aa r X i v : . [ phy s i c s . op ti c s ] O c t Photonic Phase Transitions in Certain Disordered Media
Chin Wang
1, 2 and Luogen Deng Department of Physics, Beijing Institute of Technology, Beijing, China 100081 Department of Physics, Hunan University of Arts and Science, Changde, China 415000 (Dated: November 20, 2018)In this paper we develop a generalized mode-expansion scheme for the vector lightwaves propa-gating in 3D disordered media, and find the photonic phase transition from an extended-state (ES)phase to a mixed-state (MS) phase as a localization intensity parameter G rises above 1 / √
2. Forthe disordered media consisting of plenty of uniform spherical scattering particles, we formulate thisphenomena at first in terms of actual optical and geometric parameters of the disordered media.
In spite of the fascinating behavior of the classicalphase transitions many researchers also devoted theirgreat enthusiasm to some novel phase transitions con-cerning with light or photons [1–4]. Recently, structuralstrain in photonic resonator crystals has been used to in-duce a phase transition between different light modes [2],a photonic system has been designed that may undergoa Mott insulator to superfluid quantum phase transi-tion [3], and certain common media, such as air or oxy-gen, have been proposed to support the propagation ofsteady light waves that appearing in the fermionic andliquid phases, and the transition from the former to thelatter [4]. Various phase transitions of light are essen-tially the results of delicate light-matter interaction. Afamiliar example is that the localization-delocalizationtransition of light modes in moderately disordered dielec-tric superlattices is sensitively dependent on the dielec-tric mismatch between scattering structures and back-ground [5]. However, for the more general case of fullydisordered dielectric media, the perturbation technolo-gies based on the band theory are inapplicable due tothe absence of the lattice symmetry in these dielectricstructures. Likewise, tracing the process of the multi-ple scattering of light in random scattering media maynot be a wisdom, as in this scheme one had to deal withthe excessively complicated mechanism of the dependentscattering of light, and consequently the transport meanfree path and the wavelength of photons need to be renor-malized.The study reported in this paper has been motivatedby the difficulties mentioned above. We consider that inordinary position space disordered dielectric media havecomplex representations of refractive index profiles whileusually in wave vector space sparse components, and it isan interesting issue weather some information for light lo-calization can be distilled directly from the optical struc-ture factors (OSFs) of disordered dielectric media, whichare defined by the Fourier transformation of optical po-tentials. For the idealized linear, but inhomogeneous,dielectric media we present the eigen equation of lightwaves by expanding both light fields and optical poten-tials in any set of generalized Fourier basis. Based on thisequation we derive a necessary condition supporting the strong localization, namely, the Anderson localization [6]of light, and predict the photonic phase transitions be-tween the ES phase and the MS phase. For the disorderedmedia consisting of plenty of uniform spherical scatter-ing particles, we formulate this phenomena at first, to ourbest knowledge, in terms of actual optical and geometricparameters of the disordered media.We start from the wave equation for the complex elec-tric field E of radiation in idealized linear and isotropicpolarized, but inhomogeneous, dielectric media, ∇ E − µ ∂σ E ∂t − µ ∂ ε E ∂t − ∇ ( ∇ · E ) = 0 , (1)where ε , µ and σ are, in general, the scalar position-and frequency-dependent dielectric constant, magneticpermeability and conductance, respectively. Below wewill consider media for which the magnetic permeabilityis approximately constant. The single-frequency solu-tion of the Eq.(1) is assumed to have an expression of E ( r , t ) = Ψ ( r ) e − iωt , and consequently the electric am-plitude Ψ ( r ) fulfills ∇ × ( ∇ × Ψ ( r )) − U ( ω, r ) Ψ ( r ) = µε ω Ψ ( r ) , (2)in which the optical potential U ( ω, r ), analogous to thatin Sch¨ordinger equation for electrons, is defined in thepresent paper by U ( ω, r ) = µε ω χ ( ω, r ) + iωµσ ( ω, r )= f ( ω, r ) + iωg ( ω, r ) , (3)where the χ ( ω, r ), σ ( ω, r ) and ε are the complex elec-tric susceptibility, complex conductance and permittivityof vacuum, respectively. It is worth noting that, whenthe light frequency is far away from the resonant fre-quencies of medium molecules, both f ( ω, r ) and g ( ω, r )vary with frequency more flatly than the ordinary elec-tric susceptibility and conductance do [7]. Hence theyare approximately frequency-independent for a narrowfrequency band and can be read simply as f ( r ) and g ( r ),respectively.It is a little more difficult to solve the Eq.(2) due to thecomplexity of the optical potential and the vector natureof the electric field. An elementary approach is to expandthe wave function and optical potential in an appropriateset of generalized Fourier basis functions { ϕ lmn ( r ) } , letus say, a complete set in the subspace of L functions, andto find the equations for the expansion coefficients, keep-ing in mind that it usually need three “quantum num-bers” to label the basis functions in three-dimensionalspace. For this reason f ( r ), g ( r ) and Ψ ( r ) are assumedto have following expansions, f ( r ) = X lmn c lmn ϕ lmn ( r ) , (4a) g ( r ) = X lmn d lmn ϕ lmn ( r ) , (4b) Ψ ( r ) = X lmn A lmn ϕ lmn ( r ) . (4c)Where the scalar expansion coefficients { c lmn } and { d lmn } are, respectively, referred to as a general OSFand a general optical dissipation factor (ODF), and dueto the vector nature of light each vector expansion coef-ficient A lmn includes three components A lmnx , A lmny and A lmnz . By inspection of the completeness of the gener-alized Fourier basis functions one can rationally set that ϕ hij ( r ) ϕ opq ( r ) = X lmn T lmnhij,opq ϕ lmn ( r ) , (5a) ∇ ϕ opq ( r ) = X hij Γ hijopq ϕ hij ( r ) , (5b)where T lmnhij,opq is a scalar expansion coefficient, while Γ hijopq a vector expansion coefficient. When expressions (4) for f ( r ), g ( r ) and Ψ ( r ) are substituted in the Eq.(2) andfurther use is made of the expansions (5), we reach a setof linear equations for expansion coefficients, X opq (cid:0) − Λ lmnopq − C lmnopq − iωD lmnopq + Π lmnopq · (cid:1) A opq = µε ω A lmn , (6)where we make use of definitions that Λ lmnopq = Γ hijopq · Γ lmnhij , C lmnopq = c hij T lmnhij,opq , D lmnopq = d hij T lmnhij,opq and Π lmnopq = Γ hijopq Γ lmnhij , in which the Einstein sum-mation convention with respect to repeated indiceshas been used. In the following we will chose a setof eigen solutions of Helmholtz equation, denoted by { ϕ lmn ( r ) = e i k lmn · r } , which obey periodic boundary con-ditions within a closed square box of volume V, as thebasis functions of the generalized Fourier expansions (4)and (5). Where k lmn = l πL x ˆx + m πL y ˆy + n πL z ˆz , in which ˆx , ˆy and ˆz denote three cartesian basis vectors, L x , L y and L z are three edge lengths of the imaginary box thatshould be taken large enough to contain the inhomoge-neous medium we are discussing, and l, m, n are integers.This is in fact a straightforward generalization, to thecase of three-dimensional disordered media, of the plainwave expansion of vector electromagnetic field developedfor photonic crystals [8]. As we shall limit our consid-eration to non-dissipative inhomogeneous media by re-moving the imaginary parts of the optical potentials, theEqs.(6) thus become to be (cid:2) µε ω + c − ( k lmn · k lmn ) (cid:3) A lmn + k lmn (cid:0) k lmn · A lmn (cid:1) + ′ X opq c ( l − o )( m − p )( n − q ) A opq = 0 . (7)Here putting a prime on the summation sign meansthat it excludes the term proportional to c , corre-sponding to the case of o = l , p = m and q = n .When | c | max / | ( k lmn · k lmn ) − µε ω − c | ≪
1, thesummation labeled by the prime can be neglected, here | c | max = M ax {| c lmn | | l = 0 or m = 0 or n = 0 } , whichdenotes the maximum complex modulus of that amongthe elements of the OSF in the summation.A localized optical eigenmode can be deemed to be aligh wave packet superposed by many plain-wave com-ponents exhibiting an common frequency but adjacentpropagation vectors. It is quite evident that the propa-gation vectors, k lmn ’s, with which the related plain-wavecomponents of a localized optical eigenmode have non-negligible amplitudes due to the coupling with each other through the OSF, correspond to those coupled equationsin Eqs.(7) and obey | c | max | ( k lmn · k lmn ) − k − c | > κ , (8)with 0 < κ ≪
1, here k = √ µε ω = 2 π/λ , and κ ischosen appropriately as a parameter determining whatdegree of approximation those coupled equations wouldbe saved to. From condition (8) the uncertainty of thepropagation vectors in wave vector space can be writtenas ∆ k lmn = q k + c + κ − | c | max − H ( k + c − κ − | c | max ) × q k + c − κ − | c | max , (9)where H ( x ) is the Heaviside step function. The appear-ance of a localized light wave packet with feature size of ξ requires, according to the uncertainty principle, that∆ k lmn ξ & π , in which the ξ is also a typical coherencelength of light field because of the stationary space cor-relation of the electromagnetic vibration within such awave packet. In addition we define the effective wave-lengths of the light waves in an inhomogeneous mediumas the vacuum wavelengths divided by the root meansquare refractive index, i.e., λ eff = λ / p h n ( r ) i , inwhich (cid:10) n ( r ) (cid:11) = h χ ( r ) i + 1 = c /k + 1, and the anglebrackets represents the configurational averaging of thefunction included in it. It seems natural to work with theratio of the effective wavelength to the feature size of thelocalized light wave packet as a convenient measurementfor the degree of localization. Therefore we rewrite, forany of eigen optical modes represented by a single wavepacket, the uncertainty principle in the form G ≡ ∆ k lmn p k + c & λ eff ξ , (10)where the dimensionless ratio, G , is introduced as a lo-calization intensity parameter which closely relates to theOSF.Based on the consideration that the feature size of thewave packet is of the same order of magnitude with theelastic mean free path entering the standard Ioffe-Regelcondition [9], the approximate criterion for strong local-ization of light, λ eff / ξ '
1, is likely to be satisfied onlywhen G is large enough to nearing unit, otherwise thelight field would be in extended state. In fact, the G ex-hibits a natural singular value 1 / √ G takes positive values far below 1 / √ k lmn ’s, fulfilling the inequality (8), are dis-tribute in a thin spherical shell centered at the originof k space, on which optical eigenmodes are easy to bedeveloped as modulated carrier waves with large-size en-velopes. As the G rises above 1 / √ G > / √ G ∼ λ eff /ξ is verysimilar to the Ginzburg-Landau parameter in the phe-nomenological theory of superconductivity [10], which isused to distinguish two typical classes of superconduc-tors, i.e., Type I superconductors characterized by theextended states of superconducting electrons are thosewith Ginzburg-Landau parameter less than 1 / √
2, andType II superconductors supporting the mixed states ofsuperconducting electrons those with Ginzburg-Landau parameter large than 1 / √ G > / √ G < / √ G > / √ G ∼ / √ N perfectly spherical scattering particles withboth an uniform radius a and an identical reflective in-dex n , which are immersed randomly in a backgroundmedium of dielectric constant ε . In this case we define R = 4 πa N (cid:14) (3 V ) = 4 π ˜ a N (cid:14) V = L x L y L z is the volumeof the closed square box over which the Fourier expan-sion is defined, and ˜ a = aV − a dimensionless relativeradius. The G can be written as a variety of functionsof the actual optical and geometric parameters of thisdisordered medium [12], G = 12 (cid:16) √ − H (1 − Ω) √ − Ω (cid:17) , (11)in which Ω have some equivalent explicit expressions suchas Ω( n, R, N ) = R (cid:0) n − (cid:1) R ( n −
1) 1 κ √ N , (12a)for R ≤ R max and N ≥ N min , andΩ( n, R, ˜ a ) = R (cid:0) n − (cid:1) R ( n −
1) 1 κ r R π ˜ a , (12b)for R min ≤ R ≤ R max and ˜ a ≤ [3 R /(4 πN min )] . Where R min = N min π ˜ a (cid:14) a , N min ≫ R max is the maximum filling ratio determined by the randomclose packing of the spherical particles, for which dif-ferent types of experiments have suggested an value ofabout 0 .
637 [15]. It is worth noting here that N min and κ , as they only associate with the approximation pro-cedure, can serve as adjustable parameters when fittingexperimental results. In Fig.1 we plot the contour anddensity graphs of the localization intensity parameter G following from the expressions (11) and (12). It unequiv-ocally demonstrates a photonic phase transition betweenthe ES phase and the MS phase by altering the opticaland geometric parameters of the disordered medium [12]. FIG. 1: Colour-coded graphs showing the localization inten-sity parameter, G , as a function of the refractive index n ,the rescaled filling ratio ln( R/R min ), and the relative parti-cle radius ˜ a , for the adjustable parameters taking values of κ = 0 .
005 and N min = 100. Slices (a) and (b) correspondto the cross sections at ˜ a = 0 . R = 0 . G < / √
2, as dominated mainly by the orange-red, and themixed-state (MS) phase is found on the regions of
G > / √ G , and the red dashed line corresponds to the criticalpoint where G = 1 / √
2, delineating the photonic phase tran-sition.
In conclusion, In this paper we have suggested a gener-alized mode-expansion scheme for the vector electromag-netic waves propagating in 3D disordered media. Basedon the Fourier analysis we have provided an efficient wayto estimate the coherence length ξ of the optical eigen-modes of an inhomogeneous medium, and proposed anecessary condition, G > / √
2, for the strong local-ization of light. Via changing the optical and geomet-ric parameters, such as the refractive index, the dimen-sion and the density of randomly distributed dielectricspheres, we can achieve the photonic phase transitionfrom the ES phase to the MS phase or the conversetransition. We hope that this method can be general-ized to anisotropic polarized inhomogeneous media, non-linear inhomogeneous media as well as the similar caseinvolving other wave phenomena. It remains to be seewether concepts from superconductivity theory and clas-sical fermionic systems will prove fruitful in the systemsof non-free bosons maintained by disordered media.We wish to acknowledge the supports of the NSF GrantNo.10874016 and the Program for Hunan Provincial Op-tical Key Discipline of China. [1] D.S. Wiersma, P. Bartolini, A. Lagendijk, and R. Righini,Nature. , 671 (1997).[2] H. Pier, E. Kapon and M. moser, Nature. , 880(2000).[3] A.D. Greentree, C. Tahan, J.H. Cole, and L.C.L. Hollen-berg, Nature Phys. , 856 (2006).[4] D. Novoa, H. Michinel, and D. Tommasini, Phys. Rev.Lett. , 203904 (2010).[5] S. John, Phys. Rev. Lett. , 2486 (1987).[6] P.W. Anderson, Phys. Rev. , 1492 (1958); E. Abra-hams, P.W. Anderson, D.C. Licciardello, and T.V. Ra-makrishnan, Phys. Rev. Lett. , 673 (1979).[7] R.W. Boyd, Nonlinear Optics, Third Edition(Elsevier,Singapore, 2010), p.25 and p.166; H.A. Kramers, Nature, , 673 (1924); H.A. Kramers, Nature, , 310 (1924).[8] K. M. Ho, C. T. Chan, and C. M. Soukoulis, phys. Rev.Lett. , 3152 (1990).[9] A.F. Ioffe and A.R. Regel, Prog. Semicond. , 237 (1960).[10] E.M. Lifshitz and L.P. Pitaevskii, Statistical Physics,Part 2 (Pergamon Press, London, 1980), p. 182; V.L.Ginzburg and L.D. Landau, J. Exp. Theor. Phys.U.S.S.R. , 1442(1957).[12] See the supplemental materials of this paper we will pub-lish elsewhere.[13] H. Cao, Y. Ling, J.Y. Xu and A.L. Burin, Phys. Rev. E. , R25601 (2002).[14] F. Johannes, J.B.D. Roman, S. Janos, S. Daniel, K.Claus, and K. Heinz, Nature Photon.3