The Refraction of Light
When we talk about the speed of light, we're usually talking about the speed of light in a vacuum, which is 2.99792458 x 108 m/s. When light travels through something else, such as glass, diamond, or plastic, it travels at a different speed. The speed of light in a given material is related to a quantity called the Index of Refraction, n, which is defined as the ratio of the speed of light in vacuum to the speed of light in the medium:
When light travels from one medium to another, the speed changes, as does the wavelength. The index of refraction can also be stated in terms of wavelength:
Although the speed changes and wavelength changes, the frequency of the light will be constant. The frequency, wavelength, and speed are related by:
The change in speed that occurs when light passes from one medium to another is responsible for the bending of light, or refraction, that takes place at an interface.
If you have ever half-submerged a straight stick into water, you have probably noticed that the stick appears bent at the point it enters the water. This optical effect is due to refraction. As light passes from one transparent medium to another, it changes speed, and bends. How much this happens depends on the refractive index of the mediums and the angle between the light ray and the line perpendicular (normal) to the surface separating the two mediums (medium/medium interface). Each medium has a different refractive index (see list below).
The angle between the light ray and the normal as it leaves a medium is called the angle of incidence. The angle between the light ray and the normal as it enters a medium is called the angle of refraction. If light is travelling from medium 1 into medium 2, and angles are measured from the normal to the interface, the angle of transmission of the light into the second medium is related to the angle of incidence by Snell's law, which was derived by a Dutch physicist named Willebrord van Roijen Snell (1591-1626):
Snell's law gives the relationship between angles of incidence and refraction for a wave impinging on an interface between two media with different indices of refraction. The law follows from the boundary condition that a wave be continuous across a boundary, which requires that the phase of the wave be constant on any given plane, resulting in
where and are the angles from the normal of the incident and refracted waves, respectively.
Critical Angle - Total Internal Reflection
Total Internal Reflection (TIR) is the phenomenon which involves the reflection of all the incident light off the boundary between two medium. TIR only takes place when both of the following two conditions are met:
In this introduction to TIR, we will use the example of light traveling through water towards the boundary with a less dense material such as air. When the angle of incidence in water reaches a certain critical value, the refracted ray lies along the boundary, having an angle of refraction of 90-degrees. This angle of incidence is known as the critical angle; it is the largest angle of incidence for which refraction can still occur. For any angle of incidence greater than the critical angle, light will undergo total internal reflection.
So the critical angle is defined as the angle of incidence which provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees. For the crown glass-water boundary, the critical angle is 61.0-degrees. The actual value of the critical angle is dependent upon the combination of materials present on each side of the boundary.
Let's consider two different media - creatively named medium i (incident medium) and medium r (refractive medium). The critical angle is the which gives a value of 90-degrees. If this information is substituted into Snell's Law equation, a generic equation for predicting the critical angle can be derived. The derivation is shown below.
ni • sine( ) = nr • sine (90 degrees)
ni • sine( ) = nr
sine( ) = nr/ni
= sine-1 (nr/ni) = invsine (nr/ni)
The critical angle can be calculated by taking the inverse-sine of the ratio of the indices of refraction. The ratio of nr/ni is a value less than 1.0. In fact, for the equation to even give a correct answer, the ratio of nr/ni must be less than 1.0. Since TIR only occurs if the refractive medium is less dense than the incident medium, the value of ni must be greater than the value of nr. If at any time the values for the numerator and denominator become accidentally switched, the critical angle value cannot be calculated. Mathematically, this would involve finding the inverse-sine of a number greater than 1.00 - which is not possible. Physically, this would involve finding the critical angle for a situation in which the light is traveling from the less dense medium into the more dense medium - which again, is not possible.
This equation for the critical angle can be used to predict the critical angle for any boundary, provided that the indices of refraction of the two materials on each side of the boundary are known.
TIR and the Sparkle of Diamonds
Relatively speaking, the critical angle for the diamond-air boundary is an extremely small number. Of all the possible combinations of materials which could interface to form a boundary, the combination of diamond and air provides one of the largest difference in the index of refraction values. This means that there will be a very small nr/ni ratio and subsequently a small critical angle. This peculiarity about the diamond-air boundary plays an important role in the brilliance of a diamond gemstone. Having a small critical angle, light has the tendency to become "trapped" inside of a diamond once it enters. A light ray will typically undergo TIR several times before finally refracting out of the diamond. Because the diamond-air boundary has such a small critical angle (due to diamond's large index of refraction), most rays approach the diamond at angles of incidence greater than the critical angle. This gives diamond a tendency to sparkle. The effect can be enhanced by the cutting of a diamond gemstone with a strategically planned shape. The diagram below depicts the total internal reflection within a diamond gemstone with a strategic and a non-strategic cut.
With an ideal cut, light entering through the top facet undergoes TIR a couple of times before finally exiting. When less than ideal, light entering quickly exits through the bottom of the gemstone.
DispersionEach material through which light passes will have an Index of Refraction that varies with the wavelength of the incoming light. That is, the amount of bending will increase as the wavelength decreases. This is called Chromatic Dispersion and explains why white light, which contains all wavelengths in the Visible Spectrum, is split into a rainbow when it bends through a prism.