= Phase matching in second-harmonic generation
{title2=$\Delta k=0$}
<Phase matching> makes <wave amplitudes> emitted at different depths add coherently. Consider collinear <plane waves> in a uniform transparent material of length $L$, with fixed <polarization of waves> and nonzero selected <nonlinear optical susceptibility>. Assume negligible pump depletion, negligible <absorption (electromagnetic radiation)>, an initially absent harmonic, and a wide beam for which <diffraction> is negligible. The pump's <wave amplitude> is then constant throughout the interaction.
The pump <electric field> has spatial <wave phase> $k_\omega z$; its quadratic source has spatial <wave phase> $2k_\omega z$. The freely propagating harmonic has <wave number> $k_{2\omega}$. Define the <second-harmonic phase mismatch> by
$$
\Delta k=k_{2\omega}-2k_\omega,
\qquad k_\omega=\frac{n_\omega\omega}{c},\qquad
k_{2\omega}=\frac{2n_{2\omega}\omega}{c}.
$$
For the chosen complex-field convention, the outgoing harmonic <wave amplitude>, up to an overall constant and propagation phase, is proportional to
$$
\mathcal A(L)=\int_0^L e^{i\Delta k z}\,dz
=L e^{i\Delta kL/2}\operatorname{sinc}\!\left(\frac{\Delta kL}{2}\right),
\qquad \operatorname{sinc}x=\begin{cases}\sin x/x,&x\ne0,\\1,&x=0.\end{cases}
$$
Choosing the conjugate complex-field convention reverses the exponent's sign without changing the <wave intensity>. Taking the squared <complex modulus> gives
$$
I_{2\omega}(L)=C I_\omega^2 L^2\operatorname{sinc}^2\!\left(\frac{\Delta kL}{2}\right),\qquad C>0.
$$
Here $C$ is fixed for the chosen modes and material; its dimensions accommodate the stated <wave intensity> convention. The quadratic pump dependence follows because the source <wave amplitude> is proportional to $E_0^2$, whereas $I_\omega$ is proportional to $E_0^2$. The <sinc function> comes from summing contributions along the whole material, just as a finite aperture produces <diffraction> through addition of <complex amplitudes>.
At $\Delta k=0$, the <integral> is $L$ and the harmonic <wave intensity> grows as $L^2$. At $\Delta kL=2\pi$, all those contributions cancel. This cancellation needs no <absorption (electromagnetic radiation)>: their <wave phases> wind through a full turn. At high conversion, the constant-pump assumption fails, so the $L^2$ law cannot continue indefinitely. Tight focusing, finite <optical pulses>, <absorption (electromagnetic radiation)>, and transverse separation of modes also require a more complete model.
<Optical dispersion> generally makes $n_{2\omega}\ne n_\omega$. In a suitable <crystal>, <birefringence> allows modes with different <polarization of waves> to experience different <refractive indices>, making their selected values equal. The appropriate crystal direction and allowed <nonlinear optical susceptibility> component must both work.<Nonlinear Optics and Spectroscopy by Nicolaas Bloembergen>[p. 19: momentum mismatch and birefringence compensation]{ref}
\Image[/second-harmonic-phase-matching-sinc-squared.png]
{title=Calculated <second-harmonic generation> versus <second-harmonic phase mismatch> and interaction length}
{description=Original uniform-medium, undepleted <plane wave> model with chosen $L_0=1\,\mathrm{mm}$. Both curves use the same pump and prefactor: $I_{2\omega}/(CI_\omega^2L_0^2)=(L/L_0)^2\operatorname{sinc}^2(\Delta kL/2)$. Doubling $L$ raises the matched peak fourfold and halves the mismatch to the first zero. The dimensionless horizontal coordinate is $\Delta kL_0/(2\pi)$; no curve is a fit to historical data.}
{source=second-harmonic-phase-matching-sinc-squared.py}
For a worked calculation, choose $L_0=1\,\mathrm{mm}$ and $\Delta k=\pi\,\mathrm{mm^{-1}}$. At $L=L_0$, the harmonic <wave intensity> is $4/\pi^2\simeq0.4053$ of its perfectly matched value. At $L=2L_0$ it vanishes in this model. More material can therefore reduce the output. If instead $\Delta k=0$, doubling $L$ multiplies output by four. The two curves retain that common normalization, so their different peak heights are meaningful.
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