+k1,zz) be a TE field incident

from the vacuum side. Given the translational invari-ance of the material properties along the planar in-terface, only specular reflection occurs, which impliesthat both x and y components of the wave vector kare continuous. Therefore, the reflected TE field isEref = r2E0ye

i(k1,xxk1,zz), and the transmitted TE fieldis Et = t2E0ye

i(k1,xx+k2,zz), with k22,z = (2/c2)xxxx

k21,xxx/zz . Imposing the boundary conditions on theTE modes, we have

Ein,y + Eref,y = Et,y 1 + r2 = t2,Hin,x + Href,x = Ht,x (1 + r2)k1,z = t2 k2,z

xx,

from which it follows that r2 = (xxk1,zk2,z)/(xxk1,z+k2,z). Evaluating this expression along imaginary fre-quencies = i, one obtains the TE-TE reflection am-plitude on the vacuum-MM interface [39]:

rTE,TE

2uni(i, k

) =

xxK3 xxzz

k2 + xxxx

2

c2

xxK3 +

xxzz

k2 + xxxx2c2

, (18)

where, we recall, K3 =

k2 +

2/c2 and k2 = k2x + k

2y.

Following similar steps, the TM reflection amplitude onthe vacuum-MM interface can also be derived:

rTM,TM2uni (i, k) =xxK3

xxzz

k2 + xxxx

2

c2

xxK3 +

xxzz

k2 + xxxx

2

c2

. (19)

C. Reflection matrices for biaxial, anisotropic

magnetodielectrics

In ascending order of symmetry, the crystals belongingto the triclinic, monoclinic and orthorhombic crystallo-graphic systems [33] are known as biaxial crystals, sincethey are characterized by two optic axes. In this subsec-tion we shall restrict ourselves to the orthorhombic case[40], which allows simultaneous diagonalization of and in an orthonormal basis. The calculation of the reflec-tion matrices for the other two types of biaxial metama-terials is conceptually equivalent but more cumbersomesince the material tensors cannot be brought to diagonalform in a frequency-independent basis.

Let us then consider the system described in Fig. 3,which is similar to Fig. 2 but with an orthorhombic meta-material on the right side. Assuming it is possible to pre-pare the MM in such a way that one of the eigenvectorsis perpendicular to the interface, then the diagonal basisis just {x, y, z} and the electromagnetic tensors are givenby

ij =

xx 0 00 yy 0

0 0 zz

; ij =

xx 0 00 yy 0

0 0 zz

. (20)


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x

y

k

kref

x

y

Metamaterial

z = 0

FIG. 3: An incident plane wave impinging on a biaxial meta-material with orthorhombic symmetry (see text).

Several metamaterials can be described by material ten-sors like (20); a good example is the fishnet design usedin [15].

Metamaterials with two optic axes, even those withthe simplest orthorhombic symmetry, are much harderto treat than those with an out-of-plane, uniaxial opticaxis described in the previous subsection. The reasonis that Maxwells equations do not support transversewaves for biaxial materials: neither TE nor TM wavesare solutions inside the material, and the off-diagonal el-ements of the reflection matrix do not vanish. This also

happens for uniaxial materials with in-plane optic axes,whose reflection matrix can be obtained as a particu-lar case of orthorhombic materials with yy = zz andyy = zz . The Casimir interaction between two dielec-tric semi-spaces with one in-plane optic axis was treatedin [41] and used in the experimental proposal to measurethe Casimir torque between birefrigent plates [42].

The calculation of the plane-wave solutions toMaxwells equations is simplified using a coordinate sys-tem attached to an incident wave from the vacuum side.Let a plane wave with incident wave vector k impingeon the interface forming an angle in with the normaldirection (see Fig. 3). Let (x, y, z) be the coordinate

system attached to the corresponding plane of incidence,that forms an angle with the x axis. The optical tensorsin this new coordinate system are

ij =

xx cos2 + yy sin2 (xx yy)sin cos 0(xx yy)sin cos xx sin2 + yy cos2 0

0 0 zz

;

and

ij = xx cos2 + yy sin2 (xx yy)sin cos 0(xx yy)sin cos xx sin2 + yy cos2 0

0 0 zz

.

The expressions for the incident fields are

Ein =

eTEin y + eTMin

c

(qinx

kx z)

ei(kxx+qinz

t),

(21)

Hin =

eTMin y eTEin

c

(qinx

kx z)

ei(kxx+qinz

t),

(22)

where eTEin , eTMin are given amplitudes and we definedkx = (/c)sin in and qin = (/c)cos in. The reflectedwave has a similar expression:

Eref =

eTEref y eTMref

c

(qinx

+ kx z)

ei(kxxqinz

t),

(23)

Href =

eTMref y + eTEref

c

(qinx

+ kx z)

ei(kxxqinz

t),

(24)

where we have used qref = qin. Our problem now con-sists in finding the amplitudes eTEref e

TMref , so we can con-

struct the reflection matrix (4). In order to obtain thereflection amplitudes, however, it is necessary find thetransmitted fields as well, which means that we have tosolve Maxwells equations in the metamaterial.

Let us assume plane waves

E = e(z

)ei(kxx

t)

; e = (ex , ey , ez),H = h(z)ei(kxx

t) ; h = (hx , hy , hz), (25)

as solutions to Maxwells equations in medium 2, wherewe have already deduced the x dependence from thephase-matching condition on the interface (kx is con-served across the interface). By substituting (25) intothe Faraday and Ampere-Maxwell laws

E = 1c

B

t, (1 B) = 1

c

( E)t

, (26)


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respectively, we see that the z components can be elim-inated as

ez = ckxhy/zz ; hz = ckxey/zz . (27)In order to determine the remaining x and y componentsof e and h it is convenient to introduce a vector u withcomponents u1 = ex , u2 = ey , u3 = hx and u4 = hy .

With the ansatz uj = uj(0)eiqz

we obtain the followinglinear system of equations:

L u = c

q u, (28)

where the non-zero elements of the matrix L are:

L13 = L24 = (xx yy)sin cos ,

L14 =k2xc

2

2zz yy cos2 xx sin2 ,

L23 = xx cos2 + yy sin

2 ,

L31 = L42 = (xx yy)sin cos ,

L32 = k2xc

2

2zz+ yy cos2 + xx sin2 ,

L41 = xx cos2 yy sin2 . (29)

The condition for non-trivial solutions (det(L+ q/c) =0) gives us the equation that determines the possible val-ues of q, namely

c2

2q2 A

c2

2q2 B

= C, (30)

where

A = L13L31 + L14L41,

B = L23L32 + L24L42,

C1 = L13L32 + L14L42,

C2 = L23L31 + L24L41,

C = C1C2,

whose four solutions q(m) (m = 1, 2, 3, 4) are

q(m) = c

12

A + B

(A B)2 + 4C. (31)

These solutions may be conveniently split into two pairs,according to the sign of Re q(m) - solutions with Re

q(m)

> 0 (Re q(m)

< 0) define positive (negative) prop-agating waves. If we denote the positive solutions by

m = 1, 2, we may write the general solution for u as

u(z) =

m=1,2

u(m)(0) eiq(m)z

+

m=3,4

u(m)(0) eiq(m2)z , (32)

where we have used q(3) =

q(1) and q(4) =

q(2). Itis easy to see that the refraction of a wave coming fromz < 0 can only give rise to positive propagating waves(in the sense defined above), from which we conclude thatu(3)(0) = u(4)(0) = 0. Therefore, the transmitted fieldinto the anisotropic magnetodielectric medium is

EtHt

= ei(kxx

t)

m=1,2

u(m)(0) eiq(m)z . (33)

In order to find the amplitudes u(m)(0) we have to im-pose the proper boundary conditions on the fields. In thiscase they just require the continuity of Ex , Ey , Hx , Hyacross the interface. Using (21)-(24) and (33), one derivesthe following boundary conditions:

qin(eTMin eTMref ) =

c

m=1,2

e(m)x (0),

eTEin + eTEref =

m=1,2

e(m)y (0),

qin(eTEin eTEref) =

c

m=1,2

h(m)x (0),

eTMin + eTMref =

m=1,2

h(m)y (0). (34)

This system of equations is unsolvable as it stands, giventhe large number of unknowns. It is possible, however,to use (28) to express all the transmitted amplitudes in

terms of just one, say e(m)x :

(m) e(m)y (0)

e(m)x (0)

=(q(m))2 (2/c2)A

(2/c2)C1,

(m) h(m)x (0)

e(m)x (0)

= c

L31q(m)

c

L32q(m)

(m),

(m) h(m)y (0)

e(m)x (0)

= c

L41q(m)

c

L42q(m)

(m).

Using these definitions, we can rewrite (34) as

1 0 (1) (2)cqin/ 0 (1) (2)

0 cqin/ 1 10 1 (1) (2)

eTErefeTMref

e(1)x (0)

e(2)x (0)

=

eTEincqin/eTEincqin/eTMin

eTMin

. (35)


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In order to find the reflection coefficients, we must solve(35) for the reflected amplitudes. For the sake of clarity,let us do this separately for eTMin = 0, e

TEin = 0 and for

eTMin = 0, eTEin = 0. In the first case, Cramers ruleimmediately yields

rTE,TE2 (i, k) =eTErefeTEin

=detM1detM

ikxk

, (36)

rTM,TE2 (i, k) =eTMrefeTEin

=detM2detM

ikxk

, (37)

and in the second case we have

rTE,TM2 (i, k) =eTErefeTMin

=detM3detM

ikxk

, (38)

rTM,TM2 (i, k) =eTMrefeTMin

=detM4detM

ikxk

, (39)

where M is the 4 4 matrix in (35) and

M1 =

1 0 (1) (2)

cqin/ 0 (1) (2)0 cqin/ 1 10 1 (1) (2)

,

M2 =

1 1 (1) (2)cqin/ cqin/ (1) (2)

0 0 1 10 0 (1) (2)

,

M3 =

0 0 (1) (2)

0 0 (1) (2)cqin/ cqin/ 1 1

1 1 (1) (2)

,

M4 =

1 0 (1)

(2)

cqin/ 0 (1) (2)0 cqin/ 1 10 1 (1) (2)

. (40)

III. METALLIC-BASED METAMATERIALSAND THE CASIMIR EFFECT

Metamaterials may be roughly divided into two classes.The first class consists of MMs that are partially or to-tally based on metallic structures. In this section we con-centrate on these metallic-based MMs, which were pre-viously considered by us in [21]. We study in detail theeffects of optical anisotropy on the Casimir-Lifshitz inter-action with magnetodielectric media. The second classconsists of MMs based purely on dielectric materials, thatwe shall treat in the next section.

A. Isotropic metamaterials

Before going straight to the calculations, it is necessaryto point out that metallic MMs may be also divided into

two types, which we shall characterize as (i) connectedand (ii) non-connected. As the name suggests, in theconnected MMs the metallic part is partially or totallyinterconnected throughout the metamaterial [15], whilein the non-connected it is not [13, 14]. This distinctionis important because in connected MMs there is a netconductivity contribution to the dielectric function dueto the metallic part, while in the non-connected MMs the

background is effectively non-conducting.Let us begin with the simple example of a metallic

half-space 1 in front of an isotropic, connected metallic-based metamaterial 2. For the metal we assume the usualDrude model

1() = 1 21

(2 + i1), 1() = 1, (41)

where 1 is its plasma frequency and 1 the dissipationcoefficient. For the second half-space we have to be morespecific about the MM we want to consider. In the sim-plest description isotropic, connected metallic metamate-rials may be described by a dielectric response accounting

for both a resonance and a Drude contribution:

2()= 1 (1 f) 2e

2 2e + ie f

2D

2 + iD, (42)

where e, e and e are respectively the effective electricoscillating strength, the resonance frequency, and the ef-fective dissipation parameter of the resonant part, andD and D are the Drude parameters of the metallicbackground of the MM. The filling factor f roughly quan-tifies the fraction of metallic structure present in the MM.The magnetic permeability is given by a resonant partalone:

2() = 1 2m

2 2m + im, (43)

where m, m and m are defined analogously to theirelectric counterparts. In Fig. 4 we plot the Casimir-Lifshitz force between a metallic half-space and anisotropic metallic-based planar metamaterial describedby (42) and (43) for different filling factors at zero tem-perature. We see that without the Drude contribution(f = 0) there is repulsion for a certain range of dis-tances, as long as the half-space 2 is mainly magnetic(2(i) < 2(i)). However, as we turn on a metal-lic background (f > 0), the permittivity grows strongerand reverts the previous relation for a larger and largerrange of frequencies, up to the point where the magneticactivity is no longer able to produce repulsion.

An idealization carried throughout the paper is thatboth the metal and the metamaterial are infinitely longin the z-direction. When we have slabs of finite thicknessinstead of half-spaces, the reflection coefficients change to[16]

rp,pjslab(i) = rppj (i)

1 e2Kjdj1 rp,pj 2(i)e2Kjdj

, (44)


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8

where Kj =

k2

+ j(i)j(i)2/c2, dj is the thickness

of the j-th slab, and we are assuming that both slabs aresurrounded by vacuum. From the previous expressionwe see that corrections to the half-space reflection coeffi-cients (at imaginary frequencies) are exponentially smallwhen both products K1d1 and K2d2 are sufficiently large.This basically tells us that estimates on lower bounds ford1 and d2 are actually model dependent (given that Kjdepends on the properties of medium j), so in order todiscuss those estimates we have to be more specific. Fora metal described by (41) and a wave arriving at normalincidence, we have

K1d1 1 d1 p2

2p +

2d/(d + ), (45)

where p = 2c/1, d = 2c/1 and = 2c/. Forhigh frequencies, we have d and then (45) becomes

d1 p2

2p +

2 d p

2. (46)

For typical metals p/2 is around 1020nm, so, at leastfor high frequencies, the contribution to the Casimir forceof a slab some tens of nanometers wide approaches thecontribution of a half-space. However, in the oppositelimit ( d), we have

d1 p2

d=

c40

=

2() (47)

where 0 = 21/ is the static conductivity and () isthe skin depth of the metal at imaginary frequencies.

Thus, for long wavelengths we see that d scales with ,leading to the conclusion that the half-space approxima-tion is not good for sufficiently low frequencies. Fortu-nately, for typical materials this is no source of concern,since the integration range where d holds is verysmall compared to the effective integration range, allow-ing us to push the slab approximation up to very smallfrequencies with almost no effect in the final result. Onemight wonder what happens for oblique incidence, but itis easy to see that the more oblique the incident angle isthe better the estimate for d1 holds, since K1 gets largerand larger (this only means that reflection gets easier asthe incidence angle gets larger, as physically expected).The effect of finite thickness in the Casimir effect was theobject of several papers [43], notably in [44] where a sys-tematic procedure was developed to deal with any givennumber of arbitrary slabs. The effect of finite thicknesswas also studied in the specific context of Casimir forceand metamaterials [16, 18, 45], where it was found thathaving a layer of a MM instead of a half-space reducesthe intensity of the repulsion force and also the range ofdistances where it occurs.

10-2

10-1

100

101

102

d /

-0.2

0

0.2

0.4

0.6

F/FC

f = 0

f = 10-4

f = 10-3

10-3

10-2

10-1

100

/

1

2

3

2

(i),2

(i)

2(i)

FIG. 4: The ratio F/FC for a gold half-space facing anisotropic, interconnected and silver-based metamaterial. F/Ais the Casimir force per unit area in this setup, FC/A =c2/240a4 is the Casimir force per unit area between twoperfect plane conductors, and F < 0(F > 0) corresponds to arepulsive (attractive) force. The frequency scale = 2c/is chosen as the silver plasma frequency D = 1.37 10

16

rad/sec. Parameters are: for the metal, 1/ = 0.96,1/ = 0.004, and for the metamaterial, D/ = 1, D/ =0.006, e/ = 0.04, m/ = 0.1, e/ = m/ = 0.1,e/ = m/ = 0.005. The inset shows the magnetic perme-ability 2(i) and the electric permittivity 2(i) of the MMfor the different filling factors.

B. Uniaxial Metamaterials

Electric anisotropic effects in the Casimir interactionhave been thoroughly studied in the literature [41, 42, 47,49, 50], but until recently there was no compelling reasonto study the consequences of magnetic anisotropy. Thischanged with the advent of metamaterials, and an in-vestigation of magnetic anisotropy is now in order. Thebest place to start is to consider uniaxial out-of planemetamaterials, since they constitute the simplest depar-ture from the isotropic case. This type of anisotropy is

quite common since it arises naturally when a materialis built as a stack of different layers, as is the case forseveral kinds of MMs [15, 38, 51, 52]. We are particu-larly interested in the case where the resulting medium ischaracterized by different degrees of conductivity in theplane of symmetry and in the perpendicular direction toit.

Let us begin by characterizing the electric and mag-


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9

- 0.02

0

0.1

0.2

0.3

F / FC

fx

fz

10-5 10-5

10-4

10-2 10-2

10-3 10-3

10-4

10-1 10-1

FIG. 5: The effects of uniaxial anisotropy in the Casimir forcebetween a gold semi-space and a metallic-based connectedMM with weak Drude background. The distance is fixed tod = and repulsion corresponds to negative values of F/FC.All parameters are the same as in Fig. 4 except for the fillingfactors fx and fz , which are the variables in this plot.

netic properties of our uniaxial metamaterial:

xx() = yy() = 1 (1 fx)2e,x

2 2e,x + ie,x

fx2D,x

2 + iD,x ,

zz() = 1 (1 fz)2e,z

2 2e,z + ie,z

fz2D,z

2 + iD,z,

xx() = yy() = 1 2m,x

2 2m,x + im,x,

zz() = 1 2m,z

2 2m,z + im,z, (48)

where the different filling factors fx and fz account forthe possible anisotropy in the metallic character of theMM. As we have seen in subsection II-B, the reflectionmatrix for such a metamaterial is diagonal and given by(18) and (19). For metallic-based metamaterials withlarge in-plane electric response xx(i) 1 at low fre-quencies, it is clear from Eqs. (18, 19) that anisotropyplays a negligible role in the determination of the reflec-tion coefficients when there is a dominant Drude back-ground. In order to better appreciate the effects ofanisotropy we assume henceforth a small or vanishing

Drude contribution. In Fig. 5 we show the Casimirforce for a metamaterial that has only electric anisotropy(xx = zz), which is completely coded in different fillingfactors (fx = fz, all other parameters being the same).We see that a repulsive force (F/FC) arises only for con-siderably small values of both fx and fz, from whichwe conclude that killing the Drude background in thez-direction alone is not enough to produce Casimir re-

pulsion.

C. Biaxial metamaterials

Continuing our track towards more complicated me-dia, we now tackle the biaxial orthorhombic case. Let usconsider a metamaterial characterized by the followingdielectric and magnetic functions in the basis defined byits eigenvectors (see subsection II-C):

xx() = 1 (1 fx)2e,x

2 2e,x + ie,x

fx 2D,x2 + iD,x

,

yy() = 1 (1 fy)2e,y

2 2e,y + ie,y

fy2D,y

2 + iD,y,

zz() = 1 2e,z

2 2e,z + ie,z

2D,z

2 + iD,z,

xx() = yy() = 1 2m,x

2 2m,x + im,x,

zz () = 1. (49)

We are particularly interested in the case where xx isclose to yy but in general significantly different fromzz . This means basically that the MM is only slightlyanisotropic in the plane of incidence. Our motivationin studying this particular limiting case is that it is agood approximation for certain types of metamaterials,such as those based on fishnet designs [15]. Note that weare already assuming magnetic in-plane isotropy, whichis consistent with a small electric in-plane anisotropy. Wemay then rewrite the material tensors as

ij = xx 0 0

0 xx(1 + ) 00 0 zz

,

ij =

xx 0 00 xx 0

0 0 1

, (50)

where () = (yy()xx())/xx() 1, and performthe calculations only up to first order in . The evaluationof the determinants in (36)-(39) requires the knowledge ofmatrix elements ofL defined by equation (29) and also of


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10

the solutions of equation (30). This last step is simplifiedat first order in because C/2xx (xx yy)2/2xx 2.Therefore

q(1) =

c

A ; q(2) =

c

B, (51)

and then the (m) coefficients reduce to (1) = 0 and(2) = (B

A)/C1. Performing now some straightfor-

ward calculations we get, up to O(2),

rTE,TE2 () = rTE,TE2uni

() + rTE,TE2,1 (),

rTM,TE2 () = xxxxqtmqin sin2

(qtm + qte)(qtm + xxqin)(qte + xxqin),

rTE,TM2 () = xxxxqin sin2

(qtm + xxqin)(qte + xxqin),

rTM,TM2 () = rTM,TM2uni

() + rTM,TM2,1 (), (52)

where rTE,TE2uni , rTM,TM2uni

are given respectively by (18),(19), and we have also defined

qte

xxxx2

c2 xxk2,

qtm

xxxx2

c2 xx

zzk2,

rTE,TE2,1 (2/2c2)xxxx cos2

qte(qte + xxqin)

1 + rTE,TE2uni

,

rTM,TM2,1 (qin/2)xx sin

2

qtm + xxqin

1 rTM,TM2uni

.

Now let us return to the general structure of the Lif-shitz formula. SinceR2 is not diagonal we should expect

contributions coming from the non-diagonal terms in (5),but it can be shown that they are all O(2), and thereforecan be dropped. After a few rearrangements we arrive atour final expression for the Casimir pressure:

F

A= 2

0

d

2

d2k(2)2

K3

p=TE,TM

[Iuni

+rp,p1 r

p,p2,1 e

2K3d

1 rp,p1 rp,p2unie2K3d(1 + Iuni) + O(

2)

, (53)

where

Iuni =

rp,p1 rp,p2uni

e2K3d

1 rp,p1 rp,p2unie2K3d . (54)

An easy consistency check of this result is to take thezero-anisotropy limit, which reduces immediately to theuniaxial result, as it should. A less trivial result is toobtain the non-retarded limit of expression (53), whichcan be shown to be consistent at first order with otherresults in the literature [41, 47, 48].

Let us now assume that all the in-plane anisotropyis coded in the filling factors, just like the out-of-plane

Fd

3

/hcA

d /

fx= 0.2

fx= 0.5

fx= 0.8

0.1 1 10 100

0.005

0.010

0.015

0.020

0.025

0.030

FIG. 6: The Casimir force between a gold half-space and anorthorhombic, slightly in-plane anisotropic MM for differentvalues of the filling factors fx and fy . The bands are charac-terized by a certain value offx, as shown in the legend, and acontinuum of values of fy , from fy = 0.8fx to fy = 1.2fx. Allthe other parameters involved are exactly the same as thoseused in Fig.4.

anisotropy was in the uniaxial case. In Fig. 6 we showthe effects of a slight in-plane anisotropy on the Casimirforce. Each band in the plot corresponds to a differentvalue of fx, and its width is given by a 20% variationof fy around fx. We see that the anisotropy effect ismore pronounced at small distances, because in the non-retarded limit the contribution of the electric response tothe Casimir force is maximized.

D. Dissipation effects

Let us now turn from considerations of anisotropy toother practical issues for MMs and Casimir interactions.It is known that dissipation plays an important role inmetallic-based metamaterials, especially those operatingat high frequencies. In Fig. 7 we show the effect of asimultaneous modification in the electric and magneticdissipation coefficients; it may be clearly seen that anequal change in the rates e/e and m/m favors at-traction. In the insets (a) and (b) we show respectivelythe effects of changing only the electric and magneticdissipation, that may be straightforwardly interpreted in

light of the discussion presented in [10]. From (42) wesee that an increase in e makes smaller, pushing themetamaterial slightly closer to the Boyer limit (that is,1 , 1 = 1, 2 = 1, 2 ). We should thus ex-pect an increase in the Casimir repulsion as we make elarger, and that is exactly what is observed in the inset(a). A similar reasoning may be applied to inset (b), butsince this time we are going away from the Boyer limit,repulsion diminishes as we increase m.


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11

10-1

100

101

102

d /

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

F/FC

e/e=m/m=0.1

e/

e=

m/

m=0.5

e/

e=

m/

m=2.5

10-1

100

101

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

10-1

100

101

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

(a)(a) (b)

m/

m=0.1 e/e=0.125

FIG. 7: The ratio F/FC between a gold half-space and anisotropic silver-based metamaterial for different values of thedissipation parameters. The main plot shows the effect of thesimultaneous modification of electric and magnetic dissipa-tion. Inset (a) shows the effect of electric dissipation alone for

different values of the ratio e/e= 0.1 (solid), 0.5 (dashed),2.5 (dotted). Inset (b) shows the effect of magnetic dissipa-tion alone for different values of the ratio m/m= 0.1 (solid),0.5 (dashed), 2.5 (dotted). The filling factor is f = 104 in allthree plots, and all other parameters except the dissipationcoefficients are the same as in Fig. 4.

E. Temperature effects

The effects of temperature in the Casimir force be-

tween metamaterials and dielectrics were thoroughly dis-cussed in [16], where the authors show that in this casetemperature works against repulsion, or in other words,that for sufficiently high temperatures repulsion is com-pletely overturned into attraction. In this section, wewish to extend that discussion for a metallic plate fac-ing a MM and compare the situations where the metal ismodeled by either Drude or plasma permittivities.

In Fig. 8a we show the Casimir force for different tem-peratures between a Drude metal and an isotropic MMwith no Drude background. In this case we see that tem-perature also works against repulsion, but in such a waythat keeps the repulsion window quite open for tempera-tures as high as T = 600K, allowing for repulsion at roomtemperature, at least in principle. Something even moreinteresting happens when we change the Drude metalby a plasma metal (i.e., vanishing relaxation parameter1 = 0 in eq. (41)), as shown in Fig. 8b. In this case, wesee that not only a temperature increase does not switchback the force into attraction for large distances, but itactually increases repulsion in that regime. It is possibleto explain this phenomenon in simple terms using theLifshitz formula. Let us consider the force between two

isotropic materials at a finite temperature T

F(d, )

A=

n=0

p=TE,TM

d2k(2)2

K3

rp,p1 (n) r

p,p2 (n) e

2K3d

1 rp,p1 (n) rp,p2 (n) e2K3d, (55)

where the prime in the summation means that then = 0 term is multiplied by 1/2, = 1/kBT, K3 =k2 +

2n/c

2, n = 2n/ are the Matsubara frequen-

cies, and the reflection coefficients are given by (7) and(8) with n instead of. From (55) we see that for largedistances (provided that kBT d/c 1) the n = 0 domi-nates all the others, and we may approximate the Casimirforce by

F(d, )

A=

4

p=TE,TM

dk k2

rp,p1 (0, k)r

p,p2 (0, k) e

2kd

1 rp,p

1 (0, k) rp,p

2 (0, k) e2kd

, (56)

where k = |k| and the reflection coefficients are evalu-ated at the zeroth Matsubara frequency 0 = 0. The keydifference from the setups using dieletrics or Drude met-als to the one with plasma metals is that in the formercases we have lim0 ()

2/c2 = 0, leading to

rTE,TE1 (0, k) =k kk + k

= 0, (57)

while in the latter we have

rTE,TE1 (0, k) =k

k2 + 21

k +

k2

+ 21

0. (58)

This means that for dielectrics or Drude metals facinga MM, the only contribution to (56) comes from theTM zero mode, which is always positive (given that

rTM,TM1 (0, k)rTM,TM2 (0, k) > 0). Since this term domi-

nates for large distances, we conclude then that the forceis attractive in this regime. However, for plasma metalsfacing a MM we see that both TE and TM zero modescontribute, and while rTM,TM1 (0, k)r

TM,TM2 (0, k) is posi-

tive the product rTE,TE1 (0, k)rTE,TE2 (0, k) is not, due to

the different signs ofrTE,TE1 (0, k) 0 and ofrTE,TE2 (0, k)

rTE,TE2 (0, k) =2

(0)

1

2(0) + 1 > 0. (59)

We see then that the sign of the force depends on a deli-cate balance between the TE and TM contributions, andit so happens that for our chosen parameters the TE termoverwhelms the TM term and repulsion is sustained forall distances above the crossover from attraction. Thefact that repulsion is enhanced is also easily explained,since a simple analysis shows that for large distances (56)may be put in the form C/d3, where C is a constant


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12

0.001 0.01 0.1 1 10 100

d (m)

-0.012

-0.008

-0.004

0

0.004

0.008

Fd3/

hcA

T=0KT=300KT=600K

(a)

0.001 0.01 0.1 1 10 100d (m)

-0.012

-0.008

-0.004

0

0.004

0.008

Fd

3

/hc

A

T=0KT=300KT=600K

(b)

FIG. 8: Temperature dependence of the Casimir force b e-tween a metallic plate and a metamaterial. We plot theCasimir force between a metamaterial and a Drude metal (a)

or a plasma metal (b) for different temperatures. We stressthat negative values of the force characterize repulsion, andthat all parameters are the same as the ones used in the f = 0curve of Fig. 4.

depending on the materials used. It is clear then that ifC is negative, a temperature increase can only enhancerepulsion. Our findings for temperature effects using ei-ther Drude or plasma models for the metallic plate areconsistent with the conclusions of [53] .

F. Metamaterials based on isolated metallicstructures

There are several examples of metamaterials where themetallic part is distributed in a non-connected way. In[13], for instance, the authors put forward a MM consist-ing of pairs of metallic nanopillars, regularly distributedon top of a dielectric substrate. The pairing of pillarsis necessary to create an antisymmetric resonance (when

the currents in each pillar are running in opposite direc-tions) at a certain frequency, where the electric dipolecontributions of both pillars are nearly canceled out andthe effective current loops produced by the pairs giverise to magnetic dipole contributions, resulting in a non-trivial magnetic activity.

Unfortunately, a detailed treatment of the metama-terial previously described is beyond the scope of the

present paper. It is still possible, however, to capturesome effects of metallic non-connectedness and geomet-rically built-in resonances through the use of an appro-priate toy model. In order to address the first issue, weconsider a simple MM model consisting of identical, smallmetallic spheres of radius a regularly distributed in ahost dielectric (non-magnetic) medium. Assuming thatthe metal and the dielectric are characterized respectivelyby the permittivities

2,met() = 1 22,met

2 + i2,met

2,d() = 1 Ni=1

22,i2 22,i + i2,i , (60)

and that the metallic spheres can be considered in a firstapproximation as electric dipoles, one can connect themedium effective permittivity 2,nc() to the electric po-larizability () of a given sphere through the Clausius-Mossotti formula [46]

f

a3 =

2,nc 2,d2,nc + 22,d

, (61)

where f is the metallic filling factor and we have sup-pressed the -dependence for simplicity. It is also pos-

sible to show that when (/c)a 1 (the spheres aremuch smaller than the radiation wavelength), the elec-tric polarizability may be given in terms of the dielectricfunction of the metal by the similar relation

a3=

2,met 2,d2,met + 22,d

, (62)

and by eliminating in (61) we get

2,nc() = 2,d(1 + 2f)2,met + 2(1 f)2,d

(1 f)2,met + (2 + f)2,d . (63)

This result is known as the Maxwell Garnett approxima-tion for the permittivity [35, 46], after the physicist whoderived it in the early 1900s [54]. A brief analysis showsthe main effect of having isolated metallic pieces: the pre-vious formula tends to a finite value in the zero frequencylimit, unlike (42), which describes a connected metallicMM. The effective permeability can be dealt in a similarway, and it is possible to show that in this approximationwe have simply 2,nc() = 1.

As noted earlier, formula (63) accounts only for effectsof metallic non-connectedness. In order to include the


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13

10-2

10-1

100

101

102

d /

0

0.1

0.2

0.3

0.4

F/FC 10

-310

-210

-110

010

1

/

0

2

4

6

8

10

2

(i),2

(i)

2

2

FIG. 9: The ratio F/FC for a gold half-space facing aisotropic, non-connected and gold-based metamaterial. Theparameters for the metal are 2,met/ = 0.96, 2,met/ =0.004, and for the metamaterial we have e/ = 0.34,m/ = 0.064, e/ = 0.2, m/ = 0.15, e/ = 0.04,m/ = 0.02, f = 0.1. The inset shows the permittivity andpermeability inside the MM, as given by (64), but as functionsof imaginary frequencies .

built-in electric and magnetic resonances [55], we sim-ply assume their existence in an ad hoc manner and addtheir contribution to 2,nc() and 2,nc(), respectively.Assuming those resonances can be modeled by Drude-Lorentz formulas, we have, finally,

2() = 2,b() + 2,res()

=(1 + 2f)2,met + 2(1 f)2,d

(1 f)2,met + (2 + f)2,d+ 2e

2 2e + ie2() = 1

2m

2 2m + im. (64)

The results for the Casimir force are shown in Fig. 9. Theparameters for the resonant parts 2,res() and 2,res()are roughly based on the experimental results given in[13] for a MM consisting of metallic nanopillars coveredwith a thin layer of glycerine. As indicated earlier, ourintention here is not to provide a precise description ofsuch experiments, but only to estimate how this type ofmetamaterial affects the Casimir force. The embedding

dielectric, glass BK7, is quite well described by (60) withthe parameters N = 3, 2,1/ = 1.84, 2,1/ = 1.81,2,2/ = 0.47, 2,2/ = 0.28, 2,3/ = 2,3/ = 0.014,2,1/ = 2,2/ = 2,3/ = 0. It is clearly seen that norepulsion is achieved, and the reason is that the magneticresonance created by the MM geometry is too weak tooverwhelm the electric background. In other words, theMM is mainly dielectric, leading to an attractive force.

IV. DIELECTRIC-BASED METAMATERIALSAND THE CASIMIR EFFECT

Metamaterials based exclusively on dielectrics [36, 56,57] are an interesting alternative to metallic MMs. Forone thing, they provide new possibilities for the construc-tion of negative index materials [58], since they allow forboth the permittivity and the permeability to assume

negative values in bandwidths that may be out of reachwith metallic-based MMs. In addition, dielectric-basedMMs might be interesting for Casimir force studies forthe same reason that non-connected metallic MMs mightbe: they do not present a Drude background at low fre-quencies, and this is advantageous for the observation ofmagnetic effects in the Casimir force.

The dielectrics most commonly used in the construc-tion of MMs are polaritonic crystals [59] characterizedby the dielectric function

pol() =

1 +

2pol 2pol

2 + 2pol + ipol

, (65)

where pol is a characteristic resonance of the system, is the permittivity at very high frequencies, andpol = pol

(0)/. In order to fix ideas, let us

consider a MM made of a regular array of polaritonicnanospheres of radius a embedded in an isotropic dielec-tric and non-magnetic host characterized by a dielectricfunction h. For sufficiently long wavelengths and sparsearrays, meaning x R/c 1, it is possible to use theso-called extended Maxwell-Garnett theory [35] to eval-uate the dielectric and magnetic properties of the meta-material, giving [35, 36]

emg() = h x3

3if a1x3 + 32 if a1

, emg() = x3

3if b1x3 + 32 if b1

(66)

where f is the array filling factor and a1, b1 are respec-tively the electric and magnetic dipole coefficients of thescattering matrix of a single sphere, given by [27, 60]

a1 =j1(xpol)[xj1(x)]

pol j1(x)[xpolj1(xpol)]hh(+)1 (x)[xpolj1(xpol)]

h j1(xpol)[xh(+)1 (x)]polb1 =

j1(xpol)[xj1(x)] j1(x)[xpolj1(xpol)]h(+)1 (x)[xpolj1(xpol)]

j1(xpol)[xh(+)1 (x)](67)

where j1(h+

1

) is the spherical Bessel function (Hankelfunction of the first kind) of order one, xpol = polxand the prime has the usual meaning of a derivative withrespect to the function argument. The important thingto notice here is the fact that emg may present severalresonances even when the nanospheres are purely dielec-tric, from which we conclude that in this framework wedo not have to assume an ad hoc resonant behavior; it isalready built into the theory.


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0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

/

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

(i),2

(i)

10-2

10-1

100

/

0

10

20

30

40

50

Imemg

()

10-2

10-1

100

/

0.0

0.2

0.4

0.6

0.8

1.0

Imemg

()

2

2

FIG. 10: The permittivity true(i) and permeability true(i).The parameters are = 2, pol/ = 0.4, pol/ = 0.15,pol/ = 0.001.

The usual procedure at this point would be to rotateexpressions (66) to the imaginary frequency axis and sub-stitute them into the appropriate reflections coefficients,but in this case things are not so straightforward. Im-plicit in the Lifshitz formula for imaginary frequencies itis the assumption of analyticity of () and () in theupper half-plane, a condition that emg() and emg()do not satisfy. In order to overcome this obstacle we haveto remind ourselves that expressions (66) were derived asapproximations to the true permittivity () and perme-ability () only for a given range of real frequencies,namely, for such as R/c 1. This means that while() and () must be analytic in the upper half-planedue to causality requirements, emg() and emg() arenot necessarily bound to causal behavior. In other words,

it means that the analytical continuations ofemg() andemg() into the complex plane are not necessarily closeto the continuations of () and (), and in this casethey happen to be quite different.

A possible way to proceed is to rely on the analyticproperties of() and write the Kramers-Kronig relation[37]

() = 1 +1

iP

0

dy(y) 1

y , (68)

where P stands for the Cauchy principal value, and con-sider also the analogous relation for (). Taking thereal part and evaluating it at an imaginary frequency i,

we obtain

(i) = 1 +2

0

dyyIm (y)

2 + y2, (69)

and, using the fact that () emg() [61], we have

(i) 1 + 2

0

dyyIm emg(y)

2 + y2,

(i) 1 + 2

0

dyyIm emg(y)

2 + y2. (70)

0.1 0.2 0.4 0.6 0.8 1d (m)

0

20

40

60

80

100

P(mPa)

T = 0KT = 300K (Drude)

T = 300K (Plasma)

0.4 0.6 0.8 1.0

d (m)

0.0

0.2

0.4

0.6

0.8

1.0

P(mPa)

FIG. 11: The plot of P = P(2) P(1) for different tem-peratures and models. Following our conventions throughoutthe paper, a positive force means attraction. The param-eters of P(1) are the same used in Fig. 8, in dimensionalunits they are D = 1.32 10

16rad/s, D = 5.48 1013rad/s

(D = 0 for the plasma curve), e = 4.7 1015rad/s, m =8.7 1014rad/s, e = 2.7 10

15rad/s, m = 2 1015rad/s,

e/ = 5.5 1014rad/s, m = 2.7 1014rad/s, 2,1 = 2.52 1016rad/s, 2,1 = 2.48 10

16rad/s, 2,2 = 6.4 1015rad/s,

2,2 = 3.8 1015rad/s, 2,3 = 2,3 = 1.9 10

14rad/s,

2,1 = 2,2 = 2,3 = 0, f = 0.1, and the parameters of P(2)

are exactly the same except for e = 0. The inset shows thesame plot on a different scale, since in the larger one it is notpossible to see P for large distances.

In Fig. 10 we plot (i) and (i) using approximatevalues for TlCl polaritonic spheres [56] embedded in vac-uum. We see that (i) is overwhelmingly dominant over(i), which in fact is hardly different from unity. Asthe insets show, this is basically due to a single strong

resonance, around = 0.3, that appears in emg() butnot in emg(). From these results we conclude that, de-spite the fact that some magnetic activity is created byan array of polaritonic spheres, the Casimir force in thiscase is dictated by the electric part alone and thereforeno repulsion seems possible.

V. DISCUSSION

A striking confirmation of the magnetic influence onthe Casimir force would be a measurement of repulsionbetween a metallic plate and a magnetodielectric one.

This seems unlikely in light of the examples presentedhere, but a measured reduction in the attractive forcemight nevertheless be traced back to the magnetic prop-erties of a metamaterial.

Let P(1) be the Casimir pressure between a gold half-space and a given metamaterial. If the magnetic prop-erties of the MM are turned off, keeping all otherparameters the same, the pressure will change to some


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0 2 4 6 8 10

d (m)

10-10

10-8

10-6

10-4

10-2

100

102

FIG. 12: The difference in frequency shifts caused by thepresence or absence of magnetic activity in the metamaterial.Everything is assumed to be at zero temperature. The MMis the same used in Fig. 11, and the parameters for the Rbatom are m = 1.45 1025 Kg, 0 = 4.74 10

23cm3 and0 = 2.54 10

15 rad/s, with an unperturbed trap frequencyof z = 2 229 Hz.

new value P(2). In order to check whether the differ-ence P = P(1) P(2) should be observable, we plot itscomputed value in Fig. 11 for zero and room tempera-tures, using both Drude and plasma models for the metal.The sensitivity of current experiments lies around 1 mPa,from which we conclude that detection of magnetic effectsin our setup is currently possible up to d 0.4m. Whilethis suggests a considerably large window for measure-ment, given that many experiments probe the 150 350m range quite accurately, several things must be dealtwith. First and foremost, we see that the difference be-tween the Drude and plasma predictions are consider-

ably large (as compared to the magnetic effect) above0.6m. This means that in order to ascribe changes inthe Casimir force ambiguously to magnetic effects onehas to know how to model metallic materials properly.In addition, at close distances like d 0.4m, the ef-fective medium approximation probably no longer holds,since the very structures that produce magnetic activ-ity (the metallic spheres in this example) are built onthe scale of hundreds of nanometers or larger. Thesefinite-size effects should bring significant corrections tothe Casimir force, and must be considered in a more so-phisticated analysis. Finally, there are the imperfectionsof the materials themselves, like roughness, that at those

distances play a non-negligible role. We conclude thenthat despite the fact that current experiments have inprinciple the sensitivity necessary to detect magnetic ef-fects, an actual measurement of such effects remains achallenging task.

Casimir-Polder experiments [62, 63] also provide pos-sibilities for the detection of magnetic effects. These ex-periments are able to probe larger distances than the

typical bulk-bulk measurements, which is desirable fromthe point of view of an effective medium approximation.The zero temperature Casimir-Polder potential betweena ground state atom and a material half-space is [64],

UCP(z) =

82c

0

d2(i)

0

dkke2zK3

K3

rTE,TE(i,k) 1 + 2k2c22 rTM,TM(i,k) ,(71)

where z is the distance between the atom and the half-space, K3 is defined just below (6), and r

TE,TE andrTM,TM are given by (7). (i) is the dynamic atomicpolarizability, which we assume is described reasonablywell by the single-resonance expression

(i) =0

1 + 2/20, (72)

where 0 is the static polarizability and 0 is the dom-inant atomic transition. In one type of experiment [63]the directly measured quantity is the frequency shift inthe center of mass oscillation of a Bose-Einstein conden-sate:

(z) =1

2m2z2zUCP(z

)z=z

, (73)

where m is the atomic mass and z is the unper-turbed (i.e., without Casimir-Polder forces) oscillationfrequency. The reported sensitivity for lies between105 and 104, setting the lower bound for the detec-tion of magnetic effects in the Casimir force. Let us thenconsider a Rb atom in front of the same MM used inthe previous example, and compare the frequency shiftswhen its magnetic part is turned on and off. In Fig.12 we plot the difference (z) = nm(z)m(z), wherem(z) and nm(z) are respectively the frequency shiftswhen the magnetic activity is present and absent. Wesee that in the best case scenario (sensitivity equal to105) the magnetic influence would be detectable up toaround 2.5 m; for larger distances the force is just tooweak.

As a final remark we note that, while Casimir repul-sion will likely be very difficult to observe with existingmetamaterials, the detection of magnetic effects througha slight reduction in the Casimir attraction is definitelypossible. There are still some issues to be dealt with, likethe assurance that magnetic activity is the main causeof force reduction, rather than some trivial effect like areduced filling factor. With the consistent developmentof both Casimir measurements and MMs manufacturedin the recent years, it is very reasonable to expect that aCasimir measurement of magnetic effects will be feasiblein the near future.


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VI. ACKNOWLEDGMENTS

We are greatly indebted to H.-T. Chen, R.S. Decca, N.Engheta, S.K. Lamoreaux, P.A.M. Neto, J.F. OHara,

W.J. Padilla, J.B. Pendry, V.M. Shalaev, D.R. Smithand A.J. Taylor, for very useful discussions. We also ac-knowledge the support of the U.S. Department of Energythrough the LANL/LDRD program for this work.

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http://arxiv.org/abs/0806.4752http://arxiv.org/abs/0806.4752

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