Exact solutions for the source-excited cylindrical electromagnetic waves in a nonlinear nondispersive medium

Abstract

The behavior of electromagnetic fields in nonlinear media has been a topical problem since the discovery of materials with a nonlinearity of electromagnetic properties. The problem of finding exact solutions for the source-excited nonlinear waves in curvilinear coordinates has been regarded as unsolvable for a long time. In this work, we present the first solution of this type for a cylindrically symmetric field excited by a pulsed current filament in a nondispersive medium that is simultaneously inhomogeneous and nonlinear. Assuming that the medium has a power-law permittivity profile in the linear regime and lacks a center of inversion, we derive an exact solution for the electromagnetic field excited by a current filament in such a medium and discuss the properties of this solution.

Solutions of the cylindrical nonlinear Maxwell equations

“…Over the course of the past 40 years, electromagnetic wave propagation in nonlinear media has become a wideranging problem, and significant progress has been made using analytical and numerical methods [1][2][3][4][5][6][7][8][9][10][11][12]. The studies of such a problem can be employed for various optical engineering technologies, such as wavelength conversion of light [8], noninvasive and noncontact probe of the electronic and structural properties of crystals, and information processing in wavelength-demultiplexing communication [9].…”

Classical theory of cylindrical nonlinear optics: Sum- and difference-frequency generation

“…Similarly to second-harmonic generation, SFG and DFG also come from the secondary nonlinear polarization of a nonlinear medium. Second, previous studies [14][15][16][17] of the exact solution usually focus on single-electromagnetic-wave propagation in a nonlinear medium. However, for the case of two or more waves existing in the nonlinear medium simultaneously, there are few effective methods to deal with it.…”

Nonlinear normal modes in electrodynamic systems: A nonperturbative approach

“…With appropriately chosen constants D 0 , ε 1 , and α, such an exponential constitutive relation correctly describes dielectric properties of actual media lacking a center of inversion in the case of weak nonlinearity where we can restrict ourselves to the quadratic (in E z ) correction term to the linear dependence of D z on E z (see [17,18] for more details). Note that the model of a nonlinear distributed system presented herein and its generalizations were also discussed in works [19][20][21][22][23]. In this case, the Maxwell equations read…”

“…One should only ensure the convergence of Fourier series (21) and the absence of ambiguity of the implicit solutions. For the electromagnetic fields governed by the nonlinear system of equations (1) and (2), the principle of superposition does not hold and the NNM fields lack the usual orthogonality property which takes place for linear normal modes in resonators [2].…”

“…This fact implies a quite remarkable energy orthogonality property of NNMs. The total energy W of the complicated oscillatory process, which is described by exact implicit solution (4), with E and H given by (21), is merely the sum of the NNM (or LNM) energies (14):…”