Optical properties of nanostructured metamaterials

Phys. Status Solidi B 247, No. 8, 2102–2107 (2010) / DOI 10.1002/pssb.200983941 p s sb

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basic solid state physics

Optical properties of nanostructuredmetamaterials

Ernesto Cortes1,2, Luis Mochan3, Bernardo S. Mendoza*,1, and Guillermo P. Ortiz4

1 Division of Photonics, Centro de Investigaciones en Optica, Leon, Guanajuato, Mexico2 Division de Ciencias e Ingenieras, Campus Leon, Universidad de Guanajuato, Mexico3 Instituto de Ciencias Fsicas, Universidad Nacional Autonoma de Mexico, A.P. 48-3, 62251 Cuernavaca, Morelos, Mexico4 Departamento de Fısica, Facultad de Ciencias Exactas, Naturales y Agrimensura, Universidad Nacional del Nordeste – Instituto de

Modelado e Innovacion Tecnologica, CONICET-UNNE, Av. Libertad 5500, W3404AAS Corrientes, Argentina

Received 8 October 2009, revised 14 January 2010, accepted 3 February 2010

Published online 18 June 2010

Keywords dielectric functions, nanostructured metamaterials, optical properties

* Corresponding author: e-mail [email protected], Phone: þ52 477 4414200, Fax: þ52 477 4414209

We present a very efficient recursive method to calculate the

effective optical response of nanostructured metamaterials

made up of particles with arbitrarily shaped cross sections

arranged in periodic two-dimensional arrays. We consider

dielectric particles embedded in a metal matrix with a lattice

constant much smaller than the wavelength. Neglecting

retardation our formalism allows factoring the geometrical

properties from the properties of the materials. If the conducting

phase is continuous the low frequency behavior is metallic. If

the conducting paths are nearly bloqued by the dielectric

particles, the high frequency behavior is dielectric. Thus,

extraordinary-reflectance bands may develop at intermediate

frequencies, where the macroscopic response matches vacuum.

The optical properties of these systems may be tuned by

adjusting the geometry.

Sketch of a nanostructured metamaterial slab with a dielectric-

like or metallic-like behavior depending on the frequency of the

incoming light.

� 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction Metamaterials are typically binarycomposites of conventional materials: a matrix with inclus-ions of a given shape, arranged in a periodic structure. Sincethe times of Maxwell, Lord Rayleigh, and Maxwell Garnetup to today, many authors have contributed to the calculationof the bulk macroscopic response in terms of the dielectricproperties of its constituents [1–3]. Recent technologiesallow the manufacture of ordered composite materials withperiodic structures. For instance, high-resolution electronbeam lithography and its interferometric counterpart havebeen used in order to make particular designs of nanos-tructured composites, producing various shapes with nano-metric sizes [4, 5]. Moreover, ion milling techniques arecapable of producing high quality air hole periodic and non-periodic two-dimensional (2D) arrays, where the holes can

have different geometrical shapes [6, 7]. Therefore, it ispossible to build devices with novel macroscopic opticalproperties [8]. For example, a negative refractive index hasbeen predicted and observed [9] for a periodic compositestructure of a dielectric matrix with noble metal inclusions oftrapezoidal shape [10].

Nanostructured metallic films are having an importantdevelopment as well. On one hand, the existence of surfaceplasmon-polariton (SPP) modes, excited on the metal–airinterface, yields several related phenomena such as an enha-ncement of optical transmission through sub-wavelengthholes [11–14]. Besides the single coupling to SPP modes,double resonant conditions, [15] and waveguide modes [16]seem to play an important role in the optical enhancement formetallic gratings with very narrow slits and for compound


Phys. Status Solidi B 247, No. 8 (2010) 2103

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gratings [17]. On the other hand, a very strong polarizationdependence in the optical response of periodic arrays oforiented sub-wavelength holes on metal hosts [6, 7, 18] andsingle rectangular inclusion within a perfect conductor [19]have been recently reported. These studies did not rely onSPP excitation as a mechanism to explain their opticalresults.

In this work, we obtain the macroscopic dielectricresponse of a periodic composite, using a homogenizationprocedure first proposed by Mochan and Barrera [20]. In thisprocedure, the macroscopic response of the system isobtained from its microscopic constitutive equations byeliminating the spatial fluctuations of the field with the use ofMaxwell’s equations. Besides the average dielectric func-tion, the formalism above incorporates the effects that therapidly varying Fourier components of the microscopicresponse has on the macroscopic response, i.e., the local-field effect. Similar homogenization procedures are alsofound in [18, 21–24]. However, here we show how thehomogenization of Maxwell’s equations may be done byusing Haydock’s recursive Scheme [25]. With this procedureone gains not only a tremendous speed improvement in thecalculations but also the possibility of calculating the opticalproperties of sub-wavelength three-dimensional (3D) struc-tures with rather arbitrary geometry, including interpene-trated inclusions [26]. We show that the geometry of theinclusions might lead to an extraordinary transmission and avery anisotropic optical behavior, and that the transparencywindows within metal-dielectric metamaterials appear forinclusion filling fractions slightly below the percolationthreshold of the metallic phase.

2 Theory We consider a metamaterial made of ahomogeneous host of some material a within which aperiodic lattice of arbitrarily shaped nanometric inclusionsof a material b is embedded, yielding an artificial crystal. Weassume that each region a ¼ a; b is large enough though tohave a well-defined macroscopic dielectric response eawhich we assume local and isotropic. The lattice parameter istaken to be smaller than the vacuum wavelengthl0 ¼ 2pc=v, with c the speed of light in vacuum and v thefrequency. The microscopic response is described by

www

eðrÞ ¼ ea � BðrÞeab; (1)

where eab � ea � eb and BðrÞ ¼ Bðrþ RÞ is the periodiccharacteristic function for the b regions, with fRg theBravais lattice of the metamaterial.

The constitutive equation DðrÞ ¼ eðrÞEðrÞ may bewritten in reciprocal space as

DGðqÞ ¼XG0

eGG0EG0 ðqÞ; (2)

where DðrÞ and EðrÞ are the electric and displacementfields, DGðqÞ and EGðqÞ the corresponding Fouriercoefficients with wavevectors qþ G, q the Bloch’s vectorand fGg is the reciprocal lattice. Here, eGG0 is the Fourier

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coefficient of eðrÞ corresponding to the wavevector G�G0.Ignoring retardation we may assume E is longitudinal

EG ! ELG ¼ GG � EG; (3)

where we denote the unit vectors ðqþ GÞ=jqþ Gj simplyby G, inparticular, 0 ¼ q=q.A longitudinal externalfield maybe identified withDL, which allows us to choseDL

G6¼0ðqÞ ¼ 0,i.e., we consider an external longitudinal plane wave withoutsmall scale spatial fluctuations. Substituting Eq. (3) into thelongitudinal projection of Eq. (2) allows us to solve for

EL0 ¼ qh�1

00 q � DL0 ; (4)

where we first invert

hGG0 � G � ðeGG0G0Þ; (5)

and afterwards take the 00 component. The macroscopiclongitudinal field EML is obtained from EL by eliminating itsspatial fluctuations, i.e., EML ¼ EL

0 . Similarly, DML ¼ DL0 .

Thus, from Eq. (4) we identify

e�1ML � qjq ¼ qh�1

00 q; (6)

defined through EML ¼ e�1ML � DML, as the longitudinal

projection of the macroscopic dielectric response corre-sponding to Bloch’s wavevector q.

To continue, Fourier transform the microscopicresponse, eGG0 ¼ eadGG0 � eabBGG0 , where BGG0 ¼ ð1=VÞRv d

3reiðG�G0Þ�r, V is the volume of the unit cell and v the

volume occupied by b. The geometry is characterized byBGG0 and in particular, B00 ¼ v=V � f is the filling fractionof the inclusions.

2.1 Haydock’s recursion From Eq. 5 we obtainh�1GG0 ¼ GGG0=eab, where GðuÞ ¼ ðu� HÞ�1

is a Green’sfunction corresponding to an operator H with elements

HGG0 ¼ BLLGG0 ¼ G � ðBGG0G0Þ; (7)

and where the frequency dependent spectral variableu � ð1 � eb=eaÞ�1

is analogous to a complex energy.From Eq. (6) we obtain j ¼ h0jGðuÞj0i=eab, where jGi

denotes a plane wave state with wave vector qþ G. Thisallows the use of Haydock’s recursive scheme to obtain theprojected Green’s function and thus the macroscopic res-ponse. We set j � 1i ¼ 0, j0i ¼ j0i, b0 ¼ 0 and recursivelydefine the orthonormalized states jni through

j~ni ¼ Hjn� 1i ¼ bn�1jn� 2i þ an�1jn� 1iþ bnjni;

(8)

with

an�1 ¼ hn� 1j~ni ¼ hn� 1jHjn� 1i (9)

and

b2n ¼ h~nj~ni � a2

n�1 � b2n�1: (10)


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In the basis fjnig the operatorHmay be represented by atridiagonal matrix and the inverse G�1ðuÞ � G�1

0 ðuÞ of theGreen’s function is given by the matrix

� 20

(11)

which we can write recursively in blocks as

(12)

with An ¼ ðu� anÞ and Bn ¼ ð�bn; 0; 0; � � �Þ. Here we usedcalligraphic letters to denote any matrix except 1 � 1matrices which are equivalent to scalars. Now we write Gn inblocks as

(13)

so using GnG�1n ¼ diagð1Þ we find

Rn ¼1

An � Bnþ1G�1nþ1BT

nþ1

¼ 1

An � b2nþ1Rnþ1

; (14)

where in the last step we used the fact that the vectors Bnþ1

have only one element different from zero. In this way, wesee the n-th solution is linked to the nþ 1 solution. IteratingEq. (14) we obtain G00ðuÞ ¼ R0 and then

j ¼ u

ea

1

u� a0 �b2

1

u�a1�b2

2

u�a2�b23

. ..

; (15)

Notice that Haydock’s coefficients depend only on thegeometry throughBLL

GG0 . The dependence on composition andfrequency is completely encoded in the complex valuedspectral variable u. Thus, for a given geometry we mayexplore manifold compositions and frequencies withouthaving to recalculate Haydock’s coefficients.

We should emphasize that j depends in general on thedirection of q. Calculating e�1

ML for several propagationdirections q we may obtain all the components of the fullinverse long-wavelength dielectric tensor e�1

M and from it eM .To initiate the recursion in order to obtain an and bn we

first define the following auxiliary function

’nðGÞ � hGjni: (16)

Now we project Eq. (8) into jGi and obtain

’~nðGÞ ¼ hGj~ni ¼XG0

G � ðBGG0G0Þ’n�1ðG0Þ: (17)

10 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

UsingP

g jGihGj ¼ 1 and Eq. (9) we obtain

an ¼ hnj~ni ¼XG

hnjGihGj~ni

¼XG

’�nðGÞ’~nðGÞ; (18)

and h~nj~ni ¼P

G j’~nðGÞj2 that when substituted in Eq. 10gives bn. Then from Eq. 8 we obtain

’nðGÞ ¼’~n�1ðGÞ � an�1’n�1ðGÞ � bn�1’n�2ðGÞ

bn:

(19)

Employing above equations we can recursively calculateHaydock’s coefficients an and bn starting from ’n�1ðGÞ,besides obtaining ’nðGÞ with which we can start the nextiteration till convergence is reached. We chose as the initialstate ’0ðGÞ ¼ dG0 since the macroscopic dielectric functionis given by the G ¼ 0, G0 ¼ 0 component of Green’sfunction.

We remark that since BGG0 ¼ BðG� G0Þ, Eq. (17) is aconvolution which according to Faltung’s theorem may beobtained as the product of the characteristic function BðrÞwith the inverse Fourier transform of G0’n�1ðG0Þ. This resultis of great numerical importance: by switching back andforth between real and reciprocal space we may obtainsuccessive Haydock coefficients an and bn through simplemultiplications, without performing any large matrixproducts. We can perform calculations for an arbitrarilyshaped inclusion simply by choosing the correspondingfunction BðrÞ in real space.

Finally, a fast scheme to compute the continued fractionof Eq. (15) follows from the product

pn pn�1

qn qn�1

� �

� u� a0 1

1 0

� �

� u� a1 1

�b21 0

� �� u� an 1

�b2n 0

� �; (20)

from which we obtain

j ¼ eau

limn!1

pnqn

; (21)

where in practice a large but finite n is needed to achieveconvergence of the limit.

3 Results We first compare our results to the previousformalism of Ortiz et al. [18] where a homogenization ofMaxwell’s equations was done without neglecting retar-dation. The retarded results do depend on the relative size ofthe unit cell and the wavelength of the incoming light. InFig. 1 we show the calculated normal incidence reflectivityR[27] from a semi-infinite system made of an isotropic 2D

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Figure 1 (online color at: www.pss-b.com) Reflectance R versusphoton energy for an isotropic 2D square array of cylindricalinclusion (see inset) with eb ¼ 4 and f ¼ 0:7 on a gold host. R ofgold is shown for comparison, see text for details.

Figure 2 (online color at: www.pss-b.com) Rx;y versus photonenergy for a square 2D array of isosceles triangles (see inset) witheb ¼ 4 and various values of f on a gold host. R of gold is shown forcomparison, see text for details.

Figure 3 (online color at: www.pss-b.com) R versus the photonenergy for a square 2D array of 4-point stars (see inset) with eb ¼ 4and various values of f on a gold host. R of gold is shown forcomparison.

square array of cylindrical inclusions with eb ¼ 4 on a goldhost [thus ea ¼ eðAuÞ that is taken from Ref. [28]] with fillingfraction f ¼ 0:7. When l0 � Lwith L the size of the unit cell,R as obtained in Ref. [18] disagrees with our currentcalculation. This is not surprising, as here we have neglectedretardation. However, for l0 >> L the two approaches agreewithin the numerical accuracy [18], as could be expected.Also, we notice that R for the metamaterial is rather differentfrom that of pure gold. Indeed, we see that R for the meta-material is rather low at low frequencies and becomes almostzero at frequencies where gold is opaque and stronglyreflective. We remark that for the chosen filling fraction,cylinders on different unit cells almost touch each other,nearly choking the conducting paths. Thus, the system isdielectric like except at very small frequencies, where anysmall conductance dominates the macroscopic response. Atintermediate frequencies the response of the metamaterialmatches the dielectric constant of vacuum. This behaviororiginates from the local-field effect and is determined by thegeometry of the metamaterial.

We remark that following Ref. [18] requires the solutionof a very large system of equations which took about 3 h ofCPU time using 56 processors in parallel for each of the 300energy points calculated for each spectrum in Fig. 1. Incontrast, the calculation of Haydock’s coefficients made onthe interpreted Perl Data Language (PDL) took about 3 minof a single processor, and they allow the immediatecalculation of the whole spectra shown as well as any otherspectrum for any other choice of materials. Thus, Haydock’smethod makes a huge difference in computing time.

In Figs. 2–4 we showRi (i ¼ x; y) for 2D square arrays ofprisms with assorted sections: isosceles triangles, 4- and5-point stars, with eb ¼ 4. The results are converged by using� 200 an and bn coefficients, and a real space grid of� 400 � 400 points for BðrÞ. The qualitative behavior of Ri

as a function of f is similar for the three geometries. To wit,for low f , Ri is rather similar to that of gold, as one wouldexpect. As f grows toward the percolation threshold, wenotice well-defined low energy minima where Ri deviatesfrom the metallic behavior. As in Fig. 1, their explanation is

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found in the change of behavior, from conducting at lowfrequency to dielectric at high frequencies. For the triangleand 5-point stars we see that the optical response is highlyanisotropic, i.e., Rx 6¼ Ry, since these inclusions are them-selves geometrically anisotropic, whereas for the 4-point starRx ¼ Ry ¼ R. The non-trivial behavior of R occurs atinfrared frequencies for which one would naively expectvery high values for R. This anomalous reflection is due toexcitation of resonances due to particular shape of theinclusions in the periodic array, and as for the case of thecylinders of Fig. 1, it is more apparent as f increases towardthe percolation threshold [29].


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Figure 4 (online color at: www.pss-b.com) Rx;y versus the photonenergy for a square 2D array of 5-point star inclusion (see inset) witheb ¼ 4 and various values of f on a gold host. R of gold is shown forcomparison, see text for details.

In Fig. 5 we finally show the real and imaginary parts ofthe y component of the macroscopic dielectric function eyMfor the 5-point star of Fig. 4 and an isolated 5-point star. Firstwe notice that for f ¼ 0:5 when Re½eyM� ¼ 1 at 1.5 and 1.7 eVwhere Im½eyM � is small, Ry is close to zero as seen in Fig. 4, asone should expect since the macroscopic dielectric functionis almost that of vacuum. However at 1.72 eV where againRe½eyM� ¼ 1, but now Im½eyM � is not small, Ry is close to one.Also, we can see that the Im½eyM � shows high absorption peaks(resonances) where regardless of the value of Re½eyM �, Ry is

Figure 5 (online color at: www.pss-b.com) eyM versus the photonenergy of the 5-point star system of Fig. 4 for f ¼ 0:5 and an isolated5-point star. The horizontal line is at one on the vertical scale.


close to one. For the isolated 5-point star we see that the lineshape of eyM is similar to that for f ¼ 0:5, however theimaginary part is much smaller, meaning less absorption, andmore importantly, the real part is never close to one. This inturn explains why Ry for f ¼ 0:1 is very close to that of puregold. In a sense, an isolated inclusion is similar to a systemwith low filling fraction, since in this case the inclusionswould be far from each other, like if they were isolated. Ofcourse, above analysis could be done for any direction of eMand any given system. Thus one can see that the interactionthrough the local-field effect as the inclusions are closertogether enhances the resonances seen in the Im½eM� andchanges the Re½eM � in such a way that R shows a very richspectral dependence. Also, as we move toward the percola-tion limit, Re½eM� approaches and crosses one, thus giving thehigh-transmittance effect.

Arbitrary shapes in any periodic arrangement can bevery easily investigated, as we only have to specify for eachvalue of rwithin the unit cell the valueBðrÞ ¼ 0; 1 (see insetsof Figs. 2–4). For a given geometry one can also investigatedifferent choices of ea and eb to tailor a desired optical res-ponse, as the computational time is of no concern. Forinstance in Ref. [26], Haydock’s method has been appliedto study 3D systems with different types of inclusions,obtaining anomalous transmission close to one and highlyanisotropic optical transmission in finite width thin filmsmade of cubical, cylindrical, and spherical inclusions thateven interpenetrate each other.

4 Conclusions We have developed a systematic sch-eme to calculate, the complex frequency dependent macro-scopic dielectric function eM of metamaterials in terms of thedielectric functions of the host ea and the inclusions eb, and ofthe geometry of both the unit cell and the inclusions. Startingfrom Maxwell’s equations and employing a long wavelengthapproximation we have implemented the calculation throughHaydock’s recursive method which requires rather minimalcomputing resources to obtain well converged results. Ourformalism may be employed to explore and design a tailoredoptical response. In particular, we showed that extraordinarytransparency of metamaterials is a rather generic phenomenawhenever the conducting phase percolates and the metalsurrounded inclusions display dielectric resonances. Wehope this work motivates the experimental verification of ourresults through the construction and optical characterizationof these systems.

Acknowledgements We acknowledge partial support fromCONACyT 48915-F (BMS), DGAPA-UNAM IN120909 (WLM),and FONCyT PAE-22592/2004 nodo NEA:23016 and nodoCAC:23831 (GPO).

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Phys. Rev. Lett. Submitted.[27] The normal incidence reflectivity is given by the standard

formula Ri ¼ ðffiffiffiffiffiffieiM

p� 1Þ=ð

ffiffiffiffiffiffieiM

pþ 1Þ, with i a principal

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4-point stars and 5-point stars, respectively. The percolationlimit we are referring to is that where the inclusion of a givenunit cell just touches the inclusions of the neighbouring cells.


Optical properties of nanostructured metamaterials

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