Expressions for the quasi-phase-matching (QPM) three-wave mixing (TWM) with arbitrary grating structure and phase shift are obtained in this paper, for the first time, under the small-signal approximation. The expressions can be extensively applied to the all-optical signal processing for TWM, in which the signal and pump bandwidth of the wavelength conversion in DFM and the all-optical gate (AOG) bandwidth in SFM are all optimized. The optimal results from our expressions are compared with the results from the coupled-mode equations of QPM-TWM. Compared with loss free, the propagation loss in waveguides can decrease the conversion efficiency, but only a little change for the bandwidth.
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