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 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 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.

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.

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.

Conversion of Chromatic Dispersion Values

Center wavelength:
Group velocity dispersion:				calc	(1 fs2 = 1e-30 s2)
Dispersion parameter:				calc

Enter input values with units, where appropriate. After you have modified some values, click a "calc" button to recalculate the field left of it.

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.

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.

Wavelength-dependent Refraction and Diffraction

Dispersive Pulse Broadening and Chirping

Soliton Effects

Dispersion in Nonlinear Optics

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.

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.

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.

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.

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[8]	L. Cohen, “Comparison of single-mode fiber dispersion measurement techniques”, J. Lightwave Technol. 3 (5), 958 (1985), doi:10.1109/JLT.1985.1074327
[9]	L. Thevenaz et al., “All-fiber interferometer for chromatic dispersion measurements”, IEEE J. Lightwave Technol. 6 (1), 1 (1988), doi:10.1109/50.3953
[10]	C. Dorrer, “Chromatic dispersion characterization by direct instantaneous frequency measurement”, Opt. Lett. 29 (2), 204 (2004), doi:10.1364/OL.29.000204
[11]	I. A. Walmsley et al., “The role of dispersion in ultrafast optics”, Rev. Sci. Instrum. 72 (1), 1 (2001), doi:10.1063/1.1330575
[12]	R. Paschotta, tutorial on "Passive Fiber Optics", Part 10: Chromatic Dispersion

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