Solution for the fundamental mode of a step-index lightguide with Kerr nonlinearity

The progress in the development of femtosecond fiber lasers and their applications makes the problem of modeling the high power ultrashort pulses propagation in optical fiber cables extremely important. Modern fiber lasers can generate pulses with a duration of less than ten femtoseconds with a peak power of up to tens GW. At such values of peak power nonlinear effects cannot be ignored. For fused quartz fibers and femtosecond pulse duration it can be assumed that only the Kerr nonlinearity is present. For this assumption the approximate analytical solution for the fundamental mode of an optical fiber with step-index profile was already found. This solution was obtained with Gaussian approximation method and it already takes the Kerr nonlinearity into account. However, it neglects the second degree of approximation, which is unacceptable at peak power values exceeding tens of megawatts. In this paper, we consider a solution that takes into account refractive index dependence on high-order terms of light signal peak power which occurs due to the Kerr nonlinearity. For some optical fibers studied the results of the analysis are presented, showing the dependency of the fundamental mode spot radius and propagation constant on the light signal peak power in the presence of the Kerr nonlinearity. It is shown that, with an error of less than 0.1%, the dependence of the fundamental mode propagation constant on the light signal power can be approximated by a second-order polynomial. For the considered optic fiber samples the values of the approximation coefficients were obtained.