Illustrated Physics
physics, drawn
Concept

Anomalous dispersion — where it appears

The band of frequencies in which a medium's refractive index falls as the frequency rises, which happens across every absorption line it has. It is compulsory rather than exceptional: the width of the band is the width of the line, and the same integral that ties absorption to refraction forces it.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

A peak that leaves before it should have arrived. Two pulses at the far face of a cell, both normalised to the peak the vacuum one reaches. One has crossed empty space; the other has crossed a medium with two gain lines either side of its carrier, whose group index there is -3.85 — negative, so the envelope's peak should emerge early, and it does. Measured off the two curves the advance is 616 in units where the carrier period is 2π, against 582 predicted from the group index alone; the difference is the higher-order dispersion the group index leaves out. The advance is 0.21 of the pulse's own duration, and the peak leaves the far face before the input peak has entered the near one. Nothing has outrun anything. The emergent pulse is a reshaped version of the input's leading edge, which arrived in plenty of time and already contained — for a smooth pulse — everything needed to reconstruct the rest; the medium amplifies it by 1.14× and delivers it early. Give the pulse a genuine front, a moment before which it is exactly zero, and that front travels at the speed of light in every medium there is.

The speed that carries no signal

In the right medium a pulse's peak emerges from the far side before it entered the near one. The measurement is real, it has been made, and nothing has outrun light — because the peak of a smooth pulse was never carrying any information in the first place.

waves · Wave packets Absorption and refraction, drawn as one function. The real and imaginary parts of a Lorentz oscillator's susceptibility against frequency, in units of the resonance. The imaginary part is the absorption: a symmetric line centred on the resonance, with a full width at half maximum equal to the damping — 0.05, 0.12, 0.30 here. The real part is the refraction, and it is what the same medium does to the speed of light. The two curves are not two facts about the medium: either one determines the other completely, by an integral over all frequencies, and that is a consequence of the medium responding after it is asked rather than before. Between x = 0.97 and x = 1.02 the refraction runs the wrong way — the index falls as the frequency rises, which is anomalous dispersion — and that region is exactly the width of the absorption line. Away from the line the index rises with frequency, which is ordinary dispersion and is why a prism separates colours in the order it does: every transparent material is on the low-frequency tail of an ultraviolet absorption it is not otherwise showing. A narrower line is a taller one, because the area under the absorption is fixed by how many electrons there are and by nothing else.

The answer that cannot come first

A medium absorbs at some frequencies and bends light at all of them, and those look like two separate facts to be measured separately. They are one fact. The requirement that a material respond after it is asked rather than before ties the absorption at every frequency to the refraction at every other, by an integral — and a material given the two halves independently answers before the question arrives.

waves · Attenuation Fringes that bend into hooks beside a line. The interference fringes seen in a spectrograph behind an interferometer with 10 cm of sodium vapour at 10¹⁴ atoms per cubic centimetre in one arm and a glass plate in the other, against wavelength across the D₂ line (horizontal) and position along the slit (vertical, in fringe spacings). Far from the line the plate tilts every fringe evenly. Near the line the vapour's dispersion sweeps each fringe up on one side and down on the other, and every fringe turns back on itself in a hook: the turning points sit −100 and 100 picometres from the line centre, the same for every fringe. The line itself, a few picometres wide, is a dark gap between them; the hooks lie far outside it, in vapour that is transparent.

The hooks that weigh an absorption line

A strong absorption line is black at its centre, and how black it is says almost nothing about how many atoms made it — double them and the line hardly darkens. The atoms are still there, and they still bend light: on either side of the line, where the vapour is transparent, its refractive index rises and falls by an amount proportional to their number. In 1912 Dmitry Rozhdestvensky found a way to read that index straight off a photograph. Interference fringes passed through a spectrograph curl into hooks beside the line, and the distance between the hooks, squared, counts the atoms — whatever the line's shape and however saturated its core.

optics · Dispersion

Named alongside it

The objects these essays reach for when they reach for this one.

DispersionRefractive indexAbsorptionCausalityOscillator strengthSum ruleAnalytic continuationDoppler broadeningFourier analysisFront velocityGroup velocityInterferometry

All concepts