定义：
基于X(3)非线性效应的光的相互作用

四波混频是一种基于三阶光学非线性（由X(3)系数描述）的非线性效应。当有至少两个不同频率分量的光一同在非线性介质（如光纤）中传播时就有可能发生四波混频效应。假设输入光中有两个频率分量v1和v2（v2>v1），由于差频的折射率调制的存在，会产生两个新的频率分量（如图1所示）：v3=v1-(v2-v1)=2v1-v2和v4=v2+(v2-v1)=2v2-v1。此时如果原先就存在v3或v4分量，则表现为v3或v4被放大，即这个两个频率分量经历了参量放大。 

图 1 四波混频产生新频率分量示意图
当四波混频作用涉及四个不同的频率分量时，其为非简并的四波混频。当然还存在简并的四波混频，即四波混频中的两个频率重合。例如，可以利用一个单频的泵浦作为一个临近波长的信号光的放大源，在这个四波混频过程中，每有一个光子被增加到信号光中（即实现放大）时，都会使用两个泵浦波长的光子，另外还在泵浦光波长的另一侧产生一个闲散波的光子。 

四波混频过程是对相位非常敏感的（即四波混频作用依赖于涉及到的所有光的相对相位）。当激光在光纤等介质中满足相位匹配的条件时，四波混频作用会随着传播距离的增加而有效的增强。相位匹配的条件意味着四波混频中的各个分量的频率很接近或者介质有一个合适的色散曲线。当相位严重不匹配时，四波混频作用会被大大地抑制。在固体介质中，还可以通过调节不同光束之间的方向和角度来实现相位匹配。

 光纤中的四波混频作用是与自相位调制和交叉相位调制密切相关的，这些效应都是由同一个非线性效应（克尔效应）造成的，只不过每个效应中的光的频率的简并状态不同。 

四波混频在不同的情况下有不同的作用，例如： 

1. 其会参与到一些光纤放大器（如：纳秒脉宽量级的光纤放大器）中的光谱展宽。在某些应用中，这种光谱展宽作用是很强的，甚至会产生超连续谱。超连续谱产生的过程涉及到很多种不同的非线性作用，而四波混频会在长脉冲的超连续谱产生中起到很大的作用； 
2. 基于四波混频的参量放大可以用于光纤的光学参量放大器（OPA）和光学参量振荡器（OPO）中。在这种情况下ν_1和ν_2是重叠的。相比于基于χ^((2))非线性效应的光学参量放大器和光学参量振荡器，这种基于四波混频效应的设备中的泵浦光的频率是在信号光和闲散光之间的； 
3. 四波混频作用在光纤通信中通常是有害的，特别在波分复用技术中，四波混频会造成不同信道之间的串扰和信道功率的不平衡。通过使用非等间隔的信道可以抑制该效应； 
4. 在光谱学中，四波混频通常出现在相干反斯托克斯拉曼光谱学中。在这种方法中，两个输入光波会产生一个频率略高于输入光的被探测信号。通过改变输入光束的时间延迟，还可以测量激发态寿命和退相率。 
5. 四波混频还被应用于相位共轭、全息成像和光学图像处理等技术中。

Acronym: FWM

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

More general term: nonlinear effects

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 optical frequency components propagate together in a nonlinear medium such as an optical fiber.

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

Assuming just two copropagating 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, as illustrated in Figure 1. (Alternatively, one could get the frequencies ν₅ = 2 ν₁ + ν₂ and ν₆ = ν₁ + 2 ν₂, 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 ν₃ or ν₄ 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 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 ν₁ and ν₂ often coincide. In contrast to OPOs and OPAs based on a χ⁽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