Numerical studies of the interplay between self-phase modulation and dispersion for intense plane-wave laser pulses

Abstract

A computer algorithm is presented which allows simultaneous consideration of self‐phase modulation and dispersion for predicting temporal shape changes during the propagation of plane‐wave intense light pulses. The algorithm entails considering propagation alternately in regions where only one of the two above effects is operative. It is shown for clear materials that the parameters characterizing propagation are the nonlinear index change, the wavelength λ, the relaxation time of the nonlinearity, and the disperison parameter λ³(d²n/dλ²). The thickness of material over which a pulse will significantly reshape is found to be √λ times the geometric length of the pulse divided by the square root of the product of the dispersion parameter and the maximum nonlinear index. It is demonstrated that dispersion significantly modifies the self‐steepening concept of DeMartini, Townes, Gustafson, and Kelley. Numerical simulations of propagation in CS₂ indicate that, after sufficient travel, a shock can form on the leading edge of a mathematically smooth incident pulse. This is because of the retarded response of the nonlinearity; it is found that pulse features tend to evolve which are shorter than the relaxation time of the nonlinearity. With further propagation, the entire pulse develops violent amplitude features. For a 22‐GW/cm² 5‐ps 1.06‐μm pulse in CS₂, this shock appears after a propagation distance of 15 cm. Impressed amplitude noise on the pulse is shown to intensify this instability; under the same conditions, 10% peak‐to‐peak impressed ripple triples in approximately 3 cm. Because of the recently published experiments of Ippen, Shank, and Gustafson, propagation of 5‐ps mode‐locked dye laser pulses in CS₂‐filled fibers is also considered. In this case, shocks appear at 2‐m propagation if the input peak intensity is 50 MW/cm². Furthermore, attenuation present in the fibers tends to stabilize the shape of the shock as it forms. It is shown (for pulses which have not shocked) that dispersion can also modify the Fisher‐Kelley‐Gustafson pulse compression scheme. For the case of CS₂, it is found for optimal compression that more dispersive delay is required in a compressor than would have been needed in the absence of CS₂ dispersion. When the product of the peak intensity and the propagation length is 220 GW/cm, the proper compressor dispersive delay for optimum compression can be found by multiplying the optimally compressing dispersive delay in the absence of CS₂ dispersion by the factor 1+0.1l, where l is the propagation length in cm. The calculation concerning compression of pulses emanating from the CS₂‐filled fibers was also carried out. It is found that although some temporal compression can occur, the subsequent compressed pulses are not much shorter than the temporal shock which had formed on the pulse prior to compression. We conclude that propagation distances in such experiments should be kept below the shock distance. The simulation of pulse propagation in Nd : glass laser amplifier chains is also studied, taking nonlinearity and dispersion into account. The glass dispersion and the glass nonlinearity were considered with the linear properties of the resonance (disperison and frequency‐dependent gain). No provision was made to model a nonlinear response for the resonance. In the absence of gain, pulses temporally broaden and flatten because the glass dispersion is the wrong sign to compress the chirp which develops at the temporal center of the pulse. In pumped amplifiers a sharp temporal spike forms at the center because the chirp swings the pulse center frequency through the center frequency of the amplifying transition at that time. It is demonstrated that under typical operating conditions, pulses are relatively stable to amplitude modulation.