Dispersion and Scattering of Light

Dispersion of light occurs when white light is separated into its different constituent colors because of refraction and Snell's law. White light only appears white because it is composed of every color on the visible spectrum. Although they are very close, the index of refraction for each color is unique in non-vacuous materials. These unique indices cause each wavelength to follow a different path.

White light enters a prism on the left, then is separated according to wavelength into a rainbow pattern. ^[1]

Dispersion of light is defined as follows:

Dispersion of light is the splitting of white light into its constituent colors due to the refractive index of the surface and the wavelength of the light.

Dispersion of light through a glass prism

A glass prism is used to disperse white light. The prism is a 5-faced solid, having two triangular bases and three rectangular surfaces that are inclined toward each other.

Light is sent through one of the rectangular faces, which enters the prism and exits through one of the other rectangular faces. Since different colors of light travel at different speeds, the refractive index is different for each color. As a result, when white light passes through the refracting surface of the prism, its components bend into different angles, causing the single beam of light to separate. Then, the different colors of light bend again because of the refraction caused by the second rectangular surface.

In this way, white light gets split into its component colors upon passing through a glass prism.

Polychromatic light enters a rectangular glass prism of thickness \(t=2\text{ cm}.\) If the light strikes with incident angle \(30^\circ\) in air, what is the lateral separation of violet light (\(n = 1.52\)) and red light (\(n = 1.51?\))

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Angle of refraction within the prism

Use Snell's law to find the refraction angles. For both colors, \(n_1 = 1\) and \(\theta_1 = 30^\circ\)

\[n_1\sin\theta_1 = n_2\sin\theta_2\]

\[\theta_2 = \sin^{-1}(\frac{n_1\sin\theta_1}{n_2})\]

Violet \[\theta_2 = 19.205^\circ\]

Red \[\theta_2 = 19.337^\circ\]

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Lateral displacement within the material

Each beam will travel a path diagonally through the prism. The angle of refraction forms a right triangle with the thickness and lateral displacement of the light ray.

\[\tan\theta_2 = \frac{\text{lateral displacement}}{\text{thickness}} = \frac{\ell}{t}\]

\[\ell = t \tan\theta_2\]

Violet \[\ell = (0.02 \text{ m}) \tan(19.205) = 6.97\text{ mm}\]

Red \[\ell = (0.02 \text{ m}) \tan(19.337) = 7.02\text{ mm}\]

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Lateral separation is the difference between the lateral displacements.

\[\Delta \ell = 7.02\text{ mm} - 6.97\text{ mm} = 0.05\text{ mm} = 50\text{ }\mu\text{m}\]

Plenty of seemingly mysterious natural phenomena are explained by the dispersion and scattering of light.

Rainbow formation

The formation of a rainbow is linked to the dispersion of light. Since it has a striking similarity to the dispersion of light in a prism, water droplets are sometimes called mini prisms.

A typical Rainbow formation.[2]

Water droplets are roughly spherical in nature and contain water with a refractive index that enables light to refract. When sunlight (white light) strikes water droplets suspended in air, it refracts and spreads into its constituent colors through dispersion. When sunlight touches a water droplet (at some particular angle) the light gets refracted and dispersed. Later, the refracted light undergoes total internal reflection, which causes the light rays to fall on the front side of the droplet and emerge from the back.

The path of the sunlight inside a water droplet.[3]

The rainbow pattern is made up of seven colors in a specific order. This is because the wavelength of red light is higher so it deviates the least, while the wavelength of violet is lower and deviates the most. This is why the red light is at the bottom and the violet light is at the top.

Color of the sky at different times

Sunlight reaches the earth's atmosphere and is scattered by gases and particles in the air. Blue (and violet) light is dispersed more broadly than most other colors because it travels as shorter, smaller waves. This is why the sky often appears blue.[4]

Color of the sun

As light journeys from the sun to the earth's atmosphere, violet, indigo, blue, and green lights of the spectrum get scattered because air particles increase in diameter nearer to the earth's surface. The next spectral color in terms of shortest wavelength, yellow, scatters closest to eye level, causing it to override the other spectral colors. As a result, the sun appears yellow.

Color of smoke in winter

The smoke coming from chimneys scatters the blue light the most, so it overrides the other spectral colors and the smoke appears blue.

Use of ultramarine

Ultramarine is a fluorescent substance which absorbs ultraviolet radiation from sunlight and converts it into visible light seen as violet, indigo, and blue spectral colors. Sunlight is deficient in these colors, having been scattered in the upper atmosphere. So, when sunlight falls on clothes soaked in ultramarine, the deficient sunlight again contains all the spectral colors in equal proportion on account of fluorescence. As a result, our brain perceives it as white.

[1] Image from https://en.wikipedia.org/wiki/Prism#/media/File:Prism-side-fs_PNr%C2%B00117.jpg under the creative commons attribution for reuse and modification.

[2] D-Kuru from Wikimedia Commons, Image from https://upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Double-alaskan-rainbow.jpg/400px-Double-alaskan-rainbow.jpg under the creative commons attribution for reuse and modification.

[3] Image from https://en.wikipedia.org/wiki/Rainbow#/media/File:Rainbow1.svg released under the public domain.

[4] Why is the sky blue? NASA. Retrieved 17:18, April 7, 2016, from http://spaceplace.nasa.gov/blue-sky/en/.

1. Unknown, . Dispersion prism. Retrieved May 11, 2016, from https://upload.wikimedia.org/wikipedia/commons/6/63/Dispersion_prism.jpg