Snell's Law Calculator

When light passes from one medium to another it is refracted. Snell's Law describes the degree of refraction precisely. This calculator computes the angle of refraction given the index of refraction for two media and the incident angle. It can also use the inverse relationship to solve for any one of four values when the other three are inputted. Enter three values and leave the fourth blank.

The importance of Snell's law:

Light travels at different speeds in different media. The refractive index of a medium is denoted by n and indicates how much the medium slows down light. It is the value obtained by dividing the speed of light in vacuum by the speed of light in that medium. The refractive index of vacuum is exactly 1, air is just above it at about 1.0003, water is 1.333, ordinary glass is about 1.52 and diamond is 2.417.

When a ray of light hits the interface between two media at an angle, its direction changes because of the change in speed. If the light slows down and enters a more optically dense medium, it bends toward the normal. The normal is a line perpendicular to the surface. If the light speeds up, it bends away from the normal. Snell's law relates the two angles and the two indices of refraction as follows:

where n1 and n2 are the refractive indices of the first and second medium respectively, theta1 is the angle of incidence, and theta2 is the angle of refraction. Both these angles are measured with respect to the normal, not the surface. The index of refraction itself is determined by the speed of light.

Here, c is the speed of light in a vacuum and v is the speed of light in this medium. The refractive index being a ratio of speeds has no unit.

Example calculation:

For example, let's say a light ray is traveling from air to glass and hits the surface at an angle of 30 degrees with respect to the normal. The refractive index of air is about 1.0003 while the refractive index of glass is 1.50. If we rearrange Snell's law to solve for the unknown angle of refraction, we get:

The calculation of the arcsine of 0.333 results in an angle of refraction of approximately 19.5 degrees. As the light ray enters the denser glass and its speed is reduced, it is deflected by about 10.5 degrees from the normal. The entire calculation proceeds in this manner. By releasing any fields you can carry out the calculations in the desired direction.

Backward calculation:

Since Snell's law is a single equation that relates four quantities, you can solve for any one of the quantities by rearranging the equation appropriately. There is no need to select a particular mode. The blank field indicates which quantity the calculation tool will determine.

Measuring two angles and an index of refraction to determine another index of refraction is exactly the method by which a refractometer measures the index of refraction of a liquid. So this back-calculation mode is not just a theoretical exercise from a textbook, but also something that's routinely used in laboratories.

Critical Angle and Total Internal Reflection

When light passes from a denser and slower medium to a less dense and faster one, such as from glass into air, the ray of light is deflected in a direction away from the normal. As the angle of incidence gradually increases, the refracted light bends more and more until at a particular angle it reaches the limit where it has an angle of 90 degrees. This angle is called the critical angle. If we set the angle of refraction to be 90 degrees in Snell's law, then we get the following equation:

When the critical angle is exceeded there is no way for the light ray to escape outwards. The light ray is totally reflected back into the first medium and this phenomenon is known as total internal reflection. As total internal reflection only occurs when light enters a medium with a lower refractive index from one with a higher refractive index, this calculator will only show the critical angle if the first medium is denser than the second. If the input incident angle exceeds the critical angle then rather than giving an impossible value for the angle of refraction, the calculator will instead indicate total internal reflection.

Total internal reflection is not a rare phenomenon. It allows light to be transmitted over long distances in fiber optic cables, makes the facets of cut diamonds sparkle, and creates the optical illusion that water glistens on hot roads.

Direction and extent of refraction

The direction of the refraction is determined only by which medium is denser. If light slows down it bends toward the normal line and if it speeds up it bends away from the normal line. The amount of refraction, or difference between the two angles, is the angle of deviation that this calculator shows. In the case of normal incidence, where the ray hits at exactly 0 degrees, there's no refraction at all. Both angles are zero and the light passes straight through. This is the only limitation to Snell's law you need to remember. There's nothing else to calculate for rays hitting a surface perpendicularly.

Speed, wavelength, color

The effects that occur when light slows down in a medium are not limited to the refraction of the beam. Since the frequency of light remains constant as it crosses an interface, its wavelength shortens by the same factor as its speed increases or decreases. An orange light ray with a vacuum wavelength of 600 nanometers becomes a wave with a wavelength of 400 nanometers when entering glass. Although the index of refraction for this glass is 1.5, the perceived color remains orange since the eye detects frequency and not wavelength. The "speed layer" shows the speed and wavelength of light for each of the two media.

The index of refraction changes slightly with wavelength so that different colors of light are bent by differing amounts. This phenomenon is called dispersion. It's the reason a glass prism splits white light into a rainbow spectrum and why real rainbows form from raindrops.

Brewster's Angle and Polarization

Between the two indices of refraction is another angle, called Brewster's Angle. At this angle light reflected from a surface will be completely polarized, meaning it vibrates in only one direction. It corresponds to the arctangent of the ratio of the second index of refraction over the first index of refraction, and at this angle the difference between the reflected and refracted rays is exactly 90 degrees. Polarizing sunglasses are designed to reduce this glare that is reflected from water or road surfaces, and photographers use polarizing filters for the same reason. The "Brewster's Angle" section of this calculator will calculate this angle for any given media.

Application of Snell's Law.

All lenses in glasses, cameras, microscopes and telescopes are shaped to bend light rays towards a focal point according to Snell's Law. Fiber optic cables use total internal reflection to transmit internet signals. Prisms, rainbows, the mirage phenomenon, optical illusion of a pool looking shallower than it is, and straw appearing bent in a glass of water are all caused by refraction.

This law is not just for light. It is based on the principle that waves change speed at a boundary and so it also applies to sound waves or seismic waves as long as the material is isotropic, which means that its properties are uniform in all directions. In anisotropic crystals, a ray of light can split into two rays, which cannot be described by the simple form of Snell's law.