Normal and anomalous dispersion

Normal and anomalous dispersion are the two signs of group velocity dispersion. In a medium with normal dispersion the group velocity falls as the frequency rises, so red light outruns blue; with anomalous dispersion the order is reversed and the shorter wavelengths travel faster. The sign is carried by the GVD parameter $\beta_2$ or, in fiber practice, by the dispersion parameter $D$: normal means $\beta_2 > 0$ and $D < 0$, anomalous means $\beta_2 < 0$ and $D > 0$. Most transparent glasses are normal in the visible and near infrared; bulk fused silica has $\beta_2 = +36.2$ fs²/mm at 800 nm and crosses zero at 1273 nm. Standard single-mode fiber is anomalous across the 1550 nm band, with $D \approx 17$ ps/(nm·km).

Converting between β₂ and D

The two parameters describe the same curvature of the propagation constant, one per unit angular frequency and one per unit wavelength:

$$D = -\frac{2\pi c}{\lambda^2}\,\beta_2$$ $$\beta_2 = -\frac{\lambda^2}{2\pi c}\,D$$

At 1550 nm, $\lambda^2/2\pi c = 1.275$ ps²/km per ps/(nm·km), so $D = 17$ ps/(nm·km) corresponds to $\beta_2 = -21.7$ ps²/km. Because of the minus sign, a positive $D$ is anomalous. For a bulk material the GVD follows from the index curve,

$$\beta_2 = \frac{\lambda^3}{2\pi c^2}\,\frac{d^2 n}{d\lambda^2}$$

so dispersion is normal where $n(\lambda)$ curves upward. Applied to Malitson's Sellmeier equation for fused silica, this gives $+36.2$ fs²/mm at 800 nm, $+19.0$ fs²/mm at 1030 nm, zero at 1273 nm and $-27.9$ fs²/mm, the same as $-27.9$ ps²/km, at 1550 nm. Fiber at 1550 nm is less anomalous than the bulk glass because the waveguide contribution has the opposite sign; it moves the zero-dispersion wavelength of standard fiber from 1273 nm to about 1310 nm.

What each sign does to a pulse

A short pulse contains a band of frequencies. In normal dispersion the red components move to the front and the blue to the back, so an initially unchirped pulse acquires an up-chirp, its instantaneous frequency rising from leading edge to trailing edge. In anomalous dispersion the blue components lead and the chirp is downward. Either way an unchirped pulse broadens; a 10 ps (FWHM) Gaussian pulse at 1550 nm in fiber with $\beta_2 = -21.7$ ps²/km has a dispersion length $T_0^2/|\beta_2|$ of 1.66 km and is 61 ps wide after 10 km.

A pulse that already carries a frequency chirp behaves differently. If the chirp is opposite in sign to the one the medium would impose ($C\beta_2 < 0$), the pulse first compresses, reaches a minimum width, and broadens afterward. This is the principle of the pulse compressor: a pulse broadened and up-chirped in normal-dispersion glass or fiber is recompressed by a grating pair or prism pair, which supplies anomalous dispersion. Chirped pulse amplification uses both signs on purpose: a stretcher with normal dispersion lengthens the pulse before amplification, and a grating compressor with anomalous dispersion shortens it afterward. The GVD entry gives the full broadening formula with chirp, and the Dispersion and Pulse Broadening Calculator evaluates it for fiber.

Solitons and nonlinear propagation

The Kerr effect makes a pulse impose an up-chirp on itself through self-phase modulation. In anomalous dispersion that chirp is opposed by the dispersive one, and at the right peak power the two cancel to form an optical soliton, a pulse that keeps its shape. For a fundamental sech-shaped soliton the condition is

$$P_0\,T_0^2 = \frac{|\beta_2|}{\gamma}$$

With $\beta_2 = -21.7$ ps²/km, $\gamma \approx 1.3$ /(W·km) and a 20 ps FWHM pulse ($T_0 = 11.3$ ps), the peak power is 0.13 W and the pulse energy 2.9 pJ. In normal dispersion the two chirps add, and the pulse spreads into a nearly rectangular, linearly chirped pulse that a compressor can shorten.

The sign also organizes laser design. Erbium fiber lasers at 1.55 µm can run as soliton lasers; ytterbium fiber lasers near 1 µm sit in normal dispersion and either add anomalous elements or use all-normal designs; Ti:sapphire oscillators offset their crystal with prism pairs or chirped mirrors. Supercontinuum fibers are usually pumped just on the anomalous side of the zero-dispersion wavelength, and soliton microcombs use waveguides shaped for weak anomalous dispersion.

The older meaning: anomalous index dispersion

In classical optics, "anomalous dispersion" referred to the refractive index itself rising with wavelength, $dn/d\lambda > 0$, which happens inside an absorption band and was called anomalous because it reverses the color order of a prism. The Kramers–Kronig relations tie this behavior to the absorption line. It is a different criterion from the sign of $\beta_2$: silica at 1550 nm has $dn/d\lambda < 0$, normal in the classical sense, while its group velocity dispersion is anomalous. Lens designers use a third sense: glasses of "anomalous partial dispersion", used to correct secondary color in apochromats. In fiber and pulse work the term always refers to the sign of $\beta_2$ or $D$.

Common questions

What is the difference between normal and anomalous dispersion?

The sign of the group velocity dispersion. In normal dispersion ($\beta_2 > 0$, $D < 0$) longer wavelengths travel faster; in anomalous dispersion ($\beta_2 < 0$, $D > 0$) shorter wavelengths travel faster.

Is optical fiber normal or anomalous at 1550 nm?

Standard single-mode fiber is anomalous at 1550 nm and normal below its zero-dispersion wavelength near 1310 nm.

Why do solitons need anomalous dispersion?

Self-phase modulation gives a pulse an up-chirp. Only anomalous dispersion produces the opposite chirp, so only there can the two cancel and hold the pulse shape.

References: G. P. Agrawal, Nonlinear Fiber Optics, 6th ed. (Academic Press, 2019). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019). I. H. Malitson, "Interspecimen comparison of the refractive index of fused silica," Journal of the Optical Society of America 55, 1205 (1965). E. Hecht, Optics, 5th ed. (Pearson, 2017).