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THz photonic bands of periodic stacks composed of resonant dielectric and nonlocal metal

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Abstract

The photonic band structures of superlattices composed of spatially-dispersive metal and polaritonic dielectric are theoretically investigated. The nonlocal relation between the electric current density and the electric field inside the metal layers is defined within the formalism of the Boltzmann kinetic equation, whereas the frequency dependent permittivity of the polar layers is modeled by a Lorentz-oscillator. Due to the large dielectric contrast between metal and polar components, the photonic band structure exhibits flat pass bands associated with Fabry-Perot resonances in the dielectric layers. There is also a wide stop band because of the existence of the polaritonic gap. We have compared our results with the predictions of the Drude-Lorentz model for the frequency-dependent metal permittivity. It is found that the nonlocal effect on the Fabry-Perot resonance bands is strong if their corresponding frequencies are within the interval where the difference between the impedances at both metal surfaces, predicted by the nonlocal and local formalisms, is maximal.

© 2015 Optical Society of America

1. Introduction

Periodic optical nanostructures with dielectric and metallic components belong to the meta-material systems because of their extraordinary electromagnetic properties (see, e.g., [19]). Among such properties, we should mention the negative refraction [1012], which has been observed in double negative metamaterials, whose effective permittivity and permeability are simultaneously negative. The phenomenon of negative refraction is also observed in inherently-anisotropic dielectric-metal superlattices without negative effective permeability [1315]. The optical properties of such multilayered metamaterials have been commonly described (see, e.g., [1622] and references therein) within the Drude-Lorentz local model, which establishes a local relation between the electron current density and the electric field. However, due to high mobility of conduction electrons and inhomogeneity of the electromagnetic field in metals, this relation actually takes an integral form and therefore, is nonlocal. The nonlocality (or spatial dispersion) is responsible for the anomalous skin effect in metals [23, 24]. Evidently, the simple Drude-Lorentz model absolutely ignores all these effects. Besides, in modern micro- and nanostructures, in which the size of the metallic inclusions is comparable with the skin depth of the electromagnetic field penetration, it is necessary to take into account the size effect.

In the paper [25] the electrodynamic problem for a metallic slab was solved analytically self-consistently including the nonlocal effect in the high-frequency conductivity of electrons. The current density was calculated within the semiclassical approach of the Boltzmann kinetic equation. In contrast to the Drude-Lorentz model, the kinetic approach gives rise to a nonlocal (integral) relation between the current density and the electric field. The distribution of the electromagnetic field inside the metallic slab was obtained in a closed analytical form. It has been shown that, in general, the results are qualitatively different from those obtained in the Drude-Lorentz approximation. In particular, in the high-frequency region (including the terahertz and infrared frequency range), the absorption oscillates with the radiation frequency and sample thickness. Also the absorption becomes sensitive to the Fermi velocity of electrons and depends nontrivially on the electron relaxation rate.

In a recent work [26], we have studied the photonic band structure for a one-dimensional periodic array of alternating dielectric and metal layers on the basis of the results obtained in [25]. As was shown in [26], the optic spectrum for a superlattice with very thin metallic layers exhibits narrow pass bands as a result of the strong contrast between the impedances of the dielectric and the metal. The narrow pass bands correspond to Fabry-Perot resonances in the relatively-thick dielectric layer. The metal nonlocality is well pronounced in the infrared and, therefore, the nonlocal effect upon the photonic band structure of the superlattice can be strong when the Fabry-Perot resonance bands are in that frequency range. Noticeable differences not only in the magnitude, but also in the sign and the line-shape of the real part of the Bloch wave number in the Fabry-Perot resonance bands, were found.

The spatial dispersion in metallic layers is well manifested in the THz range [25,26]. Therefore, if a superlattice is composed of a nonlocal metal and a resonant dielectric (e.g. an ionic crystal or heteropolar semiconductor), the photonic band structure can be altered at frequencies of the forbidden band, which is created from the coupling between the transverse optical phonons and the transverse electromagnetic waves inside the dielectric. The effect of the polaritonic gap on the photonic band structure for periodic systems with solely dielectric components has been studied in several works (e.g. [2739] and references therein). As is shown in [28], the photonic dispersion for E-polarized electromagnetic modes propagating in a two-dimensional (2D) photonic crystal, composed of materials with frequency-independent dielectric constants, exhibits gaps created from the periodicity of the lattice only. However, if one of the dielectric components is substituted by a resonant one, then a polaritonic gap appears in the photonic dispersion. Such a polaritonic gap turns out to be accompanied by two structural “twin” gaps because of the high dielectric contrast between components [28]. In [29, 31, 35], it was theoretically demonstrated that photonic crystals with phonon-polariton excitations exhibit a low-frequency pass band as well as flat bands directly below the transverse-optical phonon frequency ωT, which are related to localized resonance modes in the polaritonic material having a high refractive index below ωT. Just above ωT, the polaritonic material possesses a very negative permittivity and behaves as metal, i.e. expels almost all of the field. In the latter case, the presence of pass bands above ωT can be correlated with the existence of bands in the photonic dispersion for a metallodielectric crystal at such frequencies. It is also interesting the appearance of defectlike photonic modes [35] in the polaritonic gap, below the longitudinal-optical phonon frequency ωL, where the refractive index of the polaritonic material is low. Also, photonic crystals composed of polar materials have been proposed for THz metamaterials. So, 2D composites composed of dielectric-polar or polar-polar material constituents behave like a bulk homogeneous uniaxial medium with hyperbolic dispersion relation in the THz range [40, 41]. Moreover, composites of two different kinds of non-overlapping spheres, one made from inherently non-magnetic polaritonic and the other from a Drude-like material, behave as double negative metamaterials at THz frequencies [42]. As is shown there, the polaritonic spheres are responsible for the negative effective magnetic permeability, whereas the Drude-like spheres give rise to a negative effective electric permittivity.

In the present work, we study the photonic band structure for a superlattice composed of alternating resonant dielectric and metal layers. Our approach is based on the Boltzmann kinetic equation for the distribution function of the conduction electrons, which allows the calculations of the general material equation, namely the nonlocal relation between the electrical current density and the electric field inside the metallic layer. As a result, the photonic dispersion relation for the resonant-dielectric/metal array is derived in terms of the surface impedances of the metal and resonant dielectric layers. We illustrate the nonlocal effect on the photonic band structure with calculations for a GaN-Al superlattice.

2. Problem formulation. Photonic dispersion relation

We consider an array of two alternating dielectric a- and metallic b-layers (Fig. 1). Every kind of slabs has a constant thickness, da or db, respectively. Thus, the size d = da + db of a unit (a,b) cell is the period of the bi-layer stack. The materials in the unit cell are assumed to be isotropic linear nonmagnetic media. Hence, the electric and magnetic components of an electromagnetic wave of frequency ω, propagating along the x axis (growth direction of the stack), can be oriented along the y and z axis, respectively:

E(x,t)={0,E(x),0}exp(iωt),H(x,t)={0,0,H(x)}exp(iωt).

 figure: Fig. 1

Fig. 1 Scheme of the bi-layer stack

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All the a-layers are composed of a resonant dielectric, whose frequency-dependent dielectric function εa(ω) is here modeled by using a Lorentz oscillator,

εa=ε(1ωLO2ωTO2ω2ωTO2+iωv0),
where ε is the high-frequency dielectric constant determining the contribution of the valence electrons to the dielectric function, ωTO and ωLO are respectively the transverse and longitudinal optical-phonon frequencies, and ν0 denotes the damping constant. The model (2) defines refractive index na, impedance Za, wave number ka, and wave phase-shift φa of the a-layers,
na=εa,Za=na1,ka=kna,φa=kada,
with k = ω/c and c being, respectively, the wave number and the velocity of light in vacuum. Within every a-slab the electric field Ea(x) is governed by the Helmholtz equation,
Ea(x)+ka2Ea(x)=0,
where the double prime denotes the double derivative with respect to x. The magnetic field Ha(x) and derivative of the electric field within a-slabs are related by Faraday law:
ikHa(x)=Ea(x).

The general solution of Eqs. (4) and (5) for the n-th unit (a,b) cell within the standing waves representation which is the most suitable for our problem, reads

Ean(x)=iZaHan(xan)cos[ka(xbnx)]sin(kada)iZaHan(xbn)cos[ka(xxan)]sin(kada),
Han(x)=Han(xan)sin[ka(xbnx)]sin(kada)+Han(xbn)sin[ka(xxan)]sin(kada)
inside the an-layer, where xanxxbn. The coordinates xan and xbn refer to the left-hand edges of successive an - and bn -layers, respectively. The integration constants in Eqs. (6) describe the magnetic field Han(xan) at the left-hand and the magnetic field Han(xbn) on the right-hand edges of the an -layer. Note that the thicknesses of individual layers are defined as
xbnxan=da,xan+1xbn=db,xan+1xan=d.

The Maxwell equation for the electric field distribution within the metallic b-slabs has the form

Eb(x)+4πik2ωjb(x)=0.

Here we have ignored the term k2Eb(x) originated from the displacement current since in metals it is negligibly small in comparison with the current of conduction electrons up to near-IR frequency range (see, e.g., [24]). Following article [25], the relation between the electron current density jb(x) and the field Eb(x) in the metallic slab is obtained with the use of the Boltzmann kinetic equation. As a result, it gets the integral (i.e., nonlocal) form,

jb(x)=dxσ(xx)Eb(x),
σ(x)=3σDL4lω01dnx1nx2nxexp(|x|lωnx),
in contrast to the Drude-Lorentz model that is local by definition, i.e., in which the current density at the point x is specified by the electric field at the same point,
jb(x)=σDLEb(x),σDL=ωp24π(viω).

Here ωp, ν and VF are, respectively, the plasma frequency, the relaxation rate and the Fermi velocity of the conduction electrons, lω = VF/(ν−iω) stands for their effective mean free path due to both their collisions with scatters and the phase change of the electromagnetic field.

To resolve Eq. (8) with the “non-local” current (9) inside the metallic bn-layer, the electric field Eb(x) was suitably continued to the entire x axis with the use of the parity and periodicity conditions (see [25] for details),

Ebn(x)=Ebn(2xbnx),Ebn(x)=Ebn(x±2db).

This gives rise to the presentation of the electromagnetic field as a Fourier series of the normal electromagnetic modes with discrete wave number ks = πs/db,

Ebn(x)=ikHbn(xbn)dbs=cos[ks(xxbn)]ks2k2ε(ks)+ikHbn(xan+1)dbs=cos[ks(xan+1x)]ks2k2ε(ks),
Hbn(x)=Hbn(xbn)dbs=kssin[ks(xxbn)]ks2k2ε(ks)+Hbn(xan+1)dbs=kssin[ks(xan+1x)]ks2k2ε(ks),
inside the bn-layer, where xbnxxan+1. The integration constants Hbn(xbn) and Hbn(xan+1) represent the magnetic field at the corresponding layer edges. They are related to derivatives of the electric field by Faraday law which in the bn-layer reads
ikHb(x)=Eb(x).

The response of the conduction electrons in metallic slabs to the s-th electromagnetic mode is specified by its own permittivity ε(ks),

ε(ks)=ωp2ω(ω+iv)K(kslω),
which is not a metallic permittivity associated with the total electromagnetic field. Note that in Eq. (14) the traditional one contributed by the displacement current omitted in Eq. (8), is absent. Indeed, one can make sure that |ε(ks)|1 within all the frequency range up to near-IR. The mode permittivity ε(ks) depends on the mode wave number ks via the nonlocality factor K (kslω),
K(kslω)=3201(1nx2)dnx1+(kslωnx)2=32{[1kslω+1(kslω)3]arctan(kslω)1(kslω)2}.

Depending on the wave number ks, the factor (15) entirely defines the spatial dispersion effect in the mode permittivity (14). Owing to this, it is worthwhile to write down its asymptotics,

K(kslω)1(kslω)2/5,(ks|lω|)21;
K(kslω)3π/4|ks|lω,|kslω|1.

These expressions show that the nonlocal effect can be negligible only when inequality (16a) is met for all the modes contributing to the total electromagnetic field (12). Since in this case the nonlocality factor K (kslω) 1, the permittivity ε(ks) is the same for all the modes and coincides with that for the Drude-Lorentz model. As a consequence, the latter can be applied for the electrodynamic description of a metallic slab if provided by condition (16a).

The combination of Eqs. (6) and (12) with the continuity boundary conditions for the electric and magnetic fields taken at the interfaces x=xbn and x=xan+1, yields the recurrent relation describing the wave transfer through the whole n-th unit (a,b) cell,

(E(xan+1)H(xan+1))=(Q11Q12Q21Q22)(E(xan)H(xan)).

The transfer matrix Q^ has the following elements

Q11=ζ0ζdcosφaiζ02ζd2Zaζdsinφa,
Q12=ζ02ζd2ζdcosφa+iZaζ0ζdsinφa,
Q21=1ζdcosφa+iζ0Zaζdsinφa,
Q22=ζ0ζdcosφaiZaζdsinφa.

Note that the determinant of Q^-matrix equals to unity, det Q^ = Q11Q22 − Q12Q21 = 1. For a periodic stack, the transfer Q^-matrix is independent of the cell index n since all the unit cells are identical.

As one can see, the transfer matrix is specified by the phase shift φa and the impedance Za of the a-layer, as well as, by the surface impedances ζ0 and ζd of the left-hand and right-hand boundaries of the metallic b-slab,

ζ0=ikdbs1ks2k2ε(ks),ζd=ikdbs=cos(ksdb)ks2k2ε(ks).

One should emphasize the different physical meaning of the introduced impedances: while Za is the surface impedance corresponding to a half-space (with only one surface), the surface impedances ζ0 and ζd inherently belong to a layer, thus, taking into account both waves, incident onto and reflected from a given surface. Within the Drude-Lorentz model the nonlocality factor K = 1 for all normal modes, and the sums in definitions (19) can be explicitly calculated resulting in

ζ0(DL)=iZbcosφbsinφb,ζd(DL)=iZb1sinφb,
with the Drude-Lorentz optic parameters of metallic b-slabs: permittivity εb = ε(0), see (14), refractive index nb=εb, impedance Zb=nb1, wave number kb = knb, and phase shift φb = kbdb.

To proceed further, we should address the Floquet theorem that can be properly written for the electromagnetic field at the left-hand edges x=xan+1 and x=xan of an+1 and an layers,

(E(xan+1)H(xan+1))=(exp(iκd)00exp(iκd))(E(xan)H(xan)),
with κ traditionally standing for the Bloch wave number of a periodic structure. The unification of the transfer relation (17) with the Floquet theorem in the form (21) gives rise to the problem of eigenvectors and eigenvalues of the transfer Q^-matrix. As a result (see details, e.g., in the book [20]), the desired dispersion relation for the Bloch wave number κ of a one-dimensional periodic structure turns out to be determined by the trace of its unit-cell transfer matrix,
2cos(κd)=Q11+Q22.

With the use of explicit expressions (18a) and (18d), one can readily obtain

cos(κd)=ζ0ζdcosφaiZa2+ζ02ζd22Zaζdsinφa.

Thus, the optic spectrum κ(ω) of the considered periodic bi-layer stack is defined by the universal dispersion relation (23). The difference of the kinetic approach from the Drude-Lorentz model emerges merely in the metallic impedances ζ0 and ζd. The transition from the kinetic approach to the Drude-Lorentz approximation is performed by the replacement of the nonlocality factor with one (K (kslω) 1) for all of the summation indices s in definitions (19), i.e. when the general expressions (19) can be described by their asymptotics (20). In the latter case the general dispersion relation (23) is reduced to its conventional counterpart valid for a wide class of bilayer stack-structures,

cos(κd)=cosφacosφb12(ZaZb+ZbZa)sinφasinφb.

3. Analysis

In the case of a resonant-dielectric/metal superlattice, there exists a large contrast between the impedances of the a- and b-layers: the absolute value of the impedance Za is much greater than the absolute values of the metallic impedances ζ0 and ζd,

|Za|max(|ζ0|,|ζd|),
even at frequencies very close to the resonant one, ωTO. Therefore, the dispersion relation (23) for the photonic modes may have solutions for the Bloch wave number κ with |Reκ < π/d and 0 < Imκ < |Reκ| only in very narrow pass bands, associated with Fabry-Perot resonances arising in the a-layer. Indeed, when |Za0| ≫ 1 and |Zad| ≫ 1 in Eq. (23), such pass bands are close to the frequencies ωja at which the condition
ReφaReknada=jaπ,ja=1,2,3.,
is satisfied.

It should be noted that the Drude-Lorentz impedances (20) depend on three dimensionless parameters: the frequency of the electromagnetic field ω/ωp and the electron relaxation rate ν/ωp normalized to the plasma frequency ωp, as well as the ratio db of the metallic slab thickness db to the minimum skin depth δ = c/ωp in the bulk metal, which is reached in the high-frequency range νωωp, where εb=ε(0)ωp2/ω2. It is of crucial importance that the kinetic impedances (19) depend on the fourth control parameter πVF/c ≪ 1 associated with the Fermi velocity of electrons. This parameter enters the argument of the nonlocality factor K (kslω) and is responsible for the spatial dispersion effect, which is well manifested in the infrared if

ν<ω<πVFωp/c

(see details in [25]). For this reason, the nonlocal effect on the optic spectrum of photonic eigenmodes for the superlattice will be noticeable only in Fabry-Perot resonance bands appearing within such a frequency interval.

Below, we shall analyze the Fabry-Perot resonance bands of the photonic dispersion for a GaN-Al superlattice. In our numerical calculations, we use for the aluminium b-layers: ν = 0.00025ωp, the Fermi velocity VF = 2.03 × 108cm/s, ωp = 3.82 × 1015Hz, db = 4δ. The choice of GaN the a-layers is due to the fact that its reststrahlen band (ωTOωωLO) is precisely within the interval (27) of strong Al nonlocality: the transverse and longitudinal phonon frequencies are respectively ωTO = 571cm1 (17.12 THz) and ωLO = 752cm1 (22.54 THz) [43]. Other GaN parameters used here are: ε = 8.9, and ν0 = 103ωTO. For the GaN-layer, we have chosen the thickness da such that the first Fabry-Perot resonance frequency ω1 4ν, i.e. da = πc/ω1Rena(ω1).

The normalized wave frequency ω/ωp as a function of the wave number ka, which displays the polariton spectrum ka = kna(ω) for the GaN a-layer, is depicted in Fig. 2. Note that the real part of ka (red lines) is much greater than the imaginary part (blue lines) everywhere except inside the GaN reststrahlen band (ωTOωωLO) wherein the real part is negligible with respect to the imaginary one. The dark dashed lines in Fig. 2 indicate frequencies ωja corresponding to Fabry-Perot resonances (26). As is seen, for a fixed number ja there are two resonance frequencies ωja, one of them is smaller than ωTO whereas the second one is above ωLO. Besides, the lower frequencies ωja are accumulated just below the resonance frequency ωTO.

 figure: Fig. 2

Fig. 2 Polaritonic spectrum of the GaN a-layer.

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Figure 3 exhibits first four of the Fabry-Perot resonance bands situated within the lower (THz) frequency range

0<˜ω<˜ωTO.

 figure: Fig. 3

Fig. 3 Photonic band structure in the vicinities of the four lower Fabry-Perot resonances for a GaN-Al superlattice, predicted by the nonlocal (Boltzmann) and local (Drude-Lorentz) formalisms.

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The solid green and dashed red curves in Fig. 3 correspond to the photonic dispersion predicted by the formalism of the Boltzmann kinetic equation, whereas the dashed-dot blue and dashed-dot-dot black lines are obtained within the Drude-Lorentz model. According to Fig. 3, the ja-th resonance band appears below the resonant frequency ωja=jaπc/Renada. Besides, the kinetic and local pass bands are clearly distinguishable. Three main effects of spatial dispersion on the photonic band structure can be observed in Fig. 3. First of all, not only the magnitude but also the sign of real part Reκ of the Bloch wave number can change in comparison with the predictions of the local model. Second, the jumps of Reκ, which occur because Reκ is limited to the first Brillouin zone, are predicted in different Fabry-Perot resonance bands by the nonlocal and local models. Third, the minimum value of the imaginary part Imκ in the resonance bands practically does not vary from band to band within the nonlocal formalism.

Figure 4 presents the first four Fabry-Perot resonance bands that are above the longitudinal optical phonon frequency ωLO,

ωLOω.

 figure: Fig. 4

Fig. 4 Photonic band structure in the vicinities of the first four Fabry-Perot resonances above ωLO for a GaN-Al superlattice as in Fig. 3.

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Note that the distinction between the predictions of the Boltzmann kinetic approach and the Drude-Lorentz local model is not so strongly pronounced as for the first photonic bands exhibited in Fig. 3. These results show that the effect of the metal spatial dispersion on the photonic bands is well manifest at the narrow Fabry-Perot resonances if the corresponding frequencies ωja are within the interval where the difference between the surface impedances ζ0 and ζd, predicted by both nonlocal and local formalisms, is maximal. As is shown in [26] (Fig. 2 therein), in the case of Al-layers with db = 4δ, such a difference is considerable within the interval 0.001ωPω ≲ 0.005ωP.

4. Conclusion

We have studied the photonic band structure of superlattices composed of nonlocal-metallic and resonant-dielectric layers. The study is based on the semiclassical kinetic approach of the Boltzmann kinetic equation, which allows us to calculate the nonlocal (integral) relation between the electric current density and the electric field in the metal layers. Besides, we have used a Lorentz-oscillator model for describing the resonant dielectric response of the polar layers. Because of the large dielectric contrast between metal and resonant-dielectric constituents, the photonic band structure is characterized by flat Fabry-Perot resonance bands. On the other hand, the existence of the polaritonic gap leads to the appearance of a wide stop band in the photonic band structure of the superlattice. Directly below the resonance frequency of the transverse optical phonons, a high concentration of flat Fabry-Perot resonance bands is observed. Unlike the dielectric-polar superlattices, the metal-dielectric heterostructure does not exhibit a low-frequency pass band for the electromagnetic modes propagating along the growth direction. Having compared our results with those obtained within the local Drude-Lorentz model for the metal permittivity, we found that the metal spatial dispersion noticeably alters the flat Fabry-Perot resonance bands for the photonic modes. It has been shown that the nonlocal effect is particularly important if the Fabry-Perot resonances are within the interval where the difference between the impedances at both metal surfaces, predicted by the nonlocal and local formalisms, is maximal.

Acknowledgments

This work was partially supported by SEP-CONACYT (grant CB-2011-01-166382).

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30. W. Zhang, A. Hu, and N. Ming, “The photonic band structure of the two-dimensional hexagonal lattice of ionic dielectric media,” J. Phys.: Condens. Matter 9, 541–549 (1997).

31. V. Kuzmiak, A. A. Maradudin, and A. R. McGurn, “Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal,” Phys. Rev. B 55, 4298–4311 (1997). [CrossRef]  

32. W. Zhang, Z. Wang, A. Hu, and N. Ming, “The photonic band structures of body-centred-tetragonal crystals composed of ionic or metal spheres,” J. Phys.: Condens. Matter 12, 5307–5316 (2000).

33. A. Y. Sivachenko, M. E. Raikh, and Z. V. Vardeny, “Excitations in photonic crystals infiltrated with polarizable media,” Phys. Rev. A 64, 013809 (2001). [CrossRef]  

34. K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003). [CrossRef]   [PubMed]  

35. K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003). [CrossRef]  

36. T. Ovidiu and J. Sajeev, “Photonic band gap enhancement in frequency-dependent dielectrics,” Phys. Rev. E 70, 046605 (2004). [CrossRef]  

37. G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2007). [CrossRef]  

38. A. H. BaradaranGhasemi, S. Mandegarian, H. Kebriti, and H. Latifi, “Bandgap generation and enhancement in polaritonic cylinder square-lattice photonic crystals,” J. Opt. 14, 055103 (2012). [CrossRef]  

39. G. Alagappan and A. Deinega, “Optical modes of a dispersive periodic nanostructure,” Progress In Electromagnetics Research B 52, 1–18 (2013). [CrossRef]  

40. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84, 035128 (2011). [CrossRef]  

41. A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Kenanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20, 14663–14682 (2012). [CrossRef]   [PubMed]  

42. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).

43. T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

References

  • View by:

  1. R. M. Walser, “Electromagnetic metamaterials,” Proc. SPIE 4467, 1–15 (2001).
    [Crossref]
  2. F. Capolino, Theory and phenomena of metamaterials (CRC Press, 2009).
    [Crossref]
  3. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photononics 1, 41–48 (2007).
    [Crossref]
  4. C. M. Soukoulis and M. Wegener M, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
  5. J. A. Reyes-Avendao, U. Algredo-Badillo, P. Halevi, and F. Pérez-Rodríguez, “From photonic crystals to metamaterials: the bianisotropic response,” New J. Phys. 13, 073041 (2011).
    [Crossref]
  6. Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt. 15, 023001 (2013).
    [Crossref]
  7. A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express 3, 143–156 (2013).
    [Crossref]
  8. J. A. Reyes-Avendaño, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Bianisotropic metamaterials based on twisted asymmetric crosses,” J. Opt. 16, 065102 (2014).
    [Crossref]
  9. A. V. Goncharenko, E. F. Venger, Y. C. Chang, and A. O. Pinchuk, “Arrays of core-shell nanospheres as 3D isotropic broadband ENZ and highly absorbing metamaterials,” Opt. Mater. Express 4, 2310–2322 (2014).
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  10. D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).
  11. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
    [Crossref] [PubMed]
  12. J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101 (2006).
    [Crossref]
  13. V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
    [Crossref]
  14. B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
    [Crossref]
  15. A. L. Rakhmanov, V. A. Yampolskii, J. A. Fan, F. Capasso, and F. Nori, “Layered superconductors as negative-refractive-index metamaterials,” Phys. Rev. B 81, 075101 (2010).
    [Crossref]
  16. V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
    [Crossref]
  17. D. Soto-Puebla, M. Xiao, and F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
    [Crossref]
  18. X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
    [Crossref]
  19. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
    [Crossref]
  20. P. Markoš and C. M. Soukoulis, Wave Propagation. From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, NJ, 2008).
    [Crossref]
  21. B. Zenteno-Mateo, V. Cerdán-Ramírez, B. Flores-Desirena, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Effective permittivity tensor for a metal-dielectric superlattice,” Progress in Electromagnetics Research Letters 22, 165–174 (2011).
  22. A. Orlov, I. Iorsh, P. Belov, and Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express 21, 1593–1598 (2013).
    [Crossref] [PubMed]
  23. A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier, 1988).
  24. K. A. Kaner, A. A. Krokhin, and N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in the book Spatial Dispersion in Solids and Plasmas1, edited by P. Halevi, ed. (Elsevier, 1992).
  25. A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Letters 90, 623–627 (2009).
    [Crossref]
  26. A. Paredes-Juárez, D. A. Iakushev, B. Flores-Desirena, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal Effect on Optic Spectrum of a Periodic Dielectric-Metal Stack,” Opt. Express 22, 7581–7586 (2014).
    [Crossref] [PubMed]
  27. M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
    [Crossref]
  28. M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Cho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
    [Crossref]
  29. W. Zhang, A. Hu, X. Lei, N. Xu, and N. Ming, “Photonic band structures of a two-dimensional ionic dielectric medium,” Phys. Rev. B 54, 10280–10283 (1996).
    [Crossref]
  30. W. Zhang, A. Hu, and N. Ming, “The photonic band structure of the two-dimensional hexagonal lattice of ionic dielectric media,” J. Phys.: Condens. Matter 9, 541–549 (1997).
  31. V. Kuzmiak, A. A. Maradudin, and A. R. McGurn, “Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal,” Phys. Rev. B 55, 4298–4311 (1997).
    [Crossref]
  32. W. Zhang, Z. Wang, A. Hu, and N. Ming, “The photonic band structures of body-centred-tetragonal crystals composed of ionic or metal spheres,” J. Phys.: Condens. Matter 12, 5307–5316 (2000).
  33. A. Y. Sivachenko, M. E. Raikh, and Z. V. Vardeny, “Excitations in photonic crystals infiltrated with polarizable media,” Phys. Rev. A 64, 013809 (2001).
    [Crossref]
  34. K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003).
    [Crossref] [PubMed]
  35. K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
    [Crossref]
  36. T. Ovidiu and J. Sajeev, “Photonic band gap enhancement in frequency-dependent dielectrics,” Phys. Rev. E 70, 046605 (2004).
    [Crossref]
  37. G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2007).
    [Crossref]
  38. A. H. BaradaranGhasemi, S. Mandegarian, H. Kebriti, and H. Latifi, “Bandgap generation and enhancement in polaritonic cylinder square-lattice photonic crystals,” J. Opt. 14, 055103 (2012).
    [Crossref]
  39. G. Alagappan and A. Deinega, “Optical modes of a dispersive periodic nanostructure,” Progress In Electromagnetics Research B 52, 1–18 (2013).
    [Crossref]
  40. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84, 035128 (2011).
    [Crossref]
  41. A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Kenanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20, 14663–14682 (2012).
    [Crossref] [PubMed]
  42. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).
  43. T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

2014 (3)

2013 (4)

A. Orlov, I. Iorsh, P. Belov, and Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express 21, 1593–1598 (2013).
[Crossref] [PubMed]

G. Alagappan and A. Deinega, “Optical modes of a dispersive periodic nanostructure,” Progress In Electromagnetics Research B 52, 1–18 (2013).
[Crossref]

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt. 15, 023001 (2013).
[Crossref]

A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express 3, 143–156 (2013).
[Crossref]

2012 (2)

2011 (3)

C. M. Soukoulis and M. Wegener M, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).

J. A. Reyes-Avendao, U. Algredo-Badillo, P. Halevi, and F. Pérez-Rodríguez, “From photonic crystals to metamaterials: the bianisotropic response,” New J. Phys. 13, 073041 (2011).
[Crossref]

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84, 035128 (2011).
[Crossref]

2010 (1)

A. L. Rakhmanov, V. A. Yampolskii, J. A. Fan, F. Capasso, and F. Nori, “Layered superconductors as negative-refractive-index metamaterials,” Phys. Rev. B 81, 075101 (2010).
[Crossref]

2009 (1)

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Letters 90, 623–627 (2009).
[Crossref]

2007 (3)

G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2007).
[Crossref]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photononics 1, 41–48 (2007).
[Crossref]

2006 (2)

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[Crossref]

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101 (2006).
[Crossref]

2005 (3)

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[Crossref]

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[Crossref]

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).

2004 (2)

D. Soto-Puebla, M. Xiao, and F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[Crossref]

T. Ovidiu and J. Sajeev, “Photonic band gap enhancement in frequency-dependent dielectrics,” Phys. Rev. E 70, 046605 (2004).
[Crossref]

2003 (2)

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003).
[Crossref] [PubMed]

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
[Crossref]

2001 (4)

A. Y. Sivachenko, M. E. Raikh, and Z. V. Vardeny, “Excitations in photonic crystals infiltrated with polarizable media,” Phys. Rev. A 64, 013809 (2001).
[Crossref]

R. M. Walser, “Electromagnetic metamaterials,” Proc. SPIE 4467, 1–15 (2001).
[Crossref]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

2000 (1)

W. Zhang, Z. Wang, A. Hu, and N. Ming, “The photonic band structures of body-centred-tetragonal crystals composed of ionic or metal spheres,” J. Phys.: Condens. Matter 12, 5307–5316 (2000).

1997 (3)

W. Zhang, A. Hu, and N. Ming, “The photonic band structure of the two-dimensional hexagonal lattice of ionic dielectric media,” J. Phys.: Condens. Matter 9, 541–549 (1997).

V. Kuzmiak, A. A. Maradudin, and A. R. McGurn, “Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal,” Phys. Rev. B 55, 4298–4311 (1997).
[Crossref]

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[Crossref]

1996 (1)

W. Zhang, A. Hu, X. Lei, N. Xu, and N. Ming, “Photonic band structures of a two-dimensional ionic dielectric medium,” Phys. Rev. B 54, 10280–10283 (1996).
[Crossref]

1994 (1)

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Cho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[Crossref]

1993 (1)

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[Crossref]

Abrikosov, A. A.

A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier, 1988).

Acosta, M. F.

Alagappan, G.

G. Alagappan and A. Deinega, “Optical modes of a dispersive periodic nanostructure,” Progress In Electromagnetics Research B 52, 1–18 (2013).
[Crossref]

Algredo-Badillo, U.

J. A. Reyes-Avendao, U. Algredo-Badillo, P. Halevi, and F. Pérez-Rodríguez, “From photonic crystals to metamaterials: the bianisotropic response,” New J. Phys. 13, 073041 (2011).
[Crossref]

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

BaradaranGhasemi, A. H.

A. H. BaradaranGhasemi, S. Mandegarian, H. Kebriti, and H. Latifi, “Bandgap generation and enhancement in polaritonic cylinder square-lattice photonic crystals,” J. Opt. 14, 055103 (2012).
[Crossref]

Belov, P.

Bienstman, P.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
[Crossref]

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003).
[Crossref] [PubMed]

Capasso, F.

A. L. Rakhmanov, V. A. Yampolskii, J. A. Fan, F. Capasso, and F. Nori, “Layered superconductors as negative-refractive-index metamaterials,” Phys. Rev. B 81, 075101 (2010).
[Crossref]

Capolino, F.

F. Capolino, Theory and phenomena of metamaterials (CRC Press, 2009).
[Crossref]

Cardona, M.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Cerdán-Ramírez, V.

B. Zenteno-Mateo, V. Cerdán-Ramírez, B. Flores-Desirena, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Effective permittivity tensor for a metal-dielectric superlattice,” Progress in Electromagnetics Research Letters 22, 165–174 (2011).

Chan, C. T.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Cho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[Crossref]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[Crossref]

Chang, Y. C.

Chen, K.R.

Cho, K. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Cho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[Crossref]

D’Astuto, M.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Deinega, A.

G. Alagappan and A. Deinega, “Optical modes of a dispersive periodic nanostructure,” Progress In Electromagnetics Research B 52, 1–18 (2013).
[Crossref]

Díaz-Monge, F.

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Letters 90, 623–627 (2009).
[Crossref]

Economou, E. N.

A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Kenanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20, 14663–14682 (2012).
[Crossref] [PubMed]

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84, 035128 (2011).
[Crossref]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[Crossref]

Elser, J.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

Fan, J. A.

A. L. Rakhmanov, V. A. Yampolskii, J. A. Fan, F. Capasso, and F. Nori, “Layered superconductors as negative-refractive-index metamaterials,” Phys. Rev. B 81, 075101 (2010).
[Crossref]

Fan, S.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003).
[Crossref] [PubMed]

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
[Crossref]

Flores-Desirena, B.

A. Paredes-Juárez, D. A. Iakushev, B. Flores-Desirena, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal Effect on Optic Spectrum of a Periodic Dielectric-Metal Stack,” Opt. Express 22, 7581–7586 (2014).
[Crossref] [PubMed]

B. Zenteno-Mateo, V. Cerdán-Ramírez, B. Flores-Desirena, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Effective permittivity tensor for a metal-dielectric superlattice,” Progress in Electromagnetics Research Letters 22, 165–174 (2011).

Foteinopoulou, S.

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84, 035128 (2011).
[Crossref]

Gantzounis, G.

G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2007).
[Crossref]

García de Abajo, J.

Goncharenko, A. V.

Goncharenko, A.V.

Grzegory, I.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Halevi, P.

J. A. Reyes-Avendao, U. Algredo-Badillo, P. Halevi, and F. Pérez-Rodríguez, “From photonic crystals to metamaterials: the bianisotropic response,” New J. Phys. 13, 073041 (2011).
[Crossref]

Han, D.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[Crossref]

Ho, K. M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[Crossref]

Hu, A.

W. Zhang, Z. Wang, A. Hu, and N. Ming, “The photonic band structures of body-centred-tetragonal crystals composed of ionic or metal spheres,” J. Phys.: Condens. Matter 12, 5307–5316 (2000).

W. Zhang, A. Hu, and N. Ming, “The photonic band structure of the two-dimensional hexagonal lattice of ionic dielectric media,” J. Phys.: Condens. Matter 9, 541–549 (1997).

W. Zhang, A. Hu, X. Lei, N. Xu, and N. Ming, “Photonic band structures of a two-dimensional ionic dielectric medium,” Phys. Rev. B 54, 10280–10283 (1996).
[Crossref]

Huang, K. C.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
[Crossref]

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003).
[Crossref] [PubMed]

Iakushev, D. A.

Iorsh, I.

Joannopoulos, J. D.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003).
[Crossref] [PubMed]

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
[Crossref]

Juárez-Ruiz, E.

J. A. Reyes-Avendaño, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Bianisotropic metamaterials based on twisted asymmetric crosses,” J. Opt. 16, 065102 (2014).
[Crossref]

B. Zenteno-Mateo, V. Cerdán-Ramírez, B. Flores-Desirena, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Effective permittivity tensor for a metal-dielectric superlattice,” Progress in Electromagnetics Research Letters 22, 165–174 (2011).

Kafesaki, M.

Kaner, K. A.

K. A. Kaner, A. A. Krokhin, and N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in the book Spatial Dispersion in Solids and Plasmas1, edited by P. Halevi, ed. (Elsevier, 1992).

Katsarakis, N.

Kebriti, H.

A. H. BaradaranGhasemi, S. Mandegarian, H. Kebriti, and H. Latifi, “Bandgap generation and enhancement in polaritonic cylinder square-lattice photonic crystals,” J. Opt. 14, 055103 (2012).
[Crossref]

Kenanakis, G.

Kivshar, Y.

Koschny, T.

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101 (2006).
[Crossref]

Krisch, M.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Krokhin, A. A.

K. A. Kaner, A. A. Krokhin, and N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in the book Spatial Dispersion in Solids and Plasmas1, edited by P. Halevi, ed. (Elsevier, 1992).

Kuzmiak, V.

V. Kuzmiak, A. A. Maradudin, and A. R. McGurn, “Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal,” Phys. Rev. B 55, 4298–4311 (1997).
[Crossref]

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[Crossref]

Latifi, H.

A. H. BaradaranGhasemi, S. Mandegarian, H. Kebriti, and H. Latifi, “Bandgap generation and enhancement in polaritonic cylinder square-lattice photonic crystals,” J. Opt. 14, 055103 (2012).
[Crossref]

Lei, X.

W. Zhang, A. Hu, X. Lei, N. Xu, and N. Ming, “Photonic band structures of a two-dimensional ionic dielectric medium,” Phys. Rev. B 54, 10280–10283 (1996).
[Crossref]

Leszczynski, M.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Li, Z.

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt. 15, 023001 (2013).
[Crossref]

Liu, X.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[Crossref]

Makarov, N. M.

A. Paredes-Juárez, D. A. Iakushev, B. Flores-Desirena, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal Effect on Optic Spectrum of a Periodic Dielectric-Metal Stack,” Opt. Express 22, 7581–7586 (2014).
[Crossref] [PubMed]

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Letters 90, 623–627 (2009).
[Crossref]

K. A. Kaner, A. A. Krokhin, and N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in the book Spatial Dispersion in Solids and Plasmas1, edited by P. Halevi, ed. (Elsevier, 1992).

Mandegarian, S.

A. H. BaradaranGhasemi, S. Mandegarian, H. Kebriti, and H. Latifi, “Bandgap generation and enhancement in polaritonic cylinder square-lattice photonic crystals,” J. Opt. 14, 055103 (2012).
[Crossref]

Maradudin, A. A.

V. Kuzmiak, A. A. Maradudin, and A. R. McGurn, “Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal,” Phys. Rev. B 55, 4298–4311 (1997).
[Crossref]

V. Kuzmiak and A. A. Maradudin, “Photonic band structures of one- and two-dimensional periodic systems with metallic components in the presence of dissipation,” Phys. Rev. B 55, 7427–7444 (1997).
[Crossref]

Markoš, P.

P. Markoš and C. M. Soukoulis, Wave Propagation. From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, NJ, 2008).
[Crossref]

Mavidis, Ch.

McGurn, A. R.

V. Kuzmiak, A. A. Maradudin, and A. R. McGurn, “Photonic band structures of two-dimensional systems fabricated from rods of a cubic polar crystal,” Phys. Rev. B 55, 4298–4311 (1997).
[Crossref]

Merino, R. I.

Ming, N.

W. Zhang, Z. Wang, A. Hu, and N. Ming, “The photonic band structures of body-centred-tetragonal crystals composed of ionic or metal spheres,” J. Phys.: Condens. Matter 12, 5307–5316 (2000).

W. Zhang, A. Hu, and N. Ming, “The photonic band structure of the two-dimensional hexagonal lattice of ionic dielectric media,” J. Phys.: Condens. Matter 9, 541–549 (1997).

W. Zhang, A. Hu, X. Lei, N. Xu, and N. Ming, “Photonic band structures of a two-dimensional ionic dielectric medium,” Phys. Rev. B 54, 10280–10283 (1996).
[Crossref]

Moroz, A.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).

Mutlu, M.

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt. 15, 023001 (2013).
[Crossref]

Narimanov, E. E.

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[Crossref]

Nazarov, V.U.

Nelson, K. A.

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Field expulsion and reconfiguration in polaritonic photonic crystals,” Phys. Rev. Lett. 90, 196402 (2003).
[Crossref] [PubMed]

K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, “Phonon-polariton excitations in photonic crystals,” Phys. Rev. B 68, 075209 (2003).
[Crossref]

Nemat-Nasser, S.

D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).

Nori, F.

A. L. Rakhmanov, V. A. Yampolskii, J. A. Fan, F. Capasso, and F. Nori, “Layered superconductors as negative-refractive-index metamaterials,” Phys. Rev. B 81, 075101 (2010).
[Crossref]

Orera, V. M.

Orlov, A.

Ovidiu, T.

T. Ovidiu and J. Sajeev, “Photonic band gap enhancement in frequency-dependent dielectrics,” Phys. Rev. E 70, 046605 (2004).
[Crossref]

Ozbay, E.

Z. Li, M. Mutlu, and E. Ozbay, “Chiral metamaterials: from optical activity and negative refractive index to asymmetric transmission,” J. Opt. 15, 023001 (2013).
[Crossref]

Pabst, M.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Padilla, W.

D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).

Paredes-Juárez, A.

A. Paredes-Juárez, D. A. Iakushev, B. Flores-Desirena, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal Effect on Optic Spectrum of a Periodic Dielectric-Metal Stack,” Opt. Express 22, 7581–7586 (2014).
[Crossref] [PubMed]

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Letters 90, 623–627 (2009).
[Crossref]

Pavone, P.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Pendry, J. B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[Crossref]

Pérez-Rodríguez, F.

J. A. Reyes-Avendaño, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Bianisotropic metamaterials based on twisted asymmetric crosses,” J. Opt. 16, 065102 (2014).
[Crossref]

A. Paredes-Juárez, D. A. Iakushev, B. Flores-Desirena, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal Effect on Optic Spectrum of a Periodic Dielectric-Metal Stack,” Opt. Express 22, 7581–7586 (2014).
[Crossref] [PubMed]

J. A. Reyes-Avendao, U. Algredo-Badillo, P. Halevi, and F. Pérez-Rodríguez, “From photonic crystals to metamaterials: the bianisotropic response,” New J. Phys. 13, 073041 (2011).
[Crossref]

A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Letters 90, 623–627 (2009).
[Crossref]

B. Zenteno-Mateo, V. Cerdán-Ramírez, B. Flores-Desirena, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Effective permittivity tensor for a metal-dielectric superlattice,” Progress in Electromagnetics Research Letters 22, 165–174 (2011).

Pinchuk, A. O.

Podolskiy, V. A.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71, 201101 (2005).
[Crossref]

Raikh, M. E.

A. Y. Sivachenko, M. E. Raikh, and Z. V. Vardeny, “Excitations in photonic crystals infiltrated with polarizable media,” Phys. Rev. A 64, 013809 (2001).
[Crossref]

Rakhmanov, A. L.

A. L. Rakhmanov, V. A. Yampolskii, J. A. Fan, F. Capasso, and F. Nori, “Layered superconductors as negative-refractive-index metamaterials,” Phys. Rev. B 81, 075101 (2010).
[Crossref]

Ramos-Mendieta, F.

D. Soto-Puebla, M. Xiao, and F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[Crossref]

Reyes-Avendaño, J. A.

J. A. Reyes-Avendaño, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Bianisotropic metamaterials based on twisted asymmetric crosses,” J. Opt. 16, 065102 (2014).
[Crossref]

Reyes-Avendao, J. A.

J. A. Reyes-Avendao, U. Algredo-Badillo, P. Halevi, and F. Pérez-Rodríguez, “From photonic crystals to metamaterials: the bianisotropic response,” New J. Phys. 13, 073041 (2011).
[Crossref]

Reyes-Coronado, A.

Ruf, T.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Sajeev, J.

T. Ovidiu and J. Sajeev, “Photonic band gap enhancement in frequency-dependent dielectrics,” Phys. Rev. E 70, 046605 (2004).
[Crossref]

Salakhutdinov, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

Sampedro, M. P.

J. A. Reyes-Avendaño, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Bianisotropic metamaterials based on twisted asymmetric crosses,” J. Opt. 16, 065102 (2014).
[Crossref]

B. Zenteno-Mateo, V. Cerdán-Ramírez, B. Flores-Desirena, M. P. Sampedro, E. Juárez-Ruiz, and F. Pérez-Rodríguez, “Effective permittivity tensor for a metal-dielectric superlattice,” Progress in Electromagnetics Research Letters 22, 165–174 (2011).

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).

Serrano, J.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photononics 1, 41–48 (2007).
[Crossref]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Sigalas, M.

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[Crossref]

Sigalas, M. M.

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Cho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[Crossref]

Sivachenko, A. Y.

A. Y. Sivachenko, M. E. Raikh, and Z. V. Vardeny, “Excitations in photonic crystals infiltrated with polarizable media,” Phys. Rev. A 64, 013809 (2001).
[Crossref]

Smith, D.

D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[Crossref] [PubMed]

Soto-Puebla, D.

D. Soto-Puebla, M. Xiao, and F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[Crossref]

Soukoulis, C. M.

A. Reyes-Coronado, M. F. Acosta, R. I. Merino, V. M. Orera, G. Kenanakis, N. Katsarakis, M. Kafesaki, Ch. Mavidis, J. García de Abajo, E. N. Economou, and C. M. Soukoulis, “Self-organization approach for THz polaritonic metamaterials,” Opt. Express 20, 14663–14682 (2012).
[Crossref] [PubMed]

C. M. Soukoulis and M. Wegener M, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).

S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Two-dimensional polaritonic photonic crystals as terahertz uniaxial metamaterials,” Phys. Rev. B 84, 035128 (2011).
[Crossref]

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101 (2006).
[Crossref]

M. M. Sigalas, C. M. Soukoulis, C. T. Chan, and K. M. Cho, “Electromagnetic-wave propagation through dispersive and absorptive photonic-band-gap materials,” Phys. Rev. B 49, 11080–11087 (1994).
[Crossref]

M. Sigalas, C. M. Soukoulis, E. N. Economou, C. T. Chan, and K. M. Ho, “Photonic band gaps and defects in two dimensions: Studies of the transmission coefficient,” Phys. Rev. B 48, 14121–14126 (1993).
[Crossref]

P. Markoš and C. M. Soukoulis, Wave Propagation. From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University Press, NJ, 2008).
[Crossref]

Stefanou, N.

G. Gantzounis and N. Stefanou, “Propagation of electromagnetic waves through microstructured polar materials,” Phys. Rev. B 75, 193102 (2007).
[Crossref]

Suski, T.

T. Ruf, J. Serrano, M. Cardona, P. Pavone, M. Pabst, M. Krisch, M. D’Astuto, T. Suski, I. Grzegory, and M. Leszczynski, “Phonon Dispersion Curves in Wurtzite-Structure GaN Determined by Inelastic X-Ray Scattering,” Phys. Rev. Lett. 86, 906–909 (2001).

Tsai, D. P.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[Crossref]

Tuttle, G.

J. Zhou, L. Zhang, G. Tuttle, T. Koschny, and C. M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101 (2006).
[Crossref]

Vardeny, Z. V.

A. Y. Sivachenko, M. E. Raikh, and Z. V. Vardeny, “Excitations in photonic crystals infiltrated with polarizable media,” Phys. Rev. A 64, 013809 (2001).
[Crossref]

Venger, E. F.

Vier, D.

D. Smith, W. Padilla, D. Vier, S. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett.84, 4184–4187 (2000).

Walser, R. M.

R. M. Walser, “Electromagnetic metamaterials,” Proc. SPIE 4467, 1–15 (2001).
[Crossref]

Wang, Z.

W. Zhang, Z. Wang, A. Hu, and N. Ming, “The photonic band structures of body-centred-tetragonal crystals composed of ionic or metal spheres,” J. Phys.: Condens. Matter 12, 5307–5316 (2000).

Wegener M, M.

C. M. Soukoulis and M. Wegener M, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).

Wood, B.

B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B 74, 115116 (2006).
[Crossref]

Xi, Y.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[Crossref]

Xiao, M.

D. Soto-Puebla, M. Xiao, and F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004).
[Crossref]

Xu, N.

W. Zhang, A. Hu, X. Lei, N. Xu, and N. Ming, “Photonic band structures of a two-dimensional ionic dielectric medium,” Phys. Rev. B 54, 10280–10283 (1996).
[Crossref]

Xu, X.

X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
[Crossref]

Yampolskii, V. A.

A. L. Rakhmanov, V. A. Yampolskii, J. A. Fan, F. Capasso, and F. Nori, “Layered superconductors as negative-refractive-index metamaterials,” Phys. Rev. B 81, 075101 (2010).
[Crossref]

Yannopapas, V.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).

Zenteno-Mateo, B.

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X. Xu, Y. Xi, D. Han, X. Liu, J. Zi, and Z. Zhu, “Effective plasma frequency in one-dimensional metallic-dielectric photonic crystals,” Appl. Phys. Lett. 86, 091112 (2005).
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V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys.: Condens. Matter 17, 3717–3734 (2005).

W. Zhang, A. Hu, and N. Ming, “The photonic band structure of the two-dimensional hexagonal lattice of ionic dielectric media,” J. Phys.: Condens. Matter 9, 541–549 (1997).

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JETP Letters (1)

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Figures (4)

Fig. 1
Fig. 1 Scheme of the bi-layer stack
Fig. 2
Fig. 2 Polaritonic spectrum of the GaN a-layer.
Fig. 3
Fig. 3 Photonic band structure in the vicinities of the four lower Fabry-Perot resonances for a GaN-Al superlattice, predicted by the nonlocal (Boltzmann) and local (Drude-Lorentz) formalisms.
Fig. 4
Fig. 4 Photonic band structure in the vicinities of the first four Fabry-Perot resonances above ωLO for a GaN-Al superlattice as in Fig. 3.

Equations (36)

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E ( x , t ) = { 0 , E ( x ) , 0 } exp ( i ω t ) , H ( x , t ) = { 0 , 0 , H ( x ) } exp ( i ω t ) .
ε a = ε ( 1 ω L O 2 ω T O 2 ω 2 ω T O 2 + i ω v 0 ) ,
n a = ε a , Z a = n a 1 , k a = k n a , φ a = k a d a ,
E a ( x ) + k a 2 E a ( x ) = 0 ,
i k H a ( x ) = E a ( x ) .
E a n ( x ) = i Z a H a n ( x a n ) cos [ k a ( x b n x ) ] sin ( k a d a ) i Z a H a n ( x b n ) cos [ k a ( x x a n ) ] sin ( k a d a ) ,
H a n ( x ) = H a n ( x a n ) sin [ k a ( x b n x ) ] sin ( k a d a ) + H a n ( x b n ) sin [ k a ( x x a n ) ] sin ( k a d a )
x b n x a n = d a , x a n + 1 x b n = d b , x a n + 1 x a n = d .
E b ( x ) + 4 π i k 2 ω j b ( x ) = 0.
j b ( x ) = d x σ ( x x ) E b ( x ) ,
σ ( x ) = 3 σ DL 4 l ω 0 1 d n x 1 n x 2 n x exp ( | x | l ω n x ) ,
j b ( x ) = σ DL E b ( x ) , σ DL = ω p 2 4 π ( v i ω ) .
E b n ( x ) = E b n ( 2 x b n x ) , E b n ( x ) = E b n ( x ± 2 d b ) .
E b n ( x ) = i k H b n ( x b n ) d b s = cos [ k s ( x x b n ) ] k s 2 k 2 ε ( k s ) + i k H b n ( x a n + 1 ) d b s = cos [ k s ( x a n + 1 x ) ] k s 2 k 2 ε ( k s ) ,
H b n ( x ) = H b n ( x b n ) d b s = k s sin [ k s ( x x b n ) ] k s 2 k 2 ε ( k s ) + H b n ( x a n + 1 ) d b s = k s sin [ k s ( x a n + 1 x ) ] k s 2 k 2 ε ( k s ) ,
i k H b ( x ) = E b ( x ) .
ε ( k s ) = ω p 2 ω ( ω + i v ) K ( k s l ω ) ,
K ( k s l ω ) = 3 2 0 1 ( 1 n x 2 ) d n x 1 + ( k s l ω n x ) 2 = 3 2 { [ 1 k s l ω + 1 ( k s l ω ) 3 ] arctan ( k s l ω ) 1 ( k s l ω ) 2 } .
K ( k s l ω ) 1 ( k s l ω ) 2 / 5 , ( k s | l ω | ) 2 1 ;
K ( k s l ω ) 3 π / 4 | k s | l ω , | k s l ω | 1.
( E ( x a n + 1 ) H ( x a n + 1 ) ) = ( Q 11 Q 12 Q 21 Q 22 ) ( E ( x a n ) H ( x a n ) ) .
Q 11 = ζ 0 ζ d cos φ a i ζ 0 2 ζ d 2 Z a ζ d sin φ a ,
Q 12 = ζ 0 2 ζ d 2 ζ d cos φ a + i Z a ζ 0 ζ d sin φ a ,
Q 21 = 1 ζ d cos φ a + i ζ 0 Z a ζ d sin φ a ,
Q 22 = ζ 0 ζ d cos φ a i Z a ζ d sin φ a .
ζ 0 = i k d b s 1 k s 2 k 2 ε ( k s ) , ζ d = i k d b s = cos ( k s d b ) k s 2 k 2 ε ( k s ) .
ζ 0 ( D L ) = i Z b cos φ b sin φ b , ζ d ( D L ) = i Z b 1 sin φ b ,
( E ( x a n + 1 ) H ( x a n + 1 ) ) = ( exp ( i κ d ) 0 0 exp ( i κ d ) ) ( E ( x a n ) H ( x a n ) ) ,
2 cos ( κ d ) = Q 11 + Q 22 .
cos ( κ d ) = ζ 0 ζ d cos φ a i Z a 2 + ζ 0 2 ζ d 2 2 Z a ζ d sin φ a .
cos ( κ d ) = cos φ a cos φ b 1 2 ( Z a Z b + Z b Z a ) sin φ a sin φ b .
| Z a | max ( | ζ 0 | , | ζ d | ) ,
Re φ a Re k n a d a = j a π , j a = 1 , 2 , 3 . ,
ν < ω < π V F ω p / c
0 < ˜ ω < ˜ ω T O .
ω L O ω .

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