= 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 $d(z)=\sum_md_me^{imKz}$ with $K=2\pi/\Lambda$. The amplitude integral contains phases $e^{i(\Delta k+mK)z}$, so a Fourier component is useful when
$$
\boxed{\Delta k+mK\simeq0.}
$$
For 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.
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