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Proc Math Phys Eng Sci. 2019 Jan;475(2221):20180298. doi: 10.1098/rspa.2018.0298. Epub 2019 Jan 16.

Dispersive and effective properties of two-dimensional periodic media.

Author information

Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA.


We consider transverse propagation of electromagnetic waves through a two-dimensional composite material containing a periodic rectangular array of circular cylinders. Propagation of waves is described by the Helmholtz equation with the continuity conditions for the tangential components of the electric and magnetic fields on the boundaries of the cylinders. We assume that the cell size is small compared with the wavelength, but large compared with the radius a of the inclusions. Explicit formulae are obtained for asymptotic expansion of the solution of the problem in terms of the dimensionless magnitude q of the wavevector and radius a. This leads to explicit formulae for the effective dielectric tensor and the dispersion relation with the rigorously justified error of order O((q 2 + a 2)5/2).


asymptotic expansion; dispersion relation; effective tensor; periodic medium; wave propagation

[Available on 2020-01-01]

Conflict of interest statement

We declare we have no competing interests.

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