One of the most important laws in the analysis and design of prisms, and optical systems in general, is Snell's law of refraction, named for Willebrord Snell. It relates the angles of incidence and refraction at the boundary of two materials with differing refractive index (sometimes called the index of refraction). The refractive index n is defined as the ratio of the velocity of light in a vacuum c to the velocity of light in the material v_mat:

n = c / v_mat (1.1)

Since the velocity of light is reduced when traveling through optical materials, n is greater than unity. For the special case of air, which has a refractive index of approximately 1.0003, we assume the refractive index of air to be unity for most optical calculations.

Snell's law can be derived geometrically or from Fermat's principle, named for Pierre de Fermat^1,2. It is usually stated in the following form:

n sin I = n′ sin I′ (1.2)

where n is the refractive index of the incident medium, and n′ is the refractive index of the transmitting medium. I is the angle of incidence, measured relative to the boundary surface normal, and I′ is the angle of refraction at the boundary surface of the second medium (see Fig. 1.1). Snell's law is applicable to plane or curved surfaces, and both rays lie in a common plane called the plane of incidence.

A related law for reflecting surfaces is the law of reflection. It can also be derived geometrically or by using Fermat's principle. It is stated in the following form:

I = I′ (1.3)

where I is the angle of incidence, and I′ is the angle of reflection, as illustrated in Fig. 1.2. Since both incident and reflected rays are in the same medium, refractive index is not a factor in the directional change, and both rays lie in the common plane of incidence.