Analogue gravity in hyperbolic metamaterials

Transcription

1 1 Analogue gravity in hyperbolic metamaterials Igor I. Smolyaninov Department of Electrical an Computer Engineering, University of Marylan, College Park, MD 074, USA Sub-wavelength confinement of light in nonlinear hyperbolic metamaterials ue to formation of spatial solitons has attracte much recent attention because of its seemingly counter-intuitive behavior. In orer to achieve self-focusing in a hyperbolic wire meium, a nonlinear self-efocusing Kerr meium must be use as a ielectric host. Here we emonstrate that this behavior fins natural explanation in terms of analogue gravity. Wave equation escribing propagation of extraorinary light insie hyperbolic metamaterials exhibits +1 imensional Lorentz symmetry. The role of time in the corresponing effective 3D Minkowski spacetime is playe by the spatial coorinate aligne with the optical axis of the metamaterial. Nonlinear optical Kerr effect bens this spacetime resulting in effective gravitational force between extraorinary photons. In orer for the effective gravitational constant to be positive, negative self-efocusing Kerr meium must be use as a host. If gravitational self-interaction is strong enough, spatial soliton may collapse into a black hole analogue. Light propagation through hyperbolic metamaterials has attracte much recent attention ue to their ability to guie an manipulate electromagnetic fiels on a spatial scale much smaller than the free space wavelength [1-6]. Almost immeiately it was realize that nonlinear optical effects may further increase light confinement in hyperbolic

2 metamaterials base on metal nanowires an metal nanolayers immerse in a Kerr-type ielectric host (Fig.1). Several reports emonstrate existence of spatial solitons in an array of nanowires embee in a Kerr meium [7-10]. An interesting counter-intuitive feature of these solitons is that they occur only if a self-efocusing Kerr meium is use as a ielectric host. These interesting results have been obtaine base on either couple-moe theory [7,8], or on a etaile analysis of trappe states in nonlinear hyperbolic meia [9]. While efinitely vali, these approaches provie limite physical insight of this an other more general problems of soliton interaction in nonlinear hyperbolic metamaterials. In this paper we emonstrate that behavior of spatial solitons in nonlinear hyperbolic metamaterials fins natural explanation in terms of analogue gravity. Wave equation escribing propagation of extraorinary light insie hyperbolic metamaterials exhibits +1 imensional Lorentz symmetry. The role of time in the corresponing effective 3D Minkowski spacetime is playe by the spatial coorinate aligne with the optical axis of the metamaterial. Nonlinear optical Kerr effect bens this spacetime resulting in effective gravitational force between extraorinary photons. In orer for the effective gravitational constant to be positive, negative self-efocusing Kerr meium must be use as a host. While this observation explains basic finings of refs.[7-10], it has much more general consequences for soliton physics in hyperbolic metamaterials, since both classic an quantum gravity in +1 spatial imensions is known to be an exactly soluble system [11]. As a starting point, let us recall basic features of hyperbolic metamaterial moeling using +1 imensional Minkowski spacetime. Recent avances in electromagnetic metamaterials enable esign of novel physical systems which can be escribe by effective space-times having very unusual metric an topological properties. In particular, hyperbolic metamaterials offer an interesting experimental winow into physics of Minkowski spacetimes, since propagation of extraorinary light insie a hyperbolic metamaterial is escribe by wave equation exhibiting +1

3 3 imensional Lorentz symmetry [1]. The role of time in the corresponing effective 3D Minkowski spacetime is playe by the spatial coorinate, which is oriente along the optical axis of the metamaterial [13]. This spacetime may be mae causal by breaking the mirror an temporal symmetries of the metamaterial, which results in one-way light propagation along the timelike spatial coorinate [14]. Two ifferent Minkowski spacetimes may collie forming a Minkowski omain wall [15,16], while bening the effective spacetime may lea to an experimental moel of the big bang [13]. Uner certain circumstances, thermal fluctuations of the metamaterial may cause transient formation of hyperbolic regions (3D Minkowski spacetimes) insie the metamaterial [17], so that the resulting picture of multiple transient Minkowski spacetimes looks somewhat similar to cosmological multiverse. Interestingly, the physical vacuum itself behaves as a hyperbolic metamaterial when subjecte to very strong magnetic fiel [18,19]. Therefore, the Minkowski spacetime analogues mentione above appear to be quite meaningful. Despite this rich an interesting physics, all the metamaterial-base Minkowski spacetime moels escribe so far were limite in one very important respect: the effective metric was ecouple from the matter content (photons) of these spacetimes. Here we consier nonlinear optical effects which ben the effective Minkowski spacetime resulting in gravity-like interaction of extraorinary photons. We emonstrate that nonlinear optical Kerr effect results in effective gravitational force between extraorinary photons. If gravitational self-interaction is strong enough, spatial soliton may collapse into a black hole analogue. First, let us emonstrate that the wave equation escribing propagation of monochromatic extraorinary light insie a hyperbolic metamaterial oes inee exhibit +1 imensional Lorentz symmetry. A etaile erivation of this result can be foun in refs.[1,13]. We assume that the metamaterial in question is uniaxial an non-magnetic (μ=1), so that electromagnetic fiel insie the metamaterial may be separate into

4 4 orinary an extraorinary waves (vector E r of the extraorinary light wave is parallel to the plane efine by the k vector of the wave an the optical axis of the metamaterial). Since hyperbolic metamaterials exhibit strong temporal ispersion, we will work in the frequency omain an assume that in some frequency ban aroun ω=ω 0 the metamaterial may be escribe by anisotropic ielectric tensor having the iagonal components ε xx = ε yy = ε 1 >0 an ε zz = ε <0. In the linear optics approximation all the non-iagonal components are assume to be zero. Propagation of extraorinary light in such a metamaterial may be escribe by a coorinate-epenent wave function ϕ ω =E z obeying the following wave equation [1,13]: ω ϕ = ε z 1 ϕ + ε x ϕ ω + y ω ω ϕ ω c 1 (1) This wave equation coincies with the Klein-Goron equation for a massive scalar fiel ϕ ω in 3D Minkowski spacetime: ϕω 1 ϕω ϕω ω0 m* c + = ϕω = ϕ ω ε1 z ( ε ) + x y () c h in which spatial coorinate z=τ behaves as a timelike variable. Eq.() escribes worl lines of massive particles which propagate in a flat +1 imensional Minkowski spacetime [1,13]. Note that components of metamaterial ielectric tensor efine the effective metric g ik of this spacetime: g 00 =-ε 1 an g 11 =g =-ε. When the nonlinear optical effects become important, they are escribe in terms of various orer nonlinear susceptibilities χ (n) of the metamaterial: D i (1) () (3) = χij E j + χijl E j El + χijlme jel Em +... (3)

5 5 Taking into account these nonlinear terms, the ielectric tensor of the metamaterial (which efines its effective metric) may be written as (1) () (3) ε χ + χ E + χ E E +... (4) ij = ij ijl l ijlm l m It is clear that eq.(4) provies coupling between the matter content (photons) an the effective metric of the metamaterial spacetime. However, in orer to emulate gravity, the nonlinear susceptibilities χ (n) of the metamaterial nee to be engineere in some particular way. This task may be quite complicate ue to effects of spatial ispersion, which may become prominent in hyperbolic metamaterials [19]. High frequency macroscopic electroynamics of metamaterials may be escribe using two equivalent languages [0]. We can either introuce magneto-electric mouli relating (D,H) an (E,B) pairs: or assume that r tr tr D = ε E +αb, (5) r t r tr H = μ 1 B + βe, r tr D = εe an H r = B r t, while tensor ( 0, k r ε ω ) exhibits linear (o) terms in spatial ispersion: r (1) () ε ij ( ω0, k) = ε ij ( ω0,0) + γ ijl kl + γ ijlmklkm +..., (6) Connection between these two escriptions is easy to establish for time harmonic plane waves, since r c r c B = rote = iω ω r r [ k E], (7) When nonlinear optical effects nee to be taken into account, the secon choice is more convenient. Thus, we are going to assume that in the most general case all the χ (n) terms in eq.(4) may epen on the photon wave vectors. This general framework encompasses

6 6 all kins of effective gravity theories, which are much more complicate than usual general relativity. Our goal is to fin what kin of simplifications of this general framework may lea to metamaterial moels which emulate usual gravity. In the weak gravitational fiel limit the Einstein equation R k i 8πγ k 1 k = Ti δi T 4 (8) c is reuce to R πγ = Δφ = Δg 00 = T 4 00, (9) c c where φ is the gravitational potential [1]. Since in our effective Minkowski spacetime g 00 is ientifie with -ε 1, comparison of eqs. (4) an (9) inicates that all the secon orer nonlinear susceptibilities χ () ijl of the metamaterial must be equal to zero, while the thir orer terms may provie correct coupling between the effective metric an the energy-momentum tensor. These terms are associate with the optical Kerr effect. All the higher orer χ (n) terms must be zero at n>3. Inee, etaile analysis inicates that Kerr effect in a hyperbolic metamaterial leas to effective gravity. Since z coorinate plays the role of time, while g 00 is ientifie with -ε 1, eq.(9) must be translate as ( ) 16πγ 16πγ Δ ε1 = Tzz = σ 4 4 zz, (10) c c where Δ () is the D Laplacian operating in the xy plane, γ is the effective gravitational constant, an σ zz is the zz component of the Maxwell stress tensor of the electromagnetic fiel in the meium: 1 1 r r r r σ zz = DzEz + H zbz ( DE + HB) (11) 4π

7 7 Let us fin a contribution to σ zz, which is mae by a single extraorinary plane wave propagating insie the hyperbolic metamaterial. Assuming without a loss of generality that the B fiel of the wave is oriente along y irection, the other fiel components may be foun from Maxwell equations as ω c kzby = ε1ex, an kxby ε Ez Taking into account the ispersion law of the extraorinary wave ω = (1) c ω c k kx + k z y = +, (13) ε ε 1 the contribution to σ zz from a single plane wave is σ zz c B k = (14) z 4πω ε1 Thus, for a single plane wave eq.(10) may be rewritten as (0) ( ε + δε ) ( ) () 4γ B kz Δ ε1 = Δ 1 1 = kxδε1 =, (15) c ω ε 1 where we have assume that nonlinear corrections to ε 1 are small, so that we can separate ε 1 into the constant backgroun value ε (0) 1 an weak nonlinear corrections. These nonlinear corrections o inee look like the Kerr effect assuming that the extraorinary photon wave vector components are large compare to ω/c: 4γ B k 4γ B δε = z (3) 1 = χ B (16) c ω ε1kx c ω ε The latter assumption has to be the case inee if extraorinary photons may be consiere as classic particles. Eq.(16) establishes connection between the effective

8 8 gravitational constant γ* an the thir orer nonlinear susceptibility χ (3) of the hyperbolic metamaterial. Since ε xx = ε yy = ε 1 >0 an ε zz = ε <0, the sign of χ (3) must be negative for the effective gravity to be attractive. For a metal wire array metamaterial shown in Fig.1(b) the iagonal components of the ielectric tensor may be obtaine using Maxwell-Garnett approximation []: ε = ε = nε + 1 ( n) ε z m (17) ε = ε 1 x, y = nε m ε + (1 n) ε ( ε + ε ) ( 1 n) ( ε + ε m ) + nε m (18) where n is the volume fraction of the metallic phase (assume to be small), an ε m an ε are the ielectric permittivities of the metal an ielectric phase, respectively. Since - ε m >> ε, eq.(18) may be simplifie as (1 + n) ε 1 ε ~ ε (19) (1 n) Thus, we have recovere the main result of refs.[7-9]: in orer to obtain attractive effective gravity the ielectric host meium must exhibit negative (self-efocusing) Kerr effect. Extraorinary light rays in such a meium will behave as +1 imensional worl lines of self-gravitating boies an may collapse into sub-wavelength spatial solitons. Let us moify eq.() by taking into account self-efocusing Kerr effect of the ielectric host. Assuming a spatial soliton-like solution which conserves energy per unit length W~P/c (where P is the laser power), the soliton with ρ an the magnetic fiel amplitue B are relate as B ρ = P / c (0)

9 9 As a result, eq.() must be re-written as ϕω 1 ϕω ϕω ω0 + + = ϕ (1) ε1 (0) (3) ω ( χ ) P ( ε ) x y c cρ z where ε 1 (0) contribution to is the ielectric permittivity component at P=0 (note that nonlinear a black hole-like singularity at ε nε m may be neglecte). Effective metric escribe by eq.(1) has (3) ( χ ) ρ = cε (0) 1 P 1/ () Let us evaluate if this singularity may be observe in experiment. In orer to be observable, the critical value escribe by eq.() must be larger than the metamaterial structure parameter (the inter-wire istance). On the other han, light intensity must be small enough, so that higher orer nonlinear effects may be neglecte. It is obvious that negative Kerr effect in natural ielectrics is not strong enough to observe this singularity. On the other han, artificial self-efocusing ielectrics having low ε (0) may be engineere base on nanoparticle suspensions in liquis. Such suspensions are wiely available commercially. Because of the large an negative thermo-optic coefficient inherent to most liquis, heating prouce by partial absorption of the propagating beam translates into a significant ecrease of the refractive inex at higher light intensity. For example, reporte thermo-optics coefficient of water reaches Δn/ΔT= - 5.7x10-4 K -1 [3]. Moreover, introucing absorbent ye into the liqui allows for increase thermal nonlinear response [4]. The ielectric properties of nanoparticles neee to obtain an artificial low ε (0) meium may be calculate using Maxwell-Garnett approximation [5]:

10 10 ε ε l ε n ε l = α n ε + ε ε + ε l n l (3) where α n is the volume fraction of the nanoparticles (assume to be small), an ε n an (0) ε l are the ielectric permittivities of nanoparticles an liqui, respectively. A low ε meium is obtaine if ε ε n l. Therefore, either plasmonic oxies or nitraes woul be the best nanoparticle choice in the visible an near infrare ranges [6]. Base on the thermo-optics coefficient of water, Δε ~0.1 may be obtaine at quite realistic ΔT~50K. Therefore, ε (0) 1 ~0.1 may be use in eq.() to estimate the critical value of soliton raius as a function of laser power. This estimate is presente in Fig.. The critical soliton raius at P=100W equals ~0 nm, which oes not look completely unrealistic from the fabrication stanpoint. While achieving such critical values with CW lasers seems implausible, using pulse laser efinitely looks like a realistic option since thermal amage prouce by a pulse laser typically epens on the pulse energy an not the pulse power. In conclusion, we have consiere nonlinear optical effects in hyperbolic metamaterials, an emonstrate that negative nonlinear optical Kerr effect results in effective gravitational force between extraorinary photons. Our observation explains formation of subwavelength spatial solitons in metal wire array hyperbolic metamaterials base on self-efocusing ielectric host. The propose physical mechanism has much more general consequences for soliton interaction in hyperbolic metamaterials, since both classic an quantum gravity in +1 spatial imensions is known to be an exactly soluble system [11].

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13 13 Figure Captions Figure 1. (Color online) Typical geometries of nonlinear hyperbolic metamaterials: (a) multilayer metal-ielectric structure (b) metal wire array structure. The ielectric host ε exhibits nonlinear optical Kerr effect. Figure. Critical raius of spatial soliton calculate as a function of laser power for a metal wire array hyperbolic metamaterial built with an artificial self-efocusing liqui ielectric having ε (0) 1 ~0.1.

14 Fig. 1 14

15 15 Critical soliton raius (nm) Laser power (W) Fig.