Propagation of Gaussian wavelet pulses

We have developed a new formalism for the study of the time evolution of ultrashort pulses which experience gain-guiding and self-focusing. In contrast to alternative procedures describing spatially non-dispersive beam propagation, such as Hillion´s focused wave modes, this new method can be readily applied to describe transmission through stationary or moving linear optical elements. Further, our method describes the time dependence within the framework of the classical Gaussian ABCD matrices, which makes generalisation to 4 x 4 ´ray-pulse´ matrices unnecessary. With this formalism, we have treated Kerr-lensing to be equivalent to the effect of a moving lens in a straightforward manner. Additionally, we can treat time-dependent pulses with this formalism and hence we can determine the influence of the input pulse shape on these nonlinear effects. This method can be extended to an investigation of the initiation and evolution of self-modelocking. In order to deduce a traveling Gaussian beam solution, we use Maxwell´s wave equations in conjunction with the paraxial approximation