Space-time profiles of shaped ultrafast optical waveforms

A derivation of the space-time profiles of ultrafast optical waveforms shaped by filtering of spatially separated frequency components is presented. Closed form expressions for the space-time impulse response functions are given for the cases of single and double passes through a pulse shaping apparatus. For a single pass and a short unshaped pulse, diffraction by the mask filter gives rise to a translational spatial shift in the desired electric field profile that varies linearly with time along the shaped waveform. This result is completely general, and applies to frequency-domain pulse shaping with either continuous or discrete mask filters. It is also shown that double passing the apparatus does not generally reverse this effect but rather introduces further space-time coupling such as a time-varying spotsize. Examples of specific mask patterns are presented and implications for the generation of high-fidelity shaped optical waveforms are discussed.