Directed electromagnetic wave propagation in 1D metamaterial: Projecting operators method

Abstract

We consider a boundary problem for 1D electrodynamics modeling of a pulse propagation in a metamaterial medium. We build and apply projecting operators to a Maxwell system in time domain that allows to split the linear propagation problem to directed waves for a material relations with general dispersion. Matrix elements of the projectors act as convolution integral operators. For a weak nonlinearity we generalize the linear results still for arbitrary dispersion and derive the system of interacting right/left waves with combined (hybrid) amplitudes. The result is specified for the popular metamaterial model with Drude formula for both permittivity and permeability coefficients. We also discuss and investigate stationary solutions of the system related to some boundary regimes.

Interaction of orthogonal-polarized waves in 1D metamaterial with Kerr nonlinearity

“…For ordinary nonlinear Kerr material Kaplan in 1983 showed saving the arrangements of polarizations [20]. But for metamaterial the situation is different, as shown in our work [19] for unique polarization. We discover the change of arrangements of wave modes.…”

“…It's continued systematic application of projecting approach, originated from [8] for 1D metamaterial with account of both polarizations of the EM wave. The technique and results of our previous work [19] on nonlinear evolution equations of opposite directed waves with one polarization in Drude 1D-metamaterial has been developed. In this paper application projecting operators method for this case is demonstrated.…”

“…To close the system (1.5) we need to add material relations: where µ and ε are integral convolution-type operators [19]:…”

“…2 Dynamic projecting operators Doing the Fourier transformations like in [19] and plugging them into the system of equations (1.9) one have the closed system:…”

“…Doing projection operators technique, similarly with [19], finally we obtain four matrix projecting operators in t−representation by the standard general formula:…”