Propagation of exremely short pulses in non-resonant media: the total Maxwell-Duffing model

We look for travelling wave fields $$ E(x,y,z,t)= U(x,y) \cos(kz+\omega t)+ \widetilde U(x,y)\sin(kz+\omega t),\quad (x,y,z)\in\mathbb{R}^3,\, t\in\mathbb{R} $$ satisfying Maxwell's equations in a nonlinear medium which is not necessarily cylindrically symmetric. The nonlinearity of the medium enters Maxwell's equations by postulating a nonlinear material law $D=\varepsilon E+\chi(x,y, \langle |E|^2\rangle)E$ between the electric field $E$, its time averaged intensity $\langl...

We study the propagation of ultra-short short solitons in a cubic nonlinear medium modeled by nonlinear Maxwell's equations with stochastic variations of media. We consider three cases: variations of (a) the dispersion, (b) the phase velocity, (c) the nonlinear coefficient. Using a modified multi-scale expansion for stochastic systems, we derive new stochastic generalizations of the short pulse equation that approximate the solutions of stochastic nonlinear Maxwell's equation...

We prove that the short-pulse equation, which is derived from Maxwell equations with formal asymptotic methods, can be rigorously justified. The justification procedure applies to small-norm solutions of the short-pulse equation. Although the small-norm solutions exist for infinite times and include modulated pulses and their elastic interactions, the error bound for arbitrary initial data can only be controlled over finite time intervals.

We study the short pulse dynamics in the deterministic and stochastic environment in this thesis. The integrable short pulse equation is a modelling equation for ultra-short pulse propagation in the infrared range in the optical fibers. We investigate the numerical proof for the exact solitary solution of the short pulse equation. Moreover, we demonstrate that the short pulse solitons approximate the solution of the Maxwell equation numerically. Our numerical experiments prov...

Characterizing electromagnetic wave propagation in nonlinear and inhomogeneous media is of great interest from both theoretical and practical perspectives, even though it is extremely complicated. In fact, it is still an unresolved issue to find the exact solutions to the nonlinear waves in the orthogonal curvilinear coordinates. In this paper, we present an analytic method to handle the problem of electromagnetic waves propagation in arbitrarily nonlinear and particularly in...

An alternative way of visualizing electromagnetic waves in matter and of deriving the Finite Difference Time Domain method (FDTD) for simulating Maxwell's equations for one dimensional systems is presented. The method uses d'Alembert's splitting of waves into forward and backward pulses of arbitrary shape and allows for grid spacing and material properties that vary with position. Constant velocity of waves in dispersionless dielectric materials, partial reflection and transm...

We combine scattering theory, Fourier, traveling wave and asymptotic analyses together with numerical simulations to present interesting and practically useful properties of femtosecond pulse interaction with thin films. The dispersive material is described by a single resonance Lorentz model and its nonlinear extension with a cubic Duffing-type nonlinearity. A key feature of the Lorentz dielectric function is that its real part becomes negative between its zero and its pole,...

It is demonstrated in this paper that the propagation of the electric wave field in a heterogeneous medium in 3D can sometimes be governed well by a single PDE, which is derived from the Maxwell's equations. The corresponding component of the electric field dominates two other components. This justifies some past results of the second author with coauthors about numerical solutions of coefficient inverse problems with experimental electromagnetic data. In addition, since it i...

The propagation of solitary waves in a Bragg grating formed by an array of thin nanostructured dielectric films is considered. A system of equations of Maxwell-Duffing type and describing forward- and backward-propagating waves in such a grating, is derived. Exact solitary wave solutions are found, analyzed, and compared with the results of direct numerical simulations.

We consider the modes of the electric field of a cavity where there is an embedded polarized dielectric film. The model consists in the Maxwell equations coupled to a Duffing oscillator for the film which we assume infinitely thin. We derive the normal modes of the system and show that they are orthogonal with a special scalar product which we introduce. These modes are well suited to describe the system even for a film of finite thickness. By acting on the film we demonstrat...