# Refraction

## Basic

Fig.1: Refraction of a light ray

Refraction is the change of direction in which light travels as it passes from one substance to another that has a different optical density (as from air into a gemstone).

Optical density is a property that manifests itself in the slowing down of light, i.e. the higher the optical density, the lower the velocity of light. This change in velocity causes light to bend (refract), as can be seen when a spoon is put into a glass of water. Because light travels slower through water than air, the spoon appears to be bent. This bending of the light is referred to as refraction. Refraction is based on Snell's laws of refraction (see below).

In Fig.1, the angle of incidence is indicated with i and the angle of refraction by r. When light travels from air to an optically denser medium (like a gemstone), it will hit the surface at an angle to an imaginary line named the normal (NO). It will then partially enter the stone (other parts will be reflected). Due to the slowing down of light inside the stone, it will bend (refract) towards the normal.

The opposite of this is also true. Light that travels from a gemstone into an optically rarer (or less dense) medium such as air will bend away from the normal. The angle at which light is now refracted out of the stone is the same as the angle of incidence, which means it will continue along the same path as on incidence (when it entered the stone).

A gemstone’s index of refraction depends on wavelength. White light is composed of 7 spectral colors (Red, Orange, Yellow, Green, Blue, Indigo and Violet), each of which travels at different wavelengths and thus at different speeds. Therefore, the refractive index will be different for each of those colors.

In gemology, yellow light is used as the main source for measuring the refractive index of a gemstone. Yellow light was chosen because in the early days of gemology it was easily produced by salt (sodium) in a flame and was an inexpensive means of producing monochromatic light. The wavelength of sodium light lies at 589.6nm, known as the Fraunhofer D-line. When n is used to describe the index of refraction, we use nD to indicate the refractive index when measured with sodium light. This is commonly abbreviated as RI (refractive index).

The instrument of choice to obtain the refractive index of a gemstone is the refractometer.

#### The math behind Snell’s laws of refraction

Refraction follows the laws of Snell, which state that:

• The sine of the angle of incidence (i) and the sine of the angle of refraction (r) are related to each other at a fixed ratio. This ratio is known as the index of refraction. The ratio depends both on the wavelength of the light and on the substances in which this light travels.
$Index\ of\ refraction = \frac{\sin i}{\sin r}$
• The incident ray, the refracted ray and the normal all lie in the same plane.

The index of refraction is abbreviated with the letter n. As the index of refraction also relates to the velocity of light, one could write:

$n = \frac{velocity\ of\ light\ in\ air}{velocity\ of\ light\ in\ medium}$

### Causes of refraction

Explaining the causes of refraction is not an easy task, yet a pseudo-scientific approach is given below. It is based on two principles:

1. Quantum electrodynamics (to explain the slowing down of light in a denser medium)
2. Huygens' principle

#### Quantum electrodynamics

$f = \frac{V}{\lambda}$

Velocity (V) of light is dependent on wavelength (λ) and frequency (f). Frequency is the number of crests that pass a given point in one second and is expressed in hertz (Hz).

The frequency of light is the same in every medium. This means that if the velocity of light changes, the wavelength must also change. As a consequence, light traveling from air into an optically denser medium will travel at shorter wavelengths inside that medium.

At the atomic level, an incident light wave will make an electron cloud vibrate at the frequency of the wave and emit the energy of the light wave with a delay, or slowed down (a different phase velocity). Since the electron cloud vibrates at the same frequency as the incident wave, the frequency of the emitted light wave will not be altered. In order to keep the equation intact, a drop in velocity must mean a decrease in wavelength.
Thus both the velocity and the wavelength of a light wave decrease when entering a denser medium.

##### The relationship between wavelength and speed of light
Speed of light and wavelength

Light in air travels at approximately 300,000km/hour. When it enters an "optically denser medium", this light will slow down. The amount at which it slows is related to the index of refraction of that material.

Assuming the RI of the material = 2, then the light (coming from air) will be slowed down to 150,000km/hour, or half the speed of the light in air.

As there is also a direct relationship between the speed of light and wavelength, the wavelength will decrease as well. This means that a beam of red light (with a wavelength of about 700nm) will travel at a wavelength of 350nm inside the material with index of refraction = 2.
A ray of light traveling at 350nm is long-wave ultraviolet light.

#### Wavefronts (Huygens' principle)

Two wavefronts on a single wave, each one wavelength apart

Light propagates from a single source in all directions just like a pebble thrown in a pond would create waves around the center of impact. These waves (the wrinkles in the water) are at a certain distance from one and other (wavelengths). The direction of propagation is perpendicular to the wave and the wave itself is named a "wavefront".

Similar to waves in water, light also propagates in one direction with associated wavefronts. All of these wavefronts are at a distance of one wavelength apart.
Allochromatic light will have an infinite amount of wavefronts, all at different distances depending on the wavelengths.

For a basic understanding it is best to single out one lightwave.

Bending of wavefronts

When light travels through air, it travels at a speed of approx. 300.000 km/hour (actually that is the speed in a vacuum). As it encounters an optically denser medium it will be slowed down and hence the wavelenght must shorten aswell.
Thus the wavefronts on a particular lightwave are closer together inside the denser medium (as a gemstone) than they are in air. In order for the wavefront to stay intact, the wavefront must bend at the boundary between air and the gemstone (as illustrated on the left).

In the illustration you will notice that the wavelength inside the gemstone (the distance between 2 wavefronts) is shorter than the original distances in air. Because the propagation of a lightwave is perpendicular to the wavefront, the lightwave must bend aswell.
Through a series of goniometric calculations one can derive Snell's law from this.

$Snell's\ Law:\ \frac{sin\Theta1}{sin\Theta2} = \frac{n2}{n1}$

Where n1 is the index of refraction in air and n2 the refraction index inside the gemstone, Θ1 is the angle of incidence and Θ2 is the angle of refraction.

As the index of refraction of air is 1, we could rewrite that as:

$\frac{sin\Theta1}{sin\Theta2} = n$

Or:

$n = \frac{\sin(angle\ of\ incidence)}{\sin(angle\ of\ refraction)}$

## Sources

• Introduction to Optical Mineralogy 3rd edition (2004) - William D. Nesse