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An optical parametric amplifier transfers energy from a pump beam into a signal and an idler through nonlinear optics. For the common three-wave mixing process, their angular frequencies obey ${\omega }_p={\omega }_s+{\omega }_i$. A strong pump and a weaker signal enter a nonlinear medium; the signal grows and an idler is generated.

Efficient growth requires phase matching so that contributions generated along the medium reinforce. Changing crystal orientation or temperature can tune the matched frequencies. Measuring the signal with and without the pump tests amplification, while observing the corresponding idler checks the frequency-conversion mechanism. The pump supplies the extra energy; the process does not require a stored population inversion.^[ref]

Optical parametric chirped pulse amplification combines the stretch–amplify–compress sequence of chirped pulse amplification with an optical parametric amplifier. A stretched signal pulse overlaps a strong pump pulse in a nonlinear crystal. Three-wave mixing transfers pump energy into the signal and an idler; subsequent pulse compression brings the amplified signal's frequencies back together. The pump supplies the added energy, rather than a stored population inversion.^[ref]

A. Dubietis, G. Jonušauskas and A. Piskarskas reported chirped and stretched pulse parametric amplification in beta barium borate in 1992. Their paper establishes a concrete early implementation of the parametric route; its published title identifies the nonlinear crystal.^[ref]

For pump, signal and idler angular frequencies, the conversion obeysEfficient amplification also requires phase matching, so contributions generated along the crystal reinforce. For collinear propagation the corresponding wave-vector relation is $k_p=k_s+k_i$; a noncollinear geometry requires the vector relation. Crystal dispersion and the chosen geometry determine which signal bandwidth is efficiently amplified.

An original energy-accounting example uses a 515-nanometre pump and an 800-nanometre signal. Their vacuum wavelengths giveSuppose 40 percent of a 2-millijoule pump is converted into signal–idler pairs, with negligible seed energy and other losses in this chosen model. Each conversion divides photon energy according to frequency, so the added signal energy isThe idler receives the remaining 0.285 millijoule. Assigning all depleted pump energy to the signal would violate this accounting.

If a compressor transmits 80 percent of that signal energy into a 50-femtosecond Gaussian pulse, the surviving energy is 0.412 millijoule. The chirped pulse amplification energy budget givesThese are stipulated teaching values, not a reconstruction of the 1992 experiment. Stretching lowers peak power during amplification; it does not remove the need to manage pump intensity or optical damage.

A useful apparatus check varies pump–signal delay while recording signal and idler spectra and energies. This tests temporal overlap and the frequency-conversion relation. Measuring the compressed pulse separately checks whether amplified bandwidth and residual spectral phase support the claimed pulse duration. Gain alone does not demonstrate successful compression, and an energy measurement alone does not establish peak power.

An optical parametric oscillator places an optical parametric amplifier inside an optical resonator that feeds back at least one generated wave. Pump energy is converted into signal and idler waves satisfying the frequency relation of the underlying three-wave mixing process.

Oscillation begins when round-trip gain overcomes loss at an allowed resonator mode. Below that threshold, weak generated emission can still occur, but sustained oscillation is absent. Tuning the phase matching and resonator together gives a tunable source at wavelengths difficult to obtain directly from a particular laser. Pump power alone does not select the output: cavity resonances, conversion gain and optical losses also matter.

The process is sum-frequency generation, with ${\omega }_3={\omega }_1+{\omega }_2$. The same reversible three-wave mixing interaction permits the reverse conversion when the higher-frequency field supplies the pump. The second-order nonlinear polarization couples the three waves; its effective coefficient includes the crystal orientation and polarizations.

One efficiency limit is phase mismatch: contributions generated at different positions acquire different phases and can cancel. A second practical limit is spatial walk-off, which reduces the overlap of finite beams in a birefringent crystal. A beam radius $w$ and relative walk-off angle $\rho$ give an overlap length of order $w/|\tan \rho |$. Absorption and scattering provide further losses. Group-velocity mismatch matters for pulses, and pump depletion eventually invalidates an undepleted-pump calculation; none of these effects should be silently included in the ideal lossless plane-wave equations.

Define ${\kappa }_j=2d_{\mathrm{eff}}{\omega }_j/(n_{j}c)$ and hold $A_1,A_2$ constant. Direct integration of the driven equation giveswhere $\mathrm{sinc}x=\sin x/x$. The second expression has a continuous phase-matched limit. If the generated field is initially absent,For $\Delta k\ne 0$ this is proportional to ${\sin }^2(\Delta kz/2)$: the first maximum is at the nonlinear optical coherence length ${\ell }_c=\pi /|\Delta k|$, and the first subsequent zero is at $2{\ell }_c$. A seed $A_3(0)\ne 0$ adds interference with the generated amplitude and can change that pattern. At exact phase matching, the unseeded result grows as $z^2$ only while pump depletion remains negligible.

To retain depletion consistently, differentiate each intensity before inserting the coupled equations:For real $d_{\mathrm{eff}}$ in the lossless convention, put $Q=A_1{A}_2{A}_3^{\ast }{e}^{i\Delta kz}$. The result isThe signs follow the printed amplitude equations. In initial up-conversion, the newly generated $A_3$ makes $\mathrm{Im}Q<0$, so the two input intensities decrease while the sum-frequency intensity increases. Adding the derivatives gives zero because ${\omega }_1+{\omega }_2-{\omega }_3=0$. The total intensity is conserved, even though the three individual intensities change.

The Manley-Rowe relations areDividing an intensity by $\hslash {\omega }_j$ gives photon flux density. A conversion event removes one photon from each lower-frequency mode and adds one to the upper mode, or does the reverse. Therefore $C_1/\hslash$ and $C_2/\hslash$ count conserved sums of photon fluxes, while $C_3/\hslash$ counts the conserved lower-frequency difference. They are not three independent invariants: $C_3=C_2-C_1$. Energy conservation is also recovered as $I={\omega }_1{C}_2+{\omega }_2{C}_1$.

The unweighted combinations generally fail because the exchanged photons have different energies. Explicitly,The qualification “generally” matters: $D_3$ is conserved for equal lower frequencies in a distinct-mode degenerate case, and every derivative vanishes when there is no exchange. The equations establish nonconservation for generic unequal-frequency interaction, not for all special solutions.