Abstract

A rigorous homogenization theory of metamaterials—artificial periodic structures judiciously designed to control the propagation of electromagnetic (EM) waves—is developed. The theory is an amalgamation of two concepts: Smith and Pendry’s physical insight into field averaging and the mathematical framework of Whitney-like interpolation. All coarse-grained fields are unambiguously defined and satisfy Maxwell’s equations exactly. Fields with tangential and normal continuity across boundaries are associated with two different kinds of interpolation, which reveals the physical and mathematical origin of “artificial magnetism.” The new approach is illustrated with several examples and agrees well with the established results (e.g., the Maxwell–Garnett formula and the zero cell-size limit) within the range of applicability of the latter. The sources of approximation error and the respective suitable error indicators are clearly identified, along with systematic routes for improving the accuracy further. The proposed methodology should be applicable in areas beyond metamaterials and EM waves (e.g., in acoustics and elasticity).

© 2011 Optical Society of America

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  1. A. Alù, F. Bilotti, N. EnghetaL. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antenn. Propag. 55, 13–25 (2007).
    [CrossRef]
  2. K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech. 54, 135–146 (2006).
    [CrossRef]
  3. P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antenn. Propag. 54, 1654–1662 (2006).
    [CrossRef]
  4. N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
    [CrossRef] [PubMed]
  5. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
    [CrossRef] [PubMed]
  6. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
    [CrossRef] [PubMed]
  7. S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
    [CrossRef] [PubMed]
  8. A. K. Sarychev and V. M. Shalaev, Electrodynamics of Metamaterials (World Scientific, 2007).
    [CrossRef]
  9. M. G. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
    [CrossRef]
  10. C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
    [CrossRef]
  11. C. R. Simovski, “Material parameters of metamaterials (a review),” Opt. Spectrosc. 107, 726–753 (2009).
    [CrossRef]
  12. C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photon. Nanostr. Fundam. Appl. 8, 254–263 (2010).
    [CrossRef]
  13. D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B 23, 391–403 (2006).
    [CrossRef]
  14. S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).
  15. H. Whitney, Geometric Integration Theory (Princeton University, 1957).
  16. A. Bossavit, “Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism,” IEE Proc. A 135, 493–500 (1988).
    [CrossRef]
  17. A. Bossavit, Computational Electromagnetism: Variational Formulations, Complementarity, Edge Elements (Academic, 1998).
  18. P. R. Kotiuga, “Hodge decompositions and computational electromagnetics,” PhD thesis (McGill University, 1985).
  19. J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
    [CrossRef]
  20. J.-C. Nédélec, “A new family of mixed finite elements in R3,” Numer. Math. 50, 57–81 (1986).
    [CrossRef]
  21. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1984).
  22. R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. USA 106, 1693–1698 (2009).
    [CrossRef] [PubMed]
  23. D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
    [CrossRef]
  24. A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Methods in Periodic Media (North-Holland, 1978).
  25. I. Tsukerman, “Negative refraction and the minimum lattice cell size,” J. Opt. Soc. Am. B 25, 927–936 (2008).
    [CrossRef]
  26. G. Russakoff, “A derivation of the macroscopic Maxwell equations,” Am. J. Phys. 38, 1188–1195 (1970).
    [CrossRef]
  27. V. A. Markel, University of Pennsylvania, Philadelphia, Pennsylvania (personal communication, 2010).
  28. A. P. Vinogradov, “On the form of constitutive equations in electrodynamics,” Phys. Usp. 45, 331–338 (2002).
    [CrossRef]
  29. V. A. Markel, “Correct definition of the Poynting vector in electrically and magnetically polarizable medium reveals that negative refraction is impossible,” Opt. Express 16, 19152–19168(2008).
    [CrossRef]
  30. V. V. Bokut, A. N. Serdyukov, and F. I. Fedorov, “Form of constitutive equations in optically active crystals,” Opt. Spectrosc. 37, 166–168 (1974).
  31. V. M. Agranovich and V. L. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, 2nd ed. (Springer-Verlag, 1984).
  32. P. Monk, Finite Element Methods for Maxwell’s Equations (Clarendon, 2003).
    [CrossRef]
  33. J. van Welij, “Calculation of eddy currents in terms of H on hexahedra,” IEEE Trans. Magn. 21, 2239–2241 (1985).
    [CrossRef]
  34. The following exactness property is also fundamental for Whitney complexes, but is not explicitly used in the paper. Any divergence-free field in Wdiv is the curl of some field in Wcurl.
  35. G. H. Golub and C. F. Van Loan, Matrix Computations (The Johns Hopkins University, 1996).
  36. G. J. Rodin, “Higher-order macroscopic measures,” J. Mech. Phys. Solids 55, 1103–1119 (2007).
    [CrossRef]
  37. A. Moroz, “Effective medium properties, mean-field description, homogenization, or homogenisation of photonic crystals,” http://www.wave-scattering.com/pbgheadlines.html#Effective%20medium%20properties.
  38. I. Tsukerman and F. Čajko, “Photonic band structure computation using FLAME,” IEEE Trans. Magn. 44, 1382–1385(2008).
    [CrossRef]
  39. R. Gajic, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13, 8596–8605(2005).
    [CrossRef] [PubMed]
  40. I. Tsukerman, Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves (Springer, 2007).
  41. E. Tonti, “A mathematical model for physical theories,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. LII, 175–181; 350–356 (1972).
  42. L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

2010 (3)

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
[CrossRef]

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

V. A. Markel, University of Pennsylvania, Philadelphia, Pennsylvania (personal communication, 2010).

2009 (2)

R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. USA 106, 1693–1698 (2009).
[CrossRef] [PubMed]

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

2008 (5)

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[CrossRef] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

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

V. A. Markel, “Correct definition of the Poynting vector in electrically and magnetically polarizable medium reveals that negative refraction is impossible,” Opt. Express 16, 19152–19168(2008).
[CrossRef]

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

2007 (6)

G. J. Rodin, “Higher-order macroscopic measures,” J. Mech. Phys. Solids 55, 1103–1119 (2007).
[CrossRef]

A. Alù, F. Bilotti, N. EnghetaL. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antenn. Propag. 55, 13–25 (2007).
[CrossRef]

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

I. Tsukerman, Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves (Springer, 2007).

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

2006 (4)

K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech. 54, 135–146 (2006).
[CrossRef]

P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antenn. Propag. 54, 1654–1662 (2006).
[CrossRef]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

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

2005 (2)

R. Gajic, R. Meisels, F. Kuchar, and K. Hingerl, “Refraction and rightness in photonic crystals,” Opt. Express 13, 8596–8605(2005).
[CrossRef] [PubMed]

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
[CrossRef]

2003 (2)

P. Monk, Finite Element Methods for Maxwell’s Equations (Clarendon, 2003).
[CrossRef]

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

2002 (1)

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

2000 (1)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

1998 (1)

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

1996 (1)

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

1988 (1)

A. Bossavit, “Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism,” IEE Proc. A 135, 493–500 (1988).
[CrossRef]

1986 (1)

J.-C. Nédélec, “A new family of mixed finite elements in R3,” Numer. Math. 50, 57–81 (1986).
[CrossRef]

1985 (2)

P. R. Kotiuga, “Hodge decompositions and computational electromagnetics,” PhD thesis (McGill University, 1985).

J. van Welij, “Calculation of eddy currents in terms of H on hexahedra,” IEEE Trans. Magn. 21, 2239–2241 (1985).
[CrossRef]

1984 (2)

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

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

1980 (1)

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

1978 (1)

A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Methods in Periodic Media (North-Holland, 1978).

1974 (1)

V. V. Bokut, A. N. Serdyukov, and F. I. Fedorov, “Form of constitutive equations in optically active crystals,” Opt. Spectrosc. 37, 166–168 (1974).

1972 (1)

E. Tonti, “A mathematical model for physical theories,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. LII, 175–181; 350–356 (1972).

1970 (1)

G. Russakoff, “A derivation of the macroscopic Maxwell equations,” Am. J. Phys. 38, 1188–1195 (1970).
[CrossRef]

1957 (1)

H. Whitney, Geometric Integration Theory (Princeton University, 1957).

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, F. Bilotti, N. EnghetaL. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antenn. Propag. 55, 13–25 (2007).
[CrossRef]

Bensoussan, A.

A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Methods in Periodic Media (North-Holland, 1978).

Bilotti, F.

A. Alù, F. Bilotti, N. EnghetaL. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antenn. Propag. 55, 13–25 (2007).
[CrossRef]

Bokut, V. V.

V. V. Bokut, A. N. Serdyukov, and F. I. Fedorov, “Form of constitutive equations in optically active crystals,” Opt. Spectrosc. 37, 166–168 (1974).

Bossavit, A.

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

A. Bossavit, “Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism,” IEE Proc. A 135, 493–500 (1988).
[CrossRef]

Buell, K.

K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech. 54, 135–146 (2006).
[CrossRef]

Cajko, F.

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

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

Demkowicz, L.

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

Engheta, N.

A. Alù, F. Bilotti, N. EnghetaL. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antenn. Propag. 55, 13–25 (2007).
[CrossRef]

Engstrom, C.

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
[CrossRef]

Fedorov, F. I.

V. V. Bokut, A. N. Serdyukov, and F. I. Fedorov, “Form of constitutive equations in optically active crystals,” Opt. Spectrosc. 37, 166–168 (1974).

Fedotov, V. A.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Fietz, C.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
[CrossRef]

Gajic, R.

Genov, D. A.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[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 (The Johns Hopkins University, 1996).

Hingerl, K.

Ikonen, P.

P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antenn. Propag. 54, 1654–1662 (2006).
[CrossRef]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

Kotiuga, P. R.

P. R. Kotiuga, “Hodge decompositions and computational electromagnetics,” PhD thesis (McGill University, 1985).

Kristensson, G.

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
[CrossRef]

Kuchar, F.

Kurtz, J.

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

Landau, L. D.

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

Lifshitz, E. M.

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

Lions, J. L.

A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Methods in Periodic Media (North-Holland, 1978).

Liu, M.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[CrossRef] [PubMed]

Markel, V. A.

Maslovski, S. I.

P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antenn. Propag. 54, 1654–1662 (2006).
[CrossRef]

Meisels, R.

Merlin, R.

R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. USA 106, 1693–1698 (2009).
[CrossRef] [PubMed]

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

Monk, P.

P. Monk, Finite Element Methods for Maxwell’s Equations (Clarendon, 2003).
[CrossRef]

Moroz, A.

A. Moroz, “Effective medium properties, mean-field description, homogenization, or homogenisation of photonic crystals,” http://www.wave-scattering.com/pbgheadlines.html#Effective%20medium%20properties.

Mosallaei, H.

K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech. 54, 135–146 (2006).
[CrossRef]

Nédélec, J.-C.

J.-C. Nédélec, “A new family of mixed finite elements in R3,” Numer. Math. 50, 57–81 (1986).
[CrossRef]

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

Papanicolaou, G.

A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Methods in Periodic Media (North-Holland, 1978).

Papasimakis, N.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Pardo, D.

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

Paszenski, M.

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

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

Prosvirnin, S. L.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Rachowicz, W.

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

Rodin, G. J.

G. J. Rodin, “Higher-order macroscopic measures,” J. Mech. Phys. Solids 55, 1103–1119 (2007).
[CrossRef]

Russakoff, G.

G. Russakoff, “A derivation of the macroscopic Maxwell equations,” Am. J. Phys. 38, 1188–1195 (1970).
[CrossRef]

Sarabandi, K.

K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech. 54, 135–146 (2006).
[CrossRef]

Sarychev, A. K.

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

Schultz, S.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

Serdyukov, A. N.

V. V. Bokut, A. N. Serdyukov, and F. I. Fedorov, “Form of constitutive equations in optically active crystals,” Opt. Spectrosc. 37, 166–168 (1974).

Shalaev, V. M.

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

Shvets, G.

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (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. Fundam. Appl. 8, 254–263 (2010).
[CrossRef]

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

P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antenn. Propag. 54, 1654–1662 (2006).
[CrossRef]

Sjöberg, D.

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
[CrossRef]

Smith, D. R.

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

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

Tonti, E.

E. Tonti, “A mathematical model for physical theories,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. LII, 175–181; 350–356 (1972).

Tretyakov, S.

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

Tretyakov, S. A.

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

P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antenn. Propag. 54, 1654–1662 (2006).
[CrossRef]

Tsukerman, I.

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]

I. Tsukerman, Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves (Springer, 2007).

Van Loan, C. F.

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

van Welij, J.

J. van Welij, “Calculation of eddy currents in terms of H on hexahedra,” IEEE Trans. Magn. 21, 2239–2241 (1985).
[CrossRef]

Vegni, L.

A. Alù, F. Bilotti, N. EnghetaL. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antenn. Propag. 55, 13–25 (2007).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

Vinogradov, A. P.

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

Wall, D. J. N.

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
[CrossRef]

Wang, Y.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[CrossRef] [PubMed]

Wellander, N.

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
[CrossRef]

Whitney, H.

H. Whitney, Geometric Integration Theory (Princeton University, 1957).

Zdunek, A.

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

Zhang, S.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[CrossRef] [PubMed]

Zheludev, N. I.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Am. J. Phys. (1)

G. Russakoff, “A derivation of the macroscopic Maxwell equations,” Am. J. Phys. 38, 1188–1195 (1970).
[CrossRef]

Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. (1)

E. Tonti, “A mathematical model for physical theories,” Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Nat. LII, 175–181; 350–356 (1972).

IEE Proc. A (1)

A. Bossavit, “Whitney forms: a class of finite elements for three-dimensional computations in electromagnetism,” IEE Proc. A 135, 493–500 (1988).
[CrossRef]

IEEE Trans. Antenn. Propag. (2)

A. Alù, F. Bilotti, N. EnghetaL. Vegni, “Subwavelength, compact, resonant patch antennas loaded with metamaterials,” IEEE Trans. Antenn. Propag. 55, 13–25 (2007).
[CrossRef]

P. Ikonen, S. I. Maslovski, C. R. Simovski, and S. A. Tretyakov, “On artificial magnetodielectric loading for improving the impedance bandwidth properties of microstrip antennas,” IEEE Trans. Antenn. Propag. 54, 1654–1662 (2006).
[CrossRef]

IEEE Trans. Magn. (2)

J. van Welij, “Calculation of eddy currents in terms of H on hexahedra,” IEEE Trans. Magn. 21, 2239–2241 (1985).
[CrossRef]

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

IEEE Trans. Microwave Theory Tech. (1)

K. Buell, H. Mosallaei, and K. Sarabandi, “A substrate for small patch antennas providing tunable miniaturization factors,” IEEE Trans. Microwave Theory Tech. 54, 135–146 (2006).
[CrossRef]

J. Mech. Phys. Solids (1)

G. J. Rodin, “Higher-order macroscopic measures,” J. Mech. Phys. Solids 55, 1103–1119 (2007).
[CrossRef]

J. Opt. Soc. Am. B (2)

Multiscale Mod. Simul. (1)

D. Sjöberg, C. Engstrom, G. Kristensson, D. J. N. Wall, and N. Wellander, “A Floquet–Bloch decomposition of Maxwell’s equations applied to homogenization,” Multiscale Mod. Simul. 4, 149–171 (2005).
[CrossRef]

Numer. Math. (2)

J.-C. Nédélec, “Mixed finite elements in R3,” Numer. Math. 35, 315–341 (1980).
[CrossRef]

J.-C. Nédélec, “A new family of mixed finite elements in R3,” Numer. Math. 50, 57–81 (1986).
[CrossRef]

Opt. Express (2)

Opt. Spectrosc. (2)

V. V. Bokut, A. N. Serdyukov, and F. I. Fedorov, “Form of constitutive equations in optically active crystals,” Opt. Spectrosc. 37, 166–168 (1974).

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

Photon. Nanostr. Fundam. Appl. (1)

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

Phys. Rev. B (1)

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

Phys. Rev. Lett. (3)

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187(2000).
[CrossRef] [PubMed]

S. Zhang, D. A. Genov, Y. Wang, M. Liu, and X. Zhang, “Plasmon-induced transparency in metamaterials,” Phys. Rev. Lett. 101, 047401 (2008).
[CrossRef] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101, 253903 (2008).
[CrossRef] [PubMed]

Phys. Usp. (1)

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

Physica B (1)

C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B 405, 2930–2934 (2010).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

R. Merlin, “Metamaterials and the Landau–Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl. Acad. Sci. USA 106, 1693–1698 (2009).
[CrossRef] [PubMed]

Science (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980(2006).
[CrossRef] [PubMed]

Other (15)

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

S. Tretyakov, Analytical Modeling in Applied Electromagnetics (Artech House, 2003).

H. Whitney, Geometric Integration Theory (Princeton University, 1957).

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

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

P. R. Kotiuga, “Hodge decompositions and computational electromagnetics,” PhD thesis (McGill University, 1985).

A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Methods in Periodic Media (North-Holland, 1978).

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

P. Monk, Finite Element Methods for Maxwell’s Equations (Clarendon, 2003).
[CrossRef]

I. Tsukerman, Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves (Springer, 2007).

A. Moroz, “Effective medium properties, mean-field description, homogenization, or homogenisation of photonic crystals,” http://www.wave-scattering.com/pbgheadlines.html#Effective%20medium%20properties.

The following exactness property is also fundamental for Whitney complexes, but is not explicitly used in the paper. Any divergence-free field in Wdiv is the curl of some field in Wcurl.

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

V. A. Markel, University of Pennsylvania, Philadelphia, Pennsylvania (personal communication, 2010).

L. Demkowicz, J. Kurtz, D. Pardo, M. Paszenski, W. Rachowicz, and A. Zdunek, Computing with hp-Adaptive Finite Elements (Chapman & Hall/CRC, 2007), Vol. 2.

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Figures (9)

Fig. 1
Fig. 1

Sketch of the fields in the direction normal to the interface. The cell-averaged b field (dashed line) may differ from its boundary value. For simplicity, factors 4 π (present in the Gaussian system) and μ 0 (in the SI system) are not shown.

Fig. 2
Fig. 2

2D analog of the vectorial interpolation function w α (in this case, associated with the central vertical edge shared by two adjacent cells). Tangential continuity of this function is evident from the arrow plot; its circulation is equal to 1 over the central edge and to zero over all other edges.

Fig. 3
Fig. 3

2D analog of the vectorial interpolation function v β (in this case, associated with the central vertical edge). Normal continuity of this function is evident from the arrow plot; its flux is equal to 1 over the central edge and zero over all other edges.

Fig. 4
Fig. 4

Lattice (with arbitrary inclusions) serves as a “scaffolding” for the construction of coarse-grained fields. The curl-conforming fields ( E , H ) are interpolated into the cells from the edge circulations, while the div-conforming fields ( B , D ) are interpolated from the face fluxes. Only E and B are shown.

Fig. 5
Fig. 5

Effective ε for a 2D periodic array of cylinders. Curves, the proposed procedure (with nine cylindrical harmonics); markers, the MG formula.

Fig. 6
Fig. 6

Γ X Bloch bands obtained with the effective parameters (markers) versus accurate numerical simulation (solid curves). ε cyl = 9.61 , r cyl = 0.33 a .

Fig. 7
Fig. 7

Effective parameters (dashed curve, ε eff ; solid curve, μ eff ) versus frequency in the Γ X direction. ε cyl = 9.61 , r cyl = 0.33 a .

Fig. 8
Fig. 8

Contour plot of Re ( H ) . Angle of incidence π / 6 , ε cyl = 9.61 , r cyl = 0.33 a .

Fig. 9
Fig. 9

Re ( H ) along the coordinate perpendicular to the slab. The angle of incidence π / 6 . Other parameters are the same as in Figs. 7, 8.

Equations (25)

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

× e = i ω c 1 b , × b = i ω c 1 d .
E , H H ( curl , Ω ) ; B , D H ( div , Ω ) ,
q = α = 1 12 [ q ] α w α .
× W curl W div .
E α = 1 12 [ e ] α w α W curl ( [ e ] 1 12 )
B = β = 1 6 [ [ b ] ] β v β W div ( [ [ b ] ] 1 6 )
e = E + e , d = D + d , b = B + b .
b = H + 4 π m .
× ( E + e ) = i ω c 1 ( B + b ) ,
× ( H + 4 π m ) = i ω c 1 ( D + d ) .
face ( × E ) · dS = face edges E · dl = i ω c 1 face b · dS ,
× E = i ω c 1 B .
× H = i ω c 1 D .
× e = i ω c 1 b , 4 π × m = i ω c 1 d
d = ε e + ( ε E D ) , 4 π m = b + ( B H ) .
× × e ( ω / c ) 2 ε e = ( ω / c ) 2 ( ε E D ) i ω / c × ( B H ) .
Ψ e h = α c α ψ α e h ; Ψ d b = α c α ψ α d b .
ψ α D B ( r ) = ζ ( r ) ψ α E H ( r ) ,
ζ ( r ) Ψ E H ( r ) = Ψ D B ( r ) ,
ζ ( r ) = Ψ D B ( r ) ( Ψ E H ) + ( r ) .
× ( E H ) = i ω c ( B D ) = i ω c ( 0 I 3 I 3 0 ) ζ ( r ) ( E H ) ,
ε eff D 0 E 0 = cell d · dS S E 0 = S 1 cell ε d S
· ε ψ 1 = 0 ; ψ 1 ( x = ± a 2 ) = ± a 2 ; ψ 1 ( y + a ) = ψ 1 ( y ) .
[ [ d 1 ] ] = y = ± a / 2 d 1 x d y = y = ± a / 2 ε x ψ 1 d y ,
( ε x x ε x y ε y x ε y y ) ( 1 0 0 1 ) = ( [ [ d 1 ] ] 0 0 [ [ d 2 ] ] )

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