User:F G Sedgwick/fwm draft

Four-wave mixing (FWM) is the coherent interaction between four optical fields via the third order nonlinear susceptibility of a medium. The various nonlinear optical effects caused by four-wave mixing are parametric processes which can be explained mathematically by frequency mixing. If any or all of the fields have equal frequency and/or wavenumber, the interaction may be termed degenerate four-wave mixing (DFWM). Some DFWM processes are rarely refered to as such, with specific terms for these processes preferred, e.g. third-order harmonic generation, self phase modulation, etc. FWM is an important effect because the third order nonlinear susceptibility is allowed to be nonzero for all materials, regardless of symmetry. This is in contrast to the second order nonlinear susceptibility, which can only be nonzero for those materials which lack inversion symmetry.

Physics
Four-wave mixing can be compared to the intermodulation distortion in electrical systems. The time dependence of the four optical fields is always sinusoidal, the spatial dependence will be taken here to be plane waves for simplicity.


 * $$ P = \chi^{(1)} \cdot E + \chi^{(2)} \cdot EE + \chi^{(3)} \cdot EEE + ...$$

When three wavelengths (&#955;1, &#955;2, and &#955;3) interact in a nonlinear medium, they give rise to a fourth wavelength (&#955;4) which is formed by the scattering of the incident photons, producing the fourth photon. It is easier to illustrate this if we use frequency instead of wavelength.

Optical Communications
Four-wave mixing is an intermodulation distortion in optical systems, similar to the third order intercept in electrical systems.

Given inputs f1, f2, and f3, the nonlinear system will produce


 * $$ f_{1} \pm f_{2} \pm f_{3}$$

with the most damaging signals to system performance calculated as


 * $$ f_{ijk} = f_{i} + f_{j} - f_{k}, \mathrm{where}\, i \neq j \neq k $$

since these frequencies will lie close to one of the incoming frequencies.

Doing the math with the three input signals, you will find that 12 interfering frequencies are produced, 3 of which lie on one of original incoming frequencies.

Degenerate four-wave mixing
FWM is also present if only three components interact. In this case the term


 * f0=f1+f1-f2

couples three components, thus generating the so-called Degenerate Four-Wave Mixing, showing identical properties as in case of four interacting waves.

Four-Wave Mixing (FWM) is a fiber-optic characteristic that affects Wavelength Division Multiplexing (WDM) systems, where multiple optical wavelengths are spaced at equal intervals or channel spacing. The effects of FWM are pronounced with decreased channel spacing of wavelengths and at high signal power levels. High chromatic dispersion decreases FWM effects, as the signals lose coherence (physics). The interference FWM causes in WDM systems is known as interchannel crosstalk. FWM can be mitigated by using uneven channel spacing or fiber that increases dispersion.

Vier-Wellen-Mischung