Perfect optical solitons: spatial Kerr solitons as exact solutions of Maxwell's equations

Abstract

We prove that spatial Kerr solitons, usually obtained in the frame of a nonlinear Schrödinger equation valid in the paraxial approximation, can be found in a generalized form as exact solutions of Maxwell's equations. In particular, they are shown to exist, both in the bright and dark version, as TM, linearly polarized, exactly integrable one-dimensional solitons and to reduce to the standard paraxial form in the limit of small intensities. In the two-dimensional case, they are shown to exist as azimuthally polarized, circularly symmetric dark solitons. Both one-and two-dimensional dark solitons exhibit a characteristic signature in that their asymptotic intensity cannot exceed a threshold value in correspondence of which their width reaches a minimum subwavelength value.

On the limits of validity of nonparaxial propagation equations in Kerr media

“…The propagation of solitons with an amplitude of A 0 = 0.4 (i.e., well beyond the non-paraxial approximation) for the TE and TM cases is shown respectively in Figs.2(a) and 2(b). The spatial resolution of the FDTD simulations was λ/100 and the initial conditions were the exact soliton solutions for the TE and TM cases derived in [6]. The FDTD simulations confirm that also the exact TM soliton solution of the Maxwell equation is unstable.…”

“…Actually, both equations can be shown to be incapable of reproducing some exact results available in the literature about (1+1)-D spatial solitons as ab initio solutions of Maxwell's equations, both for TE [5] and TM fields. [6] This seems to indicate that writing a unique nonlinear nonparaxial propagation equation capable of going beyond the standard paraxial approximation may still be an open problem.…”

Matrix-Differential-Operator Approach to the Maxwell Equations and the Dirac Equation

“…Smalij [13] used three-parameter subgroups of the extended Poincaré group P (1, 3) to construct ansatzes of reducing the Maxwell equations to systems of ordinary differential equations and then obtained certain new exact solutions of the Maxwell equations. Ciattonic, Crosignanic, Di Porto and Yariv [2] found optical type solutions of the Maxwell equations.…”

Nonlinear interfaces: intrinsically nonparaxial regimes and effects

“…For ultra-narrow beams, a full vectorial analysis starting from the Maxwell equations can also be necessary [17,18,19] to include the tensorial refractive index dependence. Solutions to these equations in the form of bright [20] and dark [21,22] nonparaxial solitons have been reported and analysed.…”