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Metamaterial 'multiverse'

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2011 J. Opt. 13 024004

(http://iopscience.iop.org/2040-8986/13/2/024004)

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IOP PUBLISHING JOURNAL OF OPTICS

J. Opt. 13 (2011) 024004 (4pp) doi:10.1088/2040-8978/13/2/024004

Metamaterial ‘multiverse’Igor I Smolyaninov

Department of Electrical and Computer Engineering, University of Maryland, College Park,MD 20742, USA

Received 4 June 2010, accepted for publication 16 July 2010Published 16 November 2010Online at stacks.iop.org/JOpt/13/024004

AbstractOptical space in metamaterials may be engineered to mimic the landscape of amulti-dimensional universe which has regions of different topology and different effectivedimensionality. This ‘metamaterial landscape’ may include regions in which one or two spatialdimensions are compactified. Nonlinear optics of metamaterials in these regions mimics eitherU(1) or SU(2) Kaluza–Klein theories having one or more kinds of effective charges. As aresult, novel ‘photon blockade’ nonlinear optical metamaterial devices may be realized.Topology-changing phase transitions in such metamaterials lead to considerable particlecreation, perceived as flashes of light, thus providing a toy model for the birth of an individualphysical universe.

Keywords: metamaterials, nonlinear optics, Kaluza–Klein theories

(Some figures in this article are in colour only in the electronic version)

The newfound freedom of control of the local dielectricpermittivity εik and magnetic permeability μik tensors inelectromagnetic metamaterials has fuelled the recent explosionin novel device ideas based on the concept of ‘electromagneticspace’, which is different from the actual physical space,and may have non-trivial topology [1–3]. While the currentemphasis of research in this field is concentrated in the areaof novel electromagnetic devices, linear and nonlinear opticsof metamaterials may also have far-reaching implications forfundamental physics. For example, optics of metamaterials hasa unique capability to realize a table-top model of the physicalmultiverse [4].

The understanding of our physical world as a tiny fractionof a vast multi-dimensional multiverse has gained considerablerecent attention [4]. String theory, Kaluza–Klein theories [5]and many other higher-dimensional theories suggest theexistence of a landscape of vacua with diverse topologies andphysical properties. The landscape generally includes spaceswith different numbers of compactified dimensions (figure 1),with the characteristic size of the compactified dimensionbeing of the order of the Planck length. The symmetries of thiscompactified internal space define the gauge symmetries andtherefore physical laws (types of charges and their interactions)governing a particular region of the multiverse. Topology-changing transitions between vacua with different numbers ofcompact dimensions appear to be especially interesting in thismodel [6]. Such a transition may represent the birth of anindividual physical universe.

Here we demonstrate that using extraordinary waves inanisotropic uniaxial metamaterials, optical models of suchspacetimes as dS3 × S1 (3D de Sitter space with onecompactified dimension) and dS2 × S2 (2D de Sitter spacewith two compactified dimensions) may be realized. Othernon-trivial possibilities, such as the metamaterial models of4D de Sitter dS4 and anti-de Sitter AdS4 spaces [7, 8] arealso shown in figure 1. Nonlinear optics of these metamaterialspaces is shown to resemble the interaction of charges viagauge fields. Together with recent demonstrations of themetamaterial wormholes [9] and black holes [10–12] whichare supposed to connect metamaterial regions having differenteffective topologies, these observations complete the model ofthe metamaterial multiverse presented in figure 1.

Let us start by demonstrating how to produce the dS3 ×S1 electromagnetic spacetime geometry using metamaterials.Spatial geometry of this spacetime may be approximated as aproduct R2 × S1 of a 2D plane R2 and a circle S1, as shown infigure 2(a). Its line element may be written as

dl2 = dx2 + dy2 + R2 dφ2. (1)

Using the stereographic projection z = 2R sinφ/(1 + cosφ),this line element can be rewritten as

dl2 = dx2 + dy2 + dz2

(1 + z2

4R2 )2. (2)

Equation (2) indicates that we need a uniaxial anisotropicmetamaterial in order to emulate the R2 × S1 space. Let

2040-8978/11/024004+04$33.00 © 2011 IOP Publishing Ltd Printed in the UK & the USA1

J. Opt. 13 (2011) 024004 I I Smolyaninov

Figure 1. According to current understanding of the multiverse, it ispopulated by vacua of all possible dimensionalities. A spacelike slicethrough a multi-dimensional multiverse is shown. The dark regionsrepresent black holes and black branes. This is a ‘metamaterial’adaptation of the picture presented in [1].

us consider a non-dispersive (if the operating bandwidth issufficiently narrow, the dispersion can be neglected) and non-magnetic uniaxial anisotropic metamaterial with dielectricpermittivities εx = εy = ε1 and εz = ε2. The wave equation insuch a material can be written [13] as

− ∂2 �Ec2∂ t2

= ↔ε

−1 �∇ × �∇ × �E (3)

where �ε−1 = �ξ is the inverse dielectric permittivity tensorcalculated at the centre frequency of the signal bandwidth.Any electromagnetic field propagating in this material canbe expressed as a sum of the ‘ordinary’ and ‘extraordinary’contributions, each of these being a sum of an arbitrary numberof plane waves polarized in the ‘ordinary’ ( �E perpendicular tothe optical axis) and ‘extraordinary’ ( �E parallel to the planedefined by the k vector of the wave and the optical axis)directions. Let us define a ‘scalar’ extraordinary wavefunctionas ψ = Ez (so that the ordinary portion of the electromagneticfield does not contribute to ψ). Equation (3) then yields

∂2ψ

c2∂ t2= ∂2ψ

ε1∂z2+ 1

ε2

(∂2ψ

∂x2+ ∂2ψ

∂y2

). (4)

The extraordinary field will perceive the optical space as R2 ×S1 if

ε2 = n2 and ε1 = n2

(1 + z2

4R2

)2 (5)

where n is a constant. Similar consideration allows us toconstruct the R1 × S2 optical space shown in figure 2(b). Theline element may be written as

dl2 = R2(dθ2 + sin2 θ dφ2)+ dz2. (6)

Figure 2. ‘Optical space’ in anisotropic uniaxial metamaterials maymimic such topologically non-trivial 3D spaces as (a) R2 × S1, whichis a product of a 2D plane R2 and a circle S1, and (b) R1 × S2, whichis a product of a 1D line R1 and a sphere S2.

Using the stereographic projection, this line element can berewritten as

dl2 = dx2 + dy2

(1 + ρ2

4R2 )+ dz2 (7)

where ρ2 = x2 + y2. The extraordinary field will perceive theoptical space as R1 × S2 if

ε1 = n2 and ε2 = n2

(1 + ρ2

4R2 )2. (8)

This kind of metamaterial can be formed by multiple single-level quantum wells [14], so that

ε2 = ε∞ − ω2p

ω2= ε∞ −n (ω, ρ)

4πe2

mω2and ε1 = εs

(9)where n(ω, ρ) is the frequency- and coordinate-dependentcharge density. In a similar fashion, anisotropic metamaterialdescribed by equation (5) can be formed by an array ofquantum wires. Thus, optics of metamaterials gives us uniquetools to study physics in topologically non-trivial 3D spacesshown in figure 2. This unique and interesting physics isinaccessible by other means, since these spaces cannot fit intonormal Euclidean 3D space.

We are going to consider nonlinear optics of the R2 ×S1 and R1 × S2 metamaterial spaces shown in figure 2 anddemonstrate that it resembles the picture of effective chargesinteracting with each other via gauge fields. This resultis natural, since nonlinear optics of these spaces resemblesKaluza–Klein theories. A somewhat similar picture of mode

2

J. Opt. 13 (2011) 024004 I I Smolyaninov

interaction has been noted previously in the case of cylindricalsurface plasmons [15]. Let us start with nonlinear optics of theR2 × S1 metamaterial space shown in figure 2(a) and describethe effective ‘laws of physics’ which arise in this space. Theselaws are determined by the U(1) symmetry of the internal‘compactified’ S1 space. The eigenmodes of the extraordinaryfield can be written as

ψkL = eikρeiLφ (10)

where L is the quantized ‘angular momentum’ number and kis the 2D momentum. The dispersion law of these eigenmodesis

n2

c2ω2 = k2 + L2

R2. (11)

Therefore, L = 0 extraordinary photons behave asmassless 2D quasiparticles, while L �= 0 photons are massive.Let us demonstrate that, in the nonlinear optical interactionsof extraordinary photons, the ‘angular momentum’ number Lbehaves as a conserved quantized effective charge.

Nonlinear optical effects deform↔ε

−1 = ↔ξ and therefore

deform the line element (1). In the weak-field approximation

corrections to↔ε

−1are small. However, corrections to the off-

diagonal terms of↔ε

−1cannot be neglected. Therefore, the

effective metric of the deformed optical space can be writtenas

ds2 = gαβ dxα dxβ + 2gα3 dxα dφ + g33 dφ2 (12)

where the Greek indices α = 0, 1, 2 indicate coordinates ofthe (almost flat) planar 3D Minkowski spacetime: dx0 = c dt ,dx1 = dx and dx2 = dy. This 3D spacetime is populatedby extraordinary photons (described by a ‘scalar’ wavefunctionψ) which are affected by the vector field gα3 (with componentsg03 = 0, g13 = 2ξ13 dφ/dz and g23 = 2ξ23 dφ/dz) and thescalar field g33 = R2, usually called a dilaton. The dilaton fieldmay be assumed constant in the weak-field approximation. Fora given value of L, the wave equation can be written as

D̂ψ = �ψ− L2 1 − gα3gα3

g33ψ+2iLgα3 ∂ψ

∂xα+ iL

∂gα3

∂xαψ = 0

(13)where � is the covariant three-dimensional d’Alembertoperator. Equation (13) looks the same as the Klein–Gordonequation. In 3D spacetime it describes a particle of mass

m = h̄L

cg1/233

(14)

which interacts with a vector field gα3 via a quantized chargeL. The linear portion of D̂ in equation (13) describes standardmetamaterial optics in which the metamaterial plays the role ofa curvilinear background metric. On the other hand, the third-order optical nonlinearity of the form

g(3)α3 ∝ ψ∂ψ

∂φ, which means that ε(3)αz ∝ Ez

∂Ez

∂z(15)

leads to a Coulomb-like interaction of the effective chargeswith each other: extraordinary photons having L �= 0 act as

sources of the gα3 field, which in turn acts on other ‘charged’extraordinary photons having L �= 0. The effect of interactioncannot be neglected when the kinetic energy term �ψ inequation (13) becomes comparable to the potential energyterms. For example, this may occur if k ∝ Lgα3 = Lgα3

g1/233

, where

k is the momentum component in the xy plane. Therefore, inthe limit k → 0 interaction of ‘effective charges’ is importantand cannot be neglected.

Nonlinear optics of the R1 × S2 metamaterial spaceshown in figure 2(b) can be considered in a similar fashion.The effective ‘laws of physics’ arising in this case are veryinteresting because of the SU(2) symmetry of the internalS2 space. Therefore, the extraordinary photons may havetwo kinds of charges. Motion of these charges is limited tothe z direction. Since Coulomb-like interaction in the one-dimensional case does not depend on distance, interaction ofthe ‘charged’ extraordinary photons is very strong: similarto the strong interaction of quarks, ‘charged’ photons behaveas almost-free particles at short distances from each other,while at large distances the potential energy of two ‘charged’photons grows linearly with distance. It is, however, limited by

metamaterial losses and by dispersion of↔ε

−1. Nevertheless,

strong nonlinear optical interactions of extraordinary photonsin the R1×S2 metamaterial space may be used in novel ‘photonblockade’ devices, which are necessary in quantum computingand quantum communication [16].

In addition to non-trivial nonlinear optics, the describedtoy model of the metamaterial multiverse lets us study metricphase transitions [6] in which the topology of the ‘opticalspace’ changes as a function of temperature or an appliedfield. A similar topological transition may have givenbirth to our own universe. According to some theoreticalmodels [17], during the inflation our 4D universe expandedexponentially at the expense of compactification of the extraspatial dimensions. During this process a large number ofparticles had been created. Metamaterial optics is probablythe only other physical system in which a similar process canbe observed. Let us consider a topological transition fromthe R3 metamaterial space to either R2 × S1 or R1 × S2

topology. The number of photons emitted during such a metrictopology change can be calculated via the dynamical Casimireffect [18, 19]. The total energy E of emitted photons froma phase-changing volume V depends on the photon dispersionlaws ω1(k) and ω2(k) in the respective phases (see equation (3)from [18]):

E

V=

∫d3�k(2π)3

(1

2h̄ω1(k)− 1

2h̄ω2(k)

). (16)

Equation (16) is valid in the ‘sudden change’ approximation, inwhich the dispersion law is assumed to change instantaneously.The detailed discussion of the validity of this approximationcan be found in [19]. Therefore, the number of photons perfrequency interval emitted during the transition can be writtenas

dN

V dω= 1

2

(dn1

dω− dn2

dω

)(17)

3

J. Opt. 13 (2011) 024004 I I Smolyaninov

where dni/dω are the photonic densities of states inside therespective phases. Since the photonic density of states in eitherthe R2 × S1 or R1 × S2 phase is much smaller than in the R3

phase, the total number of photons emitted is

dN

V≈ 1

2

(dn

dω

) ω (18)

where (dn/dω) is the usual black-body photonic density ofstates in R3 and ω is the frequency interval in which thedielectric tensor of the uniaxial metamaterial satisfies eitherequations (5) or (8).

In conclusion, we have demonstrated that the optics ofmetamaterials presents us with new opportunities to engineertopologically non-trivial ‘optical spaces’. Nonlinear opticsof extraordinary light in these spaces resembles the Coulombinteraction of effective charges. Therefore, novel ‘photonblockade’ devices may be engineered. Topology-changingphase transitions in such metamaterials resemble the birth ofa physical universe.

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