Modeling of Metamaterials in Wave Propagation G. Leugering, E. Rohan and F. Seifrt Lehrstuhl für Angewandte Mathematik II, Universität Erlangen-Nürnberg, Germany. New Technologies Research Center, Research Institute at University of West Bohemia, Plzen, Czech Re- public. Abstract: This chapter focuses on acoustic, electromagnetic, elastic and piezo-electric wave propagation 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 refraction 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-waves 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 frequency range, the ideas rather quickly in- mal material or performance. This, at the the first flicted researcher from optics for a variety of rea- glance, sounds like the formulation of an ancient sons. Superlenses allowing nanoscale imaging and dream of man-kind. However, proper mathemati- nanophotolithography, couple light 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, resonators, lasers, switchers, waveguides tics and engineering, lead to such metamaterial- 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 macroscopic scale 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, permittivities, 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 ’Transformation Optics’, a notion pro- 3 moted by Pendry et.al. [27, 45] in optics and Green- 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 matters 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 <d ,y W˜ kl 2 can be interpreted as a Riemann metric. Transfor- > (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 energy solutions to the gion.