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

German: chromatische Dispersion

The chromatic dispersion of an optical medium is the phenomenon that the phase velocity and group velocity of light propagating in a transparent medium depend on the optical frequency. A related quantitative measure is the group velocity dispersion.

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

## Mathematical Description of Chromatic Dispersion

Chromatic dispersion of second and higher order is defined via the Taylor expansion of the wavenumber *k* (change in spectral phase per unit length) as a function of the angular frequency ω (around some center frequency ω_{0}, e.g. the mean frequency of some laser 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 s^{2}/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 fs^{2}/mm at 800 nm, or −22 fs^{2}/mm at 1500 nm.
Zero group delay dispersion is reached close to 1270 nm.

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 β_{2}.

## 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.

## Higher-order Dispersion

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.

## 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.

### Dispersive Pulse Broadening and Chirping

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

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 τ_{0}, the pulse duration is increased according to

if second-order group delay dispersion *D*_{2} is applied.
The approximation holds for the case of strong broadening, i.e., for *D*_{2} >> τ_{0}^{2}.
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 fs^{2}).
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.

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 Bragg grating pairs 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.

## Chromatic Dispersion from Polarization Effects

A special type of chromatic dispersion can result from polarization mode dispersion e.g. in optical fibers. For a certain fiber span and a given wavelength, two input principal polarization states can be determined, which exhibit somewhat different values of the group delay. Besides this differential group delay, there are contributions to the group delay dispersion with opposite signs for the two principal polarization states. Such effects can be relevant for optical fiber communications at very high bit rates.

## Measurement of Chromatic Dispersion

There are several techniques for measuring chromatic dispersion:

- The pulse delay technique [1] (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” [3] (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 [4] (e.g. white-light interferometry [5] or spectral phase interferometry [9]) 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

[1] | L. G. Cohen and C. Lin, “Pulse delay measurements in the zero material dispersion wavelength region for optical fibers”, Appl. Opt. 16 (12), 3136 (1977) |

[2] | D. N. Payne and A. H. Hartog, “Determination of the wavelength of zero material dispersion in optical fibers by pulse-delay measurement”, Electron. Lett. 13 (21), 627 (1977) |

[3] | A. Sugimura and K. Daikoku, “Wavelength dispersion of optical fibers directly measured by ‘difference method’ in the 0.8–1.6 μm range”, Rev. Sci. Instrum. 50 (3), 434 (1979) |

[4] | M. Tateda et al., “Interferometric method for chromatic dispersion measurement in a single-mode optical fiber”, IEEE J. Quantum Electron. 17 (3), 404 (1981) |

[5] | H.-T. Shang, “Chromatic dispersion measurement by white-light interferometry on metre-length single-mode optical fibres”, Electron. Lett. 17 (17), 603 (1981) |

[6] | B. Costa et al., “Phase-shift technique for the measurement of chromatic dispersion in single-mode optical fibres using LED's”, Electron. Lett. 19 (25), 1074 (1982) |

[7] | L. Cohen, “Comparison of single-mode fiber dispersion measurement techniques”, J. Lightwave Technol. 3 (5), 958 (1985) |

[8] | L. Thevenaz et al., “All-fiber interferometer for chromatic dispersion measurements”, J. Lightwave Technol. 6 (1), 1 (1988) |

[9] | C. Dorrer, “Chromatic dispersion characterization by direct instantaneous frequency measurement”, Opt. Lett. 29 (2), 204 (2004) |

[10] | I. A. Walmsley et al., “The role of dispersion in ultrafast optics”, Rev. Sci. Instrum. 72 (1), 1 (2001) |

[11] | R. Paschotta, tutorial on "Passive Fiber Optics", Part 10: Chromatic Dispersion |

(Suggest additional literature!)

See also: group velocity dispersion, group delay dispersion, zero dispersion wavelength, dispersion, waveguide dispersion, dispersion compensation, dispersion management, fibers, group velocity, refraction, refractive index, Sellmeier formula, Kramers–Kronig relations, dispersion-shifted fibers

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