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Four-wave Mixing

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Acronym: FWM

Definition: an interaction of light waves based on a χ(3) nonlinearity

German: Vierwellenmischung

Four-wave mixing is a nonlinear effect arising from a third-order optical nonlinearity, as is described with a χ(3) coefficient. It can occur if at least two different frequency components propagate together in a nonlinear medium such as an optical fiber. Assuming just two input frequency components ν1 and ν2 (with ν2 > ν1), a refractive index modulation at the difference frequency occurs, which creates two additional frequency components (Figure 1). In effect, two new frequency components are generated: ν3 = ν1 − (ν2 − ν1) = 2 ν1 − ν2 and ν4 = ν2 + (ν2 − ν1) = 2 ν2 − ν1. Furthermore, a pre-existing wave a the frequency ν3 or ν4 can be amplified, i.e., it experiences parametric amplification [3].

four-wave mixing

Figure 1: Generation of new frequency components via four-wave mixing.

In the explanation above, it was assumed that four different frequency components interact via four-wave mixing. This is called non-degenerate four-wave mixing. However, there is also the possibility of degenerate four-wave mixing, where two of the four frequencies coincide. For example, there can be a single pump wave providing amplification for a neighbored frequency component (a signal). For each photon added to the signal wave, two photons are taken away from the pump wave, and one is put into an idler wave with a frequency on the other side of the pump.

As four-wave mixing is a phase-sensitive process (i.e., the interaction depends on the relative phases of all beams), its effect can efficiently accumulate over longer distances e.g. in a fiber only if a phase-matching condition is satisfied. This is approximately the case if the frequencies involved are close to each other, or if the chromatic dispersion profile has a suitable shape. In other cases, where there is a strong phase mismatch, four-wave mixing is effectively suppressed. In bulk media, phase matching may also be achieved by using appropriate angles between the beams.

Four-wave mixing in fibers is related to self-phase modulation and cross-phase modulation: all these effects originate from the same (Kerr) nonlinearity and differ only in terms of degeneracy of the waves involved.

Four-wave mixing is relevant in a variety of different situations. Some examples are:


[1]R. L. Carman et al., “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification”, Phys. Rev. Lett. 17 (26), 1281 (1966)
[2]R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides”, IEEE J. Quantum Electron. 11 (3), 100 (1975)
[3]R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers”, IEEE J. Quantum Electron. 18 (7), 1062 (1982)
[4]D. Nodop et al., “Efficient high-power generation of visible and mid-infrared light by degenerate four-wave-mixing in a large-mode-area photonic-crystal fiber”, Opt. Lett. 34 (22), 3499 (2009)
[5]C. W. Thiel, “Four-wave mixing and its applications”, http://www.physics.montana.edu/students/thiel/docs/FWMixing.pdf

See also: nonlinearities, Kerr effect, phase matching, dispersion, supercontinuum generation, wavelength division multiplexing

Category: nonlinear optics

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