In optics, Snell's law of refraction tells us that light is refracted, or bent, when it passes from one medium to another.

History of Refraction

The Greek astronomer and geographer Ptolemy studied refraction and optics in about 140AD. Using experimental observations, Ptolemy worked out a law of refraction, which is approximately correct only for small angles. In 984 the Arabian optical engineer and mathematician, Ibn Sahl, also worked out a law of refraction.

The law of refraction was rediscovered in 1602 by the British astronomer and mathematician, Thomas Harriot. Despite this and other discoveries, Harriot remained obscure because he never published his results.

The law of refraction was again discovered by Willebrord Snell in 1621, but Snell died in 1626 before publishing his results. After independently discovering the law of refraction, Descartes published it in 1637.

Physicists refer to the law of refraction as Snell's law, despite the other independent discoveries.

Snell's Law

The angle through which light is refracted when it passes between two media depends on a property of each medium known as the index of refraction or refractive index. This angle also depends on the angle at which the light strikes the surface between the media.

Snell's law is a mathematical relationship between the refractive indices of the two media and the sines of the angles the light makes when travels in each media. The equation for Snell's law when light travels from medium 1 to medium 2 is:

n1 sin(theta1) = n2 sin(theta2)

Where n1 and n2 are the indices of refraction of each medium. Theta1 and theta2 are the angles the light makes when it strikes the surface between the media and when it leaves the surface, the incident and refracted angles. Physicists do not measure these angles from the surface between the two media. Rather physicists measure the angles from a line that is drawn perpendicular to the surface, which is called the normal. In mathematics the word normal means perpendicular.

When light strikes a lens, it is propagating in air as the first medium and glass as the second medium. Air has an index of refraction of 1 and glass might have an index of refraction of 1.5, depending on the type of glass. If the light strikes the lens at an angle of 30 degrees from the normal, we can use Snell's law to find the refracted angle, theta2. Doing the algebra:

sin(theta2) = sin(theta1)/n2 = sin(30)/1.5 = 0.33

theta2 = 19 degrees.

When the light passes from air to glass, it is refracted to a smaller angle from the normal. When the light leaves the lens it passes from glass to air and is refracted to a larger angle.

Index of Refraction and Speed of Light in a Medium

The refractive index of a medium is related to the speed of light in that medium. The speed of light in a vacuum is one of the fundamental constants of nature and according to Einstein's special relativity the ultimate speed limit in the universe. However when light travels in any other medium it slows down. The index of refraction in a medium is the ratio of the speed of light in a vacuum to the speed of light in that medium.

Fermat's Least Time Principle

In addition to experiments, Snell's law can be found from Fermat's least time principle. The light is refracted at an angle so that it travels from a point in the first medium to a point in the second medium in the least possible amount of time.

Further Reading

Knight, R.D., Physics for Scientists and Engineers with Modern Physics, Pearson, 2004.

Feynman, R.P., Leighton, R.B., and Sands, M. The Feynman Lectures on Physics, Addison-Wesley, 1964.

Optics & Total Internal Reflection

Reflection of Light in Optics