Optical parametric amplifier

Remarks

Assume perfect phase matching, an undepleted pump, and no idler at the input. Then the signal amplitude grows as $\cosh(g z)$ along the medium, where the gain $g$ is proportional to $\chi^{(2)}$ and to the pump amplitude. The idler amplitude grows as $\sinh(g z)$.

Example: a pump at $\omega_p$ and a signal at $\omega_s = 0.6\,\omega_p$ in a phase-matched crystal give an idler at $\omega_i = 0.4\,\omega_p$, and both the signal and the idler grow.

Non-example: second-harmonic generation in the same crystal is not parametric amplification. There energy flows from the low frequency $\omega$ to $2\omega$, and no weak input wave is amplified.

An optical parametric amplifier placed in a resonant cavity is an optical parametric oscillator.