Angular Dispersion

When white light passes through a glass prism, it fans out into a rainbow of colours. This spreading is called dispersion, and the precise measure of how much different colours separate from one another is called angular dispersion. At its heart, angular dispersion describes how strongly the angle of refraction — and therefore the angle of deviation from the original path — depends on the wavelength of light. Different wavelengths are bent by different amounts because the refractive index of a material is not the same for all wavelengths, a property known as chromatic dispersion. Think of a prism as a wavelength-sorting machine. Violet light (around 400 nm) travels more slowly in glass than red light (around 700 nm), so violet is refracted more strongly. The angular dispersion between any two wavelengths is simply the difference in their deviation angles. Quantifying this tells engineers how well a prism or diffraction grating separates colours, which is essential for designing spectrometers, rainbow cameras, and even correcting the colour fringing (chromatic aberration) that blurs images in camera lenses. Angular dispersion is not just a curiosity of glass prisms. It appears wherever light interacts with a dispersive medium or a periodic structure: in diffraction gratings, raindrops producing rainbows, and optical fibres carrying internet data. Mastering this concept links together Snell's law, the wavelength-dependence of refractive index, and practical instrument design, making it a cornerstone of optics.

Key takeaways

Refractive index depends on wavelength
Deviation by a prism
Defining angular dispersion
Dispersive power
Angular dispersion in diffraction gratings

Refractive index depends on wavelength

The refractive index n of a material is defined as n = c/v, where c is the speed of light in vacuum and v is the speed in the material. For most transparent materials (glass, water), shorter wavelengths experience a larger refractive index than longer wavelengths. This is called normal dispersion. The Cauchy relation n(λ) ≈ A + B/λ² is a simple empirical description: as wavelength λ decreases, n increases, so violet is bent more than red.

Deviation by a prism

For a thin prism of apex angle A (in radians, kept small), the angle of deviation δ for a ray is approximately δ ≈ (n − 1)A. Because n is larger for violet than red, δ is larger for violet. The absolute deviation of each colour depends on its own refractive index, so every wavelength exits the prism at a slightly different angle.

Defining angular dispersion

Angular dispersion dδ/dλ is the rate of change of deviation angle with wavelength. For a thin prism: dδ/dλ = (dn/dλ) · A. The quantity dn/dλ is the dispersion of the glass — how rapidly the refractive index changes with wavelength. A steeply sloped n(λ) curve (high dispersion glass, such as flint glass) gives large angular dispersion; a flat curve (like crown glass) gives small angular dispersion. The apex angle A acts as a simple multiplier.

Dispersive power

To compare materials fairly, physicists use the dispersive power ω = (n_V − n_R)/(n_Y − 1), where subscripts V, R, Y refer to violet, red, and yellow (mean) wavelengths respectively. A high dispersive power means the material separates colours strongly relative to its overall bending ability. The reciprocal of dispersive power is the Abbe number V = (n_Y − 1)/(n_V − n_R), widely tabulated for optical glasses. Flint glass has a low Abbe number (high dispersion); crown glass has a high Abbe number (low dispersion).

Angular dispersion in diffraction gratings

For a diffraction grating with slit spacing d, the grating equation is d sin θ = mλ, where m is the order of diffraction. Differentiating with respect to λ gives angular dispersion dθ/dλ = m/(d cos θ). This shows that higher orders m and finer gratings (smaller d) give greater angular dispersion, allowing finer wavelength resolution. This is why high-resolution spectrometers use fine gratings rather than prisms.

The Key Equation

\frac{d\delta}{d\lambda} = \frac{dn}{d\lambda} \cdot A

d\delta/d\lambda — Angular dispersion (rad m⁻¹): Rate at which the deviation angle changes per unit change in wavelength.
dn/d\lambda — Chromatic dispersion of the material (m⁻¹): How rapidly the refractive index n changes with wavelength λ; negative for normal dispersion (n decreases as λ increases).
A — Apex angle of prism (rad): The angle at the tip of the prism through which light is refracted; must be small for the thin-prism approximation to hold.

Worked Example

Problem. A thin glass prism has an apex angle of 6° (0.1047 rad). For the glass used, the refractive index is n = 1.650 for violet light (λ = 400 nm) and n = 1.610 for red light (λ = 700 nm). Calculate: (a) the deviation of violet and red rays, and (b) the angular dispersion between violet and red.

\Delta\delta = \delta_V - \delta_R = 0.0042 \text{ rad} \approx 0.24°

Answer. Angular dispersion between violet and red = 0.0042 rad (≈ 0.24°); dispersion coefficient ≈ 1.40 × 10⁴ rad m⁻¹.

Figure

Diagram

Origins · 1 minIsaac Newton and how this was discovered

Isaac Newton

1643–1727 · English

Newton performed the definitive prism experiments (c. 1666–1672) that proved white light is a mixture of colours and that a prism does not colour light but merely separates wavelengths already present. His careful measurements of deviation angles for different colours established the empirical foundation for angular dispersion and led to his corpuscular theory of light and colour, published in Opticks (1704). Newton initially kept his prism experiments secret for years, fearing ridicule. When he finally published, they triggered one of the most heated scientific controversies of the 17th century — Robert Hooke accused him of plagiarism, sparking a feud that lasted until Hooke's death.

Refraction and Snell's Law — Angular dispersion arises because Snell's law applies differently to each wavelength; understanding refraction is the essential prerequisite.
Refractive Index — The wavelength-dependence of refractive index (n varies with λ) is the direct physical cause of dispersion and angular dispersion.
Diffraction Gratings — Gratings produce angular dispersion through diffraction; their dispersion formula dθ/dλ = m/(d cos θ) is a direct analogue of prism angular dispersion.
Chromatic Aberration — Chromatic aberration in lenses is caused by angular dispersion; achromatic doublets are designed by balancing dispersive powers of two glasses.
Total Internal Reflection — The critical angle also depends on wavelength via the refractive index, linking total internal reflection to the same dispersion physics.