Prism and Dispersion

Newton was twenty-two. The plague had closed Cambridge, and he had gone home to Woolsthorpe, where he did not have much to do besides think. He got himself a cheap triangular glass prism — the sort sold at Sturbridge Fair — and held it up to a ray of sunlight coming through a hole in his shutter. On the far wall, a narrow band of colours appeared: red at one end, through orange, yellow, green, blue, indigo, to violet at the other. Everyone before him had seen such bands and shrugged — the colours, they thought, were some kind of mixture the glass added to the beam. Newton wanted to know where they came from.

His critical experiment, two years later, was the experimentum crucis: he put a second prism in the path of the coloured band, arranged to recombine it. Out came white light, unchanged. He had shown that the colours were already there in the sunlight, and the prism had only spread them out because each colour refracted by a different angle. White light is not a colour; it is all the colours at once. A prism is not an adder; it is a separator.

Every rainbow you have ever seen over the Nilgiris after a May shower, every peacock-coloured sheen of spilled petrol on a Kolkata street, every chromatic haze in the old telescope at the Kodaikanal Solar Observatory, every "VIBGYOR" song from primary school — it all goes back to that single insight. This chapter builds Newton's understanding from Snell's law: how a prism deviates light, why the deviation is a minimum when the path is symmetric, how to measure a glass's refractive index from that minimum, and how varying index with wavelength produces the rainbow.

Geometry of a prism — the two refractions

A prism is a transparent solid with a triangular cross-section. Sunlight enters one face, the glass bends it, it crosses the body of the prism, hits the second face, and bends again as it exits. The angle between the two refracting faces (the third face is the base, usually ignored) is called the apex angle or refracting angle, A.

Deriving the deviation formula

Four angles are in play inside the geometry: i_1, r_1, r_2, i_2. The first and last are outside the glass (air side); the middle two are inside.

Step 1. At the first face, Snell's law gives

(taking n_{\text{air}} = 1).

Step 2. At the second face, Snell's law gives

Step 3. Look at the triangle formed by P_1, P_2, and the apex. The angles at P_1 and P_2 (inside the prism) are (90° - r_1) and (90° - r_2), respectively — because r_1 and r_2 are measured from the normals, and the normals are perpendicular to the faces.

The sum of angles in the apex triangle is 180°, so

Why: the apex angle A of the prism equals the sum of the refraction angles inside. This little geometric fact is the reason the deviation formula works out so cleanly — it ties the four angles into a single equation.

Step 4. Now look at the deviation. At the first face, the ray turns from direction along i_1 (outside) to along r_1 (inside), changing direction by (i_1 - r_1). At the second face, it turns from r_2 (inside) to i_2 (outside), changing by (i_2 - r_2).

The total change in direction — the deviation — is

Why: each face contributes to the total turn by the amount the ray bends at that face. The sum of the two bends is the total deviation, and using the result from Step 3 that r_1 + r_2 = A, the formula collapses to one line involving only the external angles and the apex.

This is the foundational equation for prism optics. Every subsequent result — minimum deviation, refractive index from minimum deviation, the dispersion of white light — flows from here.

Minimum deviation

Experimentally, as you rotate the prism about a vertical axis (or equivalently, change i_1 while keeping the prism fixed), the deviation \delta varies. A graph of \delta versus i_1 looks like a shallow U — it drops, reaches a minimum, and rises again. The minimum value \delta_m occurs at a specific angle of incidence, and at that special angle, i_1 = i_2 and r_1 = r_2 = A/2.

Why minimum deviation occurs at symmetric passage

Claim: \delta is minimum when i_1 = i_2.

Proof by symmetry. Start with a ray that passes through the prism with some i_1 and emerges at some i_2, giving a deviation \delta. By the time-reversal symmetry of light (Fermat's principle: light paths are reversible), reverse the ray. It now enters at i_2 on the right face and exits at i_1 on the left face, giving the same deviation \delta.

So the function \delta(i_1) has the property that the same \delta is achieved at two values of i_1 — one with i_2 > i_1, one with i_2 < i_1. The only place these two values can coincide is at i_1 = i_2, which must therefore be the extremum. Differentiating \delta(i_1) shows the extremum is a minimum, not a maximum. So

Derivation of the refractive-index formula. At minimum deviation, i_1 = i_2 \equiv i_m and the deviation formula becomes \delta_m = 2i_m - A, so

Also r_1 = r_2 = A/2. Snell's law at the first face gives

Solving for n:

Why: this formula converts a measurable external quantity — the minimum deviation \delta_m — and a known apex angle A into the refractive index n. It is the operating equation of every classroom spectrometer and was the standard method of measuring n for over two centuries.

This is the classroom spectrometer formula. Set up a prism on a rotating table, send a narrow beam of monochromatic light (a sodium lamp, say — yellow at 589 nm) through it, rotate the prism to find the minimum deviation, measure \delta_m with an optical goniometer, and plug in. For a 60° prism with \delta_m = 37.2°:

— the refractive index of crown glass, to three decimal places, from three measured angles.

Dispersion — why white light splits

So far the refractive index n has been one number. But n depends on the wavelength (colour) of the light. This is dispersion.

How n varies with \lambda

A good approximate fit for transparent glasses in the visible spectrum is Cauchy's dispersion formula:

where A and B are empirical constants of the glass. The key fact is that the 1/\lambda^2 term is positive: shorter wavelengths (violet, blue) see a larger n than longer wavelengths (red). So a prism deviates violet more than red.

Typical crown glass data:

The refractive index changes by about 0.009 across the visible spectrum — a small number, but enough to spread white light into a spectrum several degrees wide when passed through a 60° prism.

Cauchy's formula as physics, not just fit

Why does n decrease with wavelength? The atomic electrons of the glass oscillate in response to the incoming electric field. Their natural oscillation frequencies are in the ultraviolet. For visible light, the electron's response lags the driving field by an amount that depends on how close the light's frequency is to the natural frequency. Shorter-wavelength (higher-frequency) visible light is closer to the UV resonance, so the electrons respond more and the dielectric polarisation is larger, raising n.

The full theory — the Lorentz oscillator model — gives a formula with a single pole at the resonance frequency. Expanding this formula in powers of 1/\lambda^2 recovers Cauchy's formula as the first two terms. So Cauchy is not an empirical fit — it is the low-order physics, correct to a few percent across the visible spectrum.

Saptavarna — the prism spectrum

When a beam of white sunlight enters a prism symmetrically, each wavelength exits at a slightly different angle. The result is a fan of colours:

Angular dispersion and dispersive power

For a thin prism (apex angle A \lesssim 10° — the small-angle limit), the deviation simplifies to

Why: for small A, \sin(A/2) \approx A/2 and \sin((A+\delta_m)/2) \approx (A+\delta_m)/2. Snell gives (A+\delta_m)/2 = n(A/2), so A + \delta_m = nA, hence \delta_m = (n-1)A. For small prisms this is independent of the angle of incidence to leading order.

The angular dispersion — the angular separation between violet and red — is the difference of the two deviations:

For a thin prism made of crown glass (n_V = 1.523, n_R = 1.514, A = 10°):

So a 10° thin crown-glass prism spreads white light into a 0.09°-wide fan — a narrow rainbow, but enough to resolve sodium D lines if the light is narrow-band to begin with.

The dispersive power \omega measures how much a given medium spreads colours, relative to how much it bends the mean-colour ray:

where n_Y is the mid-visible (yellow) index. For crown glass: \omega \approx 0.009/0.517 = 0.017. For dense flint glass: \omega \approx 0.020/0.650 = 0.031. Flint glass spreads colours twice as much as crown, at the same mean deviation — useful when you want strong dispersion (spectrometers) and avoided when you don't (telescope objectives, which need all colours to focus at one point).

Achromatic prism pair — cancelling dispersion while keeping deviation

A simple trick combines crown and flint glasses: place a crown prism (low dispersion) and a flint prism (high dispersion) back-to-back, with the flint flipped, such that their dispersions cancel but their deviations partially add. The condition for zero net dispersion is

Simultaneously, the net deviation is \delta = (n-1)_{\text{crown}} A_{\text{crown}} - (n-1)_{\text{flint}} A_{\text{flint}} (the flint deviates in the opposite sense). Choose the apex angles so the first equation is satisfied; the second gives a non-zero residual deviation. Result: all colours bend by the same angle — the prism deviates the light without splitting it.

This is the same trick used in achromatic doublet lenses (the crown-flint pair glued together in every decent telescope and binocular objective). The reason the Refractor at the Kodaikanal Solar Observatory can image the Sun in crisp white light is that its objective is an achromatic doublet — two lenses of different glasses, shaped so their dispersions cancel.

Worked examples

Common confusions

• "The prism adds the colours to the light." Newton's experiment disproved this over three centuries ago. The colours are already in the white light; the prism only separates them. Recombine the colours and you get white light back — which is why a second prism (inverted relative to the first) cancels the dispersion.

• "Red bends more than blue because red has more energy." The opposite. Blue/violet bends more because its refractive index is larger — and its wavelength is smaller, closer to the UV resonance of the glass's electrons. Photon energy is proportional to frequency, so blue photons do have more energy — but that is a consequence, not the cause, of the refractive-index dispersion.

• "Minimum deviation means the light barely bends." No — minimum deviation is the smallest deviation for that prism at that wavelength, but it is still typically 30°–50° for a 60° prism. "Minimum" is relative to other orientations of the prism, not to zero.

• "A rainbow is made by a prism." A rainbow is made by water droplets in the air, each of which acts as a tiny combination of refraction and total internal reflection. The droplet is nearly spherical; light enters it, refracts, reflects once off the back, and refracts again on exit. The dispersion happens at the two refractions, and the specific geometry of a sphere picks out the primary rainbow angle at about 42° from the antisolar point.

• "All prisms disperse the same way." No. The wavelength-dependence of n is a property of the specific glass. Crown glass disperses less than flint. Fused silica disperses less than crown. Certain synthetic materials (extra-low-dispersion or ED glass) disperse much less still, and are used in expensive camera lenses precisely because they minimise chromatic aberration.

• "You need sunlight to see a rainbow." You need any broadband light source and water droplets or a prism. A lamp post in fog can produce a faint rainbow (though asymmetric); a garden sprinkler in afternoon sun will produce a full rainbow if you stand with your back to the sun.

Where this leads next

• Refraction of Light — Snell's Law — the foundation on which all prism geometry is built. Every prism result is two applications of Snell's law and one triangle equation.
• Total Internal Reflection — when the ray inside the prism hits a face at an angle beyond the critical angle, it reflects instead of refracting. This is the physics of prism periscopes and the Porro prisms inside binoculars.
• Thin Lenses — a lens can be thought of as a smoothly varying prism. The chromatic aberration of a simple lens is the dispersion of a prism applied at every point of the lens surface.
• The Electromagnetic Spectrum — prism dispersion in the visible is the bridge from the general concept of different wavelengths (all the bands) to the specific seven-colour spectrum your eye can detect.
• Reflection of Light — the partner law that governs the reflection part of the rainbow derivation inside a raindrop.