Approximate Treatment of the Nonlinear Waveguide Equation in the Regime of Nonlinear Self-Focus

The nonlinear waveguide equation is studied quasi-analytically for insights into the effect of waveguide designs. Equations governing stationary mode and transient process are successfully derived. A number of new understandings of the nonlinear process at high peak power in optical waveguides are developed which will be critical for fiber laser research and developments at extremely high peak powers. The first key finding is that the critical power for nonlinear self-focus is independent of waveguide parameters. The second key finding is that the nonlinear guided stationary mode is a stable solution of a nonlinear waveguide below the critical power for nonlinear self-focus and has a reduced mode size dependent on the optical power and V value of the waveguide. The third key finding is that the transient process scale with Rayleigh range similar to those in bulk media and are adiabatic, i.e., the optical power remains in the fundamental mode. The fourth key finding is that a larger V value increases the transient process and self-focus distance, and the self-focus distance is proportional to square root of the optical power and independent of V value at higher power levels, i.e., it becomes more like a bulk medium at peak powers above ten times critical power levels. B integrals are calculated for various amplifiers, taking into account the impact of the gradual collapse of beam size along the amplifier.