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Quasi-phase-matching compensates phase mismatch by deliberately modulating the nonlinear coefficient, rather than demanding that the material wavevectors themselves exactly match. In a periodically poled ferroelectric crystal, reversing domains changes the sign of the relevant second-order coefficient. Flipping it when the nonlinear interaction would begin to undo previous conversion restores constructive accumulation over many domains.

Write $d(z)={\sum }_m{d}_m{e}^{imKz}$ with $K=2\pi /\Lambda$. The amplitude integral contains phases $e^{i(\Delta k+mK)z}$, so a Fourier component is useful whenFor first-order compensation, $\Lambda =2\pi /|\Delta k|$ and each domain has width $\Lambda /2={\ell }_c$. This is twice the single-domain coherence length for the complete sign pattern, not a full period equal to ${\ell }_c$.

For ideal equal-width domains switching between $+d$ and $-d$, the resonant first Fourier coefficient has magnitude $2d/\pi$. The effective amplitude coefficient is therefore reduced by $2/\pi$, and conversion intensity by $4/{\pi }^2$, relative to a perfectly phase-matched uniform crystal using that same $d$. Nevertheless, quasi-phase matching can permit a larger tensor coefficient or wavelengths and polarizations inaccessible to ordinary birefringent phase matching, more than compensating that idealized reduction.

Periodically poled lithium niobate is a practical example. Such gratings support second-harmonic and sum-frequency generation, difference-frequency conversion, optical parametric oscillation and pair generation. The period and temperature select the operating wavelengths; finite length gives a finite mismatch acceptance. Domain duty-cycle errors, nonuniform temperature, absorption and finite spatial or temporal overlap limit the result. Chirped or apodized patterns can reshape bandwidth, but do not make these physical limits disappear.

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]