Quasi-phase-matching ($\Delta k+mK=0$)

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.

Periodically poled lithium niobate is lithium niobate whose ferroelectric domain orientation is reversed periodically along an optical interaction path. Reversal changes the sign of the relevant second-order nonlinear optical susceptibility, supplying the spatial modulation needed for quasi-phase-matching.

For a uniform collinear conversion mismatch $\Delta k$, a first-order poling period $\Lambda$ can compensate it through $\Delta k-2\pi /\Lambda \simeq 0$, with signs chosen consistently. The generated contributions then reinforce over many domains instead of repeatedly canceling. The poling period, temperature and input wavelengths determine the useful conversion band; domain defects and finite length affect efficiency. ^[ref]