Operator approach to effective medium theory to overcome a breakdown of Maxwell Garnett approximation

Vladislav Popov, Andrei Lavrinenko, Andrey Novitsky

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We elaborate on an operator approach to effective medium theory for homogenization of the periodic multilayered structures composed of nonmagnetic isotropic materials, which is based on equating the spatial evolution operators for the original structure and its effective alternative. We show that the zeroth-, first-, and second-order approximations of the operator effective medium theory correspond to electric dipoles, chirality, and magnetic dipoles plus electric quadrupoles, respectively. We discover that the spatially dispersive bianisotropic effective medium obtained in the second-order approximation perfectly replaces a multilayered composite and does not suffer from the effective medium approximation breakdown that happened near the critical angle of total internal reflection found previously in the conventional effective medium theory. We establish the criterion of the validity of the conventional effective medium theory depending on the ratio of unit-cell length to the wavelength, the number of unit cells, and the angle of incidence. The operator approach to effective medium theory is applicable for periodic and nonperiodic layered systems, being a fruitful tool in the fields of metamaterials and subwavelength nanophotonics.
Original languageEnglish
Article number085428
JournalPhysical Review B
Issue number8
Number of pages10
Publication statusPublished - 2016


  • Condensed Matter Physics
  • Electronic, Optical and Magnetic Materials
  • Artificial electromagnetic wave materials and structures
  • Mathematical analysis
  • electric moments
  • electromagnetic metamaterials
  • mathematical operators
  • multilayers
  • quadrupole moments
  • operator approach
  • effective medium theory
  • Maxwell Garnett approximation
  • homogenization
  • periodic multilayered structure
  • nonmagnetic isotropic materials
  • spatial evolution operators
  • electric dipoles
  • chirality
  • magnetic dipoles
  • electric quadrupoles
  • second order approximation
  • metamaterials
  • subwavelength nanophotonics


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