= Second-harmonic generation
{wiki}
= Frequency doubling
{synonym}
= SHG
{c}
{synonym}
<Second-harmonic generation> converts an incident optical <frequency> $\nu$ into $2\nu$ through a material's nonlinear response. In the <photon> description of an ideal lossless conversion, two input <photons> supply one output <photon>:
$$
2h\nu=h(2\nu),\qquad \omega=2\pi\nu.
$$
Here $h$ is the <Planck constant>. The relation states <conservation of energy>; the material mediates the interaction. It does not say that each input <photon> independently splits its <wavelength> in half.
For <vacuum> <wavelengths>, $\lambda_{0,\omega}=c/\nu$ and $\lambda_{0,2\omega}=c/(2\nu)$, where $c$ is the <speed of light>. Hence
$$
\lambda_{0,2\omega}=\frac{\lambda_{0,\omega}}{2},\qquad
\frac{\lambda_{\mathrm{med},2\omega}}{\lambda_{\mathrm{med},\omega}}=\frac{n_\omega}{2n_{2\omega}}.
$$
The second relation uses the <refractive index> for each propagating mode and <frequency>. Within a medium with <optical dispersion>, the <wavelength> need not halve. For an invented example, $\lambda_{0,\omega}=800\,\mathrm{nm}$, $n_\omega=1.5$, and $n_{2\omega}=1.6$ give $400\,\mathrm{nm}$ in <vacuum>, but in the material the two <wavelengths> are $533.33\,\mathrm{nm}$ and $250\,\mathrm{nm}$; their ratio is $0.46875$.
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