Three-wave mixing

An interaction among three waves mediated by a second-order nonlinear optical susceptibility. In sum-frequency generation, two input angular frequencies satisfy ${\omega }_3={\omega }_1+{\omega }_2$. The same coupling permits reverse conversion. Phase matching determines whether successive generated contributions reinforce; the Manley-Rowe relations express the associated lossless exchange constraints.

Conservation relations for resonant, lossless wave mixing. For sum-frequency generation with ${\omega }_3={\omega }_1+{\omega }_2$, the intensities obey $I_1^{\prime }/{\omega }_1=I_2^{\prime }/{\omega }_2=-I_3^{\prime }/{\omega }_3$. Thus $I_1/{\omega }_1+I_3/{\omega }_3$ and $I_2/{\omega }_2+I_3/{\omega }_3$ are constant. Their difference gives a third, dependent invariant. Adding the intensity derivatives yields conservation of energy.

The exchange interpretation divides each intensity by its photon energy $\hslash {\omega }_j$. One conversion removes a photon from each lower-frequency mode and adds one to the upper mode. For classical coupled-wave equations, differentiating the squared complex amplitudes gives the same common exchange rate multiplied by $({\omega }_1,{\omega }_2,-{\omega }_3)$. Loss or externally replenished pumps require corresponding source terms.

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]

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.

Difference-frequency generation produces a wave at the positive difference ${\omega }_i={\omega }_p-{\omega }_s$ of two incident angular frequencies, through a second-order nonlinear optical susceptibility. The product of two oscillating electric fields contains both sum and difference frequencies, and the corresponding polarization density drives the generated wave.

For plane waves in a uniform medium, coherent buildup is favored by ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$, the relevant phase matching condition. Two shorter-wavelength inputs can thereby generate longer-wavelength infrared radiation. Collecting the output after removing both incident frequencies distinguishes frequency generation from scattered pump light.^[ref]

A nonlinear optics process producing a field at the sum of two input angular frequencies. The driven electric polarization contains a term proportional to the product of the input complex amplitudes. With undepleted plane-wave inputs, integration along a crystal gives an output amplitude proportional to $L{e}^{i\Delta kL/2}\mathrm{sinc}(\Delta kL/2)$; the sinc function therefore determines the ideal length and phase mismatch dependence. A nonzero initial output field also interferes with this generated field.