Prisms, Visible Spectrum, and Dispersion

Dispersion is the separation of white light into its component colors because the refractive index of a material depends on wavelength; a glass prism refracts violet light more than red, producing the visible spectrum in the order red, orange, yellow, green, blue, indigo, violet (ROYGBIV). In brief: in glass, blue and violet light travel more slowly than red, so they are bent more. The visible spectrum covers wavelengths from about 380 nm (violet) to 750 nm (red). A rainbow is formed by refraction, internal reflection and dispersion in water droplets. A second prism can recombine the colors into white light.

Rainbows, the colored fringes at the edge of cheap lenses and the beam of colors from a CD all involve dispersion. This guide explains how prisms deviate and disperse light, describes the visible spectrum and the rainbow, compares colors in a table, works through deviation calculations and corrects common mistakes.

How does a prism bend light?

A prism is a transparent block with two plane faces inclined to each other at the apex angle A, usually 60° for a triangular glass prism. A ray bends toward the normal when it enters the glass and away from the normal when it leaves through the second face. Since the faces are not parallel, the two bendings add up, and the ray is deviated toward the base of the prism by the angle of deviation δ. The deviation is smallest, the angle of minimum deviation δ_min, when the ray passes symmetrically through the prism. Then n = sin[(A + δ_min) ÷ 2] ÷ sin(A ÷ 2). For a thin prism with small angles, δ ≈ (n − 1)A.

What is dispersion?

In a transparent material, the refractive index is a little different for each wavelength (normal dispersion): n increases as the wavelength decreases. White light is a mixture of all the colors. In a prism, each color is deviated by a different angle, with violet bent most and red bent least, and the beam spreads into a spectrum. In a vacuum, all colors travel at the same speed, so there is no dispersion. Dispersion is often a nuisance, leading to chromatic aberration in simple lenses (colored fringes), which is reduced by combining lenses of different glasses (achromatic doublets). Dispersion is also used in spectroscopes to analyze light from stars and lamps, as each element emits or absorbs light at characteristic wavelengths.

What is the visible spectrum?

Visible light is the small part of the electromagnetic spectrum that the eye can detect: wavelengths of roughly 380 to 750 nm, frequencies of about 4 × 10¹⁴ Hz to 8 × 10¹⁴ Hz. Beyond the violet end is the ultraviolet, and beyond the red is the infrared, which Herschel found in 1800 using a thermometer. The order of the colors is red, orange, yellow, green, blue, violet (Newton named indigo too, to make seven colors, like the seven notes of the musical scale). The colors of the spectrum are monochromatic: they cannot be split further by a second prism. An object’s color is determined by which wavelengths it reflects: a red apple reflects red light and absorbs the others.

How does a rainbow form?

Sunlight enters a raindrop, is refracted and dispersed, is reflected once from the back of the drop, and is refracted again as it leaves. The returning light is concentrated near an angle of about 42° from the direction opposite to the Sun for red, and about 40° for violet. An observer sees a circular arc, with red on the outside and violet on the inside, when the Sun is behind them and the rain in front. A fainter secondary bow, outside the primary, is formed by two internal reflections, with the colors in reverse order.

Colors of the spectrum compared

Worked example: deviation of red and violet light

A glass prism has an apex angle of 60°. For the glass, the refractive index is 1.514 for red light (656 nm) and 1.530 for violet light (405 nm). These are typical values for a borosilicate crown glass.

1. Minimum deviation for red: sin[(60° + δ) ÷ 2] = n sin 30° = 1.514 × 0.500 = 0.757. So (60° + δ) ÷ 2 = 49.2°, giving δ_red = 38.4°.
2. Minimum deviation for violet: sin[(60° + δ) ÷ 2] = 1.530 × 0.500 = 0.765. So (60° + δ) ÷ 2 = 49.9°, giving δ_violet = 39.8°.
3. Angular spread of the spectrum: 39.8° − 38.4° = 1.4°. At a screen 2.0 m away, the width of the spectrum is about 2.0 × tan(1.4°) = 2.0 × 0.0244 = 0.049 m, about 5 cm.
4. Thin prism approximation (A = 10°): δ_red = (1.514 − 1) × 10° = 5.14°, δ_violet = (1.530 − 1) × 10° = 5.30°; the spread is 0.16°.
5. Speeds in glass: v = c ÷ n. For red: 3.00 × 10⁸ ÷ 1.514 = 1.98 × 10⁸ m/s; for violet: 3.00 × 10⁸ ÷ 1.530 = 1.96 × 10⁸ m/s, so violet light is slower.

How was the spectrum discovered?

Many people had seen colors produced by prisms, but it was thought that the glass added the colors to the light. In 1666 to 1672, Isaac Newton showed, in his experiments with two prisms (the experimentum crucis), that white light is a mixture of colors: a second prism did not change a single color, and a prism and lens could recombine the colors into white. Joseph von Fraunhofer discovered dark lines in the solar spectrum in 1814, and Gustav Kirchhoff and Robert Bunsen showed in 1859 that these lines identify chemical elements, the beginning of spectroscopy.

Which mistakes are common?

Mistake 1: Thinking that the prism adds colors to the light

It separates the colors that are already present in white light.

Mistake 2: Saying that red light is bent the most

Violet light is bent the most in a normal prism, and red the least.

Mistake 3: Believing that colors travel at different speeds in a vacuum

Only in media. In a vacuum, all colors travel at c.

Mistake 4: Confusing the order of the rainbow colors

In a primary rainbow, red is on the outer edge and violet on the inner edge.