Anderson localization of classical waves in weakly scattering metamaterials Ara A. Asatryan1, Sergey A. Gredeskul2,3, Lindsay C. Botten1, Michael A. Byrne1, Valentin D. Freilikher4, Ilya V. Shadrivov3, Ross C. McPhedran5, and Yuri S. Kivshar3 1Department of Mathematical Sciences and Centre for Ultrahigh-bandwidth Devices for Optical Systems (CUDOS), University of Technology, Sydney, NSW 2007, Australia 2 Department of Physics, Ben Gurion University of the Negev, Beer Sheva, 84105, Israel 3 Nonlinear Physics Center and CUDOS, Australian National University, Canberra, ACT 0200, Australia 4 Department of Physics, Bar-Ilan University, Raman-Gan, 52900, Israel 5School of Physics and CUDOS, University of Sydney, Sydney, NSW 2006, Australia 0 1 Westudythepropagationandlocalization ofclassicalwavesinone-dimensionaldisorderedstruc- 0 turescomposedofalternatinglayersofleft-andright-handedmaterials(mixedstacks)andcompare 2 them with structures composed of different layers of the same material (homogeneous stacks). For weaklyscatteringlayers,wehavedevelopedaneffectiveanalyticalapproachandhavecalculatedthe n transmission length within a wide range of the input parameters. This enables us to describe, in a a J unified way, the localized and ballistic regimes as well as the crossover between them. When both refractive index and layer thickness of a mixed stack are random, the transmission length in the 9 long-waverangeofthelocalized regimeexhibitsaquadraticpowerwavelengthdependencewithdif- 2 ferentcoefficientsofproportionalityformixedandhomogeneousstacks. Moreover,thetransmission ] length of a mixed stack differs from the reciprocal of the Lyapunov exponent of the corresponding n infinite stack. In both the ballistic regime of a mixed stack and in the near long-wave region of a n homogeneous stack, the transmission length of a realization is a strongly fluctuating quantity. In - the far long-wave part of the ballistic region, the homogeneous stack becomes effectively uniform s i and the transmission length fluctuations are weaker. The crossover region from the localization to d the ballistic regime is relatively narrow for both mixed and homogeneous stacks. In mixed stacks . t with only refractive-index disorder, Anderson localization at long wavelengths is substantially sup- a pressed, with the localization length growing with wavelength much faster than for homogeneous m stacks. The crossover region becomes essentially wider and transmission resonances appear only in - much longer stacks. The effects of absorption on one-dimensional transport and localization have d also been studied,both analytically and numerically. Specifically,it isshown that thecrossover re- n gionisparticularlysensitivetolosses,sothatevensmallabsorptionnoticeablysuppressesfrequency o dependent oscillations in the transmission length. All theoretical predictions are in an excellent c agreement with theresults of numerical simulations. [ 4 PACSnumbers: 42.25.Dd,42.25.Fx v 2 6 I. INTRODUCTION it was shown that the presence of a single defect led to 3 theappearanceofalocalizedmode. Ametamaterialwith 0 manypoint-likedefectswasconsideredinRef.11 whereit . Metamaterialsare artificialstructures having negative 2 was demonstrated that even weak microscopic disorder refractive indices for some wavelengths1. While natural 1 mightleadto asubstantialsuppressionofwavepropaga- materials having such properties are not known, it was 9 tionthroughametamaterialoverawidefrequencyrange. 0 the initial paper2 that sought to realize artificial meta- : materials which triggered the rapidly increasing interest The next steps in this direction focused on the study v in this topic. Over the past decade, the physical proper- of localization in metamaterials. Anderson localization i X tiesofthesestructures,andtheirpossibleapplicationsin is one of the most fundamental and fascinating phe- r modern optics and microelectronics, have received con- nomenon of the physics of disorder. Predicted in the a siderable attention (see e.g. Refs3–6). The reasons for seminal paper12 for spin excitations, it was extended to such interest are their unique physical properties, their the case of electrons and other one-particle excitations ability to overcome the diffraction limit1,2, and their in solids, as well as to electromagnetic waves (see, e.g., potential role in cloaking7, the suppression of sponta- Refs.13–18), becoming a paradigm of modern physics. neous emission rate8, and the enhancement of quantum Anderson localization results from the interference of interference9, etc. multiplyscatteredwaves,manifestingitselfinamostpro- Until recently, most studies considered only ideal sys- nounced way in one-dimensional systems14,16, in which tems and did not address the possible effects of disor- all states become localized19 so that the envelope of der. However, real metamaterials are always disordered, each state decays exponentially away from a randomly at least, in part, due to fabrication errors. Accordingly, located localization center14. The rate of this decay is the study ofdisorderedmetamaterialsis notjustanaca- non-randomandiscalledthe Lyapunovexponent, γ,the demic question but is also relevant to their application. reciprocal of which determines the size of the area of lo- The first step in this direction was made in Ref.10 where calization. 2 Inafinite,butsufficientlylong,disorderedsample,the Inthis paper,westudythe wavetransmissionthrough localizationmanifestsitselfinthefactthatthefrequency disordered M- and H-stacks of a finite size composed of dependent transmission amplitude is (typically) an ex- a weakly scattering right- and left-handed layers. In the ponentially decreasing function of the sample size. The frameworkoftheweakscatteringapproximation(WSA), average of this decrement is a size-dependent quantity, wehavedevelopedaunifiedtheoreticaldescriptionofthe whose inverse (i.e., reciprocal) is termed the transmis- transmission and localization lengths over a wide wave- sion length, l . In the limit as the sample becomes of length range, allowing us to explain the pronounced dif- N infinite length, the decrement tends to a constant non- ferenceinthetransmissionpropertiesofM-andH-stacks randomvalue. Thereciprocalofthisvaluedeterminesan- at long wavelengths. other characteristic spatial scale of the localized regime, When both refractive index and layer thickness of the which is the localization length, l. It is commonly ac- mixed stack are random, the transmission length in the ceptedinboththesolidstatephysicsandopticalcommu- long wavelength part of the localized regime exhibits a nities, that the inverse of the Lyapunov exponent, γ−1, quadraticpowerlawdependence onwavelengthwithdif- and the localization length, l are always equal. While ferent constants of proportionality for mixed and homo- this is true for media with a continuous spatial distri- geneous stacks. Moreover, in the localized regime, the bution of the random dielectric constant, in the case of transmission length of a mixed stack differs from the re- randomly layered samples, the situation, as we show in ciprocal of the Lyapunov exponent of the corresponding this paper, is more complicated. In particular, the in- infinite stack (to the best of our knowledge, in all one- verse of the Lyapunov exponent by itself, calculated, for dimensionaldisorderedsystemsstudiedtillnowthesetwo example, in Ref.20, does not provide comprehensive in- quantities always coincide). formation about the transport properties of disordered Both M- and H-stacks demonstrate a rather narrow media. Furthermore, it is unlikely that it can be mea- crossover from the localized to the ballistic regime. The sured directly, at least in the optical regime. H-stack in the near ballistic region, and the M-stack in the ballistic region are weakly scattering disordered The first study of localization in metamaterials was stacks,whileinthefarballisticregion,theH-stacktrans- presented in Ref.21 where wave transmission through an mits radiation as an effectively uniform medium. alternating sequence of air layers and metamaterial lay- We also consider the effects of loss and show that ab- ers of random thicknesses was studied. Localized modes sorption dominates the effects of disorder at very short withinthegapwereobservedanddelocalizedmodeswere and very long wavelengths. The crossover region is par- revealeddespitetheone-dimensionalnatureofthemodel. ticularly sensitive to losses, so that even small absorp- A more general model of alternating sequences of right- tion suppresses oscillations in the transmissionlength as andleft-handedlayerswithrandomparameterswasstud- a function of frequency. ied in Ref.22. There, it has been shown that in mixed All ofthe theoreticalresults mentionedabovearecon- stacks (M-stacks) with fluctuating refractive indices, lo- firmed by, and are shown to be in excellent agreement calization of low-frequency radiation was dramatically with, the results of extensive numerical simulation. suppressed so that the localization length exceeded that In M-stacks with only refractive index disorder, An- for homogeneous stacks (H-stacks), composed solely of derson localization and transmission resonances are ef- right-orleft-handed slabs,bymany ordersof magnitude fectively suppressed and the crossover region between and scaling as l λ6 or even higher powers of wave- ∝ the localizedandballistic regimesis ordersofmagnitude length (in what follows we refer to this result as the greater. A more detailed study of the λ6-anomaly shows λ6 anomaly), in contrast to the well-known dependence thatthegenuinewavelengthdependenceofthetransmis- l λ2 observed in H-stacks23. As noted in Ref.22, a ∝ sionlengthisnotdescribedbyanypowerlawandrather possiblephysicalexplanationofthisisthesuppressionof is non-analytic in nature. phase accumulation in M-stacks, related to the opposite Inwhatfollows,Sec. II presentsa detaileddescription signs of the phase andgroupvelocities in left- andright- of our model. Section III is devoted to the analytical handed layers. Scaling laws of the transmission through studies of the problem, while the results of numerical a similar mixed multilayered structure were studied in simulations and a discussion of these are presented in Ref.24. There, it was shown that the spectrally averaged Sec. IV. transmissioninafrequencyrangearoundthefullytrans- parentresonantmode decayedwiththe number oflayers much more rapidly than in a homogeneousrandom slab. II. MODEL Localization in a disordered multilayered structure com- prising alternating random layers of two different left- handed materials was considered in Ref.25, where it was A. Mixed and homogeneous stacks shown that within the propagation gap, the localization length was shorterthan the decay length in the underly- Weconsideraone-dimensionalalternatingM-stack,as ing periodic structure, andthe opposite ofthat observed shown in Fig. 1. It comprises disordered mixed left- (L) in the corresponding random structure of right-handed and right- (R) handed layers, which alternate over its layers. length of N layers, where N is an even number. The 3 thicknesses of each layer are independent random values Although localization in disordered H-stacks with with the same mean value d. In what follows,allquanti- right-handed layers has been studied by many ties with the dimension of length are measured in units authors18,23,26–28, here we consider this problem in of d. In these units, for the thicknesses of a layerwe can its most general form and show that the transmission write properties of disordered H-stacks are qualitatively the same for stacks comprised of either solely left- or right-handed layers. d =1+δ(d), j j (d) B. Transmission: localized and ballistic regimes where the fluctuations of the thickness, δ , j = 1,2,... j are zero-mean independent random numbers. We take We introduce the transmission length l of a finite N random configuration as 1 dj 1 ln T N TN l =− N| | , (1) æææ æææ N (cid:28) (cid:29) RN whereTN istherandomtransmissioncoefficientofasam- RH LH RH LH pleofthelengthN.Asaconsequenceoftheself-averaging of ln T /N, N | | N N-1 2m 2m-1 4 3 2 1 ln T 1 1 FIG. 1: (Color online) Structuregeometry. lim | N| = lim = . (2) N→∞ N N→∞lN l the magnetic permeability for right-handed media to be Thismeansthatforasufficientlylongstack(inthelocal- µj = 1 and for metamaterials to be µj = 1, while the ized regime) the transmissioncoefficient is exponentially − dielectric permittivity is small T exp( N/l). N | |∼ − In what follows, we consider stacks composed of lay- ers with low dielectric contrasts, i.e, δν,d 1, so that εj =±(1+δj(ν)±iσj)2, theFresnelreflectioncoefficientsofeac|hin|te≪rface,andof each layer, are much smaller than 1. Here, a thin stack where the upper and lower signs respectively correspond comprising a small number of layers is almost transpar- tonormal(right-handed)andmetamaterial(left-handed) ent. In this, the ballistic regime, the transmissionlength layers. The refractive index of each layer is then takes the form ν = (1+δ(ν))+iσ , 1 R 2 j ± j j h| N| i, (3) l ≈ 2N N (ν) where all δ and absorption coefficients of the slabs, σ 0, arej independent random variables. With this, involving the average reflection coefficient29, which is j the≥impedance of each layer relative to the background valid in the case of lossless structures. This follows di- (free space) is rectlyfromEq. (1)byvirtueoftheconservationrelation- ship, R 2+ T 2 =1. Thus, in the ballistic regime, N N | | | | Zj = µj/εj =1/(1+δj(ν)±iσj), R 2 2N, N b. (4) q h| N| i≈ b ≪ with the same choice of the sign. Webeginwiththegeneralcasewhenbothtypesofdis- wherethelengthbinthisequationistermedtheballistic order (in refractive index and in thickness) are present. length. Two particular cases, each with only one type of disor- Accordingly,instudiesofthe transportofthe classical der, are rather different. In the absence of absorption, waves in one-dimensional random systems, the following the M-stack with only thickness disorder is completely spatial scales arise in a natural way: transparent,a consequenceof Z 1. However,the case of only refractive-indexdisorderjis≡intriguingbecause, as • lN —thetransmissionlengthofafinitesample(1), is shown below, such mixed stacks manifest a dramatic l — the localization length (2), and suppression of Anderson localization in the long wave • region22. b — ballistic length (4). • 4 Note that in the case of absorbing stacks (σ = σ > 0), Here, β = kd ν , k = 2π/λ, and λ denotes the dimen- j j j j the right hand side of Eq. (1) defines the attenuation sionless free space wavelength. While the sign of the length, l , which incorporates the effects of both disor- phase shift across each slab Re(β ) varies according to att j der and absorption. the handedness of the material, the Fresnel interface co- In what follows, we show that contrary to commonly efficient ρ given by j accepted belief, the quantities γ−1, l, and b are not nec- essarily equal, and, under certain situations, can differ noticeably from each other. Z 1 j ρ = − , (9) Inthispaper,westudymainlythetransmissionlength j Z +1 j defined above by Eq. 1. This quantity is very sensi- tive to the size ofthe system andtherefore is best suited depends only on the relative impedance of the layer Z , j to the description of the transmission properties in both a quantity whose real part is positive, irrespective of the the localized and ballistic regimes. More precisely, the handedness of the material. transmissionlengthcoincideseitherwiththelocalization Equations(5)-(9)aregeneralandprovideanexact de- length or with the ballistic length, respectively in the scription of the system and will be used later for direct cases of comparatively thick (localized regime) or com- numerical simulations of its transmission properties. paratively thin (ballistic regime) stacks. That is, It was mentioned previously that we consider the spe- cialcaseofweakscatteringforwhichthereflectionfroma singlelayeris small. i.e., r 1. This occurs either for l N l j | |≪ ≫ weak disorder, or for strong disorder provided that the l . N ≈ wavelength is sufficiently long. The transmission length b N b.  ≪ then follows from Anotherargumentsupportingourchoiceofthetransmis- sion length as the subject of investigation is that it can be found directly by standardtransmissionexperiments, ln T 2 =2RelnT , (10) N N | | while measurements of the Lyapunov exponent callfor a much more sophisticated arrangement. and requiresthe following first orderapproximationsde- rived from Eqs. (5) and (6): III. ANALYTICAL STUDIES lnTn = lnT1,n−1+lntn+Rn−1rn, (11) A. Weak scattering approximation Rn = rn+Rn−1t2n. (12) The theoretical analysis involves the calculation of In deriving Eq. (12), we omit the first-order term the transmission coefficient using a recursive procedure. R2 t2r since it contributes only to the second order Consider a stack which is a sequence of N layers enu- n−1 n n of lnT already after the first iteration. Then, by sum- merated by index n from n = 1 at the rear of the stack n ming up logarithmic terms (11), we obtain through to n = N at the front. The total transmis- sion (T ) and reflection (R ) amplitude coefficients of n n the stack satisfy the recurrence relations N N N j−1 Tn−1tn lnTN = lntj + rj−m+1rj t2p. (13) Tn = 1 Rn−1rn, (5) Xj=1 mX=2jX=m p=jY−m+2 − R = r + Rn−1t2n (6) This equation enables us to derive a general expres- n n 1 Rn−1rn sion for the transmission length l (N) which is valid in − T all regimes (see the next Section). However,the ballistic for n = 2,...,N, in which both the input and output length b, according to Eq. (4) (and the average reflec- media are free space. In Eqs (5) and (6), the amplitude tion coefficient as well), is determined only by the total transmission (t ) and the reflection (r ) coefficients of a j j reflection amplitude R in the case of lossless strutures. single layer are given by N In the ballistic regime, this amplitude coefficient, to the necessary accuracy, is given by ρj(1 e2iβj) r = − , (7) j 1 ρ2e2iβj − j N (1 ρ2)eiβj R = r . (14) t = − j . (8) N j j 1−ρ2je2iβj Xj=1 5 B. Mixed stack In the lossless case (σ = 0), the first term on the right handsideofEq. (18)correspondstothe so-calledsingle- 1. General Approach scattering approximation, which implies that multi-pass reflections are neglected so that the total transmission coefficient is approximated by the product of the single From here on, we assume that the random variables layer transmission coefficients, i.e., δ(ν) of left-handed or right-handed layers, δ(d), and σ j j j are identically distributed according to the correspond- ing probability density functions. This enables us to ex- N press all of the required quantities via the transmission T 2 t 2. N j | | → | | andreflectionamplitudesofasingleright-handedorleft- j=1 Y handed layer, t , r , and also to calculate easily all of r,l r,l the necessary ensemble averages. In the case of very long stacks (i.e., as the length TheaverageofthefirstterminEq. (13)canbewritten N ), we can replace the arithmetic mean, → ∞ as N−1 N ln t , by its ensemble average ln t . On the j=1 | j| h | |i otherhand,inthislimitthereciprocalofthetransmission P length coincides with the localization length. Using the N N N energy conservation law, r 2+ t 2 = 1, which applies lntj = lntr + lntl . | j| | j| * + 2 h i 2 h i intheabsenceofabsorption,theinversesingle-scattering j=1 X localization length may be written as Next,wesplitthesecondtermofEq. (13)intotwoparts 1 1 = r2 (19) N N j−1 l 2h| | i rj−m+1rj t2p =NR1+NR2, (cid:18) (cid:19)ss m=2j=m p=j−m+2 andisproportionaltothemeanreflectioncoefficientofa X X Y singlerandomlayer14,30. Thecorrespondingmodification where of Eq. (18) then reads R1 = N1 mN=/21j=N2mrj−2m+1rjp=jj−−21m+2t2p, l1N = h|rr|2i+4 h|rl|2i −Re(hR1i+hR2i). (20) X X Y N/2 N j−1 Here, the first term corresponds to the single-scattering 1 R2 = N rj−2mrj t2p, approximation, while the next two terms take into ac- m=1j=2m−1 p=j−2m+1 counttheinterferenceofmultiplyscatteredwavesaswell X X Y asthedependenceofthetransmissionlengthonthestack comprisingcontributionstothedepletionofthetransmit- size. Note that Eq. (20) is appropriate only for lossless ted field due to two pass reflections respectivelybetween structures. Inthepresenceofabsorption,Eq. (18)should slabsofdifferentmaterials(i.e., ofoppositehandedness), be used instead. andbetweenslabsofthesamematerial(i.e.,oflikehand- edness). Averaging these expressions, we obtain 2. Transmission length 1 (BN 1)(1+B2) hR1i = ArAl 1 B2 + N−(1 B2)2 , (15) Fromthispointon,weassumethatthestatisticalprop- (cid:20) − − (cid:21) ertiesoftheright-handedandleft-handedlayersareiden- A2B +A2B N 2(BN 1) = r l r l + − ,(16) tical. As a consequence of this symmetry, the following hR2i 2N 1 B2 (1 B2)2 relationsholdforanyreal-valuedfunctiong ineitherthe (cid:20) − − (cid:21) lossless or absorbing cases: where ∗ ∗ g(t ) = g(t ) , g(r ) = g(r ) . (21) Aτ =hrτi, Bτ =ht2τi, τ =l,r, B2 =BlBr. (17) h r i h l i h r i h l i Therefore, A and B are the complex conjugates of A The resulting transmission length is determined by the r r l and B , and B2 is real quantity, as are both averages equation l and . 1 2 hR i hR i Accordingly,asaconsequenceofthe left-rightsymme- 1 ln t + ln t try (21), the transmission length of a M-stack depends r l = h | |i h | |i +Re( + ). (18) − l 2 hR1i hR2i only on the properties of a single right-handed layerand N 6 maybe expressedinterms ofthree averagedcharacteris- In summary, tics: r , ln t ,and t2 (inwhichweomitthesubscript r). h i h | |i h i l(λ), N ¯l(λ), ≫ With these observations, the transmission length of a l (λ) finite length M-stack may be cast in the form: N ≈b(λ), N ¯l(λ),  ≪ with the transition between the two ranges of N being 1 = 1 + 1 1 f(N,¯l). (22) determined by the crossoverlength. lN l (cid:18)b − l(cid:19) While in the lossless case, the ballistic regime occurs whenthestackismuchshorterthanthe crossoverlength where (N ¯l(λ)), it is important to note that, in the local- ≪ ization regime, the opposite inequality is not sufficient andthe necessaryconditionfor localizationis N l(λ). 1 r 2+Re r 2 t2 ∗ ≫ In what follows, we consider samples of an intermediate = ln t |h i| h i h i , (23) l −h | |i− 1 t2 2 length, i.e., ¯l(λ) N l(λ). −|h(cid:0) i| (cid:1) ≪ ≪ ForaM-stackoffixedsizeN,theparametergoverning and thetransmissionisthewavelength,andtheconditionsfor the localized and ballistic regimes should be formulated in the wavelength domain. To do this, we introduce two 1 1 2/¯l b = l − 1 exp( 2/¯l) × cthhearraecltaetrioisntsicwavelengths,λ1(N)andλ2(N), definedby − − r 2+Re r 2 t2 ∗ r 2 |h i| h i h i |h i| (24) 1−|h(cid:0)t2i|2 (cid:1) − 2 ! N =l(λ (N)), N =¯l(λ (N)). (27) 1 2 are, as we will see below, the inverse localization and It can be shown that the long wavelengthregion, λ inverse ballistic lengths. The function f(N,¯l) is defined ≪ λ (N), corresponds to localization where the transmis- as 1 sion length coincides with the localization length, while in the extremely long wavelength region, λ λ (N), 2 ≫ ¯l N the propagationis ballistic,with the transmissionlength f(N,¯l)= 1 exp , (25) given by the ballistic length b. That is, N − − ¯l (cid:20) (cid:18) (cid:19)(cid:21) and introduces a new characteristic length termed the l(λ), λ λ (N), 1 crossover length ≪ l (λ) (28) N ≈ b(λ), λ λ (N).  ≫ 2 1 ¯l = , (26) Whenλ (N)<λ (N),thereexistsanintermediaterange −ln t2 1 2 |h i| of wavelengths, λ (N) < λ < λ (N), which will be dis- 1 2 which arises in the calculations in a natural way and, cussed below. as will be demonstrated below, plays an important role To better understand the physical meaning of the ex- in the theory of the transport and localization in one- pressions (23) and (24) for the localization and ballis- dimensional random systems. Equations (22)-(24) com- tic lengths, we will consider ensembles of random con- pletely describe the transmissionlengthofamixedstack figurations in which the fluctuations δ(ν) and δ(d) are j j in the weak scattering approximation. distributed uniformly over the intervals [ Q ,Q ] and ν ν − Obviously, the characteristic lengths l(λ), b(λ), and [ Q ,Q ]respectively,withσ =0. Theaveragequanti- d d j ¯l(λ) appearing in Eq. (22) are functions of wavelength. t−ies that arise in Eqs. (23) and (24) are presented in the Usingstraightforwardcalculations,itmaybeshownthat Appendix. Theseformulaeallowforcalculationswithan the first two always satisfy the inequality l(λ) > b(λ), accuracy of order O(Q2) for arbitrary Q . For the sake ν d while, as we will see, the crossoverlength is the shortest of simplicity, we assume that the fluctuations of the re- of the three, i.e., b(λ) > ¯l(λ) in the long wavelength fractive index and thickness are of the same order, i.e., region. Q Q so that the dimensionless parameter ν d ∼ Inthecaseofafixedwavelengthλandastacksoshort that N ¯l(λ), the expansion of the exponent in Eq. Q2 ≪ ζ =2 d (25) yields f 1, in which case the transmission length Q2 → ν approaches b(λ). Correspondingly, for a sufficiently long stack, N ¯l(λ), f 0 and the transmission length is of order of unity. We also neglect the contribution of assumes th≫e value of l→(λ). terms of order higher than Q2. d 7 The short wavelength asymptotic of the localization ways. The localization length is defined by Eq. (2) via length is then the transmittivity on a realization, while the Lyapunov exponent describes an exponential growth of the enve- 12 lope of a currentless solution far from a given point in l(λ)= . (29) Q2 which the solution has a given value14. We have calcu- ν latedthelongwaveasymptoticoftheM-stackLyapunov In the long wavelength limit, we obtain the following exponentusingthewellknowntransfermatrixapproach, asymptoticformsforthecorrespondingsinglelayeraver- assuming the same statistical properties of the modulus ages: of dielectric constant and the thickness of both left- and right-handed layers, ikQ2 k2Q2 5ik3Q2 r ν ν ν, (30) h i ≈ 6 − 2 − 9 k2Q2 π2d2ǫ2 ǫ2 ln t ν, (31) γ − , (38) h | |i ≈ − 6 ≈ 2λ2 ǫ ikQ2 ǫ= (1+δ(ν))2, d=1+δ(d). t2 1+2ik+ ν h i ≈ 3 − Inthecaseofrectangulardistributionsofthefluctuations 2k2 5k2Q2ν + 2k2Q2d. (32) ofthe dielectric constantsandthicknesses,this resultre- − − 3 3 duces to Substitution of these expansions into Eq. (24) yields the long wavelength asymptotic of the ballistic length γ 2π2Q2ν = 13+ζ > 1 (39) ≈ 3λ2 l 1+ζ l 3λ2 b(λ) . (33) (weneglectedthesmallcorrectionsproportionaltoQ2 ≈ 2π2Q2ν 1). Thus, the disorderedM-stack in the long wavelendg≪th This asymptotic can be calculated directly from Eq. region presents a unique example of a one-dimensional (4) with the average reflection coefficient, determined disorderedsysteminwhichthelocalizationlengthdiffers from Eq. (14), being from the reciprocal of the Lyapunov exponent. The long wavelength asymptotic of the Lyapunov ex- ponent γ λ−2 and the asymptotic of the localization ∝ R 2 =N r2 r 2 +N2 Re r 2. (34) length, l λ2,Eq. (36)haveratherclearphysicalmean- | N| | | −|h i| h i ing. Inde∝ed, in the limit λ , the propagating wave D E (cid:16)D E (cid:17) → ∞ is insensitive to disorder since the disorder is effectively Thesamesubstitutionsintothisequation(34)givethe averaged over distances of the order of the wavelength. average total reflection coefficient This means that Andersonlocalizationis absentand the Lyapunov exponent γ l−1 vanishes. For large but fi- Nk2Q2 N2k4Q4 nite wavelengths (i.e.,∝small wavenumbers k = 2π/λ), R 2 ν + ν. (35) | N| ≈ 3 4 the Lyapunov exponent is small and, assuming that its D E dependence on the wavenumber is analytic, we can ex- In the ballistic regime, the final term is negligibly small. panditinpowersofk. Thisexpansioncommenceswitha This, together with Eq. (4), againresults in the value of termoforderk2 sincetheLyapunovexponentisrealand the ballistic length given in Eq. (33). the wavenumber enters the field equations in the form Substitutingthelongwavelengthexpansions(30)–(32) (ik). Accordingly, in the long wavelength limit, γ k2 into Eqs. (23) and (26), we derive the following asymp- and so l λ2. ∝ totic forms for the localization length ∝ The behavior of the Lyapunovexponent givenby (38) does not depend on the handness of the layers; compare 3λ2 3+ζ this equation with the corresponding long wave asymp- l(λ) , (36) totic obtained in Refs.15,25 for homogeneous stacks com- ≈ 2π2Q2 1+ζ ν posed of only positive or only negative layers. However, it crucially depends on the propagating character of the and the crossover length field within a given wavelength region. When this re- gion is within the gap of the propagation spectrum of 3λ2 1 the effective ordered medium, the frequency dependence ¯l(λ)≈ 2π2Q2 4(3+ζ). (37) of the Lyapunov exponent differs from λ−2. It takes ν ∼ place, for example, for a stack composed of single nega- In seeking to compare the result (36) for the localiza- tivelayers,whereonlyoneoftwocharacteristics(ε, µ)is tionlengthwiththecorrespondinglongwaveasymptotic negative25. of the reciprocal Lyapunov exponent, it is important to The main contribution to the ballistic length (33) is note that these two quantities are defined in different due to the final term in the right hand side of Eq. (24), 8 whichcorrespondstothesingle-scatteringapproximation Therefore, the corresponding analogue of the function f discussed at the end of the Sec. IIIB1. While the bal- (25) preserves all necessary limiting properties (see the listic length follows from the single scattering approxi- next Sec. IIIB3). mation, the calculation of the localization length (36) is more complex and requires that interference due to the multiple scattering of wavesmust be takeninto account. 3. Homogeneous stacks In the case under consideration (Q Q and ζ 1), ν d ∼ ∼ thetwocharacteristicwavelengthsλ (N)andλ (N)(27) In this Section, we consider a H-stack composed en- 1 2 for an M-stack of a fixed length size N take the form tirelyofnormalmateriallayersnotingthatthebehaviour ofaH-stackofmetamaterial(left-handed)layersaloneis exactlythesame,aresultwhichmaybeobtaineddirectly 2N 1+ζ from Eq. (18) by replacing each l by r, after which any λ (N) = πQ , (40) 1 νs 3 3+ζ reference to the index r may be omitted. The transmis- sion length of an H-stack is then ζ λ (N) = 2πQ 2N 1+ , (41) 2 ν s 3 1 1 1 1 t2 N (cid:18) (cid:19) = + Re r 2 −h i , (42) l l N h i (1 t2 )2 andareofthesameorderofmagnitude. Thismeansthat N (cid:20) −h i (cid:21) thetransmissionlengthl coincideswiththelocalization where the inverse localization length l is N length (36) for N l(λ)), and with the ballistic length (33) in the case wh≫en N ≪¯l(λ). 1 r 2 The crossover between the localized and ballistic = ln t Re h i . (43) l −h | |i− 1 t2 regimes occurs for λ1(N) . λ . λ2(N), with the two −h i bounds being proportional to Q √N and thus growing NowweconsideraH-stackcomposedofweaklyscatter- ν as √N. For long stacks, the transmission length in the ing layers. To simplify the discussion, we consider only crossover region can be described with high accuracy refractive index disorder (i.e., Q = 0). In this case, d by the general equations (22) and (25), where the bal- the asymptotic behavior of the localizationlength in the listic length b(λ), the localization length l(λ), and the short and long wavelength limits are crossover length ¯l(λ) are replaced by their asymptotic forms (33)–(37). In the case of solely refractive index 12 disorder (Q 0) when all thicknesses are set to unity, , λ 0, the ballisticdle→ngth and the shortwavelengthasymptotic l(λ)= Q2ν → (44) oofftthheeilroccaoluiznatetiropnarletnsgfothr aconinMci-dsteawckithwitthhebliomthitirnegfrvaacltuivees  3λ2 , λ . 2π2Q2 →∞ indexandthicknessdisorder(33)and(29). However,the ν corresponding limiting values of the long wavelength lo- The main contribution to the localizationlength is re- calization length and crossover lengths have nothing to latedtothefirstterminEq. (43). Thus,thelocalization dowiththegenuinebehaviourofthetransmissionlength length of the H-stack in the long wavelength region is (see Sec. IVB1). This means that the weak scatter- describedcompletelybythesinglescatteringapproxima- ing approximation fails to describe the long wavelength tion and coincides with the ballistic length (33) of the asymptoticsofthe transmissionlengthinboththe local- M-stack. izationandcrossoverregions. Thisisdiscussedingreater Using the transfer matrix approach, we can also cal- detail below. culate the long wavelength asymptotic of the Lyapunov Eqs. (15)–(18) (or Eqs. (22)–(26)) in the symmetric exponentforaH-stack. Itisdescribedbythesameequa- case) completely determine the behaviour of the trans- tion(38)astheasymptoticfortheM-stack,thuscoincid- mission length for a mixed stack composed of weakly ingwiththeasymptoticofthereciprocalLyapunovexpo- scattering layers. Although ¯l has been introduced as a nent. In recent work20, this coincidence was established crossoverlength,the entireregionN ¯l doesnotneces- analytically in a wider spectral region. However,the nu- ≫ sarilysupportlocalization. Correspondingly,theballistic merical calculations intended to confirm this result are regime may exist outside the region N ¯l. rather unconvincing. Indeed, the numerically obtained ≪ All of the results obtained under the symmetry as- plots demonstrate strong fluctuations (of the same order sumption (21) are qualitativelyvalid in the generalcase. as the mean value) of the calculated quantity, while the Indeed, the existence of the crossover (28) is related to genuine Lyapunov exponent is non-random and should the exponential dependence in Eq. (25) with a real and be smooth without any additional ensemble averaging positivecrossoverlength¯l. Whentheassumptionofsym- mentioned by the authors. metrynolongerholds,thislengthtakesacomplexvalue. If we cast t2 N in the second term of Eq. (42) in the However, the quantity B in Eq. (16) satisfies (by its form exp Nhlnit2 we see that the crossover length of h i definition in Eq. (17)) the evident inequality B < 1. the H-stack is | | (cid:0) (cid:1) 9 ¯l = lnht2i −1, 1 = 2π2Q2ν + Nπ2Q4ν. (47) b (λ) 3λ2 18λ2 f (cid:12) (cid:12) with its long wavelength(cid:12) asym(cid:12)ptotic, according to Eq. (32), being In the case of a relatively short H-stack NQ2/12 1, ν ≪ thecontributionofthefirsttermintherighthandsideof this equation dominates, and hence the transition from λ ¯l(λ)= . (45) theneartothe farsubregionsisnotaccompaniedbyany 4π change in the analytical dependence on the wavelength. The ballistic length is thus described by the same wave- Here, the crossover length is proportional to the wave- length dependence over the entire ballistic region length, in stark contradistinction to the situation for M- stacks, in which the crossover length is proportional to λ2. 3λ2 To consider this further, we define the characteristic b(λ)= , λ (N) λ. (48) 2π2Q2 1 ≪ wavelengths λ (N) and λ (N) by the expressions (27). ν 1 2 For a H-stack, these lengths are For sufficiently long H-stacks NQ2/12 1, in the far ν ≫ longwavelengthregion,thesecondtermisdominantand so 2N λ (N) = πQ , 1 ν 3 r λ (N) = 4πN. (46) 18λ2 2 b (λ)= . (49) f Nπ2Q4 Evidently,thesecondcharacteristicwavelengthisalways ν much larger than the first λ2(N) λ1(N). As a con- Thus, the wavelength dependence of the ballistic length ≫ sequence, the long wavelength region, where the ballis- of a sufficiently long H-stack is tic regime is realized, can be divided into two subre- gions. Thenearsubregion(moderatelylongwavelengths) is bounded by the characteristic wavelengths 3λ2 , λ (N).λ.λ (N), 2π2Q2 1 2  ν b(λ)= (50) λ1(N).λ.λ2(N).  18λ2 , λ (N).λ. Nπ2Q4 2 Tbahlelismtiacinrecgoionntr,ibbnu,tiiosndtuoetthoetbhaellifisrtsict lteenrmgthininEtqh.e(n4e3a)r. The sameresult foνr the ballistic length also follows Thus the ballistic length bn(λ) has the same wavelength fromEq. (4)withEq. (14),inthecaseofahomogeneous dependence as the localization length l(λ) (44) stack, yielding bn(λ)= 2π32λQ22, |RN|2 =N |r|2 −|hri|2 +N2|hri|2. (51) ν D E (cid:16)D E (cid:17) In the long wave limit this leads to andiswelldescribedbythe single scatteringapproxima- tion (19). For a H-stack , the transition from the local- ized to the ballistic regime at λ λ1(N) is not accom- Nk2Q2 N2k2Q4 panied by any change of the wav∼elength dependence of RN 2 = ν + ν, λ1(N) λ, (52) | | 3 36 ≪ the transmissionlength. This change can occur at much D E longerwavelengthsλ λ2(N)inthefarlongwavelength The final term in Eq. (51) differs from the final term ∼ subregion. inEq. (34)and,therefore,incontrasttotheM-stack,its To derive the ballistic length bf in the far long wave- contribution to the total reflection coefficient can be of length region, we may proceed in a similar manner to the order of, or larger than, that of the first term. This that outlined in the case of a M-stack and expand the together with Eq. (4) is equivalent to the result in Eqs. exponent t2 N = exp Nln t2 in Eq. (42). However, (48) and (50), and is applicable to short and long stacks h i h i because t2 in this expression is complex, the situation respectively. h i (cid:0) (cid:1) is more complicated than was the case for the M-stack. Thefarlongwavelengthballisticasymptotic(49)hasa In particular, the first two terms of the expansion do simplephysicalinterpretation. Indeed,inthissubregion, not contribute to the ballistic length. Taking account of the wavelength essentially exceeds the stack size and so thesecondorderleadstothe followingexpressionforthe we may consider the stack as a single weakly scattering ballistic length, b (λ), in the far long wavelength region uniformlayerwithaneffectivedielectricpermittivityε . f eff 10 Inthis case,the ballistic lengthofthe stackaccordingto Eq. (58), together with Eqs (53), (54), leads imme- Eq.(4) is diately to the far long wavelength ballistic length (49). We emphasize that because of the effective uniformity 2N of the H-stack in the far ballistic region, the transmis- b = , (53) f R 2 sion length on a single realization is a less fluctuating eff | | quantity. In contrast, transmission length for H-stacks where fluctuates strongly in the near ballistic region, as indeed it does over the entire ballistic region for M-stacks. Tocharacterizetheentirecrossoverregionbetweenthe ikN R = (ε 1) (54) two ballistic regimes (50) in greater detail, we return to eff eff 2 − the general formula (42), in which we represent t2 N as is the long wavelength form of the reflection amplitude exp(Nln t2 ). Then, in accordance with Eq. (h32i), we h i for a uniform right-handed stack of the length N and can write constant dielectric permittivity ε (with µ = 1). In the case of a uniform left-handed sefftack, the value of the ht2iN ≈exp{N(2ik−k2Q2ν)≈(1−Nk2Q2ν)exp(2ikN). reflection coefficient should be (61) As a consequence, instead of the result in Eq. (47), we ikN obtain R = (ε +1), (55) eff eff − 2 in which the value of ε is now negative (with µ= 1). 1 2π2Q2 Q4 2πN eff − = ν + ν sin2 . (62) To calculate the effective parameters, we use Eq.(14). b (λ) 3λ2 72N λ n Neglecting small fluctuations of the layer thickness, the single layer reflection amplitude in the long wavelength We note that, strictly speaking, the expansion (61) is limit reads valid inside the interval √6λ (N) λ λ (N). 1 2 ≪ ≪ The second term in Eq. (52) represents standard os- cillations of the reflection coefficient of a uniform slab of ik rj = 2 (εj −1). (56) efiqnuitaetisoinze.coIinnctihdeesfawriltohng(4w7)a.veTlehnugst,hEliqm.it(5λ2)≫deλs2c,ritbheiss theballisticlengthoverpracticallythewholeballisticre- The total reflection amplitude for a stack of length N is gion λ λ (N). Moreover,taking into account that the 1 ≫ longwavelengthasymptoticofthelocalizationlengthco- N N ik ikN 1 incides with that of the near long wavelength ballistic R = (ε 1)= ε 1 , (57) N j j 2 − 2 N −  length,weseethatthe righthandsideofEq. (52)serves j=1 j=1 X X as an excellent interpolation formula for the reciprocal   corresponding to the effective dielectric permittivity de- of the transmission length 1/lN(λ) of a sufficiently long termined by the expression stack (NQ2ν/12≫1) over the entire long wavelength re- gion λ 1. ≫ N 1 ε = ε . (58) eff j N C. Comparison of the transmission length j=1 X behaviour in M- and H- stacks For sufficiently long stacks, the right hand side of this equation can be replaced by the ensemble average εj Away from the transition regions, the transmission h i and hence lengthcanexhibitthreetypesoflongwavelengthasymp- toticsdescribedbytherighthandsidesofEqs. (33),(36), and(49). Thefirst(33)correspondstothesinglescatter- Q2 ε = ε 1+ ν. (59) ing approximationwherethe inversetransmissionlength eff j h i≈ 3 is proportional to the average reflection coefficient of a This expressionis similar to that of the two-dimensional single random layer. In the absence of absorption, this case for disordered photonic crystals reported in Ref.31. characterizes both the localization and ballistic lengths. The same considerations as above, but for a homoge- The second asymptotic form (36) takes into account the neous stack composed entirely with disordered metama- interference of multiply scattered waves and describes terial slabs, also lead to an effective value of the permit- the localization length. The third asymptotic (49) cor- tivity, responds to transmission through a uniform slab with an effective dielectric constant given by Eq. (59) and is relevant only in the ballistic regime. Q2 In the case of a M-stack,the first twoexpressions (33) ε = ε 1+ ν . (60) eff h ji≈− 3 and (36) characterize the ballistic and localized regimes (cid:18) (cid:19)