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# Four-wave Mixing

Acronym: FWM

Definition: an interaction of light waves based on a <$\chi^{(3)}$> nonlinearity

More general term: nonlinear effects

German: Vierwellenmischung

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Four-wave mixing is a nonlinear effect arising from a third-order optical nonlinearity, as is described with a <$\chi^{(3)}$> coefficient. It can occur if at least two different optical frequency components propagate together in a nonlinear medium such as an optical fiber.

Assuming just two copropagating input frequency components <$\nu_1$> and <$\nu_2$> (with <$\nu_2 \gt \nu_1$>), 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: <$\nu_3 = \nu_1 - (\nu_2 - \nu_1) = 2 \nu_1 - \nu_2$> and <$\nu_4 = \nu_2 + (\nu_2 - \nu_1) = 2 \nu_2 - \nu_1$>, as illustrated in Figure 1. (Alternatively, one could get the frequencies <$\nu_5 = 2 \nu_1 + \nu_2$> and <$\nu_6 = \nu_1 + 2 \nu_2$>, but that is less common because it can hardly be phase-matched e.g. in a fiber.) Furthermore, a pre-existing wave a the frequency <$\nu_3$> or <$\nu_4$> can be amplified, i.e., it experiences parametric amplification .

## Degenerate and Nondegenerate 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.

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. The modulational instability can also be interpreted as an effect of four-wave mixing.

## Phase Matching

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 (which is influenced by chromatic dispersion but also by nonlinear phase shifts).

Phase matching is approximately given if the frequencies involved are very 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.

## Relevance of Four-wave Mixing

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

• It can be involved in strong spectral broadening in fiber amplifiers e.g. for nanosecond pulses. For some applications, this effect is made very strong and then called supercontinuum generation. Various nonlinear effects are involved here, and four-wave mixing is particularly important in situations with long pump pulses.
• The parametric amplification by four-wave mixing can be utilized in fiber-based optical parametric amplifiers (OPAs) and oscillators (OPOs). Here, the frequencies <$\nu_1$> and <$\nu_2$> often coincide. In contrast to OPOs and OPAs based on a <$\chi^{(2)}$> nonlinear medium, such fiber-based devices have a pump frequency between that of signal and idler.
• Four-wave mixing can have important deleterious effects in optical fiber communications, particularly in the context of wavelength division multiplexing (WDM), where it can cause cross-talk between different wavelength channels, and/or an imbalance of channel powers. One way to suppress this is avoiding an equidistant channel spacing.
• On the other hand, four-wave mixing may be employed in a WDM telecom system for wavelength channel translation. Here, an input signal together with continuous-wave pump light at some other wavelength is injected into a piece of fiber (possibly a highly nonlinear fiber), which leads to the generation of a output signal at another optical frequency – the input optical frequency mirrored at the pump frequency.
• Four-wave mixing is applied for laser spectroscopy, most commonly in the form of coherent anti-Stokes Raman spectroscopy (CARS), where two input waves generate a detected signal with slightly higher optical frequency. With a variable time delay between the input beams, it is also possible to measure excited-state lifetimes and dephasing rates.
• Four-wave mixing can also be applied for phase conjugation, holographic imaging, and optical image processing.

### Bibliography

  R. L. Carman et al., “Observation of degenerate stimulated four-photon interaction and four-wave parametric amplification”, Phys. Rev. Lett. 17 (26), 1281 (1966); https://doi.org/10.1103/PhysRevLett.17.1281  R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides”, IEEE J. Quantum Electron. 11 (3), 100 (1975); https://doi.org/10.1109/JQE.1975.1068571  R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers”, IEEE J. Quantum Electron. 18 (7), 1062 (1982); https://doi.org/10.1109/JQE.1982.1071660  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); https://doi.org/10.1364/OL.34.003499  L. Drescher et al., “Extreme-ultraviolet spectral compression by four-wave mixing”, Nature Photonics 15, 263 (2021); https://doi.org/10.1038/s41566-020-00758-8

## Questions and Comments from Users

2020-05-03

Is it true that the phase mismatch HAS to be zero? I keep reading that it must be zero but, then also read that this is only needed for high efficiency.

Indeed, such nonlinear processes can in principle occur for any phase mismatch, but they are then often highly inefficient. How strong the difference is, depends on the circumstances – essentially, on the accumulated (integrated) phase mismatch (in units of radians) over the whole propagation length.

2020-06-19

Why do we consider exactly four frequency components? Why not more or fewer than four?

Certain optical nonlinearities – in particular, the Kerr nonlinearity – provide a mechanism for four frequency components to interact with each other. It cannot be more than four, since that would require a higher-order nonlinear process. It can be less, however, if we have frequency degeneracies.

2023-04-24

We are running a system with channels (50 GHz spacing). With FWM, what will happen: will it create a 97th channel? Will it create other channels out of the C band?

Yes, you will have influences at other wavelengths along that grid, particularly 50 GHz below the lowest and 50 GHz above the highest one – and channel cross-talk. But how severe that will be, depends on the circumstances.

2023-09-19

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