Geometric Optics: Refraction

Total Internal Reflection

Imagine a beam of light in a dense medium incident on an interface with a vacuum (or any less dense medium). For small angles of incidence we know that some of the beam will be reflected and some transmitted into the vacuum. If we increase the angle of incidence, however, the angle the transmitted beam makes with the normal, θ_t, increases at a greater rate. We know this because from Snell's Law we have:

sinθ_i =

sinθ_t

Since the transmitting medium is less dense,

< 1, so θ_t > θ_i. Eventually it reaches the situation where θ_t = 90^o and the transmitted beam just grazes the surface. If θ_i is increased still further, the transmitted beam disappears. This is illustrated in :

The incident angle at which this occurs is called the critical angle, θ_c, and is given by:

sinθ_c =

sin 90^o =

When θ_I > θ_c all the light is reflected inside the dense medium. This is known as total internal reflection (TIR). TIR is employed to great benefit in technologies such as optic fibers in which a laser beam is shined down a narrow transparent fiber; total internal reflection maintains the laser signal within the fiber, irrespective of the shape into which it is bent.

The Fresnel Equations

Although we know that a wave incident on an interface will be partially reflected and partially transmitted along certain paths, we do not know what proportion of the energy of the ray will take each path. We can use Maxwell's Equations to help us solve this problem. Consider the diagram , in which the electric field is perpendicular to the plane of incidence.

Figure %: Directions of fields of reflected and transmitted rays.

Maxwell's Equations imply certain boundary conditions. The first is that components of the electric field perpendicular to the interface must be continuous across the interface, hence:

. The other relevant boundary condition in this case is that the perpendicular component of the magnetic field divided by the permeability,

. Thus:

where the positive direction is that of increasing x. Recall from that |

. We can use these three equations to find the two ratios

and

. The first of these is known as r_âä¥, the amplitude reflection coefficient, and the second as t_âä¥, the amplitude transmission coefficient. These equations tell us the relative amplitudes of the reflected and transmitted rays for light incident on an interface at some angle θ_i. When the permeabilities of the two media are taken to be both equal to μ₀, which is usually true to a good approximation, we find: