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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Editor: Henry van Driel
  • Vol. 28, Iss. 3 — Mar. 1, 2011
  • pp: 577–586

Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation

Igor Tsukerman  »View Author Affiliations

JOSA B, Vol. 28, Issue 3, pp. 577-586 (2011)

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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

OCIS Codes
(260.2110) Physical optics : Electromagnetic optics
(050.1755) Diffraction and gratings : Computational electromagnetic methods
(050.2065) Diffraction and gratings : Effective medium theory
(160.3918) Materials : Metamaterials
(350.4238) Other areas of optics : Nanophotonics and photonic crystals
(050.5298) Diffraction and gratings : Photonic crystals

ToC Category:
Diffraction and Gratings

Original Manuscript: November 17, 2010
Manuscript Accepted: December 16, 2010
Published: February 28, 2011

Igor Tsukerman, "Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation," J. Opt. Soc. Am. B 28, 577-586 (2011)

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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.

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