定义：
光束在传播时由于介质的克尔效应导致的在其自身强度作用下的非线性相位调制

由于克尔效应，在介质中（如光纤）传播的高光强的光会对其自身产生一个非线性的相位延迟，该非线性的相位延迟在时域上有着与其自身光强形状一样的形状。这种效应可以通过折射率的非线性变化来表达： 

其中n_2是介质的非线性系数，I是光强。自相位调制是克尔效应在时域上由时域强度变化的导致的相位延迟，相应的在空间上克尔效应还会导致截向光强分部的自聚焦效应。 

目录

1. 对于脉冲光的影响
2. 半导体中由载流子密度变化导致的自相位调制
3. 锁模激光器中的自相位调制
4. 级联非线性中的自相位调制

对于脉冲光的影响
对于在介质中传播的脉冲光，由于随时间变化的光强变化，克尔效应会在脉冲上加上一个随时间变化的相位延迟。由于这个效应，一个初始无啁啾的脉冲会变成一个啁啾脉冲，也就是在时域上频率即时发生变化的脉冲。 

图 1 一个初始无啁啾的脉冲在经历了自相位调制后的即时频率。在脉冲再其中心部分表现为上啁啾 
一个光束半径为w的高斯光束在介质中传播了L的距离后，其单位光功率导致的相位变化可以有以下公式表述： 

在有些情况下也会省去距离L，来表征单位长度单位光功率导致的相位变化。 
由自相位调制引起的与时间相关的相位变化同时还会导致光谱的变形。在非线性系数为正的情况下，如果初始脉冲无啁啾或者上啁啾，自相位调制会导致光谱展宽，而如果初始脉冲为下啁啾则可能会发生光谱上的压缩。

对于自相位调制较强的情况，光谱还会有一个强烈的振荡（如图2所示）。这种振荡特性在本质上是由即时频率经历了一个很强的偏移导致的，所以一般会有两个时间上的量对于同一频率分量的傅里叶积分产生贡献。因此对于不同的频率，这两个频率可能会相加也可能会相消。 

Figure 2 初始脉宽为1ps的无啁啾脉冲在经历了强的自相位调制（最大的相位变化高达20rad）后的光谱。其频谱（或光谱）产生了强烈的振荡。 
对于具有反常色散的光纤，由自相位调制产生的啁啾会被色散作用补偿，该效应会导致光孤子的形成。表现为在无损耗的光纤中，即使有自相位调制效应的作用，基阶光孤子的谱宽会在传播过程中保持不变。 

半导体中由载流子密度变化导致的自相位调制
也有除克尔效应以外的其它效应也会导致随强度变化而变化的相位延迟效应，这些效应也被叫做自相位调制效应。在半导体激光器和半导体光放大器中，高信号强度会降低载流子的密度，从而引起折射率的变化，因此单位传播距离的相位也会发生变化。

与由克尔效应导致的自相位调制效应相比，这两者有一个重要的区别：这种由载流子变化导致的相位变化不会简单的随时域形状的变化，因为载流子浓度不会因时域强度的即时变化而即时地发生变化。这种效应只会在脉冲宽度小于载流子的弛豫时间时比较显著，而载流子的弛豫时间通常在皮秒到几纳秒的范围内。 

锁模激光器中的自相位调制
自相位调制对于锁模飞秒激光器有很重要的影响。虽然在长的激光谐振腔中，连空气的克尔效应也会对激光器产生影响，但是自相位调制对于激光器最主要的影响是来自于增益介质中的克尔非线性效应。如果没有色散的影响，非线性相移的影响会尤为严重，以至于不能维持激光器的稳定工作。孤子锁模对于锁模激光器而言是一个稳定解，因为在这种情况下自相位调制和色散会达到平衡，就与光纤中的光孤子相类似。 

级联非线性中的自相位调制
级联的χ^((2))非线性也会导致强的自相位调制。这也就意味着，非相位匹配的非线性相互作用导致的频率倍增（倍频），但同时也会导致反向的频率变换。实际上，在这种情况下，只有很少的功率会转换为其它波长，但是对于原有波长的相位变化是巨大的。

Acronym: SPM

Definition: nonlinear phase modulation of a beam, caused by its own intensity via the Kerr effect

More general term: nonlinear optical effects

Opposite term: cross-phase modulation

Due to the Kerr effect, high optical intensity in a medium (e.g. an optical fiber) causes a nonlinear phase delay of the propagating light. That phase delay which has the same temporal shape as the optical intensity. In simple models, the effect is described as a nonlinear change in the refractive index:

with the nonlinear index n₂ and the optical intensity I. In the context of self-phase modulation, the emphasis is on the temporal dependence of the phase shift, whereas the transverse dependence for some beam profile leads to the phenomenon of self-focusing.

Of course, the nonlinear phase delay comes in addition to the linear phase delay.

Note that the description of self-phase modulation with a time-dependent refractive index is somewhat simplified, and accurate only for not too short pulses. In more extreme situations, extended models are required, which also take into account the effect of self-steepening – a change of the temporal pulse shape even in the absence of chromatic dispersion. That may be interpreted as a nonlinear modification of the group velocity. There can also be higher-order nonlinear effects, leading to more complicated propagation phenomena.

Effects on Optical Pulses

Time-dependent Phase Shift

If an optical pulse is transmitted through a medium, the Kerr effect causes a time-dependent phase shift according to the time-dependent pulse intensity. In this way, an initial unchirped optical pulse acquires a so-called chirp, i.e., a temporally varying instantaneous frequency.

Figure 1: Instantaneous frequency of an initially unchirped pulse which has experienced self-phase modulation. The central part of the pulse exhibits an up-chirp.

For a Gaussian beam with beam radius w in a homogeneous medium with length L, the on-axis phase change per unit optical power is described by the proportionality constant

(In some cases, it may be more convenient to omit the factor L, obtaining the phase change per unit optical power and unit length.)

Note that 2 times smaller coefficients sometimes occur in the literature, which can have two different reasons:

• An incorrect (2 times too low) equation for the peak intensity of a Gaussian beam is sometimes used. With that, one underestimates the on-axis phase change.
• One may calculate the overall nonlinear phase shift of a Gaussian-shaped waveguide mode, where one gets a kind of averaging between higher- and lower-intensity parts. That reduces the nonlinear phase shift by a factor of 2.

Spectral Changes

The time-dependent phase change caused by SPM is associated with a modification of the optical spectrum:

• If the pulse is initially unchirped or up-chirped, SPM leads to spectral broadening (an increase in optical bandwidth).
• Spectral compression can result if the initial pulse is downchirped (always assuming a positive nonlinear index).

For strong SPM, the optical spectrum can exhibit strong oscillations (see Figure 2). The reason for the oscillatory character is essentially that the instantaneous frequency undergoes strong excursions, so that in general there are contributions from two different times to the Fourier integral for a given frequency component. Depending on the exact frequency, these contributions may constructively add up or cancel each other (see also Ref. [7]).

Figure 2: Spectrum of an initially unchirped 1-ps pulse which has experienced strong self-phase modulation (with a peak nonlinear phase shift of 20 rad), leading to characteristic oscillations of the power spectral density. The simulation has been done with the software RP Fiber Power.

Spatial Effects

For a propagating laser beam, the optical intensity is not only time-dependent, but also spatially dependent. The nonlinear phase changes are then also spatially dependent, and that leads to nonlinear self-focusing.

Self-phase Modulation in Optical Fibers

In optical fibers, SPM can be the dominant effect on an ultrashort pulse if the peak power is high (leading to strong SPM) while the chromatic dispersion is weak, so that the pulse duration remains approximately constant. Figure 3 shows an example case where that assumption is well fulfilled within the first 30 mm of fiber; here, the overall spectral width rises about linearly with the propagation distance. Thereafter, it grows faster because anomalous dispersion leads to pulse compression and thus to an increased peak power and an enhanced nonlinear interaction.

Figure 3: Propagation of an ultrashort pulse in a fiber, where self-phase modulation is the dominant effect on the pulse within the first 30 mm of propagation distance. The numerical simulation (which takes into account chromatic dispersion, SPM, self-steepening and stimulated Raman scattering) has been done with the RP Fiber Power software.

In optical fibers with anomalous chromatic dispersion, the chirp from self-phase modulation may be compensated by dispersion; this can lead to the formation of solitons. In the case of fundamental solitons in a lossless fiber, the spectral width of the pulses stays constant during propagation, despite the SPM effect.

In optical fibers with normal dispersion, a modulational instability can occur. That can also contribute to pulse break-up in supercontinuum generation.

Self-phase Modulation in Semiconductors via Carrier Density Changes

The term self-phase modulation is occasionally used outside the context of the Kerr effect, when other effects cause intensity-dependent phase changes. In particular, this is the case in semiconductor lasers and semiconductor optical amplifiers, where a high signal intensity can reduce the carrier densities, which in turn lead to a modification of the refractive index and thus the phase change per unit length during propagation. Comparing that effect with SPM via the Kerr effect, there is an important difference: such carrier-related phase changes do not simply follow the temporal intensity profile, because the carrier densities do not instantly adjust to modified intensities. That aspect is pronounced for pulse durations below the relaxation time of the carriers, which is typically in the range of picoseconds to a few nanoseconds.

Self-phase Modulation in Mode-locked Lasers

Self-phase modulation has important effects in mode-locked femtosecond lasers. It results mainly from the Kerr nonlinearity of the gain medium, although for very long laser resonators even the Kerr nonlinearity of air can be relevant [5]. Without chromatic dispersion, the nonlinear phase shifts can be so strong that stable operation is no longer possible: the laser would not reach a steady state with well-defined pulses. In that case, (quasi-)soliton mode locking [4] is a good solution, where a balance of self-phase modulation and dispersion is utilized, similar to the situation of solitons in fibers.

Self-phase Modulation via Cascaded Nonlinearities

Strong self-phase modulation can also arise from cascaded χ⁽²⁾ nonlinearities. Basically this means that a not phase-matched nonlinear interaction leads to frequency doubling, but with subsequent backconversion. In effect, there is little power conversion to other wavelengths, but the phase changes on the original wave can be substantial. This effect may also be used to compensate self-phase modulation from other origins [6].

Bibliography