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We analyse the behaviour of the spectrum of the system of Maxwell equations of electromagnetism, with rapidly oscillating periodic coefficients, subject to periodic boundary conditions on a “macroscopic” domain
We consider the case where the contrast between the values of the coefficients in different parts of their periodicity cell increases as the period of oscillations
goes to zero. We show that the limit of the spectrum as
contains the spectrum of a “homogenized” system of equations that is solved by the limits of sequences of eigenfunctions of the original problem. We investigate the behaviour of this system and demonstrate phenomena not present in the scalar theory for polarized waves.
We study the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities
have a high degree of contrast between each other. We assume that the ratio between the permittivities of the components with low and high values of
is of the order
is the period of the medium. We determine the asymptotic behaviour of the electromagnetic response of such a medium in the “homogenization limit”, as
, and derive the limit system of Maxwell equations in
. Our results extend a number of conclusions of a paper by Zhikov [On gaps in the spectrum of some divergent elliptic operators with periodic coefficients. St. Petersburg Math. J.16(5) (2004), 719–773] to the case of the full system of Maxwell equations.
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