Dispersion of light

Why colours separate. White light — sunlight, or light from a filament bulb — is a mixture of wavelengths from about $400\ \text{nm}$ (violet) to $700\ \text{nm}$ (red). A transparent medium such as glass does not have one single refractive index; it has a slightly different $n$ for each wavelength, always a little higher for shorter wavelengths in the visible range. This is normal dispersion:

where: $n_{\text{violet}}$ and $n_{\text{red}}$ are the refractive indices of the prism glass for violet and red light respectively. Because Snell's law uses a different $n$ for each colour, each colour refracts by a different amount at both faces of the prism, and a single incident white ray leaves as a fan of separated colours ordered violet-to-red by increasing wavelength — the familiar VIBGYOR spectrum.

Angular dispersion. The spread of the spectrum is measured by the angle between the two extreme emergent rays (violet and red), for the same angle of incidence on the prism:

where: $\theta$ is the angular dispersion, and $\delta_V$, $\delta_R$ are the angles of deviation of violet and red light through the prism. A bigger $\theta$ means a more spread-out, easier-to-resolve spectrum. Two things push $\theta$ up: a larger prism (refracting) angle $A$, and a glass whose refractive index changes more strongly with wavelength — its dispersive power. Flint glass, for instance, disperses light more than crown glass of a similar mean refractive index, which is why flint-glass prisms are preferred in spectrometers.

Mean deviation and the "yellow standard". Since $\delta$ varies continuously from violet to red, physicists quote a single representative value, the mean deviation, taken near the middle of the visible range — conventionally the sodium D-line (yellow, $589\ \text{nm}$). A glass's quoted refractive index ("the" $n$ of a glass, without further qualification) is really $n$ for yellow light for exactly this reason.

Dispersion has no long derivation on the ISC syllabus, but the single relation used above — bigger prism angle and higher dispersive power give more spread — is easy to justify with the small-angle (thin) prism deviation formula. For a thin prism of small angle $A$, the deviation for a wavelength of refractive index $n$ is approximately $\delta \approx (n - 1)A$, obtained the same way as the ordinary minimum-deviation prism formula but under the thin-prism approximation. Applying this separately to violet and red light with their own indices $n_V$ and $n_R$:

So the angular dispersion is, to this approximation,

where: $n_V - n_R$ is the difference in refractive index between violet and red (the glass's dispersive strength), and $A$ is the prism angle. This single line explains both qualitative rules quoted in the syllabus: doubling $A$ doubles $\theta$ for the same glass, and swapping in a glass with a larger $n_V - n_R$ (a "more dispersive" glass, such as flint over crown) spreads the spectrum further for the same $A$ — without needing $\theta = \delta_V - \delta_R$ to be evaluated from full Snell's-law deviation at each face.

Rainbows, briefly. The same wavelength-dependence of $n$ operates inside a spherical raindrop. Sunlight entering a drop refracts (and disperses), undergoes one total internal reflection off the back inner surface, and refracts again on exiting — two dispersive refractions plus a reflection send different colours out at slightly different angles, so an observer sees red from higher drops and violet from lower ones, painting the familiar arc. A primary rainbow uses one internal reflection; a fainter secondary rainbow, seen higher up with the colour order reversed, uses two. No raindrop-geometry calculation is required at this level — only this refraction-reflection-refraction mechanism.

A thin glass prism has angle $A = 5.0°$. For its glass, $n_V = 1.523$ for violet light and $n_R = 1.513$ for red light. Find the angular dispersion produced by this prism, and compare it with the earlier deviation-based example where $\theta = 2°$ for a prism whose deviations were measured directly.

Step 1. Identify the quantities needed for $\theta \approx (n_V - n_R)A$: the refractive-index difference $n_V - n_R = 1.523 - 1.513 = 0.010$, and the prism angle $A = 5.0°$.

Step 2. Substitute into the thin-prism angular dispersion relation. $\theta \approx (n_V - n_R)A = 0.010 \times 5.0° = 0.050°$.

Step 3. Compare orders of magnitude. This thin-prism value, $0.050°$, is far smaller than the $2°$ angular dispersion quoted earlier for a prism measured at near-minimum deviation with a much larger effective angle and non-thin geometry. The thin-prism formula is an approximation for small $A$; real spectrometer prisms with $A$ around $60°$ and measured at minimum deviation give the larger, exam-quoted values of $\theta$ (like the earlier $2°$ example), not the small-angle estimate.

Answer: for this thin prism, $\theta \approx 0.050°$; the two examples are consistent in trend (more dispersive glass or larger $A$ gives larger $\theta$) even though their absolute values differ because one uses the thin-prism approximation and the other uses full measured deviations.

• Thinking colours differ only in speed, not deviation. It is the wavelength-dependent refractive index — and hence the deviation angle at the prism faces — that separates colours, not their speed considered alone.
• Confusing dispersion with ordinary refraction. A single wavelength bending through a prism is refraction, not dispersion; dispersion specifically means a mixture of wavelengths separating because each has its own $n$.
• Believing red bends more than violet. For ordinary (normal dispersion) glass in the visible range, $n_{\text{violet}} > n_{\text{red}}$, so violet always deviates most — this fixes the VIBGYOR order at the emergent face.
• Applying $\theta \approx (n_V - n_R)A$ to a large-angle prism at minimum deviation. That formula assumes a thin prism (small $A$); for typical spectrometer prisms, compute $\delta_V$ and $\delta_R$ separately from the full prism formula and subtract, as in the worked example's comparison.

Ask the tutor: curious why flint glass disperses more than crown glass at a similar refractive index, or want to combine angular dispersion with the full prism deviation formula for a large-angle prism?