定义：脉冲时域上的光强形状为抛物线形的脉冲

抛物线型脉冲的光脉冲（通常的超短脉冲）在时域上其光强不仅仅在邻近脉冲中心的地方具有抛物线的形状，而且在脉冲边缘光强接近于零的地方也符合抛物线的形状。在理想的情况下，该脉冲的光强分布如下所示： 

图1给出了一个近似抛物线型的脉冲，这种脉冲可以通过一定情况下的光纤放大器产生。抛物线型脉冲具有一些特殊的性能，这些性能对于在光纤放大器的产生高能量的超短脉冲是尤其重要的。 

图1：近抛物线型脉冲的时域分布，其可以通过在光纤放大器中放大一个高斯型脉冲产生。 
上啁啾的抛物线型脉冲可以利用光纤放大器产生，脉冲在其中不仅仅经历光学增益，同时还受到克尔非线性和正常色散的作用。在这种情况下，具有啁啾的抛物线型脉冲是长传播长度长度下的渐近解。在具有足够大的放大系数的情况下，经过足够长的光纤传播后，将所得的脉冲的形状将不再依赖于种子脉冲的形状或脉宽，而峰值功率和脉宽在经过很擦汗那个的传播长度之后只依赖于种子脉冲的能量。这就是自相似的脉冲传播情况，在这种情况下，虽然脉冲的脉宽发生了变化，但抛物线型脉冲的形状并不会发生改变。因此，这种脉冲有时被称为自相似子（similaritons）。脉冲脉宽为： 

其中g是指数下的增益系数，其全带宽（不是半高全宽）为 

两者均随着脉冲能量Ep的立方根增长，使得脉冲的时间带宽积随着EP2/3增长。 
该脉冲具有一个接近线性的上啁啾，即一个线性上升的瞬时频率，而瞬时频率的增长速率是与脉冲能量无关的： 

随着传播长度的增长，峰值功率随EP2/3成比例的上升。啁啾率（以Hz/ ps为单位）接近与恒定值，仅由光纤色散和单位长度的增益决定的。谱宽则正比于脉冲持续时间，或者说随EP1/3成比例上升。 

图2：脉冲脉宽在一个具有无限增益带宽的放大器中的演化过程。蓝色和红色实线分别给出了脉冲持续时间在所述放大器和压缩器后的脉冲宽度。虚线则给出了渐近值，在这种情况下，放大倍率约为20dB。 
受输入脉冲参数的限制，特别是脉冲脉宽和能量的限制，抛物线演化过程只会在一定量的线性增长后发生。因此，即使在脉冲能量增加了几个数量级，脉冲脉宽和带宽也不一定上升几个数量级。如图2所示，其所述的种子脉冲能量为10个pJ，脉宽为500 fs。最初，所述的放大是几乎线性的，但随后带宽和脉宽都开始显著上升。由于带宽的增加，导致压缩后脉冲比初始脉冲更短。 
需要注意的是一个实际中的光纤放大器的有限增益带宽限制了抛物线型脉冲放大过程的脉冲能量。而图2中的参数，就已经远远超出了这一限制。 

抛物线型脉冲放大的亮点 
抛物线型脉冲的特性导致了当光纤放大器被设计为“抛物线型放大器时”时具有了对超短脉冲的放大的诸多吸引人的特性： 

1. 避免了脉冲分裂。放大所产生的脉冲在时域和频域上都不具有显著的旁瓣。 
2. 这种放大器中种子源并不需要精确设计的脉冲参数。只要该放大器的放大系数足够大，这些参数就都不重要。种子脉冲的脉宽可以比输出脉冲的脉宽宽很多。但是脉宽也不应该太长，否则作为渐近解的抛物线型脉冲需要很长的过程才能达到。 
3. 光纤参数（色散，非线性和增益）也不是非常严格的。 
4. 由于放大后的脉冲是具有近似线性啁啾的，它是比较容易就能通过脉冲压缩获得接近变换极限脉冲的。但是，压缩器的高阶色散不宜过多。 
5. 放大期间的啁啾是有益的因素，因为对变换极限脉冲的放大可能导致过度的非线性相移。这种情况类似与啁啾脉冲放大，但脉冲的展宽量较小。 

这种放大的限制来自以下作用： 

1. 有限的增益带宽限制了可能的脉冲能量。对于标准的模场面积的掺镱光纤，其能量只在数十nJ，远低于饱和能量。因此，抛物线型脉冲只能提取光纤所存储能量的一小部分。这一作用使得这样的放大器适合于具有非常高的重复频率，但脉冲能量并不十分大的情况。 
2. 原则上，被放大的脉冲会在较长的波长产生较高的拉曼增益，从而产生一个较强的斯托克斯脉冲，导致原来的脉冲被扭曲。但是，抛物线型脉冲放大器通常不会产生那么搞的拉曼增益。 
3. 光纤和压缩器的较高阶色散不应太强。 
4. 放大器噪声在某些情况下也会限制放大器的能量。 

最好通过数值模拟脉冲放大的传播过程来获得最佳的放大器设计。 
抛物线型脉冲放大的原理也适用于锁模光纤激光器（自相似光纤激光器）。在这种情况下，循环传播脉冲的光谱宽度是强烈振荡：放大期间它会增加，通过一些光滤波器来重置每个循环。这种方法使得有可能通过廉价的光纤激光器来获得较高的脉冲能量。

Definition: pulses with a parabolic intensity profile

More general term: light pulses

A parabolic pulse is a light pulse (usually an ultrashort pulse) that has a temporal intensity profile with a parabolic shape – not only near the pulse center, but also well towards the wings up to the point where the intensity goes to zero. In the ideal case, the pulse would have an intensity profile according to

Figure 1 shows an approximately parabolic pulse, as it can result from pulse propagation in a fiber amplifier under certain circumstances. Parabolic pulses have a number of remarkable properties, which have turned out to be particularly important in the context of fiber amplifiers for generating high-energy ultrashort pulses.

Figure 1: Temporal profile of a nearly parabolic pulse, such as can be generated in a fiber amplifier fed with Gaussian pulses. Such simulations can be made with the software RP Fiber Power.

Up-chirped parabolic pulses can be generated in a fiber amplifier where the amplified pulses not only experience optical gain, but also the Kerr nonlinearity and normal chromatic dispersion. In this situation, a chirped parabolic pulse is the asymptotic solution for long propagation lengths [3, 8]. After a sufficiently long propagation length, associated with a sufficiently large amplification factor, the shape of the resulting pulses no longer significantly depends on the shape or duration of the seed pulses, and the peak power and duration after a long propagation length depend only on the seed pulse energy. This is a case of self-similar pulse propagation, since the parabolic pulse shape, although not the pulse duration, is preserved. Therefore, such pulses are sometimes called similaritons. The pulse duration parameter is

where g is the exponential gain coefficient, and the full bandwidth (not the full width at half maximum) is

Both grow with the cubic root of the pulse energy E_p, so that the time–bandwidth product of the pulses grows in proportion to E_p^2/3.

In the asymptotic regime, the pulses have a very nearly linear up-chirp, i.e., a linearly rising instantaneous frequency, with a growth rate of the instantaneous frequency which is independent of the pulse energy:

With increasing propagation length, the peak power rises in proportion to E_p^2/3. The chirp rate (in THz/ps) approaches a constant value, which is determined only by the fiber dispersion and gain per unit length. The spectral bandwidth rises in proportion to the pulse duration, i.e., in proportion to E_p^1/3.

Figure 2: Evolution of the pulse duration in a hypothetical amplifier with infinite gain bandwidth. The solid curves show the pulse duration in the amplifier (blue) and after a compressor with optimized GDD (red). The dashed curve shows the asymptotic values, which in this case are reached after ≈ 20 dB of amplification.

Depending on the input pulse parameters, in particular on the pulse duration and energy, the parabolic evolution may set in only after some amount of linear amplification. Therefore, the pulse duration and bandwidth do not necessarily rise by orders of magnitude even if the pulse energy is increased by several orders of magnitude. This is shown in Figure 2, where the seed pulses had 10 pJ energy and 500 fs duration. Initially, the amplification is nearly linear, but then both the bandwidth and pulse duration begin to rise significantly. Due to the increased bandwidth, dispersive compression of the pulses after fiber propagation can lead to a pulse duration significantly below that of the initial pulses.

Note that the limited gain bandwidth of a real fiber amplifier limits the pulse energy range in which parabolic pulse amplification is possible. The parameters in Figure 2 go far beyond that limit, making the simulation somewhat unrealistic in that sense. Numerical simulations can of course take into account a finite gain bandwidth and can thus be used to test to which extent self-similar pulse propagation is practically possible.

Attractions of Parabolic Pulse Amplification

The remarkable properties of parabolic pulses lead to a number of technologically attractive features of fiber amplifiers for ultrashort pulses, when they are designed as “parabolic amplifiers”:

• Wave breaking is safely avoided. The pulses generated have no significant side lobes in the temporal or spectral shape.
• Such an amplifier does not require a seed source with carefully chosen pulse parameters, as these are not important provided that the amplification factor is large enough. The seed pulses may be significantly longer than the output pulses. They should not be too long, however, as otherwise the asymptotic solution is only slowly approached.
• The fiber parameters (chromatic dispersion, nonlinearity and gain) are also not critical. (This is remarkably different to schemes based on soliton effects.)
• As the amplified pulses have a nearly linear chirp, it is relatively easy to obtain nearly transform-limited pulses by dispersive pulse compression. However, higher-order chromatic dispersion of the compressor should not be excessive.
• The chirp during amplification can be an important advantage, because the amplification of transform-limited pulses could lead to excessive nonlinear phase shifts. The situation is therefore similar to that of chirped-pulse amplification, but with a lower amount of pulse stretching.

Limitations arise from the following effects:

• The limited gain bandwidth effectively limits the possible pulse energy. For ytterbium-doped fibers with standard mode areas, it is of the order of some tens of nJ – far below the saturation energy. Therefore, with a parabolic pulse one can only extract a minor part of the stored energy in the fiber. This regime is thus most suitable for amplifying pulse trains with very high repetition rates but moderate pulse energies.
• In principle, the amplified pulse could generate a high Raman gain at some longer wavelengths, so that a strong Stokes pulse would be formed, and the original pulse would be distorted. However, a high enough Raman gain for that problem is usually not reached in a parabolic pulse amplifier.
• Higher-order chromatic dispersion of the fiber and a compressor should not be too strong.
• Amplifier noise may also be relevant under certain circumstances.

To find an optimized design of a parabolic amplifier, it is advisable to study its behavior with numerical pulse propagation modeling, including the potentially disturbing effects mentioned above.

The principle of parabolic pulse amplification has also been applied to mode-locked fiber lasers [6] (similariton fiber lasers). Here, the spectral width of the circulating pulse is strongly oscillating: it increases during amplification but is reset in every round trip by some optical filter. This approach makes it possible to reach significantly higher pulse energies with potentially cheap fiber laser sources.

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