Simulation of Hamiltonian light-beam propagation in nonlinear media

Abstract

The simulation of nonlinear wave propagation in the ray regime, i.e., in the limit of geometrical optics, is discussed. The medium involved is nonlinear, which means that the field amplitudes affect the constitutive parameters (e.g., dielectric constant) involved in the propagation formalism. Conventionally, linear ray propagation is computed by the use of Hamilton's ray equations whose terms are derived from the appropriate dispersion equation. The formalism used to solve such a set of equations is the Runge-Kutta algorithm in one of its variants. In the present case of nonlinear propagation, a proper dispersion equation must first be established from which the rays can be computed. Linear ray tracing with Hamilton's ray theory allows for the computation of ray trajectories and wave fronts. The convergence or divergence of rays suggests heuristic methods for computing the variation of amplitudes. Here, terms appearing in the Hamiltonian ray equations involve field amplitudes, which themselves are determined by the convergence (or divergence) of the rays. This dictates the simultaneous computation of a beam comprising many rays, so it is necessary to modify the original Runge-Kutta scheme by building into it some iteration mechanism such that the process converges to the values that take into account the amplitude effect. This research attempts to modify the existing propagation formalism and apply the new algorithm to simple problems of nonlinear ray propagation. The results display self-focusing effects characteristic of nonlinear optics problems. The influence of weak losses on the beam propagation and its self focusing is also discussed. Some displayed results obtained by simulating the modified formalism seem to be physically plausible and are in excellent agreement with experimental results reported in the literature.

Application-Oriented Relativistic Electrodynamics (2)

“…In free space c −2 = µ o ε o hence if the value of the invariant (56) is set to zero, it becomes the well-known Lorentz condition (in K space). However, in material media (56) ceases to be the Lorentz condition. This is a point that might cause some confusion, especially in view of the fact that the Lorentz condition is a gauge transformation invariant, as explained in many of the textbooks cited above.…”

“…Paramount are the phenomena of harmonic generation, which one finds also in nonlinear lumped elements (e.g., magnetic materials which become saturated when flux increases, or electronic devices possessing nonlinear voltage-current characteristic curves), and new wave-specific phenomena such as self-focusing. In the latter, due to the field dependent constitutive parameters, the wave, depending on the intensity profile, "creates for itself" a "lens", thus a self-focusing phenomenon appears, e.g., see [56]. Once the tools of the Minkowski four-vectors and the associated Fourier transforms are at our disposal, it can be introduced in a compact and consistent manner.…”

Electrodynamics, Topsy-Turvy Special Relativity, and Generalized Minkowski Constitutive Relations for Linear and Nonlinear Systems

“…For additional literature references and related work see [19][20][21][22][23][24][25][26][27][28]. A postulated model [29], applied to numerical simulation of rays in nonlinear media, was used in conjunction with experimental data [30] given in the literature, and close agreement of the experimental and simulation results was found. Our prototype linear model for constitutive relations is given by (23), or its spectral equivalent in one of the forms (24) or (27), whichever is more convenient at a given time, or the spatiotemporal convolution integral equivalent (28).…”

“…But this special case refers once more to a homogeneous medium! If we reject forms like (30), (33) as inadequate for describing physically meaningful constitutive relations, the unavoidable conclusion is that spatiotemporal constitutive relations like (29) are invalid. This sweeping conclusion still allows for approximation methods like the Hamiltonian ray theory mentioned above.…”