## Nonlocal effect on optic spectrum of a periodic dielectric-metal stack |

Optics Express, Vol. 22, Issue 7, pp. 7581-7586 (2014)

http://dx.doi.org/10.1364/OE.22.007581

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### Abstract

On the basis of the formalism of the Boltzmann kinetic equation for the distribution function of the conduction electrons, the photonic band structure of binary dielectric-metal superlattice is theoretically studied. Using the constitutive nonlocal relation between the electrical current density and the electric field inside the metallic layer, the dispersion equation for photonic eigenmodes in the periodic stack is analytically expressed in terms of the surface impedances at the interfaces of the metal and dielectric layers. In the case of very thin metallic layers, the optic spectrum for the superlattice exhibits narrow pass bands as a result of the strong contrast between the impedances of the dielectric and the metal. The narrow pass bands are attributed 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. Our results for the photonic spectrum have been compared with those obtained within the local Drude-Lorentz model. Noticeably differences not only in the the magnitude, but also in the sign of the real part of the Bloch wave number in the Fabry-Perot resonance bands, have been found.

© 2014 Optical Society of America

## 1. Introduction

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

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

4. 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]

8. 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]

## 2. Problem Formulation: Basic Relations

*a*- and metallic

*b*-layers, see Fig. 1. Every kind of slabs has the constant thickness,

*d*or

_{a}*d*, respectively. Thus, the size

_{b}*d*=

*d*+

_{a}*d*of a unit (

_{b}*a*,

*b*) cell is also the period of the bi-layer stack. The dielectric

*a*-slabs are specified by permittivity

*ε*, permeability

_{a}*μ*, corresponding refractive index

_{a}*Z*=

_{a}*μ*/

_{a}*n*, wave number

_{a}*k*=

_{a}*n*and wave phase shift

_{a}k*φ*=

_{a}*k*(

_{a}d_{a}*k*=

*ω*/

*c*). We assume the permeability of the metallic

*b*-layers to be constant

*μ*, their other optic parameters will be introduced below.

_{b}*ω*propagates perpendicularly to the stack with the electric and magnetic components as displayed in Fig. 1,

**E**(

*x*,

*t*) = {0,

*E*(

*x*),0} exp(−

*iωt*),

**H**(

*x*,

*t*) = {0,0,

*H*(

*x*)} exp(−

*iωt*). In such a geometry, within every

*a*- or

*b*-layer the relation between the electric

*E*(

*x*) and magnetic

*H*(

*x*) fields reads

*ikμ*

_{a}_{,}

_{b}H_{a}_{,}

*(*

_{b}*x*) =

*E′*

_{a}_{,}

*(*

_{b}*x*), where the prime implies the derivative with respect to

*x*.

*a*-slabs the electric field

*E*(

_{a}*x*) obeys the one-dimensional Helmholtz equation. Its general solution for the

*n*-th unit (

*a*,

*b*) cell can be presented as a superposition of forward and backward traveling plane waves, inside

*a*-layer, where

_{n}*x*

_{an}⩽

*x*⩽

*x*

_{bn}. Here

*x*

_{an}and

*x*

_{bn}refer to the left-hand edges of successive

*a*- and

_{n}*b*-layers, respectively. The thicknesses of individual layers are defined as

_{n}*x*

_{bn}

*− x*

_{an}=

*d*and

_{a}*x*

_{an+1}−

*x*

_{bn}=

*d*. In accordance with the results of [3

_{b}3. A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. **90**, 623–627 (2009). [CrossRef]

*E*(

_{b}*x*) relevant to the nonlocal effect is given by

*b*-layer, where

_{n}*x*

_{bn}⩽

*x*⩽

*x*

_{an+1}. Expression (1b) represents a Fourier series of the normal electromagnetic modes with discrete wave number

*k*=

_{s}*πs*/

*d*. The interaction of

_{b}*s*-th mode with the conduction electrons of metallic slabs is specified by its own permittivity

*ε*(

*k*), which is

_{s}*not a permittivity associated with the total electromagnetic field*. The mode permittivity

*ε*(

*k*) depends on the mode wave number

_{s}*k*via the

_{s}*nonlocality factor 𝒦*(

*k*), Here

_{s}l_{ω}*ω*,

_{p}*ν*and

*V*are, respectively, the plasma frequency, relaxation rate and the Fermi velocity of the electrons;

_{F}*l*=

_{ω}*V*/(

_{F}*ν − iω*) implies the effective mean free path of the electrons due both to their collisions with scatters and to the phase change of the electromagnetic field.

*k*, the factor (3) entirely defines the spatial dispersion effect in the mode permittivity (2). Owing to this, it is worthwhile to write down its asymptotics, Equation (4) shows that the nonlocal effect can be negligible only when condition (4a) is met for all the modes contributing to Eq. (1b). Since in this case

_{s}*ε*(

*k*) is the same for all the modes and coincides with that for the Drude-Lorentz model, the latter can be applied for the electrodynamic description of a metallic slab.

_{s}*x*=

*x*

_{bn}and

*x*=

*x*

_{an+1}, yields the recurrent relation describing the wave transfer through the whole

*n*-th unit (

*a*,

*b*) cell, The transfer matrix

*Q̂*has the following elements Note that the determinant of

*Q̂*-matrix equals to unit, det

*Q̂*= 1. For a periodic stack, the transfer

*Q̂*-matrix is independent of the cell index

*n*since all the unit cells are identical.

*φ*and the impedance

_{a}*Z*of the dielectric

_{a}*a*-layer, as well as, by the surface impedances

*ζ*

_{0}and

*ζ*of the left-hand and right-hand boundaries of the metallic

_{d}*b*-slab, Note the different physical meaning of the introduced impedances: while

*Z*is the surface impedance corresponding to a half-space (with only one surface), the surfaces impedances

_{a}*ζ*

_{0}and

*ζ*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

_{d}*𝒦*= 1 for all normal modes, and the sums in Eq. (7) can be explicitly calculated resulting in with the optic parameters of metallic

*b*-slabs: impedance

*Z*=

_{b}*μ*/

_{b}*n*, phase shift

_{b}*φ*=

_{b}*k*, wave number

_{b}d_{b}*k*=

_{b}*n*, refractive index

_{b}k*ε*=

_{b}*ε*(0), see Eq. (2).

*κ*of a one-dimensional periodic structure is determined by the trace of its unit-cell transfer matrix, 2cos(

*κd*) =

*Q*

_{11}+

*Q*

_{22}. With the use of Eq. (6), one can readily obtain Thus, the optic spectrum

*κ*(

*ω*) of the dielectric-metal periodic stack is defined by the universal dispersion relation (9). The difference of the kinetic approach from the Drude-Lorentz model emerges merely in the metallic impedances

*ζ*

_{0}and

*ζ*. The transition from the kinetic approach to the Drude-Lorentz approximation is performed by the replacement

_{d}*𝒦*(

*k*) → 1 for all of the summation indices

_{s}l_{ω}*s*in Eq. (7), i.e. when the general expressions (7) can be properly described by their asymptotics (8). In the latter case Eq. (9) degenerates into the conventional dispersion relation valid for a wide class of dielectric bilayer stack-structures.

## 3. Analysis

*Z*, and metallic,

_{a}*ζ*

_{0}and

*ζ*, impedances, the dispersion Eq. (9) for the photonic modes can have solutions for the Bloch wave number

_{d}*κ*with |Re(

*κ*)| <

*π*/

*d*and 0 < Im(

*κ*) < |Re(

*κ*)| only in pass bands associated with Fabry-Perot resonances emerging in the dielectric

*a*-layer. Indeed, when |

*Z*/

_{a}*ζ*

_{0}| ≫ 1 and |

*Z*/

_{a}*ζ*| ≫ 1 in Eq. (9), the solutions with minimal values of Im

_{d}*κ*are found in very narrow bands, being close to the frequencies

*ω*at which the Fabry-Perot resonance condition

_{j}*φ*=

_{a}*jπ*(

*j*= 1,2,3,...) is fulfilled.

*ω*/

*ω*and the electron relaxation rate

_{p}*ν*/

*ω*normalized to the plasma frequency

_{p}*ω*, as well as the ratio

_{p}*d*/

_{b}*δ*of the metallic slab thickness

*d*to the minimum skin depth

_{b}*δ*=

*c*/

*ω*in the bulk metal, which is reached in the high-frequency range

_{p}*ν*≪

*ω*≪

*ω*, where

_{p}*πV*/

_{F}*c*≪ 1 associated with the Fermi velocity of electrons. This parameter enters the argument of the nonlocality factor

*𝒦*(

*k*) and is responsible for the spatial dispersion effect, which is well manifested in the infrared if

_{s}l_{ω}*ν*<

*ω*< (

*πV*/

_{F}*c*)

*ω*(see details in [3

_{p}**90**, 623–627 (2009). [CrossRef]

*a*-layers, our results are valid not only when

*ε*and

_{a}*μ*are positive constants. In general, they can be of complex values and/or frequency dispersive. However, for simplicity, we restrict our further analysis to the consideration of a vacuum-aluminum superlattice.

_{a}*ζ*

_{0}and

*ζ*, Eq. (7), used for calculating the photonic bands from Eq. (9). The metal parameters are:

_{d}*d*= 4

_{b}*δ*,

*V*= 2.03 × 10

_{F}^{8}cm/s,

*ω*= 3.82 × 10

_{p}^{15}s

^{−}^{1}and

*ν*= 0.00025

*ω*. As seen in Fig. 2, the differences of

_{p}*ζ*

_{0}(

*ω*) and

*ζ*(

_{d}*ω*) from those predicted by the local Drude-Lorentz model are of the order of 10

^{−4}that is the order of the real parts of the impedances themselves. In Fig. 3, we present the four lower Fabry-Perot resonance bands (panels a–d) for a vacuum-aluminum superlattice, whose

*a*-layer in the unit (

*a*,

*b*) cell has a thickness

*d*such that the frequency

_{a}*ω*

_{1}=

*πc*/

*d*for the first Fabry-Perot resonance in the

_{a}*a*-layer coincides with the frequency

*ω*= 9.708×10

^{−4}

*ω*of the absolute minimum of the difference

_{p}*j*-th resonance band appears below the frequency

*ω*=

_{j}*jπc*/

*d*. Besides, one can observe various effects of spatial dispersion on the photonic band structure. First, the kinetic and local pass bands are clearly distinguishable. Both the magnitude and the sign of real part Re

_{a}*κ*of the Bloch wave number can change in comparison with the predictions of the local model. Second, the jumps of Re

*κ*, which occur because Re

*κ*is confined to the first Brillouin zone, are different for the nonlocal and local models. Finally, the minimum value of the imaginary part Im

*κ*in the resonance bands practically does not vary from band to band within the nonlocal formalism.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. A. Abrikosov, |

2. | E. A. Kaner, A. A. Krokhin, and N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in |

3. | A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, and F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. |

4. | 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 |

5. | D. Soto-Puebla, M. Xiao, and F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A |

6. | P. Markoš and C. M. Soukoulis, |

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

8. | A. Orlov, I. Iorsh, P. Belov, and Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express |

**OCIS Codes**

(160.4760) Materials : Optical properties

(260.2030) Physical optics : Dispersion

(260.3060) Physical optics : Infrared

(160.3918) Materials : Metamaterials

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Metamaterials

**History**

Original Manuscript: January 9, 2014

Revised Manuscript: March 13, 2014

Manuscript Accepted: March 16, 2014

Published: March 25, 2014

**Citation**

Alejandro Paredes-Juárez, Denis A. Iakushev, Benito Flores-Desirena, Nykolay M. Makarov, and Felipe Pérez-Rodríguez, "Nonlocal effect on optic spectrum of a periodic dielectric-metal stack," Opt. Express **22**, 7581-7586 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-7581

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### References

- A. A. Abrikosov, Fundamentals of the Theory of Metals (Elsevier, 1988).
- E. A. Kaner, A. A. Krokhin, N. M. Makarov, “Spatial dispersion and surface electromagnetic absorption in metals,” in Spatial Dispersion in Solids and Plasmas, P. Halevi, ed. (Elsevier, 1992), Vol. 1, Chap. 2, pp. 161–214.
- A. Paredes-Juárez, F. Díaz-Monge, N. M. Makarov, F. Pérez-Rodríguez, “Nonlocal effects in the electrodynamics of metallic slabs,” JETP Lett. 90, 623–627 (2009). [CrossRef]
- 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 (1997). [CrossRef]
- D. Soto-Puebla, M. Xiao, F. Ramos-Mendieta, “Optical properties of a dielectric-metallic superlattice: the complex photonic bands,” Phys. Lett. A 326, 273–280 (2004). [CrossRef]
- P. Markoš, C. M. Soukoulis, Wave Propagation] From Electrons to Photonic Crystals and Left-Handed Materials (Princeton University, 2008).
- J. Elser, V. A. Podolskiy, I. Salakhutdinov, I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007). [CrossRef]
- A. Orlov, I. Iorsh, P. Belov, Y. Kivshar, “Complex band structure of nanostructured metal-dielectric metamaterials,” Opt. Express 21, 1593–1598 (2013). [CrossRef] [PubMed]

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