 Solution

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.