OPTICAL DISPERSION

§09.1 handed you a refractive index as one number: n for glass, n for water, a scalar constant that slowed light down and bent it at boundaries. That was a lie of convenience. The real object is a function — n(λ) — and in every transparent material, that function is not flat. Red light sees one value, blue sees another, and the difference is the whole story of how a prism works, how a rainbow forms, and why every decent camera lens has between six and sixteen glass elements instead of one.

The microscopic reason, already sketched in §09.1: bound electrons in the material ring at their own frequencies, and a driven oscillator lags its drive by a phase that depends on how close the drive sits to resonance. Shorter wavelengths sit closer to the electronic UV resonance of most glasses than longer ones, so they accumulate more phase lag per unit length — which is exactly what a higher refractive index is. That monotonic slope — n larger for smaller λ, well away from any absorption band — is normal dispersion. Its sign flips inside an absorption band; the name for that is anomalous dispersion. In the visible, crown glass and water and your prescription lenses all live on the normal side.

Before the microscopic story, the phenomenon. Isaac Newton, 1672, sent sunlight through a hole in his Cambridge shutter onto a glass prism. The emerging beam wasn't a white disc shifted sideways — it was a spectrum: a stretched band of red, orange, yellow, green, blue, violet, fanning out by roughly 2°. Prisms had been known for a century; the received opinion, going back to Aristotle, was that the prism modified white light — coloured it by some staining action from the glass.

Newton settled that with a second prism. He let the first prism's spectrum fall on a board with a small hole. The hole isolated one colour — a narrow band of red. He passed that monochromatic beam through a second prism. Nothing changed. Red stayed red. A second prism cannot split red further, because there is nothing inside red to split. So:

The second-prism move is the experimental backbone of every dispersion argument since. It is also the first decisive demonstration that a beam of light is an ensemble of modes that linear optics can disentangle — the same move, dressed up, is how a diffraction grating works and how Fraunhofer read the absorption lines of the Sun in 1814.

Augustin Cauchy, 1836, wanted a formula. He noticed that across the visible — well away from the ultraviolet resonance — n(λ) is well-approximated by a short power series in 1/λ²:

Two terms are plenty for most optical glasses — C vanishes at the precision of handbook tables, and even for steep flints the cubic term is a small correction. The coefficients are pure fits: A is the asymptotic index at infinite wavelength, B encodes the curvature, and they are tabulated in every glass catalogue to four decimal places. Schott's N-BK7 crown glass, for example, lives at (A, B) = (1.5046, 4.2×10⁻³ µm²) — plug in λ = 589 nm (the sodium D line, the classical reference) and you get n = 1.517, exactly the handbook value. That is the calculation the prism scene above runs, once per wavelength.

To talk about how strongly a glass disperses without the two-coefficient chatter, opticians use a single scalar — Ernst Abbe's number:

The three reference wavelengths are Fraunhofer's d-line (helium, 587.6 nm), F-line (hydrogen, 486.1 nm), and C-line (hydrogen, 656.3 nm). The numerator is the index's departure from vacuum; the denominator is the index's spread across the visible. Higher V = weaker dispersion. Crowns cluster near V ≈ 60; dense flints near V ≈ 25. A lens designer needs both numbers to pick a glass: n for the focusing power, V for the colour discipline.

The families are chemistry: crowns are silicate plus alkali (K, Na) plus B₂O₃; flints add lead oxide, which raises n but drops V because the heavy Pb atoms put their resonances closer to the visible. Lanthanum glasses swap in La₂O₃ and sit at high n with surprisingly high V — focusing power without the chromatic penalty of lead. Fluorite (CaF₂) is the aristocrat: V ≈ 95, vanishing dispersion, which is why high-end telephoto lenses list "one element of fluorite" in their specs at eye-watering markup.

An achromatic doublet pairs a positive crown with a negative flint so the focal lengths add but first-order chromatic aberration cancels. The Dollond condition (1758): f_crown · V_flint = −f_flint · V_crown — two Abbe numbers glued together by a sign.

The clearest confirmation of all of this is overhead, roughly once per rain shower. A rainbow is a dispersion experiment at cloud scale: millions of ~1 mm spheres of water, each one acting as a miniature prism that also reflects internally. The geometry was solved by René Descartes in 1637 and polished by Newton.

Trace a ray into a spherical drop. It refracts at the front surface (air → water), reflects once off the back, and refracts out the front again (water → air). The total deviation — the angle between the outgoing and incoming rays — is a function of the impact parameter b (how far off-centre the ray enters). Descartes computed D(b) and found it has a minimum at cos²θ_i = (n² − 1)/3, where θ_i is the angle of incidence at the first surface. At that minimum, drops at many different impacts all send light into the same exit angle — a caustic, a piling-up of rays that makes the bow bright only near one specific angle from the antisolar point.

For n = 1.333 (red), that angle is ≈ 42.4°. For n = 1.344 (violet), it is ≈ 40.5°. Red ends up on the outside of the arc; violet on the inside. That is the primary rainbow.

The secondary rainbow, visible on perhaps one bow in three, comes from rays that reflect twice off the back before exiting. An extra bounce changes the Descartes formula to cos²θ_i = (n² − 1)/8, which gives ≈ 51° for water. The colour order is reversed — red on the inside, violet on the outside — because the second bounce flips the sorting. The dark gap between the two bows is Alexander's dark band: no geometry sends primary or secondary rays into it.

This is the chromatic specialisation of a much more general idea. In classical mechanics §23 — dispersion and group velocity you met dispersion as the statement that ω(k) is not linear: a pulse spreads because its Fourier components travel at different speeds. Optical dispersion is exactly that, with the wavelength dependence packed into n(λ) and reflected into ω(k) via k = n(λ)·ω/c. Near an absorption band, anomalous dispersion can briefly send the phase velocity above c — not a relativity violation, because the signal velocity stays ≤ c.

Downstream: achromats and apochromats in §09.6 geometric optics, and the fact that fibre-optic cable budgets include a "chromatic dispersion" line-item because a pulse that starts Gaussian at the transmitter arrives smeared at the receiver after 100 km of silica. A prism separates colours because n depends on λ; a lens softens an image because the same does; a rainbow paints the sky because a raindrop is just a sphere of material that won't treat red and violet the same way.