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, which can make light bend in unusual ways.

History

Ptolemy in Alexandria, Egypt, found a relationship about refraction angles, but it wasn’t accurate for larger angles. He thought he had the right answer, partly by changing his data to fit his ideas.

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 design special shapes for lenses that could focus light perfectly. Later, Ibn al-Haytham almost found the same rule in his book Book of Optics from 1021, but he didn’t finish the work.

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 the shortest path.

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 vice versa, 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 derived in different ways.

Derivation from Fermat's principle

Snell's law comes from Fermat's principle, which says that light takes the path that takes the least time. By looking at the derivative of the optical path length, we find the path the light takes. Imagine a beach (with a lower refractive index) and the sea (with a higher refractive index). The fastest way for a rescuer on the beach to reach someone in the sea follows Snell's law.

When light moves from one material to another, we call the place it enters 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 moves faster in materials with a lower refractive index. We can calculate the time it takes for light to travel using these angles and the indexes of the materials. By finding the smallest possible time, we get Snell's law.

Derivation from Huygens's principle

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

Derivation from Maxwell's equations

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

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 because the math would need a value greater than one, which isn’t possible. 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 causes light to spread out into different colors when it passes through materials like glass or water. This spreading is what creates rainbows and other colorful effects in nature.

This color spreading can cause problems in optical instruments, such as telescopes, by blurring the images. Special lenses were later invented to reduce this blurring and improve clarity.

Lossy, absorbing, or conducting media

See also: Mathematical descriptions of opacity

When light travels through certain materials, special things happen. The way we describe these materials includes complex numbers, which means the angle at which light bends and the direction of the light waves can act differently. This makes the light wave uneven, and the light gets weaker as it moves through the material. The weakening happens in a way that depends on part of the material’s properties.