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Home , Metamaterial cloaking

Modeling of in Propagation

G. Leugering, E. Rohan and F. Seifrt

Lehrstuhl für Angewandte Mathematik II, Universität Erlangen-Nürnberg, Germany. New Center, Research Institute at University of West Bohemia, Plzen, Czech Re- public.

Abstract: This chapter focuses on acoustic, electromagnetic, elastic and piezo-electric through heterogenous layers. The motivation is provided by the demand for a better understanding of meta-materials and their possible construction. We stress the analo- gies between the mathematical treatment of phononic, photonic and elastic meta-materials. Moreover, we treat the cloaking problem in more detail from an analytical and simulation oriented point of view. The novelty in the approach presented here is with the interlinked homogenization- and optimization procedure.

INTRODUCTION as ’negative Poisson’ ratio in elastic material foams, negative ’mass’ and ’negative indices’ for The terminology ’metamaterials’ refers to ’beyond the forming of band-gaps in acoustic and optical de- conventional material properties’ and consequently vices, respectively. those ’materials’ typically are not found in nature. Thus given acoustic, elasto-dynamic, piezo-electric It comes as no surprise that research in this area, or electromagnetic wave propagation in a non- once the first examples became publicly known, has homogeneous medium and given a certain merit undergone an exponential growth. Metamaterials function describing the desired material-property or are most often man-made, are engineered materi- dynamic performance of the body involved, one als with a wide range of applications. Starting in wants to find e.g. the location, size, shape and the area of micro- where one aims at cloak- material properties of small inclusions such that ing objects from electromagnetic waves in the in- the merit function is increased towards an opti- visible range, the ideas rather quickly in- mal material or performance. This, at the the first flicted researcher from for a variety of rea- glance, like the formulation of an ancient sons. allowing nanoscale imaging and dream of man-kind. However, proper mathemati- nanophotolithography, couple to the nanoscale cal modelling, thorough mathematical analysis to- yielding a family of negative-index-material(NIM)- gether with a model-based optimization and sim- based devices for nanophotonics, such as nanoscale ulation can, when accompanied by experts in op- antennae, , lasers, switchers, waveguides tics and , lead to such - and finally cloaking are just the most prominent fas- concepts and finally to products. cinating fields. Nano-structured materials are char- acterized by ’ultra-fine microstructure’. There are at Designing optimal microstructures can be seen from least two reasons why downscaling the size of a mi- two aspects. Firstly, inclusions, their size, positions crostructure can drastically influence its properties. and properties are considered on a finite, say, nano- ’First, as grain size gets smaller, the proportion of scale and are subject to shape, topology and material atoms at grain boundaries or on surfaces increases optimization. Secondly, such potential microstruc- rapidly. The other reason is related to the fact that tures are seen from the in form of many physical phenomena (such as dislocation gen- some effective or averaged material. This brings in eration, ferromagnetism, or quantum confinement the notion and the theory of homogenization of mi- effects) are governed by a characteristic length. As crostructures. The interplay between homogeniza- the physical scale of the material falls below this tion and optimization becomes, thus, most promi- length, properties change radically’(see [44]). nent. Metamaterial properties, therefore, emerge under Besides the optimal design approach to metama- the controlled influence of microstructures. Inclu- terial, in particular in the context of negative re- sions on the nano-scale together with their material fraction indices, , permeabilities, there properties and their shape are to be designed in or- is another fascinating branch of research that con- der to fulfill certain desired material properties, such centrates on ’’, a notion pro- 3 moted by Pendry et.al. [27, 45] in optics and - where Br(x0) := x R : x x0 r and such that { 2 | | } leaf et.al. [16] in the more mathematically inclined g satisfies: for a,b with 0 < a < b, g C2([0,b]), 2 literature. We refrain from attempting any recol- g(0)=a, g(b)=b and g (r) > 0, r [0,b] This 0 8 2 lection of major contribution to this field and refer transformation maps the punctuated three-space into to these survey articles ([27, 45, 16]) and the refer- a spherical ring with inner radius a and outer ra- ences therein. In order to be more specific and be- dius b, such that the exterior of the ball Bb(0) is cause in this contribution we will not dwell on this left unchanged. We consider the ball K := Ba(0) approach on any research level, we give a brief ac- as the cloaked object, the layer x : a < x b { | | } count of the underlying idea. as the cloaking layer and the union as the spheri- Cloaking problem and metamaterials: transfor- cal cloak. The shape of the cloak can be arbitrary, however. Examples for spherical cloaks are g(r) := mation method b a a r +a (linear) or g(r) := 1 + p(r b) r + b b In order to keep as simple as possible, we a (quadratic) ⇥ ⇤ consider the following classical problem We consider a similar construction as above, but — s—u = 0, in W, now for many cloaked objects located at point ci,i = · (1) 1,...,N: (u = f , on ∂W. We have the Dirichlet-to-Neumann map (DtN) f (y) := c + g ( y c )(yˆ c ), i i | i| i x = F(y) := for y Bb (ci), i = 1,...,N Ls ( f ) := n s—u ∂W. (2) 8 i · | 2 3 N <>y, for y R Bb (ci)=: W˜ , Calderón’s problem is then to reconstruct s from 2 0 \{[i=1 i } (6) Ls ! For smooth and isotropic s this is possi- :> ( , ( )) ble. Thus, in that case the Cauchy data f Ls f where the cloaked objects are now uniquely determine s. Therefore, no cloaking is possible with smooth variations of the material! In 3 Ki := x R : x ci ai ,i = 1,...N (7) the heterogeneous an-isotropic case, we may con- { 2 | | } N sider a diffeomorphism F : W W with F ∂W = I K = K is the entire cloaked object. The cloaked ! | [i=1 i and then make a change of variables y = F(x) s.t. subregions are supposed to be separated: u = v F 1. The so-called push forward is defined as mindist (Bb (ci),Bb (c j)) > 0, i = j, i, j = 1,...,N i j 8 6 jk 1 jk (8) (F s) (y) := S (x) x=F 1(y) ⇤ detDFjk | 3 n j k (3) The domains of interest are now: W0 := R jk ∂F ∂F pq 3 \ S (x) := (x) (x)s (x). c1,...,cN , W := R K. F( ) is only piecewise  p q { } \ · p,q=1 ∂x ∂x smooth with singularities across ∂K. We notice that g ( y c ) g ( y c j ) g ( y c ) j j 0j | | j j | | dkl + 2 | 3| Ls = LF s , (4) y c j y c j y c j · ⇤ | | | | | | DF(y)kl = 8 ✓ ◆ (y c j) (y c j) ,y B (c j) where DFjk denotes the Jacobi-matrix of F (DF = > · k l 2 bJ —FT ). The idea behind is that the coefficients s (9) mations into curvilinear coordinates are classic in : mechanics, see e.g. Gurtin[17]. Thus, transforma- We have the determinant D(y)=detDF(y) tions into curvilinear coordinates correspond one- 2 g j( y c j ) to-one with transformation between different mate- g0 ( y c j ) | | , j | | y c j rials. The construction of a transformation that al- D(y)=8 | | (10) y Bb j (c⇣j), j = 1,...,⌘ N lows for cloaking is as follows. > 2 x y <1,y W˜ Denotex ˆ := x , yˆ := y and define the mapping F : 2 3 | 3| | | > R 0 R Ba(0) It is obvious:> that s = F s is degenerate along the \{ }! \{ } ⇤ ⇤ x = x(y)= f (y) := g( y )yˆ, boundary ∂K. Thus, in order to properly pose a self- | | adjoint extension of the corresponding Laplace(- x = F(y) := for 0 < y b, (5) 8 | | Beltrami-)operator, we need to work in weighted <>x = x(y) := y, for y > b, | | spaces. :> The idea above is extended to the phononic and evident that even from the point of view of transfor- the photonic situation. In particular treating the mation optics the appearance of singular behaviour Maxwell system in its time-harmonic form the at the boundary of the region to be cloaked indicates transformed system reads as that microstructures may genuinely occur. Indeed, a second approach [16] is based on a truncation of — E = jkµ(x)H, — H = jke(x)H + Je e, µ to such tensors, say e , µ that are uniformly ⇥ ⇥ R R (11) (in x) bounded above and below. When R 1 they ! tend to e, µ, respectively. It is shown in [16] that where e, µ are given by: it is possible to match these tensors eR, µR by peri- 1 1 odic microstructured material in the cloak in the ho- e = DT Fe DF, µ = DFT µ DF (12) mogenization limit. The result shows that utopian D(y) 0 D(y) 0 ’metamaterial’ constructed by an approximation to The material matrices e, µ are again degenerate at exact cloaking can be ’realized’ via homogenization ∂K! of periodic microstructures within the cloaking re- In order to obtain finite solutions to the gion. This is a very encouraging result that needs to Maxwell system, one needs to work in weighted be further exploited. spaces. For cloaking, one requires energy conser- Metamaterials via homogenization vation. Introduce weighted scalar products In this contribution we want to discuss the theme 1 2 1 ¯2 1 2 of object cloaking by ’homogenized metamaterials’. (E ,E )W,E := E eE dx, (H ,H )W,H · We are aiming at designing coating layers contain- WZ (13) ing microstructure which are ’wrapped’ around an = H1 µH¯2dx object. The coated object may be subject to acoustic · WZ or electromagnetic incoming waves. We want to sur- vey and present new results applying the method of and require local energy conservation. To this end homogenization and at the same time thin-domain define the local energy for an open bounded sub- approximation to such nano-structured layers. We domain O W ⇢ investigate the resulting effective transmission con- dition and represent the cloaking problem as an op- E eE¯ dx+ H µH¯ dx < •. · · (14) timization problem or a problem of exact controlla- WZ WZ bility, the controls being shape, topology and ma- terial parameters for the inclusions constituting the A solution satisfies the Maxwell system in the dis- microstructure. tributional sense and has finite local energy. One In the context of mathematical modeling, there are obtains two boundary (over-determined i.g.) condi- many connections and analogies between tions on ∂K and optics. Below we summarize some recent in- vestigations on homogenization of periodically het- E n = 0, H n = 0, on ∂K+, ⇥ ⇥ (15) erogeneous structures exposed to inciding acoustic, (— E) n = 0, (— H) n = 0, on ∂K , ⇥ · ⇥ · or electromagnetic waves. Namely the following is- This procedure of defining cloaking transformations sues are discussed: is rather general and applies also to elliptic systems, Phononic metamaterials which may exhibit 2-d and 3-d elasticity, elasto-dynamics and the time- • negative effective mass for certain frequency dependent Maxwell equations. Thus, formally, from ranges (the so called band-gaps). a purely mathematical point of view, the problem of cloaking can be regarded as analytically solved. The Homogenized ’acoustic sieve’ problem; there • fundamental question however remains: How can the periodic perforation of a rigid layer (the ob- the transformed material tensors be realized ? stacle) influences the acoustic impedance of the Indeed, this problem is widely open. There is an ap- discontinuity interface. proach to approximate the cloaking transforms by less singular mappings in particular by inflating a In analogy to the ’phononic’ metamaterials, • ball rather than a point to a ring-shaped domain. But the ’photonic’ ones may provide frequency- still, the material could not be realized so far and dependent magnetic permeability which may further analysis is in order. On the positive side it is become even negative for some . As a central theme of this contribution is re- which enables one to reduce significantly the com- • lated to the cloaking problem, we discus the plexity of modeling such structures. The complexity optical transmission on thin heterogeneous sur- is due to “detailed ” associated with de- face. The homogenization of such structure scription of piecewise defined material coefficients leads to a model resembling the homogenized (properties), which at the end may lead to an in- acoustic sieve problem. tractable numerical problem featured by millions of unknowns and huge data to be treated. “Averaging” In all of the above cases combinations of ’classical’ of the material properties, based on the asymptotic materials and geometrical arrangement of the het- analysis and the representative volume element (the erogeneities gives rise to ’new’ materials – meta- representative periodic cell) leads to the “homoge- materials – characterized by their effective proper- nized medium” described by the effective material ties which makes their behaviour qualitatively dif- parameters, so that the whole structure can be de- ferent from any of the individual components. Es- scribed with a few data. pecially the geometrical influence of materials’ mi- In this section we demonstrate how the homogeniza- crostructures is challenging and inspires the meta- tion approach (see e.g. [1, 13, 14, 15, 41] for general material optimal design. We consider the cloak- references) can be used to approximate ing problem formulated as the optimization prob- properties in strongly heterogeneous media. In the lem parametrized by the homogenized metamaterial case of phononic and photonic materials, the disper- structure, i.e. by geometry of the heterogeneities sion (and thereby the possible occurrence of band distributed in the cloaking layer. gaps) is retained even in the homogenized medium, The optimization problem will also be considered in due to special scaling of material properties of one the context finite diameter material inclusion, thus of the material components. without homogenization. For the interlacing of op- timization and optimization and optimal control see Kogut and Leugering [20, 21, 22, 23] PHONONIC MATERIALS – ELASTIC AND PIEZOELECTRIC WAVES Topology optimization for the cloaking problem We now consider an elastic medium formed by pe- Instead of transformation techniques and the method riodic structures involving very soft substructures. of optimizing micro-structures before or after ho- Thus, the material properties, being attributed to mogenization one may look directly into material material constituents vary periodically with the lo- optimization of coated objects. Indeed, given a re- cal position. Throughout the text all the quantities gion to be cloaked by a layer with material inclu- varying with this microstructural periodicity are la- sions or ’holes’, one may want to use topology op- beled with superscript e , where e is the characteris- timization and shape optimization in order to find tic scale of the microstructure. Typically e can be such optimal ’micro-structures’. More precisely, the considered as the ratio between the microstructure concept of material interpolation (SIMP) [5] can be size and the incident wave length. used in order to detect material densities of a given class of materials around the object. Moreover, Periodic strongly heterogeneous material the concept of topological derivatives or topologi- The material properties are associated to the peri- cal sensitivities can be used to check as to whether odic geometrical decomposition which is now in- at a given point in the cloaking region an inclusion troduced. We consider an open bounded domain should be considered. Once the location is detected W R3 and the reference (unit) cell Y =]0,1[3 with a subsequent shape sensitivity analysis followed by ⇢ an embedded inclusion Y Y, whereby the matrix shape variation will then assign the optimal shape 2 ⇢ part is Y =Y Y . Let us note, that Y may be defined of that inclusion. Variations of this theme will be 1 \ 2 as a parallelepiped, the particular choice of the unit discussed in this contribution. cube is just for ease of explanation. Using the refer- ence cell we generate the decomposition of W as the HOMOGENIZATION FOR MOD- union of all inclusions (which should not penetrate ELING OF METAMATERIALS IN ∂W), having the size e, ACOUSTIC AND ELECTROMAG- ⇡ NETIC WAVE PROPAGATION e W2 = e(Y2 + k) , Homogenization of periodically heterogeneous k Ke (16) 2[ e structures is a well accepted mathematical tool where K = k Z e(k +Y2) W , { 2 | ⇢ } whereas the perforated matrix is We = W We . Also can be treated for a general case of boundary con- 1 \ 2 we introduce the interface Ge = We We , so that ditions, for simplicity we restrict the model to the 1 \ 2 W = We We Ge . description of clamped structures loaded by volume 1 [ 2 [ Properties of a three dimensional body made of the forces. Assuming a harmonic single-frequency vol- elastic material are described by the elasticity ten- ume forces, e sor cijkl, where i, j,k = 1,2,...,3. As usually we e F(x,t)=f(x)eiwt , (19) assume both major and minor symmetries of cijkl e e e (cijkl = c jikl = ckli j). We assume that inclusions are occupied by a “very where f =(fi),i = 1,2,3 is its local amplitude and soft material” in the sense that the coefficients of the w is the frequency. We consider a dispersive dis- e elasticity tensor in the inclusions are significantly placement field with the local magnitude u smaller than those of the matrix compartment, how- e e iwt ever the material density is comparable in both the U (x,w,t)=u (x,w)e . (20) compartments. Such structures exhibit remarkable band gaps. Here, as an important feature of the mod- This allows us to study the steady periodic re- 2 sponse of the medium, as characterized by displace- eling based on asymptotic analysis, the e scaling e of elasticity coefficients in the inclusions appears. ment field u which satisfies the following boundary This strong heterogeneity in elasticity coefficients value problem: is related to the geometrical scale of the underlying microstructure (possibly another w2re ue divs e = re f in W, involving “soft” and “hard” materials). The follow- (21) e = , ing ansatz is considered: u 0 on ∂W e e r1 in We , where the stress tensor s =(sij) is expressed in re (x)= 1 e e r2 in We , terms of the linearized strain tensor e =(eij) by the 2 e e e ⇢ (17) Hooke’s law s = c ekl(u ). Problem (21) can be c1 in We , ij ijkl ce (x)= ijkl 1 formulated in a weak form as follows: Find ue ijkl e2c2 in We . 1 2 ⇢ ijkl 2 H0(W) such that

Extension for piezoelectric materials. 2 e e e e Properties of a three dimensional body made of the w r u v + cijklekl(u )eij(v)= W · W piezoelectric material are described by three ten- Z Z (22) 1 sors: the elasticity tensor ce , the tensor = f v for all v H0(W) , ijkl W · 2 e Z dij and the piezoelectric coupling tensor gki j, where 1 i, j,k = 1,2,...,3. The following additional symme- where H0(W) is the standard Sobolev space of vec- e e e e tries hold: dij = d ji and gki j = gkji. torial functions with square integrable generalized In analogy with the purely elastic case, the scaling derivatives and with vanishing trace on ∂W, as re- 2 e of material coefficients by e is considered in W2, quired by (21)2. The weak problem formulation except of the density: (22) is convenient for the asymptotic analysis us- ing the two-scale convergence [1], or the unfolding r1 in We , re (x)= 1 method of homogenization [13]. r2 in We , ⇢ 2 Extension for piezoelectric materials. In addi- 1 e tion, a synchronous harmonic excitation by volume e cijkl in W1, cijkl(x)= 2 2 e charges with a single frequency w can be considered e cijkl in W2, ⇢ q˜(x,t)=q(x)eiwt , where q is the magnitude of the 1 e (18) e gki j in W1, distributed volume charge. Accordingly, we should gki j(x)= 2 2 e e gki j in W2, expect a dispersive piezoelectric field with magni- ⇢ e e d1 in We , tudes (u ,j ) de (x)= ij 1 ij e2d2 in We . ⇢ ij 2 u˜ e (x,w,t)=ue (x,w)eiwt , e e iwt Modeling the stationary waves j˜ (x,w,t)=j (x,w)e . We consider stationary wave propagation in the Then the periodic response of the medium is charac- medium introduced above. Although the problem terized by field (ue ,je ) which satisfies the follow- ing boundary value problem: notations:

2 e e e e 2 y y w r u divs = r f in W, aY2 (u, v)= cijklekl(u)eij(v), ZY2 divDe = q in W, (23) d (f, y)= d2 ∂ yf∂yy, ue = 0 on ∂W, Y2 kl l k ZY2 e (26) j = 0 on ∂W, 2 y y gY2 (u, y)= gki jeij(u)∂k y, ZY2 e e where the stress tensor =( ) and the electric 2 s sij r (u, v)= r u v, e Y2 · displacement D are defined by constitutive laws ZY2

e e e e e whereby analogous notations are used when the in- sij = cijklekl(u ) gki j∂kj , tegrations apply over Y1. e e e e e (24) Dk = gki jekl(u )+dkl∂lj . Elastic medium. Frequency–dependent homoge- nized mass involved in the macroscopic momentum The problem (23) can be weakly formulated as fol- equation are expressed in terms of eigenelements e e 1 1 r r 1 lows: Find (u ,j ) H (W) H (W) such that (l ,j ) R H (Y2), r = 1,2,... of the elastic 2 0 ⇥ 0 2 ⇥ 0 spectral problem which is imposed in inclusion Y2 r 2 e e e e with j = 0 on ∂Y2: w r u v + cijklekl(u )eij(v) W · W Z Z 2 y r y r 2 r 1 e e cijklekl(j )eij(v)=l r j v v H0(Y2) , gki jeij(v)∂kj = f v , · 8 2 · ZY2 ZY2 ZW ZW e e e r2jr js = d . g eij(u )∂ky + dkl∂lj ∂ky = qy , rs ki j Y2 · ZW ZW ZW Z (25) (27)

1 1 To simplify the notation we introduce the eigenmo- for all (v,y) H0(W) H0 (W). r r 2 ⇥ mentum m =(mi ), The homogenized model mr = r2jr. (28) Y2 Due to the strong heterogeneity in the elastic Z (and other piezoelectric) coefficients, the homoge- The effective mass of the homogenized medium is nized model exhibits dispersive behaviour; this phe- represented by mass tensor M⇤ =(Mij⇤ ), which is nomenon cannot be observed when standard two- evaluated as scale homogenization procedure is applied to a 1 1 w2 medium without scale-dependent material parame- ( 2)= r r Mij⇤ w rdij  2 r mi m j ; ters, as pointed out e.g. in [3]. In [4] the unfold- Y Y Y r 1 w l | | Z | | ing operator method of homogenization [13] was (29) applied with the strong heterogeneity assumption (17), (18) We shall now record the resulting homog- The elasticity coefficients are computed just using enized equations, as derived in [4], which describe the same formula as for the perforated matrix do- the structure behaviour at the “macroscopic”scale. main, thus being independent of the inclusions ma- They involve the homogenized coefficients which terial: depend on the characteristic responses at the “mi- 1 1 y kl kl ij ij croscopic” scale. Cijkl⇤ = cpqrsers(w + P )epq(w + P ) , Y Y Below it can be seen that the “frequency– | | Z 1 (30) dependent” mass coefficients are determined just by material properties of the inclusion and by the mate- where Pkl =(Pkl)=(y d ) and wkl H1(Y ) are rial density r1 in the matrix, whereas the elasticity i l ik 2 # 1 the corrector functions satisfying (and other piezoelectric) coefficients are related ex- clusively to the matrix material occupying the per- c1 ey (wkl + Pkl)ey (v)=0 v H1(Y ) . forated domain. pqrs rs pq 8 2 # 1 ZY1 For brevity in what follows we employ the following (31) a (jr, v) g (v, pr)=l rr (jr, v) Y2 Y2 Y2 v H1(Y ), 8 2 0 2 g (jr, y)+d (pr, y)=0 y H1(Y ), Y2 Y2 8 2 0 2 (33) with the orthonormality condition imposed on eigenfunctions jr: Fig. (1): Weak band gaps (white) and strong band r s r s r r s ! r gaps () computed for an elastic composite aY2 (j , j )+dY2 (p , p )=l rY2 (j , j ) = l drs. with L-shaped inclusions, the green bands are prop- (34) agation zones. Moreover, if q 0 in (23) , then the following prob- 6⌘ 2 lem must be solved: findp ˜ H1(Y ), the unique so- 2 0 2 lution satisfying

1 dY (p˜, y)= y y H (Y2) . (35) 2 8 2 0 ZY2

The homogenized mass Mij⇤ (w) is evaluated using the same formula (29), as in the elastic case. Further new coefficients Qi⇤(w) are introduced using the so- lution of (35) 1 w2 ( )= r ( r, ), Qi⇤ w  2 r mi gY2 j p˜ (36) Y r 1 w l | | describing influence of the volume charge on the Fig. (2): The first eigenmode of the L-shaped mechanical loading. clamped elastic inclusion. The piezoelectric coefficients of the homogenized medium are defined in terms of the corrector basis functions satisfying the microscopic auxiliary prob- Above 1(Y ) is the restriction of 1(Y ) to the Y- H# 1 H 1 lems: periodic functions (periodicity w.r.t. the homolo- ij ij 1 1 gous points on the opposite edges of ∂Y). 1. Find (c ,p ) H (Y1) H (Y1), i, j = 2 # ⇥ # The global (homogenized) equation of the homog- 1,...,3 such that (the notation corresponds to enized medium, here presented in its differential that introduced in (26)) form, describes the macroscopic displacement field u: a cij + Pij, v g v, pij = 0 , Y1 Y1 g cij + Pij, y + d pij, y = 0 , ⇢ Y1 Y1 2 ∂ 1 1 w M⇤ (w)u j + C⇤ ekl(u)= M⇤ (w) f j , v H#(Y1), y H# (Y1) , ij ∂x ijkl ij 8 2 8 2 j (37) (32) where Pij =(Pij)=(y d ); Heterogeneous structures with finite scale of hetero- k j ik geneities exhibit the frequency band gaps for certain 2. Find (ck,pk) H1(Y ) H1(Y ), i, j = 1,...,3 2 # 1 ⇥ # 1 frequency bands. In the homogenized medium, the such that wave propagation depends on the positivity of mass tensor M⇤(w); this effect is explained below. a ck, v g v, pk + Pk = 0 , Piezoelectric medium. In the piezoelectric medium, Y1 Y1 g ck, y + d pk + Pk, y = 0 , the spectral problem analogous to (27) com- ⇢ Y1 Y1 prises the additional constraint arising from elec- v H1(Y ), y H1(Y ) , 8 2 # 1 8 2 # 1 tric charge conservation (23)2: find eigenelements (38) r r r r 1 r 1 [l ;(j , p )], where j H0(Y2) and p H0 (Y2), 2 2 k r = 1,2,... , such that where P = yk. Using the corrector basis functions just defined the 2. strong band gap – All eigenvalues of Mij⇤ (w) homogenized coefficients are expressed, as follows: are negative: then homogenized model (32), or (40) does not admit any wave propagation; 1 kl kl ij ij Cijkl⇤ = aY1 c + P , c + P + Y 3. weak band gap – Tensor Mij⇤ (w) is indefinite, | | 1 ⇣ ⌘ i.e. there is at least one negative and one pos- + d pkl, pij , Y Y1 itive eigenvalue: then propagation is possible | | ⇣ ⌘ only for waves polarized in a manifold deter- 1 k k i i k i D⇤ = d p + P , p + P + a c , c , mined by eigenvectors associated with posi- ki Y Y1 Y1 | | h ⇣ ⌘ ⇣ ⌘i tive eigenvalues. In this case the notion of 1 ij ij k ij k wave propagation has a local character, since Gki⇤ j = gY c + P , P + dY p , P . Y 1 1 the “desired wave polarization” may depend on | | h ⇣ ⌘ ⇣ ⌘i (39) the local position in W. The global equation describes the macroscopic field In Fig. (1) we introduce a graphical illustration of of displacements u and of j the band gaps analyzed for an elastic material with L-shaped inclusions (its eigenmode fig. (2)). When- ever inclusions (considered in 2D) are symmetric 2 ∂ w Mij⇤ (w)u j + Cijkl⇤ ekl(u) Gki⇤ j∂kj = w.r.t. more than 1 axis of symmetry, only strong ∂x j band gaps exist, see Fig. (3). This may not be the = M⇤ (w) Q⇤(w)q , case for piezoelectric materials; in Fig. (4) we il- ij i ∂ lustrate dispersion and the weak band gaps Gki⇤ jeij(u)+Dkl⇤ ∂lj = q . obtained for a homogenized piezoelectric compos- ∂xk (40) ite with circular inclusions. Usually the band gaps are identified from the dis- Further related work on the sensitivity analysis can persion diagrams. For the homogenized model the be found in [32, 34]. dispersion of guided plane waves is analyzed in the standard way, using the following ansatz: Band gap prediction j(wt x k ) u(x,t)=u¯ e j j , As the main advantage of the homogenized mod- (41) j(wt x k ) els (32) and (40), by analyzing the dependence j(x,t)=j¯ e j j , w M⇤(w) one can determine distribution of the ! where ¯ is the displacement polarization vector (the band gaps; it was proved in [4] that there exist u wave amplitude), ¯ is the electric potential ampli- frequency intervals Gk, k = 1,2,... such that for j k k k+1 tude, k j = n j{, n = 1, i.e. n is the incidence direc- w G ]l ,l [ at least one eigenvalue of ten- | | 2 ⇢ tion, and is the wave number. The dispersion anal- sor M (w) is negative. Those intervals where all { ij⇤ ysis consists in computing nonlinear dependencies eigenvalues of M are negative are called strong, or ij⇤ ¯ = ¯( ) and = ( ). For this one substitutes full band gaps. In the latter case the negative sign u u w { { w (41) into the homogenized model (40); on introduc- of the mass changes the hyperbolic type of the wave ing projections of the homogenized tensors into the equation to the elliptic one, therefore any waves can- direction of the wave propagation, not propagate. In the “weak” bad gap situation only waves with certain polarization can propagate, as Gik = Cijkl⇤ n jnl , gi = Gki⇤ jn jnk , z = Dkl⇤ nlnk , (42) explained below. The band gaps can be classified w.r.t. the polariza- and substituting in (40), we obtain tion of waves which cannot propagate; the polariza- 2 2 2 w Mij⇤ (w )u¯ j + { (Giku¯k gij¯)=0 , (43) tion is determined in terms of the eigenvectors of 2 { (gku¯k + zj¯)=0 . Mij⇤ (w). Given a frequency w, there are three cases to be distinguished according to the signs of eigen- In (43) we can eliminate j¯ (assuming {2 = 0), thus r 6 values g (w), r = 1,2,3 (in 3D), which determines the dispersion analysis reduces to the “standard elas- the “positivity, or negativity” of the mass: tic case” where the acoustic tensor is modified, thus ( ) 2 2 2 1. propagation zone – All eigenvalues of Mij⇤ w w Mij⇤ (w )u¯ j + { Hiku¯k = 0 , are positive: then homogenized model (32), or (44) where H = G + g g /z (40) admits wave propagation without any re- ik ik i k striction of the wave polarization; is analyzed as follows Fig. (3): Dispersion curves for guided waves in Fig. (4): Dispersion curves for piezoelectric mate- composites with circular inclusions: elastic mate- rial. rial, only strong band gaps. Different angles of wave incidence displayed by different colours. crostructure. The detailed analysis was presented in [38]. a b r for all w [w ,w ] and w l r compute We consider the acoustic medium occupying do- • 2 b b 62 { } eigenelements (h ,w ): main WG which is subdivided by perforated plane + G0 in two disjoint subdomains W and W, so that 2 2 b b b G + w Mij⇤ (w )w j = h Hikw , b = 1,2,3; W = W W G , see Fig. (7). Denoting by p k [ [ 0 (45) the acoustic pressure field in W+ W , in a case of [ no convection flow, the acoustic waves in WG are if hb > 0, then {b = hb , • described by the following equations (w is the fre- else w falls in an acousticp gap, wave number is quency of the incident wave), • not defined. 2 2 2 + c — p + w p = 0 in W W , in general [ (46) In heterogeneous media the polarizations + boundary conditions on ∂WG , of the two waves (outside the band gaps) are not mutually orthogonal, which follows easily from the supplemented by the transmission conditions on in- b 2 fact that w are M⇤(w )–orthogonal. More- { }b terface G0 — these present the key issue of this sec- over, in the presence of the piezoelectric coupling, tion. The boundary conditions on G0 will be spec- which introduces another source of , the + ified later on. Let p and p be the traces of p on standard orthogonality is lost even for heteroge- ∂W+ G and on ∂W G , respectively. \ 0 \ 0 neous materials with “symmetric inclusions” (cir- The standard treatment of the acoustic transmission cle,hexagon, etc.), in contrast with elastic structures on a sieve-like perforation G0 results in the relation- where these designs preserve the standard orthogo- + ship between jump p p and normal derivatives + nality. ∂ p ∂ p + = , More details on the band gap properties and their ∂n ∂n relationship to the dispersion of guided waves were p+ discussed in [35, 30, 10]. The sensitivity analysis ∂ wr + + = j (p p), for the optimization problem was discussed in [31, ∂n Z (47) ∂ p wr + 32, 34, 33]. = j (p p ) , ∂n Z ACOUSTIC TRANSMISSION ON + + PERFORATED INTERFACES where n and n are the outward unit normals to W and W, respectively, w is the frequency, r is the In this section we present an example which il- density and Z is the transmission impedance. This lustrates, how homogenization can be employed to quantity incorporates many physical aspects of the describe acoustic transmission between two halfs- transmission, namely the geometry – the design of paces separated by an interface that establishes a mi- the perforation. In [38] a homogenized transmission dilation zoom: yα = x α /ε ε + x Γδ 3 z I+ y ε + + Ω Ω Γ * δ δ/2 δ δ δ Y s δ xα Ω δ yα Γ0 − − ε S s<1 Ω δ Γ δ Sε Γ− − δ δ ε Iy

Fig. (6): Layer Wd embedding the rigid obstacles e dilatation e unfolding W W Y ⇤ periodically distributed. Obstacles should not ap- d ! ! proach the fictitious boundaries Gd±, thus s << 1. Fig. (5): Left: global problem imposed in entire do- main WG before homogenization of the layer W . d equation in We and Neumann condition on ∂W Right: representative cell of the periodic structure. d d The dark patterns represent the obstacles in the fluid. 2 2 ed 2 ed e c — p + w p = 0 in Wd , ed 2 ∂ p ed c = jwg ± on G± , conditions were proposed which describe the acous- ∂nd d (49) tic impedance of the interface characterized by a pe- ∂ ped = 0 on ∂Se ∂W• , riodically perforated obstacle embedded in a layer ∂nd d [ d of thickness d. In Figure (5) we illustrate such a G + + where c = w/k is the speed of propagation layer Wd embedded in W = W W Wd G±. d [ d [ [ d and by nd we denote the normal vector outward to Periodic perforation and acoustic problem in the Wd . transmission layer Homogenized transmission conditions 2 Let G0 R be an open bounded subdomain of the ⇢ The asymptotic analysis of system (49) results in an plane spanned by coordinates xa , a = 1,2 and con- + equation which describes an acoustic wave propa- taining the origin. Further let Gd and Gd be equidis- + gating in the layer as a response to the incident wave tant to G0 with the distance d/2 = dist(G0,Gd )= e acoustic momentum g ±. The following assumption dist(G ,G). We introduce layer W = G ] 0 d d 0⇥ is important. / , / [ 3 d 2 d 2 R , an open domain representing the Let us introduce shifted fluxesg ˆe L2(G ) such ⇢ ± 2 0 transmission layer bounded by ∂Wd which is split thatg ˆe (x¯)=ge (x ) where x G are homolo- ± ± ± ± 2 ± as follows, see Fig. (6) gous points associated tox ¯ G , i.e.x ¯ =(x¯ ,0) and 2 0 a x x¯ =(0,0, 1/2). We assume + • ± ± ∂Wd = Gd Gd ∂Wd , [ [ e 0 2 d gˆ ± * g ± weakly in L (G0) , (50) G± = G e~ , (48) d 0 ± 2 3 • 1 e+ e 2 ∂W = ∂G0 ] d/2,d/2[ , gˆ + gˆ * 0 weakly in L (G ) , (51) d ⇥ e 0 consequently g0 g0+ = g0 . This equality where d > 0 is the layer thickness and e~3 =(0,0,1), ⌘ see Fig. (6). The acoustic medium occupies domain means continuity of the normal momentum, which We = W Se , where Se is the solid rigid obstacle is consistent with the consequence of (47). d d \ d d which in a simple layout has a form of the period- The homogenized coefficients governing the acous- ically perforated sheet with the thickness sd, s < 1 tic transmission are introduced below using so and with e characterizing the scale of the periodic called corrector functions defined in the reference 2 3 Se pe- periodic cell Y =]0,1[ ] 1/2,+1/2[ R . The perforation; thus, d is obtained by the usual ⇥ ⇢ acoustic medium occupies the domain Y = Y S, riodic lattice extension of the solid unit structure. ⇤ \ where S Y is the solid (rigid) obstacle. For clarity For passing to the limit e 0 we consider a pro- ⇢ ! we use notation I =]0,1[2 and I =] 1/2,+1/2[. portional scaling between the period length and the y z thickness, so that d = he, where h > 0 is fixed. The upper and lower boundaries are translations of (I ,0); we define I+ = y ∂Y : z = 1/2 and Acoustic problem in the layer. We assume a y y { 2 } d I = y ∂Y : z = 1/2 . By H1 (Y) we denote monochromatic wave propagation in layer W . The y { 2 } #(1,2) total acoustic pressure, ped satisfies the Helmholtz the space of H1(Y) functions which are “1-periodic” L in coordinates ya , a = 1,2; in this paper such func- tions will be called “transversely Y-periodic”. In [38] the homogenization of problem (49) was considered in detail. As the result, the homoge- Γ Ω+ w Γw R nized transmission conditions were obtained, being Γ0 expressed in terms of the interface mean acoustic Γ − Γ 0 1 in Ω r out pressure p H (G0), and the fictitious acoustic 2 0 2 transverse velocity g L (G ); these quantities sat- Γw 2 0 isfy the following PDE system in weak form: Fig. (7): The domain and boundary decomposition x 0 x 2 0 x 0 A ∂ p ∂ q f ⇤w p q + jw B ∂ qg = 0 , of the global acoustic problem considered. This lay- ab b a a a ZG0 ZG0 ZG0 out is inspired by [8] jw D ∂ x p0y + w2 Fg0y = b b ZG0 ZG0 1 + lows: jw (p p)y , e0 G0 2 Z c b b a a (52) A = a⇤ p + y , p + y , ab Y Y | | 2 ⇣ ⌘ 1 2 Y ⇤ c ( ) ( ) = | | 1 for all q H G0 and y L G0 , where f ⇤ Y h Da = Ba = a⇤ (x, ya ) , (55) 2 2 | | Y Y is the porosity related to the layer thickness. We | | remark that while (52) is the direct consequence of 1 1 F = g±(x) . (49) for e 0, additional constraint (52) arises due Iy ! 2 | | to coupling the “outer acoustic problem” imposed in WG W with the one imposed in the layer. A Structure of the global problem \ d quite analogous treatment is employed in the elec- The coupled system (52) described above constitute tromagnetic transmission problem described in Sec- the transmission condition in a global problem con- tion . Equations (52) involve the homogenized co- sidered. As an example, we shall present a model of efficients Aab,Ba ,Da and F expressed in terms of b an acoustic duct with perforated (rigid) plate. the local corrector functions p and x. Let us consider the domain of WG, as in (46), where The homogenized coefficients, A,B,F are deter- G the outer boundary ∂W = Gin Gout Gw consists mined by the solution of the local corrector prob- [ [ of the planar surfaces Gin, Gout and the channel walls lems. To simplify the notation, we introduce Gw, see Fig. (7). On Gin we assume an incident wave jknl xl jwt ˆ y 1 of the formp ˜(x,t)=pe¯ · e , where (nl) is the —q =(∂a q,h ∂zq), outward normal vector of W, on Gout we impose the a⇤ (p, x)= —ˆ p —ˆ x radiation condition of the Sommerfeld type, so that Y · ZY ⇤ y y 1 (53) ∂ p = + , jw p + c = 2jw p¯ on Gin , ∂a p∂a x 2 ∂zp∂zx ∂n Y ⇤ h Z ✓ ◆ ∂ p + = , (56) g±(x)= x x . jw p c 0 on Gout + ∂n Iy Iy Z Z ∂ p = 0 on G . The two following local corrector problems are de- ∂n w fined: Find pb ,x H1 (Y)/R such that 2 #(1,2) The interface condition has the following form, see illustration in Fig. (8), b 1 a⇤ p + y , f = 0 , f H (Y), b = 1,2 , Y b 8 2 #(1,2) c2 ∂ p = jwg ⇣ ⌘ Y ∂n+ 0 on G , (57) ( , )= ( ) , 1 ( ) , 2 ∂ p 0 aY⇤ x f | 2| g± f f H#(1,2) Y ( c = jwg0 hc 8 2 ∂n (54) ∂ p where ∂n = n± —p are the normal derivatives on ± · + see Fig. (9) where function x is displayed for three G0 w.r.t. normals outward to W and W, respec- different microstructures. The homogenized coeffi- tively. Thus, transmission conditions on the inter- a cients are expressed in terms of p and x, as fol- face G0 involve the transversal acoustic momentum Mic. A[(m/s)2] B[m] F[s2] 5 5 + #1 1.155 10 0 1.391 10 · 5 · 5 Ω #2 1.704 10 0.251 1.324 10 + 0 · 5 · 5 + 0+/− p #3 2.186 10 0.897 4.265 10 Γ0 p g · · Γ0 − Γ p− Table 1: Comparison of homogenized transmission 0 parameters for different microstructures. − Ω global response can be characterized by the trans- mission loss TL = 20log p¯ G / p G , wherep ¯ | | in | | | out | is the incident plane wave, see (56). The transmis- Fig. (8): Illustration of the transmission condition sion losses for the waveguide with perforations #1, obtained by the homogenization of the perforated #2 and #3 are shown in Fig. (10). On the horizontal interface. Normal derivatives of the acoustic pres- axis there is the wave number k (k = w/c) multi- sure are continuous, being proportional to g0. plied by length L of the “expansion chamber” (see

Mic. #1 Mic. #2 Mic. #3 Fig. (7)). The resulting acoustic pressures in the waveguide are displayed in Fig. (11). The numerical results were obtained for acoustic speed c = 343m/s and scale parameter e0 = 0.035, which e.g. for the microstructure type #1 means that the thickness of the perforated plate is 7mm. According to this study the perforation design seems to have quite important Fig. (9): Distribution of x in Y ⇤. influence on the global behaviour of the acoustic pressure field, as viewed by the transmission losses. g0 satisfying This is a motivation for an optimal perforation prob- lem, see [29, 24]. 0 2 0 0 ∂a (Aab∂b p )+w f ⇤ p ∂a (Ba g )=0 on G0 , 40 2 0 1 + Mic #1 jhwBb + w Fg = jw (p p) on G0 , Mic #2 e0 35 Mic #3 0 Aab∂b p = 0 on ∂G0 , 30 (58) 25 where ∂G0 is the edge of the obstacle G0 and f ⇤ = 20 Y ⇤ / Y is the layer porosity (depending on param- | | | | 15 eter h). This is the differential form of integral iden- Transmission loss tities (52) that were developed in [38] using asymp- 10 totic analysis. 5 Numerical illustration 0 0 2 4 6 8 10 12 In Table 1 we introduce homogenized transmission k⋅ L parameters A,B,F for 2D microstructures #1,#2 and #3 displayed in Fig. (9); whenever the microstruc- Fig. (10): Transmission losses for different perfora- ture is symmetric w.r.t. the vertical axis of Y, coef- tion types. ficient B vanishes and, as the consequence, the sur- face wave is decoupled from the transversal momen- tum. ELECTROMAGNETIC WAVES IN We shall now illustrate that the global macroscopic PHOTONIC CRYSTALS response is very sensitive to the specific geometry of the perforation. The following numerical example In analogy with the photonic crystals (materials) shows the global response of a waveguide contain- treated in Section , homogenization was employed ing the homogenized transmission layer. The geom- to describe dispersion of optical waves in strongly etry of the waveguide is depicted in Figs. (7). The heterogeneous periodic materials, cf. Helmholtz equations hold

2 2 1 — E + k E = e —r jwµJ , — E = r/e , e · —2H + k2H = — J , — H = 0 , ⇥ e · Mic. #1; k L = 5 (60) · where k is the wave number characterized by the material:

k2 = w2µb = w2µ(e + js/w) . (61) Mic. #2; k L = 5 · The vectorial Helmholtz equations (60) present three independent scalar “componentwise” equa- tions, however they are coupled by the divergence conditions, which makes the analysis more difficult. To simplify construction of the solutions to (60), the Mic. #3; k L = 5 · vector potentials are introduced. Two standard cases can be treated:

1. Electric Hertz potential. Let us consider the special case Je = 0, thereby r = 0. Then by (60) it follows that — E = 0. The electric Mic. #3; k L = 1 1 · · Hertz potential E = — AE then satisfies (60) , ⇥ 1 Fig. (11): Modulus of the acoustic pressure in W for which yields k L = 5 (1 in the last picture). For this 2D compu- · —2 E + k2 E = —f , (62) tation a finite element mesh comprising 820 quadri- A A lateral elements was used. where —f is any scalar differentiable function. 2. Magnetic Hertz potential of the magnetic Helmholtz equation for harmonic waves field. Let H = — AH , where AH is the Hertz ⇥ potential. Then (59)2 yields Here we recall the possible description of electro- 2 H 2 H magnetic fields in heterogeneous materials using the — A + k A = Je + —y , (63) Hertz potential (cf. [2]). Maxwell equation for harmonic waves. We as- where y is any scalar differentiable function. sume monochromatic wave of frequency w and am- Transmission conditions. Let G be the interface plitudes and standing for magnetic and elec- H E separating two subdomains W and W where in tric Fields, respectively, which satisfy the Maxwell 1 2 each the material parameters are constant. From the equations: integral form of the Maxwell equations the follow- ing transmission conditions can be derived, see e.g. — H =( jwe + s)E + J , ⇥ e [2], — E = jwµH , ⇥ (59) — (eE)=r , [n E]G = 0 , [n H]G = 0 , (64) · ⇥ ⇥ — (µH)=0 , where [ ] is the jump of on G and n is normal · • G • vector to G. where is the current associated with external Je Two-dimensional model for a heterogeneous sources of , r is the volume elec- E medium. Let us consider A = v~e3, so that~e3 is the tric , e is the electric (a normal of the plane transversal to the fibres aligned real number), µ is the magnetic permeability (a real with ~e3 and characterizing the heterogeneities, and number) and s is conductivity which is zero in vac- v = v(x1,x2) is the scalar potential of the transversal uum (a real number). electric H-mode (TE-H-mode). Now (62) reduces Let us assume for a while, that the material is homo- to the scalar Helmholtz equation geneous, i.e. (e, µ,s) are constants. Then either E, 2 2 or H can be eliminated from system (59), so that the — v + k v = ∂3f . (65) In what follows we may put —f = 0, thus ∂3f = 0, to Wk can be introduced. We shall see that there ex- (cf. [2]). Further we consider two materials occupy- ists a continuous field u such that ing two disjoint domains W1 and W2, separated by µk 1 1 interface G, so that W = W1 G W2. For this spe- v = u = u = u in Wk (70) [ [ 2 e w2 + js w w2b cial case we rewrite (64) , noting that n ~e = 0 and kk k k k 1 · 3 also~e3 (—n)=0: · where bk = ek + jsk/w and v satisfies (68). Sub- E stitution (70) is well defined provided w > 0 and [n E]G =[n — A ]G ⇥ ⇥ ⇥ ek = 0. Now we are allowed to apply this substitu- E E E 6 =[—(n A ) (—n) A ∂nA ]G (66) tion in (68) to obtain the following modified system · · = ~e3[∂nv]G , 1 — —u + w2µ u = ∂ g in W , k = 1,2 , · b k 3 k where ∂n is the normal derivative. Then we employ ✓ k ◆ (59)2 in (64)2: some b.c. on ∂W , 1 1 1 transmission cond.: [ ∂nu]G = 0 on G , [n H]G = [ n — E]G b ⇥ jw µ ⇥ ⇥ [u]G = 0 on G , 1 1 2 E = [ n — A ]G (67) jw µ ⇥ (71) 2 1 k where in (71) b = b on G ∂W . Obviously, con- =~e3 [ v]G = 0 , 3 k \ k jw µ tinuity on G follows by (71)3 and (71)4 preserves continuity of the co-gradients. where (64) was employed. Thus, for the time- harmonic response featured by the frequency w and Remark 1. Notation: Alternatively we can rewrite the TE-mode, the Maxwell equations yields the fol- (71) using the relative permittivity and permeability. lowing system Let e0, µ0 be the permittivity and permeability of the r r r , then µk = µk µ0, ek = ek e0 and bk = bk e0, 2 2 r r — v + k v = 0 in Wk, k = 1,2 , where bk (w)=ek + jsk/(we0). On introducing the some b.c. on ∂W , wave number k0 = wpe0µ0, (71)1 can be rewritten (assuming g = 0) transmission cond.: [∂nv]G = 0 on G , 2 k 1 2 r [ v] = 0 on G , — —u + k µ u = 0 in Wk, k = 1,2 . G · b r 0 k µ ✓ k ◆ (68) (72) where denotes the co-normal derivative, i.e. = For magnetically inactive materials µr 1, there- ∂n ∂n k ⇡ n —. The complex wave number k is defined lo- fore alternatively · cally by the material parameters; we consider them 1 piecewise constant in W, in particular — —u + w2µ u = 0 in W , k = 1,2 , · (nr )2 0 k ✓ k ◆ (µ ,e ,s ) x W (73) (µ,e,s)(x)= 1 1 1 2 1 , (69) (µ2,e2,s2) x W2 ⇢ 2 r r where nk = bk /e0 is the refraction index. where (µ ,e ,s ), k = 1,2 are constants. k k k p Meanwhile the boundary conditions on ∂W are not 4 specified; importantly, when a part of ∂W is attached Remark 2. Alternatively one can consider the so to a perfect conductor, then ∂ v = 0 on this part. called transversal magnetic E-mode (TM-E-mode), n H It is worth noting that solutions to (68) have contin- on introducing A = w~e3, in analogy with the TE- uous co-normal derivative on G, but the traces of v H-mode. This applies in particular for Je = je~e3, on G are discontinuous. In the next section we mod- thus —2w + k2w = j + ∂ y . ify the formulation represented by (68) to get rid of e 3 these discontinuities. The transmission conditions on G are By virtue of the piecewise constant material proper- ties (69) piecewise-defined rescaling of v restricted [∂nw]G = 0, [µw]G = 0 , so that for µ constant in whole domain the solution the infinity, see [9]. Note that at any interface sep- w is smooth and continuous on G; typically this is arating the inhomogeneities the standard interface satisfied by a class of optical materials where µ = µ0 condition of the type (71)3 applies. . In [9] it was proved mathematically that the artificial can be obtained by homogenization (i.e. 4 by asymptotic analysis) of the following problem Photonic crystals 1 Photonic crystals and magnetically active materi- — —ue + w2µe ue = 0 in R2, · b e als became a quite interesting field of material sci- ✓ ◆ ence due to vast applications in optical technolo- 1 (75) ∂ usce jwµ0usce = O(1/pk0r) gies (waveguides, optical fibres, special ...). b 0 r There is a rich literature facing this subject, see e.g. when r +• , [9][28][45]. ! In this section we aim to demonstrate the modelling where uinc is the incident wave and usce = ue uinc analogy between acoustic waves in phononic mate- is the scattered field. We shall here recall the model rials and the electromagnetic waves in the photonic of homogenized material (metamaterial which will ones. Therefore, we shall focus on the homogeni- allow us to see the analogies between the homog- sation approach which consists in replacing a com- enization of the phononic crystals (acoustic waves) posite with a large number of periodic microstruc- and the photonic ones (electromagnetic waves). tures by a limit homogeneous material. Such a treat- Homogenized coefficients. In analogy with the con- ment is relevant for the modelling of the periodic struction of mass tensor Mij⇤ in (29) using eigensolu- structures presented by photonic crystals. As Bou- tions of (27), the effective permeability is expressed chitté and Felbacq proposed [9] in the case of pe- in terms of eigensolutions of the problem: find cou- k k 1 riodic photonic crystals made of “strongly hetero- ples (l ,w ) R H (Y2), k = 1,2,... 2 ⇥ 0 geneous composites” ( i.e., with permittivity co- k k k 1 efficients strongly different in the inclusions and —w —f = l w f , f H0 (Y2), Y · Y 8 2 in the matrix), the limit homogenized permeability Z 2 Z 2 (76) k l is negative for certain , thus yielding w w = dkl . the existence of band gaps. More precisely, they ZY2 showed that when the ratio between permeability of Now the effective permeability is computed as fol- the inclusions and permeability of the background lows: is of the order of the square of the size of the mi- 1 2 µ Y1 + µ Y2 crostructures, then the band-gaps phenomenon ap- µ⇤(w)= | | | |+ Y pears. Historically this observation motivated the | | 2 2 homogenization approach applied to elastic waves, 2 1 w k + µ k 2 2 2 w , as reported above. Y Â l /(b µ ) w Y k I+ Z 2 Periodic structure with large contrasts in permit- | | 2 ✓ ◆ tivity. Let us consider a periodic structure, as gen- where I = k wk > 0 . e + { | } erated in (16), characterized by permeability µ (x) ZY2 e (77) and complex permittivity b (x) given as piecewise constant functions The effective permittivity becomes a 2 2 symmetric 1 e ⇥ µ in W1, tensor: µe (x)= µ2 in We , 2 1 i j 8 0 2 A⇤ = — (h + y ) — (h + y ) , (78) < µ in R W, ij 1 ⇠ y i · y j \ (74) b ZY1 b 1 in We , : 1 i = H1(Y ) b e (x)= e2b 2 in We , where h # 1 , being Y-periodic, satisfies the 8 2 following identities: b 0 in R2 W < \ i 1 and assume that for < no inclusion intersects —y(h + yi) —yy = 0 y H# (Y1) , i = 1,2 ,. :e e0 ⇠ · 8 2 ZY1 ∂W. Further we may assume that the heterogeneous (79) medium occupying domain W is subject to an inci- dent wave imposed in R2 W with the Sommerfeld Homogenized photonic materials. The limit analy- \ radiation condition applied on the scattered field in sis of the heterogeneous medium leads to the model + of homogenized medium which is characterized by Γδ effective (homogenized) material parameters. One can show that ue (x) in (75) two-scale converges (cf. the unfolding method of homogenization [13]) to Γ0 δ u(x)+c2(y)uˆ(x,y), where c2 is the characteristic function of Y2 andu ˆ(x,y) are the non-vanishing os- cillations in the inclusions. u is the “macroscopic” solution satisfying − Γδ ε 2 — A⇤ — u + w µ⇤(w)u = 0 , in W , x · · x 1 Fig. (12): Illustration of a section through the ficti- —2u + w2µ0u = 0 , in R2 W , b 0 \ tious layer in which the heterogeneous structure is 1 embedded. The black parts represent perfect con- n A⇤ —xu n —u+ = 0 on ∂W , · · · b 0 ductors, in the “void” part the material coefficients are the same as those outside the layer; The colour u+ u = 0 on ∂W , (grey) regions are occupied by different materials. usc u uinc satisfies (75) , ⌘ (80) where n is a normal vector on ∂W and u ,u+ are the interior and exterior values on ∂W, respectively. Remark 3. Here we consider the TE-H-mode, i.e. Thus the solution is continuous on ∂W. the two-dimensional restriction of the electromag- Photonic band gaps. The homogenized medium netic wave propagation (65), which is characterized by scalar function v = v(x1,x3), thus ∂2v 0. Such represented by µ⇤(w) and Aij⇤ is the magnetic ac- ⌘ tive metamaterial with possibly negative permeabil- a situation is relevant whenever the heterogeneous structure is generated in 3D independently of co- ity µ⇤(w) < 0 for some w. This effect features oc- currence of band gaps, in analogy with the phononic ordinate x2 (e.g. by fibrous graining aligned with material described above in the text, where the x2-axis). For generality we shall keep 3D descrip- acoustic band gaps are indicated by negative effec- tion w.r.t. coordinates (x1,x2,x3)=(xa ,x3), where a = 1,2 refers to the in-plane position in G0 only. tive mass M⇤(w). However, due to the TE-H-mode restriction, only ELECTROMAGNETIC WAVE gradients w.r.t. x1 and x3 coordinates do not vanish, TRANSMISSION ON HETEROGE- therefore in the sequel one may consider a = 1. NEOUS LAYERS AND CLOAKING In the “ad hoc 2D” treatment, G0 is just a line, In analogy with the acoustic transmission problem whereas Wd is a two-dimensional domain spanned reported in Section , we discus the electromag- by coordinates x1,x3. netic wave transmission through periodically heterogeneous layer. We consider a strip W R3 with the thickness d ⇢ d > 0 generated by a planar surface G0 and bounded + 4 by Gd and Gd, see Fig. (12); the same notation is From similar studies of elliptic problems in thin lay- used as that introduced in Section . In general, the ers having a periodic microstructure it is well known strip may contain perfect conducting material; we that different limit models are obtained when com- denote by Se We union of all such conductor (e.g. muting e 0 (the period of heterogeneities) and d ⇢ d ! realized by fibrous graining) which also constitute d 0 (the thickness). Here we consider fixed pro- ! the periodic pattern in the strip; length of the period portion d = he, h > 0. in xa , a = 1 is e, see Remark 3; the pattern is de- fined by the 2D section spanning coordinates x1,x3, so that interfaces of the graining between different Non-homogenized layer – problem formulation materials have the form of general infinite cylinders. he dielectric material with finite conductivity occu- pies domain We = W Se . The problem of the TE- d d \ d mode radiation will be imposed in the perforated do- We can define the boundary value problem for the e main Wd . rescaled potential, see (71), and consider the Neu- + mann conditions on Gd±: z t Γ L + Ξ 1 ed + 2 ed = e , — 2 —u µw u 0 in Wd t t · b ! ed x 1 1 − ed d Ξ ∂n±u = jwg± on Gd± , b0 t.E t − d 0 1 Γ L where g± = g (x )+eg ±(x ,x/e) , ± a a d so that g± d , Fig. (13): Illustration of the integral form of the in- ⇠I+ I ⇡ Z y [ y duction law. ed e ∂nu = 0 on ∂Sd , ed ed u ,∂nu periodic on opposite • Above the equivalence between the l.h.s. expres- sides of ∂Wd , sions follows from the general transmission condi- (81) tion (66) which in 2D situation of the TE-H-mode [ ] = = 1 yields t E G 0. Let k l and consider the in- where g ± is the fluctuation part. The perfect con- · 6 tegral over Gkl = ∂Sk ∂Sl which appears in the ductor in Se results in the zero Neumann condition \ d l.h.s. of (82)1: due to the opposite orientation, on the associated perforation boundary. It is worth tk = tl on G , the following holds: e kl recalling that bed is piecewise constant in Wd and e-periodic in x1 (for fibrous structure relevant to the tk EkdG + tl EldG = TE-mode analysis bed(x1,x3) is independent of x2). ∂S G · ∂S G · Z k\ kl Z l \ kl (83) In any case we assume that material on Gd± is ho- = [t E]G dG = 0 , mogeneous, thus bed = b0 is a constant (whatever · kl ZGkl possibly a complex number). Due to (71)3,4 the so- lution ued is smooth and the transmission conditions which yields the equivalence between the l.h.s. in are satisfied automatically. (82)1 and (82)2. In the 2D situation, due to the TE-mode assumption, Induction law constraint (82) yields the following constraint

For stating the boundary conditions on Gd±, as ex- plained below, the induction low is needed to define ( t ∂ v +t ∂ v)dG = k2v , (84) 1 3 3 1 a suitable scaling of the Neumann fluxes. Z∂S ZS Let S R2 be a planar surface spanned by co- 2 where (t ,0,t ) is the tangent of ∂S and v~e is the ordinates x ,x , bounded by ∂S , and let us con- 1 3 2 1 3 electric Hertz potential for the TE-mode. Note that sider decomposition S = k Sk using a finite num- (84) holds also on “perforated” domains S ⇤ S ber of mutually non-overlapping subdomains Sk, ⇢ S when the perforation represents perfect conductors; k = 1,2,...; in each S the medium is assumed k this is the simple consequence of the homogeneous to be homogeneous. For zero external current, i.e. E Neumann conditions on the part of ∂S ⇤ attached to Je = 0, and using the electric Hertz potential A the E the conductors (the “holes”). Maxwell equations (59)1,2 yield H =(s jwe)A E We now consider Wd S = WdL =(x+] and — E = jwµ(s jwe)A in each Sk. Fur- ⇥ L/2,L/2[) ] d/2,d/2[ where x G is such that ther let tk be the tangent unit vector associated with ⇥ 2 0 k (x+] L/2,L/2[) G0. Boundary of WdL is as fol- closed oriented curve ∂Sk and let E be the trace ⇢ lows, see Fig. (13): on Sk of E defined in Sk. On integrating in Sk and then using the summation over all subdomains, one + + ∂W = G G X X , obtains subsequently (µk,ek,s k are local material dL dL [ dL [ d [ d G± G± , (85) constants valid in Sk): dL ⇢ d X± =(x L/2) ] d/2,d/2[ . d ± ⇥ tk EkdG = µk(jwsk + w2ek) AE , ∂Sk · Sk [k Z [k Z Using substitution (70) in (84) we obtain 2 E t EdG = µ(jws + w e)A . 1 1 ∂S · S 2 Z Z ∂n±u + ∂n±u = µw u , (86) (82) b G X b We 0 Z d±L Z d± ed Z d⇤L where We = W We and where sign matches that in the homogenized layer represented by G, it d⇤L dL \ d⇤ ± the integration over G± or X±. It is important to is necessary to derive the relationship between the dL d + e limit traces u and u of the bulk field in W± on note, that Wd⇤L and Xd± are proportional to d; this d | | | | G± for d 0 on one hand and the corresponding observation was respected in the definition of gd± in ! ed (81). limit traces on Gd± on the other hand. Let u be the smooth extension over all perforations due to the The homogenized transmission condition f perfect conductors. The traces from Wd± satisfy The homogenized transmission condition is defined ed d d in terms of the homogenized coefficients which in- f∂3u = f u G+ dG f u G dG Wd G+ | d G | d volve the corrector functions in the integral form. Z Z d Z d In what follows we explain, how the transmission f d,e 0 + ! f(u u)dG , ! condition can be evaluated, for its detailed deriva- ZG0 tion we refer to [37]. Here we shall just summa- (89) rize the main steps of the homogenization procedure for any f L2(W ) constrained by ∂ f = 0. We which is quite analogous to the result obtained for 2 0 3 the acoustic problem reported above. shall now consider a finite thickness d0 > 0 of the An important ingredient of the analysis is the di- layer. The l.h.s. in (89) can also be written as e0d0 lation procedure, the affine mapping transforming d0 W f∂3u (we recall the use of smooth exten- domain Wd on W = G0 ] 1/2,1/2[ which, thus, sion ue0d0 to entire W ) . We consider the following ⇥ R d0 is independent of = h . The material structure g d e approximation for e < e0 : in the layer is periodic being generated by repre- g sentative cell Y in analogy with the acoustic prob- 1 ∂ue d f∂ ue0d0 e f lem discussed in Section where the role of the fluid 0 3 ⇡ 0 z ZW ZW e ∂ is now played by the dielectric material situated in e 1 g e 0 ∂u Y ⇤, whereas the obstacles now represent the super- = ! e0 f (90) ! G0 ⇠Y ∂z ! conducting material. Z Z e Based on the a priori estimate of the solution to 1 1 1 = e0 f u dGy u dGy , (81), one obtains the convergence result (in the I + ZG0 y ZIy ZIy sense of the two-scale convergence). There exist | |  0 2 1 2 2 u L (G ) and u L (G ) H (Y) such that for all f L (G0), see (87)3, hence using (89) 2 0 2 0 ⇥ #(1,2) 2 (denoting ue the solution of (81) on the dilated do- + 1 1 1 main W) the following two-scale limits hold: f(u u)dG = e0 f u dGy u dGy . G G I I+ I Z 0 Z 0 | y| Z y Z y ue 2 u0 (91) ! e 2 x 0 y 1 Continuity or a jump of potential normal derivative ∂a u ∂a u + ∂a u , a = 1,2 (87) ! on G0? In the limit situation, e 0, one can prove 1 2 ! ∂ ue ∂ u1 using the induction law constrain (86) that e z ! z ∂u+ ∂u Below we introduce the corrector basis functions (92) + + =[∂nu]G0 = 0 , which enable to express the “microscopic” func- ∂n ∂n 1 u ∂u± tion in terms of the “macroscopic” quantities where are traces from W± of the normal deriva- u0 and g0; these are involved in the homogenized ∂n± ∂a tives on interface G0. Helmholtz equation arising from (81)1. However, an alternative treatment is possible. We Coupling the interface layer response with outer may adapt the spirit of handling the potential jump fields. In the limit situation the domain Wd degener- [u]G0 . For this we divide (86) by d and approximate ates into the “mid-surface” (plane) . Let the layer G0 for a small d0 > 0, which yields Wd is embedded in W0 where the scattered field can be observed, 1 1 ∂ 1 2 ± u + ± ∂1u µw u . e d0b0 G± d ∂z X be ⇡ W + Z L Z ± Z L⇤ W0 = W W W , W± W = /0 , (88) d [ d [ d d \ d (93) + e where also Wd and Wd are disjoint. In order to be Above domains G±,X±,WL⇤ are obtained by the e able to couple the exterior problem in W W with thickness dilatation (cf. [15],[38])of G±,X±,W . 0 \ d d d d⇤L + Iy As the consequence of Remark 3, in fact a,b = 1 0 and A,B are only scalar values. Also ∂2u = 0 due to the TE-H-mode restriction. − * Jump condition. Using decomposition (96), from Iz Y (90) for a.a. x G0 we obtain + 2 Iz + 1 1 u u = e g±(u ) − 0 I Iy | y| 1 a x 0 0 = e0 g±(p )∂a u + jwg±(x)g Fig. (14): Reference cell Y. Iy | | = d B ∂ x u0 + jwFg0 , 0 a a (100) Since for e 0 the second l.h.s. term vanishes, he ! limit of (93) results in where (note Y = I I and I = 1) | | | z|| y| | z| 1 ∂u+ ∂u 1 2 0 1 1 + + = [∂nu]G0 = r⇤µw u d0b0 ∂n ∂n d0b0 F = aY⇤ (x, x)= . (101)  Y h Y g±(x) for a.a. x G , | | | | 2 0 (94) Using auxiliary problems (97) one can verify that 1 where r⇤ = Y ⇤ / Y . ( , )= ( , a )= ( a ) , | | | | aY⇤ x ya aY⇤ x p g± p Corrector basis functions. We employ notation in- h troduced in Section , however now the bilinear form 1 1 a hence Ba = aY⇤ (x, ya )= g±(p ) Y h Iy Yz aY⇤ is modified: | | | || | 1 a 1 = g±(p ) , a⇤ (u, v)= —ˆ u —ˆ vdy, (95) h Iy Y ˜ · | | ZY ⇤ b (102) ˜ where b(y) is defined piecewise constant in Y ⇤. Due which was employed in (100). a 1 to linearity, we may define p ,x H# (Y) such that Complete homogenized interface conditions. They 2 0 1 a x 0 0 involve the in-plane limit electric Hertz potential u u = p ∂a u + jwxg , (96) (see the transformation (70)), the transformed tan- gential electric field components, g+ = g0 + e g1+ and they satisfy the following auxiliary problems: 0 and g = g0 + e g1 related to faces G+ and G , 0 b 1 respectively, where the fluctuating part is relevant aY⇤ p + yb , f = 0 , f H# (Y) , 8 2 for a given layer thickness d0 = e0h > 0. There ⇣ ⌘ 1 1 is now discussion concerning the fluctuation parts aY⇤ (x, f)= g±(f) , f H# (Y) . 1s h 8 2 e0g , s =+, . (97) 1. Let us consider the perfect continuity of normal Macroscopic wave equation on G0. The macro- derivatives according to (92). This is satisfied scopic equation governs the surface wave propaga- 2 (in the sense of weak limits in L (G0)) for the tion. The limit of the Helmholtz equation reads as following two situations: 0 0 ∂a Aab∂b u + jw∂a (Ba g )+ a) for “the true limit case”, e0 = 0, so that 1 d 0 2 0 jw 1s (92) holds for any g ± (sinceg ˆ ± * g +µw r⇤u = g 2 1 ± Â s weakly in L (G0)). In this case g ± is to h s=+, ⇠Iy Z be defined in (98). (98) b) for the zero average in (98), i.e. assuming in G, where the homogenized coefficients are 1s G± g = 0 . (103) 1 b a ⌘ Â ⇠Is A = a⇤ p + y , p + y , s=+, Z y ab Y Y b a | | ⇣ ⌘ (99) 1 1 In this case functions g ± are not present B = a⇤ (x, y ) . a Y Y a in the limit model. | | 2. Let us now consider (94). Since the bound- Cloaking problem aries G are not related to any structural (mate- d± The cloaking problem consists in finding model pa- rial) discontinuity, the normal derivatives must G rameters related to some subdomain W W such be continuous. Thus, for e0 > 0, the exter- ⇢ + that an object Wc W is not visible outside W, nal field gradients represented by Te (g )= ⇢ + G 0 1+ i.e. the incident wave imposed in W = W W is g (xa )+e0g (xa ,y) are related to ∂n±u± by \G not perturbed by a refracted field on Gs ∂W . The G ⇢ + + 2 0 1+ medium parameters in W are defined as piecewise ∂ + u = jwb (g + e0 g ) , n |G 0 ⇠+ constant functions (pcw. const. func.): ZIy (104) domain: parameters description: 2 0 1 ∂ u = jwb ( g + e g ) , n |G 0 0 ⇠ of the medium: ZIy + + + + These “external field boundary conditions” can Wd ,W b0 , µ0 const. be substituted in (94), therefore W W b , µ const. d \ c 0 0 , + W Wc b0 µ0 const. 1 ∂u ∂u \ Wc b, µ pcw. const. func. 2 + + = d0b ∂n ∂n e ed ed 0  Wd b , µ pcw. const. func. jw 0 1+ 0 1 = g + e0 g g + e0 g We shall discus the following alternative definition d ⇠+ ⇠ 0 ✓ ZIy ZIy ◆ of the cloaking problem with heterogeneous trans- ! 2 0 mission layer: = r⇤µw u , (105) 1. the d-formulation – the layer is not homoge- , WG = W W W+ (disjoined subdo- hence the constraint nized d d d [ [ G mains) and the observation manifold Gs ∂W 1+ 1 0 ⇢ G± g + g = jwr⇤µhu a.e. on G0 . is located far away from Wd. ⌘⇠+ ⇠ ZIy ZIy (106) 2. the homogenized formulation with the far- field cloaking effect, i.e. the layer is repre- We shall consider either (103) holds, so that the sented by homogenized material distributed on + fluctuating parts are irrelevant in the limit situation, G0 = ∂W ∂W and the manifold Gs is de- \ or (106) holds, which is an additional constraint. fined as above. Therefore, the following problem is meaningful: + + 3. the homogenized formulation with the Let u and u are given on faces G and G of thin , in this case the cloak- heterogeneous interface (with the thickness d << strong cloaking effect 0 ing effect is examined on the “exterior surface” 1) which is represented by surface (line in 2D – the of G+, thus no scattered field component is ob- relevant case) G0 in the homogenized form. De- served in W+. noting by U#(G0) the space of periodic functions on G0, which is the consequence of periodic condi- In general there is the scattered field in W+ given 0 0 tions (81)5, we find u U#(G0) and fluxes g ,G± as usc = u uinc, i.e as the subtraction of the total 2 2 2 L (G0) such that: and the incident field. A physically reasonable mea- sure of the cloaking effect is the extinction function A ∂ xu0∂ x v0 + jw B g0∂ x v0 ab b a a a defined for a cylindric particle of unit length as: ZG0 ZG0 2 0 0 jw 0 0 1 jg µw r⇤u v G±v = 0 v U#(G0) , ext inc sc sc inc Q∂W = Re n du u + u u dl . G0 h G0 8 2 d · kinc Z Z ⇢Z∂W ✓ ◆ x 0 0 1 + q B ∂ u + jwFg = q(u u) a a ZG0 d0 ZG0 q L2(G ) , 8 2 0 0 G± jwz r⇤µhu = 0 a.e. on G , (108) 0 0 (107) where kinc is the incidence wavenumber, d is the ef- where z0 = 0,1 in (107)3, according to the case fective diameter of the cross-sectional area of the (103) and (106), respectively. particle projected onto a plane perpendicular to the 1 direction of propagation d and g = jk + , n is the follows. 2R G inc sc outer normal unit vector, R is the radius of ∂W . minQWs (u ,u ) s.t. b,µ The extinction function will be derived and its struc- 8 d+ d d (110) (u ,u ,u ) satisfies (109) ture explained in the next section. > < (b, µ) U , Far field cloaking observation for non- 2 ad homogenized layer. The cloaking structure is > where U:ad has to be specified. In particular, b, µ are situated in domain W which is locally periodic d fixed on the object to be cloaked (Wc) an can be cho- in the sense we discussed above. The global sen out of a set of materials in W Wc =: W . The domain, WG, consists of three disjoint parts: d \ d,c G + so-called free material optimization problem would W = W W W, see Fig (15). The objects d [ d [ d amount to require to conceal are located in W, whereas on Gs d • 3 the cloaking effect is evaluated using extinction Uad := a L (W design;S ) al a au,tr a V { 2 |    } function (108). 3 for positive semi-definite matrices a0,au S . Ex- We assume that in + the material is homoge- 2 Wd istence of solutions and approximation properties neous (material parameters labeled by subscript 0), with respect to H-convergence have been shown in whereas in W W the material is heterogeneous d [ d a different context by Haslinger, Kocvara, Leuger- in general. However, to be consistent with the as- ing, Stingl[18]. However, the realization of H-limits sumption considered in the next paragraph, we re- is well known to be a nontrivial problem. See = = = quire that µ µ0±, b b0± and s s0± on the re- however [18] for a numerical approximation anal- spective interfaces . The state problem has the Gd± ysis. The application of free material optimization following structure: to the cloaking problem (110) is under way. An alternative to treat the cloaking problem for (109) 1 2 d+ 2 d+ + is to parametrize the material properties as well + — u + w µ0u = 0 in Wd , b0 as the shapes of the inclusions and possible holes

1 d 2 d in the layer Wd,c and view the problem as a non- — —u + w µu = 0 in W , · b d linear finite dimensional constrained optimization ✓ ◆ 1 problem in reduced form, in which the the prob- — —ud + w2µud = 0 in W , lem (109) is solved for the given data and parameter · b d ✓ ◆ set. In particular on the level of a suitable finite- element-discretization one can derive sensitivities of standard transmission conditions: the cost-function with respect to the parameters by fairly standard means. Again, the numerical treat- ∂ (ud+ ud )=0 on G+ , n d ment is under way. d d ∂n(u u )=0 on Gd , Far field cloaking observation for homogenized G + d+ d + layer. We consider the domain W = W W u u = 0 on Gd , + [ [ G0, where G0 = ∂W ∂W can be curved as the d d \ u u = 0 on G , straightforward generalization of the transmission d layer model. Therefore, we shall introduce the boundary conditions: the local (t,n) for any X G X 2 0 where t and n are, respectively, the coordinates in sc sc G ∂nu gu = 0on∂W , the tangential and normal directions w.r.t. curve G0 at position X. As above, the objects to conceal are G where located in W, see Fig. (15). On the rest of ∂W , the radiation condition can be prescribed. The total G usc = ud+ uinc. field in Gs ∂W is obtained by solving the follow- ⇢ + (109) ing problem (we assume that in W the medium is homogeneous, possibly air):

The cloaking effect can be achieved by minimiza- 1 2 + 2 + + — u + w µ0u = 0 in W , tion of Qext(uinc,usc). b + Gs 0 (111) The corresponding optimization problem can be 1 2 — —u + w µu = 0 in W , treated as a free material optimization problem as · b ✓ ◆ transmission conditions – Neumann type:

+ + + + 0 ∂n u = ∂ u = jwb g on G0 , n 0 (112) 0 ∂n u = ∂nu = jwb0g on G0 , wave transmission through the layer – jump control:

0 0 2 0 + ∂t A∂t u + jwBg + w µr⇤u = 0 on G0 , Ω δ incident wave Ω 0 2 0 jw + δ jwB∂t u + w Fg = (u u) Γ d0 area to be concealed s on G0 , object − Ω δ (113) boundary conditions:

∂ usc gusc = 0on∂WG. (114) n Above in the wave transmission condition we em- Fig. (15): Illustration to the cloaking problem for- ployed (107) with G = 0, i.e. z = 0. mulation: for finite thickness layer W , (109). Do- ± 0 d As well as in the previous case, in this situation, the main W contains the object to be cloaked by sur- cloaking effect can be achieved by minimization of face G containing the metamaterial. Cloaking ef- 0 Qext (uinc,usc). In other words, one is looking for the fect is evaluated on G . Gs s solutions of the following problem

ext inc sc minQG (u ,u ) s.t. b,µ s 8 d+ d d (115) (u ,u ,u ) satisfies (111) (114) > < (A,B,F,b, µ) U , 2 ad > where:> the optimization is with respect to a class of admissible functions A,B,F appearing in the trans- mission condition and µ,b as before. In order to understand in particular the transmission conditions along G0 in (113) we focus on

0 0 2 0 ν Ω+ ∂t A∂t u + jwBg + w µr⇤u = 0 on G0 , τ incident wave 0 2 0 jw + jwB∂t u + w Fg = (u u) area to be concealed Γ Γ d0 0 s on G0. − object Ω (116) The first equation contains a Laplace-Beltrami- Helmholtz equation on G0. Indeed, we define the operator

T : L2(G ) L2(G ), Fig. (16): Illustration to the cloaking problem for- A 0 ! 0 mulation: for the homogenized layer represented by D(T ) := u H1(G ) A∂ u H1(G ) , (117) A { 2 # 0 | t 2 0 } G0, (111) T u := ∂ A∂ u A t t The operator TA is self-adjoint and positive semi- definite with discrete . The equation to solve is now

2 0 T u + w µr⇤u = jw∂ Bg . A t following coupled system: wave in cloaked region: ν incident wave τ Ω+ 1 2 — —u + w µu = 0in W b Γ ✓ ◆ (121) area to be concealed 0 u = g0 , Ω− object ∂n jwb0 on G0 wave transmission through the layer:

0 0 2 0 ∂t A∂t u + jwBg + w µr⇤u = 0 on G0 , Fig. (17): Illustration to the problem formulation 0 2 0 jw inc (121). Domain W contains the object to be cloaked jwB∂t u + w Fg = (u u) d0 by surface G0 containing the metamaterial. on G0 . (122) We introduce the resolvent R(l,T ) :=(lI T ) 1 A A In fact the cloaking condition (120) can be viewed of TA at a point l r(TA). With this notation the 2 0 as an exact controllability constraint with variables first equation in (116) can be solved for u as fol- (A,B,F), the coefficients of the homogenized trans- lows. mission through the heterogeneous layer, as con-

0 2 0 trols. This exact controllability problem can be u = jwR(w µr⇤,T )∂ Bg , (118) A t solved for special scenarios. However, in general we cannot expect exact controllability, and therefore while the second equation in (116) turns into the controllability constraint has to be relaxed by an appropriate optimization with penalty. In general, the flux g0 obtained by solving (121), 2 0 0 1 + B∂t R(w µr⇤)∂t Bg +Fg = (u u), on G0. (122) i.e. as the State Problem solution, is not con- jw sistent with the incident wave assumed in W+; it fits (119) the assumption of “no reflection”, when Equation (119) is an integral equation of the sec- 0 ond kind which admits a unique solution g . If one + inc 0 0 = k u + jwb0g , a.e. on G0 , then inserts g0 into the Neumann conditions of (112) n one obtains a nonlocal transmission condition along therefore, the cloaking effect can be approached by G0 which contains the functions A,B,F, µ as mate- the following minimization: rial parameters to be used in the optimization. The optimization problem (115) has not yet been fully min Y(g0,A,B,F) , explored. This will be subject to a forthcoming pub- A,B,F (123) lication. Strong form of the cloaking problem. We keep the + inc 0 0 where Y = kn u + jwb0g G0 , s.t. g solves the domain WG = W+ W G , the objects to conceal k k inc [ [ 0 State Problem (121) with (122) for given u . are located in W, as before. The incident wave is Coefficients (A,B,F) can be handled by designing imposed in W+. We impose the incident wave in inc the microstructure in cell Y. W+; let u be the local amplitude of the plane wave, then Remark 4. In general, there is the jump on G0, [u] = u+ u 0. u0 involved in (122) is an in- G0 6⌘ ternal variable which is relevant only if B 0 on G ; 6⌘ 0 0 + + + inc otherwise (122) reduces to jwb g = ∂ u = ∂ u = k u , (120) 0 n n n 2 0 jw inc w Fg + (u u)=0 on G0 . where u is the trace of u on ∂W G and k is d0 ± ± \ 0 n the projection of the wave vector to the unit out- 1+ In this case the problem (121), (122) reduces to a ward normal n. Above + g = 0 applies due to Iy Helmholtz-problem the form of the incident wave.R As the consequence, 1 g = 0 results by G± 0, see (106) and (107). 1 2 Iy ⌘ — —u + w µu = 0 in W RWe consider the problem imposed in W, being de- b 0 0 8 inc fined in terms of triplet (u,u ,g ) which satisfies the < ∂n u + au = au on G0 : with local Robin-type boundary condition on G0. faces and may vary in W2. Since we will solve the The cloaking constraint then also reduces to just an- Helmholtz equation on a finite computational do- other boundary condition on G0. This leads to an main we have to define appropriate boundary con- overdetermined boundary value problem which may ditions. These conditions should prevent occurrence or not may have a solution. of non-physical reflections from the artificial bound- ary (i. e. the outer boundary should be transpar- 4 ent for the scattered field or the boundary condi- TOPOLOGY OPTIMIZATION FOR tions should absorb the scattered wave, that’s why THE CLOAKING PROBLEM in the following we will call them absorbing bound- In this section we would like to demonstrate the ary conditions). There are various ways in which topology optimization method to design a cloaking such conditions can be chosen, we have used a.b.c. layer such that the given object will become less vis- of first order for it’s simplicity, these conditions re- ible. tains sparsity of the finite element system matrix, on Let us consider a small object (i.e. a nanoparticle the other hand, they do not prevent reflections for composed from a given material). Our aim is to de- all directions of incidence. The total rescaled elec- sign a topology of a cloaking layer (composite of tric Hertz potential u may be decomposed into the the matrix medium and a medium with a low refrac- incident and the scattered field tive index) in such a way that for an observer (sen- inc sc inc jkincd x sor) present behind the particle, the particle becomes u = u + u ,u = e · , (124) in some sense (specified by a cost function) invisi- ble. Propagation of the electromagnetic waves in where d is the direction of propagation of the inci- the composite is described by the Helmholtz equa- dent wave. Furthermore we observe tion (as defined in the previous sections). The ge- —uinc = jkincduinc. (125) ometry of the problem is described by figure (18). The state equation is considered in a circular domain The absorbing b.c. give the relation between the W = 3 W with the boundary ∂W. We place a par- [i=1 i scattered field and its derivative in the direction of ticle (characterized by a complex ) the outer normal on the boundary in the middle of the computational domain. Its body is included in the set W1. The particle is coated by sc sc ∂nu gu = 0on∂W, (126) a shell (W2). And the core-shell is in turn embedded into a matrix medium (W3). 1 where g = jk + . The Helmholtz equation has 2R than the following form

1 — —u + k2µ u = 0inW, · b 0 r 8 ✓ r ◆ > 1 > ∂nu = 0onG, (127) > br G <>  Ω1 [u]G = 0onG, sc sc ∂nu gu = 0on∂W, Ω > 2 > > where: b = n2 is the complex relative permittivity Ω3 r (square of the refractive index). ∂Ω Remark 5. The total (or also scattered) po- tential u(usc) depends generally on the frequency w L and on the direction of propagation d = 2 (cosa,sina),a S, where L = w ,...,w is a 2 { 1 n} set of given frequencies, S = a ,...,a is the set Fig. (18): Description of geometry for the strong { 1 m} form of cloaking problem of angles of incidence. 4 The refractive index is supposed to be constant in To obtain the weak form of Helmholtz equation subdomains W ,W , but is changing across inter- we multiply (127) by the test function v H(W), 1 3 1 2 where H(W) is the standard Sobolev space The absorbed energy rate W abs may be decomposed into the incident energy rate (identically zero), ex- ∂v H(W)=W 1,2 = v v, L2(W),i = 1,2 . tincted and scattered energy rates | ∂xi 2 ⇢ W abs = W inc +W ext W sc (134) (128) We apply the Green’s theorem, further we use (124), Extinction efficiency is then defined as (127) and (125). Then the weak formulation may 1 4 Qext = W ext, (135) be written as follows GIinc Find usc H(W) such that for all v H(W) holds where Iinc is incident irradiance - magnitude of the 2 2 a(usc,v)= f (v). of the incident wave ⇢ (129) 1 inc Iinc = Sinc = Re Einc H (136) where a sesquilinear form a : H H C is defined | | 2| ⇥ | ⇥ ! n o as and G = Ld is cross-sectional area of the particle 1 projected onto a plane perpendicular to the direc- a(usc,v)= —usc—vdS + k2µ uscvdS tion of propagation (d is the diameter of the shelled b 0 r ZW r ZW particle). 1 + guscvdl In the following we will formulate the extinction ef- sc Z∂W br ficiency in terms of the state variable u . The mag- (130) netic end electric field intensities for a homogeneous and non-absorbing medium ( = const > 0) may be and the operator f ( ) is the operator of the right hand b · rewritten as follows side f : H C ! 1 1 1 E = — (ue3)= e3 —u, (137) f (v)= —uinc—vdS k2µ uincvdS w2b ⇥ w2b ⇥ 0 r ZW br ZW 1 j 1 H =(s jew) ue3 = ue3. (138) + n djkincuincvdl v H(W). w2b w · 8 2 Z∂W br (131) Then the Poynting vector may be rewritten as fol- lows (noting that e —u = 0) 3 · Cost functional 1 S = Re j(e —u) ue , 2w3b { 3 ⇥ ⇥ 3} Our aim is to minimize the so-called extinction ef- (139) 1 ficiency. That is a function that reflects energy loss = Re ju—u . due to the inserted particle. 2w3b { } Energy flux at any point of space is represented by The incident irradiance is then given by (using the Poynting vector (125)) 1 1 kinc S = Re E H . (132) Iinc = Re Einc Hinc = . (140) 2 ⇥ 2 ⇥ 2w3b n o In the following we will define the energy that is Using (139) also extinction energy rate is obtained scattered, absorbed and extincted per unit length of as (using (127)4 and again (125)) the cylinder L. We will ignore effects of the ends of L the cylinder. Now imagine a fictive cylinder around ext sc inc inc sc W = 3 Re ju —u + ju —u ndl, 2w b ∂W · the particle (in our concept it will be represented by Z n o the boundary of the computational domain ∂W). We L inc inc sc sc inc = 3 Re n dk u u + jgu u dl. define net rate W abs at which the electromagnetic en- 2w b ∂W · Z n o ergy crosses ∂W (141) Using (135) the final formula for the extinction effi- W abs = L S ndl. (133) · ciency is obtained as Z∂W abs abs 1 jg If W > 0 energy is absorbed in W, if W < 0 Qext = Re n duincusc + uscuinc dl . d · kinc energy is created in W (not considered in the follow- ⇢Z∂W ✓ ◆ ing). (142) Min-max problem The Method of Moving Asymptotes (MMA) is used to solve the preceding problem. One additional re- The aim of the optimization is to minimize values of formulation of (146) is necessary the cost functional for a selected interval of frequen- cies. It can be achieved by the worst scenario ap- min c (147) proach: we shall minimize the cost functional value rˆ Uad for the worst case frequency. 2 We would like to find an optimal distribution of two subject to: g isotropic materials characterized with refractive in- dices n ,n . This leads to the discrete optimization, h 0, i = 1,...,n, j = 1,...,m, 0 1 i, j  which is generally a very difficult problem. One g 0,  possibility to handle this problem is to introduce re- 0 r 1, e E,  e  8 2 laxation of the material (the SIMP method, [5]). We (148) define pseudo density function r(x) Uad 2 where n(r(x),w)=n (w)+(n (w) n (w))r(x)p, 0 1 0 h = Y(usc ),w L,a S p > 1, i, j wi,a j i 2 j 2 = ,..., , = ,..., , 1 for i 1 n j 1 m (149) Uad = r(x)dS r⇤, 1 W2 W  g = Âe E re r⇤. ⇢| | Z 2 card(E) 2 0 r(x) 1,x W ,   2 2 The MMA method requires knowledge of the gradi- (143) ent of the cost functional which is obtained via the sensitivity analysis. Sensitivity analysis of similar where Uad is the admissible set, r⇤ is the maximal problems is provided in a detailed way in [40] or fraction of the material with refractive index n1 that [32]. may be included in the design layer. The worst sce- The main task is the solution of the adjoint equa- nario approach may be formulated as follows tions (that are in fact optimality conditions of the sc Lagrangian L of our problem), the equations are min max Y(uw,a ), (144) r Uad w [w ,w ],a [a ,a ] 2 2 1 n 2 1 m formally defined as follows where Y is the cost functional depending on the Find w H(W) such that for all v H(W) holds state variable. 2 2 (d y(usc) jd y(usc)) v + a(v,w)=0. For the finite element analysis we have to define the ⇢ Re Im · (150) discrete form of the previous problem. Let E be a set of indices of finite elements in the design subdo- Then the final sensitivity of the cost functional for main W2. Then the refractive index for every finite a given frequency w and an angle of incidence a is element in E is defined as follows formulated as p ne(w)=n0(w)+(n1(w) n0(w))re , p > 1, sc sc dy = dL (r,u ,w)=dr (a(u ,w) f (w)) rˆ( )= r c ( ),rˆ U x  e e x ad = d (a(u,w)), e E 2 r 2 3 p 1 g = 2ne(re) p(n1 n0)r —u—wdS. U = card(E) r r⇤, e ad  e WD e E  Z ⇢ 2 (151) g 0 r 1 e E ,  e  8 2 (145) Implementation and results The discretization of the state equations was done by where ce is a characteristic function of the finite el- the classical approach of the finite element method ement e in W2, card(E) is the amount of finite ele- ments in the design layer. The problem (144) is then (for details we recommend the well known book in then reformulated as follows Zienkiewicz et.al. [47]). The state equation is solved by the finite element method using isopara- sc min max Y(uw,a ). (146) metric, bilinear, hexahedral finite elements ( an in- rˆ U w L,a S 2 ad 2 2 troduction is given by Jianming Jin in [19]). g In all following examples the extinction efficiency was minimized (Y = Qext), although the efficiency would be also a good alternative, since

Qext = Qabs + Qsc (152) and we observed the decrease of the extinction was mainly due to lower scattering than absorption. On figure (19) we may observe a particle with higher refractive index (2.1) that is surrounded by the layer with refractive index given by the pseudo density r = 0.3. Dark in the shell corresponds to the matrix material (n = 1.31), by the color low Fig. (21): Design - iteration 9. refractive index material is represented (n = 0.95, that is more or less air). We see that the design evolves to two oval inclusions ((24)), which main- tains more than 60 % decrease in extinction.

Fig. (22): Design - iteration 12.

Fig. (19): Initial design - iteration 0.

Fig. (23): Design - iteration 14.

optimization method. On Figures. (26), (27) the contour lines and initial shape of 3 layers with piece- Fig. (20): Design - iteration 6. wise constant refractive index are defined. The ge- ometry of such structure could be parametrized and The extinction efficiency curves for particular iter- optimized in a similar way as was published in [39], ations are displayed on figure (25). The pink inter- [40]. rupted curve corresponds to the bare particle. Finally the optimal design for two, three and four The inclusions in the final design (24) have no clear angles of incidence is displayed on figures (28), (29) interface with respect to the matrix medium. The and (30). Of course the decrease of extinction is not production of such shell is out of reach of nowa- so huge as in the previous simulation, but we still days . Our suggestion is to use the opti- get improvement approximately 20-40 %. The com- mal topology design as an initial guess for the shape plicated structures that develop give us hint back to Fig. (24): Design - iteration 18. Fig. (27): Contour layers.

Extinction efficiency

0.7 only core iter 0 0.6 iter 6 iter 9 0.5 iter 12 iter 14 0.4 iter 18 0.3

0.2

0.1

0 400 450 500 550 600 650 700 750 800 λ [nm] Fig. (28): Optimal design for 2 directions, Fig. (25): Cost functional values for particular iter- S = 1/4p,1/4p . { } ations

Fig. (29): Optimal design for 3 directions, Fig. (26): Contour lines.. S = 1/4p,0,1/4p . { } the previous section (Fig. (15)). We believe that ing theory of meta-materials for wave propagation the optimally designed micro-structure would re- is within reach. It turns out that micro- or nano- duce the extinction even more significantly than has structured layers play an important role in obtaining been shown on Fig. (25). meta-properties, like cloaking and band-gap phe- nomena. Similarly, micro-structures appear in aux- CONCLUSION etic elastic materials, like metallic or ceramic foams. As we have amply demonstrated, meta-materials In order to achieve results that lead to a an actual in the acoustic, electromagnetic, elastic and piezo- mechanical, acoustic or electromagnetical device, electric context can be approached by quite anal- further research has to be conducted. In particular, ogous mathematical methods. Therefore, a unify- post-processing and interpretation tools have to be [11] R. Cimrman and E. Rohan, Three- phononic materials. ac- cepted : Appl. Comp. Mech., Univ. West Bohemia, Pilsen, (2009). [12] D. Cioranescu, A. Damlamian, G. Griso and D. Onofrei, The peri- odic unfolding method for perforated domains and neumann sieve models, J Math Pure Appl. 2008; 89: 248-277. [13] D. Cioranescu, A. Damlamian and G. Griso, The periodic un- folding method in homogenization, SIAM J Math Anal 2008; 40: 1585-1620. [14] D. Cioranescu and P. Donato, An introduction to homogenization. No. 17 in Oxford Lecture Series in Mathematics and its Applica- tions. Oxford University Press, Oxford, 1999. [15] D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticu- lated Structures, Springer, Appl. Math. Ser. 136, New York, 1999. [16] A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Approx- imate quantum and acoustic cloaking, arXiv:0812.1706v1[math- ph] 9 Dec 2008. [17] M.E. Gurtin, Configurational forces as basic concept in contin- uum , Springer-Verlag, Applied Mathematical Fig. (30): Optimal design for 4 directions, 137, 2000. S = 1/2p, 1/3p,1/3p,1/2p . [18] J. Haslinger, M. Kocvara, G. Leugering and M. Stingl, Multidisci- { } plinary free material optimization, submitted 2009. [19] J. Jin, The finite element method in electromagnetics, John Wiley & Sons, Inc., New York, 2002. developed in order to transfer the numerical results [20] P. Kogut and G. Leugering, S-homogenization of optimal controls problems in Banach spaces, Math Nachr 2002; 233-234: 141-169. into practice. [21] P. Kogut and G. Leugering, Homogenization of optimal control problems in variable domains. Principle of the fictitious homoge- nization, Asympt Anal 2001; 26: 37-72. ACKNOWLEDGEMENT [22] P.I. Kogut and G. Leugering, Homogenization of constrained op- timal control problems for one-dimensional elliptic equations on The notion ’Engineering of Advanced Materials’ periodic graphs, ESAIM, Control Optim Calc Var 2009;15: 471- 498. implicitly announcing a program all by itself, is [23] P.I. Kogut and G. Leugering, Homogenization of Dirichlet opti- also the title of a research cluster within the Ger- mal control problems with exact partial controllability constraints, Asympt Anal 2008; 57: 229-249. man Excellence Initiative funded by the Deutsche [24] V. Lukešand E. Rohan, Computational analysis of acoustic trans- Forschungsgemeinschaft (DFG). The current con- mission through periodically perforated interfaces, Appl Comput Mech 2009; 3: 111-120. tribution was initiated in the Center of ’Multi- [25] B. Miara, E. Rohan, M. Zidi and B. Labat, Piezomaterials for scale Modeling and Simulation’ run by the first au- bone regeneration design - homogenization approach, J Mech Phys Solids 2005; 53, 2529-2556. thor. Second author has been visiting the institute [26] G. W. Milton and J. R. Willis, On modifications of Newton’s sec- a number of times, while the third author is em- ond law and linear continuum elastodynamics, Proc Royal Soc. A 2007; 483: 855-880. ployed there. The second author has been also sup- [27] J.B. Pendry, D. Schurig and D.R. Smith, Calculation of of material ported by projects of the Czech Foundation properties and tracing in transformation optics, Opt Eng 2006; 14: 9794. 101/07/1417 and by project MSM 4977751301 of [28] J.B. Pendry, Negative refraction makes perfect lens, Phys. Rev. the Czech Ministry of Education. Letters 2000; 85: 3966-3969. [29] E. Rohan and V. Lukeš, Sensitivity analysis for the optimal perfo- ration problem in acoustic transmission, Appl Comput Mech 2009; 3: 163-176. [30] E. Rohan and R. Cimrman, Dispersion properties in homogenized REFERENCES piezoelectric phononic materials, In: Proceedings of the confer- ence Engineering Mechanics 2009, Svratka 2009. [1] G. Allaire, Homogenization and two-scale convergence. SIAM J [31] E. Rohan and B. Miara, Modelling and sensitivity analysis for de- Math Anal 1992; 23: 1482-1518. sign of phononic crystals, PAMM, Proc Appl Math Mech 2006; 6: [2] T. S. Angell and A. Kirsch. Optimization Methods in Electromag- 505-506. netic Radiation, Springer-Verlag New York, 2004. [32] E. Rohan and B. Miara, Sensitivity analysis of acoustic wave [3] J.L. Auriault and G. Bonnet, Dynamique des composites elastiques propagation in strongly heterogeneous piezoelectric composite, In: periodiques, Arch Mech 1985; 37: 269-284. Topics on Mathematics for Smart Systems, World Scientific Pub- [4] A. Ávila, G. Griso, B. Miara and E. Rohan, Multi-scale modelling lishers, 2006: 139-207. of elastic waves – theoretical justification and numerical simulation [33] E. Rohan and B. Miara, Shape sensitivity analysis for material op- of band gaps, SIAM Multi Model Simul 2008; 7: 1-21. timization of homogenized piezo-phononic materials, In: Proceed- [5] M. B. Bendsoe and O. Sigmund, Topology optimization, Theory, ings of the WCSMO-8, Lisbon, Portugal 2009. Methods and Applications, Springer-Verlag, Berlin, Heidelberg, [34] E. Rohan and B. Miara, Homogenization and shape sensitivity 2003 and 2004. of microstructures for design of piezoelectric bio-materials, Mech [6] C.R. Bohren and D.R. Huffman, Absorption and Scattering of Adv Materials Struct 2006; 13: 473-485. Light by Small Particles, Wiley-VCH, Weinheim, 2004. [35] E. Rohan, B. Miara and F. Seifrt, Numerical simulation of acous- [7] A. S. Bonnet-Bendhia, D. Drissi and N. Gmati, Simulation of muf- tic band gaps in homogenized elastic composites, Int J Engin Sci fler’s transmission losses by a homogenized finite element method, 2009; 47: 573-594. J Comput Acous 2004; 12: 447-474. [36] E. Rohan et.al., On nonsmooth sensitivity of phononic band gaps [8] A. S. Bonnet-Bendhia, D. Drissi and N. Gmati, Mathematical anal- in elastic structures, in preparation. ysis of the acoustic diffraction by a muffler containing perforated [37] E. Rohan and G. Leugering, Homogenization of particle-cloaking ducts, Math Model Meth Appl Sci 2005; 15: 1059-1090. layer in EM radiation, in preparation. [9] G. Bouchitte and D. Felbacq, Homogenization near [38] E. Rohan and V. Lukeš, Homogenization of the acoustic transmis- and artificial magnetism from , C.R.Acad.Sci, Ser. 2004; sion through perforated layer, J Appl Math Phys, (to appear 2009). 1339: 377-382. [39] F. Seifrt, G. Leugering and E. Rohan, Shape optimization for [10] R. Cimrman and E. Rohan, Three phase phononic materials, Appl. the Helmholtz equation, PAMM, Proc Appl Math Mech 2008; 8: Comput Mech 2009; 3: 5-16. 10705-10706. [40] F. Seifrt, G. Leugering and E. Rohan, Tracking optimization for propagation of electromagnetic waves for multiple wavelengths, Appl Comp Mech 2008; 2: 369-378, Univ. of West Bohemia, Pilsen. [41] E. Sanchez-Palencia, Non-homogeneous media and vibration the- ory, Lecture Notes in Physics 127, Springer, Berlin, 1980. [42] A.P. Seyranian, E. Lund and N. Olhoff, Multiple eigenvalues in structural optimization problems, Struct Opt 1994; 8: 207-227. [43] O. Sigmund and J. S. Jensen, Systematic design of phononic band- gap materials and structures by topology optimization, Phil Trans Royal Soc London, A 2003; 361: 1001-1019. [44] V. Shalaev, Optical negative-index metamaterials, Nature - ics 2007; 1, www.nature.com/naturephotonics [45] D.R. Smith and J.B. Pendry, Homogenization of metamaterials by field averaging, J Opt Soc Am 2006; 23: 391-403. [46] E. Wadbro and M. Berggren, Topology optimization of an acoustic horn, Comput Meth Appl Mech Engrg 2006; 196: 420-436. [47] O.C. Zienkiewicz, R.L. Taylor and J.Z. Zhu, The Finite Elements Method: Its Basis and Fundamentals, Elsevier, Burlington, 2005.