Homogenization of time harmonic Maxwell equations: the effect of interfacial currents

Abstract

Communicated by H. AmmariWe consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time-harmonic current source. We perform the two-scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. the variables with the hat symbol .O/ denote reference values and the variables with the prime superscript denote dimensionless values, respectively. Let O " and O be the reference values of " and : " D O "" 0 , D O 0 . Remind that both " and are dimensionless parameters in the Gaussian system of units. We choose O J D O O E then, in the dimensionless variables, equation (1) becomes curl 0

Homogenization of time-harmonic Maxwell’s equations in nonhomogeneous plasmonic structures

“…To our knowledge, this combination of two effective parameters, one coming from a bulk property and another expressing the property of a hypersurface, has not occurred in most of the previous homogenization results; see, e.g., [1,39]. An exception is the homogenization of Maxwell's equations in a two-phase composite material carried out in [4], which we mentioned above; see also the discussion in Sections 1.4, and 5.…”

“…As mentioned above, our work is related to that in [4] which also analyzes the effect of surface currents in the homogenization of the time harmonic Maxwell equations.…”

“…In [4], however, the authors focus on a different geometric and physical setting, which involves two-phase materials with certain inclusions. Various regimes, which depend on the strength of the interfacial currents, the wavelength and the skin depth, are studied in [4].…”

“…In [4], however, the authors focus on a different geometric and physical setting, which involves two-phase materials with certain inclusions. Various regimes, which depend on the strength of the interfacial currents, the wavelength and the skin depth, are studied in [4]. Both the mathematical formulation and the homogenized equations that we derive in the present paper are closely related to the case referred to as the strong interface layer in [4] (see Theorem 2 in [4]).…”

“…Using (2.10) and Young's inequality, we notice that (3.2) gives 4) and the use of (2.10) and (3.3) combined with (3.4) in turn implies…”