f. s. s. rosa, d. a. r. dalvit and p. w. milonni- casimir-lifshitz theory and metamaterials

8/3/2019 F. S. S. Rosa, D. A. R. Dalvit and P. W. Milonni- Casimir-Lifshitz Theory and Metamaterials

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arXiv:0803.2908v1[quant-ph]19Mar2008

Casimir-Lifshitz Theory and 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: March 19, 2008)

Based on a generalization of the Lifshiftz theory we calculate Casimir forces involving magne-todielectric and possibly anisotropic metamaterials, focusing on the possibility of repulsive forces.It is found that Casimir repulsion decreases with magnetic dissipation, and even a small Drudebackground in metallic-based metamaterials acts to make attractive a Casimir force that wouldotherwise be predicted to be repulsive. The sign of the force also depends sensitively on the de-gree of optical anisotropy of the metamaterial, and on the form of the frequency-dependency of themagnetic response.

PACS numbers: 42.50.Ct, 12.20.-m, 78. 20.Ci

The growing importance of both Casimir effects [1] andmetamaterials [2] raises questions concerning the natureof Casimir forces between metamaterials. The rathercomplicated but highly successful Lifshitz theory [3] ofCasimir forces assumes isotropic and nonmagnetic me-dia. Calculations based on its generalization to magne-todielectric media suggest that Casimir forces involvingmetamaterials could have some very interesting proper-ties, e.g., left-handed metamaterials might lead to repul-sive Casimir forces (quantum levitation) [4, 5]. Meta-materials, however, typically have narrow-band magneticresponse and are anisotropic, and so questions naturallyarise concerning the validity of such predictions for realmetamaterials. Such questions are addressed here.

The Lifshitz theory, as in all of macroscopic QED,is based on the assumption that the media can be ap-proximated as continua described by permittivities andpermeabilities. This requires that field wavelengths con-

tributing significantly to the force be large comparedto length scales characterizing the metamaterial struc-tures and separations. The original Lifshitz formulais easily generalized to the case of magnetically per-meable, anisotropic media. One approach is based onthe general scattering formula describing the Casimir in-teraction energy between two bodies in vacuum. Con-sider media 1 and 2 occupying the half-spaces z < 0and z > d, respectively, and separated by vacuum (re-gion 3). The zero-temperature Casimir interaction en-ergy is given by E(d) = (/2)

0

d log detD, where

D = 1 R1eKdR2eKd [6]. Here R1,2 are reflectionoperators at the interfaces z = 0 and z = d, respectively,

and eKd represents a one-way propagation between thetwo media. Assuming the media are spatially homoge-neous, only specular reflection occurs, and the resultingCasimir force per unit area A between the two media is

F(d)

A= 2

0

d

2

d2k(2)2

K3 TrR1 R2e2K3d

1R1 R2e2K3d,

(1)where R1,2 are the 2 2 reflection matrices, k is the

transverse wavevector, and K3 =

k2 +

2/c2. A posi-

tive (negative) value of the force corresponds to attrac-tion (repulsion). Note that, based on the well-known an-alytic properties of the permittivities and permeabilities,the reflection matrices here are evaluated at imaginaryfrequencies = i. For general anisotropic media thesereflection matrices are defined as

Rj =

rssj (i, k) r

spj (i, k)

rpsj (i, k) rppj (i, k)

(j = 1, 2), (2)

where rabj is the ratio of a reflected field with b-polarization divided by an incoming field with a-polarization, the indices s, p corresponding respectivelyto perpendicular and parallel polarizations with respectto the plane of incidence. In the case of isotropic mediathe off-diagonal elements vanish, the diagonal elementsare given by the usual Fresnel expressions in terms ofthe electric permittivity (i) and magnetic permeability(i), and Eq.(1) then leads to the usual Lifshitz formula

for the force [3].One possible route towards Casimir repulsion invokes

non-trivial magnetic permeability [7]. Casimir repulsionwas predicted by Boyer [8] for perfectly conducting andperfectly permeable parallel plates, but it may also occurbetween real plates as long as one is mainly (or purely)non-magnetic and the other mainly (or purely) magnetic[9]. The latter possibility has been considered unphysi-cal [10], since naturally occurring materials do not showstrong magnetic response at near-infrared/optical fre-quencies (corresponding to gaps d = 0.1 10 m, forwhich Casimir forces are typically measured), which isusually assumed necessary to realize repulsion (but see

below). However, recent progress in nano-fabrication hasresulted in metamaterials with magnetic response in thevisible range of the spectrum [11], fueling the hope forquantum levitation [4, 5]. In the following we discussthese expectations and consider key roadblocks in thepursuit of metamaterials-based Casimir repulsion.

Quantum levitation and metamaterials It has beenshown that a perfect-lens slab sandwiched between twoperfect conducting planar plates implies a Casimir repul-sion between the two plates [5]. The condition () =
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() = 1 on which this interesting prediction is basedis incompatible with the Kramers-Kronig relations forcausal, passive materials (Im() > 0 and Im() > 0),which do not allow () = () = 1 at all frequen-cies. Of course we can have the real parts of () and() approximately equal to 1 near a given resonancefrequency of the metamaterial (with non-zero imaginaryparts), but this is not a necessary condition for obtaining

repulsion. Given that the integral in Eq.(1) is dominatedby low-frequency modes < c/d, it implies that a repul-sive force is in principle possible for a passive left-handedmedium as long as the low-frequency magnetic response(i) along frequencies is sufficiently larger than (i).In such a situation the repulsion is a consequence of thelow-frequency behavior of (i) and (i) and not of thefact that the medium happens to be left-handed in a nar-row band about some real resonant frequency.

An alternative scenario was also considered in [5],where it was argued from the Lifshitz formula thatCasimir repulsion can occur with active (amplify-ing) metamaterials, satisfying the perfect-lens condition(i) = (i) = 1 at frequencies < c/d. In our viewthe use of the Lifshitz formalism for active materials re-quires further analysis to account, for instance, for theinevitable amplification of noise in active media, whichis not taken into account in the Lifshitz formula (1)[12].

Repulsion via metallic-based metamaterials Appli-cation of the Lifshitz formula requires the knowledge ofj(i) and j(i) for a large range of frequencies, upto the order of c/d. Such functions can be eval-uated via the Kramers-Kronig (KK) relations in termsof optical responses j() and j() at real frequen-cies. The KK relations imply that dispersion data are

required over a large range of frequencies , typically inthe low-frequency range for metals. This point is veryimportant, as it shows that knowledge of the optical re-sponse of a metallic-based metamaterial near a resonanceis not sufficient for the computation of Casimir forces:the main contribution to (i) and (i) typically comesfrom frequencies lower than the resonance frequency. Al-though somewhat counterintuitive, this fact explains therecent experiment [16] in which it was found that a sharpchange of the optical response of a hydrogen-switchablemirror at optical frequencies c/d did not apprecia-bly change the Casimir force. The fact that the dominantcontribution to the force comes from low frequencies over

bandwidths wide compared to typical metamaterial res-onances also implies that repulsive forces are in principlepossible without the requirement that the metamaterialresonance should be near the frequency scale defined bythe inverse of the gap d of the Casimir cavity.

Let us consider the simple example of a metallic half-space 1 in front of a metallic metamaterial 2. For themetal we assume the usual Drude model, 1() = 1 21/(

2 + i1) ; 1() = 1 , where 1 is its plasma fre-quency and 1 the dissipation coefficient. Typical meta-

10-2

10-1

100

101

102

d /

-0.002

-0.001

0

0.001

0.002

0.003

0.004

0.005

Fd

3/

hcA

f = 0

f = 10- 4

f = 10- 3

10-3

10-2

10-1

100

/

1

2

3

(i),(i)

res

(i)

FIG. 1: Casimir force between a gold half-space and a silver-based planar metamaterial for different filling factors. Thefrequency scale = 2c/ is chosen as the silver plasma fre-quency D = 1.37 10

16 rad/sec. Parameters are: for themetal, 1/ = 0.96, 1/ = 0.004, and for the metamate-rial, D/ = 1, D/ = 0.006, e/ = 0.04, m/ = 0.1,e/ = m/ = 0.1, e/ = m/ = 0.005. The inset showsthe magnetic permeability res(i) and the electric permittiv-

ity 2

(i) for the different filling factors.

materials have resonant electromagnetic response at cer-tain frequencies; in the simplest description, the resonantbehavior can be modelled by a Drude-Lorentz model,

res() , res() = 12e,m

2 2e,m + ie,m, (3)

which e (m) is the electric (magnetic) resonance fre-quency, and e (m) is the metamaterial electric (mag-netic) dissipation parameter. For metamaterials that are

partially metallic, such as split-ring resonators (operat-ing in the GHz-THz range) and fishnet arrays (operatingin the near-infrared/optical) away from resonance, it isreasonable to assume that the dielectric function has aDrude background response in addition to the resonantpart. For frequencies = i, the total dielectric permit-tivity of half-space 2 can then be modeled as

2(i) = 1 + f2D

2 + D+ (1 f)

2e2 + 2e + e

, (4)

where f is a filling factor that accounts for the fractionof metallic structure contained in the metamaterial, andD, D are the Drude parameters of this metallic struc-

ture. The total magnetic permeability is given by theresonant part alone, Eq. (3). As the Drude backgroundclearly overwhelms the resonant contribution res for lowfrequencies , it contributes substantially to the Lifshitzforce. In Fig. 1 we plot the Casimir-Lifshitz force be-tween a metallic half-space and a metallic-based planarmetamaterial for different filling factors at zero temper-ature. (The effect of finite temperature is to decreasethe amount of repulsion [4]). Without the Drude con-tribution (f = 0) there is repulsion for a certain range


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3

104

103

102

104

103

102

1

0

1

2

3

4

5

x 103

0

1

2

3

4

x 103

hcAF4

f f

a)

0.1

0.1

1

10

1

10

0.015

0.01

0.005

0

0.005

0.01

10

5

0

5

x 103

m

m

e

e

hcA

F4

b)

0.1

FIG. 2: Anisotropy and the Casimir force: a) effects of dielec-tric anisotropy for a metallic-based metamaterial with weakDrude background; b) combined effects of dielectric and mag-netic anisotropy for a dielectric-based (f = 0) metamaterial.The distance is fixed to d = . The perpendicular electricand magnetic oscillator strengths in b)are fixed and equal tothose used in Fig. 1. All other parameters in a) and b) arethe same as in Fig. 1.

of distances, as long as half-space 2 is mainly magnetic(2(i) < 2(i)). However, even a small amount ofmetallic background (f > 0) can spoil the possibility ofrepulsion for any separation (see also [14]). In view ofthese observations, metamaterials with negligible Drudebackground, including those based on polaritonic pho-tonic crystals [15] or on dielectric structures [16] show-ing magnetic response, seem more likely candidates forCasimir repulsion.

Given that dissipation plays an important role in

metallic-based metamaterials, especially those operatingat high frequencies, we conclude this section by not-ing that, as a general rule, dissipation tends to reduceCasimir repulsion. For example, increasing the ratiose/e = m/m from 0.01 to 1 decreases the magnitudeof Casimir repulsion by approximately 50% at d/ = 1(for f = 104, e/ = 0.04, m/ = 0.1, and all otherparameters as in Fig. 1).

Anisotropy Most metamaterials have anisotropicelectric and/or magnetic activity, depending on the di-

rection and polarization of the incident radiation [17].To take anisotropy into account when calculating theCasimir force we require in Eq. (1) the appropriate re-flection matrices. Since the general case is rather in-volved, we treat here a particular case in which medium1 is isotropic and medium 2 has uniaxial anisotropy per-pendicular to the interfaces (the more general case willbe treated elsewhere). There are several metamateri-

als consisting of very thin plane layers of alternatingmaterials [18], and which could be described by per-mittivity and permeability tensors, = diag(, , )and = diag(, , ), where , are given by (4)and , by (3) with respective anisotropic parameters.Anisotropy generally implies a mixing of linear polariza-tions, which means that the off-diagonal elements of thereflection matrix (2) will generally not vanish. However,the case under consideration is simple enough that westill have rsp2 (i, k) = r

ps2 (i, k) = 0, while the diago-

nal elements of the reflection matrix are [19]

rss2 =

K3

k

2

+ 2

/c2

K3 +

k2 + 2/c2

, (5)

rpp2

=K3

k2 +

2/c2

K3 +

k2

+ 2/c2. (6)

For metallic-based metamaterials with large in-planeelectric response (i) 1 at low frequencies, it isclear from Eqs. (5, 6) that anisotropy plays a negligi-ble role in the determination of the reflection coefficientswhen there is a dominant Drude background. In orderto better appreciate the effects of anisotropy we assumehenceforth a small or vanishing Drude contribution. InFig. 2a we show the Casimir force for a metamaterialthat has only electric anisotropy ( = ), which iscompletely coded in different filling factors (f = f).We see that a repulsive force arises only for considerablysmall values of both f and f, from which we concludethat the Drude contribution must be isotropically sup-pressed in order to have Casimir repulsion. Things be-come more interesting for dielectric-based (f = f = 0)metamaterials, since in that case the electric and mag-netic contributions are more similar. Accordingly, Fig.2b shows the effect of electric and magnetic anisotropy

on the Casimir force when they are present solely interms of oscillator strengths. We chose parameters inFig.2b such that the force is repulsive for the isotropic

case e/e =

m/m = 1. As expected, repulsion is

enhanced by increasing the ratio of magnetic anisotropy

m/m to the electric anisotropy

e/e , while there is

a crossover to attraction when such a ratio is decreased.

Modelling the magnetic response Having discussedhow low-frequency contributions to the dielectric permit-tivity of a metamaterial affect the Casimir force, let us


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4

10-2

10-1

100

101

102

d /

-0.001

0

0.001

0.002

0.003

0.004

0.005

Fd3

/

hcA

res

(i)

SRR

(i)

10-3

10-2

10-1

100

/

1

2

3

(i)

2(i)

FIG. 3: Casimir force between a half-space filled with goldand a metamaterial whose magnetic response is described ei-ther by res(i) or by SRR(i). The inset shows the behaviorof both permeabilities as well as of the underlying permittiv-ity 2(i). Parameters are: f = 10

4, C = 0.25, and all theother ones are the same as in Fig. 1.

now investigate how it is affected by the magnetic per-

meability. Up to now we have characterized the mag-netic response of a metamaterial by the Drude-Lorentzexpression (3), which has been widely used in the contextof Casimir physics with metamaterials. However, it isworth pointing out that calculations based on Maxwellsequations in a long-wavelength approximation for split-ring resonator (SRR) metamaterials result in a slightlydifferent form [20]:

SRR() = 1 C2

2 2m + im, (7)

where C < 1 is a parameter depending on the geome-

try of the split ring. The crucial difference between (3)and (7) is the 2 factor appearing in the numerator ofthe latter, a consequence of Faradays law. Althoughclose to the resonance both expressions give almost iden-tical behaviors, they differ in the low-frequency limit.Moreover, for imaginary frequencies, res(i) > 1 whileSRR(i) < 1 for all [21]. Given that all passive mate-rials have (i) > 1, we conclude that Casimir repulsionis impossible for any metamaterials, such as SRRs, de-scribed approximately by (7), since the electric responsewould always dominate the magnetic one, even withoutDrude background (see Fig. 3).

Conclusions We have considered several aspects of

the Casimir force between an ordinary medium and ametamaterial. An important conclusion of this work isthat low frequencies , less than c/d, make the most sig-nificant contribution to the force (1), and that the band-width of the dominant low range is much larger thanthe widths of realistic material resonances. For this rea-son repulsive Casimir forces are possible even without astrong magnetic response in the optical, a condition thathas generally been assumed to be necessary for Casimirrepulsion at micron separations. Similarly, a sufficiently

strong low-frequency magnetic response permits quan-tum levitation of metamaterials. This possibility appliesin particular to metamaterials that are left-handed athigher frequencies, but, in contrast to other recent work[5], does not require an amplifying medium. Dissipationand Drude backgrounds reduce the magnitude of repul-sive forces or change an otherwise repulsive force to anattractive one. Given that Drude backgrounds are key

roadblocks for quantum levitation, non metallic-basedmetamaterials are more likely candidates to achieve thegoal of Casimir repulsion.

[1] For recent reviews of both theoretical and experimentalwork see, for instance, S.K. Lamoreaux, Rep. Prog. Phys.68, 201 (2005) and M. Bordag et al., Phys. Rep. 353, 1(2001) and references therein.

[2] Reviews of work on metamaterials have been given, forinstance, by S.A. Ramakrishna, Rep. Prog. Phys. 68, 449(2005). See also the special issue IET Microwaves, Anten-

nas and Propagation 1, no. 1 (2007) eds. G. Eleftheriadesand Y. Vardaxoglou.

[3] E.M. Lifshitz, Sov. Phys. JETP 2, 73 (1956).[4] C. Henkel and K. Joulain, Europhys. Lett. 72, 929

(2005).[5] U. Leonhardt and T.G Philbin, New J. Phys. 9, 254

(2007).[6] E. I. Kats, Sov. Phys. JETP 46, 109 (1977). See also A.

Lambrecht et al., New J. Phys. 8, 243 (2006).[7] Casimir repulsion is also possible between non-magnetic

media, as long as the inequality 1(i) < 3(i) < 2(i)holds. This is the basis for superfluid creeping out ofbeakers, and for frictionless bearings. See F. Capasso etal., IEEE J. Select Topics Quantum Electron. 13, 400

(2007) and references therein.[8] T.H. Boyer, Phys. Rev. A 9, 2078 (1974).[9] O. Kenneth et al., Phys. Rev. Lett. 89, 033001 (2002).

[10] D. Iannuzzi and F. Capasso, Phys. Rev. Lett. 91, 029101(2003).

[11] V.M. Shalaev, Nature Photonics 1, 41 (2007).[12] In fact in the recent paper by C. Raabe and D.-K. Welsch,

arXiv:0710.2867v1, it is argued for just these same rea-sons that the prediction of a repulsive force in Ref. [5] isincorrect.

[13] D. Iannuzzi et al., Proc. Nat. Acad. Sci. USA 101, 4019(2004).

[14] I.G. Pirozhenko and A. Lambrecht, arXiv:0801.3223.[15] K.C. Huang et al., Appl. Phys. Lett. 85, 543 (2004).[16] J.A. Schuller et al., Phys. Rev. Lett. 99, 107401 (2007).

[17] See, for instance, S. Linden et al., Science 306, 1351(2004); A.V. Kildishev et al., J. Opt. Soc. Am. B 23,423 (2006).

[18] See, for instance, J. Schilling, Phys. Rev. E 74, 046618(2006); T. Tanaka et al., Phys. Rev. B 73, 125423 (2006).

[19] See, for instance, L. Hu and S.T. Chui, Phys. Rev. B 66085108 (2002).

[20] J. B. Pendry et al., IEEE Trans. Microwave Theory Tech.47, 2075 (1999). High-frequency corrections to this for-mula that would ensure SRR() = 1 are not found toaffect our numerical results or conclusions.
http://arxiv.org/abs/0710.2867http://arxiv.org/abs/0801.3223http://arxiv.org/abs/0801.3223http://arxiv.org/abs/0710.2867


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[21] Unlike (i), (i) can be less than 1 without violatingKramers-Kronig relations. See, for instance, L.D. Lan-dau, E.M. Lifshitz and L.P. Pitaevskii, Electrodynamics

of Continuous Media (Elsevier, Oxford, 2007).

f. s. s. rosa, d. a. r. dalvit and p. w. milonni- casimir-lifshitz theory and metamaterials

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