## Surface waves in three-dimensional electromagnetic composites and their effect on homogenization |

Optics Express, Vol. 21, Issue 9, pp. 10412-10421 (2013)

http://dx.doi.org/10.1364/OE.21.010412

Acrobat PDF (1723 KB)

### Abstract

Reflection and transmission of electromagnetic waves at the boundaries of periodic composites (electromagnetic/optical metamaterials) depends in general on both bulk and surface waves. We investigate the interplay of these two contributions using three-dimensional full-wave numerical simulations and a recently developed non-asymptotic homogenization theory.

© 2013 OSA

## 1. Introduction

1. J. D. Baena, L. Jelinek, R. Marques, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A **78**, 013842 (2008) [CrossRef] .

*h*. Let us assume that a slab of sufficiently large width

*L*(that is,

*L*=

*Nh*where

*N*≫ 1) can be accurately described by some effective medium parameters. On the other hand, for a slab containing only one layer of elementary cells (

*N*= 1), this description is expected to be substantially different or inapplicable. The composites used in practice will, most likely, have some intermediate value of

*N*. This raises the questions of how large the number of elementary layers should be for the effective medium description to be applicable, what errors are incurred by using this description for a given finite

*N*, and whether the effective medium parameters can depend on

*N*. It should be noted that many simulations have revealed that the width effects are unimportant in slabs containing only a few layers of elementary cells [8

8. C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B **81**, 035320 (2010) [CrossRef] .

10. T. Paul, C. Menzel, W. Smigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B **84**, 115142 (2011) [CrossRef] .

11. C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B **77**, 035126 (2008) [CrossRef] .

12. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B **65**, 195104 (2002) [CrossRef] .

13. X. Chen, B. I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E **71**, 046610 (2005) [CrossRef] .

8. C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B **81**, 035320 (2010) [CrossRef] .

9. C. Menzel, C. Rockstuhl, R. Iliew, F. Lederer, A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “High symmetry versus optical isotropy of a negative-index metamaterial,” Phys. Rev. B **81**, 195123 (2010) [CrossRef] .

14. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B **77**, 195328 (2008) [CrossRef] .

15. I. Tsukerman, “Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation,” J. Opt. Soc. Am. B **28**, 577–586 (2011) [CrossRef] .

17. I. Tsukerman, “Nonlocal homogenization of metamaterials by dual interpolation of fields,” J. Opt. Soc. Am. B **28**, 2956–2965 (2011) [CrossRef] .

15. I. Tsukerman, “Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation,” J. Opt. Soc. Am. B **28**, 577–586 (2011) [CrossRef] .

17. I. Tsukerman, “Nonlocal homogenization of metamaterials by dual interpolation of fields,” J. Opt. Soc. Am. B **28**, 2956–2965 (2011) [CrossRef] .

*λ*/

*h*≲ 5 [8

8. C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B **81**, 035320 (2010) [CrossRef] .

18. V. A. Markel and J. C. Schotland, “On the sign of refraction in anisotropic non-magnetic media,” J. Opt. **12**, 015104 (2010) [CrossRef] .

21. V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E **85**, 066603 (2012) [CrossRef] .

*h*is the lattice period, see below). We note that the practical utility of nonlocal parameters, if these can be reasonably introduced at all, is debatable and, in any event, the onset of nonlocality tends to destroy all physical effects that enable the frequently discussed applications of metamaterials, such as super-resolution. We, therefore, do not investigate the short-wavelength spectral region or the effects of nonlocality in detail, but rather focus on the influence of the surface wave, which is present at both long and short waves.

2. C. R. Simovski, “Material paremeters of metamaterials (a review),” Opt. Spectrosc. **107**, 766–793 (2009) [CrossRef] .

4. C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. **13**, 103001 (2011) [CrossRef] .

**81**, 035320 (2010) [CrossRef] .

9. C. Menzel, C. Rockstuhl, R. Iliew, F. Lederer, A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “High symmetry versus optical isotropy of a negative-index metamaterial,” Phys. Rev. B **81**, 195123 (2010) [CrossRef] .

14. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B **77**, 195328 (2008) [CrossRef] .

*ε*.

2. C. R. Simovski, “Material paremeters of metamaterials (a review),” Opt. Spectrosc. **107**, 766–793 (2009) [CrossRef] .

## 2. Surface waves – theory

*z*> 0. An incident wave would induce a polarization field

**P**(

**r**) inside the inclusions (

**P**= 0 in vacuum). It is known that the normal modes in an

*infinite*composite have the form of the Bloch waves. This applies to every Bloch-periodic function, and in particular to

**P**(

**r**). However, the Bloch wave is not the normal modes of a semi-infinite composite. Therefore, the polarization induced in such a composite is a superposition of the Bloch wave

**P**

*(*

_{B}**r**) and an additional surface wave

**P**

*(*

_{S}**r**).

10. T. Paul, C. Menzel, W. Smigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B **84**, 115142 (2011) [CrossRef] .

21. V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E **85**, 066603 (2012) [CrossRef] .

21. V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E **85**, 066603 (2012) [CrossRef] .

**r**∈ Ω

_{tot}, where the latter is the spatial region occupied by all inclusions,

*χ*= (3/4

*π*)(

*ε*− 1)/(

*ε*+ 2) is the coupling parameter,

*ε*is the dielectric permittivity of the inclusions at the working frequency, and

**F**

*(*

_{S}**r**) is a free term, which is completely determined by the transmitted Bloch wave. The function

**F**

*(*

_{S}**r**) was computed explicitly in Ref. [21

**85**, 066603 (2012) [CrossRef] .

**k**

*=*

_{i}**x̂**

*k*+

_{ix}**ẑ**

*k*is the incident wave vector which satisfies

_{iz}*XYZ*, the plane of incidence is

*y*= 0 and the interface is located in the plane

*z*= 0. Then where

**p**= (2

*π*/

*h*)(

**x̂**

*n*+

_{x}**ŷ**

*n*) (

_{y}*n*,

_{x}*n*= 0, ±1, ±2,...) are the reciprocal lattice vectors with zero component along the

_{y}*z*-axis, ⊗ denotes tensor product,

*q*is the

_{z}*z*-component of the transmitted Bloch wave vector (the

*x*-component is the same as in the incident wave, so that we can write

**q**=

**x̂**

*k*+

_{iz}**ẑ**

*q*), and Finally,

_{z}**P**

^{˜}

*(*

_{B}**k**) is the Fourier transform of the transmitted Bloch wave of polarization

**P**

*(*

_{B}**r**) taken over an elementary cell. The transform is defined as follows. Consider an elementary cubic cell

*C*centered at the point

_{n}**r**

*. Then for*

_{n}**r**∈

*C*, we can write

_{n}**P**

*(*

_{B}**r**) = exp(

*i*

**q**·

**r**

*)*

_{n}**P**

_{cell}(

**r**

*+*

_{n}**R**), where the Cartesian components of

**R**satisfy −

*h*/2 ≤

*R*≤

_{α}*h*/2. The Fourier transform is defined as The transmitted wave is completely defined by the geometry of the cell and the amplitude of the incident wave. Specifically, it was shown in [21

**85**, 066603 (2012) [CrossRef] .

**q**and the corresponding function

**P**

_{cell}(

**R**) in an absorbing medium. This is conceptually different from the similar problem encountered in conventional photonic crystals composed of transparent and nondispersive dielectrics; in the latter case, the problem can be reduced to a real symmetric eigenproblem whilst the problem encountered here is not even Hermitian. In addition to computing the Bloch wave, one also faces the problem of solving the integral equation Eq. (1), which can have resonant and therefore non-perturbative solutions. Thus, instead of computing the surface wave directly, we have adopted in this paper an approach based on computing the total field and polarization inside the composite, which includes both the direct and reflected Bloch waves (in the case of a finite slab), as well as the surface wave contributions.

*Q*

**in (3) have nonzero imaginary parts. Therefore, the function**

_{p}**F**

*(*

_{S}**r**) decays exponentially away from the interface. The same is true in this case for the surface wave of polarization,

**P**

*(*

_{S}**r**). In the homogenization limit, the exponential decay is fast. Indeed, in the limit

*h*→ 0, we have (for

**p**≠ 0):

*Q*

**→**

_{p}*i*|

**p**|,

**k**→

_{p}**p**+

*i*

**ẑ**|

**p**|, exp[

*i*(

*Q*

**−**

_{p}*q*)

_{z}*h*] − 1 → −1|

**g**|

*h*. With these limits taken into account, the surface wave takes the following form: It can be seen that

**E**

*(*

_{S}**r**) decays exponentially on the scale of

*h*and is, therefore, evanescent, as any SPP.

## 3. Effective medium theory

15. I. Tsukerman, “Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation,” J. Opt. Soc. Am. B **28**, 577–586 (2011) [CrossRef] .

17. I. Tsukerman, “Nonlocal homogenization of metamaterials by dual interpolation of fields,” J. Opt. Soc. Am. B **28**, 2956–2965 (2011) [CrossRef] .

**b**,

**h**,

**e**and

**d**in the composite to define the coarse-grained fields

**B**,

**H**,

**E**,

**D**. It should be emphasized that by

**b**,

**h**,

**e**, and

**d**are not “truly microscopic”, e.g., atomic-scale fields, but rather rapidly varying fields on the scale of tens of nanometers. These fields still obey macroscopic Maxwell’s equations in the composite with spatially-varying material parameters. In contrast, the “macroscopic” or, as we refer to them, coarse-grained fields

**B**,

**H**,

**E**,

**D**are obtained by appropriate interpolation procedures and experience spatial variations on larger spatial scales. Note that in intrinsically-nonmagnetic composites,

**b**=

**h**identically.

*ℒ*is sought between the field pairs (

**E**,

**H**) and (

**D**,

**B**). In a suitable “canonical” basis, the operator

*ℒ*becomes a generalized material tensor with a leading 6 × 6 block relating (

**D**,

**B**) to (

**E**,

**H**). Note that

*ℒ*can also contain a “nonlocal” block that relates the coarse-grained fields to field variations over a cell [17

**28**, 2956–2965 (2011) [CrossRef] .

**e**,

**d**,

**h**and

**b**in the composite can be approximated by a superposition of suitable basis functions [15

**28**, 577–586 (2011) [CrossRef] .

**28**, 2956–2965 (2011) [CrossRef] .

**28**, 577–586 (2011) [CrossRef] .

**28**, 2956–2965 (2011) [CrossRef] .

- Choose a set of
*N*approximating modes. - Choose a set of
*M*and^{EH}*M*degrees of freedom (d.o.f.s) for the (^{DB}**E**,**H**) and (**D**,**B**) pairs, respectively. The d.o.f. will in general include the mean values of the tangential components of**E**,**H**and of the normal components of**D**,**B**; in addition, the mean values of some derivatives of**E**,**H**may be included. By increasing the number of d.o.f., one trades higher accuracy for a greater level of nonlocality in the characterization of the material. Typically for 3D problems,*M*= 6 (three mean values for each of the two fields) but^{DB}*M*≥ 6. Note that nonlocal d.o.f.s may be included in addition to the mean values.^{EH} - For each mode
*m*= 1, 2,...*M*, compute its respective d.o.f. (the mean boundary values of the tangential components of**E**,**H**for this mode, etc.) Assemble the d.o.f. for the**E**,**H**fields into the*m*th column of matrix*W*and the d.o.f. for the^{EH}**D**,**B**fields into the*m*th column of matrix*W*. Ultimately, matrix^{DB}*W*is of dimension^{EH}*M*×^{EH}*N*and matrix*W*is^{DB}*M*×^{DB}*N*(typically 3 ×*N*).

**28**, 577–586 (2011) [CrossRef] .

**28**, 577–586 (2011) [CrossRef] .

**28**, 577–586 (2011) [CrossRef] .

**28**, 2956–2965 (2011) [CrossRef] .

*γ*that comes from the least squares fit of the material relation (8) is defined as where || · || is a suitable matrix norm.

## 4. Numerical results

*h*. A spherical gold particle of radius

*a*<

*h*/2 is located at the center of each cell; the rest of the cell is assumed to be air. The sample under investigation is a plane-parallel slab that contains a finite number

*N*of cells in the

*z*-direction but is infinite in the

*x*- and

*y*-directions. The crystallographic axes of the composite coincide with the axes of the laboratory frame

*xyz*.

24. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm. **181**, 687 (2010) [CrossRef] .

*z*direction (Fig. 1), the electromagnetic fields inside the composite satisfy Bloch-periodic boundary conditions. Therefore, the computational domain is effectively reduced to

*N*elementary cells arranged as shown in Fig. 1(a). In all simulations reported below, the radius of the sphere is

*a*= 20 nm, the lattice unit is

*h*= 80 nm, so that the filling factor is

*f*= 4

*πa*

^{3}/3

*h*

^{3}≈ 0.07. We consider the spectral interval 300 nm ≤

*λ*≤ 900 nm. The unit of the FDTD cubic grid is Δ = 2.857 nm. In MEEP, boundary conditions at the exterior boundary are enforced by adding perfectly matched layers (PMLs). Additionally, MEEP allows one to apply Bloch-periodic boundary conditions in the

*x*- and

*y*-directions. The Lorentz-Drude model with six pole expansion terms is used for modeling gold particles. The material parameters are taken from [25

25. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. **37**, 5271 (1998) [CrossRef] .

**x̂**, ±

**ŷ**, and ±

**ẑ**and two transverse polarizations for each incidence direction). One can, of course, take advantage of the symmetries of the lattice cell and reduce the number of the actual FDTD simulations. For a cubic lattice of spheres, only one basis function needs to be computed numerically for a cell in the bulk and four functions for a cell at the surface; other basis functions can be obtained just by symmetry and rotation. In general, the number of basis functions needed to obtain an accurate homogenization result and to quantify the errors incurred may be greater than 12.

26. S. Ha, A. A. Sukhorukov, D. K. B., L. C. Botten, C. M. de Sterke, and Y. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. **34**, 3776 (2009) [CrossRef] [PubMed] .

*E*

_{0}exp(

*iK*) to the field values

_{B}z_{n}*E*at the cell boundaries

_{y}*z*

_{1},

*z*

_{2}, …

*z*

_{7}of the seven inner layers. Color plots of the total field, its Bloch component and the surface wave are pictured in Fig. 2 for

*λ*= 20

*h*/3. The slab is illuminated with a plane wave propagating along the +

*z*direction. As seen in Fig. 2(c), the surface wave decays rapidly in the direction normal to the surface.

*L*= 5 layers and compare the effective parameters of a cell adjacent to the surface (layer

*l*= 1; cyan circles in Fig. 3) with those of a cell in the center of the slab (

*l*= 3; green diamonds in Fig. 3). An appreciable difference between the two cases is observed, which indicates a noticeable effect of the surface wave in this case. The ”in-the-basis error”

*γ*(9) is plotted in Fig. 3(d).

*L*for which the effective parameters computed using the “central” cell no longer depend on

*L*. The effective parameters shown in Fig. 3 for

*L*= 5 (green diamonds) and

*L*= 9 (red triangles) are also compared to those obtained from Lewin’s theory [27] (solid blue line). The discrepancy between the

*L*= 5 and

*L*= 9 cases is small, which indicates that, for the parameters considered,

*L*= 5 is sufficient to represent bulk samples.

*T*) and reflection (

*R*) coefficients of a composite sample is illustrated in Fig. 4 where we compare

*T*and

*R*of a homogenized slab to the “brute force” FDTD simulations in the composite structure with spherical inclusions present. We used the effective parameters obtained by our theory and also by Lewin’s theory (see plots of the respective effective parameters in Fig. 3). When our homogenization procedures are used, the overall agreement is quite good (this has already been observed in [16

16. A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: Homogenization by dual field interpolation,” Phys. Rev. E **84**, 016609 (2011) [CrossRef] .

*k*

_{0}, Lewin’s prediction for the permeability deviates from unity asymptotically as

*f*and

_{v}*n*are the volume fraction and the index of refraction of the particles. This correction tends to be relatively small for the parameters considered, as was already noted [16

_{p}16. A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: Homogenization by dual field interpolation,” Phys. Rev. E **84**, 016609 (2011) [CrossRef] .

*R*and, especially, of

*T*can be very sensitive to the refractive index but can in some cases tolerate a moderate error in the impedance.

## 5. Summary and discussion

**28**, 577–586 (2011) [CrossRef] .

**28**, 2956–2965 (2011) [CrossRef] .

2. C. R. Simovski, “Material paremeters of metamaterials (a review),” Opt. Spectrosc. **107**, 766–793 (2009) [CrossRef] .

4. C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. **13**, 103001 (2011) [CrossRef] .

**81**, 035320 (2010) [CrossRef] .

9. C. Menzel, C. Rockstuhl, R. Iliew, F. Lederer, A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “High symmetry versus optical isotropy of a negative-index metamaterial,” Phys. Rev. B **81**, 195123 (2010) [CrossRef] .

14. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B **77**, 195328 (2008) [CrossRef] .

*local*approximations of parameters, one can solve Maxwell’s equations numerically using standard methods but with position-dependent parameter tensors. Secondly, extended material tensors that represent nonlocal effects [17

**28**, 2956–2965 (2011) [CrossRef] .

## Acknowledgments

## References and links

1. | J. D. Baena, L. Jelinek, R. Marques, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A |

2. | C. R. Simovski, “Material paremeters of metamaterials (a review),” Opt. Spectrosc. |

3. | C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B |

4. | C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. |

5. | R. V. Craster, J. Kaplunov, E. Nolde, and S. Guenneau, “High-frequency homogenization for checkerboard structures: defect modes, ultrarefraction, and all-angle negative refraction,” J. Opt. Soc. Am. A |

6. | S. Guenneau and F. Zolla, “Homogenization of three-dimensional finite photonic crystals,” Prog. Electromagnetic Res. |

7. | Y. Liu, S. Guenneau, and B. Gralak, “A route to all frequency homogenization of periodic structures,” (2012). ArXiv:1210.6171v2. |

8. | C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev. B |

9. | C. Menzel, C. Rockstuhl, R. Iliew, F. Lederer, A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “High symmetry versus optical isotropy of a negative-index metamaterial,” Phys. Rev. B |

10. | T. Paul, C. Menzel, W. Smigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B |

11. | C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. P. Meyrath, and H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B |

12. | D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

13. | X. Chen, B. I. Wu, J. A. Kong, and T. M. Grzegorczyk, “Retrieval of the effective constitutive parameters of bianisotropic metamaterials,” Phys. Rev. E |

14. | C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B |

15. | I. Tsukerman, “Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation,” J. Opt. Soc. Am. B |

16. | A. Pors, I. Tsukerman, and S. I. Bozhevolnyi, “Effective constitutive parameters of plasmonic metamaterials: Homogenization by dual field interpolation,” Phys. Rev. E |

17. | I. Tsukerman, “Nonlocal homogenization of metamaterials by dual interpolation of fields,” J. Opt. Soc. Am. B |

18. | V. A. Markel and J. C. Schotland, “On the sign of refraction in anisotropic non-magnetic media,” J. Opt. |

19. | H. Raether, |

20. | P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B |

21. | V. A. Markel and J. C. Schotland, “Homogenization of Maxwell’s equations in periodic composites,” Phys. Rev. E |

22. | R. Hiptmair, “Discrete Hodge operators,” Num. Math. |

23. | R. Hiptmair and I. Tsukerman, “Non-asymptotic homogenization of electromagnetic metamaterials via discrete Hodge operators with Trefftz calibration,” to appear in |

24. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comp. Phys. Comm. |

25. | A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. |

26. | S. Ha, A. A. Sukhorukov, D. K. B., L. C. Botten, C. M. de Sterke, and Y. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. |

27. | L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Elec. Eng. |

**OCIS Codes**

(160.1245) Materials : Artificially engineered materials

(260.2065) Physical optics : Effective medium theory

**ToC Category:**

Metamaterials

**History**

Original Manuscript: December 10, 2012

Revised Manuscript: March 8, 2013

Manuscript Accepted: March 21, 2013

Published: April 22, 2013

**Citation**

Xiaoyan Y.Z. Xiong, Li Jun Jiang, Vadim A. Markel, and Igor Tsukerman, "Surface waves in three-dimensional electromagnetic composites and their effect on homogenization," Opt. Express **21**, 10412-10421 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-9-10412

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### References

- J. D. Baena, L. Jelinek, R. Marques, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A78, 013842 (2008). [CrossRef]
- C. R. Simovski, “Material paremeters of metamaterials (a review),” Opt. Spectrosc.107, 766–793 (2009). [CrossRef]
- C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B405, 2930–2934 (2010). [CrossRef]
- C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt.13, 103001 (2011). [CrossRef]
- R. V. Craster, J. Kaplunov, E. Nolde, and S. Guenneau, “High-frequency homogenization for checkerboard structures: defect modes, ultrarefraction, and all-angle negative refraction,” J. Opt. Soc. Am. A28, 1032–1040 (2011). [CrossRef]
- S. Guenneau and F. Zolla, “Homogenization of three-dimensional finite photonic crystals,” Prog. Electromagnetic Res.27, 91–127 (2011). [CrossRef]
- Y. Liu, S. Guenneau, and B. Gralak, “A route to all frequency homogenization of periodic structures,” (2012). ArXiv:1210.6171v2.
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