Geometric interpretation of four-wave mixing

Abstract

The nonlinear phenomenon of four-wave mixing (FWM) is investigated using a method, where, without the need of calculus, both phase and amplitudes of the mixing fields are visualized simultaneously, giving a complete overview of the FWM dynamics. This is done by introducing a set of Stokes-like coordinates of the electric fields, which reduce the FWM dynamics to a closed two-dimensional surface, similar to the Bloch sphere of quantum electrodynamics or the Pointcaré sphere in polarization dynamics. The coordinates are chosen so as to use the gauge invariance symmetries of the FWM equations which also give the conservation of action flux known as the Manley-Rowe relations. This reduces the dynamics of FWM to the one-dimensional intersection between the closed two-dimensional surface and the phase-plane given by the conserved Hamiltonian. The analysis is advantageous for visualizing phase-dependent FWM phenomena which are found in a large variety of nonlinear systems and even in various optical communication schemes.

Application of Manley‐Rowe Relation in Analyzing Nonlinear Interactions Between Planetary Waves and the Solar Semidiurnal Tide During 2009 Sudden Stratospheric Warming Event

“…The term "nonlinear interaction" in the current study specifically refers to the weakly nonlinear interaction or weak (wave) turbulence (e.g., L'Vov et al, 2009), also known as wave-wave interaction, wave mixing, resonant nonlinear interaction, or amplitude modulation (e.g., Nguyen et al, 2016;Ott et al, 2013;Saleh & Teich, 2007;Teitelbaum & Vial, 1991). Different from strong nonlinearity where turbulence cascades energy continuously in wave number k space following power laws (e.g., Kolmogorov, 1991), in the weakly nonlinear interaction, energy dispersion entangles a finite number of modes with discrete k. Mathematically, the linear combination A a ψ a + A b ψ b of two harmonic solutions of a linear system ψ a = e i ωatÀkaÁrþϕ a ð Þ and ψ b = e i ω b tÀk b Árþϕ b ð Þ is also a solution.…”

The coupled 1:2 resonance in a symmetric case and parametric amplification model

“…Applied problem statement(parametric amplification model). The resonant FWM model is an effective method that is applied for fiber-optic parametric amplifiers which are described by a resonant Hamiltonian system [5,16,18,22]. Usually this model is considered in two cases: degenerate and non-degenerate (see the second paragraph of Subsection 4.1, for the detailed explanations on what is degenerate and non-degenerate cases for FWM).…”

“…This system is a Hamiltonian system for the four canonical couples (z j ,z j ) [5,16,18,22], with the Hamiltonian…”

“…Here we will try to find some boundary conditions for phase space and maximum transfer energy. In many papers of physics journals, the scientists considered the dynamics of this problem (for instance see [5,7,15,16,18,19,22]). For example in [16,18,19], some results on the analysis of phase portraits and the solution of the optimization problem are presented.…”

“…As we mentioned above, the major problem is the existence of trajectories under maximum transfer energy. Based on application, with these findings, we can use a proper example to phase-sensitive amplification (see [16,18,19]). Also we can improve the detuned parameters, initial conditions, and level energies.…”