Advertisement

Homogenization of Maxwell Equations—Macroscopic and Microscopic Approaches

  • Arkadi ChipoulineEmail author
  • Franko Küppers
Chapter
Part of the Springer Series in Optical Sciences book series (SSOS, volume 211)

Abstract

We consider as a starting point a system of microscopic MEs in the following form: \( \left\{ \begin{aligned} & \text{rot}\, \vec {e} \;= \frac{{i\omega}}{c}\vec {h} \\ & \text{div} \vec {h} \;= 0 \\ & \text{div} \,\vec {e} \;= 4\pi \rho \\ & \text{rot} \,\vec {h} \;= - \frac{{i\omega}}{c}\vec {e} + \frac{{4\pi}}{c}\vec {j} \\ \end{aligned} \right.\quad \quad \left\{ \begin{aligned} &\rho \quad\;= \sum\limits_{i} {q_{i}\delta\left( {\vec {r} - \vec {r}_{i} } \right)} \\ & \vec {j} \quad\;= \sum\limits_{i} {\vec {v}_{i} q_{i}\delta\left( {\vec {r} - \vec {r}_{i} } \right)} \\ & \frac{{\text{d}\vec {p}_{i} }}{\text{d}t} = q_{i} \vec {e} + \frac{{q_{i} }}{c}\left[ {\vec {{v_{i} }} *\vec {h} } \right]. \\ \end{aligned} \right. \)

References

  1. 1.
    L.D. Landau, E.L. Lifshitz, Electrodynamics of Continuous Media, 2nd edn. (Pergamon Press, New York, 1960) (Chapter IX)Google Scholar
  2. 2.
    V. Dubovik, S. Shabanov, in Essays on the Formal Aspects of Electromagnetic Theory, ed. by A. Lakhakia. The Gauge Invariance, Toroid Order Parameters and Radiation in Electromagnetic Theory, vol. 399 (World Scientific, Singapore, New Jersey, London, Hong-Kong, 1993)CrossRefGoogle Scholar
  3. 3.
    J. Schwinger, L.L. De-Raad, K. Milton, W. Tsai, Classical Electrodynamics (Perseus Books, Reading, MA, 1998)Google Scholar
  4. 4.
    P. Mazur, B. Nijboer, On the statistical mechanics of matter in an electromagnetic field. I. Physica XIX, 971 (1953)CrossRefGoogle Scholar
  5. 5.
    G. Rusakoff, A derivation of the macroscopic Maxwell equations. Am. J. Phys. 38(10), 1188 (1970)CrossRefGoogle Scholar
  6. 6.
    A. Maradudin, D.L. Mills, Phys. Rev. B 7, 2787 (1973)CrossRefGoogle Scholar
  7. 7.
    J.D. Jackson, Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999)Google Scholar
  8. 8.
    S. Maslovsky, Electrodynamics of composite materials with pronounced spatial dispersion. Ph.D. thesis (St. Petersburg Polytechnic University, Russia, 2004)Google Scholar
  9. 9.
    A. Vinogradov, A. Aivazyan, Scaling theory of homogenization of the Maxwell equations. Phys. Rev. E 60, 987 (1999)CrossRefGoogle Scholar
  10. 10.
    C. Simovski, Weak spatial dispersion in composite media. Polytechnika (St. Petersburg, 2003) (in Russian)Google Scholar
  11. 11.
    Y. Svirko, N. Zheludev, M. Osipov, Layered chiral metallic microstructures with inductive coupling. APL 78, 498 (2001)Google Scholar
  12. 12.
    N. Zheludev, The road ahead for metamaterials. Science 328, 582 (2010)CrossRefGoogle Scholar
  13. 13.
    N. Papasimakis, V. Fedotov, K. Marinov, N. Zheludev, Gyrotropy of a metamolecule: wire on a torus. PRL 103, 093901 (2009)CrossRefGoogle Scholar
  14. 14.
    A.N. Grigorenko, A.K. Geim, H.F. Gleeson, Y. Zhang, A.A. Firsov, I.Y. Khrushchev, J. Petrovic, Nanofabricated media with negative permeability at visible frequencies. Nature 438, 335–338 (2005)CrossRefGoogle Scholar
  15. 15.
    V. Shalaev, W. Cai, U. Chettiar, H.-K. Yuan, A. Sarychev, V. Drachev, A. Kildishev, Negative index of refraction in optical metamaterials. Opt. Lett. 30, 3356 (2005)CrossRefGoogle Scholar
  16. 16.
    A. Vinogradov, Electrodynamics of Compound Media (Scientific and Educational Literature Publisher, Russian Federation, 2001). ISBN 5-8360-0283-5 (in Russian)Google Scholar
  17. 17.
    C. Simovski, Material parameters of metamaterials (a review). Opt. Spectrosc. 107, 726 (2009)CrossRefGoogle Scholar
  18. 18.
    R. Clausius, Mechanische Warmetheorie, vol. 2, 2nd edn. (Braunschweig, 1878), p. 62Google Scholar
  19. 19.
    O. Mossotti, Mem. Soc. Sci. Modena 14, 49 (1850)Google Scholar
  20. 20.
    H. Lorentz, Proc. Acad. Sci. Amsterdam 13, 92 (1910)Google Scholar
  21. 21.
    P. Ewald, Ann. Phys. 64, 2943 (1921)Google Scholar
  22. 22.
    D. Bruggeman, Ann. Phys. Lpz. 24, 636 (1935)CrossRefGoogle Scholar
  23. 23.
    C. Simovski, Radiotekh. Elecktron. 52, 1031 (2007)Google Scholar
  24. 24.
    G. Shvets, Photonic approach to make a surface wave accelerator. AIP Conf. Proc. 647, 371 (2002)CrossRefGoogle Scholar
  25. 25.
    P. Belov, R. Marques, M. Silveirinha, I. Nefedov, C. Simovski, S. Trtyakov, Strong spatial dispersion in wire media in the very long wavelength limit. Phys. Rev. B 70, 113103 (2003)CrossRefGoogle Scholar
  26. 26.
    M. Silveirinha, Nonlocal homogenization model for a periodic array of epsilon-negative rods. Phys. Rev. E 73, 046612 (2006)CrossRefGoogle Scholar
  27. 27.
    E. Pshenay-Severin, A. Chipouline, J. Petschulat, U. Huebner, A. Tuennermann, T. Pertsch, Optical properties of metamaterials based on asymmetric double-wire structures. Opt. Express 19, 6269 (2011)CrossRefGoogle Scholar
  28. 28.
    C. Simovski, On electromagnetic characterization and homogenization of nanostructured metamaterials. J. Opt. 13, 013001 (2011)CrossRefGoogle Scholar
  29. 29.
    A. Sarychev, V. Shalaev, Electrodynamics of Metamaterials (World Scientific, Singapore, 2007)Google Scholar
  30. 30.
    D. Smith, D. Shurig, PRL 90, 077405 (2003)CrossRefGoogle Scholar
  31. 31.
    P. Belov, C. Simovski, Phys. Rev. E 72, 026615 (2005)CrossRefGoogle Scholar
  32. 32.
    J.D. Joannopoulos, R.D. Meade, J.N. Winn, Photonic Crystals: Molding the Flow of Light (1995). ISBN-13: 978-0-691-03744-8Google Scholar
  33. 33.
    C. Rockstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T. Meyrath, H. Giessen, Phys. Rev. B 77, 035126 (2008)CrossRefGoogle Scholar
  34. 34.
    M. Born, K. Huang, Dynamic Theory of Crystal Lattices (Oxford University Press, Oxford, 1954)Google Scholar
  35. 35.
    M. Silveirinha, Time domain homogenization of metamaterials. Phys. Rev. B 77, 035126 (2011)Google Scholar
  36. 36.
    C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, F. Lederer, Validity of effective material parameters for optical fishnet metamaterials. Phys. Rev. B 81, 035320 (2010)CrossRefGoogle Scholar
  37. 37.
    M. Bredov, V. Rumyantcev, I. Toptygin, Classical Electrodynamics (Nauka, 1985) (in Russian)Google Scholar
  38. 38.
    I.B. Zeldovich, Electromagnetic interaction with parity violation. JETP 33, 1531 (1957)Google Scholar
  39. 39.
    L. Mandelshtam, Full collection of publications. Publ. Acad. Sci. USSR 1, 162 (1957) (in Russian)Google Scholar
  40. 40.
    T. Kaelberer, V.A. Fedotov, N. Papasimakis, D.P. Tsai, N.I. Zheludev, Science 330, 1510 (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Microwave Engineering and PhotonicsTechnical University of DarmstadtDarmstadtGermany
  2. 2.Department of Electrical Engineering and Information TechnologiesTechnical University of DarmstadtDarmstadtGermany

Personalised recommendations