Towards an experimental realization of affinely transformed linearized QED vacuum via inverse homogenization
aa r X i v : . [ phy s i c s . op ti c s ] S e p Towards an experimental realization of affinelytransformed linearized QED vacuum via inversehomogenization
Tom G. Mackay School of Mathematics and Maxwell Institute for Mathematical SciencesUniversity of Edinburgh, Edinburgh EH9 3JZ, UK and
NanoMM — Nanoengineered Metamaterials GroupDepartment of Engineering Science and MechanicsPennsylvania State University, University Park, PA 16802–6812, USA
Akhlesh Lakhtakia NanoMM — Nanoengineered Metamaterials GroupDepartment of Engineering Science and MechanicsPennsylvania State University, University Park, PA 16802, USA and
Materials Research InstitutePennsylvania State University, University Park, PA 16802, USA
Abstract
Within the framework of quantum electrodynamics (QED), vacuum is a nonlinear medium whichcan be linearized for a rapidly time-varying electromagnetic field with a small amplitude subjected toa magnetostatic field. The linearized QED vacuum is a uniaxial dielectric-magnetic medium for whichthe degree of anisotropy is exceedingly small. By implementing an affine transformation of the spatialcoordinates, the degree of anisotropy may become sufficiently large as to be readily perceivable. Theinverse Bruggeman formalism can be implemented to specify a homogenized composite material (HCM)which is electromagnetically equivalent to the affinely transformed QED vacuum. This HCM can arisefrom remarkably simple component materials; for example, two isotropic dielectric materials and twoisotropic magnetic materials, randomly distributed as oriented spheroidal particles.
Keywords: quantum electrodynamics, vacuum birefringence, homogenization, inverse Bruggeman formal-ism
Classical vacuum is a linear medium for which the principle of superposition holds. Consequently, lightpropagation in classical vacuum is unaffected by the presence of a magnetostatic field. However, within the E–mail: [email protected] E–mail: [email protected] × T symbol. The 3 × I . The permittivity and permeability of classical vacuumare written as ǫ = 8 . × − F m − and µ = 4 π × − H m − , respectively. We consider vacuum under the influence of a magnetostatic field B s = | B s | ˆ B s . In classical vacuum, thepassage of light is unaffected by B s , as reported by an inertial observer. However, this is not the casefor the QED vacuum. The QED vacuum is a nonlinear medium which can be linearized for rapidly time–varying plane waves. Thereby, for propagation of light, QED vacuum is represented by the anisotropicdielectric–magnetic constitutive relations [3] D = ǫ ǫ • EB = µ µ • H ) . (1)The relative permittivity and relative permeability dyadics of the QED vacuum have the uniaxial forms [8] ǫ = (cid:0) − a | B s | (cid:1) (cid:16) I − ˆ B s ˆ B s (cid:17) + (cid:0) a | B s | (cid:1) ˆ B s ˆ B s µ = 11 − a | B s | (cid:16) I − ˆ B s ˆ B s (cid:17) + 11 − a | B s | ˆ B s ˆ B s , (2)where the constant a = 6 . × − H − kg − m s . The constitutive dyadics (2) were derived by Adler[3] from the Heisenberg–Euler effective Lagrangrian of the electromagnetic field [9, 10].Since a is exceedingly small, the value of a | B s | is also exceedingly small in comparison with unityfor typical values of | B s | . For example, in the PVLAS experiment | B s | = 5 T typically [5], which yields a | B s | = 1 . × − . Accordingly, the degree of anisotropy represented by the constitutive dyadics (2)is also exceedingly small. In order to achieve degrees of anisotropy that could be realistically attained in acontrolled manner for a practical simulation of QED vacuum, we implement the affine transformation x x ′ ≡ J • x (3)of the spatial coordinates. The transformation dyadic J = p (cid:16) I − ˆ B s ˆ B s (cid:17) + q ˆ B s ˆ B s (4)2mploys p = s (1 − a | B s | ) (1 + 20 a | B s | )(1 − δ | B s | ) (1 + 5 δ | B s | ) q = 1 − a | B s | − δ | B s | (5)and the scalar parameter δ >
0. For definiteness, we fix δ = 0 .
02. Thus, the affine-transformed relativepermittivity and permeability dyadics are given as [11] ǫ ′ ≡ ǫ J • ǫ • J T (6)= (1 − δ | B s | ) (cid:16) I − ˆ B s ˆ B s (cid:17) + (1 + 5 δ | B s | ) ˆ B s ˆ B s (7) ≡ ǫ ′ t (cid:16) I − ˆ B s ˆ B s (cid:17) + ǫ ′ s ˆ B s ˆ B s (8)and µ ′ ≡ µ J • µ • J T (9)= 1 − δ | B s | (1 − a | B s | ) (cid:16) I − ˆ B s ˆ B s (cid:17) + 1 + 5 δ | B s | − a | B s | (1 + 120 a | B s | ) ˆ B s ˆ B s (10) ≡ µ ′ t (cid:16) I − ˆ B s ˆ B s (cid:17) + µ ′ s ˆ B s ˆ B s . (11)Notice that for the range | B s | ∈ (0 ,
10) T, the denominators of both terms on the right side of Eq. (10) areboth approximately equal to unity, and therefore ǫ ′ ≈ µ ′ . The components ǫ ′ s and ǫ ′ t are linearly dependentupon | B s | , as illustrated in Fig. 1. The plots of µ ′ s,t versus | B s | are practically identical to those of ǫ ′ s,t . Let us now turn to the question: How can one specify an HCM which is a uniaxial dielectric-magneticmaterial with relative permittivity dyadic ǫ HCM ≡ ǫ ′ and relative permeability dyadic µ HCM ≡ µ ′ ? Inorder to answer this question, we make use of the well-established Bruggeman homogenization formalism[12, 13].Suppose we consider the homogenization of four component materials, labelled a , b , c and d . Two ofthe components ( a and b , say) are isotropic dielectric materials while the other two ( c and d ) are isotropicmagnetic materials. Thus, the component materials are specified by(i) the relative permittivities ǫ a , ǫ b , ǫ c and ǫ d , with ǫ c = ǫ d = 1; and(ii) the relative permeabilities µ a , µ b , µ c and µ d , with µ a = µ b = 1.The four component materials are randomly distributed with volume fractions f a , f b , f c , f d ∈ (0 , f d = 1 − f a − f b − f c . All four component materials consist of identically oriented spheroidal particles. Thesymmetry axis for all these spheroidal particles lies parallel to ˆ B s . Accordingly, the surface of each spheroidrelative to its centre is prescribed by the vector r s = ρ ℓ U ℓ · ˆ r, (12)wherein the shape dyadic U ℓ = (cid:16) I − ˆ B s ˆ B s (cid:17) + U ℓ ˆ B s ˆ B s , ( ℓ = a, b, c, d ) , (13)3s real symmetric [14] and positive definite, the radial unit vector is ˆ r , and the linear measure ρ ℓ is requiredto be small compared to the electromagnetic wavelengths under consideration. The shape parameter U ℓ > < U ℓ < τ HCM = τ HCMt (cid:16) I − ˆ B s ˆ B s (cid:17) + τ HCMs ˆ B s ˆ B s , ( τ = ǫ, µ ) . (14)For the particular case of the uniaxial dielectric-magnetic HCM involved here, full details of the numericalprocess of computing the dyadics ǫ HCM and µ HCM , from a knowledge of ǫ a,b,c,d , µ a,b,c,d , U a,b,c,d and f a,b,c,d ,are provided elsewhere [6].Conventionally, homogenization formalisms are used to estimate the constitutive parameters of HCMs,based on a knowledge of the constitutive and morphological parameters of their component materials andtheir volume fractions. In contrast, here our goal is to estimate the constitutive and morphological parametersas well as the volume fractions of the component materials which give rise to a HCM such that ǫ HCM coincideswith ǫ ′ and µ HCM coincides with µ ′ . We do so via an inverse implementation of the Bruggeman formalism.Formal expressions of the inverse Bruggeman formalism are available [15], but in some instances these can beill-defined [16]. In practice, the inverse formalism may be more effectively implemented by direct numericalmethods [17]. Note that certain constitutive parameter regimes have been identified as problematic for theinverse Bruggeman formalism [18], but these regimes are not the same as those considered here.We consider the following three different implementations of the inverse Bruggeman formalism. In eachimplementation, four scalar parameters are to be determined.I. The relative permittivities ǫ a,b and the relative permeabilities µ c,d are assumed to be known, and allspheroidal particles have the same shape, i.e., U a = U b = U c = U d ≡ U . We then determine thecommon shape parameter U and the volume fractions f a , f b and f c .II. The relative permittivities ǫ a,b and the relative permeabilities µ c,d are assumed to be known, and thevolume fractions f a,b,c are fixed. We then determine the shape parameters U a , U b , U c and U d .III. The shape parameters U a,b,c,d and the volume fractions f a,b,c are fixed. We then determine the relativepermittivities ǫ a,b and relative permeabilities µ c,d .To describe the inversion of the Bruggeman formalism, let us focus on implementation I as a repre-sentative example, the inversion processes for implementations II and III being analogous. Suppose that (cid:8) ˘ ǫ HCMs , ˘ ǫ HCMt , ˘ µ HCMs , ˘ µ HCMt (cid:9) are forward Bruggeman estimates of the HCM’s relative permittivity andrelative permeability parameters which are computed for physically reasonable ranges of the parameters U and f a,b,c ; i.e., U ∈ ( U − , U + ) and f a,b,c ∈ (cid:16) f − a,b,c , f + a,b,c (cid:17) . Next:(1) Let f a = ( f − a + f + a ) / f b = (cid:0) f − b + f + b (cid:1) /
2, and f c = ( f − c + f + c ) /
2. For all U ∈ ( U − , U + ), determinethe value U † which yields the minimum value of the scalar quantity∆ = h (cid:18) ˘ ǫ HCMs − ǫ ′ s ǫ ′ s (cid:19) + (cid:18) ˘ ǫ HCMt − ǫ ′ t ǫ ′ t (cid:19) + (cid:18) ˘ µ HCMs − µ ′ s µ ′ s (cid:19) + (cid:18) ˘ µ Brt − µ ′ t µ ′ t (cid:19) i / . (15)(2) Let U = U † , f b = (cid:0) f − b + f + b (cid:1) /
2, and f c = ( f − c + f + c ) /
2. For all f a ∈ ( f − a , f + a ), determine the value f † a which yields the minimum value of ∆.(3) Let U = U † , f a = f † a , and f c = ( f − c + f + c ) /
2. For all f b ∈ (cid:0) f − b , f + b (cid:1) , determine the value f † b whichyields the minimum value of ∆.(4) Let U = U † , f a = f † a , and f b = f † b . For all f c ∈ ( f − c , f + c ), determine the value f † c which yields theminimum value of ∆. 4he steps (1)–(4) are then repeated, with f † a , f † b , and f † c being the fixed values of f a,b,c in step (1), f † b and f † c being the fixed values of f b,c in step (2), and f † c being the fixed value of f c in step (3), until the value of∆ becomes acceptably small. Numerical illustrations of the implementations I–III are provided in Figs. 2–4. For all results presented, thedegree of convergence of the numerical schemes which provide the inverse Bruggeman estimates was < < . ǫ a = 4, ǫ b = 0 . µ c = 3 . µ d = 0 .
4. The computed common shape parameter U and volume fractions f a,b,c areplotted versus | B s | . While the volume fractions vary little as | B s | is increased from 1 to 2.5 T, thecommon shape parameter increases exponentially.II. The constitutive parameters of the component materials were again taken to be ǫ a = 4, ǫ b = 0 . µ c = 3 . µ d = 0 . f a = 0 . f b = 0 . f c = 0 .
21. The computed four shape parameters U a,b,c,d are plotted versus | B s | . All four shapeparameters increase uniformly as | B s | is increased from 1 to 2 T. This reflects the fact that the degreeof anisotropy of the HCM is required to increase as | B s | increases.III. Lastly, in Fig. 4 the common shape parameter is fixed at U = 5 while the volume fractions are fixed at f a,b,c = 2 .
5. The computed constitutive parameters ǫ a,b and µ c,d are plotted versus | B s | . In this case, ǫ a turns out to be approximately the same as µ c . And similarly ǫ b turns out to be approximately thesame as µ d . While ǫ a and µ c decrease uniformly as | B s | is increased from 1 to 3 T, the opposite is trueof ǫ b and µ d . By means of the inverse Bruggeman formalism, an HCM may be specified which is electromagneticallyequivalent to the QED vacuum subject to a spatial affine transformation. The affinely transformed QEDvacuum retains the same uniaxial dielectric-magnetic form as the un-transformed QED vacuum, but the degeeof anisotropy is greatly exaggerated by means of the affine transformation. By reversing the transformationrepresented by eq. (3), the properties of QED vacuum may be inferred from those of the HCM.For illustration, the inverse homogenization formulation presented here was based on four isotropic com-ponent materials. However, the desired HCM could also be realized by alternative inverse homogenizationformulations. For example, the HCM could arise from only two component materials. These two compo-nents materials could be either both isotropic dielectric-magnetic materials distributed as oriented spheroidalparticles or both uniaxial dielectric-magnetic materials (with parallel symmetry axes) distributed as spher-ical particles [13, 19]. However, the four-component formulation presented here involves the simplest ofcomponent materials and allows a large degree of freedom in choosing their constitutive parameters.Finally, let us note that the relative permittivities and relative permeabilities of the component materialsneeded for the HCM, as presented in Figs. 2–4, are not at all infeasible. Indeed, present-day technology allowsfor the possibility of materials with a considerably wider range of constitutive parameters to be engineered[20, 21, 22].
Acknowledgment:
AL thanks the Charles Godfrey Binder Endowment at Penn State for partial financialsupport of his research activities. 5 eferences [1] J. D. Jackson, Classical Electrodynamics, 3rd ed., Wiley, New York, NY, USA, 1999, pp. 9-13.[2] S. L. Adler, J. Phys. A: Math. Theor. 40 (2007) F143; correction: 40 (2007) 5767.[3] S. L. Adler, Ann. Phys. (NY) 67 (1971) 599.[4] E. Iacopini, E. Zavattini, Phys. Lett. B 85 (1979) 151.[5] E. Zavattini, G. Zavattini, G. Ruoso, E. Polacco, E. Milotti, M. Karuza, U. Gastaldi, G. Di Domenico,F. Della Valle, R. Cimino, S. Carusotto, G. Cantatore, M. Bregant, Phys. Rev. Lett. 96 (2006) 110406.See also Editorial Note, Phys. Rev. Lett. 99 (2007) 129901.[6] T. G. Mackay, A. Lakhtakia, Phys. Rev. B 83 (2011) 195424.[7] T. G. Mackay, A. Lakhtakia, http : // arxiv . org / abs / . [8] A. Lakhtakia, T. G. Mackay, Electromagnetics 27 (2007) 341.[9] W. Heisenberg, H. Euler, Z. Phys. 98 (1936) 714.[10] J. Schwinger, Phys. Rev. 82 (1951) 664.[11] M. Yan, W. Yan, M. Qiu, Prog. Opt. 52 (2009) 261.[12] W. S. Wiglhofer, A. Lakhtakia, B. Michel, Microw. Opt. Technol. Lett. 15 (1997) 263; correction: 22(1999) 221.[13] T. G. Mackay, A. Lakhtakia, Electromagnetic Anisotropy and Bianisotropy: A Field Guide, WorldScientific, Singaore, 2010, chap. 6.[14] A. Lakhtakia, Microw. Opt. Technol. Lett. 27 (2000) 175.[15] W. S. Weiglhofer, Microw. Opt. Technol. Lett. 28 (2001) 421.[16] E. Cherkaev, Inverse Problems 17 (2001) 1203.[17] T. G. Mackay, A. Lakhtakia, J. Nanophoton. 4 (2010) 041535.[18] S. S. Jamaian, T. G. Mackay, J. Nanophoton. 4 (2010) 043510.[19] T. G. Mackay, W. S. Weiglhofer, J. Opt. A: Pure Appl. Opt. 2 (2000) 426.[20] A. Al`u, M. Silveirinha, A. Salandrino, N. Engheta, Phys. Rev. B 75 (2007) 155410.[21] G. Lovat, P. Burghignoli, F. Capolino, D. R. Jackson, IET Microw. Antennas Propagat. 1 (2007) 177.[22] M. N. Navarro-C´ıa, M. Beruete, I. Campillo, M Sorolla, Phys. Rev. B 83 (2011) 115112.6 È B s È H T L Ε ¢ s , t Figure 1: The relative permittivity parameters ǫ ′ s (blue, dashed) and ǫ ′ t (red, solid) plotted versus | B s | (T). Ε a = Ε b = Μ c = Μ d = È B s È H T L f a , b , c U Figure 2: Implementation I. The common shape parameter U (thick solid, red) and volume fractions f a (dashed, green), f b (broken dashed, blue), and f c (thin solid, blue) plotted versus | B s | (T). The relativepermittivities ǫ a = 4, ǫ b = 0 . µ c = 3 . µ d = 0 . a = Ε b = Μ c = Μ d = f a = f b = f c = È B s È H T L U a , b , c , d Figure 3: Implementation II. The shape parameters U a (thick solid, red), U b (dashed, green), U c (brokendashed, blue), and U d (thin solid, blue) plotted versus | B s | (T). The relative permittivities ǫ a = 4, ǫ b = 0 . µ c = 3 . µ d = 0 .
4; and the volume fractions f a = 0 . f b = 0 .
25 and f c = 0 . U = f a , b , c = È B s È H T L Ε a , Μ c Ε b , Μ d Figure 4: Implementation III. The relative permittivities ǫ a (thick solid, red) and ǫ b (dashed, green) and therelative permeabilities µ c (broken dashed, blue) and µ d (thin solid, blue) plotted versus | B s | (T). The shapeparameter U = U a,b,c,d = 5 and volume fractions f a,b,c = 0 ..