定义：透明介质中的相速度随频率变化的特性。

色散（Chromatic dispersion）的数学描述 
二阶及高阶色散的定义是由波数k（单位长度的光相位）泰勒展开为角速度ω（中心频率 ω0附近，例如，一些激光脉冲的平均频率）的函数而得到的： 

其中不同项的意义为： 

1. 零阶项代表普通的相移。 
2. 一阶项为群速度的倒数（也就是单位长度的群时延），描述总体时间延迟：

1. 二阶项代表二阶色散或者单位长度的群时延色散（GDD）：

1. 三阶项表示单位长度的三阶色散（TOD）：  

二阶色散的单位为 s2/m。它是群速度的倒数对角频率的微分：  

例如，二氧化硅的群速度色散在800 nm时为+36 fs2/mm，或者在1550 nm时为−22 fs2/mm。零群延迟色散在1270nm 附近可以得到。 
计算介质的各阶色散时将折射率用Sellmeier公式表示出来比较方便。列表索引数据不太适合，因为数值微分对损耗非常敏感。 

正常和反常色散 
正常色散（ k > 0）中群速度随着频率增大而减小，在可见波段大多数透明介质都具有正常色散。反常色散（ k < 0）发生在更长的波长情况下，例如，二氧化硅中，波长大于零色散波长约为1300 nm时。 这里需要强调下色散的符号。超快光学中，色散采用与k相同的符号。而在光纤通信中则采用相反的符号，色散由色散参数表示： 
 
单位为ps/(nm km)。不同的符号来源于一种情况是对频率的微分，而另一种情况则是对波长微分。并且二者的转化因子依赖于波长。 
正常色散和反常色散区域之间存在零色散波长。该波长附近的区域比较特殊，不仅仅表现为是其弱色散脉冲展宽。 

高阶色散 
三阶及更高阶色散被称为高阶色散。处理非常宽的光谱时，需要考虑四阶甚至五阶、六阶的色散。最终，当需要考虑很多阶色散时，泰勒展开的方法在这种情况下无太大意义。采用数值模型直接计算相位随频率的变化更方便。 

色散（Chromatic dispersion）的影响 
折射和衍射过程随波长变化

常遇见的现象为色散引起随波长变化的折射，因此会产生彩虹。类似的，衍射光栅中的衍射随波长变化可以将不同频率组分的光在空间上分离开。 

色散脉冲展宽和啁啾
色散对脉冲的传输有很重要的影响，因为脉冲具有有限的谱宽（带宽），因此会使频率组分以不同的速度传输。例如，正常色散情况下，高频部分的群速度更小，因此产生正啁啾，而反常色散则产生负啁啾。 
群速度随频率变化也会对脉冲长度产生影响。如果初始脉冲是无啁啾的高斯脉冲，脉冲长度为τ0，那么脉冲长度增大到： 
 

这时考虑了二阶群延迟色散β2 。以上近似在强展宽情况下也适用，也就是当 β2 >> τ02时。较短的输入脉冲可以得到变长的输出脉冲。这就是增大脉冲带宽的效应。 
例如，初始无啁啾的800nm波长，长度为30 fs的脉冲在二氧化硅（+360 fs2）中传输10 mm后展宽为45 fs。而传输10 cm后，脉冲长度变为334 fs。 
具有相反的色散则可以用来压缩脉冲（色散脉冲压缩）。这在啁啾脉冲放大时非常有用。根据色散的符号和大小以及光峰值功率等因子，不同的装置都可以用于脉冲压缩。例如棱镜对，衍射光栅对，啁啾镜，啁啾布拉格光栅和色散光纤。 
高阶色散会使脉冲形状产生更加复杂的变化。色散补偿不仅需要补偿最低阶的色散，还需要补偿高阶色散从而使脉冲接近与变换极限脉冲。 
 

图1：由具有不同大小的群延迟色散（GDD）产生的色散脉冲展宽效应得到的出射脉冲长度与初始脉冲长度关系曲线。短脉冲对色散更加敏感。当脉冲长度的平方小于群延迟色散时，会产生显著的脉冲展宽。 

孤子效应
仅色散效应会引起脉冲展宽，而当色散效应与克尔非线性效应结合起来会形成孤子，这样可以用于孤子锁模激光器中产生超短脉冲。但是，这需要在很大的波长范围内严格控制色散，也就是需要考虑高阶色散项。 

非线性光学中的色散
非线性光学中，尤其是非线性频率转换过程中，色散会产生多方面的影响，主要分为下面三类： 

1. 非线性参量过程中的相位匹配选择 
2. 通过群速度失配（参阅时间走离，有限的相位匹配带宽）限制短脉冲的有效相互作用长度 
3. 通过色散脉冲展宽限制非线性相互作用长度，最终脉冲变得太长以至于峰值功率太低，不能有效的进行非线性相互作用。而且，群速度失配会引入其他的限制效应。 

光学元件的色散（Chromatic dispersion）效应 
色散也可以定义为光学元件的色散，而不是介质中。这时需要对元件总相位延迟做泰勒展开（而不是对波数，也就是单位长度的相位延迟），然后得到总的群延迟色散（单位为秒的平方），而不是如上所述的单位长度的色散。 
光学元件的色散（Chromatic dispersion）可能仅仅来自于其组成部件的色散，有时很大的一部分色散来自于干涉效应。例如，Gires-Tournois干涉仪由于存在干涉效应引起光学谐振腔中的色散。其它类型的干涉仪中存在相同的情形。当传播长度差别很大时，干涉效应产生的色散非常大。 
几何效应也会产生色散（Chromatic dispersion）。例如，棱镜对通常用来对锁模激光器进行色散补偿。其中，由于棱镜表面的折射依赖于波长，路径长度依赖于波长从而产生色散（Chromatic dispersion）。激光器谐振腔中包含布儒斯特角度的光学元件，所以存在类似效应，另外可以采用布拉格光栅进行色散脉冲压缩。 

波导色散 
以上的讨论都是采用的平面波假设。实际中，情形与之前所述可能差别较大，尤其是在波导中。这时通常不需要考虑波矢（k的矢量）的幅值，而是需要考虑β的值（传播常数的虚部），表征传播方向上单位长度的相位变化。β值受波导影响（尤其是当模式直径为几个波长甚至更小时），因此色散也受其影响（参阅波导色散）。这在光纤中很重要，尤其是在具有非常小有效模式面积的光子晶体光纤中。有些情况下，波导色散会使总色散即使在可见光波段也是反常色散，虽然这时二氧化硅的材料色散是处于正常色散区域的。在通信应用中，需要设计光纤控制其色散性质，也就是色散位移光纤。 

偏振效应产生的色散（Chromatic dispersion） 
光纤中的偏振模式色散是一种特别的色散类型。对于给定光纤长度和波长，存在两个方向上的主偏振态，它们的群延迟是不同的。除了群延迟不同，两主偏振态的群延迟色散的符号是相反的。光纤通信中比特速率很高时需要考虑这一效应。 

色散测量 
可以采用下列方法测量色散（Chromatic dispersion）： 

1. 脉冲延迟技术，测量具有不同中心波长的脉冲的传播时间（群延迟）差。这需要采用几百米（甚至几千米）的光纤。通过对所得的数据微分可以得到色散。 
2. 相移技术或者差分方法（也用于光纤）：强度受到正弦调制的光束在光纤中传输，对比输入和输出功率的相位。通过该相位可以计算群延迟，通过测量不同的波长可以得到色散。 
3. 调谐波长被动锁模激光器谐振腔的色散可以通过测量脉冲重复速率随波长改变的变化，这样可以得到随波长变化的群延迟。 
4. 可以采用不同类型的干涉仪（例如，白光干涉仪或者光谱相位干涉仪）可以测量色散元件引起的相位延迟。可以采用相位的数值微分得到色散特性。这一方法通常用于测量色散激光反射镜和光纤中的色散。

Definition: the frequency dependence of the phase velocity in a transparent medium

More specific terms: normal and anomalous dispersion, material dispersion, waveguide dispersion, group delay dispersion, third-order dispersion, principal dispersion

The chromatic dispersion of an optical material is the phenomenon that the phase velocity and group velocity of light propagating in a transparent medium depend on the optical frequency. That dependency results mostly from the interaction of light with electrons of the medium, and is related to absorption in some spectral regions; see the article on Kramers–Kronig relations. A quantitative measure is the group velocity dispersion.

The attribute “chromatic” is used to distinguish that type of dispersion from other types, which are relevant particularly for optical fibers: intermodal dispersion and polarization mode dispersion.

Chromatic dispersion can also occur from geometrical effects, for example; see below the section on chromatic dispersion of optical components.

Mathematical Description of Chromatic Dispersion

An early measure for the magnitude of chromatic dispersion was the Abbe number V_D, introduced by Ernst Abbe:

The denominator is also called the principal dispersion. The Abbe number depends on the refractive indices at only three different wavelengths:

• 486.1 nm (blue Fraunhofer F line from hydrogen)
• 589.2 nm (orange Fraunhofer D line from sodium)
• 656.3 nm (red Fraunhofer C line from hydrogen)

Large values of the Abbe number indicate low chromatic dispersion and vice versa. Such values can be used for the design of achromatic optical elements, but can of course give only a quite rough indication.

The modern way of quantifying chromatic dispersion is based on a Taylor expansion of the wavenumber k (change in spectral phase per unit length) as a function of the angular frequency ω. The expansion is made around some center frequency ω₀, e.g. the mean angular frequency of some light pulses:

where the terms corresponding to the different orders have the following meaning:

• The zero-order term describes a common phase shift.
• The first-order term contains the inverse group velocity (i.e., the group delay per unit length) and describes an overall time delay without an effect on the pulse shape:

• The second-order (quadratic) term contains the second-order dispersion or group delay dispersion (GDD) per unit length:

• The third-order (cubic) term contains the third-order dispersion (TOD) per unit length:

Second-order dispersion is often specified in units of s²/m. It is the derivative of the inverse group velocity with respect to angular frequency:

As an example, the group delay dispersion of silica is +36 fs²/mm at 800 nm, or −22 fs²/mm at 1500 nm. Zero group delay dispersion is reached close to 1270 nm.

Dispersion of third and higher order is called higher-order dispersion. When dealing with very broad optical spectra, one sometimes has to consider dispersion up to the fourth or even fifth and sixth order. Ultimately, the concept of Taylor expansion loses its value in this regime, where many dispersion orders have to be considered. It is therefore often more convenient e.g. in numerical modeling to work directly with a table of frequency-dependent phase changes.

The dispersion of various orders for a medium can most conveniently be calculated if the refractive index is specified with a kind of Sellmeier formula. Tabulated index data are less suitable, since the numerical differentiation is sensitive to noise.

For light propagating in waveguides such as optical fibers, one considers the phase constant β instead of the wavenumber k. The second-order dispersion, for example, is then given as the second derivative of β with respect to angular frequency. This is often called β'' or β₂.

Normal and Anomalous Dispersion

One distinguishes normal dispersion (for k'' > 0) and anomalous dispersion (for k'' < 0). Normal dispersion, where the group velocity decreases with increasing optical frequency, occurs for most transparent media in the visible spectral region. Anomalous dispersion sometimes occurs at longer wavelengths, e.g. in silica (the basis of most optical fibers) for wavelengths longer than the zero-dispersion wavelength of ≈ 1.3 μm.

Great care is recommended when the sign of dispersion is specified. The ultrafast optics community identifies that sign with the sign of k''. The opposite sign is usually used in optical fiber communications, where the dispersion is often specified with the dispersion parameter

in units of picoseconds per nanometer and kilometer (ps/(nm km)). The different signs result from using a frequency derivative in one case and a wavelength derivative in the other. Note also that the conversion factor depends on the wavelength.

Between wavelength regions with normal and anomalous dispersion, there is a zero dispersion wavelength. The region around this wavelength can be special in some respects, not only concerning weak dispersive pulse broadening.

Effects of Chromatic Dispersion

Wavelength-dependent Refraction and Diffraction

A frequently encountered phenomenon is that dispersion causes wavelength-dependent refraction, which is responsible, e.g., for the occurrence of rainbows. Similarly, wavelength-dependent diffraction at a diffraction grating allows the spatial separation of different frequency components of light. That is exploited in grating spectrometers, for example.

Dispersive Pulse Broadening and Chirping

Chromatic dispersion has an important impact on the propagation of light pulses, because a pulse always has a finite spectral width (bandwidth), so that dispersion can cause frequency-dependent phase changes. Consequently, its frequency components propagate with different group velocities. Normal dispersion, for example, leads to a lower group velocity of higher-frequency components, and thus to a positive chirp (instantaneous frequency rising over time), whereas anomalous dispersion creates negative chirps.

Note that the spectral shape (i.e., the frequency dependence of the power spectral density of the optical field) is never changed by chromatic dispersion, since only phase changes are caused. However, if chromatic dispersion occurs together with optical nonlinearities, e.g. in an optical fiber, it can indirectly affect the spectral shape, since the phase changes modify the temporal shape and thus the effect of nonlinearities.

Polychromatic light, e.g. corresponding to ultrashort light pulses, may be described with complex amplitudes A in the time or frequency domain. If it propagates through a dispersive medium, the resulting change of complex amplitudes can be described with the following equations. The simpler description is found in the frequency domain, where second- and third-order dispersion generate phase changes proportional to the second or third power of frequency offset, respectively:

In the time domain, those factors are replaced with time derivatives:

The frequency dependence of the group velocity also has an effect on the pulse duration. If the pulse is initially unchirped, dispersion in a medium will always increase its duration (dispersive pulse broadening). For an originally unchirped Gaussian pulse with the duration τ₀, which experiences only second-order dispersion (= group delay dispersion) D₂, the pulse duration is increased according to

The approximation holds for the case of strong broadening, i.e., for D₂ >> τ₀². In that regime, shorter input pulses lead to longer output pulses. This is an effect of the increased pulse bandwidth.

As a numerical example, an initially unchirped 30-fs pulse at 800 nm becomes stretched to 45 fs after 10 mm of propagation in silica (with +360 fs²). After 10 cm, the pulse duration would increase to 334 fs.

Dispersion of the opposite sign can be used later to recompress the pulse (dispersive pulse compression). This is important, e.g., in the context of chirped-pulse amplification. Depending on the sign and amount of required dispersion, and also on other factors such as the optical peak power, different devices are used as dispersive compressors. Examples are prism pairs, pairs of diffraction gratings, chirped mirrors or other dispersive mirrors, chirped Bragg gratings, and dispersive glass fibers.

Figure 1: Output pulse duration versus initial pulse duration for dispersive pulse broadening with different levels of group delay dispersion (GDD). Note that shorter pulses are increasingly sensitive to dispersion. Substantial broadening occurs when the square of the pulse duration is smaller than the group delay dispersion.

Higher-order dispersion causes more complicated changes of the pulse shape. A challenge in dispersion compensation is often to compensate not only the lowest (second) order of dispersion, but also higher orders for approaching the transform limit.

Soliton Effects

Although dispersion alone leads to pulse broadening, dispersion combined with the Kerr nonlinearity can give rise to the formation of solitons and in this way assist the generation of extremely short pulses of light with a soliton mode-locked laser. However, this usually requires the careful management of dispersion over a wide wavelength range, i.e., taking care also of higher-order terms.

Dispersion in Nonlinear Optics

In nonlinear optics, particularly concerning nonlinear frequency conversion, chromatic dispersion has manifold influences, which can be grouped into three categories:

• It determines the options for phase matching of parametric nonlinear processes.
• It limits the effective interaction length for short pulses via the group velocity mismatch (→ temporal walk-off, limited phase-matching bandwidth).
• It can also limit the nonlinear interaction length via dispersive pulse broadening: eventually, the pulses become so long that their peak power is too low for efficient nonlinear interactions. However, the group velocity mismatch normally introduces the more stringent limitation.

Chromatic Dispersion of Optical Components

Dispersion can also be defined for optical components rather than media. In that case, one performs the Taylor expansion as shown above for the total phase delay of the component (rather than for the wavenumber, i.e., the phase delay per unit length), and obtains the total group delay dispersion (in units of seconds squared) rather than the dispersion per unit length.

The chromatic dispersion of an optical component may simply result from the dispersion of its parts, but in some cases large amounts of dispersion can arise from interference effects. For example, a Gires–Tournois interferometer generates dispersion by interference effects in an optical resonator, and not primarily via material dispersion. The same happens in other types of interferometers. The amount of dispersion arising from such effects can be huge when large differences in propagation lengths are involved.

Chromatic dispersion can also result from geometric effects. An important example is that of a prism pair as is often used for dispersion compensation in mode-locked lasers. Here, chromatic dispersion arises from wavelength-dependent path lengths, caused by wavelength-dependent refraction at prism surfaces. Similar effects occur in laser resonators containing Brewster-angled optical components, and in pairs of diffraction gratings as used for dispersive pulse compression.

Waveguide Dispersion

The discussion above is based on the assumption of plane waves. In practice, significant deviations from this situation can occur, in particular in the context of waveguides. Here, the quantity of interest is usually not the magnitude of the wave vector (k vector) (which anyway is no longer well defined), but rather the value β (the imaginary part of the propagation constant), which specifies the phase change per unit length in the propagation direction. As β is influenced by the waveguide (particularly for mode diameters of only a few wavelengths or even less), the dispersion is also affected (→ waveguide dispersion). This is important e.g. in optical fibers, and particularly in photonic crystal fibers with very small effective mode areas. In some cases, waveguide dispersion makes the overall dispersion anomalous even in the visible wavelength region, where the material dispersion of silica alone is clearly in the normal dispersion regime. For telecom applications, fiber designs are often made for tailored dispersion properties, resulting in, e.g., dispersion-shifted fibers.

Measurement of Chromatic Dispersion

There are several techniques for measuring chromatic dispersion:

• The pulse delay technique [2] (for fibers) is based on measuring the difference in propagation time (group delay) for pulses with different center wavelengths. This is typically done using hundreds of meters (or even some kilometers) of a fiber. The dispersion is obtained by differentiation of these data.
• The phase shift technique or “difference method” [4] (also for fibers): a light beam with a sinusoidally modulated intensity is sent through a fiber, and the phases of the oscillations of input and output power are compared. The group delay can be calculated from that phase, and the dispersion can be measured by performing the measurement at different wavelengths.
• Dispersion in the resonator of a wavelength-tunable passively mode-locked laser can be measured by monitoring changes in the pulse repetition frequency when the laser wavelength is changed, as this reveals the wavelength-dependent group delay.
• Different types of interferometry [5] (e.g. white-light interferometry [6] or spectral phase interferometry [10]) can be used to measure the phase delay caused by a dispersive component. The dispersion properties can be obtained from this phase by numerical differentiation. The method is normally used for dispersion measurements on dispersive laser mirrors and sometimes for fibers.

Bibliography