Efficient homogenization procedure for the calculation of optical properties of 3D nanostructured...

Efficient homogenization procedure forthe calculation of optical properties of

3D nanostructured composites

W. Luis Mochan,1,4 Guillermo P. Ortiz,2,5 and Bernardo S. Mendoza3,6

1Instituto de Ciencias Fısicas, Universidad Nacional Autonoma de Mexico, Apdo. Postal 48-3,62251 Cuernavaca, Morelos, Mexico

2Departamento de Fısica, Facultad de Ciencias Exactas, Naturales y Agrimensura,Universidad Nacional del Nordeste - Instituto de Modelado e Innovacion Tecnologica,

CONICET-UNNE, Corrientes, Argentina3Department of Photonics, Centro de Investigaciones en Optica, Leon, Guanajuato, Mexico

[email protected]@exa.unne.edu.ar

[email protected]

Abstract: We present a very efficient recursive method to calculate theeffective optical response of metamaterials made up of arbitrarily shapedinclusions arranged in periodic 3D arrays. We apply it to dielectric particlesembedded in a metal matrix with a lattice constant much smaller than thewavelength of the incident field, so that we may neglect retardation andfactor the geometrical properties from the properties of the materials. If theconducting phase is continuous the low frequency behavior is metallic, andif the conducting paths are thin, the high frequency behavior is dielectric.Thus, extraordinary-transparency bands may develop at intermediatefrequencies, whose properties may be tuned by geometrical manipulation.

© 2010 Optical Society of America

OCIS codes: (350.4238) Nanophotonics and photonic crystals; (160.3918) Metamaterials;(160.4236) Nanomaterials; (260.2065) Effective medium theory.

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#131887 - $15.00 USD Received 19 Jul 2010; revised 30 Aug 2010; accepted 3 Sep 2010; published 4 Oct 2010(C) 2010 OSA 11 October 2010 / Vol. 18, No. 21 / OPTICS EXPRESS 22119

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rials,” Phys. Rev. B 79, 245132 (2009).28. R. Haydock, “The recursive solution of the Schrodinger equation,” Solid State Phys. 35, 215 (1980).29. S. L. Alder, “Quantum theory of the dielectric constant in real solids,” Phys. Rev. 126, 413 (1962).30. N. Wiser, “Dielectric constant with local field effects included,” Phys. Rev. 129, 62 (1963).31. W. L. Mochan and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. i. general

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1. Introduction

Metallic films with sub-wavelength nanometric holes may display an extraordinarily largetransmittance at near infrared frequencies for which the metal is opaque and light waves are not


expected to propagate within the holes [1, 2]. The anomalous transmission and other extraor-dinary optical properties of nano-structured metallic films open the possibility of tailored de-sign for many applications that include hyperlens far-field-subdiffraction imaging [3–5], cloak-ing [6], optical antennas [7,8], and circular polarizers [9]. Thus, understanding the electromag-netic properties of this sometimes called plasmonic metamaterials has become important.

Many different approaches have been proposed to describe the extraordinary optical proper-ties of some structured systems [10–26]. Some resonances have been identified with surface-plasmon-polaritons (SPP’s) that may be excited by light after being scattered by the system.

In a recent work [27] the frequency-dependent complex macroscopic dielectric-response ten-sor εM

i j (ω) of 2D-periodic lattices of cylindrical inclusions with arbitrarily shaped cross sectionsembedded within metallic hosts were obtained in the local limit and enhanced transparency wasobtained without invoking explicitly a SPP mechanism. In this paper we develop Haydock’srecursive scheme [28] to obtain the macroscopic dielectric response of 2D and 3D periodicmetamaterials in the long wavelength limit. Our method yields a speed improvement of severalorders of magnitude over that of Ref. [27], which allows previously untractable calculations for3D structures with arbitrary geometry, such as interpenetrated inclusions made out of disper-sive and dissipative components. We show that the geometry of the inclusions and of the latticemight lead to very anisotropic optical behavior and to a very generic enhanced transmittancefor metal-dielectric metamaterials whenever there are only poor conducting paths across thewhole sample.

2. Theory

We consider a periodic lattice of arbitrarily shaped nanometric inclusions (b) embedded withina homogeneous material (a). We assume that each region α = a,b is large enough to have awell defined macroscopic response εα which we assume local and isotropic, but much smallerthan the free wavelength λ0 = 2πc/ω with c the speed of light in vacuum and ω the frequency.The microscopic response is described by

ε(rrr) = εa −B(rrr)εab (1)

where εab ≡ εa − εb and B(rrr) = 0 or 1 is the characteristic function for the b regions, whichwe assume periodic, B(rrr + RRR) = B(rrr), with {RRR} the Bravais lattice of the metamaterial. Theconstitutive equation DDD(rrr) = ε(rrr)EEE(rrr) may be written in reciprocal space as

DDDGGG(qqq) = ∑GGG′

εGGGGGG′EEEGGG′′′(qqq), (2)

where DDD(rrr) is the displacement field and EEE(rrr) the electric field, DDDGGG(qqq) and EEEGGG(qqq) the corre-sponding Fourier coefficients with wavevector qqq+GGG, qqq is the conserved Bloch’s vector and {GGG}the reciprocal lattice. Here, εGGGGGG′′′ is the GGG−GGG′′′ Fourier coefficient of ε(rrr). We consider now alongitudinal external field EEEex(rrr) = −∇φ ex(rrr) and we neglect retardation within the small unitcell, so we may assume that the total electric field is longitudinal

EEEGGG → EEELGGG = GGGGGG ·EEEGGG, (3)

where we simplify our notation denoting (qqq+GGG)/|qqq+GGG| by GGG. We remark that our final resultsapply as well to transverse fields. As ∇×EEEex = ∇×DDDL = 000 and ∇ ·EEEex = ∇ ·DDDL = 4πρex, wemay identify EEEex with DDDL which we chose as a plane wave with wavevector qqq without smalllengthscale fluctuations, DDDL

GGG �=000(qqq) = 0. Substituting Eq. (3) into the longitudinal projection ofEq. (2) allows us to solve for

EEEL000 = qqqη−1

000000 qqq ·DDDL000, (4)


where we first invertηGGGGGG′′′ ≡ GGG · (εGGGGGG′′′GGG′′′) (5)

and afterwards take its 000000 component. The macroscopic longitudinal field EEELM is obtained from

EEEL by eliminating its spatial fluctuations, i.e., EEELM = EEEL

000. Similarly, DDDLM = DDDL

000. Thus, fromEq. (4) we identify the longitudinal projection of the macroscopic dielectric response

εεε−1ML ≡ qqqξ qqq = qqqη−1

000000 qqq, (6)

defined through EEELM = εεε−1

ML ·DDDLM. A more formal derivation of this result, first obtained for

natural crystals [29, 30], may be found in [31].We should emphasize that ξ in Eq. (6) depends in general on the direction of qqq. Calculating

εεε−1ML = qqqqqq ·εεε−1

M · qqqqqq for several propagation directions qqq [32] we may obtain all the componentsof the full inverse long-wavelength dielectric tensor εεε−1

M and from it εεεM . Properties such asreflectance and transmittance may then be calculated using standard formulae [33, 34].

Equation (6) is the starting point for a very efficient numerical calculation of the macroscopicresponse of nanostructured metamaterials, as developed in the next section.

3. Haydock’s recursive method

To calculate efficiently the macroscopic response, we start by taking the Fourier transform ofEq. (1), εGGGGGG′′′ = εaδGGGGGG′′′ − εabBGGGGGG′′′ , where

BGGGGGG′′′ ≡ BGGG−GGG′ = (1/Ω)∫

vd3r ei(GGG−GGG′′′)·rrr (7)

describes the geometry of the inclusions which occupy the volume v within the unit cell ofvolume Ω. In particular, B000000 = v/Ω ≡ f is the filling fraction of the inclusions. From Eq. (5)we obtain

η−1GGGGGG′′′ =

1εab

[uδGGGGGG′ −BLLGGGGGG′ ]−1 =

1εab

GGGGGGG′ , (8)

where GGGGGGG′′′(u) are the matrix elements of a Green’s operator G , i.e., the resolvent of the oper-ator (u−H ), corresponding to a Hamiltonian H with elements

HGGGGGG′′′ ≡ BLLGGGGGG′′′ = GGG · (BGGGGGG′′′GGG′′′), (9)

and the spectral variable u(ω) ≡ (1− εb(ω)/εa(ω))−1 plays the role of energy.From Eqs. (6) and (8) we obtain ξ = G000000(u)/εab = 〈000|G (u)|000〉/εab, where |000〉 represents the

state corresponding to a plane wave with wave vector qqq, and we denote in general the state cor-responding to a plane wave with wave vector qqq + GGG as |GGG〉. This allows the use of Haydock’srecursive scheme [28] to obtain the projected Green’s function and thus the macroscopic re-sponse. We set |−1〉 = 0, |0〉 = |000〉, b0 = 0 and recursively define the states |n〉 through

|n〉 = H |n−1〉 = bn−1|n−2〉+an−1|n−1〉+bn|n〉. (10)

The coefficients an and bn can be recursively obtained by demanding that the states |n〉 benormalized and orthogonal to each other, yielding an−1 = 〈n− 1|n〉 = 〈n− 1|H |n−1〉 andb2

n = 〈n|n〉−a2n−1 −b2

n−1.From Eq. (9) we notice that the action of H includes first of all a multiplication with a

vector (in cartesian space) operator with elements GGGδGGGGGG′ , which is trivially done in reciprocalspace. Afterwards we have to multiply with a scalar operator with elements BGGGGGG′ . However,from Eq. (7), this product is actually a convolution. Thus, a fast Fourier transform allows us to


perform this operator in real space simply as a multiplication with the characteristic functionB(rrr). Finally, we transform back into reciprocal space and apply an inner product with GGGδGGGGGG′ .This way, by going back and forth between real and reciprocal space we can calculate theaction of H performing only trivial multiplications with diagonal matrices. We implementedthe calculation of Haydock’s coefficients using the Perl Data Language [35, 36].

In the basis {|n〉} the Hamiltonian is represented by a symmetric tridiagonal matrix withmain diagonal elements {an} and sub/sup-diagonal elements {bn}. Then, we may write G−1 =u−H recursively in blocks as

G−1n =

(An Bn+1

BTn+1 G−1

n+1

), (11)

with An = (u− an) and Bn = (−bn,0,0, · · ·), and where G ≡ G0. Here we used calligraphicletters to denote any matrix except 1×1 matrices which are equivalent to scalars.

Now we write Gn in blocks as

Gn =(

Rn Pn

Qn Sn

), (12)

where

Rn =1

An −Bn+1G−1n+1B

Tn+1

=1

An −b2n+1Rn+1

, (13)

obtained by inverting Eq. (11). Iterating Eq. (13) backwards we obtain G000000(u) = R0 given re-cursively through the continued fraction

ξ =R0

εab=

uεa

1

u−a0 − b21

u−a1−b22

u−a2−b23

...

. (14)

which may be evaluated very efficiently. Further details of this calculation in the 2D case maybe found in Ref. [37].

Notice that Haydock’s coefficients depend only on the geometry through BLLGGGGGG′′′ . The depen-

dence on composition and frequency is completely encoded in the complex valued spectralvariable u. Thus, for a given geometry we may explore manifold compositions and frequencieswithout recalculating Haydock’s coefficients. Furthermore, our method is equally suited to thecalculation of the properties of systems made up of non-dispersive, transparent materials aswell as dispersive, opaque and dissipative media; we simply have to provide the appropriate(complex) value for u.

4. Results

To test our calculation scheme, in Fig. 1(a) we show the macroscopic response εM calculated fora simple cubic lattice of dielectric spheres, with dielectric function εb = 4, embedded within adielectrically inert host εa = 1 for various filling fractions f . We compare the results of our for-mulation based on Haydock’s recursive scheme (H) with the standard Maxwell-Garnett (MG)and Bruggeman (B) effective medium theories for 3D. As expected, the three theories coincideat low filling fractions, but not surprisingly, they differ as f increases, with our results beingbetween MG which lies below and B which lie above. Bruggeman theory treats both a and bmedia on an equivalent basis, while medium a plays the role of the host and medium b that of


1

1.2

1.4

1.6

1.8

2

2.2

0 0.1 0.2 0.3 0.4 0.5

ε M

Filling fraction f

3D B

MGH

(a)

0 0.1 0.2 0.3 0.4 0.51

1.2

1.4

1.6

1.8

2

2.2

ε M

Filling fraction f

2D

B

MG

H

(b)

Fig. 1. (a) Macroscopic dielectric function εM for a simple cubic array of spherical inclu-sions with response εb = 4 within a host with εa = 1 as a function of filling fraction fcalculated with our Haydock’s recursive method (H), with Maxwell-Garnett’s (MG) andwith Bruggeman’s formulae (B). (b) εM normal to the optical axis of a square array ofsquare prisms with diagonals along the sides of the unit cell as a function of f for thetheories H, MG and B in 2D.

inclusions in both Maxwell-Garnett theory and our’s. Thus the MG results are closer to oursthan B. Similarly, in Fig. 1(b) we show the response εM normal to the optical axis of a systemcomposed of a 2D square array of square prisms whose diagonals are oriented along the con-ventional unit cell sides, i.e., the prisms are rotated 45◦ with respect to the unit cell. As in the3D case, we took εb = 4 and εa = 1. We compare our theory with the 2D MG and B formulae.As in the 3D example, the three formulations coincide in the dilute f → 0 limit. For larger f ,MG yields the lowest εM and B the largest. At the percolation limit f = 0.5, at which the edgesof nearby prisms touch each other, the geometry does not distinguish between inclusions andhost. At this limit, our results coincide with the symmetric Bruggeman 2D formula. Thus, inthis particular 2D case, B lies closer to our results than MG. Furthermore, due to the symmetryof this system at f = 0.5, Keller’s theorem [38–40] implies that εM =

√εaεb, which is exactly

fulfiled by both B and H theories.In a recent paper [27] an alternative, essentially exact but much more numerically expen-

sive method to calculate the macroscopic response of metamaterials was developed and it wasapplied to manifold 2D systems. The results were satisfactorily tested against previous calcu-lations and validated through the fulfilment of Keller’s reciprocal theorem [38–40]. We haverepeated all the calculations in that paper and have verified that our results agree in the longwavelength limit and are about four orders of magnitude faster. The speed enhancement is muchlarger in 3D. Further 2D calculations with the present formulation were presented in Ref. [37].

In Fig. 2(a) we show the normal-incidence reflectance of an isotropic semi-infinite metama-terial consisting of a simple cubic lattice of small cubical voids embedded with filling fractionf = 0.6 within a Drude conductor with plasma frequency ωp and a low dissipation parame-ter Γ = 0.01ωp. The low frequency reflectance is almost unity but attains a deep minimum atω ≈ 0.51ωp and becomes very small for 0.74 < ω/ωp < 0.91, it attains a peak at 0.97ωp and


0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1 1.2

R,T

ω/ωp

(a)

R lowT lowT highTef lowTef high

0.2 0.4 0.6 0.8 1 1.2-1

0

1

2

ε M

ω/ωp

(b)

Re[εM

]Im[ε

M]

Fig. 2. (a) Normal-incidence reflectance R of a semi-infinite metamaterial and transmittanceT of a d = 200 nm film made of a model Drude conductor with low (Γ = 0.01ωp) andhigh (Γ = 0.1ωp) dissipation parameters with a simple cubic lattice of cubical cavities offilling fraction f = 0.6, as a function of the frequency ω , together with the correspondingtransmittance Teff of an homogeneous Drude film with an effective width deff = (1− f )d.(b) Macroscopic dielectric response of the low dissipation metamaterial. The vertical linesindicate the frequencies at which εM ≈ 1.

finally decreases at high frequencies. This behavior may be understood by looking at Fig. 2(b),which shows the corresponding macroscopic dielectric function. We remark that at low frequen-cies εM is negative and large corresponding to a metallic behavior, as the interstitial conductingphase percolates. Nevertheless, at larger frequencies there is a series of resonances originatedfrom the coupled plasmon modes of the cubic inclusions [41]. Thus, the macroscopic responsebecomes dielectric like and approaches at some intermediate frequencies the vacuum valueεV = 1. In the figure we have indicated the frequencies ω ≈ 0.80ωp and 0.86ωp for whichRe[εM] = 1 and the metamaterial would become perfectly transparent were it not for its smallIm[εM]. Similarly, at ω ≈ 0.52ωp there is a resonance below which Re[εM] approaches 1, cor-responding to a sharp minimum in the reflectance followed by a sharp maximum due to reso-nant absorption. Above the last resonance, at 0.94ωp, there is a narrow frequency band whereRe[εM] < 0, corresponding to a maximum in R. Finally, asymptotically [εM] → 1 and R → 0.

The figure also shows the transmittance T of a d = 200 nm wide film made of the same meta-material. We notice that below ωp it is almost null except for a band where it takes appreciablevalues and which corresponds to the frequencies where the reflectance R of the correspondingsemi-infinite system becomes small. The interpretation of the optical properties of a film isencumbered by the interference between the multiply reflected waves and their decay as theycross the film. Thus, T shows a very rich structure originated from resonant oscillations in theimaginary part of k = (ω/c)

√εM below ωp and Fabry-Perot resonances above ωp. Increasing

dissipation diminishes both types of oscillations, as illustrated in the figure for Γ = 0.1ωp with-out eliminating the transmission band. Finally, as a reference we show the transmittance of lowand high dissipation homogeneous Drude films of effective width deff = (1− f )d = 80 nm, sothat it contains the same amount of metal as the metamaterial. Notice that for both cases thetransmittance of the metamaterial can be enhanced by about two orders of magnitude abovethat of the effective film.

In Fig. 3(a) we show the transmittance T of a film made of a simple cubic lattice of spher-ical dielectric inclusions with response εb = 4 within an Au host [42]. We chose the radius asr = 0.6a, with a the lattice parameter, so the spheres overlap their neighbors. We have normal-ized the results to the transmittance Teff of an effective homogeneous Au film of width deff, inorder to emphasize the transmittance enhancement due to the metamaterial geometry. Severalenhancement peaks between one and two orders of magnitude are visible in the transmittance


0

10

20

30

40

50

60

0 0.5 1 1.5 2 2.5

2480 1240 826 620 496

T/T

ef

photon-energy hω (eV)

λ0 (nm)

(a)T = Tx = Ty

1 1.2 1.4 1.6 1.8 2 2.2 2.40

50

100

150

2001240 1033 885 775 689 620 563 516

Tα/T

ef

photon-energy hω (eV)

λ0 (nm)

(b)Ty

10 × Tx

Fig. 3. Normal-incidence transmittance Tα for α = x,y polarization vs. frequency ω for200 nm Au films with faces normal to the z axis with an embedded lattice of dielectricinclusions, normalized to the transmittance Teff of a homogeneous Au film with the sameamount of metal. (a) Simple cubic lattice of spheres of radius r = 0.6a with a the latticeparameter with εb = 4. (b) Simple orthorhombic lattice of z-oriented cylinders with radiusr = 0.53ax, height h = 0.9az and dielectric response εb = 2 with lattice parameters ax = az

and ay = 1.15ax.

spectrum, corresponding to the excitation of coupled multipolar plasmon resonances within theregion where the metal is opaque.

In Fig. 3(b) we show the normalized transmittance Tα/Teff (α = x,y) for plane polarized lightnormally incident on a 200 nm film lying on the xy plane made of a simple orthorhombic latticeof z-oriented dielectric cylinders with radius r = 0.53ax, height h = 0.9az dielectric functionεb = 2 and lattice parameters ay = 1.15ax and az = ax within an Au host. There is a hugeanisotropy, with a peak enhancement of Ty almost two orders of magnitude larger than that ofTx. For this geometry there is an overlap between neighbor cylinders along x, so the system is abetter low frequency conductor along y.

We should discuss some limitations of our theory. Our assumption of periodicity is not toorestrictive, as the efficiency of our scheme permits calculations for complex unit cells withmany inclusions. Our use of the long-wavelength approximation does impose a bound on thesize a of the unit cell. Retardation corrections are expected to be of order (a/λ0)2 and wehave verified through comparisons to exact calculations in 2D [27] that for a ≤ λ0/5 they maybe considered negligible for many purposes. The interesting regions in Figs. 3(a) and 3(b) liebelow ∼ 2 eV, where our macroscopic response would be accurate up to a ∼ 100 nm, a scaleeasily attainable with current fabrication techniques. Nevertheless, our theory does neglect thepresence of transition layers of width dt ∼ a close to the metamaterial boundaries which arenot well described by a local, position independent macroscopic response εM , as inclusionscloser to the surface than their size would be sliced by the surface, and the effect of the surfaceon the microscopic fluctuations of the field extendes a distance ∼ a. Similar transition regionsare present at ordinary crystalline and spatially dispersive materials and are known to producecorrections to the optical properties of semi-infinite media of the order of dt/λ0 [43]. For a free-standing subwavelength thin film of width d , the corresponding corrections can become of theorder of dt/d. In our case, both corrections would be of only a few percent if the inclusionswere of the order of a few nanometers.

In summary, we developed a formalism that allows very efficient calculations of the macro-scopic response of 3D nano-structured periodic metamaterials. We applied it to films madeof various lattices of dielectric inclusions with assorted shapes within opaque metallic hostsand found an extraordinary enhancement of the transmittance as a generic property whenever


the metamaterial is conducting at low frequencies and has dielectric like resonances at largerfrequencies.

Acknowledgment

This work was supported DGAPA-UNAM (IN120909 (WLM)), CONACyT (48915-F (BMS)and by ANPCyT-UNNE (204 y 190-PICTO-UNNE-2007 (GPO)).


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