Four-wave Mixing

Author: the photonics expert Dr. Rüdiger Paschotta (RP)

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 [3].

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 neighbouring 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.

Spontaneous Processes

If we have two pump fields at different optical frequencies, classically be would obtain optical amplification in some range of optical frequencies (determined by phase matching, see below), but would not generate any signal or idler without having any inputs for those. In reality, however, some spontaneous four wave mixing is possible, where signal and idler start with vacuum fluctuations (vacuum noise). Despite the quantum fluctuations being weak, this can lead to substantial signal and idler output powers if there is a high parametric gain of tens of decibels, for example.

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.

Phase-matching of four-wave mixing processes is often substantially influenced by nonlinear phase changes (→ Kerr effect) caused by the involved high optical intensities. This leads to phase-matching conditions which also involve optical intensities and not only wavenumbers.

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.

Simulation of Four-wave Mixing

Like other nonlinear effects, the consequences of four-wave mixing can well be simulated with numerical models. This is often vital, since it is not possible otherwise to obtain a solid quantitative and even qualitative understanding, as required for device or system optimization.

Different simulation techniques (types of simulation models) are suitable for different situations. Some examples:

• In the context of ultrashort pulses, chromatic dispersion and other nonlinearities can crucially influence the outcome. One needs to fully consider the complex amplitudes of pulses in the time or frequency domain.
• For optical fiber communications, particularly at high data transmission rates, similar techniques as for ultrashort pulses are required, possibly in combination with statistical methods, e.g. to evaluate the impact on the achievable bit error rate.
• For longer light pulses, a simpler treatment can be used, possibly only considering optical powers but not amplitudes, after evaluating the properties of phase matching.

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