Dispersion by Prism

When a narrow beam of white light passes through a glass prism, it fans out into a beautiful band of colours — red, orange, yellow, green, blue, indigo, and violet — in that precise order. This separation is called dispersion, and it arises because white light is not a single colour but a mixture of many wavelengths. The prism does not create the colours; it reveals them by bending each wavelength by a slightly different amount as light enters and again as it exits the glass. The physical reason for dispersion is that the refractive index of glass (and most transparent materials) is not a fixed number — it depends on the wavelength of light. Violet light, with its shorter wavelength, is slowed more inside glass than red light, which has a longer wavelength. A higher refractive index means a greater bending angle at each surface, so violet is deviated most and red least. This wavelength-dependence of refractive index is called chromatic dispersion, and it is a fundamental property of any dispersive medium. Dispersion by a prism has enormous practical importance. It is the mechanism behind rainbows, the working principle of spectrometers used to identify chemical elements from starlight, and the source of the unwanted 'chromatic aberration' in simple lenses. Understanding dispersion quantitatively — using Snell's law applied separately to each wavelength — is the key to designing optical instruments that either exploit or correct for this effect.

Key takeaways
  • White Light as a Mixture of Wavelengths
  • Refraction at the First Surface
  • Refraction at the Second Surface
  • Cauchy's Equation and Wavelength Dependence
  • Angle of Minimum Deviation

White Light as a Mixture of Wavelengths

Visible light spans wavelengths from roughly 380 nm (violet) to 700 nm (red). White light contains all these wavelengths simultaneously. Think of it like a chord in music: a single chord contains many individual notes played at once — a prism is the instrument that separates those notes.

Refraction at the First Surface

When light strikes the first face of the prism at an angle, Snell's law (n₁ sin θ₁ = n₂ sin θ₂) applies to each wavelength independently. Because n₂ is slightly larger for violet than for red, violet bends more toward the normal inside the glass. The rays immediately begin to separate — the prism is already 'sorting' the colours inside the glass.

Refraction at the Second Surface

At the second face, each wavelength refracts again as it exits into air. Because the second surface is tilted relative to the first (this is what makes it a prism rather than a flat slab), the angular separation between colours is amplified rather than cancelled. The total deviation for each colour is the sum of the deviations at both surfaces, and the difference in deviation between violet and red is the angular dispersion of the prism.

Cauchy's Equation and Wavelength Dependence

The variation of refractive index with wavelength is captured empirically by Cauchy's equation: n(λ) = A + B/λ². Here A and B are positive constants specific to the glass, and λ is the wavelength. Because B/λ² decreases as λ increases, n is smaller for red (large λ) and larger for violet (small λ). This confirms mathematically why violet is bent more than red.

Angle of Minimum Deviation

For a prism of apex angle A, there is a special angle of incidence at which a given wavelength passes through symmetrically — entering and exiting at equal angles. At this condition the ray suffers its minimum deviation δ_min. The refractive index for that wavelength can be determined very precisely from the measured minimum deviation, making this configuration essential in laboratory spectroscopy.

The Key Equation

n=sin⁡ ⁣(A+δmin⁡2)sin⁡ ⁣(A2)n = \frac{\sin\!\left(\dfrac{A + \delta_{\min}}{2}\right)}{\sin\!\left(\dfrac{A}{2}\right)}
  • n — Refractive index (dimensionless): The ratio of the speed of light in vacuum to its speed in the prism material; varies with wavelength, causing dispersion.
  • A — Apex angle (degrees or radians): The angle at the top of the prism between its two refracting faces. Typically between 30° and 60° in laboratory prisms.
  • \delta_{\min} — Minimum deviation angle (degrees or radians): The smallest possible angle by which the prism deflects a ray of a given wavelength; occurs when the ray travels symmetrically through the prism.

Worked Example

Problem. A glass prism has an apex angle A = 60°. A monochromatic yellow beam passes through at minimum deviation and the minimum deviation angle is measured to be δ_min = 40°. Calculate the refractive index of the glass for yellow light.

n=sin⁡50°sin⁡30°=0.7660.500≈1.53n = \frac{\sin 50°}{\sin 30°} = \frac{0.766}{0.500} \approx 1.53

Answer. The refractive index of the glass for yellow light is approximately 1.53, a typical value for crown glass.

Figure
Diagram
AGlass prismWhitelightRedOrangeYellowGreenBlueVioletBaseViolet bends most; red bends leastDispersion by a Triangular Prism
Origins · 1 minSir Isaac Newton and how this was discovered
SI
Sir Isaac Newton
1643–1727 · English

In 1666, Newton performed his famous 'crucial experiment' using two prisms: the first spread white light into a spectrum, and a second prism recombined the colours back into white light. This conclusively proved that white light is a mixture of colours and that the prism does not add colour to light — it merely separates wavelengths that are already present. Newton also introduced the word 'spectrum' and identified seven colours (he added indigo to match musical octaves). His work laid the foundation for the science of spectroscopy. Newton was so secretive about his prism experiments that he blocked out his room with curtains, allowing only a tiny hole for sunlight — his 'experimentum crucis' was performed in near-total darkness.

  • Refraction of Light — Dispersion is caused by refraction at each prism surface; Snell's law applied wavelength-by-wavelength is the mathematical foundation of dispersion.
  • Refractive Index — The wavelength-dependence of the refractive index (chromatic dispersion) is the direct physical cause of colour separation in a prism.
  • Wave Nature of Light — Understanding wavelength as the property that distinguishes colours is essential for explaining why different colours refract by different amounts.
  • Rainbow Formation — Rainbows are a natural example of dispersion, where water droplets act like tiny prisms, separating sunlight into a circular spectrum.
  • Spectrometer and Spectroscopy — The spectrometer uses a prism (or diffraction grating) to disperse light and measure wavelengths precisely, directly applying the minimum-deviation formula.