Anomalous dispersion

Anomalous dispersion is a frequency range in which the refractive index n(ω) decreases with increasing frequency, i.e., dn/dω < 0, opposite to the normal-dispersion behaviour of transparent glasses in the visible. It always occurs in the vicinity of an absorption resonance — whenever the driving frequency straddles a natural oscillation frequency of the material (an electronic transition, a molecular vibration, a lattice phonon), the Lorentz-model derivation of n(ω) predicts a resonant dip-and-peak structure, with normal dispersion on either side and anomalous dispersion in the narrow absorbing region.

The phenomenon was first observed in strongly absorbing dyes by Leroux (1862) and Christiansen (1870), before Hendrik Lorentz's 1878–1880 electron-oscillator theory gave the quantitative explanation. In the anomalous-dispersion band the phase velocity v_p = c/n can exceed c — since n can be less than 1 in that range — and the group velocity v_g = c/n_g can become very large, become negative, or exceed c. This does not violate special relativity because the signal velocity (the leading edge of a genuine information-carrying pulse) remains at or below c; anomalous-dispersion phase velocities exceeding c are a kinematic property of the phase-surface geometry, not a real propagation speed for a new signal. Modern optics exploits anomalous dispersion in Kramers–Kronig analyses, in chirped-pulse amplification of femtosecond lasers, and in negative-refractive-index metamaterials.