Abstract

The simultaneous effects of self-phase modulation, self-focusing, and diffraction on the propagation of ultrafast pulses in homogeneous and graded-index media are described. Approximate analytical solutions for the Gaussian diameter, temporal pulse shape, and pulse phase are derived. Numerical calculations of the supercontinuum generated by picosecond and femtosecond pulses are performed. The pulse phase, shape, and frequency modulation are shown to differ strongly from the plane-wave theory of self-phase modulation. At low intensity, the phase of a diffracting beam is given by p(z) = w₀t−kz + n(z), where n(z) = arotan (z/z₀) is a corrective longitudinal phase term due to diffraction. Our work shows that for a finite beam size or graded-index media, both the nonlinearity and diffraction contribute to this phase. We show that the sign of the time-dependent phase changes as a function of pulse energy. In a certain energy range red-shifted and blue-shifted frequencies are generated at the trailing edges of the pulse, respectively. Therefore, the pulse compression of ultrafast pulses may be obtained simultaneously with spectral broadening in the nonlinear medium without the use of grating pairs or prisms.

© 1988 Optical Society of America