定义：
锁模激光器的光谱的某些边带，由孤子脉冲在激光谐振器中循环时的周期性扰动造成

在一个孤子锁模激光器中，孤子脉冲会在激光谐振腔中循环。如果谐振腔中色散和非线性的影响较弱，由于色散和非线性的离散性，脉冲会经历一些微弱的周期性扰动。一些额外的干扰还可能是由于谐振腔内的周期性的损耗或放大。

 这种定期发生的干扰会使得孤子同一个色散波一起传播。这往往不会产生很大的影响，因为只有孤子会经历非线性，使得孤立和色散波的相对相位是不断变化的。然而，在光纤激光器中有时会发生有谐振耦合（或准相位匹配），使得在某些频率时色散波和光孤子之间的相位会在每次循环中变化2π的整数倍。这会在孤子光谱上形成一些窄峰（参见图1）。这样的窄峰被称为凯利边带。从它们的光谱中的位置，可以得到一些光纤的色散信息。 

图 1 孤子光纤激光器的输出的频谱，其具有明显的凯利边带。 
锁模固体激光器中通常不会发哼凯利边带，但在孤子光纤激光器中则经常能够看到。这是因为，光纤激光器的腔内非线性和色散（参阅锁模光纤激光器）要比固体激光器大很多。在一个孤子光纤激光器中的非线性相移与脉冲脉宽成反比。因此，通过增加脉冲能量而减小脉冲脉宽可以使得凯利边带更为明显。当激光的凯利边带十分显著时，孤子脉冲变得不稳定。强的凯利边带表明了该脉冲脉宽已经接近其最小值。

Definition: sidebands in the spectrum of certain mode-locked lasers, related to a periodic disturbance of soliton pulses in the laser resonator

In a soliton mode-locked laser, a quasi-soliton pulse is circulating in the laser resonator. If the effects of chromatic dispersion and nonlinearities per resonator round trip are weak, the pulse essentially experiences their average values plus some periodic disturbance due to the discrete nature of dispersion and nonlinearity. Additional disturbances can result from the periodically occurring losses and amplification in the laser resonator.

The mentioned periodically occurring disturbances couple the soliton to a copropagating dispersive wave. This often does not have strong effects, since the relative phase of soliton and dispersive wave is constantly changing due to the nonlinearity which is experienced only by the soliton. However, particularly in fiber lasers a kind of resonant coupling (or quasi-phase matching) can occur, when for some optical frequencies the relative phase of soliton and dispersive wave changes by an integer multiple of 2π per resonator round trip. This results in the formation of narrow peaks superimposed on the soliton spectrum (see Figure 1). Such peaks are called Kelly sidebands. From their positions in the optical spectrum, information on the spectral profile of the chromatic dispersion of the fiber can be retrieved.

Figure 1: Optical spectrum of the output of a soliton fiber laser, exhibiting Kelly sidebands.

Kelly sidebands are usually not observed in mode-locked bulk lasers, but often in soliton fiber lasers. This is because the intracavity nonlinearity and dispersion is much larger in fiber lasers (→ mode-locked fiber lasers). The nonlinear phase shift per resonator round trip in a soliton fiber laser is inversely proportional to the pulse duration. Therefore, Kelly sidebands become much more pronounced as the pulse duration is decreased e.g. by increasing the pulse energy. When the laser is driven too far into this regime, the soliton pulses become unstable. Strong Kelly sidebands thus indicate that the pulse duration is near the minimum possible value.

For mitigating problems with excessive Kelly sidebands, one should try to reduce the nonlinear phase shift per resonator round-trip, so that the disturbance of the circulating soliton is reduced. For example, one may operate the laser with a longer pulse duration by increasing the amount of anomalous chromatic dispersion, or reduce the pulse energy by pumping the laser less strongly.

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