Abstract

We analyze the time harmonic Maxwell equations in a complex three-dimensional geometry. The scatterer $\Omega\subset\mathbb{R}^3$ contains a periodic pattern of small wire structures of high conductivity, and the single element has the shape of a split ring. We rigorously derive effective equations for the scatterer and provide formulas for the effective permittivity and permeability. The latter turns out to be frequency dependent and has a negative real part for appropriate parameter values. This magnetic activity is the key feature of a left-handed metamaterial.

MSC codes

  1. 35B27
  2. 35Q60
  3. 35Q61
  4. 78M35
  5. 78M40

Keywords

  1. Maxwell equations
  2. metamaterials
  3. homogenization
  4. two-scale convergence
  5. artificial magnetism
  6. negative index

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Information & Authors

Information

Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 717 - 750
ISSN (online): 1540-3467

History

Submitted: 3 January 2009
Accepted: 8 December 2009
Published online: 3 March 2010

MSC codes

  1. 35B27
  2. 35Q60
  3. 35Q61
  4. 78M35
  5. 78M40

Keywords

  1. Maxwell equations
  2. metamaterials
  3. homogenization
  4. two-scale convergence
  5. artificial magnetism
  6. negative index

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