Operator Approach to Effective Medium Theory to Overcome a Breakdown of Maxwell Garnett Approximation

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Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation

Popov, Vladislav; Lavrinenko, Andrei; Novitsky, Andrey

Published in: Physical Review B

Link to article, DOI: 10.1103/PhysRevB.94.085428

Publication date: 2016

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Citation (APA): Popov, V., Lavrinenko, A., & Novitsky, A. (2016). Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation. Physical Review B, 94(8), [085428]. https://doi.org/10.1103/PhysRevB.94.085428

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PHYSICAL REVIEW B 94, 085428 (2016)

Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation

Vladislav Popov* Department of Theoretical Physics and Astrophysics, Belarusian State University, Nezavisimosti avenue 4, 220030, Minsk, Republic of Belarus

Andrei V. Lavrinenko† DTU Fotonik, Technical University of Denmark, rsteds Plads 343, DK-2800 Kongens Lyngby, Denmark

Andrey Novitsky‡ Department of Theoretical Physics and Astrophysics, Belarusian State University, Nezavisimosti avenue 4, 220030, Minsk, Republic of Belarus and DTU Fotonik, Technical University of Denmark, rsteds Plads 343, DK-2800 Kongens Lyngby, Denmark (Received 2 June 2016; revised manuscript received 8 August 2016; published 26 August 2016) We elaborate on an operator approach to effective medium theory for homogenization of the periodic multilayered structures composed of nonmagnetic isotropic materials, which is based on equating the spatial evolution operators for the original structure and its effective alternative. We show that the zeroth-, first-, and second-order approximations of the operator effective medium theory correspond to electric dipoles, chirality, and magnetic dipoles plus electric quadrupoles, respectively. We discover that the spatially dispersive bianisotropic effective medium obtained in the second-order approximation perfectly replaces a multilayered composite and does not suffer from the effective medium approximation breakdown that happened near the critical angle of total internal reflection found previously in the conventional effective medium theory. We establish the criterion of the validity of the conventional effective medium theory depending on the ratio of unit-cell length to the wavelength, the number of unit cells, and the angle of incidence. The operator approach to effective medium theory is applicable for periodic and nonperiodic layered systems, being a fruitful tool in the fields of metamaterials and subwavelength nanophotonics.

DOI: 10.1103/PhysRevB.94.085428

I. INTRODUCTION There are different approaches to find material parameters of a metamaterial, which can be grouped into the retrieval and Metamaterials is the effective approach in condensed- homogenization methods. A retrieval method allows finding matter physics for designing novel composite materials with material parameters using reflection and transmission data exceptional electromagnetic (acoustic) properties that do not [18,19]. These data can be obtained experimentally and/or occur naturally [1]. The first demonstration of the negative from numerical simulations. Although this technique has refractive index behavior predicted by Veselago [2] was done certain drawbacks and constraints, e.g., ambiguity of the with split-ring resonators employed as meta-atoms [3]. Cur- retrieval, difficulties with gathering the phase information in rently, there is an extended nomenclature of various metallic, experiments, etc., these problems can be, in principle, solved. graphene, and high-index dielectric inclusions applied for For example, an alternative wave-propagation retrieval method designing metamaterials in microwave, terahertz, infrared, and [20,21] does not suffer from the ambiguity and can be applied optical ranges [4–17]. to thick metamaterials. Usually, the parameters are retrieved Typically, to treat a periodic composite as a continuous accurately away from resonances, but in the vicinity retrieval medium, the working wavelength should be much greater than suffers from enhanced errors and noises. In Ref. [22]the the lattice period. Then fields within the unit cell are local, generalized retrieval method functioning near resonances was and material properties are determined by the polarization and proposed. magnetization vectors, i.e., by the dielectric permittivity and Plenty of homogenization methods have appeared over magnetic permeability tensors. The procedure of substituting the last decade as a reaction to the absence of a universal an inhomogeneous structure with its homogeneous analog, procedure applicable to a vast number of cases [23]. To which performs effectively the same, is called homogenization. overcome the limitations of the standard homogenization the As sketched in Fig. 1(a), the homogenized medium can first-principles homogenization theory for periodic metamate- acquire bianisotropic properties, which can include, in general, rials, the effective material parameters of which are spatially chirality and spatial dispersion. dispersive, was proposed in Ref. [24]. The multipole approach developed in Refs. [25,26] allows us to describe analytically bulk and planar metamaterials. The microscopic approach *[email protected] based on the dynamics of the classical electrons was used †[email protected] in Ref. [27] to describe the asymmetric transmission in ‡[email protected] planar chiral metamaterials [17]. Reference [28] deals with

(a) b near the critical angle of total internal reﬂection deﬁned for the (b) effective medium. As reported in Ref. [35], the transmission kt q of a multilayered structure strongly depends on periodicity, the εin kz order of the layers, and their parity, although such a dependence d ε is supposed to be nonexistent when the EMA is applied, in particular for deep-subwavelength dielectric multilayers. Metal-dielectric planar multilayered systems can be con- d ε d d sidered effectively anisotropic (uniaxial) media possessing

2 either closed ellipsoid or open hyperboloid isofrequency N

z slabs surfaces. In the latter case the multilayers, called hyperbolic metamaterials, enable a number of distinctive properties, including guiding evanescent waves (with wave vectors much bigger than those in the corresponding uniform dielectric), hyperlensing, and enhancement of the spontaneous emission ε μ α β rate [10,38–42]. The EMA is applicable for multilayered hyperbolic metamaterials, too; however certain constraints d must be acknowledged [43]. It should be noted that, typically,

d εout homogenization approaches have been developed for layered z structures to catch some specific features. For example, the high-frequency homogenization is applicable near the points z of topological transition from the elliptic to hyperbolic regime [44]. However, a general and consistent homogenization FIG. 1. (a) Bilayer unit cell of a multilayer dielectric structure approach for multilayers is still missing. (top) and its homogenized counterpart with bianisotropic effective In this paper, we use an operator technique to develop an medium parameters (bottom). (b) Periodic multilayered dielectric operator effective medium approximation (OEMA), in which structure with N bilayer unit cells. the zeroth order renders the conventional EMA (Maxwell Gar- nett theory), while higher orders ameliorate the accurateness of material tensors. The approach is grounded on the theory electromagnetostatic homogenization of materials with mi- presented in Ref. [45], which is adapted to find the effective croscopically bianisotropic inclusions. A rigorous homoge- material tensors of the homogenized multilayered structure. In nization theory via the Whitney interpolation developed in Sec. II we briefly discuss the propagation of plane waves in Ref. [29] allows us to reproduce the well-known Maxwell bianisotropic media using the spatial evolution operator tech- Garnett formula and introduce artificial magnetism. nique developed in Refs. [46–48]. It was used, in particular, Local homogenization can be insufficient for correct de- for analyses of planar waveguides modes [49] and retrieving scription of the wave propagation in a metamaterial even in bianisotropic material tensors from transmission and reflection its validity range. In this case, the spatial dispersion should spectra [50,51]. In Sec. III we outline the OEMA and show how be taken into consideration, and homogenization becomes to accurately determine material parameters of the effective nonlocal. Although material parameters still depend on wave medium. The zeroth-order OEMA for multilayered dielectric characteristics in nonlocal theories, the latter provide better systems is considered in Sec. IV. This approach is limited correspondence to the exact solutions, and therefore, their to only electric dipoles. In Sec. V we discuss the first-order usage is justified for linear [24,30] and nonlinear media [31]. OEMA and its connection with the chiral properties of the Nonlocality can be used for dispersion engineering [32,33] multilayer. In the second-order OEMA, consequently, mag- in transformation optics formalism applied to the spectral netic dipoles and electric quadrupoles are taken into account domain and for investigation of anisotropy induced by spatial (Sec. VI). In Sec. VII we demonstrate that application of the dispersion [34]. When spatial dispersion is a shortcoming second-order OEMA allows avoiding the EMA breakdown for some metamaterial-based application, one can apply the near the critical angle of total internal reflection. We also nonlocal homogenization approach to find out how to reduce identify the criterion for the zeroth-order OEMA validity, undesired dispersion consequences. In the present paper we which depends on the unit-cell thickness to wavelength deal with nonlocal homogenization for a multilayered system. ratio, the number of unit cells, and the angle of incidence. Different materials forming the unit cell of a metamaterial Section VIII advances application of the OEMA to frequency- can have a complex shape. However, it is instructive to use a dispersive metal-dielectric multilayers. Comparison with the simple toy model which can provide the main features of the results of the latest high-frequency homogenization approach metamaterial function. Such a simple model can be a periodic [44] and conclusions are also provided here. multilayered system, as demonstrated in Fig. 1(b).The homogenization in this case is given by the Maxwell Garnett II. PLANE WAVES IN MULTILAYERED BIANISOTROPIC (quasistatic) approach, which is called the effective medium MEDIA approximation (EMA). Recently, the homogenization of the subwavelength dielectric layered structures has attracted close We consider a plane monochromatic electromagnetic wave attention in connection with the discovered EMA breakdown with circular frequency ω propagating in a bianisotropic slab [35–37]. The EMA breakdown typically is quite pronounced characterized by dielectric permittivity tensorε ˆ, magnetic

085428-2 OPERATOR APPROACH TO EFFECTIVE MEDIUM THEORY . . . PHYSICAL REVIEW B 94, 085428 (2016) permeability tensorμ ˆ , and gyration pseudotensorsα ˆ and In the case of a homogeneous bianisotropic medium βˆ [52], matrix Mˆ is constant, thus allowing us to write down the fundamental solution of the system by means of the matrix D(ω,r) = εˆ(ω)E(ω,r) + αˆ (ω)H(ω,r), exponential [47,49] B(ω,r) = βˆ(ω)E(ω,r) + μˆ (ω)H(ω,r). (1) W(z) = exp[ik0zMˆ ]W(0), (4)

Here E, H, D, and B are, respectively, the strengths of electric where W(0) is the initial field at z = 0. The field transmitted and magnetic fields, electric displacement, and magnetic by the first slab is the initial field for the second one, the second induction vectors. Under the oblique incidence the wave vector slab produces the initial field for the third slab at its output, = can be decomposed into two components, tangential kt k0b and so on. Therefore, the field transmitted by the stack of N = = and longitudinal kz k0ηq, where k0 ω/c is the wave slabs is of the form number in vacuum, b lies in the plane of incidence, and q is the unit vector normal to the slab interfaces (see Fig. 1). N = ˆ = ˆ Since the field is translation invariant in the plane (x, y), W(D) exp[ik0dj Mj ]W(0) P W(0), (5) j=1 it can be written in the form E(r) = E(z)exp(ik0b · r), thus reducing the Maxwell equations to the system of the four = N ˆ first-order ordinary differential equations for the tangential where D j=1 dj is the thickness of the stack. Tensor P T can be called a spatial evolution operator because it describes field components W = (Ht ,q × E) , where T denotes the the spatial evolution of the initial field W(0). transpose operation. Here two tangential fields, Ht = IˆH = T T In Fig. 1(b) the periodic structure of 2N alternating (Hx ,Hy ) and q × E = (−Ey ,Ex ) , are introduced, where Iˆ = 1ˆ − q ⊗ q is the projector onto the plane orthogonal to q dielectric slabs with permittivities ε1 and ε2 and thicknesses (two-dimensional unit tensor), 1ˆ is the three-dimensional unit d1 and d2 is considered. Oblique incidence of a plane wave in tensor (Kronecker’s delta tensor), and ⊗ stands for the tensor dielectric material εin is assumed. The transmitted wave exits to the semi-infinite dielectric medium εout. For the parameters (outer) product defined as (u ⊗ v)i,j = ui vj , i,j = 1,2,3. The system of four differential equations can be rewritten given in the caption of Fig. 2 the effective parameters in the (0) = (0) = as a single equation for unknown field vector W: zeroth-order OEMA are equal to ε|| 3 and ε⊥ 5/3. Then the critical angle of total internal reflection for a TE wave = ◦ dW(z) is αc 60 , which corresponds√ to the normalized tangential = ik Mˆ W(z), (2) = = dz 0 wave number kt /k0 bc 3. Fresnel’s (amplitude) transmission operator can also be ˆ where Mˆ is the 4 × 4 matrix, whose representation as a block written in terms of the spatial evolution operator P as [47] matrix reads [47] −1 − Iˆ Aˆ Bˆ tˆ = 2 γˆ Iˆ Pˆ 1 γˆ , (6) Mˆ = . in γˆ in Cˆ Dˆ out

whereγ în andγ ôut are the surface impedance tensors of the In the case of bianisotropic media described by material input and output media shown in Fig. 1(b). For instance,γ în equations (1), the 2 × 2 blocks Aˆ, Bˆ , Cˆ , and Dˆ are equals × × × Aˆ = q αˆ Iˆ + q εˆq ⊗ v + (b + q αˆ q) ⊗ v , − 2 3 1 = εin b ⊗ + 1 ⊗ × × × × × × γîn a a b b. (7) ˆ =− + ⊗ + + ⊗ 2 2 2 B q εˆq q εˆq q v4 (b q αˆ q) q v2, b εin b εin − b Cˆ = Iˆμˆ Iˆ + Iˆμˆ q ⊗ v + (−a + Iˆβˆq) ⊗ v , 1 3 The output magnetic field equals H(tr) = tˆH(inc), where H(inc) ˆ =−ˆ ˆ × + ˆ ⊗ × + − + ˆ ˆ ⊗ × t t t D Iβq Iμˆ q q v2 ( a Iβq) q v4, (3) is the magnetic field of the incident wave. Then the power transmission coefficient for a TE wave reads × × where q is the tensor dual to vector q [47,52,53][(q )ik = −1 (tr)2 εij k qj , where εij k is the antisymmetric Levi-Civita tensor and Re η H T = out t , (8) summation over repeated indices from 1 to 3 is assumed], − 2 Re η 1 H(inc) a = b × q, and in t where η = ε − b2 and η = ε − b2 are the longi- v = δ (β qαˆ Iˆ − ε qμˆ Iˆ − β a), in in out out 1 q q q q tudinal components of the input and output wave vectors v2 = δq (βq qεÎˆ − εq qβÎˆ − εq a), normalized to k0, respectively. Fresnel’s transmission coefficient t for the TE-polarized v = δ (α qμˆ Iˆ − μ qαˆ Iˆ + μ a), 3 q q q q wave passing through the N-period multilayer [see Fig. 1(b)] v4 = δq (αq qβÎˆ − μq qεIˆ + αq a), equals − δ = (ε μ − α β ) 1,ε= qεˆq, −1 q q q q q q ηin + ηout P1P3 t = P2 + , (9) μq = qμˆ q,αq = qαˆ q,βq = qβˆq. 2ηout 2ηout

085428-3 POPOV, LAVRINENKO, AND NOVITSKY PHYSICAL REVIEW B 94, 085428 (2016)

FIG. 2. Transmission of s- (TE-) polarized light through the N-period stack of slabs calculated exactly, in the zeroth-order√ OEMA, in = = − the first-order OEMA, and in the second order√ OEMA vs the number of unit cells N for (a)√ subcritical (εout 4, b 3 0.01) and = = = = (b) critical angles of incidence (εout 4, b 3),√ (c) output permittivity εout (N 25, b 3), and (d) the number of unit cells for the impedance-matched output medium (εout = 3, b = 3 − 0.01). Parameters are ε1 = 5, ε2 = 1, d1 = d2 = 10 nm, εin = 4, radiation wavelength λ = 500 nm. The first layer of the stack is slab 1 in (a), (b), and (d) and slab 2 in (c). where The idea behind the OEMA is a conventional routine to replace the periodic structure of thickness Nd by a homo- sin(NKB d) geneous slab (effective medium) of the same thickness Nd, P1 = ,P2 = cos(NKB d), sin(KB d) where N and d are the number of unit cells and thickness of a P 11 − P 22 single unit cell, respectively. In general, the effective medium P =−P 12 − η η P 21 + (η − η ) TE TE. is bianisotropic, thus having dielectricε ˆ and magnetic 3 TE in out TE in out 2 eff μˆ eff tensors and gyration pseudotensorsα ˆ eff and βêff as (10) material parameters. With the ideal homogenization we cannot distinguish the initial layered medium and effective medium = + Here d d1 d2 is the thickness of the unit cell. The because the electromagnetic field outside the homogeneous dispersion equation for the Bloch wave number KB and slab and inhomogeneous medium are the same. This can 11 12 21 22 elements PTE, PTE, PTE, and PTE of the TE-wave evolution be achieved owing to the equality of the spatial evolution operator bPˆb/b2 are given in Appendix A. operators for homogeneous and inhomogeneous samples over a period:

III. OPERATOR EFFECTIVE MEDIUM APPROXIMATION exp[ik dMˆ ] = exp[ik d Mˆ ]exp[ik d Mˆ ]. (11) The EMA is usually derived using the Maxwell Garnett 0 eff 0 2 2 0 1 1 (quasistatic) approach. Then a homogeneous effective medium replacing the multilayered structure has quite simple material It should be noted that the OEMA through the equality (11) parameters of an anisotropic medium. The coordinate-free is applicable even for a single period, i.e., for nonperiodic approach of such homogenization generalized for an arbitrary structures, too. Thus, in no way we are restricted by the number of anisotropic layers deﬁned through complex non- number of periods N in the process of homogenization. In symmetrical tensorsε, ˆ μˆ composing the unit cell are derived in the above discussion we assume an external illumination of [54]. In this section we apply an operator approach developed the multilayer system. OEMA can also be applied when the in Ref. [45] to improve the accuracy of the EMA. We will refer light source is placed at the boundaries between periods, i.e., to this method as operator effective medium approximation. at z = jd (j = 1,2,...,N − 1).

085428-4 OPERATOR APPROACH TO EFFECTIVE MEDIUM THEORY . . . PHYSICAL REVIEW B 94, 085428 (2016)

Matrix Mˆ eff can be derived using the logarithm of the αˆ eff = α||Iˆ + α⊥q ⊗ q, and βˆeff = β||Iˆ + β⊥q ⊗ q.MatrixMˆ operator in the zeroth-order EMA reads × β⊥b⊗a ε⊥b⊗b 1 α||q − ε||Iˆ − (0) δ⊥ δ⊥ Mˆ eff = ln(exp[ik0d2Mˆ 2]exp[ik0d1Mˆ 1]) (12) Mˆ = , eff μ⊥a⊗a × α⊥a⊗b (16) ik0d μ||Iˆ − −β||q − δ⊥ δ⊥ or through the Baker-Campbell-Hausdorff series [45] where δ⊥ = ε⊥μ⊥ − α⊥β⊥. By introducing matrices Mˆ into Mˆ eff = ρMˆ 1 + (1 − ρ)Mˆ 2 Eq. (14), we get a system of algebraic equations ik d × − ⊗ = + 0 ρ(1 − ρ)[Mˆ ,Mˆ ] α||q (β⊥/δ⊥)b a 0, 2 2 1 ε||Iˆ − (ε⊥/δ⊥)b ⊗ b = ρ(ε1Iˆ − b ⊗ b) (k d)2 − 0 ρ(1 − ρ)(ρ[[Mˆ ,Mˆ ],Mˆ ] + − ˆ − ⊗ 12 2 1 1 (1 ρ)(ε2I b b), μ||Iˆ − (μ⊥/δ⊥)a ⊗ a = ρ(Iˆ − a ⊗ a/ε ) + (1 − ρ)[[Mˆ 1,Mˆ 2],Mˆ 2]) +··· , (13) 1

+ (1 − ρ)(Iˆ − a ⊗ a/ε2), where ρ = d1/d and [Mˆ 1,Mˆ 2] is a commutator. Equation (13) × is the expansion with respect to the powers of k0d. Since M is − β||q − (α⊥/δ⊥)a ⊗ b = 0. (17) the 4 × 4 matrix, Eq. (13) represents the system of 16 algebraic equations, which are not sufficient to uniquely determine 36 The solution of this system of equations brings us to the = ˆ = = ˆ material parameters: each of the tensorsε ˆ ,ˆμ ,ˆα , and βˆ nongyrotropic (α ˆ eff βeff 0) nonmagnetic (μ ˆ eff 1) uni- eff eff eff eff axial medium as the zeroth-order effective medium: has, in general, nine components. Hence, there exists a group of equivalent effective media which are able to describe the − −1 (0) = + − (0) = ρ + 1 ρ same inhomogeneous structure. ε|| ρε1 (1 ρ)ε2,ε⊥ . (18) ε1 ε2 It should be noted that such expansion over k0d was also used in other works [30,55]. In Ref. [55], the authors Thus, the zeroth-order material parameters are those obtained 0 consider the zero-order term (k0d) corresponding to the using the quasistatic Maxwell Garnett approach [40]. With local effective parameters and outline the way to introduce the assumption of the form of the effective media tensors we higher-order corrections to effective medium parameters. In determined them uniquely because eight unknown material Ref. [30], the expansion up to the second order is applied to parameters can be found from 16 equations of the system the three-dimensional metamaterials to describe the effects of (14). It should be noted that the wrong ansatz of the effective artificial chirality and spatial dispersion. medium tensors can result in the contradictory system of algebraic equations. A dielectric multilayered system in the lowest order of IV. ZEROTH-ORDER OEMA: ELECTRIC DIPOLES the OEMA is just a dielectric described by the average The Baker-Campbell-Hausdorff series (13)servesasthe electric dipole moment (medium polarization). Parameters exact homogenization. If k0d is small, the series can be safely (18) depend on neither the number of unit cells nor the location cut. The power of k0d defines the order of approximation: the of the materials in the unit cells. n nth-order approximation corresponds to (k0d) in Eq. (13). Fresnel’s transmission coefficient of the multilayered struc- (0) Then the zeroth-order approximation of Mˆ eff can be written as ture in the zeroth-order approximation t can be written in follows: awaysimilartoEq.(9), but the quantities P are replaced by (0) (0) P : Mˆ = ρMˆ 1 + (1 − ρ)Mˆ 2. (14) eff sin(Nk η(0)d) P (0) = 0 ,P(0) = cos(Nk η(0)d), ˆ 1 (0) 2 0 If the wave and material parameters in M1,2 are not peculiar, the sin(k0η d) zeroth-order approximation is conventionally valid at d λ, i sin(k η(0)d) where λ is the radiation wavelength. (0) = 0 − (0) 2 − P3 [ (η ) ηinηout], (19) Let us find material parameters from Eq. (14) in the case of η(0) alternating isotropic dielectric slabs with matrices Mˆ 1 and Mˆ 2. where η(0) = ε(0) − b2 is the longitudinal component of the By substitutingε ˆ1,2 = ε1,21,ˆ μ ˆ 1,2 = 1,ˆ α ˆ 1,2 = βˆ1,2 = 0into || Eq. (3), one obtains matrices Mˆ 1,2 as wave vector in the zeroth- order OEMA normalized to k0. Despite the validity condition of EMA d = d + d ˆ − ⊗ 1 2 0 ε1,2I b b λ, the zeroth-order approximation can fail when either the Mˆ = . (15) 1,2 ˆ − a⊗a number of unit cells exceeds a certain value [see Figs. 2(a) and I ε 0 1,2 2(b)] or the ambient medium is impedance matched [see the The effective medium can be regarded as a general example in Figs. 2(c) and 2(d)]. bianisotropic one. Then its parameters appear not to be unique. However, the physical insight can significantly reduce V. FIRST-ORDER OEMA: CHIRALITY the number of material parameters. In fact, the stack of isotropic slabs has a preferred direction specified by the Matrix Mˆ in the first-order OEMA vector q. Then the effective medium material tensors can (1) ik0d = ˆ + ⊗ = ˆ + ⊗ Mˆ = ρMˆ 1 + (1 − ρ)Mˆ 2 + ρ(1 − ρ)[Mˆ 2,Mˆ 1] (20) be assumed asε êff ε||I ε⊥q q,ˆμeff μ||I μ⊥q q, eff 2

085428-5 POPOV, LAVRINENKO, AND NOVITSKY PHYSICAL REVIEW B 94, 085428 (2016) breaks the symmetry with respect to the sequence of the slabs. The first two approximations of OEMA work well only Starting the stack from layer 2 one exchanges Mˆ 1 and Mˆ 2 and for a small number of unit cells N [Fig. 2(a)]. Further, both therefore changes the sign of the last term. approximations fail, with the zeroth-order OEMA being even Since the commutator closer to the exact solution than the first order, when ε = ε , in out i.e., when chirality owing to the same ambient media does not ik0d Aˆ 0 ρ(1 − ρ)[Mˆ ,Mˆ ] = 0 (21) appear. 2 1 0 −Aˆ 2 0 The obtained first-order correction for the material param- is a matrix with nonzero diagonal blocks, the effective eters results in, however, the second-order corrections for the ˆ (1) wave number of the plane wave in a homogenized medium medium matrix Meff can have nonzero gyration pseudotensors because the Fresnel equation gets the correction α2 or α2 αˆ eff and βêff. Thus, the chiral properties of the effective 1 2 material are built up only in the first-order approximation and depending on the wave polarization. So the spatially dispersive chiral parameters should result in the second-order effect, and therefore are quite small as k0d 1. Keepingμ ˆ eff = 1ˆ and a permittivity tensor as in Eq. (18), we shall find the gyration the second-order OEMA should be used for more accurate prediction of the reflection and transmission. pseudotensors. Constantε êff andμ ˆ eff require invariant Bˆ and It should be noted that, alternatively, chiral material pa- Cˆ in Eq. (3), that is, qαˆ effIˆ = qβêffIˆ = Iαˆ effq = Iβˆ effq = 0. Imposing conditions α = β = 0, one derives the blocks of rameters can be assigned with a Hermitian permittivity tensor q q = + × = T ˆ ˆ = × ˆ ˆ =−ˆ ˆ × of the formε êff εˆsym ih , whereε ˆsym εˆsym and h is a matrix Meff: A q αˆ effI and D Iβeffq . Finally, we = = derive the gyration pseudotensors as nonunitary vector, while αeff βeff 0. Such a form ofε ˆ is often used to describe the Faraday optical activity induced by a ⊗ b b ⊗ a αˆ =−βˆT = Iˆαˆ Iˆ =−q×Aˆ = α + α , the magnetic field [52]. eff eff eff 0 1 b2 2 b2 (22) VI. SECOND-ORDER OEMA: MAGNETIC DIPOLES AND where ELECTRIC QUADRUPOLES 2 Material tensors of the second-order OEMA can be = ik0d = ik0d = b − α1 σ, α2 σf (b),f(b) 1, searched like those of a bianisotropic crystal. Since the third 2 2 εr term in Eq. (13) maintains when the sequence of the slabs σ = ρ − ρ ε − ε . (1 )( 2 1) (23) changes to the opposite one, the second order does not affect = −1 + −1 −1 the chiral properties. The gyration pseudotensors can be kept as Here εr (ε1 ε2 ) is the reduced permittivity. When in Eq. (22), but the permittivity and permeability tensors have the stack starts from layer 2 (1 ↔ 2),α ˆ eff and βêff change their 2 signs. to be changed to incorporate proportional to (k0d) terms: Eigenwaves of the effective medium of the first order are 2 (0) (k0d) TE- and TM-polarized plane waves. Although the obtained ε|| = ε + σf (b)˜ε||, || 6 constitutive relations are very similar to those for uniaxial structures [56], the coupling between the electric and magnetic 2 − = (0) − (k0d) (0)2 2ρ 1 − f (b) fields is different for TE and TM waves, i.e., α = α .It ε⊥ ε⊥ σε⊥ , 1 2 6 εr ε˜⊥ is curious that this result is in contrast to that obtained 2 in Ref. [30], where it was shown that the structure under (k0d) μ|| = 1 + σ (2ρ − 1), 6 consideration is a genuine uniaxial medium. 2 It should be noted that the effective gyrotropy is surface (k0d) ε1 ε2 induced because it can be removed by transferring half of μ⊥ = 1 − σ ρ − (1 − ρ) , (25) 6 ε ε the last slab to the place before the first layer, which makes 2 1 the structure symmetric. Volume gyrotropy, in principle, can where also be achieved, e.g., using a three-layer unit cell without an −1 inversion center. ρ 1 − ρ (1) ε˜|| = ρε1 − (1 − ρ)ε2, ε˜⊥ = − . (26) Parameters Pj in the transmission coefficients (9)ofthe ε1 ε2 TE-polarized wave in the first-order approximation are equal to Corrections introduced in this section depend on the tangential wave number kt = bk ; hence, the spatial dispersion sin(Nk η(1)d) 0 P (1) = 0 ,P(1) = cos(Nk η(1)d), is taken into account. The spatial dispersion effect manifests 1 (1) 2 0 sin(k0η d) that the structure is thick compared with the radiation (1) wavelength. One expects that the higher-order multipole (1) i sin(k0η d) (0) 2 P = [−(η ) − ηinηout + (ηin − ηout)α1], moments besides the electric dipole will be excited. In fact, 3 η(1) an artificial magnetic response is observed in Eq. (25), and (24) it is not spatially dispersive. The correction to the dielectric permittivity tensor originates from the electric quadrupole (1) = (0) − 2 + 2 where η ε|| b α1 is the normalized longitudinal moment, which depends on the tangential wave number wave number of the wave. The dependence of η(1) versus introducing a spatial dispersion. Values of the magnetic dipole b is the dispersion equation of the effective medium in the and electric quadrupole moments are of the same order, as first-order approximation. anticipated for a nonmagnetic object.

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The eigenwaves in the second-order OEMA are TE- and matched and no dependence on N is detected. The second- TM-polarized plane waves like in the first-order OEMA, while order OEMA still functions perfectly. the medium can be treated as a uniaxial medium with In order to find out the reasons for the EMA breakdown different coupling constants α1,2. we analyze the transmission coefficient T for the incident The second-order OEMA yields the following parameters TE-polarized electromagnetic wave. Fresnel’s transmission Pj to substitute into Eq. (9) for TE-polarized light: coefficient of the multilayered structure composed of N layer pairs is given by Eq. (9). It should be compared with sin(Nk η(2)d) (0) P (2) = 0 ,P(2) = cos(Nk η(2)d), the transmission coefficient t of the zeroth-order effective 1 (2) 2 0 sin(k0η d) medium of thickness Nd, which follows from Eq. (9) when (0) (2) P = P according to Eq. (19). iμ|| sin(k η d) α P (2) = 0 (η − η ) 1 − η η (0) 3 (2) in out in out The difference between coefficients t and t is apparently η μ|| (0) = caused by the discrepancy in coefficients Pj and Pj (j α2 − (η(2))2 1,2,3). Since we consider subwavelength structures, it is + 1 , μ2 instructive for further analysis to use approximation k0d 1. || We can write the difference in transmission T = T − T (0) as = − (0) where the dimensionless longitudinal wave number equals a linear function of deviations δPj Pj Pj (see Appendix (2) 2 B for details): η = α + ε||μ|| − μ||/μ⊥b2. 1 + (0) (0) ηin ηout T =−2T Re t δP2 2ηout VII. SECOND-ORDER OEMA WITHOUT A BREAKDOWN t(0) As shown in Ref. [35], the periodic multilayered structure, + (0) + (0) P3 δP1 P1 δP3 . (27) parameters of which are indicated in the caption of Fig. 2, 2ηout violates the main predictions of the effective medium ap- If the terms on the right-hand side are small, the zeroth-order proximation (zeroth-order OEMA in this paper). We shall OEMA works well. Otherwise, the zeroth-order OEMA cannot demonstrate that the elaborated corrections to the material be applied, and the phenomenon is called the EMA breakdown parameters solve the problem of the EMA breakdown. [35]. We aim to derive the conditions of the breakdown. For the subcritical angle of incidence the zeroth- and first- Each of the quantities in the transmission difference (27) order OEMAs keep the periodic dependencies characteristic is calculated in Appendix B, establishing the dependence on to the exact solution but quantitatively differ, as shown in (0) k0d, N, and η. By substituting the values of P and δPj into Fig. 2(a). Taking into consideration the effects of the structure’s j Eq. (27) in approximation k0d 1, we have chirality, magnetic dipole, and electric quadrupole moments, = − + 2 we restore the correct behavior of transmission T for any T A(ηout ηin)σ (k0d) O[(k0d) ] (28) number of unit cells N. Moving to the critical angle, the for η(0) = 0 and transmission in the zeroth- and first-order OEMAs changes = { − 2 + 3 } drastically [see Fig. 2(b)]. In this case, the zeroth-order T N B(ηout ηin)σ (k0d) O[(k0d) ] (29) effective medium does not transmit electromagnetic waves in for η(0) → 0, where A and B are the coefficients derived in accordance with the phenomenon of the total internal reflection 3 Appendix B. The term proportional to N(k0d) arises due to except for the range of small N when the waves instead tunnel (0) 3 the wave-number deviation Kd = (KB − k0η )d ∼ (k0d) through the thin structure. These tunneling waves result in the (see Appendix A). It is less than other terms stemming transmission corresponding to that calculated accurately. from chirality and spatial dispersion. In the case η(0) = 0 In Fig. √2(c) we show the transmission at the critical angle the transmission difference linearly depends on k d as it = 0 (i.e., b 3) as a function of the permittivity εout of the is commonly adopted. But when η(0) → 0, the transmission output medium. The effective permittivity of the multilayer difference grows with the number of unit cells N, yielding the (0) is ε|| = 3. When εout 3, the effective permittivity of the great deviation, in accordance with Fig. 2(a). system “periodic structure and output medium” is not greater If εin = εout (or ηin = ηout), the term proportional to σ in = than 3. Hence, the total internal reflection occurs, and T 0. the deviation δP3 vanishes (see Appendix B). This means 3 For εout > 3 the system is transmissive with the greatest value that δP3 ∼ (k0d) and the EMA works better. In fact, the (0) 2 at εout ≈ ε|| . The sharp peak in Fig. 2(c) that cannot be caught transmission difference is proportional to T ∼ (k0d) for (0) 3 (0) by the zeroth-order OEMA is reproduced by the second-order η = 0 and T ∼ N(k0d) for η → 0. OEMA even though the second-order corrections of material Thus, the criterion for the zeroth-order OEMA to be valid parameters are small by default and have no peculiarities with is no longer k0d 1but respect to b. This means that the resonance exists only in sin[Nη(0)k d](k d) transmission near the critical angle and is extremely sensitive 0 0 1 (30) (0) to the parameters of the system. Transmission in the peculiar η ηout (0) = case of the impedance-matched output medium (εout = ε|| )is for εin εout and shown in Fig. 2(d). It should be noticed that the behavior of sin[Nη(0)k d](k d)2 T is totally different for the exact solution and in the zeroth- 0 0 1 (31) (0) order OEMA. T is constant in the zeroth-order OEMA because η ηout the multilayer and output substrate are effectively impedance for εin = εout.

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In the first-order OEMA η(1) differs from η(0) by the term 2 3 proportional to (k0d) . Therefore, Kd ∼ (k0d) is not more precise than that in the zeroth-order OEMA. At the same time 3 δP3 ∼ (k0d) , whose estimation is more accurate than in the zeroth order for εin = εout. Thus, the first-order OEMA is better than the zeroth one only if εin = εout. Figure 2(d) supports our conclusion: the first-order OEMA has oscillating behavior as the exact solution, while the zeroth-order approximation exhibits a constant value. 5 In the second-order OEMA Kd ∼ (k0d) and δP3 ∼ 5 3 (k0d) because the terms proportional to (k0d) cancel each other. Then there is no peculiarity of the transmission at η(0) = 5 0 and T ∼ N(k0d) . This approximation can be violated if ∗ ∗ −5 N Kd ∼ 1, i.e., N ∼ (k0d) . For the parameters in Fig. 2 we estimate N ∗ ∼ 1000.

VIII. DISCUSSION AND CONCLUSION The second-order OEMA is applicable within an extremely wide range of multilayered nanostructures. In what follows we show that it can describe wave propagation in frequency- dispersive media as well. We compare the transmission in the zeroth-, first-, and second-order OEMAs with those calculated accurately and using the high-frequency homogenization [44]. In the stack of metal-dielectric slabs, only metal is assumed to have the Drude-model dispersion. When the validity conditions of the ordinary EMA hold true [see Fig. 3(a)], the high-frequency homogenization and zeroth-order OEMA are FIG. 3. Transmission of s- (TE-) polarized light through the very close but differ heavily from the exact solution. The latter metal-dielectric structure calculated exactly, in the zeroth-order is indistinguishable from the second-order EMA, although the OEMA, in the first-order OEMA, in the second-order OEMA, and first-order OEMA is insufficient. When thickness d is not much with the high-frequency homogenization (HFH) approach vs normal- less than the wavelength [see Fig. 3(b)], the second-order EMA ized tangential wave number b = kt /k0 for (a) k0d = 0.24 (d1 ======is not a valid approximation anymore. Nevertheless, it catches d2 0.2c/ωp, ω 0.6ωp, εin 1,εout 0.241) and (b) k0d 2 = = = = = the resonances of the exact solution, thus exhibiting similar (d1 d2 2c/ωp, ω 0.5ωp, εin 1,εout 0.215). Parameters are = − 2 2 = = behavior. Other homogenization models considered cannot be ε1 1 ωp/ω , ε2 2.25, N 10. used at all outside their applicability ranges. The operator effective medium approximation is a general method to homogenize an arbitrary multilayered system. It is valid for a small number of slabs in the system, e.g., two correction of the material parameters dramatically changes slabs, as well as for any periodic and nonperiodic structure. transmission. The slabs of these structures can be anisotropic, magnetic, and In conclusion, we have found the effective medium tensors chiral. OEMA does not require the development of a special in different orders of the operator effective medium approx- technique for another composition of layers. The method is imation. Each order corresponds to the artificial multipole straightforward, but the technical difficulties should be solved moments. We have revealed that the second-order OEMA to succeed in finding material tensors of the effective medium. accurately describes the periodic multilayered structure even Although the idea of the OEMA is given in Ref. [45], in such peculiar conditions as total internal reflection and a we have analyzed different orders of the effective medium dispersive hyperbolic metamaterial. OEMA in the zeroth-order approximation for bilayer unit cells in detail. We have approximation can be used to find local parameters of a revealed that the orders of the approximation correspond to the multilayered structure composed of anisotropic slabs. appearance of the artificial multipole moments in the structure, with the value k0d being the dimensionless size parameter as in scattering theory [57]. When k d is small, the electric 0 ACKNOWLEDGMENTS dipole moment is important. Greater k0d produces higher- order artificial multipole moments. We apply the second-order We are grateful to P. Belov, S. Maslovski, and the anony- OEMA to avoid the EMA breakdown in the vicinity of the mous reviewer for fruitful discussions. We acknowledge the critical angle of the total internal reflection found earlier [35]. Belarusian Republican Foundation for Fundamental Research The breakdown in the zeroth-order OEMA is unavoidable (Grant No. F16R-049) for financial support. Partial financial because transmission near the critical angle is very sensitive to support from the Villum Fonden via the DarkSILD project is the exact values of the effective parameters: even a very small acknowledged.

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11 12 21 22 APPENDIX A: COEFFICIENTS PTE, PTE, PTE, AND PTE where AND BLOCH WAVE NUMBER K B + (0) + (0) ηin ηout δP1P3 P1 δP3 The evolution operator Pˆ of a bilayer unit cell for a TE- τ = δP2 + . (B2) 2η 2η polarized electromagnetic wave with ﬁelds (b/b) · W = (b · out out · T Ht /b,a Et /b) should be projected onto unit vector b/b as Then the transmission difference T = T − T (0) takes the 11 12 form bPˆb PTE PTE PˆTE = = . (A1) η η 2 21 22 = in | |2 − in | (0)|2 =− (0) b PTE PTE T t t 2T Re(t0τ). (B3) ηout ηout ˆ The elements of PTE are of the form T in the explicit form is demonstrated by Eq. (27). k (0) 11 = − z2 Let us calculate Pj and δPj in approximation k0d 1. PTE cos(kz1d1) cos(kz2d2) sin(kz1d1)sin(kz2d2), (0) (0) kz1 In the expressions we can treat sin(k0η d) ≈ k0η d,but (0) (0) = (0) i Nk0η d is large, and sin(Nk0η d) Nk0η d except in 12 (0) P = [sin(kz2d2) cos(kz1d1)kz2 + sin(kz1d1) the case of η → 0. TE k 0 Then × cos(kz2d2)kz1], (0) sin(Nk0η d) (0) = 21 = + (0) (0) for η 0, PTE ik0[sin(kz2d2) cos(kz1d1)/kz2 sin(kz1d1) P = k0η d 1 (0) = × N for η 0, cos(kz2d2)/kz1], (0) (0) k cos(Nk0η d)forη = 0, 22 = − z1 P (0) = PTE cos(kz1d1) cos(kz2d2) sin(kz1d1)sin(kz2d2), 2 (0) = kz2 1forη 0, (A2) −ik d(η(0)2 + η η )forη(0) = 0, (0) = 0 in out P3 (B4) 2 − (0) = where kjz = k0 εj − b (j = 1,2) is the longitudinal compo- ik0dηinηout for η 0, nent of the wave vector in the jth layer. When k d 1, one obtains and 0 2 2 2 2 pN K + 2 (0) = k d + k d (0) O[(N Kd) ]forη 0, 11 = − z1 1 z2 2 − 2 + 4 δP = k0η PTE 1 kz2d1d2 O[(k0d) ], 1 3 6 (0) 2 O[N (k0d) ]forη = 0, i ⎧ 12 = 2 + 2 + 3 ⎨− sin(Nk η(0)d)N Kd PTE kz1d1 kz2d2 O[(k0d) ], 0 k0 = +O[(N Kd)2]forη(0) = 0, δP2 ⎩ 21 = + + 3 2 6 (0) PTE ik0(d1 d2) O[(k0d) ], O[N (k0d) ]forη = 0, k2 d2 + k2 d2 2 22 z1 1 z2 2 2 4 σ (k0d) 3 P = 1 − − k d1d2 + O[(k0d) ]. δP = (η − η ) + O[(k d) ], (B5) TE 2 z2 3 out in 2 0 (A3) where The Bloch wave number K follows from the dispersion B sin(Nk η(0)d) equation p = cos(Nk η(0)d) − 0 . (B6) 0 (0) Nk0η d 11 + 22 PTE PTE cos[KB d] = . (A4) (0) = 2 The transmission difference for η 0is For k d 1 this equation reduces to 2 0 T = 2T0Re(t0τ1)N Kd + 2T0Re(t0)τ2k0d + O[(k0d) ] η(0)2k2d2 (B7) cos[K d] = 1 − 0 + O[(k d)4], (A5) B 2 0 and for η(0) → 0isoftheform (0) 3 whose solution is KB d = k0η d + O[(k0d) ]. Therefore, (0) 2 3 the wave-number difference equals Kd = KB d − k0η d = T = 2T0Re(t0)τ3N(k0d) + O[N(k0d) ], (B8) 3 O[(k0d) ]. where APPENDIX B: TRANSMISSION COEFFICIENT η + η η(0)2 + η η τ = in out sin(Nk η(0)d) + ip in out , DIFFERENCE 1 0 (0) 2ηout 2ηoutη Fresnel’s transmission coefﬁcient (9) can be written as η − η τ = in out σ sin(Nk η(0)d), a linear function of the deviations from the zeroth-order 2 4η η(0) 0 = − (0) = | || | out approximation δPj Pj Pj (j 1,2,3), if δPj Pj : ηin − ηout τ3 = σ. (B9) t = t0(1 − t0τ), (B1) 4ηout

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