Chromatic dispersion

Chromatic dispersion is the dependence of an optical signal's travel time on its wavelength. Every modulated signal occupies a band of wavelengths, and because the group velocity of light in glass or in a waveguide changes with wavelength, the parts of that band arrive at slightly different times; a pulse leaves narrow and arrives wide. The underlying quantity is the group velocity dispersion ${\beta }_2$; chromatic dispersion is the name for its effect on signals, and in fiber practice the term is usually applied to the dispersion parameter $D$.

The dispersion parameter

$D$ is the change in group delay per unit fiber length per unit wavelength,

$$D=\frac{d}{d\lambda }\text{\hspace{0.17em}⁣}\left(\frac{1}{v_g}\right)=-\frac{2\pi c}{{\lambda }^2}{\beta }_2$$

in ps/(nm·km). A signal of spectral width $\mathrm{\Delta }\lambda$ that crosses a length $L$ therefore acquires a spread in arrival time

$$\mathrm{\Delta }\tau =\mathrm{\mid }D\mathrm{\mid }L\mathrm{\Delta }\lambda$$

and the product $DL$, the accumulated dispersion in ps/nm, is the figure that transceiver data sheets specify a tolerance against. The wavelength derivative $S=dD\mathrm{/}d\lambda$ is the dispersion slope, in ps/(nm²·km). Positive $D$ (${\beta }_2<0$) is called anomalous dispersion: shorter wavelengths travel faster. Negative $D$ is normal dispersion.

For standard single-mode fiber ITU-T G.652 does not specify $D$ directly. It specifies the zero-dispersion wavelength ${\lambda }_0$ and the slope there, $S_0$, and the dispersion follows from

$$D(\lambda )=\frac{S_0}{4}(\lambda -\frac{{\lambda }_0^4}{{\lambda }^3})$$

with ${\lambda }_0$ between 1300 and 1324 nm and $S_0$ no more than 0.092 ps/(nm²·km). The Recommendation gives the formula for the O and E bands; applied at 1550 nm it is an extrapolation, and it is how link calculations usually estimate C-band dispersion when only ${\lambda }_0$ and $S_0$ are known. With ${\lambda }_0=1310$ nm and $S_0=0.092$ it gives 17.46 ps/(nm·km) at 1550 nm and 21.59 at 1625 nm; across the permitted range of ${\lambda }_0$ at the maximum slope the 1550 nm value runs from 16.67 to 18.01.

The dispersion-shifted and non-zero dispersion-shifted entries explain why the latter replaced the former, and the zero-dispersion wavelength entry covers the behavior near ${\lambda }_0$.

Material and waveguide contributions

The dispersion of a guided mode is, to good approximation, the sum of a material term, from the curvature of the glass's refractive index with wavelength, $D_M=-(\lambda \mathrm{/}c)d^{2}n\mathrm{/}d{\lambda }^2$, and a waveguide term from the way the mode's effective index changes as the mode spreads further into the cladding at longer wavelengths. From Malitson's Sellmeier equation for fused silica, $D_M$ crosses zero at 1273 nm and is 3.6 ps/(nm·km) at 1310 nm and 21.9 at 1550 nm. The waveguide term of a conventional step-index fiber is negative, and it moves the zero of a G.652 fiber up to around 1310 nm; the difference between the silica value and the fiber value at 1550 nm, about −4.5 ps/(nm·km), is the waveguide term together with the effect of the germania-doped core. Dispersion-shifted and non-zero dispersion-shifted fibers use core profiles with a much larger negative waveguide term. In silicon and silicon nitride waveguides the waveguide term dominates and can take either sign, which is the subject of the GVD entry. In hollow-core fiber the material term almost disappears; the article Hollow-Core Fiber: Below the Rayleigh Floor gives its dispersion budget.

What it does to a link

A 10 Gb/s signal with a 0.1 nm spectrum crossing 80 km of G.652 fiber at 1550 nm acquires $\mathrm{\Delta }\tau =17.46\times 80\times 0.1=140$ ps, longer than the 100 ps bit period, which is why uncompensated 1550 nm intensity-modulated links at that rate run out of reach in a few tens of kilometers, and sooner with a chirped source (see frequency chirp). The two dominant modern cases avoid the problem in opposite ways. Datacenter optics such as 400GBASE-FR4 place their lanes at 1271, 1291, 1311 and 1331 nm around the zero; the worst case for a G.652 fiber is $\mathrm{\mid }D\mathrm{\mid }=5.19$ ps/(nm·km) at 1271 nm when ${\lambda }_0=1324$ nm, which is 10.4 ps/nm over 2 km, and the Datacenter Link Budget Explorer converts that into spread per unit interval. Coherent links in the C-band accept 17,460 ps/nm over 1000 km and remove it in the receiver's DSP with a fixed digital filter, as described in What the Coherent DSP Actually Does. Between those cases lie the optical methods covered under dispersion compensation. Dispersion also interacts with nonlinearity: it spoils the phase matching of four-wave mixing between WDM channels, and in the anomalous regime it can balance self-phase modulation.

The Dispersion and Pulse Broadening Calculator evaluates $D$ and $S$ at any wavelength from ${\lambda }_0$ and $S_0$ with the G.652 formula (with presets for G.652, G.653, G.655, dispersion-compensating fiber and an O-band case), plots $D(\lambda )$ from 1200 to 1700 nm, and gives the accumulated dispersion $DL$ with the pulse broadening and bit-rate limits that follow from it.

References: ITU-T Recommendation G.652, Characteristics of a single-mode optical fibre and cable. ITU-T Recommendation G.653, Characteristics of a dispersion-shifted, single-mode optical fibre and cable. ITU-T Recommendation G.655, Characteristics of a non-zero dispersion-shifted single-mode optical fibre and cable. G. P. Agrawal, Fiber-Optic Communication Systems, 5th ed. (Wiley, 2021). I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," Journal of the Optical Society of America 55, 1205 (1965). IEEE Std 802.3, clause 151 (400GBASE-FR4).