Snell's law

The law says that the ratio of the sines of two specific angles—the angle at which the light hits the surface (angle of incidence) and the angle at which it bends as it enters the new material (angle of refraction)—is equal to the ratio of the refractive indices of the two materials. In simpler terms, this means that light bends more or less depending on the properties of the materials it is moving between.

Snell's law comes from a bigger idea called Fermat's principle of least time, which tells us that light always takes the path that requires the least time to travel. This helps explain why light bends when it moves between materials—it’s finding the fastest route! This law also works in special materials called meta-materials.

History

Ptolemy in Alexandria, Egypt, noticed how light changes direction when it passes from one material to another, but his findings weren’t always correct.

The law we now call Snell’s law was first discovered by the Persian scientist Ibn Sahl in Baghdad in 984. He used it to create special lens shapes that could focus light perfectly. Later, Ibn al-Haytham almost found the same rule in his book Book of Optics from 1021.

The rule was rediscovered by Thomas Harriot in 1602, but he didn’t share his findings. In 1621, the Dutch astronomer Willebrord Snellius worked out a similar version, though he also didn’t publish it. René Descartes later came up with the same idea using different reasoning in his essay La Dioptrique. Another scientist, Pierre de Fermat, found the same answer by thinking about how light travels.

In his book Geometry, Descartes used Snell’s idea to solve old math problems. Later, Christiaan Huygens explained Snell’s rule by thinking about light as waves.

With new science, Snell’s law was updated. In 2008 and 2011, special surfaces were made to change how light bends.

Explanation

Snell's law helps us understand how light bends when it moves from one material to another, like from air to water. When light goes from a place where it moves slowly to a place where it moves quickly, or the other way around, it changes direction. We measure these directions using a line called the normal, which stands straight up from the surface the light crosses.

The way light bends depends on how much slower it moves in each material. For example, light slows down in water compared to air, so it bends toward the normal when entering water. If it moves from water back to air, it bends away from the normal. This bending works both ways—if you reverse the direction of the light, it will bend the same way it did before.

Derivations and formula

Snell's law can be found in different ways.

Derivation from Fermat's principle

Snell's law comes from Fermat's principle, which says that light chooses the path that takes the least time. By studying the optical path length, we learn how light travels. Imagine a beach (with a lower refractive index) and the sea (with a higher refractive index). The quickest route for a runner on the beach to reach someone in the sea follows Snell's law.

When light moves from one material to another, the point it enters is called point O. The angle the light makes when it hits the surface is the angle of incidence (θ₁), and the angle after it passes through is the angle of refraction (θ₂).

Light travels faster in materials with a lower refractive index. Using these angles and the indexes of the materials, we can find the time it takes for light to travel. By finding the smallest possible time, we get Snell's law.

Derivation from Huygens's principle

Snell's law can also come from how light waves spread out and interact with each other.

Derivation from Maxwell's equations

Another way to understand Snell's law uses the basic rules that describe how electric and magnetic fields work.

Derivation from conservation of energy and momentum

We can also get Snell's law by thinking about how energy and motion stay the same when light moves between materials.

Vector form

We can find the direction of reflected and refracted light without using angles, just by using vectors that point in certain directions.

Total internal reflection happens when light tries to move from a material with a higher refractive index to one with a lower index at a steep angle. In these cases, the light reflects completely instead of passing through. The largest angle at which light can still pass through is the critical angle.

For example, light moving from water to air at a 50° angle cannot pass through. The critical angle for water to air is about 48.6°.

Dispersion

Main article: Dispersion (optics)

In many materials, the speed of light changes depending on its color or wavelength. This makes light spread out into different colors when it goes through things like glass or water. This spreading creates rainbows and other pretty colorful effects in nature.

This color spreading can sometimes cause problems in tools that use light, like telescopes, by making the pictures blurry. Special lenses were made later to help fix this and make the pictures clearer.

Lossy, absorbing, or conducting media

See also: Mathematical descriptions of opacity

When light moves through some materials, interesting things can happen. These materials have special properties that change how light behaves. This can make the light weaker as it travels through the material.