Approximate solution to nonlinear pulse propagation in optical fibers

Soliton pulse propagation in optical fibers is now currently exploited. Since the original proposal by Hasegawa and Tappert1 of using solitons in optical communication, many other applications have been proposed including the control of a laser by a soliton formed in an optical fiber.2 Generally, in applications, the initial pulse parameters are not exactly equal to those required to generate pure soliton solutions. In such a case, the nonlinear Schrodinger equation must be solved numerically. However, to extract the salient features of the solution, it is of interest to search for a simple approximate solution. Such a solution is presented here. It consists in a hyperbolic secant with a variable width and linearly chirped phase. In a first-order approximation, the nonlinear Schrodinger equation is reduced to a nonlinear differential equation for the width, which can be integrated easily. This also yields simple relations for the linear chirp coefficient and peak amplitude. These results are then compared with the numerical solutions obtained by the beam propagation method. A good agreement is observed for the amplitude and phase factor up to energy of the order of the energy of an N = 2 soliton solution. The results obtained here by the approximate solution are also similar to those obtained by Anderson3 using a variational approach based on Gaussian trial functions. Our simple model could be useful for some application (for example, modeling the soliton laser), and it provides a better understanding of the interplay between dispersion and nonlinear effects.