Directed Electromagnetic Pulse Dynamics: Projecting Operators Method

Abstract

In this article, we consider a one-dimensional model of electromagnetic pulse propagation in isotropic media, taking into account a nonlinearity of the third order. We introduce a method for Maxwell's equation transformation on the basis of a complete set of projecting operators. The operators correspond to wave dispersion branches including the direction of propagation. As the simplest result of applying the method, we derive a system of equations describing one-dimensional dynamics of pulses of opposite directions of propagation without dispersion account. The corresponding self-action equations are extracted. Next we introduce a dispersion and show how the operators change.Through applications in such a manner, the generalized short pulse equations (SPE) of Shafer and Wayne are obtained for both propagation directions. The effects of the unidirectional pulse interaction are discussed.

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

“…The main aim of the work is very similar to the recent [39] and [37] we do want to derive an evolution equation for the mentioned conditions with the minimal simplifications. The methodical differences and results are highlighted and discussed.…”

“…Following the technique described in [37], we arrive at 2 × 2 eigenvalue problem. Let us look for such matrices P (i) , i = 1, 2 that P (i) Ψ = Ψ i would be eigenvectors of the evolution matrix (35).…”

“…For nonlinear Kerr materials [44,45], the third-order nonlinear part of polarisation [40,37] has the form:…”

Ultrashort Opposite Directed Pulses Dynamics with Kerr Effect and Polarization Account

“…Following the general scheme in Ref. 20 we showed in our previous work, 14) new variables, Eqs. (23) and (24) may be introduced into Eq.…”

“…The projection operator method is a powerful tool, which was initially set for solving the Cauchy problem, and allows for detailed analysis of a field and its modes. Applying the projection operators to the one-dimensional model of the light pulse propagation has led to the generalized SP equation, 14) where the physical form of cofactors has been shown and the interaction between waves in two directions has been taken into consideration. The case of interaction of two counterpropagating waves was the topic of experimental studies, 15) and proof of the interaction of a counterpropagating pulse polarization has been presented.…”