Homogenization of nonlocal wire metamaterialvia a renormalization approach

A. I. Căbuz,1,* A. Nicolet,2 F. Zolla,2 D. Felbacq,3 and G. Bouchitté4

1CNRS, Institut Fresnel, Campus de St. Jérôme, 13013 Marseille, France2Université d’Aix-Marseille, Institut Fresnel, Campus de St. Jérôme, 13013 Marseille, France

3Université de Montpellier II, Groupe d’Étude des Semiconducteurs, 34095 Montpellier, France4Université de Toulon et du Var, BP 132, 83957 La Garde CEDEX, France

*Corresponding author: alexcabuz@gmail.com

Received December 6, 2010; revised February 13, 2011; accepted February 28, 2011;posted March 8, 2011 (Doc. ID 138747); published April 27, 2011

It is well known that defining a local refractive index for a metamaterial requires that the wavelength be large withrespect to the scale of its microscopic structure (generally the period). However, the converse does not hold. Thereare simple structures, such as the infinite, perfectly conducting wire medium, that remain nonlocal for arbitrarilylarge wavelength-to-period ratios. In this work, we extend these results to the case of finite wire media with finiteconductivity, using a two-scale renormalization approach.We show that the nonlocality of the homogeneousmod-el is so extreme that the permittivity is geometry dependent. Its accuracy is tested and confirmed numerically viafull-vector three-dimensional finite element calculations. Moreover, lossy finite wire media exhibit large absorp-tion with small reflection, while their low fill factor allows considerable freedom to control other characteristics ofthe metamaterial, such as its mechanical, thermal, or chemical robustness. © 2011 Optical Society of America

OCIS codes: 160.3918, 160.3900, 290.4210.

1. INTRODUCTIONThe effective medium theory of artificial metallodielectricstructures goes back to the beginning of the twentiethcentury, with the work of Maxwell-Garnett [1] and Wiener(reproduced in [2]). These and subsequent effective mediumtheories focused on disordered media where only partial in-formation about the microscopic structure was available. Amajor step forward was made with the work of Kock in the1940s [3]. This time, Lorentz theory [4] was used to design ar-tificial effective media in a bottom-up fashion as an array ofscatterers. In the 1970s, more mathematically sophisticatedmethods emerged, where instead of seeking a limiting effec-tive medium (equivalent in some suitably defined sense to thestructure of interest), one obtains a limiting equation system[5,6] for the macroscopic electromagnetic field in a givenstructure [7–10].

In recent years, the advent of negative index metamaterialsand composites has led to increased interest in effectivemedium theories. The most popular by far is, of course, theLorentz theory approach, being the most accessible and intui-tively appealing [11]. However, the usefulness of Lorentz the-ory is much diminished when one is interested in materials inwhich the size of objects is much larger than the distancesseparating them or materials that are strongly nonlocal or inwhich the scatterers are strongly coupled, leading to behaviorof a collective nature [12,13]. Contrary to common intuition,nonlocal behavior persists, in certain structures, even whenthe wavelength is much larger than the characteristic scaleof the structure; an excellent example is the wire mediumstudied by Belov et al. [14–16]. In such situations, the Lorentzmodel is no longer useful, and more sophisticated techniquesare required.

The finite wire medium has already been studied (althoughfor the perfectly conducting case) by Belov, Silveirinha, andothers [17–21] and more recently by Lemoult et al. [22–24].However, these results focused on the application of themedium as an imaging device. Indeed, Lemoult et al. makeno mention of the effective medium problem or of homoge-neous quantities. Thus, the problem of the engineering anddesign of an imaging device based on a finite wire structureis an independent and separate issue from the theoretical pro-blem of the effective medium theory of such structures. Thehomogenization problem that concerns us here may be statedas follows: replacing the finite wire structure with an equiva-lent homogeneous medium of the same size, characterized bya constitutive relation linking the polarization field P and theelectric field E. The results of Belov, Silveirinha, and othersproceed from the assumption that the finite wire mediummay be characterized by the same constitutive relation as theinfinite medium, with an additional boundary condition to ac-count for the finiteness of the structure. However, this is dif-ficult to justify theoretically because the extreme nonlocalityof the structure implies automatically that, by truncating theinfinite medium to obtain a finite slab, one automatically mustalso truncate the constitutive relation, leading to a geometry-dependent constitutive relation. In other words, this is amediumwhose permittivity depends on its size and shape. Ourapproach allows us to obtain a constitutive relation that ex-hibits the geometry dependence explicitly.

Our effective medium model of the finite conductivity finitewire medium (the freestanding “bed-of-nails” structure, Fig. 1)is based on a two-scale renormalization approach. Instead ofletting the wavelength tend to infinity, as is customary ineffective medium theories, we keep it fixed and let other geo-metrical parameters tend to zero. The advantage of this ap-proach is that it leaves us the possibility of keeping some

Căbuz et al. Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B 1275

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of the geometrical parameters fixed (in this case, the wirelength L), leading to a new type of partial homogenizationscheme. To put it less formally, we would like to homogenizewhile keeping the thickness fixed with respect to the wave-length, which prevents us from letting λ tend to infinity, sothe only remaining option is to make all the other dimensions(the wire radius and the period) tend to zero. Unlike commonpractice in much of the metamaterials literature, we include adiscussion of the model’s domain of applicability.

2. EFFECTIVE MEDIUM MODELThe structure under study is a square biperiodic array of thinwires, of length L, radius r, and conductivity σ. Note that ourrenormalization approach does work irrespective of whetherthe structure is of finite extension along the x or y direction.We note the period d0 and the wavelength λ. The renormaliza-tion (depicted in Fig. 1) involves a limiting process that is gov-erned by the parameter η ¼ drenorm. For the real, physicalstructure, we have η≡ d0. The three quantities η, rη, and1=ση tend simultaneously to zero. The asymptotics of the othertwo parameters, ση and rη, with respect to η are described byfixed parameters κ and γ according to the following relations:

κ ¼ πr2ησηε0ωη2

; ð1Þ

1γ ¼ η2 log

� ηrη

�; ð2Þ

where ω is the angular frequency of the electromagnetic field.In other words, the conductivity is renormalized inversely tothe fill factor θη ¼ πr2η

η2 , while the radius is renormalized suchthat the expression η2 logðrηη Þ remains constant.

While these expressions may at first seem obscure, theyhave simple intuitive interpretations. The first requires thecurrent density to remain constant during the renormaliza-tion. Notice that κ is nothing other than the volume averageof the imaginary part of the permittivity. Also, recall that thestatic admittance per unit length of a circular wire is given by

Ywire ¼ πr2σ ð3Þand that the number of wires per unit area is given by 1=η2.The second expression requires the average internal capaci-tance of the wires to remain constant during renormalization.This feature is known to be essential for their asymptotic be-havior (e.g., see [25,26]). One may object that the expressionon the right side of Eq. (2) is valid for infinitely long wires,whereas we are working with wires of finite length. Indeed,as shown below, the model fails for wires that are not longwith respect to the period, but that configuration is best trea-ted with the Lorentz approach anyway [27], placing it outsideour present scope.

The essential quantities in the rescaling process are there-fore the geometric quantities rη and η, the material quantity ση,and the field quantities Eη and Hη. To these one must also adda quantity characterizing the all-important electric field in thewires. This is noted Fη, it is nonzero only inside the wires andis given by

Fη ¼κθη

Eη ¼σηε0ω

Eη:

Fη has the units of electric field, and in the microscopic, in-homogeneous picture, it is clearly proportional to the currentdensity. In the macroscopic, homogeneous picture, however,it corresponds to the polarization density P. More precisely,we have limη→0 Fη ¼ P=iε0.

Since the homogenization process is designed to conservethe current density in the medium JzðzÞ, the polarization den-sity is a macroscopic homogeneous quantity that is related tothe microscopic (i.e., nonhomogeneous) current in the wiresIðzÞ via the relation

JzðzÞ ¼IðzÞd2

¼ ∂PzðzÞ∂t

¼ −iωPzðzÞ: ð4Þ

The question to be answered now becomes: what happensin the limit η → 0? The answer is that the fields converge (in aprecise sense described in [28]) to the unique solution of thefollowing system:

8>>>>><>>>>>:

∇ × E

∇ ×H

∂2Pz

∂z2þ�k20 þ 2iπγ

κ

�Pz

∂Pz

∂z

¼ iωμ0H¼ −iωε0

�E þ Pz

ε0 z

�

¼ −2πγε0Ez; z ∈ ½−L=2; L=2�¼ 0; z ∈ f−L=2; L=2g

; ð5Þ

where k0 ¼ 2π=λ. Before solving the system, let us first seewhat it tells us on a more intuitive level.

All field quantities above are effective, homogeneous quan-tities, which have meaning when the wires have been replacedwith a homogeneous effective medium with an electric polar-ization density equal to P. The equation that gives P is an in-homogeneous Helmholtz equation where the source term isgiven by the z component of the electric field Ez. The polar-ization satisfies Neumann conditions at the upper and lowerinterfaces of the slab. In general, it is not continuous therebecause Maxwell’s equations impose the continuity of the nor-mal component of the displacement field D≡ ε0E þ P; conse-quently, any jump in E must be canceled by an equivalentjump in P=ε0. The dependence of P on E, i.e., the constitutive

Fig. 1. Freestanding bed-of-nails structure and the renormalizationprocess. The conducting fibers occupy a region Ω⫅ R2, are orientedin the z direction, and the structure is periodic in the x–y plane. Tworenormalized structures are shown, corresponding to η1 and η2, re-spectively, with η1 > η2, ση1 < ση2 , and rη1=η1 > rη2=η2 [see Eqs. (1)and (2)]. The real, physical structure corresponds by definition toη ¼ d0. The length L and the wavelength λ remain fixed, i.e., weare homogenizing in the x–y plane only.

1276 J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011 Căbuz et al.

relation, takes the form of an integral. In this case, we are deal-ing with a one-dimensional inhomogeneous Helmholtz equa-tion, but this situation is slightly complicated by the factthat it is valid on a bounded domain only (the thickness L

of the slab). The polarization field has the form

Pzðx; z0Þ ¼ −2πγε0Z

L=2

−L=2gðz; z0ÞEzðx; zÞdz; ð6Þ

where gðz; z0Þ is the Green function of the Helmholtz operatoron the bounded domain z ∈ ð− L

2 ;L2Þ. It takes the form (see

Appendix A)

gðz; z0Þ ¼1

K sinðKLÞ cos�K

�z< þ L

2

��cos

�K

�z> −

L

2

��;

where K2 ¼ k20 þ 2iπγκ , z< ¼ minðz; z0Þ, and z> ¼ maxðz; z0Þ.

Equation (6) is clearly a nonlocal constitutive relation becausethe value of the polarization field at a position z0 depends onvalues of the electric field at positions different from z0. More-over, it depends on the geometry (i.e., size) of the medium viathe slab thickness L.

When the imaginary part of K is large, the integral abovedrops off quickly. In particular, when L < 1=imagðKÞ, the non-local behavior dominates, which is the case for the structurespresented here. In the limit of small conductivity (and hencesmall κ), the polarization becomes local for sufficiently largewavelengths. In the opposite limit, for infinite conductivityand infinitely long wires, the integral covers all space (in thez direction) and the material is nonlocal, even in the long-wavelength regime. In fact, this can be seen immediately bydoing a Fourier transform on the third equation of Eq. (5)(with κ → ∞):

Pz ¼−2πγε0k20 − k2z

Ez;

which gives

ε ¼ 1 −2πγ

k20 − k2z:

This is consistent with the findings of Belov et al. [14–16].It is interesting to point out that a valuable alternate point of

view of this system to the one presented here can be obtainedby an analysis that would draw an analogy between themacroscopic phenomena studied here and the quantum the-ory of the electron gas for which there exists a vast and richliterature. In that sense, the behavior studied here can be for-mulated in terms of an electron gas with a certain effectivedensity, an effective electron mass, and an effective boundarycondition. This approach is, indeed, the one used by the verypioneers of the field of metamaterials [25,26,29,30]. From thisangle, some of the quantities whose meaning might seem ob-scure in our approach become transparent. For instance, K isclearly a quantity that has units of inverse length. It defines ascale that may be compared to L in order to determinewhether, in a particular structure, the nonlocal behavior dom-inates. That scale may be interpreted as analogous to a meanfree path. Similar analogies may be obtained from the otherrelevant quantities, with the mention that, in our case, these

quantities depend on macroscopic features of the structure,such as the wire radius, period, or conductivity. In this way,one may say that, when one engineers a conducting meta-material structure, one is actually engineering an artificialtype of electron gas as a purely geometrical effect in a macro-scopic structure. For further ideas in this direction, we directthe reader to the relevant literature [31–35].

Until now, the discussion has been independent of the ac-tual shape of the domain Ω (Fig. 1). From this point on, how-ever, for purposes of illustration, we specialize to the caseΩ ¼ R2, which is an infinite two-dimensional freestandingbed of nails of thickness L, period d0, wire radius r, and con-ductivity σ. The effective medium is therefore a homogeneousslab parallel to the x–y plane and of thickness L.

3. NUMERICAL RESULTSWe now proceed to test the homogeneous model by compar-ing it with three-dimensional (3D) full vector simulations ofthe structure; i.e., we must compare the reflection, transmis-sion, and absorption coefficients and the current distributionof the homogeneous problem with those of the original wiremetamaterial. The solution to the homogeneous problem isobtained by integrating Eq. (5) as described in Appendix B.

The 3D full vector simulations of the metamaterial weredone using the Comsol Multiphysics finite element method[36] software package. The periodicity was implementedusing Floquet–Bloch conditions [37] in the two periodic direc-tions (x and y) and absorbing perfectly matched layers(PMLs) [38] in the positive and negative z directions. The lin-earity of the materials in the structure was used to treat theincident field (a plane wave) as a localized source within theobstacle, as detailed in [39,40]. In other words, the computa-tion is used to obtain the scattered field only, and the totalfield is obtained by summing it with the plane wave incidentfield. The Comsol/Matlab scripts of the models used to pro-duce Figs. 2 and 3 are available as online support materialfor the readers’ convenience.

Figures 2 and 3 show good agreement between the effectivemediummodel and the finite element simulation. Note that the

Fig. 2. Transmission (solid), reflection (dotted–dashed), and absorp-tion (dashed) efficiency curves comparing the finite element solution(dot markers) and the effective medium solution (no markers) as afunction of the angle of incidence. The structure has a conductivityσ ¼ 8ðΩmÞ−1, period d0 ¼ 0:01m, and dimensionless parametersL=d0 ¼ 120, λ=d0 ¼ 20, r=d0 ¼ 0:1, and δ=r ¼ 4:6, where δ is the skindepth of the wire. The coarse mesh and the very large number of de-grees of freedom (approximately 2.8 million) explains the approxi-mate nature of the energy conservation (× markers) of the finiteelement model.

Căbuz et al. Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B 1277

current density behavior near the boundaries differs betweenthe effective mediummodel and the finite element model. Thisis due to the fact that, in the macroscopic, homogeneous sce-nario, one speaks of a polarization field obeying Neumannboundary conditions, as discussed earlier. However, in the mi-croscopic scenario, we have a free conductor-carrying currentinduced by an external electric field. Since, in our geometry atthe given wavelength, the capacitance of the wire end points isvery small, the accumulation of charge will be correspond-ingly small, leading to an almost continuous normal compo-nent of the electric field (and, therefore, also current).Numerically, it seems as if the current goes to zero at the wireend points, even though this is not strictly exact. Nevertheless,since, in the homogeneous limit, the boundary condition of thecurrent is of Neumann type, the convergence of the renorma-lization process is clearly nonuniform near the boundaries. Inphysical terms, the fact that the model breaks down for smallL=d0 ratios (short wires) is due to the fact that one of the as-sumptions of the model, Eq. (2), fixes the capacitance of thewires per unit length. Hence, it is to be expected that it nolonger holds for short wires.

It must also be pointed out that the parameters of the par-ticular structure chosen for the illustration in Figs. 2 and 3were forced upon us by practical constraints: finite ele-ment meshing of thin, long, circular wires requires very largeamounts of computer memory and time. Simulation of wiresthinner than r=d0 ¼ 0:05 is prohibitive. Consequently, to ex-plore a wider domain of the parameter space, we have takenadvantage of the fact that the structures in which we are in-terested have r ≪ d0 and δ ≫ r. Such thin conducting struc-tures can be simulated much more efficiently as lines of zerothickness [41] (i.e., edges, in the finite element formulation)carrying current and exhibiting an equivalent linear imped-ance. This approach gives excellent results with a fraction ofthe computing power and enables us to model realistic struc-tures that would otherwise be inaccessible.

For instance, Figs. 4 and 5 show the results of calculationsfor a structure of Toray T300® carbon fibers [42] with a con-ductivity of σ ¼ 5:89 · 104ðΩmÞ−1 and a radius of 3:5 μm. Thewires have an aspect ratio of L=r ¼ 2:28 × 105, which is farbeyond what would have been accessible by meshing the in-terior of the wires. The finite element model of Fig. 2 (curveswith markers), in which the interior of the wires is meshed, is

a problem with approximately 2.8 million degrees of freedom,which requires at least 42GB of available random-accessmemory (RAM) to solve. By comparison, the model of Fig. 4(curves with markers), in which the wires are modeled as cur-rent carrying edges, is a problem of approximately 62,000 de-grees of freedom, which requires less than 1GB of availableRAM and can therefore be solved on any fairly recent desktopcomputer.

Figures 2–5 illustrate the behavior that is typical of the mod-el. The agreement remains good up to high incidence anglesand over a large wavelength domain. The structure is trans-parent in normal incidence. For increasingly oblique anglesof incidence, the absorption increases more or less gradually,depending on the thickness L. The reflection is generally low,although it increases when approaching grazing incidence.The low reflection may be explained by the small radii ofthe wires: their extremities have low capacitance, hence they

Fig. 3. Squared magnitude of the current density [see Eq. (4)] for theeffective medium solution of Eq. (A1) (dashed curve) and the finiteelement solution (solid curve) as a function of position within the wirestructure (which is positioned in z ∈ ð0; LÞ). The structure is the sameas in Fig. 2, illuminated at an angle of incidence θ ¼ 40° from the top.

Fig. 4. Transmission (solid), reflection (dotted–dashed), and absorp-tion (dashed) efficiency curves comparing the finite element solution(dot markers) and the effective medium (no markers) as a function ofthe angle of incidence. The wire conductivity is that of Toray T300®carbon fibers σ ¼ 5:89 · 104ðΩmÞ−1. The structure has period d0 ¼0:01m and dimensionless parameters L=d0 ¼ 80, λ=d0 ¼ 20, r=d0 ¼3:5 · 10−4, and δ=r ¼ 15. Energy conservation of the finite elementmodel (× markers) is respected to within better than 1% for mostangles of incidence. The departure around 80° is explained by thepoor performance of the PMLs when close to grazing incidence.

Fig. 5. Squared magnitude of the current density for the effectivemedium model (dashed curve) and the finite element solution (solidcurve) as a function of position within the slab (which is positionedin z ∈ ð0; LÞ). The structure is the same as in Fig. 4, illuminated at anangle of incidence θ ¼ 40° from the top. Note that the surface areasunder the two curves (in this figure as well as Fig. 3) are the samebecause they are proportional to the Joule dissipation rates, which areseen to be equal from Fig. 4 (and Fig. 2) at the given angle of incidence.

1278 J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011 Căbuz et al.

exhibit very little charge accumulation, leading to an almostcontinuous normal component of the electric field. Certainconfigurations exhibit very low reflection for almost all anglesof incidence (see Fig. 6 and 7) around λ ¼ 1:2m. The currentdensity decreases roughly exponentially within the structuredue to absorption.

We must also point out an essential feature of Figs. 3 and 5.In the microscopic picture, the structure response consists ofa z-dependent current in the wires, giving a current density ineach cell, JzðzÞ, while, in the macroscopic picture, the re-sponse consists of a polarization density, PzðzÞ. One expectsthat the first converge, in some sense, to the time derivative ofthe second, i.e., to the polarization current density. This istrue, but the “sense” in which this convergence happens ishighly nontrivial. The nonlocality of the medium meansthat the two pictures, the microscopic and the macroscopic

(homogeneous), cannot be identified in a local, or pointwise,sense, but only in a global, or integral, sense. More specifi-cally, although the curves differ, their integrals do not, asFigs. 2 and 4 indicate (recall that Joule absorption is propor-tional to the integral of the square of the current density).Hence, the apparent disagreement between the curves inFigs. 3 and 5 is in no way an artefact or shortcoming ofour particular effective medium approach, but an intrinsic fea-ture of any nonlocal effective medium model. As a matter offact, if the agreement in these figures had been perfect, itwould have been an indication of an inconsistency in our non-local description! This also explains why we have chosen toplot the square of the current density, rather than the currentdensity itself.

4. DOMAIN OF VALIDITYThe boundaries of the domain of validity of the model are gi-ven by four dimensionless parameters: the ratio of the skindepth to the radius in the wires δ=r, the ratio of the wire lengthto the period L=d0, the ratio of the wavelength to the periodλ=d0, and the ratio of the wire radius to the period r=d0.

The skin depth must be larger than the radius due to thefact that the impedance used in defining κ [Eq. (3)] is the staticimpedance that differs from the quasistatic value by an imagi-nary inductive term iωμ=8π (e.g., see [43]). Requiring this termto be negligible is equivalent to requiring that δ2=r2 ≫ 1. More-over, in the rescaling process, the skindepth/radius ratio isgiven by

δηrη

¼ λη

ffiffiffiffiffiffiffiffi1

2πκ

r:

Since η approaches zero in the rescaling process, it is naturalto expect the homogeneous model to be valid when the skindepth is large compared to the radius.

In addition, recall that the definition of γ in Eq. (2) fixes thecapacitance of the wires to the value for thin, long wires. Con-sequently, we expect the model to hold for large L=d0 andsmall r=d0. To these, we must add the general requirementfor all effective medium models: the wavelength must be largecompared to the period.

Because of the large (four-dimensional) parameter space,an exhaustive numerical exploration of the wire structureis not feasible in a reasonable time frame. Still, our study hasmade it possible to broadly determine the boundaries of thedomain of applicability of the effective medium model.Roughly, one must have λ=d0 ⪆ 7 − 12, δ=r ⪆ 4 − 8, L=d0 ⪆20 − 30, and r=d0 ⪅ 1=10. Our (a fortiori limited) numerical ex-ploration of the parameter space suggests that the skin depth-to-radius ratio is often the main limiting factor, particularlywhen considering highly conducting wires.

5. CONCLUSIONWe have tested numerically the effective medium theory ofthe freestanding bed-of-nails structure, whose rigorous math-ematical foundation is described in [28]. We have found goodagreement between the transmission, reflection, and absorp-tion efficiencies between the effective medium model and a3D finite element model for a broad range of angles ofincidence and wavelengths. The current density in the realstructure is homogenized to become the polarization current

Fig. 6. Transmission (solid), reflection (dotted–dashed), and absorp-tion (dashed) efficiency curves comparing the finite element solution(dot markers) and the effective medium (no markers) as a functionof the angle of incidence. The structure has a conductivity σ ¼1000 ðΩmÞ−1, period d0 ¼ 0:01m, and dimensionless parametersL=d0 ¼ 50, λ=d0 ¼ 8, r=d0 ¼ 0:002, and δ=d0 ¼ 13. The reflection re-mains low for angles of incidence of up to 80° even as the Joule ab-sorption reaches almost 100% for θ > 60°. Energy conservation isindicated by the × markers.

Fig. 7. Transmission (solid), reflection (dotted–dashed), and absorp-tion (dashed) efficiency curves comparing the finite element solution(dot markers) and the effective medium (no markers) as a function ofwavelength. Energy conservation for the finite element model is la-beledwith ×markers. The structure has a conductivity σ ¼ 3000 ðΩmÞ−1(in the semiconductor domain), period d ¼ 0:01m, and dimensionlessparameters L=d ¼ 60 and r=d ¼ 0:003, and the angle of incidence isθ ¼ 70°. δ=r runs approximately from 4 to 25 from left to right overthe domain of the plot. The model fails around λ⪅ 0:1m ¼ 10d.

Căbuz et al. Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. B 1279

density of the effective medium model. The medium is non-local, meaning that the polarization field depends on the elec-tric field over a region of finite size. That dependence is givenby Eq. (6). The nonlocality also explains why the wire currentdoes not converge locally (i.e., pointwise, in a strong sense) tothe polarization current (Figs. 3 and 5) but only globally (i.e.,weakly, in an integral sense). This nonlocal behavior alsomeans that the permittivity depends on the wave vector, soit can no longer be seen, strictly, as a property of the medium,but rather, as a property of a given wave propagating in thestructure [44,45].

Equation (6) is the main result of our approach. Its geome-try dependence (via L) illustrates how, in cases of extremelynonlocal media, such as the wire medium, a finite structurecan be qualitatively different from the infinite structure. Theeffective medium corresponding to a given finite object willtherefore depend, in a nonobvious way, on its size and shape.This type of phenomenon is always relevant whenever dealingwith structures containing charges that are free on similarscales as the size of the object. It has also been studied in thecontext of metal nanoparticles with sizes comparable to theelectron mean free path [46,47]. The effective permittivity ofsuch objects needs to be corrected to take into account theirsize and shape for the same reasons that our constitutive re-lation Eq. (6) depends on the thickness of our wire structure.

The freestanding bed-of-nails structure is a medium exhibit-ing high absorption with low reflection. It requires a very lowfilling fraction of conducting material, but exhibits near-perfect absorption over a wide range of angles of incidencefor sufficiently large thicknesses. The low filling fraction isuseful because it allows the engineer to fill the space betweenthe wires with materials satisfying other design constraints,such as mass density, or mechanical, chemical, or thermalrobustness. The geometries studied here are transparent atnormal incidence, but this aspect can easily be rectified byslanting the wires by about 20° with respect to the upperand lower boundaries. This design may therefore be usedto obtain a near-perfect electromagnetic absorber for all an-gles of incidence in a very straightforward way and with con-siderable freedom in the resulting mechanical, thermal, orchemical properties of the structure.

APPENDIX AWe require the Green function for the problem (see [48],Chapter 2):

p00 þ α2p ¼ βEz with p0ð−L=2Þ ¼ p0ðL=2Þ ¼ 0: ðA1Þ

For the purpose of this Appendix, it is convenient to considerthat the structure is positioned between −L=2 and L=2. TheGreen function satisfies the equation

g00 þ α2g ¼ δz0 ; z0 ∈

�−L

2;L

2

�ðA2Þ

and may be written as

gðz; z0Þ ¼ Cu1ðz<Þu2ðz>Þ ðA3Þ

with

z< ¼ minðz; z0Þ; z> ¼ maxðz; z0Þ

such that

when z ∈

�−L

2; z0

�; g ¼ Cu1ðzÞu2ðz0Þ;

when z ∈

�z0;

L

2

�; g ¼ Cu1ðz0Þu2ðzÞ:

Replacing form Eq. (A3) into Eq. (A2), one finds that gmustbe continuous at z0, its derivative must have a jump disconti-nuity of 1, and the two functions u1 and u2 must be sinusoidalof wave constant α:

u1ðzÞ ¼ A cosðαðzþ L=2ÞÞ u2ðzÞ ¼ B cosðαðz − L=2ÞÞ:

By imposing the boundary conditions Eq. (A1), we obtain

u01ð−L=2Þ ¼ 0 u0

2ðL=2Þ ¼ 0;

and, by requiring g0 to have a jump discontinuity of 1 at z0, weobtain

ABC ¼ 1α sinðαLÞ ;

giving finally

gðz; z0Þ ¼1

α sinðαLÞ cosðαðz< þ L=2ÞÞ cosðαðz> − L=2ÞÞ:

APPENDIX BWe now proceed to solve the homogeneous limit Eq. (5). Forconvenience, we position it in z ∈ ð0; LÞ. Since we are dealingwith a system with translational invariance, a slab, we cansplit the problem into two independent polarization cases:TE, where the electric field is in the x–y plane, and TM, wherethe magnetic field is in the x–y plane. However, since we areconsidering thin wires (small volume fraction), the structurewill be transparent to TE waves. Therefore, we only have toconsider TM waves. We choose a coordinate system so thatthe plane of incidence is the x–z plane, with angle of incidenceθ, in which case our unknowns will be Hy and Pz. The transla-tion invariance allows us to seek solutions of the form

Hy ¼ uðzÞeiαx Pz ¼ pðzÞeiαx

with α ¼ k0 sin θ. Inserting these into Eq. (5), we obtain a sys-tem of equations for u and p:

8<:

u00ðzÞþ ðk20 − α2ÞuðzÞp00ðzÞþ

�k20 þ 2iπγ

κ − 2πγ�pðzÞ

¼ αωpðzÞ¼ 2παγ

ω uðzÞ; z∈ ½0;L� ðB1Þ

with the important boundary conditions p0 ¼ 0 at z ¼ 0 andz ¼ L and u and u0 continuous everywhere.

The objective is now to obtain the transfer matrix T of theslab, which relates the field u and its derivative u0 at thebottom and the top of the slab:

1280 J. Opt. Soc. Am. B / Vol. 28, No. 5 / May 2011 Căbuz et al.

�uðLÞu0ðLÞ

�¼ T

�uð0Þu0ð0Þ

�: ðB2Þ

Once T is known, the reflection and transmission coefficientsr and t can be obtained immediately from

r ¼ e−2iβL AþBA−B

and t ¼ 2e−iβLA−B

A≡ T11 − iβT12 and B≡T21 − iβT22

iβ ; ðB3Þ

where β ¼ k0 cos θ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffik20 − α2

q.

We begin by integrating Eq. (B1). Noting δ2 ¼ k20 þ 2iπγκ − 2πγ

for readability, we rewrite the system as

W 00ðzÞ ¼ −MWðzÞ; ðB4Þ

where

WðzÞ ¼�uðzÞpðzÞ

�

M ¼�

β2 −αω−

2παγω δ2

�:

The matrix M can be diagonalized M ¼ QDQ−1 with D ¼diagðK2

u; K2pÞ so Eq. (B4) can be rewritten as Q−1W 00ðzÞ ¼

−DQ−1WðzÞ. Since Q is constant and known, it can be inte-grated directly, and the general solution is then obtained asa sum of plane waves:

Q−1WðzÞ ¼�Aþu expðiKuzÞ þ A−

u expð−iKuzÞAþp expðiKpzÞ þ A−

p expð−iKpzÞ�: ðB5Þ

Once the integration performed, obtaining T is now only amatter of algebraic manipulation. u and p are now expressedin terms of the elements of the matrix Q and the coefficientsAþu , A−

u, Aþp , and A−

p . However, recall that we are not interesteddirectly in these coefficients, but in the matrix T . Since thatmatrix does not depend directly on p, the first step is to elim-inate the Aps from the equation system. This is done by mak-ing use of the boundary conditions. By differentiating thebottom equation of Eq. (B5), we can obtain p0 as

p0 ¼ iKuQ21ðAþu e

iKuz − A−ue

−iKuzÞþ iKpQ22ðAþ

p eiKpz − A−

pe−iKpzÞ:

Setting this to zero at z ¼ 0, L, we can obtain the Aps in termsof the Aus. Noting vectors

Au ¼�Aþu

A−u

�Ap ¼

�Aþp

A−p

�;

we introduce the matrix

C ¼ −KuQ21

KpQ22

12i sinðKpLÞ

×�eiKuL − e−iKpL e−iKpL − e−iKuL

eiKuL − eiKpL eiKpL − e−iKuL

�

so that

Ap ¼ CAu:

We are now in a position to expressWðzÞ in terms of Au alone.Equation (B5) can be rewritten as

WðzÞ ¼ QEðzÞAu; ðB6Þwhere EðzÞ is defined as

EðzÞ ¼�

eiKuz e−iKuz

C11eiKpz þ C21e

−iKpz C12eiKpz þ C22e

−iKpz

�:

Equation (B6) contains (within its first row) the expressionfor u. However, to obtain the transfer matrix T , we alsorequire u0. We simply differentiate Eq. (B6) to obtain

W 0ðzÞ ¼ QE0ðzÞAu: ðB7Þ

By combining the first rows of Eqs. (B6) and (B7), we are ina position to construct the matrix GðzÞ such that

�uðzÞu0ðzÞ

�¼ GðzÞAu:

By writing this equation at z ¼ 0 and z ¼ L, we obtain

�uðzÞu0ðzÞ

�¼ GðLÞGð0Þ−1

�uð0Þu0ð0Þ

�:

Comparing with Eq. (B2), we obtain the result we seek,

T ¼ GðLÞGð0Þ−1;

leading to the reflection and transmission coefficientsvia Eq. (B3).

To summarize, we are now capable of modeling a structurewith a given d, r, σ, and L at a given incident field wavelength λin the following way. We first obtain the two rescaling pa-rameters κ and γ for the given structure using Eqs. (1) and(2). Then, we integrate Eq. (A1) to obtain the reflectionand transmission coefficients.

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