## Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials |

Optics Express, Vol. 21, Issue 14, pp. 16514-16527 (2013)

http://dx.doi.org/10.1364/OE.21.016514

Acrobat PDF (11228 KB)

### Abstract

The feature of spatial dispersion in periodic layered metamaterials is theoretically investigated. An effective medium model is proposed to derive the nonlocal effective permittivity tensor, which exhibits drastic variations in the wave vector domain. Strong spatial dispersion is found in the frequency range where surface plasmon polaritons are excited. In particular, the nonlocal effect gives rise to additional waves that are identified as the bonding or antibonding modes with symmetric or antisymmetric surface charge alignments. Spatial dispersion is also manifest on the parabolic-like dispersion, a non-standard type of dispersion in the medium. The associated negative refraction and backward wave occur even when the effective permittivity components are all positive, which is considered a property not available in the local medium.

© 2013 OSA

## 1. Introduction

28. P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B **67**, 113103 (2003) [CrossRef] .

29. C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E **70**, 046616 (2004) [CrossRef] .

30. F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C **112**, 17983–17987 (2008) [CrossRef] .

*effective*permittivity that reveals the spatial dispersion or nonlocal effect in an explicit manner [32

32. 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] .

34. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B **86**, 115420 (2012) [CrossRef] .

35. 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] .

36. A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B **79**, 245127 (2009) [CrossRef] .

37. X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. **97**, 073901 (2006) [CrossRef] [PubMed] .

38. E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett. **105**, 223901 (2010) [CrossRef] .

## 2. Effective medium model

*x*direction, one with the dielectric constant

*ε*

_{1}and thickness

*a*

_{1}, and the other with

*ε*

_{2}and

*a*

_{2}, as schematically shown in Fig. 1. Assume that the electromagnetic wave is of the form

*e*

^{i(k·r−ωt)}. Let the wave vector lie on the

*xz*plane, that is,

**k**= (

*k*, 0,

_{x}*k*), without loss of generality. The dispersion relation of the periodic layered structure is given by [31] for TM and TE polarizations, respectively, where

_{z}*a*=

*a*

_{1}+

*a*

_{2}, and

*k*

_{0}=

*ω/c*. Here, TM and TE refer to transverse magnetic, where

**E**= (

*E*, 0,

_{x}*E*) and

_{z}**H**= (0,

*H*, 0), and transverse electric, where

_{y}**E**= (0,

*E*, 0) and

_{y}**H**= (

*H*, 0,

_{x}*H*), respectively.

_{z}### 2.1. Nonlocal effective permittivity tensor

*a*≪

*λ*= 2

*π/k*

_{0}, so that the layered structure can be regarded as an

*effective*medium, characterized by the effective permittivity tensor: where

*k*,

_{x}*k*, and

_{z}*k*

_{0}up to fourth order, truncating the higher order terms, and rearranging the expansions in the forms of Eqs. (4) and (5), respectively, we have where are the

*quasistatic*effective permittivities based on the Maxwell-Garnett mixing rule [41

41. J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London **203**, 385 (1904) [CrossRef] .

*f*

_{1}=

*a*

_{1}/

*a*and

*f*

_{2}=

*a*

_{2}/

*a*, and the material parameters,

*ε*

_{1}and

*ε*

_{2}.

*y*and

*z*directions. The medium properties, therefore, depend on the polarization. Note also that

*n*= 1, 2) appear in the effective permittivities [cf. Eqs. (6) and (8)], meaning that the underlying effective medium is

*spatially dispersive*or

*nonlocal*. The nonlocal effective permittivities differ from the quasistatic ones by the factors depending on the structure and material parameters. In addition, there are no

*k*and

_{x}*k*terms in the effective permittivities, which is a consequence of

_{z}*inversion symmetry*[42

42. R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. **171**, 1065–1074 (1968) [CrossRef] .

*a*≪

*λ*. The benefit of the present approach is to express the effective permittivities in compact formulas. A more rigorous approach to the effective parameters may resort to homogenization based on the field averaging [33

33. A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B **84**, 115438 (2011) [CrossRef] .

34. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B **86**, 115420 (2012) [CrossRef] .

### 2.2. Characteristic frequencies

*ω*is the plasma frequency of the metal, the quasistatic effective permittivities are given as where are the

_{p}*zero*frequency of

*pole*frequency of

*ω*

_{0}(

*ω*

_{∞}) is reduced (raised) as the ratio

*f*

_{1}/

*f*

_{2}increases. The zero frequency of

*ω*, which is the same as that of

_{p}*ε*

_{2}. In the quasistatic limit, the layered structure is regarded as a local anisotropic medium, which behaves like a plasmonic material in the parallel (to the metal-dielectric interface) direction, with a

*reduced*plasma frequency

*ω*

_{0}[cf. Eq. (14)], and like an ionic crystal in the perpendicular direction, with a transverse resonance frequency

*ω*

_{∞}and a longitudinal resonant frequency

*ω*[cf. Eq. (15)].

_{p}*ω*

_{0}, and

*ω*

_{0}and the other is close to

*ω*. In addition, there are two pole frequency branches of

_{p}*ω*

_{1}being the lowest frequency (at

*k*= 0), and the other is located at higher frequencies, with

_{z}*ω*

_{2}being the highest frequency (also at

*k*= 0). They are close to either

_{z}*ω*

_{0}or

*ω*

_{∞}, depending on the fraction of dielectric (or metal) in the unit cell. Figure 2(a) is an example of the variations of

*ω*

_{1}and

*ω*

_{2}, along with

*ω*

_{0}and

*ω*

_{∞}, with respect to

*f*

_{1}.

*a*, the wavelength

*λ*= 2

*π/k*

_{0}, and the plasma wavelength

*λ*= 2

_{p}*πc/ω*. In order for the effective medium to be valid,

_{p}*a/λ*= (

*a/λ*)(

_{p}*ω/ω*) should be, in principle, much smaller than unity. In practice, this condition is attained when

_{p}*a/λ*is small and

_{p}*ω*is below

*ω*. The parameters used in this article will be carefully arranged so that

_{p}*a/λ*is considered small enough, yet still feasible in fabrication with the modern nanotechnology.

## 3. In-plane effective permittivities

*ω*and

*k*or

_{x}*k*for the periodic metal-dielectric layered structure with

_{z}*f*

_{1}>

*f*

_{2}, based on the effective medium model (cf. Sec. 2) with the expansions of the wave vector components to eighth order. Another example of

*f*

_{1}<

*f*

_{2}is shown in Fig. 4. In either case, the effective permittivity

*k*as well as the frequency

_{x}*ω*. For periodic structures, the largest value of |

*k*| is determined by the lattice period:

_{x}*π/a*, and

*k*| increases. The nonlocal effect exhibited by

_{x}*k*, which is not restricted by the geometry,

_{z}*k*= 0). For a given frequency,

_{z}*k*|, which is considered a typical feature of

_{z}*nonlocal resonance*that occurs in a spatially dispersive medium. At

*k*= 0, the onset of resonance occurs at

_{z}*ω*

_{1}and

*ω*

_{2}. A similar feature can be observed in the quasistatic effective permittivity

*ω*

_{∞}). As

*k*≠ 0, the resonance frequencies gradually move toward each other. In the metal-dielectric layered structure, such resonance, as will be shown later, is attributable to the excitation of surface plasmon polaritons.

_{z}### 3.1. Weak spatial dispersion range: 0 < ω < ω_{1}

*ω*

_{1}is the lowest frequency of the lower pole branch of

*ω*

_{0}for

*f*

_{1}>

*f*

_{2}and close to

*ω*

_{∞}for

*f*

_{1}<

*f*

_{2}[cf. Fig. 2(a)]. The dispersion relation [cf. Eq. (4)] is dominated by the

*xy*plane be an interface between vacuum and the effective medium. For a wave incident from vacuum, with the wave vector lying on the

*xz*plane, a forward wave with negative refraction will occur in the medium. This feature is similar as in a uniaxially anisotropic medium with opposite signs of the permittivity components, the one normal to the interface (between vacuum and the medium) being negative [40

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. **37**, 259–263 (2003) [CrossRef] .

### 3.2. Strong spatial dispersion range: ω_{1} < ω < ω_{2}

*f*

_{1}>

*f*

_{2}[cf. Fig. 3]. Here,

*ω*

_{2}is the highest frequency of the higher pole branch of

*ω*

_{∞}for

*f*

_{1}>

*f*

_{2}[cf. Fig. 2(a)]. At small |

*k*|, both

_{z}*k*|,

_{z}*k*, both of the allowed

_{x}*k*, either positive or negative, can be real. For a wave incident from vacuum onto the layered structure, the splitting of wave in the structure is expected to occur [24

_{z}24. A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B **84**, 045424 (2011) [CrossRef] .

*f*

_{1}<

*f*

_{2}, on the other hand,

*ω*

_{2}is close to

*ω*

_{0}[cf. Fig. 2(a)]. At small |

*k*|, both

_{z}*k*, only one allowed

_{x}*k*, either positive or negative, is real. Therefore, there is only one eigenwave, with the parabolic-like dispersion. Regarding the wave propagation characteristics, this range is further divided into two subranges:

_{z}#### 3.2.1. Below surface plasma frequency: *ω*_{1} < *ω* < *ω*_{sp}

_{sp}

**Example 1:**

*f*

_{1}>

*f*

_{2}. Figure 5(a) is an example showing the equifrequency contours, along with the wave vectors and Poynting vectors, for a larger dielectric fraction. The contour is characterized by a quartic curve in the

*xy*plane: where

*a*and

*b*are constants, and

*ε*is a small positive number. The parabolic-like character of the quartic curve comes from the

*x*

^{4}term at larger

*x*, which corresponds to the nonlocal effect in the present problem. The elliptic-like character, on the other hand, is dominant at smaller

*x*. Note that the contour normal (denoted by the green arrow) with the parabolic-like dispersion is oriented toward the same side of the incident wave vector (with respect to the interface normal), leading to negative refraction. The corresponding eigenmode, plotted in Fig. 6(a), shows a typical feature of

*surface plasmon polariton*, with the fields highly concentrated on the metal-dielectric interfaces. In particular, this mode has a symmetric alignment of surface charges and is identified as the

*bonding mode*. The colors of surface charges in the figure have been exaggerated to show more clearly the alignment pattern. Note also that this mode is located (marked by the red dot) near the pole of

*k*| increases, there is a large discrepancy between

_{z}40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. **37**, 259–263 (2003) [CrossRef] .

**Example 2:**

*f*

_{1}<

*f*

_{2}. Figure 8(a) is an example for a smaller dielectric fraction. In this case, only one eigenwave with negative refraction exists. The character of the wave, however, is different from the counterpart for

*f*

_{1}>

*f*

_{2}[cf. Fig. 5(a)]. The equifrequency contour is characterized by a similar quartic curve in the

*xy*plane as in Eq. (22), but with the minus sign on the right side: The parabolic-like character remains, while the elliptic-like character disappears. The corresponding eigenmode also shows a typical feature of surface plasmon polariton, but with an antisymmetric alignment of surface charges on the metal-dielectric interfaces, which is identified as the

*antibonding mode*, as shown in Fig. 9(a). The feature of negative refraction is consistent with that in a uniaxially anisotropic medium with opposite signs of the permittivity components:

40. P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. **37**, 259–263 (2003) [CrossRef] .

*ω*<

*ω*

_{1}(cf. Sec. 3.1). The spatial dispersion associated with this mode, however, is still strong. The effective permittivity

*f*

_{1}>

*f*

_{2}[cf. Fig. 7(a)]. The eigenmode is also located near the pole, where

#### 3.2.2. Above surface plasma frequency: *ω*_{sp} < *ω* < *ω*_{2}

_{sp}

*ω*

_{1}<

*ω*<

*ω*(cf. Sec. 3.2.1). The wave associated with the parabolic-like dispersion, however, is very much different. Two illustrative examples with different dielectric (metal) fractions in the unit cell are given below:

_{sp}**Example 3:**

*f*

_{1}>

*f*

_{2}. Figure 5(b) is an example showing the equifrequency contours, along with the wave vectors and Poynting vectors, for a larger dielectric fraction. Due to the anomalous frequency dispersion,

*k*for the parabolic-like dispersion is chosen to be negative, so that the energy flows away from the interface (+

_{z}*z*direction). Otherwise, the principle of causality will be violated. The wave with the parabolic-like dispersion is therefore a backward wave with ordinary refraction, rather than the forward wave with negative refraction as in the range below

*ω*[cf. Fig. 5(a)]. The eigenmode in Fig. 6(b) is shown to be an antibonding mode of surface plasmon polariton, with an antisymmetric alignment of surface charges on the metal-dielectric interfaces. Note that the antibonding mode has a higher frequency than the respective bonding mode [cf. Fig. 6(a)]. This feature is similar as in a metal film in vacuum [39

_{sp}39. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**, 539–554 (1969) [CrossRef] .

*δω*, the patterns of

**37**, 259–263 (2003) [CrossRef] .

**Example 4:**

*f*

_{1}<

*f*

_{2}. Figure 8(b) is an example for a smaller dielectric fraction. In this case, there is only one eigenwave with the parabolic-like dispersion, as in the range below

*ω*[cf. Fig. 8(a)]. The eigenwave, however, is a backward wave with ordinary refraction, as in the range above

_{sp}*ω*for

_{sp}*f*

_{1}>

*f*

_{2}[cf. Fig. 5(b)]. The character of the wave is different from either case. The eigenmode in Fig. 9(b) is shown to be a bonding mode of surface plasmon polariton, with a symmetric alignment of surface charges on the metal-dielectric interfaces. Note that the bonding mode has a higher frequency than the respective antibonding mode [cf. Fig. 9(a)]. This feature is similar as in an insulating film between two semi-infinite metals [39

39. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**, 539–554 (1969) [CrossRef] .

**37**, 259–263 (2003) [CrossRef] .

*ω*. The effective permittivity

_{sp}### 3.3. Weak spatial dispersion range: ω_{2} < ω < ω_{p}

*ω*<

_{sp}*ω*<

*ω*

_{2}(cf. Sec. 3.2.2). The spatial dispersion, however, is weak in this range. The effective permittivities do not change much with the wave vector components.

### 3.4. Weak spatial dispersion range: ω > ω_{p}

## 4. Out-of-plane effective permittivity

*f*

_{1}>

*f*

_{2}and

*f*

_{1}<

*f*

_{2}, the out-of-plane effective permittivity

*k*| and |

_{z}*k*|,

_{x}*ω*

_{0}. This frequency is very close to the cutoff frequency

*ω*

_{3}for TE polarization, the lowest frequency for TE waves to propagate in the effective medium [cf. Fig. 2(b)]. For larger |

*k*| and |

_{z}*k*|,

_{x}### 4.1. Below cutoff frequency: ω < ω_{3}

*k*and

_{z}*k*, based on the effective medium model [cf. Sec. 2] for

_{x}*ω*<

*ω*

_{3}. Note that

*k*| and |

_{z}*k*|, which becomes more negative as |

_{x}*k*| increases and changes to positive as |

_{z}*k*| increases (the contour of

_{x}*ω*

_{3}, there are no waves allowed to propagate in the effective medium. The spatial dispersion or nonlocal effect is not relevant.

### 4.2. Above cutoff frequency: ω > ω_{3}

*ω*>

*ω*

_{3}. Note that

*k*| and |

_{z}*k*|, which may change to negative as |

_{x}*k*| increases and becomes more positive as |

_{z}*k*| increases. Unlike the in-plane effective permittivities, where the nonlocal effect comes mainly from

_{x}*k*, the nonlocal effect associated with the out-of-plane effective permittivity basically comes from the dependence on

_{z}*k*. This effect is usually weak as |

_{x}*k*| is bound by

_{x}*π/a*. The equifrequency contours can be characterized by a quartic curve in the

*xy*plane: where

*r*is a constant and

*ε*is a small positive number. At smaller

*x*and

*y*, the contours are nearly circles, while at larger

*x*and

*y*, the contours are a bit of distorted along the

*y*axis. The dispersion relation is elliptic-like, with a normal frequency dispersion. The eigenwave is a forward wave with ordinary refraction. The spatial dispersion exhibited by

*k*| and thus no waves exist there. The effect due to

_{z}*k*, however, is weak, as |

_{x}*k*| is bound by

_{x}*π/a*.

## 5. Summary on spatial dispersion

## 6. Concluding remarks

## Acknowledgments

## References and links

1. | J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev. |

2. | L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, |

3. | V. M. Agranovich and V. L. Ginzburg, |

4. | J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev. |

5. | S. I. Pekar, “Theory of electromagnetic waves in a crystal with excitons,” J. Phys. Chem. Solids |

6. | R. Ruppin, “Reflectivity of a nonlocal dielectric with an excitonic surface potential,” Phys. Rev. B |

7. | B. Chen and D. F. Nelson, “Wave propagation of exciton polaritons by a wave-vector-space method,” Phys. Rev. B |

8. | G. E. H. Reuter and E. H. Sondheimer, “The theory of the anomalous skin effect in metals,” Proc. R. Soc. A-Math. Phys. Eng. Sci. |

9. | K. L. Kliewer and R. Fuchs, “Anomalous skin effect for specular electron scattering and optical experiments at non-normal angles of incidence,” Phys. Rev. |

10. | R. Ruppin, “Optical properties of a spatially dispersive cylinder,” J. Opt. Soc. Am. B |

11. | R. Ruppin, “Extinction properties of thin metallic nanowires,” Opt. Comm. |

12. | A. A. Maradudin and D. L. Mills, “Effect of spatial dispersion on the properties of a semi-infinite dielectric,” Phys. Rev. B |

13. | A. R. Melnyk and M. J. Harrison, “Resonant excitation of plasmons in thin films by elecromagnetic waves,” Phys. Rev. Lett. |

14. | W. E. Jones, K. L. Kliewer, and R. Fuchs, “Nonlocal theory of the optical properties of thin metallic films,” Phys. Rev. |

15. | A. R. Melnyk and M. J. Harrison, “Theory of optical excitation of plasmons in metals,” Phys. Rev. B |

16. | R. Ruppin, “Optical properties of small metal spheres,” Phys. Rev. B |

17. | V. Yannopapas, “Non-local optical response of two-dimensional arrays of metallic nanoparticles,” J. Phys.-Condes. Matter |

18. | R. J. Pollard, A. Murphy, W. R. Hendren, P. R. Evans, R. Atkinson, G. A. Wurtz, A. V. Zayats, and V. A. Podolskiy, “Optical nonlocalities and additional waves in epsilon-near-zero metamaterials,” Phys. Rev. Lett. |

19. | J. M. McMahon, S. K. Gray, and G. C. Schatz, “Nonlocal optical response of metal nanostructures with arbitrary shape,” Phys. Rev. Lett. |

20. | J. M. McMahon, S. K. Gray, and G. C. Schatz, “Optical properties of nanowire dimers with a spatially nonlocal dielectric function,” Nano Lett. |

21. | S. Raza, G. Toscano, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Unusual resonances in nanoplasmonic structures due to nonlocal response,” Phys. Rev. B |

22. | B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hubner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. |

23. | S. I. Pekar and O. D. Kocherga, |

24. | A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B |

25. | M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Phys. Rev. B |

26. | D. F. Nelson, “Generalizing the Poynting vector,” Phys. Rev. Lett. |

27. | M. A. Vladimir and N. G. Yu, “Spatial dispersion and negative refraction of light,” Phys. Usp. |

28. | P. A. Belov, R. Marques, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B |

29. | C. R. Simovski and P. A. Belov, “Low-frequency spatial dispersion in wire media,” Phys. Rev. E |

30. | F. J. Garcia de Abajo, “Nonlocal effects in the plasmons of strongly interacting nanoparticles, dimers, and waveguides,” J. Phys. Chem. C |

31. | L. Brillouin, |

32. | J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. |

33. | A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B |

34. | A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B |

35. | B. Wood, J. B. Pendry, and D. P. Tsai, “Directed subwavelength imaging using a layered metal-dielectric system,” Phys. Rev. B |

36. | A. Fang, T. Koschny, and C. M. Soukoulis, “Optical anisotropic metamaterials: Negative refraction and focusing,” Phys. Rev. B |

37. | X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. |

38. | E. Verhagen, R. de Waele, L. Kuipers, and A. Polman, “Three-dimensional negative index of refraction at optical frequencies by coupling plasmonic waveguides,” Phys. Rev. Lett. |

39. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

40. | P. A. Belov, “Backward waves and negative refraction in uniaxial dielectrics with negative dielectric permittivity along the anisotropy axis,” Microw. Opt. Technol. Lett. |

41. | J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London |

42. | R. M. Hornreich and S. Shtrikman, “Theory of gyrotropic birefringence,” Phys. Rev. |

**OCIS Codes**

(230.4170) Optical devices : Multilayers

(240.6680) Optics at surfaces : Surface plasmons

(260.2065) Physical optics : Effective medium theory

**ToC Category:**

Metamaterials

**History**

Original Manuscript: April 30, 2013

Revised Manuscript: June 26, 2013

Manuscript Accepted: June 26, 2013

Published: July 2, 2013

**Citation**

Ruey-Lin Chern, "Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials," Opt. Express **21**, 16514-16527 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-14-16514

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### References

- J. J. Hopfield and D. G. Thomas, “Theoretical and experimental effects of spatial dispersion on the optical properties of crystals,” Phys. Rev.132, 563–572 (1963). [CrossRef]
- L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Electrodynamics of Continuous Media, 2nd ed. (Butterworth-Heinenan, 1984).
- V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons (Springer-Verlag, 1984). [CrossRef]
- J. J. Hopfield, “Theory of the contribution of excitons to the complex dielectric constant of crystals,” Phys. Rev.112, 1555–1567 (1958). [CrossRef]
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