定义：
是一种布拉格类型的色散型反射镜，其中布拉格波长具有空间变化。

啁啾反射镜是一种电介质反射镜，通常用在锁模激光器中用于色散补偿。啁啾反射镜另一个重要的特性是比传统的布拉格反射镜具有更宽的反射带宽。 

色散啁啾反射镜的原理 
啁啾反射镜设计最基本的就是布拉格波长不是常数，而是在结构中是空间变化的，因此不同波长的光进入反射镜结构的深度不同，因此具有不同的群时延。 
但是，直接基于这一原理的设计也是不行的：群时延会存在很强的振荡，甚至得到更强的群时延色散。这一效应可以通过优化层结构在一定程度上可以解决，但是非常困难，因为优化过程需要在多维空间进行（由于层数很多），存在很大数目的局部最优的参数，其中大多数都不是符合标准的设计。 

图1：啁啾反射镜的工作原理。长波长的光能够在反射镜结构中穿透更深，因此存在更大的群时延。这会引起反常色散。 
后来，人们发现这种振荡来自于下列两个原因： 

• 前表面（与空气接触的表面）发生菲涅尔反射，引起类似于Gires-Tournois干涉仪中的强的附加色散。 
• 两相反方向的波在空气中的耦合为0，而在该结构中突然变为一有限值会引起阻抗失配。 这两个问题都可以采用双啁啾设计进行消除，这种设计具有两个特性： 
• 通过改变占空因数，即高折射率和低折射率层之间光学厚度的比值，两相反方向光之间的耦合缓慢增加。（占空因数大于或者小于50%时，有效反射率会减小。） 
• 在双啁啾区域顶端采用附加的抗反射结构可以消除菲涅尔反射（图1中未显示）。 

即使不采用数值优化，双啁啾设计也能得到与理想目标近乎匹配的色散曲线。然后再采用数值优化，即精细调节层厚度值可以进一步改进。 

图2：经过数值优化的啁啾反射镜的反射和色散性质。虚线是理想的色散曲线，可见在很宽的波长范围内是非常匹配的。如果采用更多的层数，可以实现更宽的波长范围。 

图3：图2中啁啾反射镜的电场穿透情况。 

锁模激光器的应用 
在需要几个工作周期工作时需使用很大带宽的锁模激光器，设计具有相应的很宽反射带宽的反射镜和在该区域内具有合适的色散非常困难。限制带宽的因素通常也是制作具有很小剩余反射率的大带宽的抗反射结构时存在的困难。这一问题可以采用背面涂层（BASIC）啁啾反射镜来解决。这种设计的关键是使这种啁啾反射镜结构与玻璃衬底相接触，而不是与空气接触；空气-玻璃界面就位于一个不同的位置，因此该界面处的剩余反射率效应可以通过采用玻璃片的楔形结构进行消除。 
双啁啾反射镜（DCMs）通常用在锁模激光器中用于色散补偿，尤其是当脉冲长度小于20fs时。它们通常不仅可以补偿恒定的群时延色散，还可以补偿高阶色散。但是，双啁啾反射镜能够补偿的色散是有范围的。一种解决方案是采用多个反射镜的合理组合，这样不同反射镜的色散可以部分抵消，然后将啁啾反射镜与棱镜对结合起来使用。另外一个问题来自于严格的制作允差；至少膜层厚度需要制备的非常精准，精确至几个纳米。仍然存在的群时延随波长的摆动可以通过不同反射镜的结合来进一步减小，因为摆动会部分的相互抵消。 

半导体啁啾反射镜 
双啁啾反射镜也可以采用半导体反射镜。这种反射镜会产生更高的色散，但是带宽更小。它们可以用于补偿锁模激光器中的色散，并且器件尺寸很小，即使脉冲长度很长需要很大的反常色散用于孤子锁模的情况下也是如此。

Definition: Bragg-type dispersive mirrors with a spatial variation of the Bragg wavelength

More general term: dielectric mirrors

A chirped mirror is a kind of dielectric dispersive mirror with a spatial variation of the layer thickness values. Such mirrors are used for dispersion compensation in mode-locked lasers, for example. Another interesting feature of such mirrors is that they make it possible to achieve a broader reflection bandwidth than ordinary Bragg mirrors. Due to the many dimensions of possible optimization, chirped mirrors are often sold as custom optics.

Principles of Dispersive Chirped Mirrors

The basic idea of chirped mirror designs [1] is that the Bragg wavelength is not constant but varies within the structure (along the propagation direction), so that light at different wavelengths penetrates to a different extent into the mirror structure and thus experiences a different group delay.

However, a naïve design directly based on this idea would not work: it would exhibit strong oscillations of the group delay and even more so of the group delay dispersion. This disturbing effect can to some extent be mitigated by numerical optimization of the layer structure, but this is difficult because the optimization has to be done in a multi-dimensional space (resulting from the large number of layers) where a huge number of local optima exist, most of which do not correspond to satisfactory designs.

Figure 1: Operation principle of a chirped mirror. Light with a long wavelength penetrates deeper into the mirror structure and thus experiences a larger group delay. This leads to anomalous chromatic dispersion.

It was found [5] that the disturbing oscillations have two origins:

• There is a Fresnel reflection at the front face (the interface to air), which leads to strong additional dispersion as in a Gires–Tournois interferometer.
• The sudden “switching” of the coupling of counterpropagating waves from zero in air to a finite value in the structure causes a kind of impedance mismatch.

Both problems can be eliminated with a double-chirped design, which has two additional features:

• The coupling of counterpropagating waves is turned on smoothly by also varying the duty cycle, i.e., the ratio of optical thickness of high and low index layers. (For a duty cycle above or below 50%, the effective reflectance of a layer pair is reduced.)
• The Fresnel reflection is removed with an additional anti-reflection structure on top of the double-chirped section (not shown in Figure 1).

Even without numerical optimization, double-chirped designs can have a dispersion profile which relatively nicely matches the design goal. Further refinement is then achieved with numerical optimization, i.e., with fine tuning of the layer thickness values.

Figure 2: Reflection and dispersion properties of a numerically optimized chirped mirror design, developed with the software RP Coating. The dashed curve shows the target dispersion, which is fairly precisely matched in a broad wavelength range. Even broader wavelength ranges are possible, when a larger number of layers is used.

Figure 3: Field penetration into the chirped mirror of Figure 2. It is apparent that within the wavelength range 1000–1200 nm, the field penetrates more deeply into the structure for longer wavelengths. The larger group delay for longer wavelengths corresponds to anomalous dispersion.

Remaining oscillations in the group to latest version can be further reduced by using suitable pairs of dispersive mirrors, or alternatively by using the same kinds of mirrors with slightly different incidence angles [15].

Application in Mode-locked Lasers

For mode-locked lasers with an ultrabroad bandwidth, as required for operation in the few-cycle regime, it is challenging to design mirrors with the corresponding ultrabroad reflection bandwidth, combined with proper chromatic dispersion over the most of that range. The factor limiting the bandwidth achievable is in most cases the difficulty of making anti-reflection structures with very small residual reflectance over a large bandwidth. This problem can be solved with so-called backside coated (BASIC) chirped mirrors [11]. The key idea of such a design is to interface the chirped mirror structure with a glass substrate rather than with air; the air–glass interface is then at a different location, and the effects of the residual reflectance of that (also AR-coated) surface can be eliminated by using a wedge shape for that glass piece.

Double-chirped mirrors (DCMs) are often used for dispersion compensation in mode-locked lasers, particularly for those with pulse durations below ≈ 20 fs. They are typically designed not only to compensate a constant group delay dispersion, but also to correct higher-order dispersion. However, there are limits concerning how much dispersion (and in particular higher-order dispersion) can be compensated with a double-chirped mirror. Possible solutions are to use a suitable combination of several mirrors, where the dispersion errors from different mirrors partially cancel each other, and to combine chirped mirrors with a prism pair. Another challenge arises from the tight fabrication tolerances; at least some of the layers typically have to be fabricated with a precision of the thickness of a few nanometers. The remaining wiggles in the group delay versus wavelength can be further reduced by using appropriate combinations of mirrors where the wiggles at least partially cancel each other.

Chirped Semiconductor Mirrors

It has been shown [10] that double-chirped mirror designs can also be used with semiconductor mirrors. Such mirrors can generate a much higher amount of dispersion, although in a much smaller bandwidth. They can be used for compensating the dispersion in a mode-locked laser with a single compact device even when e.g. a long pulse duration requires a high amount of anomalous dispersion for soliton mode locking.

Bibliography