Abstract

We present a very efficient recursive method to calculate the effective optical response of metamaterials made up of arbitrarily shaped inclusions arranged in periodic 3D arrays. We apply it to dielectric particles embedded in a metal matrix with a lattice constant much smaller than the wavelength of the incident field, so that we may neglect retardation and factor the geometrical properties from the properties of the materials. If the conducting phase is continuous the low frequency behavior is metallic, and if the conducting paths are thin, the high frequency behavior is dielectric. Thus, extraordinary-transparency bands may develop at intermediate frequencies, whose properties may be tuned by geometrical manipulation.

© 2010 Optical Society of America

1. Introduction

Metallic films with sub-wavelength nanometric holes may display an extraordinarily large transmittance 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 extraordinary optical properties of nano-structured metallic films open the possibility of tailored design for many applications that include hyperlens far-field-subdiffraction imaging [35], cloaking [6], optical antennas [7, 8], and circular polarizers [9]. Thus, understanding the electromagnetic properties of this sometimes called plasmonic metamaterials has become important.

Many different approaches have been proposed to describe the extraordinary optical properties of some structured systems [1026]. 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 tensor ɛijM(ω) of 2D-periodic lattices of cylindrical inclusions with arbitrarily shaped cross sections embedded within metallic hosts were obtained in the local limit and enhanced transparency was obtained without invoking explicitly a SPP mechanism. In this paper we develop Haydock’s recursive scheme [28] to obtain the macroscopic dielectric response of 2D and 3D periodic metamaterials in the long wavelength limit. Our method yields a speed improvement of several orders of magnitude over that of Ref. [27], which allows previously untractable calculations for 3D structures with arbitrary geometry, such as interpenetrated inclusions made out of dispersive and dissipative components. We show that the geometry of the inclusions and of the lattice might lead to very anisotropic optical behavior and to a very generic enhanced transmittance for metal-dielectric metamaterials whenever there are only poor conducting paths across the whole sample.

2. Theory

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

ɛ(r)=ɛaB(r)ɛab
where ɛabɛaɛb and B(r) = 0 or 1 is the characteristic function for the b regions, which we assume periodic, B(r + R) = B(r), with {R} the Bravais lattice of the metamaterial. The constitutive equation D(r) = ɛ(r)E(r) may be written in reciprocal space as
DG(q)=GɛGGEG(q),
where D(r) is the displacement field and E(r) the electric field, DG(q) and EG(q) the corresponding Fourier coefficients with wavevector q + G, q is the conserved Bloch’s vector and {G} the reciprocal lattice. Here, ɛGG′ is the GG′ Fourier coefficient of ɛ(r). We consider now a longitudinal external field Eex(r) = −∇ϕex(r) and we neglect retardation within the small unit cell, so we may assume that the total electric field is longitudinal
EGEGL=G^G^EG,
where we simplify our notation denoting (q + G)/|q + G| by Ĝ. We remark that our final results apply as well to transverse fields. As ∇ × Eex = ∇ × DL = 0 and ∇ · Eex = ∇ · DL = 4πρex, we may identify Eex with DL which we chose as a plane wave with wavevector q without small lengthscale fluctuations, DG0L(q)=0. Substituting Eq. (3) into the longitudinal projection of Eq. (2) allows us to solve for
E0L=q^η001q^D0L,
where we first invert
ηGGG^.(ɛGGG^)
and afterwards take its 00 component. The macroscopic longitudinal field EML is obtained from EL by eliminating its spatial fluctuations, i.e., EML=E0L. Similarly, DML=D0L. Thus, from Eq. (4) we identify the longitudinal projection of the macroscopic dielectric response
ɛML1q^ξq^=q^η001q^,
defined through EML=ɛML1DML. 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 q. Calculating ɛML1=q^q^ɛM1q^q^ for several propagation directions [32] we may obtain all the components of the full inverse long-wavelength dielectric tensor ɛM1 and from it ɛM. Properties such as reflectance 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 macroscopic response 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 of Eq. (1), ɛGG′ = ɛaδGG′ɛabBGG′, where

BGGBGG=(1/Ω)vd3rei(GG)r
describes the geometry of the inclusions which occupy the volume v within the unit cell of volume Ω. In particular, B00 = v/Ω ≡ f is the filling fraction of the inclusions. From Eq. (5) we obtain
ηGG1=1ɛab[uδGGBGGLL]1=1ɛab𝒢GG,
where 𝒢GG′ (u) are the matrix elements of a Green’s operator 𝒢̂, i.e., the resolvent of the operator (uℋ̂), corresponding to a Hamiltonian ℋ̂ with elements
GGBGGLL=G^(BGGG^),
and the spectral variable u(ω) ≡ (1 – ɛb(ω)/ɛa(ω))−1 plays the role of energy.

From Eqs. (6) and (8) we obtain ξ = 𝒢00(u)/ɛab = 〈0|𝒢̂(u)|0〉/ɛab, where |0〉 represents the state corresponding to a plane wave with wave vector q, and we denote in general the state corresponding to a plane wave with wave vector q + G as |G〉. This allows the use of Haydock’s recursive scheme [28] to obtain the projected Green’s function and thus the macroscopic response. We set |−1〉 = 0, |0〉 = |0〉, b0 = 0 and recursively define the states |n〉 through

|n˜=^|n1=bn1|n2+an1|n1+bn|n.
The coefficients an and bn can be recursively obtained by demanding that the states |n〉 be normalized and orthogonal to each other, yielding an−1 = 〈n – 1|ñ〉 = 〈n – 1|ℋ̂|n – 1〉 and bn2=n˜|n˜an12bn12.

From Eq. (9) we notice that the action of ℋ̂ includes first of all a multiplication with a vector (in cartesian space) operator with elements ĜδGG′, which is trivially done in reciprocal space. Afterwards we have to multiply with a scalar operator with elements BGG′. 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 function B(r). Finally, we transform back into reciprocal space and apply an inner product with ĜδGG′. This way, by going back and forth between real and reciprocal space we can calculate the action of ℋ̂ performing only trivial multiplications with diagonal matrices. We implemented the 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 with main diagonal elements {an} and sub/sup-diagonal elements {bn}. Then, we may write 𝒢̂–1 = uℋ̂ recursively in blocks as

𝒢n1=(Ann+1T|n+1𝒢n+11),
with An = (uan) and n = (−bn, 0, 0, ⋯), and where 𝒢̂𝒢̂0. Here we used calligraphic letters to denote any matrix except 1 × 1 matrices which are equivalent to scalars.

Now we write 𝒢n in blocks as

𝒢n=(Rn𝒬n|𝒫n𝒮n),
where
Rn=1Ann+1𝒢n+11n+1T=1Anbn+12Rn+1,
obtained by inverting Eq. (11). Iterating Eq. (13) backwards we obtain 𝒢00(u) = R0 given recursively through the continued fraction
ξ=R0ɛab=uɛa1ua0b12ua1b22ua2b32.
which may be evaluated very efficiently. Further details of this calculation in the 2D case may be found in Ref. [37].

Notice that Haydock’s coefficients depend only on the geometry through BGGL. The dependence on composition and frequency is completely encoded in the complex valued spectral variable u. Thus, for a given geometry we may explore manifold compositions and frequencies without recalculating Haydock’s coefficients. Furthermore, our method is equally suited to the calculation of the properties of systems made up of non-dispersive, transparent materials as well 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 ɛM calculated for a simple cubic lattice of dielectric spheres, with dielectric function ɛb = 4, embedded within a dielectrically inert host ɛa = 1 for various filling fractions f. We compare the results of our formulation 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 coincide at low filling fractions, but not surprisingly, they differ as f increases, with our results being between MG which lies below and B which lie above. Bruggeman theory treats both a and b media on an equivalent basis, while medium a plays the role of the host and medium b that of inclusions in both Maxwell-Garnett theory and our’s. Thus the MG results are closer to ours than B. Similarly, in Fig. 1(b) we show the response ɛM normal to the optical axis of a system composed of a 2D square array of square prisms whose diagonals are oriented along the conventional unit cell sides, i.e., the prisms are rotated 45° with respect to the unit cell. As in the 3D case, we took ɛb = 4 and ɛa = 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 ɛM and B the largest. At the percolation limit f = 0.5, at which the edges of nearby prisms touch each other, the geometry does not distinguish between inclusions and host. At this limit, our results coincide with the symmetric Bruggeman 2D formula. Thus, in this particular 2D case, B lies closer to our results than MG. Furthermore, due to the symmetry of this system at f = 0.5, Keller’s theorem [3840] implies that ɛM=ɛaɛb, which is exactly fulfiled by both B and H theories.

 

Fig. 1 (a) Macroscopic dielectric function ɛM for a simple cubic array of spherical inclusions with response ɛb = 4 within a host with ɛa = 1 as a function of filling fraction f calculated with our Haydock’s recursive method (H), with Maxwell-Garnett’s (MG) and with Bruggeman’s formulae (B). (b) ɛM normal to the optical axis of a square array of square prisms with diagonals along the sides of the unit cell as a function of f for the theories H, MG and B in 2D.

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In a recent paper [27] an alternative, essentially exact but much more numerically expensive method to calculate the macroscopic response of metamaterials was developed and it was applied to manifold 2D systems. The results were satisfactorily tested against previous calculations and validated through the fulfilment of Keller’s reciprocal theorem [3840]. We have repeated all the calculations in that paper and have verified that our results agree in the long wavelength limit and are about four orders of magnitude faster. The speed enhancement is much larger 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 metamaterial consisting of a simple cubic lattice of small cubical voids embedded with filling fraction f = 0.6 within a Drude conductor with plasma frequency ωp and a low dissipation parameter Γ = 0.01ωp. The low frequency reflectance is almost unity but attains a deep minimum at ω ≈ 0.51ωp and becomes very small for 0.74 < ω/ωp < 0.91, it attains a peak at 0.97ωp and 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 frequencies ɛM is negative and large corresponding to a metallic behavior, as the interstitial conducting phase percolates. Nevertheless, at larger frequencies there is a series of resonances originated from the coupled plasmon modes of the cubic inclusions [41]. Thus, the macroscopic response becomes dielectric like and approaches at some intermediate frequencies the vacuum value ɛV = 1. In the figure we have indicated the frequencies ω ≈ 0.80ωp and 0.86ωp for which Re[ɛM] = 1 and the metamaterial would become perfectly transparent were it not for its small Im[ɛM]. Similarly, at ω ≈ 0.52ωp there is a resonance below which Re[ɛM] approaches 1, corresponding to a sharp minimum in the reflectance followed by a sharp maximum due to resonant absorption. Above the last resonance, at 0.94ωp, there is a narrow frequency band where Re[ɛM] < 0, corresponding to a maximum in R. Finally, asymptotically [ɛM] → 1 and R → 0.

 

Fig. 2 (a) Normal-incidence reflectance R of a semi-infinite metamaterial and transmittance T of a d = 200 nm film made of a model Drude conductor with low (Γ = 0.01ωp) and high (Γ = 0.1ωp) dissipation parameters with a simple cubic lattice of cubical cavities of filling fraction f = 0.6, as a function of the frequency ω, together with the corresponding transmittance 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 lines indicate the frequencies at which ɛM ≈ 1.

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The figure also shows the transmittance T of a d = 200 nm wide film made of the same metamaterial. We notice that below ωp it is almost null except for a band where it takes appreciable values and which corresponds to the frequencies where the reflectance R of the corresponding semi-infinite system becomes small. The interpretation of the optical properties of a film is encumbered by the interference between the multiply reflected waves and their decay as they cross the film. Thus, T shows a very rich structure originated from resonant oscillations in the imaginary part of k=(ω/c)ɛM below ωp and Fabry-Perot resonances above ωp. Increasing dissipation diminishes both types of oscillations, as illustrated in the figure for Γ = 0.1ωp without eliminating the transmission band. Finally, as a reference we show the transmittance of low and high dissipation homogeneous Drude films of effective width deff = (1 – f)d = 80 nm, so that it contains the same amount of metal as the metamaterial. Notice that for both cases the transmittance of the metamaterial can be enhanced by about two orders of magnitude above that of the effective film.

In Fig. 3(a) we show the transmittance T of a film made of a simple cubic lattice of spherical dielectric inclusions with response ɛb = 4 within an Au host [42]. We chose the radius as r = 0.6a, with a the lattice parameter, so the spheres overlap their neighbors. We have normalized the results to the transmittance Teff of an effective homogeneous Au film of width deff, in order to emphasize the transmittance enhancement due to the metamaterial geometry. Several enhancement peaks between one and two orders of magnitude are visible in the transmittance spectrum, corresponding to the excitation of coupled multipolar plasmon resonances within the region where the metal is opaque.

 

Fig. 3 Normal-incidence transmittance Tα for α = x, y polarization vs. frequency ω for 200 nm Au films with faces normal to the z axis with an embedded lattice of dielectric inclusions, normalized to the transmittance Teff of a homogeneous Au film with the same amount of metal. (a) Simple cubic lattice of spheres of radius r = 0.6a with a the lattice parameter with ɛb = 4. (b) Simple orthorhombic lattice of z-oriented cylinders with radius r = 0.53ax, height h = 0.9az and dielectric response ɛb = 2 with lattice parameters ax = az and ay = 1.15ax.

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

We should discuss some limitations of our theory. Our assumption of periodicity is not too restrictive, as the efficiency of our scheme permits calculations for complex unit cells with many inclusions. Our use of the long-wavelength approximation does impose a bound on the size a of the unit cell. Retardation corrections are expected to be of order (a/λ0)2 and we have verified through comparisons to exact calculations in 2D [27] that for aλ0/5 they may be considered negligible for many purposes. The interesting regions in Figs. 3(a) and 3(b) lie below ∼ 2 eV, where our macroscopic response would be accurate up to a ∼ 100 nm, a scale easily attainable with current fabrication techniques. Nevertheless, our theory does neglect the presence of transition layers of width dta close to the metamaterial boundaries which are not well described by a local, position independent macroscopic response ɛM, as inclusions closer to the surface than their size would be sliced by the surface, and the effect of the surface on the microscopic fluctuations of the field extendes a distance ∼ a. Similar transition regions are present at ordinary crystalline and spatially dispersive materials and are known to produce corrections 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 the order of dt/d. In our case, both corrections would be of only a few percent if the inclusions were of the order of a few nanometers.

In summary, we developed a formalism that allows very efficient calculations of the macroscopic response of 3D nano-structured periodic metamaterials. We applied it to films made of various lattices of dielectric inclusions with assorted shapes within opaque metallic hosts and 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 larger frequencies.

Acknowledgments

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

References and links

1. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).

2. J. Weiner, “The physics of light transmission through subwavelength apertures and aperture arrays,” Rep. Prog. Phys. 72, 0644011 (2009).

3. L. Chen and G. P. Wang, “Pyramid-shaped hyperlenses for three-dimensional subdiffraction optical imaging,” Opt. Express 17, 3903 (2009).

4. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).

5. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystal: theory and simulations,” Phys. Rev. B 74, 075103 (2006).

6. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 1111051 (2007).

7. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 2668021 (2007).

8. W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).

9. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

10. B. Hou, H. Wen, Y. Leng, and W. Wen, “Enhanced transmission of electromagnetic waves through metamaterials,” Appl. Phys. A 87, 217 (2007).

11. L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 2439051 (2005).

12. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).

13. P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to difraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907 (1982).

14. H. Lochbihler and R. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32, 3459 (1993).

15. H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795 (1994).

16. H. Ghaemi, T. Thio, D. Grupp, T. Ebbesen, and H. Lezec, “Surface plasmon enhanced optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).

17. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

18. D. Grupp, H. Lezec, T. Ebbesen, K. Pellerin, and T. Thio, “Surface plasmon enhanced optical transmission through subwavelength holes,” Appl. Phys. Lett 77, 1569 (2000).

19. S. Darmanyan and A. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003).

20. J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).

21. Q.-H. Park, J. H. Kang, J. W. Lee, and D. S. Kim, “Effective medium description of plasmonic metamaterials,” Opt. Express 15, 6994 (2007).

22. A. Agrawal, Z. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express 16, 9601 (2008).

23. Q. Cao and P. Lalanne, “Negative role of surface plasmon in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031 (2002).

24. M. Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606 (1999).

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27. G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and W. Mochán, “Effective dielectric response of metamaterials,” Phys. Rev. B 79, 245132 (2009).

28. R. Haydock, “The recursive solution of the Schrödinger 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).

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31. W. L. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. i. general formalism,” Phys. Rev. B 32, 4984 (1985).

32. That the long-wavelength response ɛM is independent of the direction of q → 0 allows us to use the results of longitudinal calculations to solve optical (i.e., transverse) problems.

33. R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).

34. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7th ed.

35. K. Glazebrook, J. Brinchmann, J. Cerney, C. DeForest, D. Hunt, T. Jenness, T. Luka, R. Schwebel, and C. Soeller, “The perl data language v.2.4.4,” Available from http://pdl.perl.org.

36. K. Glazebrook and F. Economou, “Pdl: The perl data language,” Dr. Dobb’s Journal (1997). http://www.ddj.com/184410442.

37. E. Cortés, W. L. Mochán, B. S. Mendoza, and G. P. Ortiz, “Optical properties of nano-structured metamaterials,” Phys. Status Solidi B 247, 2102 (2010).

38. J. B. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” J. Appl. Phys. 34, 991 (1963).

39. J. B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548 (1964).

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41. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732 (1975).

42. P. Johnson and R. Christy, “Optical constant of noble metals,” Phys. Rev. B 6, 4370 (1972).

43. W. L. Mochán and R. G. Barrera, “Intrinsic surface-induced optical anisotropies of cubic crystals: local-field effect,” Phys. Rev. Lett. 55, 1192 (1985).

References

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  1. L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).
  2. J. Weiner, “The physics of light transmission through subwavelength apertures and aperture arrays,” Rep. Prog. Phys. 72, 0644011 (2009).
  3. L. Chen and G. P. Wang, “Pyramid-shaped hyperlenses for three-dimensional subdiffraction optical imaging,” Opt. Express 17, 3903 (2009).
  4. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).
  5. A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystal: theory and simulations,” Phys. Rev. B 74, 075103 (2006).
  6. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 1111051 (2007).
  7. L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 2668021 (2007).
  8. W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).
  9. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).
  10. B. Hou, H. Wen, Y. Leng, and W. Wen, “Enhanced transmission of electromagnetic waves through metamaterials,” Appl. Phys. A 87, 217 (2007).
  11. L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 2439051 (2005).
  12. J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
  13. P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to difraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907 (1982).
  14. H. Lochbihler and R. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32, 3459 (1993).
  15. H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795 (1994).
  16. H. Ghaemi, T. Thio, D. Grupp, T. Ebbesen, and H. Lezec, “Surface plasmon enhanced optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).
  17. L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).
  18. D. Grupp, H. Lezec, T. Ebbesen, K. Pellerin, and T. Thio, “Surface plasmon enhanced optical transmission through subwavelength holes,” Appl. Phys. Lett 77, 1569 (2000).
  19. S. Darmanyan and A. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003).
  20. J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
  21. Q.-H. Park, J. H. Kang, J. W. Lee, and D. S. Kim, “Effective medium description of plasmonic metamaterials,” Opt. Express 15, 6994 (2007).
  22. A. Agrawal, Z. Vardeny, and A. Nahata, “Engineering the dielectric function of plasmonic lattices,” Opt. Express 16, 9601 (2008).
  23. Q. Cao and P. Lalanne, “Negative role of surface plasmon in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031 (2002).
  24. M. Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606 (1999).
  25. M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66, 1951051 (2002).
  26. E. Popov, M. Neviére, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).
  27. G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and W. Mochán, “Effective dielectric response of metamaterials,” Phys. Rev. B 79, 245132 (2009).
  28. R. Haydock, “The recursive solution of the Schrödinger 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. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. i. general formalism,” Phys. Rev. B 32, 4984 (1985).
  32. That the long-wavelength response ɛM is independent of the direction of q → 0 allows us to use the results of longitudinal calculations to solve optical (i.e., transverse) problems.
  33. R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).
  34. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7th ed.
  35. K. Glazebrook, J. Brinchmann, J. Cerney, C. DeForest, D. Hunt, T. Jenness, T. Luka, R. Schwebel, and C. Soeller, “The perl data language v.2.4.4,” Available from http://pdl.perl.org .
  36. K. Glazebrook and F. Economou, “Pdl: The perl data language,” Dr. Dobb’s Journal (1997). http://www.ddj.com/184410442 .
  37. E. Cortés, W. L. Mochán, B. S. Mendoza, and G. P. Ortiz, “Optical properties of nano-structured metamaterials,” Phys. Status Solidi B 247, 2102 (2010).
  38. J. B. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” J. Appl. Phys. 34, 991 (1963).
  39. J. B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548 (1964).
  40. J. Nevard and J. B. Keller, “Reciprocal relations for effective conductivities of anisotropic media,” J. Math. Phys. 26, 2761 (1985).
  41. R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732 (1975).
  42. P. Johnson and R. Christy, “Optical constant of noble metals,” Phys. Rev. B 6, 4370 (1972).
  43. W. L. Mochán and R. G. Barrera, “Intrinsic surface-induced optical anisotropies of cubic crystals: local-field effect,” Phys. Rev. Lett. 55, 1192 (1985).

2010 (1)

E. Cortés, W. L. Mochán, B. S. Mendoza, and G. P. Ortiz, “Optical properties of nano-structured metamaterials,” Phys. Status Solidi B 247, 2102 (2010).

2009 (3)

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and W. Mochán, “Effective dielectric response of metamaterials,” Phys. Rev. B 79, 245132 (2009).

J. Weiner, “The physics of light transmission through subwavelength apertures and aperture arrays,” Rep. Prog. Phys. 72, 0644011 (2009).

L. Chen and G. P. Wang, “Pyramid-shaped hyperlenses for three-dimensional subdiffraction optical imaging,” Opt. Express 17, 3903 (2009).

2008 (1)

2007 (6)

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 1111051 (2007).

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 2668021 (2007).

W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).

B. Hou, H. Wen, Y. Leng, and W. Wen, “Enhanced transmission of electromagnetic waves through metamaterials,” Appl. Phys. A 87, 217 (2007).

Q.-H. Park, J. H. Kang, J. W. Lee, and D. S. Kim, “Effective medium description of plasmonic metamaterials,” Opt. Express 15, 6994 (2007).

2006 (1)

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystal: theory and simulations,” Phys. Rev. B 74, 075103 (2006).

2005 (1)

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 2439051 (2005).

2004 (1)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).

2003 (1)

S. Darmanyan and A. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003).

2002 (1)

Q. Cao and P. Lalanne, “Negative role of surface plasmon in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031 (2002).

2001 (1)

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

2000 (2)

D. Grupp, H. Lezec, T. Ebbesen, K. Pellerin, and T. Thio, “Surface plasmon enhanced optical transmission through subwavelength holes,” Appl. Phys. Lett 77, 1569 (2000).

E. Popov, M. Neviére, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).

1999 (2)

M. Treacy, “Dynamical diffraction in metallic optical gratings,” Appl. Phys. Lett. 75, 606 (1999).

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).

1998 (1)

H. Ghaemi, T. Thio, D. Grupp, T. Ebbesen, and H. Lezec, “Surface plasmon enhanced optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).

1994 (1)

H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795 (1994).

1993 (1)

1985 (3)

W. L. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. i. general formalism,” Phys. Rev. B 32, 4984 (1985).

J. Nevard and J. B. Keller, “Reciprocal relations for effective conductivities of anisotropic media,” J. Math. Phys. 26, 2761 (1985).

W. L. Mochán and R. G. Barrera, “Intrinsic surface-induced optical anisotropies of cubic crystals: local-field effect,” Phys. Rev. Lett. 55, 1192 (1985).

1982 (1)

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to difraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907 (1982).

1980 (1)

R. Haydock, “The recursive solution of the Schrödinger equation,” Solid State Phys. 35, 215 (1980).

1975 (1)

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732 (1975).

1972 (1)

P. Johnson and R. Christy, “Optical constant of noble metals,” Phys. Rev. B 6, 4370 (1972).

1964 (1)

J. B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548 (1964).

1963 (2)

J. B. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” J. Appl. Phys. 34, 991 (1963).

N. Wiser, “Dielectric constant with local field effects included,” Phys. Rev. 129, 62 (1963).

1962 (1)

S. L. Alder, “Quantum theory of the dielectric constant in real solids,” Phys. Rev. 126, 413 (1962).

1951 (1)

M. Treacy, “Dynamical diffraction explanation of the anomalous transmission of light through metallic gratings,” Phys. Rev. B 66, 1951051 (2002).

1842 (1)

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).

Agrawal, A.

Alder, S. L.

S. L. Alder, “Quantum theory of the dielectric constant in real solids,” Phys. Rev. 126, 413 (1962).

Atkinson, R.

W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).

Bade, K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

Barrera, R. G.

W. L. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. i. general formalism,” Phys. Rev. B 32, 4984 (1985).

W. L. Mochán and R. G. Barrera, “Intrinsic surface-induced optical anisotropies of cubic crystals: local-field effect,” Phys. Rev. Lett. 55, 1192 (1985).

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7th ed.

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 1111051 (2007).

Cao, Q.

Q. Cao and P. Lalanne, “Negative role of surface plasmon in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031 (2002).

Chan, C. T.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 2439051 (2005).

Chen, L.

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 1111051 (2007).

Christy, R.

P. Johnson and R. Christy, “Optical constant of noble metals,” Phys. Rev. B 6, 4370 (1972).

Cortés, E.

E. Cortés, W. L. Mochán, B. S. Mendoza, and G. P. Ortiz, “Optical properties of nano-structured metamaterials,” Phys. Status Solidi B 247, 2102 (2010).

Darmanyan, S.

S. Darmanyan and A. Zayats, “Light tunneling via resonant surface plasmon polariton states and the enhanced transmission of periodically nanostructured metal films: an analytical study,” Phys. Rev. B 67, 035424 (2003).

Decker, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

Depine, R.

Dickson, W.

W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).

Ebbesen, T.

D. Grupp, H. Lezec, T. Ebbesen, K. Pellerin, and T. Thio, “Surface plasmon enhanced optical transmission through subwavelength holes,” Appl. Phys. Lett 77, 1569 (2000).

H. Ghaemi, T. Thio, D. Grupp, T. Ebbesen, and H. Lezec, “Surface plasmon enhanced optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).

Ebbesen, T. W.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

Engheta, N.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystal: theory and simulations,” Phys. Rev. B 74, 075103 (2006).

Enoch, S.

E. Popov, M. Neviére, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).

Evans, P.

W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).

Fuchs, R.

R. Fuchs, “Theory of the optical properties of ionic crystal cubes,” Phys. Rev. B 11, 1732 (1975).

Gansel, J. K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

García-Valenzuela, A.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).

García-Vidal, F.

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).

García-Vidal, F. J.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

Ghaemi, H.

H. Ghaemi, T. Thio, D. Grupp, T. Ebbesen, and H. Lezec, “Surface plasmon enhanced optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).

Grupp, D.

D. Grupp, H. Lezec, T. Ebbesen, K. Pellerin, and T. Thio, “Surface plasmon enhanced optical transmission through subwavelength holes,” Appl. Phys. Lett 77, 1569 (2000).

H. Ghaemi, T. Thio, D. Grupp, T. Ebbesen, and H. Lezec, “Surface plasmon enhanced optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).

Haydock, R.

R. Haydock, “The recursive solution of the Schrödinger equation,” Solid State Phys. 35, 215 (1980).

Hecht, B.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).

Hou, B.

B. Hou, H. Wen, Y. Leng, and W. Wen, “Enhanced transmission of electromagnetic waves through metamaterials,” Appl. Phys. A 87, 217 (2007).

Johnson, P.

P. Johnson and R. Christy, “Optical constant of noble metals,” Phys. Rev. B 6, 4370 (1972).

Kang, J. H.

Keller, J. B.

J. Nevard and J. B. Keller, “Reciprocal relations for effective conductivities of anisotropic media,” J. Math. Phys. 26, 2761 (1985).

J. B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548 (1964).

J. B. Keller, “Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders,” J. Appl. Phys. 34, 991 (1963).

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 1111051 (2007).

Kim, D. S.

Lalanne, P.

Q. Cao and P. Lalanne, “Negative role of surface plasmon in the transmission of metallic gratings with very narrow slits,” Phys. Rev. Lett. 88, 0574031 (2002).

Lee, H.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).

Lee, J. W.

Leng, Y.

B. Hou, H. Wen, Y. Leng, and W. Wen, “Enhanced transmission of electromagnetic waves through metamaterials,” Appl. Phys. A 87, 217 (2007).

Lezec, H.

D. Grupp, H. Lezec, T. Ebbesen, K. Pellerin, and T. Thio, “Surface plasmon enhanced optical transmission through subwavelength holes,” Appl. Phys. Lett 77, 1569 (2000).

H. Ghaemi, T. Thio, D. Grupp, T. Ebbesen, and H. Lezec, “Surface plasmon enhanced optical transmission through subwavelength holes,” Phys. Rev. B 58, 6779 (1998).

Lezec, H. J.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

Linden, S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

Liu, Z.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).

Lochbihler, H.

H. Lochbihler, “Surface polaritons on gold-wire gratings,” Phys. Rev. B 50, 4795 (1994).

H. Lochbihler and R. Depine, “Highly conducting wire gratings in the resonance region,” Appl. Opt. 32, 3459 (1993).

Martínez-Zérega, B. E.

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and W. Mochán, “Effective dielectric response of metamaterials,” Phys. Rev. B 79, 245132 (2009).

Martín-Moreno, L.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

Mendoza, B. S.

E. Cortés, W. L. Mochán, B. S. Mendoza, and G. P. Ortiz, “Optical properties of nano-structured metamaterials,” Phys. Status Solidi B 247, 2102 (2010).

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and W. Mochán, “Effective dielectric response of metamaterials,” Phys. Rev. B 79, 245132 (2009).

Mochán, W.

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and W. Mochán, “Effective dielectric response of metamaterials,” Phys. Rev. B 79, 245132 (2009).

Mochán, W. L.

E. Cortés, W. L. Mochán, B. S. Mendoza, and G. P. Ortiz, “Optical properties of nano-structured metamaterials,” Phys. Status Solidi B 247, 2102 (2010).

W. L. Mochán and R. G. Barrera, “Electromagnetic response of systems with spatial fluctuations. i. general formalism,” Phys. Rev. B 32, 4984 (1985).

W. L. Mochán and R. G. Barrera, “Intrinsic surface-induced optical anisotropies of cubic crystals: local-field effect,” Phys. Rev. Lett. 55, 1192 (1985).

Nahata, A.

Nevard, J.

J. Nevard and J. B. Keller, “Reciprocal relations for effective conductivities of anisotropic media,” J. Math. Phys. 26, 2761 (1985).

Neviére, M.

E. Popov, M. Neviére, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).

Novotny, L.

L. Novotny, “Effective wavelength scaling for optical antennas,” Phys. Rev. Lett. 98, 2668021 (2007).

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).

O’Connor, D.

W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).

Ortiz, G. P.

E. Cortés, W. L. Mochán, B. S. Mendoza, and G. P. Ortiz, “Optical properties of nano-structured metamaterials,” Phys. Status Solidi B 247, 2102 (2010).

G. P. Ortiz, B. E. Martínez-Zérega, B. S. Mendoza, and W. Mochán, “Effective dielectric response of metamaterials,” Phys. Rev. B 79, 245132 (2009).

Park, Q.-H.

Pellerin, K.

D. Grupp, H. Lezec, T. Ebbesen, K. Pellerin, and T. Thio, “Surface plasmon enhanced optical transmission through subwavelength holes,” Appl. Phys. Lett 77, 1569 (2000).

Pellerin, K. M.

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

Pendry, J.

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).

Pendry, J. B.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).

L. Martín-Moreno, F. J. García-Vidal, H. J. Lezec, K. M. Pellerin, T. Thio, J. B. Pendry, and T. W. Ebbesen, “Theory of extraordinary optical transmission through subwavelength hole arrays,” Phys. Rev. Lett. 86, 1114 (2001).

Pollard, R.

W. Dickson, G. Wurtz, P. Evans, D. O’Connor, R. Atkinson, R. Pollard, and A. Zayats, “Dielectric-loaded plasmonic nanoantenna arrays: A metamaterial with tunable optical properties,” Phys. Rev. B 76, 115411 (2007).

Popov, E.

E. Popov, M. Neviére, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).

Porto, J.

J. Porto, F. García-Vidal, and J. Pendry, “Transmission resonances on metallic gratings with narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).

Reinisch, R.

E. Popov, M. Neviére, S. Enoch, and R. Reinisch, “Theory of light transmission through subwavelength periodic hole arrays,” Phys. Rev. B 62, 16100 (2000).

Reyes-Coronado, A.

R. G. Barrera, A. Reyes-Coronado, and A. García-Valenzuela, “Nonlocal nature of the electrodynamic response of colloidal systems,” Phys. Rev. B 75, 184202 (2007).

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

Salandrino, A.

A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystal: theory and simulations,” Phys. Rev. B 74, 075103 (2006).

Sanda, P.

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to difraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907 (1982).

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. 91, 1111051 (2007).

Sheng, P.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 2439051 (2005).

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to difraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907 (1982).

Stepleman, R.

P. Sheng, R. Stepleman, and P. Sanda, “Exact eigenfunctions for square-wave gratings: application to difraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907 (1982).

Sun, C.

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315, 1686 (2007).

Thiel, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

Thio, T.

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That the long-wavelength response ɛM is independent of the direction of q → 0 allows us to use the results of longitudinal calculations to solve optical (i.e., transverse) problems.

L. Novotny and B. Hecht, Principles of Nano-Optics (Cambridge University Press, New York, 2006).

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science325, 1513 (2009).

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Figures (3)

Fig. 1
Fig. 1 (a) Macroscopic dielectric function ɛM for a simple cubic array of spherical inclusions with response ɛb = 4 within a host with ɛa = 1 as a function of filling fraction f calculated with our Haydock’s recursive method (H), with Maxwell-Garnett’s (MG) and with Bruggeman’s formulae (B). (b) ɛM normal to the optical axis of a square array of square prisms with diagonals along the sides of the unit cell as a function of f for the theories H, MG and B in 2D.
Fig. 2
Fig. 2 (a) Normal-incidence reflectance R of a semi-infinite metamaterial and transmittance T of a d = 200 nm film made of a model Drude conductor with low (Γ = 0.01ωp) and high (Γ = 0.1ωp) dissipation parameters with a simple cubic lattice of cubical cavities of filling fraction f = 0.6, as a function of the frequency ω, together with the corresponding transmittance 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 lines indicate the frequencies at which ɛM ≈ 1.
Fig. 3
Fig. 3 Normal-incidence transmittance Tα for α = x, y polarization vs. frequency ω for 200 nm Au films with faces normal to the z axis with an embedded lattice of dielectric inclusions, normalized to the transmittance Teff of a homogeneous Au film with the same amount of metal. (a) Simple cubic lattice of spheres of radius r = 0.6a with a the lattice parameter with ɛb = 4. (b) Simple orthorhombic lattice of z-oriented cylinders with radius r = 0.53ax, height h = 0.9az and dielectric response ɛb = 2 with lattice parameters ax = az and ay = 1.15ax.

Equations (14)

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ɛ ( r ) = ɛ a B ( r ) ɛ a b
D G ( q ) = G ɛ GG E G ( q ) ,
E G E G L = G ^ G ^ E G ,
E 0 L = q ^ η 00 1 q ^ D 0 L ,
η GG G ^ . ( ɛ GG G ^ )
ɛ M L 1 q ^ ξ q ^ = q ^ η 00 1 q ^ ,
B GG B G G = ( 1 / Ω ) v d 3 r e i ( G G ) r
η GG 1 = 1 ɛ a b [ u δ GG B GG L L ] 1 = 1 ɛ a b 𝒢 GG ,
GG B GG L L = G ^ ( B GG G ^ ) ,
| n ˜ = ^ | n 1 = b n 1 | n 2 + a n 1 | n 1 + b n | n .
𝒢 n 1 = ( A n n + 1 T | n + 1 𝒢 n + 1 1 ) ,
𝒢 n = ( R n 𝒬 n | 𝒫 n 𝒮 n ) ,
R n = 1 A n n + 1 𝒢 n + 1 1 n + 1 T = 1 A n b n + 1 2 R n + 1 ,
ξ = R 0 ɛ a b = u ɛ a 1 u a 0 b 1 2 u a 1 b 2 2 u a 2 b 3 2 .

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