Abstract

The proposed homogenization theory—a major extension of the recently published methodology [J. Opt. Soc. Am. B 28, 577 (2011)]—yields an extended second-order material tensor consolidating the usual 36 local material parameters and additional ones that rigorously quantify nonlocality. The local part of the tensor relates the mean values of pairs of coarse-grained fields, while the nonlocal part relates the mean values to variations of the fields. The theory is based on a direct analysis of fields in a lattice cell rather than on an indirect retrieval of material parameters from transmission/reflection data. There are no heuristic assumptions and no artificial averaging rules. Nontrivial magnetic behavior, if present, is a logical consequence of the theory. The approximations involved and the respective errors are clearly identified. Illustrative examples of resonant structures with high-permittivity inclusions are given.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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  38. 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]

2011 (6)

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

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102(R) (2011).
[CrossRef]

C. Scheiber, A. Schultschik, O. Bíró, and R. Dyczij-Edlinger, “A model order reduction method for efficient band structure calculations of photonic crystals,” IEEE Trans. Magn. 47, 1534–1537 (2011).
[CrossRef]

I. Babuška and R. Lipton, “Optimal local approximation spaces for generalized finite element methods with application to multiscale problems,” Multiscale Model. Simul. 9, 373–406(2011).
[CrossRef]

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011).
[CrossRef] [PubMed]

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]

2010 (3)

V. A. Markel, “On the current-driven model in the classical electrodynamics of continuous media,” J. Phys. 22, 485401(2010).
[CrossRef]

C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photon. Nanostr. 8, 254–263 (2010).
[CrossRef]

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

2009 (3)

C. R. Simovski, “On material parameters of metamaterials (review),” Opt. Spectrosc. 107, 726–753 (2009).
[CrossRef]

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610(2009).
[CrossRef]

D. Felbacq, B. Guizal, G. Bouchitté, and C. Bourel, “Resonant homogenization of a dielectric metamaterial,” Microw. Opt. Technol. Lett. 51, 2695–2701 (2009).
[CrossRef]

2008 (2)

I. Tsukerman, “Negative refraction and the minimum lattice cell size,” J. Opt. Soc. Am. B 25, 927–936 (2008).
[CrossRef]

I. Tsukerman and F. Čajko, “Photonic band structure computation using FLAME,” IEEE Trans. Magn. 44, 1382–1385 (2008).
[CrossRef]

2007 (3)

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[CrossRef]

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, H.-K. Yuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Metamagnetics with rainbow colors,” Opt. Express 15, 3333–3341 (2007).
[CrossRef] [PubMed]

2006 (2)

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).
[CrossRef]

I. Tsukerman, “A class of difference schemes with flexible local approximation,” J. Comp. Phys. 211, 659–699 (2006).
[CrossRef]

2005 (2)

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]

D. Felbacq and G. Bouchitté, “Theory of mesoscopic magnetism in photonic crystals,” Phys. Rev. Lett. 94, 183902 (2005).
[CrossRef] [PubMed]

2004 (1)

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

2003 (1)

A. Plaks, I. Tsukerman, G. Friedman, and B. Yellen, “Generalized finite element method for magnetized nanoparticles,” IEEE Trans. Magn. 39, 1436–1439 (2003).
[CrossRef]

2002 (2)

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]

A. P. Vinogradov, “On the form of constitutive equations in electrodynamics,” Phys. Usp. 45, 331–338 (2002).
[CrossRef]

1997 (1)

I. Babuška and J. M. Melenk, “The partition of unity method,” Int. J. Num. Meth. Eng. 40, 727–758 (1997).
[CrossRef]

Agranovich, V. M.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, 2nd ed. (Springer-Verlag, 1984).

Alù, A.

A. Alù, “Restoring the physical meaning of metamaterial constitutive parameters,” Phys. Rev. B 83, 081102(R) (2011).
[CrossRef]

Aydin, K.

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610(2009).
[CrossRef]

Babuška, I.

I. Babuška and R. Lipton, “Optimal local approximation spaces for generalized finite element methods with application to multiscale problems,” Multiscale Model. Simul. 9, 373–406(2011).
[CrossRef]

I. Babuška and J. M. Melenk, “The partition of unity method,” Int. J. Num. Meth. Eng. 40, 727–758 (1997).
[CrossRef]

Bíró, O.

C. Scheiber, A. Schultschik, O. Bíró, and R. Dyczij-Edlinger, “A model order reduction method for efficient band structure calculations of photonic crystals,” IEEE Trans. Magn. 47, 1534–1537 (2011).
[CrossRef]

Bossavit, A.

A. Bossavit, Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements (Academic, 1998).

Bouchitté, G.

D. Felbacq, B. Guizal, G. Bouchitté, and C. Bourel, “Resonant homogenization of a dielectric metamaterial,” Microw. Opt. Technol. Lett. 51, 2695–2701 (2009).
[CrossRef]

D. Felbacq and G. Bouchitté, “Theory of mesoscopic magnetism in photonic crystals,” Phys. Rev. Lett. 94, 183902 (2005).
[CrossRef] [PubMed]

Bourel, C.

D. Felbacq, B. Guizal, G. Bouchitté, and C. Bourel, “Resonant homogenization of a dielectric metamaterial,” Microw. Opt. Technol. Lett. 51, 2695–2701 (2009).
[CrossRef]

Bozhevolnyi, S. I.

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]

Braun, J.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011).
[CrossRef] [PubMed]

Cai, W.

Cajko, F.

I. Tsukerman and F. Čajko, “Photonic band structure computation using FLAME,” IEEE Trans. Magn. 44, 1382–1385 (2008).
[CrossRef]

Chen, X.

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]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Chettiar, U. K.

Cui, T. J.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[CrossRef]

de Silva, V. C.

Demmel, J. W.

J. W. Demmel, Applied Numerical Linear Algebra (SIAM, 1997).
[CrossRef]

Drachev, V. P.

Dressel, M.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011).
[CrossRef] [PubMed]

Dyczij-Edlinger, R.

C. Scheiber, A. Schultschik, O. Bíró, and R. Dyczij-Edlinger, “A model order reduction method for efficient band structure calculations of photonic crystals,” IEEE Trans. Magn. 47, 1534–1537 (2011).
[CrossRef]

Felbacq, D.

D. Felbacq, B. Guizal, G. Bouchitté, and C. Bourel, “Resonant homogenization of a dielectric metamaterial,” Microw. Opt. Technol. Lett. 51, 2695–2701 (2009).
[CrossRef]

D. Felbacq and G. Bouchitté, “Theory of mesoscopic magnetism in photonic crystals,” Phys. Rev. Lett. 94, 183902 (2005).
[CrossRef] [PubMed]

Fietz, C.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

Friedman, G.

A. Plaks, I. Tsukerman, G. Friedman, and B. Yellen, “Generalized finite element method for magnetized nanoparticles,” IEEE Trans. Magn. 39, 1436–1439 (2003).
[CrossRef]

Giessen, H.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011).
[CrossRef] [PubMed]

Ginzburg, V. L.

V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, 2nd ed. (Springer-Verlag, 1984).

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations (Johns Hopkins Univ., 1996).

Gompf, B.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011).
[CrossRef] [PubMed]

B. Gompf, Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57 Stuttgart 70550, Germany, and R. Vogelgesang, Max-Planck Institute for Solid State Research, Heisenbergstrasse 1 Stuttgart 70569, Germany (private communication, 2011).

Grzegorczyk, T. M.

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]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Guizal, B.

D. Felbacq, B. Guizal, G. Bouchitté, and C. Bourel, “Resonant homogenization of a dielectric metamaterial,” Microw. Opt. Technol. Lett. 51, 2695–2701 (2009).
[CrossRef]

Hiptmair, R.

R. Hiptmair, A. Moiola, and I. Perugia, “Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations,” preprint IMATI-CNR Pavia, http://www-dimat.unipv.it/perugia/PREPRINTS/TDGM.pdf (2011).

Huang, D.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[CrossRef]

Hübner, U.

B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011).
[CrossRef] [PubMed]

Kildishev, A. V.

Kong, J. A.

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]

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Li, Z.

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610(2009).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).

Lipton, R.

I. Babuška and R. Lipton, “Optimal local approximation spaces for generalized finite element methods with application to multiscale problems,” Multiscale Model. Simul. 9, 373–406(2011).
[CrossRef]

Liu, R.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[CrossRef]

Markel, V. A.

V. A. Markel, “On the current-driven model in the classical electrodynamics of continuous media,” J. Phys. 22, 485401(2010).
[CrossRef]

V. A. Markel, Radiology and Bioengineering, University of Pennsylvania, 3400 Spruce St., Philadelphia, Pa., 19104 (private communication, 2010–2011).

Markoš, P.

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]

Melenk, J. M.

I. Babuška and J. M. Melenk, “The partition of unity method,” Int. J. Num. Meth. Eng. 40, 727–758 (1997).
[CrossRef]

Moiola, A.

R. Hiptmair, A. Moiola, and I. Perugia, “Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations,” preprint IMATI-CNR Pavia, http://www-dimat.unipv.it/perugia/PREPRINTS/TDGM.pdf (2011).

Ozbay, E.

Z. Li, K. Aydin, and E. Ozbay, “Determination of the effective constitutive parameters of bianisotropic metamaterials from reflection and transmission coefficients,” Phys. Rev. E 79, 026610(2009).
[CrossRef]

Pacheco, J.

X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608 (2004).
[CrossRef]

Pendry, J. B.

Perugia, I.

R. Hiptmair, A. Moiola, and I. Perugia, “Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations,” preprint IMATI-CNR Pavia, http://www-dimat.unipv.it/perugia/PREPRINTS/TDGM.pdf (2011).

Plaks, A.

A. Plaks, I. Tsukerman, G. Friedman, and B. Yellen, “Generalized finite element method for magnetized nanoparticles,” IEEE Trans. Magn. 39, 1436–1439 (2003).
[CrossRef]

Pors, A.

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]

Sarychev, A. K.

A. K. Sarychev and V. M. Shalaev, Electrodynamics of Metamaterials (World Scientific, 2007).
[CrossRef]

Scheiber, C.

C. Scheiber, A. Schultschik, O. Bíró, and R. Dyczij-Edlinger, “A model order reduction method for efficient band structure calculations of photonic crystals,” IEEE Trans. Magn. 47, 1534–1537 (2011).
[CrossRef]

Schultschik, A.

C. Scheiber, A. Schultschik, O. Bíró, and R. Dyczij-Edlinger, “A model order reduction method for efficient band structure calculations of photonic crystals,” IEEE Trans. Magn. 47, 1534–1537 (2011).
[CrossRef]

Schultz, S.

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]

Shalaev, V. M.

Shlens, J.

J. Shlens, “A tutorial on principal component analysis,” http://www.snl.salk.edu/~shlens/pca.pdf (2009).

Shvets, G.

C. Fietz and G. Shvets, “Homogenization theory for simple metamaterials modeled as one-dimensional arrays of thin polarizable sheets,” Phys. Rev. B 82, 205128 (2010).
[CrossRef]

Silveirinha, M. G.

M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[CrossRef]

Simovski, C. R.

C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photon. Nanostr. 8, 254–263 (2010).
[CrossRef]

C. R. Simovski, “On material parameters of metamaterials (review),” Opt. Spectrosc. 107, 726–753 (2009).
[CrossRef]

Smith, D. R.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
[CrossRef]

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).
[CrossRef]

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]

Soukoulis, C. M.

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]

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photon. Nanostr. 8, 254–263 (2010).
[CrossRef]

Tsukerman, I.

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

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]

I. Tsukerman and F. Čajko, “Photonic band structure computation using FLAME,” IEEE Trans. Magn. 44, 1382–1385 (2008).
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B. Gompf, J. Braun, T. Weiss, H. Giessen, M. Dressel, and U. Hübner, “Periodic nanostructures: spatial dispersion mimics chirality,” Phys. Rev. Lett. 106, 185501 (2011).
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[CrossRef]

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A. Plaks, I. Tsukerman, G. Friedman, and B. Yellen, “Generalized finite element method for magnetized nanoparticles,” IEEE Trans. Magn. 39, 1436–1439 (2003).
[CrossRef]

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Zhao, B.

R. Liu, T. J. Cui, D. Huang, B. Zhao, and D. R. Smith, “Description and explanation of electromagnetic behaviors in artificial metamaterials based on effective medium theory,” Phys. Rev. E 76, 026606 (2007).
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Figures (12)

Fig. 1
Fig. 1

Key parts of the proposed methodology. Two types of interpolation are used to obtain the coarse-grained fields with tangential and normal continuity. The electromagnetic field inside the cell, with all its microstructure, is approximated as a superposition of basis modes. Material parameters are linear relationships between the pairs of coarse-grained fields.

Fig. 2
Fig. 2

(From [5].) 2D analog of the vectorial interpolation function w α (in this case, associated with the central vertical edge shared by two adjacent cells). The tangential continuity of this function is evident from the arrow plot; its circulation is equal to one over the central edge and to zero over all other edges.

Fig. 3
Fig. 3

(From [5].) 2D analog of the vectorial interpolation function v β (in this case, associated with the central vertical edge). The normal continuity of this function is evident from the arrow plot; its flux is equal to one over the central edge and zero over all other edges.

Fig. 4
Fig. 4

Linear relation between the coarse-grained fields. E, H are interpolated to ensure tangential continuity; D, B are normally continuous (1D rendition for simplicity). Operator L is, in general, multidimensional. In a canonical basis, L contains a 6 × 6 submatrix of local electrodynamic parameters.

Fig. 5
Fig. 5

Structure of the matrix equations for the material tensor. See text for details.

Fig. 6
Fig. 6

Re ( H ) in the vicinity of a slab with resonant inclusions; p mode. Radius of the inclusions relative to the cell size r / a = 0.25 ; their permittivity ε incl = 200 + 5 i (as in [35]). (a) Passband, λ / a = 11 and (b) bandgap, λ / a = 9 . Inset, magnification of a few cells.

Fig. 7
Fig. 7

Transmission coefficient (with respect to H) of a five-layer slab as a function of the vacuum wavelength [(a) absolute value and (b) phase]. r / a = 0.25 ; ε incl = 200 + 5 i . Triangles, direct finite difference simulation [24]; lines, slab with effective parameters. Normal incidence.

Fig. 8
Fig. 8

Effective parameters of the metamaterial with resonant inclusions.

Fig. 9
Fig. 9

In-the-basis error γ for the metamaterial with resonant inclusions.

Fig. 10
Fig. 10

Examples of Bloch modes in a nanoparticle trimer. r / a = 0.125 ; ε incl = 200 + 5 i for all three particles. (a)  λ / a = 4.6 and (b)  λ / a = 8 . The jagged contour lines are only an artifact of the grid-based drawing; the FLAME solution [23, 24, 25] itself is very accurate.

Fig. 11
Fig. 11

Diamonds, in-the-basis error γ for the particle trimer; triangles, the matrix norm of column 4 of the material tensor; circles, same for column 5. (Columns 4 and 5 characterize the nonlocal response.)

Fig. 12
Fig. 12

Effective parameters of the particle trimer: solid lines, real parts; dashed lines, imaginary parts; diamonds, ε x x ; triangles, ε y y ; squares, μ.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

E = α = 1 12 [ e ] α w α , H = α = 1 12 [ b ] α w α ,
D = β = 1 6 [ [ d ] ] β v β , B = β = 1 6 [ [ b ] ] β v β ,
× W curl W div .
× E = i ω c 1 B ; × H = i ω c 1 D .
Ψ e h = α c α ψ α e h + δ e h ; Ψ d b = α c α ψ α d b + δ d b .
η W E H = l . s . W D B η = W D B W E H + ,
γ = W D B η W E H / W D B ,
e B ( z ) = E PER ( z ) exp ( i K B z ) x ^ i ω c 1 b B ( z ) = y ^ ( E PER ( z ) + i K B E PER ( z ) ) exp ( i K B z ) .
ω 2 μ eff ε eff = K B 2 ,
( μ eff / ε eff ) 1 2 = e B / b B = ω / K B .

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