Phase mismatch ($\Delta k$)

The residual wave vector difference in a wave-conversion process. For collinear sum-frequency generation, choose $\Delta k=k_1+k_2-k_3$. Contributions separated by $z$ then differ by wave phase $\Delta kz$. At phase matching this residual is zero. The sign convention changes some amplitude phases but not the unseeded conversion factor ${\mathrm{sinc}}^2(\Delta kL/2)$.

The distance over which a constant phase mismatch accumulates phase difference $\pi$. In unseeded, undepleted sum-frequency generation, the generated intensity first reaches a maximum at this length and returns to zero at $2{\ell }_c$. This length concerns relative phases of interacting waves, rather than the temporal optical coherence length of one source.