Prism Dispersion

How a prism makes a rainbow

White light contains every visible wavelength at once. When it meets the slanted face of a glass prism, it slows down and bends — refraction, governed by Snell's law. The trick is that glass does not bend every wavelength by the same amount: the refractive index n is slightly larger for short wavelengths (violet) than for long ones (red). That wavelength dependence of n is what we call dispersion.

A prism refracts the beam twice — once entering, once leaving — and because the two faces are not parallel, the small per-wavelength differences add up instead of cancelling. By the time the light exits, violet has been deviated by a degree or two more than red, and the colors fan out into a spectrum: red, orange, yellow, green, blue, violet.

Snell's law and the deviation angle

At each glass surface the ray obeys Snell's law:

n₁ sin θ₁ = n₂ sin θ₂

For a prism with apex angle A and refractive index n (air ≈ 1), tracing the ray through both faces gives the total deviation D. The deviation is smallest at the angle of minimum deviation D_min, reached when the ray travels symmetrically through the prism. There:

n = sin((A + D_min) / 2) / sin(A / 2)

Since n depends on wavelength, each color has its own D_min. The difference between the violet and red deviations is the angular dispersion of the prism. For a thin prism (small A), the deviation simplifies to D ≈ (n − 1)A, so the spread between two colors is roughly ΔD ≈ (n_violet − n_red)A — directly proportional to how much n changes across the visible band.

Modeling the index: Cauchy, Sellmeier, Abbe

Two empirical equations describe how n varies with wavelength λ in the transparent region:

Cauchy:    n(λ) ≈ A + B/λ² + C/λ⁴
Sellmeier: n²(λ) = 1 + Σ Bᵢ λ² / (λ² − Cᵢ)

Cauchy is a handy low-order fit; Sellmeier is physically grounded (each term is a resonance) and is the form glass catalogs publish. Both capture normal dispersion: n decreases as λ increases, so violet always bends more than red, away from any absorption band.

Lens designers compress all of this into one number, the Abbe number:

V_d = (n_d − 1) / (n_F − n_C)

where the d-line is 587.6 nm (helium), F is 486.1 nm and C is 656.3 nm (hydrogen). High V means weak dispersion; low V means strong dispersion.

Glass	n_d (587.6 nm)	Abbe V_d	Dispersion
Fused silica	1.4585	67.8	Very low
BK7 crown	1.5168	64.2	Low
Water (liquid)	1.333	~55.6	Low–moderate
SF2 flint	1.6477	33.8	High
SF11 dense flint	1.7847	25.8	Very high
Diamond	2.4175	~55.3	Huge index, modest V (intense "fire")

Real numbers for BK7

For BK7 crown glass evaluated from its Sellmeier coefficients, the index across the visible spectrum is:

Color	Wavelength	n (BK7)	Relative bending
Violet	400 nm	1.5337	Bends most
Blue	486 nm (F)	1.5224	↓
Green	546 nm	1.5187	↓
Yellow	588 nm (d)	1.5168	↓
Orange	656 nm (C)	1.5143	↓
Red	700 nm	1.5131	Bends least

The whole visible range changes n by only about 0.021 — yet that tiny variation, doubled by the prism's two faces and amplified by the apex angle, is enough to throw the colors apart by a couple of degrees and produce a fully resolved rainbow on a wall a metre away.

JavaScript — tracing a prism spectrum

// Cauchy index for BK7 (λ in micrometers): n ≈ A + B/λ² + C/λ⁴
function indexBK7(lambda_nm) {
  const L = lambda_nm / 1000;           // nm → µm
  const A = 1.5046, B = 0.0042, C = 0.00006;
  return A + B / (L * L) + C / (L * L * L * L);
}

// Deviation of a ray through a prism (apex A, ray traced both faces)
function deviation(nGlass, apexDeg, incidenceDeg) {
  const A   = apexDeg * Math.PI / 180;
  const i1  = incidenceDeg * Math.PI / 180;
  const r1  = Math.asin(Math.sin(i1) / nGlass);     // entering: air → glass
  const r2  = A - r1;                               // geometry of the prism
  const i2  = Math.asin(nGlass * Math.sin(r2));     // leaving: glass → air
  const D   = i1 + i2 - A;                          // total deviation
  return D * 180 / Math.PI;                          // degrees
}

const apex = 60, incidence = 50;
for (const [name, lam] of [['violet',400],['green',546],['red',700]]) {
  const n = indexBK7(lam);
  console.log(`${name} (${lam} nm): n=${n.toFixed(4)}, D=${deviation(n,apex,incidence).toFixed(2)}°`);
}
// violet (400 nm): n=1.5332, D≈40.10°
// green  (546 nm): n=1.5194, D≈38.88°
// red    (700 nm): n=1.5134, D≈38.36°  → ~1.7° violet-to-red spread

// Index from a measured minimum deviation (spectrometer method)
function indexFromMinDeviation(apexDeg, dMinDeg) {
  const A = apexDeg * Math.PI / 180;
  const D = dMinDeg  * Math.PI / 180;
  return Math.sin((A + D) / 2) / Math.sin(A / 2);
}
console.log(indexFromMinDeviation(60, 38.9).toFixed(4)); // ≈ 1.5197

// Abbe number from three line indices
function abbe(nF, nd, nC) { return (nd - 1) / (nF - nC); }
console.log(abbe(1.5224, 1.5168, 1.5143).toFixed(1)); // ≈ 63.8 (BK7-like)

Where prism dispersion shows up

• Spectroscopy. Prism spectrographs spread starlight or lab sources into a spectrum to read emission and absorption lines; high throughput and no overlapping diffraction orders.
• Rainbows and halos. Raindrops and ice crystals disperse sunlight; the primary rainbow peaks near 42°, the secondary near 51° with reversed color order.
• Chromatic aberration. Every simple lens is a weak prism at its edges, so it focuses violet and red at different planes — corrected by achromatic doublets combining crown and flint glass.
• Gemstone "fire." Diamond's huge index plus moderate dispersion sends flashes of separated color from a cut stone; deliberately maximized by the facet design.
• Pulse compression and fiber optics. The same wavelength-dependent index makes short laser pulses smear out in glass (group-velocity dispersion); prism pairs and chirped mirrors compensate it.
• Atmospheric optics. Air itself disperses light, so a star near the horizon shows a faint vertical spectrum — the reason astronomers use atmospheric dispersion correctors.

Prism vs. grating

	Prism (refraction)	Grating (interference)
Governing relation	Snell + n(λ)	d sin θ = mλ
Bends most	Violet (short λ)	Red (long λ), higher orders
Linearity	Nonlinear, material-dependent	Near-linear in λ
Orders	Single beam, no overlap	Multiple orders can overlap
Throughput	High (one beam)	Light split among orders
Typical use	Bright, low-resolution work	High-resolution spectrographs

Common mistakes

• Thinking the glass adds color. Newton's two-prism experiment proved a prism only separates colors already in white light; a second prism recombines them back to white.
• Assuming red bends most. In a prism, violet bends most (higher n). It's a diffraction grating that bends red most — opposite physics.
• Confusing dispersion with refraction. Refraction is the bending itself; dispersion is the wavelength dependence of that bending. A medium with constant n would refract but never disperse.
• Forgetting the apex angle and incidence both matter. The color spread scales with the apex angle and is minimized at the angle of minimum deviation, not at normal incidence.
• Using n at a single wavelength for a lens. Ignoring dispersion is what produces chromatic aberration; designers must track at least the F, d, and C lines.
• Treating air as dispersion-free. Air's index varies enough with wavelength to smear a low-altitude star into a tiny spectrum — small, but real.