Total Internal Reflection - Derivation of Evanescent Wave

Derivation of Evanescent Wave

An important side effect of total internal reflection is the propagation of an evanescent wave across the boundary surface. Essentially, even though the entire incident wave is reflected back into the originating medium, there is some penetration into the second medium at the boundary. The evanescent wave appears to travel along the boundary between the two materials, leading to the Goos-Hänchen shift.

If a plane wave, confined to the xz plane, is incident on a dielectric with an angle and wavevector then a transmitted ray will be created with a corresponding angle of transmittance as shown in Fig. 1. The transmitted wavevector is given by:

If, then since in the relation obtained from Snell's law, is greater than one.

As a result of this becomes complex:

The electric field of the transmitted plane wave is given by and so evaluating this further one obtains:

and

.

Using the fact that and Snell's law, one finally obtains

where and .

This wave in the optically less dense medium is known as the evanescent wave. Its characterized by its propagation in the x direction and its exponential attenuation in the z direction. Although there is a field in the second medium, it can be shown that no energy flows across the boundary. The component of Poynting vector in the direction normal to the boundary is finite, but its time average vanishes. Whereas the other two components of Poynting vector (here x-component only), and their time averaged values are in general found to be finite.