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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 12 — Jun. 11, 2007
  • pp: 7720–7729
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Towards -1 effective index with one-dimensional metal-dielectric metamaterial: a quantitative analysis of the role of absorption losses

Jinlong Zhang, Haitao Jiang, Boris Gralak, Stefan Enoch, Gérard Tayeb, and Michel Lequime  »View Author Affiliations


Optics Express, Vol. 15, Issue 12, pp. 7720-7729 (2007)
http://dx.doi.org/10.1364/OE.15.007720


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Abstract

We propose a theoretical study of the optimization of one dimensional metal-dielectric metamaterials in order to approach -1 effective optical index. Taking into account actual values of dielectric constants of metal (silver) and dielectrics (HfO2, GaP), and taking advantage of the dispersion relation of Bloch modes, we get a silver/HfO2 metamaterial with suitable parameters that possesses a near −1 effective optical index for all angles of incidence at a visible wavelength for H-polarized light (i.e. the magnetic field is parallel to the interfaces). The absorption losses of materials appear to be a crucial factor that affects the effective properties of the metamaterial. We show that the losses not only decrease the transmission of the stack, but also change the negative refraction effect. Then, we propose another silver/GaP structure design that is less sensitive to losses. When considering finite thickness structures, and with adequate thickness for the terminating layers, it is possible to achieve a high transmittance of the structure. A near −1 effective index and high transmittance metal-dielectric metamaterial may pave the way to the realization of negative refraction in the visible or ultraviolet wavelength range.

© 2007 Optical Society of America

1. Introduction

Negative refraction was originally suggested by V.G. Veselago in a pioneering work devoted to the study of left-handed (LH) materials (homogenous material whose permittivity and permeability are both negative) [1

1. V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–517 (1968). [CrossRef]

]. In a seminal paper, Sir J. Pendry predicted that in addition to producing a negative refraction effect, a LH material would make it possible to build a perfect lens that is a lens not limited by the Rayleigh criterion [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

]. Obviously, such a result has attracted an important attention from the scientific community.

Although nature does not provide us with such a material, at least in the optical range, nowadays negative refraction has been experimentally verified in some metamaterials [3

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

, 4

4. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004). [CrossRef] [PubMed]

]. There are also many articles reporting theoretical and experimental study of the negative refraction in photonic crystals (PCs) [5–7

5. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000). [CrossRef]

]. However it is still a formidable challenge to create metamaterials that have suitable properties in the optical range of wavelength.

A different approach has been suggested: for the H-polarization case a metallic layer, in the electrostatic limit, may act as a perfect lens [2

2. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

]. Using such a simple structure super resolution has been experimentally exhibited [8–10

8. D. O. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127–2134 (2005). [CrossRef] [PubMed]

].

Recently, several groups have reported studies of the negative refraction and sub-wavelength imaging using one-dimensional metal-dielectric metamaterials (1DMD) [11–14

11. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

]. Obviously, 1D metamaterials are easier to fabricate and can achieve negative refraction in optical or even ultra-violet wavelength. Moreover, it has been shown that they could have better properties than a single layer [11

11. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

].

In this letter, we show that with proper design, nearly -1 effective index for all angles of incidence at a visible wavelength can be achieved using this 1DMD metamaterial. When the influence of the dissipation of the metallic and dielectric layers is considered, not only the transmission is decreased, but also the negative refraction angle (and thus the effective index) is affected.

2. Lossless materials

The structure of 1DMD is shown in Fig. 1, it is composed of alternating layers of metal and dielectric, periodic with respect to the z-axis, and invariant along the x and y directions.

Fig. 1. 1DMD metamaterial with a symmetric arrangement. The dashed line shows the unit cell.

Firstly, for the sake of simplicity, we consider a lossless Drude model to describe the dielectric function of the metallic layers, ε2 = 1 - ω2 P / ω2, ωp is the bulk plasma frequency of the metal (with λp = 2πc/ωp =250nm). A parametric study has led us to choose ε1 = 4.6 , d 1 =37.5nm (0.15 λp), d 2 =25nm (0.1 λp), and assume that the permeability is constant (μ 1 = μ 2 = 1). Note that we assume a exp(-iωt) time dependency. In the following, we will consider a symmetric unit cell as it has been shown that it allows obtaining higher transmittance [14

14. M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and sub-wavelength focusing in the visible range using transparent metallo-dielectric stacks,” Opt. Express , 15, 508–523 (2007). [CrossRef] [PubMed]

].

Fig. 2. (a). Band structure of 1DMD metamaterial. The purple zones represent the pass bands. The green dashed line represents the light line in free space. The black solid line indicates the frequency at which neff = -1 . (b) EFCs for the second band, the labels indicate the frequencies in unit of 2πc/d . Black curve: EFC of free space for ω = 0.375×2πc /d .

For this structure, we have computed the band structure and equal frequency contours (EFCs) using transfer matrix formalism (see Fig. 2) [5

5. B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000). [CrossRef]

]. In present paper, we only consider the H-polarization case.

Following the analysis developed in Ref. [15

15. P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

], the dispersion relation of 1DMD is given by:

cos(Kzd)=cos(α1d1)cos(α2d2)α12ε22+α22ε122α1α2ε1ε2sin(α1d1)sin(α2d2)
(1)

where Kz is the z-component of the Bloch wave vector and kx its x-component , d = d 1 +d 2, αi=k02εiμikx,(i=1,2), and k 0 = ω/c = 2π/λ is the modulus of wave vector in free space. The existence of a propagative Bloch mode requires |cos(Kzd)| ≤ 1.

One can see from Fig. 2(b), that at frequency ω = 0.375×2πc/d, the EFC is almost circular (indicating that the metamaterial can be considered as an isotropic medium or more precisely as a uni-axial material as we consider only one case of polarization) and neff can be well defined at this frequency. Note that the circular EFC is not centered at the origin but on the point (kx = 0,Kz = π/d) [see Figs. 2(b)]. However, if the interface delimiting the structure is perpendicular to the z-axis, then this position of the EFC has no effect on the refractive properties of the structure. Also note that the EFC’s radius decreases when the frequency v g = grad k (ω) is in the opposite direction of the wave vector, and thus shows negative refraction behavior.

We found that for ω = 0375×2πc/d2 =-0.778), the radius of EFC is almost equal to neff of this stack is nearly -1 at this frequency.

In order to analyze the negative refraction effect of 1DMD structure, we did some numerical experiments to show the field when a Gaussian beam illuminates the 1DMD metamaterial slab (a structure with a finite thickness). Note that in the considered model, the Gaussian beam is limited only along one direction, thus the incident magnetic field is given by the following relation:

Hiy=dkxexp[i(kxx+kizz)]ϕ(kx)
(2)

where

ϕ(kx)=g2πexp[g2(kxkix)24]
(3)

The incident beam centered about ki=x̂kix + ẑkiz=x̂k 0 sin θi + ẑ k 0 cos θi, θi is the incident angle, g represents the width of the waist, in the following simulation work, we choose g = 2λ.

As an example, we consider a five periods structure (total thickness is 312.5 nm, with 125 nm of metal) surrounded by free space. Figure 3 shows the time-averaged power density calculated using the transfer matrix method [16

16. J. A. Kong, B. L. Wu, and Y. Zhang, “A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability,” Microwave Opt. Technol. Lett. 33, 136–139 (2002). [CrossRef]

]. The Gaussian beam is incident from left with resp. angle of 30° and 45° [resp. Figs. 3(a) and 3(b)]. The beam is refracted at the interface according to Snell’s law (similarly to a homogenous material with neff close to -1). The transmittance is significant in a wide range of incident angle (see Fig. 8, for the lossless silver/HfO2 structure, i.e. the black solid curve). The arrows in the figure are the loci of the maximum incident, refracted and transmitted light.

Fig. 3. Time-averaged power density distribution for negative refraction of a Gaussian beam by a 5-periods stack (H polarization). (a) incident angle 30° (b) incident angle 45°. The arrow line in the map shows the locus of the maximum incident, refracted and transmitted fields. We use arbitrary units for the power density in all the maps.

From the field map, we have evaluated the refraction angle as being respectively 30.1° and 45.0° when the incident angle is 30° and 45°. It is in accordance with the results deduced from the dispersion relationship: respectively 30.4° and 45.5° [see Fig. 2(b)]. This confirms that the effective index of this stack is close to -1.

3. Effects of the absorption losses on the negative refraction

In order to take into account realistic material parameters, we now consider the losses in the metal and dielectric layers. At the operating wavelength (λ = 333.4 nm), the dielectric constants for silver is εsilver =-0.778 + 0.3i [17

17. P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

], the dielectric layer is assumed to be HfO2Hfo2 =4.6 + 0.016i) [18

18. R. Thielsch, A. Gatto, J. Heber, and N. Kaiser, “A comparative study of the UV optical and structural properties of SiO2, Al2O3, and HfO2 single layers deposited by reactive evaporation, ion-assisted deposition and plasma ion-assisted deposition,” Thin. Solid. Films 410, 86–93 (2002). [CrossRef]

]. We study the complex band structure of this metamaterial, the real component corresponds to the wave number (one can check that the average energy flow direction is given by v g = grad real(k) (ω)), while the imaginary part determines the absorption [19

19. 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 (1996). [CrossRef]

].

Fig. 4. (a). The red dashed curve is the real component of the Bloch vector of 1DMD stack with losses. The black and blue curves are the same as Fig.2 (b), i.e. for lossless materials. The dot line illustrates the conservation of the parallel component of the wave vector. (b) Imaginary component of the Bloch vector. The red dashed curve represents the case with losses and the black case with lossless materials.

Figure 4 shows the complex Bloch vector of 1DMD metamaterial taking into account losses. The real component is distorted when compared with the dispersion curve for the lossless case, especially when the incident angle is increased (large values of kx). The red arrow indicates the refraction direction (35.6°) with an incident angle of 45° for absorptive materials, whereas the blue arrow indicates the refraction direction in the lossless case. Thus, the angle of refraction is greatly affected by the presence of losses.

In Fig. 4(b), for propagative waves (i.e. -1< kx/k 0 <1), note that the imaginary part of the Bloch wave vector is null when considering lossless materials while not when absorption is not zero, this obviously results in a decrease of transmission.

Fig. 5. Identical to Fig. 3(b) but with losses in the metallic and dielectric layers.

The time-averaged power density distribution in Fig. 5 confirms the influence of the losses discussed above. The refraction angle calculated according to the field distribution is about 31°. The little difference with the angle deduced from the dispersion relation could be attributed to the fact that, first, we consider a finite thickness metamaterial and, also, to the finite width of the Gaussian beam and thus its non-null angular width.

As the losses obviously influence the refraction angle and transmittance in the metamaterial mentioned above (Silver/HfO2 metamaterial), we tried to find another structure (Silver/GaP metamaterial) that is less sensitive to loss, and still possesses a near -1 effective optical index.

From mathematical studies, it appears that the suitable transformation providing the relationship between non absorptive and absorptive dielectrics is the so-called “complex dilation” [20

20. A. Tip, A. Moroz, and J. M. Combes, “Band structure of absorptive photonic crystals,” J. Phys. A: Math. Gen. 33, 6223–6252 (2000). [CrossRef]

, 21

21. J.-M. Combes, B. Gralak, and A. Tip, “Spectral properties of absorptive photonic crystals,” in Contemporary Mathematics 339, Ed. P. Kuchment, 1–13 (2003). [CrossRef]

]. In practice, this transformation consists in performing a phase-variation for the dielectric constant of silver and GaP. Consequently, if this phase-variation is small, then the effect of losses is expected to be small also. From these arguments, we designed a structure with the dielectric constant of silver εsilver such that the ratios |Im(εsilver)/Re(εsilver)| and |Im(εGaP)/Re(εGaP)| are as small as possible.

The metamaterial is made of silver and GaP layers and the operating wavelength has been found to be 459 nm, ε1 = εsilver =-5.19 + 0.23 i (the data is measured in our lab, in order to be much closer to the experimental process), ε2GaP =14.6 + 0.1i [22

22. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1985).

] and d 1=46.7nm, d 2 =26.7 nm. Figure 6 shows the dispersion curve of this metamaterial (EFC).

Fig. 6. Identical to Fig. 4, but for the silver/GaP metamaterial.

Since the period of this type of metamaterial is thicker than the period of the silver/HfO2 metamaterial, we consider a 4-periods stack in order to have a similar total thickness (the total thickness of metal is 186.8 nm, the total thickness of dielectric is 106.8 nm and the total thickness of the metamaterial slab is 293.6 nm). Figure 7 shows the time-averaged power density distribution with an incident angle equal to 45°. One can see that the refraction angle change is hardly visible when considering losses or not. The calculated refraction angle is 39.2° for the metamaterial without losses, and 33.3° with losses which is comparable to the values obtained from the dispersion relation.

Fig. 7. Time-averaged power density distribution of negative refraction with the 4-periods silver/Gap metamaterial without or with losses as the angle of incidence is 45°.
Fig. 8. Transmittance of silver/HfO2 and silver/GaP metamaterials. The solid curves are for 5-periods silver/HfO2 metamaterial, the dashed curves are for 4-periods silver/GaP metamaterial. The black two are the results without loss, the red are with losses.

Figure 8 shows the transmittance through the two different metamaterials (silver/HfO2 and silver/GaP) with respect to the parallel wave vectors kx . Comparing the transmittances of the two types of metamaterials shows that it is almost unity in a wide range of kx when losses are not considered. However, when losses are taken into account, the transmittance is less sensitive to the angle of incidence for the silver/GaP metamaterial.

4. Conclusion

In conclusion, we have shown that near -1 effective index can be achieved using 1DMD metamaterials with a proper choice of parameters. Considering realistic material parameters, we have presented a study of the complex band structure and shown that the absorption losses in metallic layers not only decrease the transmittance, but also change the negative refraction angle in the metamaterial (and thus its effective optical index). Then, we have proposed another 1DMD metamaterial less sensitive to the losses. Therefore a near -1 effective index 1DMD stack with high transmittance can be realized using classical film deposition fabrication technology.

References and links

1.

V. G. Veselago, “Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities,” Sov. Phys. Usp. 10, 509–517 (1968). [CrossRef]

2.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

3.

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

4.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306, 1351–1353 (2004). [CrossRef] [PubMed]

5.

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000). [CrossRef]

6.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696 (2000). [CrossRef]

7.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: Negative refraction by photonic crystals,” Nature (London) 423, 604–605 (2003). [CrossRef]

8.

D. O. Melville and R. J. Blaikie, “Super-resolution imaging through a planar silver layer,” Opt. Express 13, 2127–2134 (2005). [CrossRef] [PubMed]

9.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-Diffraction-Limited Optical Imaging with a Silver Superlens,” Science 308, 534–537 (2005). [CrossRef] [PubMed]

10.

T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313, 1595 (2006). [CrossRef] [PubMed]

11.

S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. 50, 1419–1430 (2003).

12.

P. A. Belov and H. Yang, “Subwavelength imaging at optical frequencies using a transmission device formed by a period layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006). [CrossRef]

13.

H. Shin and S. H. Fan, “All-angle negative refraction and evanescent wave amplification using one-dimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89, 151102 (2006). [CrossRef]

14.

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and sub-wavelength focusing in the visible range using transparent metallo-dielectric stacks,” Opt. Express , 15, 508–523 (2007). [CrossRef] [PubMed]

15.

P. Yeh, A. Yariv, and C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977). [CrossRef]

16.

J. A. Kong, B. L. Wu, and Y. Zhang, “A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability,” Microwave Opt. Technol. Lett. 33, 136–139 (2002). [CrossRef]

17.

P. B. Johnson and R. W. Christy, “Optical constants of noble metals,” Phys. Rev. B 6, 4370–4379 (1972). [CrossRef]

18.

R. Thielsch, A. Gatto, J. Heber, and N. Kaiser, “A comparative study of the UV optical and structural properties of SiO2, Al2O3, and HfO2 single layers deposited by reactive evaporation, ion-assisted deposition and plasma ion-assisted deposition,” Thin. Solid. Films 410, 86–93 (2002). [CrossRef]

19.

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 (1996). [CrossRef]

20.

A. Tip, A. Moroz, and J. M. Combes, “Band structure of absorptive photonic crystals,” J. Phys. A: Math. Gen. 33, 6223–6252 (2000). [CrossRef]

21.

J.-M. Combes, B. Gralak, and A. Tip, “Spectral properties of absorptive photonic crystals,” in Contemporary Mathematics 339, Ed. P. Kuchment, 1–13 (2003). [CrossRef]

22.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1985).

OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(310.6860) Thin films : Thin films, optical properties

ToC Category:
Metamaterials

History
Original Manuscript: April 20, 2007
Revised Manuscript: May 18, 2007
Manuscript Accepted: May 28, 2007
Published: June 7, 2007

Citation
Jinlong Zhang, Haitao Jiang, Boris Gralak, Stefan Enoch, Gérard Tayeb, and Michel Lequime, "Towards -1 effective index with one-dimensional metal-dielectric metamaterial: a quantitative analysis of the role of absorption losses," Opt. Express 15, 7720-7729 (2007)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7720


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References

  1. V. G. Veselago, "Electrodynamics of substances with simultaneously negative electrical and magnetic permeabilities," Sov. Phys. Usp. 10, 509-517 (1968). [CrossRef]
  2. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  3. R. A. Shelby, D. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  4. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic response of metamaterials at 100 Terahertz," Science 306, 1351-1353 (2004). [CrossRef] [PubMed]
  5. B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A 17, 1012-1020 (2000). [CrossRef]
  6. M. Notomi, "Theory of light propagation in strongly modulated photonic crystals: refractionlike behavior in the vicinity of the photonic band gap," Phys. Rev. B 62, 10696 (2000). [CrossRef]
  7. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, "Electromagnetic waves: Negative refraction by photonic crystals," Nature (London) 423, 604-605 (2003). [CrossRef]
  8. D. O. Melville, R. J. Blaikie, "Super-resolution imaging through a planar silver layer," Opt. Express 13, 2127-2134 (2005). [CrossRef] [PubMed]
  9. N. Fang, H. Lee, C. Sun, and X. Zhang, "Sub-Diffraction-Limited Optical Imaging with a Silver Superlens," Science 308, 534-537 (2005). [CrossRef] [PubMed]
  10. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, "Near-field microscopy through a SiC superlens," Science 313, 1595 (2006). [CrossRef] [PubMed]
  11. S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, "Imaging the near field," J. Mod. Opt. 50, 1419-1430 (2003).
  12. P. A. Belov and H. Yang, "Subwavelength imaging at optical frequencies using a transmission device formed by a period layered metal-dielectric structure operating in the canalization regime," Phys. Rev. B 73,113110 (2006). [CrossRef]
  13. H. Shin and S. H. Fan, "All-angle negative refraction and evanescent wave amplification using one-dimensional metallodielectric photonic crystals," Appl. Phys. Lett. 89, 151102 (2006). [CrossRef]
  14. M. Scalora, G. D'Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, A. Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, "Negative refraction and sub-wavelength focusing in the visible range using transparent metallo-dielectric stacks," Opt. Express,  15, 508-523 (2007). [CrossRef] [PubMed]
  15. P. Yeh, A. Yariv, and C. S. Hong, "Electromagnetic propagation in periodic stratified media. I. General theory," J. Opt. Soc. Am. 67, 423-438 (1977). [CrossRef]
  16. J. A. Kong, B. L. Wu, Y. Zhang, "A unique lateral displacement of a Gaussian beam transmitted through a slab with negative permittivity and permeability," Microwave Opt. Technol. Lett. 33,136-139 (2002). [CrossRef]
  17. P. B. Johnson and R. W. Christy, "Optical constants of noble metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
  18. R. Thielsch, A. Gatto, J. Heber, N. Kaiser, "A comparative study of the UV optical and structural properties of SiO2, Al2O3, and HfO2 single layers deposited by reactive evaporation, ion-assisted deposition and plasma ion-assisted deposition," Thin. Solid. Films 410, 86-93 (2002). [CrossRef]
  19. V. Kuzmiak, 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 (1996). [CrossRef]
  20. A. Tip, A. Moroz and J. M. Combes, "Band structure of absorptive photonic crystals," J. Phys. A: Math. Gen. 33, 6223-6252 (2000). [CrossRef]
  21. J.-M. Combes B. Gralak and A. Tip, "Spectral properties of absorptive photonic crystals,' in Contemporary Mathematics 339, Ed. P. Kuchment, 1-13 (2003). [CrossRef]
  22. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1985).

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