= Quasi-phase-matching
{title2=$\Delta k+mK=0$}
{wiki}
Compensation of <phase mismatch> by a spatial modulation of the coupling coefficient. Write $d(z)=\sum_m d_m e^{imKz}$. The driven amplitude contains $e^{i(\Delta k+mK)z}$, so the selected Fourier component accumulates coherently when $\Delta k+mK=0$. For first-order compensation the full spatial period is $2\pi/|\Delta k|$, twice the <nonlinear optical coherence length>.
An ideal equal-duty pattern alternating $+d$ and $-d$ has first Fourier coefficient of magnitude $2d/\pi$. Hence its conversion intensity is $4/\pi^2$ of a uniformly phase-matched crystal using that same coefficient, under the same undepleted plane-wave assumptions. The modulation can nevertheless allow a larger tensor coefficient or an otherwise inaccessible wavelength. <Periodically poled lithium niobate> implements such sign reversals using ferroelectric domains.
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