25.5 Dispersion: The Rainbow and Prisms

Dispersion and Rainbows

Dispersion of white light

White light is a combination of all visible wavelengths. Dispersion is the process of separating white light into its component colors by passing it through a material like a prism or water droplet.

Different wavelengths travel at different speeds inside a medium. Because of this, each color bends by a slightly different amount when it enters or exits the material. Shorter wavelengths (violet, blue) slow down more and refract at steeper angles, while longer wavelengths (red, orange) refract less.

The separated colors form a continuous spectrum in the order ROYGBIV: red, orange, yellow, green, blue, indigo, violet. Red sits at one end with the longest visible wavelength, and violet sits at the other with the shortest.

Angular dispersion refers to how much the refraction angles differ between the longest and shortest wavelengths. A material with high angular dispersion spreads the colors apart more dramatically.

Wavelength and refraction relationship

Refraction occurs when light crosses a boundary between two materials with different optical densities. The refractive index ($n$) of a material tells you how much light slows down in that material compared to its speed in a vacuum. A higher refractive index means the light travels more slowly.

The key point for dispersion: a material's refractive index is not the same for every wavelength. Violet light typically has a slightly higher refractive index than red light in the same material, which is why violet bends more.

Snell's Law describes how much light bends at a boundary:

$n_1 \sin \theta_1 = n_2 \sin \theta_2$

• $n_1$ and $n_2$ are the refractive indices of the first and second media
• $\theta_1$ is the angle of incidence (measured from the normal to the surface)
• $\theta_2$ is the angle of refraction

Since $n_2$ is slightly different for each wavelength, $\theta_2$ comes out slightly different for each color. That small difference in angle is what produces the spread of colors you see after a prism.

Total internal reflection is a related concept: when light inside a medium hits the boundary at an angle greater than the critical angle, it reflects entirely back into the medium instead of passing through. This plays a role inside rainbow-forming water droplets.

Formation of rainbows

Rainbows form when sunlight interacts with water droplets suspended in the atmosphere. The process combines refraction, dispersion, and internal reflection in a single droplet:

1. Sunlight enters the front surface of a water droplet and refracts. Because the refractive index of water varies with wavelength, the colors begin to separate (dispersion).
2. The separated light travels to the back surface of the droplet and reflects off it (internal reflection).
3. The reflected light travels back to the front surface and refracts again as it exits, spreading the colors even further apart.

The angle between the incoming sunlight and the light reaching your eyes determines where you see the rainbow. For a primary rainbow, this angle is about 42° from the anti-solar point (the point directly opposite the sun from your perspective). Red appears on the outer edge and violet on the inner edge.

A secondary rainbow sometimes appears at about 51° from the anti-solar point. It forms from light that reflects twice inside each droplet before exiting. That extra reflection makes it fainter and reverses the color order, with violet on the outside and red on the inside.

Rainbow brightness and clarity depend on the size and uniformity of the water droplets, as well as your position relative to the sun and the rain.

Additional optical phenomena

• Chromatic aberration is an unwanted effect in lenses caused by dispersion. Because a lens refracts different wavelengths by different amounts, the colors don't all focus at the same point, producing color fringing in images.
• Diffraction gratings separate light into colors using a different mechanism than prisms. Closely spaced grooves cause different wavelengths to constructively interfere at different angles, producing a spectrum.
• Polarization restricts light waves to oscillate in a single plane. While not directly related to dispersion, polarizing filters are sometimes used alongside prisms and gratings in optical instruments.