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Four-wave mixing (FWM), including phase-insensitive amplification (PIA, no idler input) and phase-sensitive
amplification (PSA, with idler input), has been exploited for a wide range of applications in all-optical signal processing,
e.g. in-line amplification, wavelength conversion, optical-phase conjugation (OPC), amplitude limiter, phase
regeneration, etc. In a lot of those applications, the input signal is pulse. In this paper, the FWM-based signal processing
with pulsed signal input is analytically analyzed. A set of exact analytical expressions for the output amplitude and phase
is derived, taking into account the pump depletion and the self- and cross-phase modulation (SPM and XPM) from
pumps, signal and idler. For PIA, analytical analysis shows that the generated idler pulse is far away from the phase
conjugation of the input signal, and that the output signal is not an amplified replica of the input signal. These
phenomena are due to both the phase distortion caused by SPM/XPM from signal/idler, and the temporal envelope
broadening resulting from pump depletion. Besides, in the frequency domain, the spectral inversion between signal/idler
will be destroyed. The induced phase distortions of idler pulses in dual pumps case are found to be much smaller than
those of single pump case. For PSA, neglecting the SPM/XPM from signal pulse, chirp-free output signal pulse will be
obtained. However, the SPM/XPM from signal will cause frequency chirping to the output. For a pulse-sequence,
through the SPM/XPM, amplitude-fluctuation will cause phase-jitter thus destroy the phase regeneration. We found that
in some PSA configurations, the frequency chirping and phase-jitter mentioned above will be suppressed. Similar to PIA
case, the output signal pulse of PSA may be broadened due to pump depletion. In both PIA and PSA applications, the
implications of the phase and shape distortions in all-optical signal processing by FWM are discussed.
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