Brewster’s angle and total reflection

There are two useful properties of reflection that we discuss here. Reflection at Brewster’s angle can be used to polarize an electromagnetic wave. This is useful in design of lasers and is important to know when setting up an optical experiment. Secondly, total internal reflection is a very useful property of light that can be used to design experiments.

 

The Brewster angle

The electric field intensity of the reflected wave is either in phase or p radians out of phase with the incident wave, depending on whether sin(qt - qi)cos(qt + qi) is greater than or less than zero. There is no reflected wave when sin(qt - qi)cos(qt + qi) = 0, that is when qi + qt = p/2.

When this condition is met there is no reflected wave for electromagnetic radiation polarized parallel to the plane of incidence (p polarized). The conditions of continuity and the interface are met by two waves - the incident and the transmitted - instead of three. This angle is called the Brewster angle. It is also called the polarizing angle since an unpolarized wave incident on an interface at this angle is reflected as a polarized wave with its E vector normal to the plane of incidence. This property is very useful in laser design since a Brewster window on an ion tube or a mirror can be used to polarize the radiation inside the laser cavity.

Example: Calculate the Brewster angle for radiation in air (n1 = 1) that impinges on a quartz laser window (n2 = 1.45).

Sample problem solution

 

Total internal reflection

If n1 > n2 and if qi is sufficiently large, Snell's law

leads to the apparently absurd result that sin qt is greater than 1. If we imagine a wave propagating in medium 1, then for angles greater than the critical angle, all of the radiation is reflected. The critical angle for which sin qt = 1 and qt = 90o, is

Although there is no refraction at angles larger than the critical angle, there is still an interaction of electromagnetic radiation across the interface. This is known as an evanescent wave and absorption by species on the other side of the interface (dielectric medium 2) can give rise to attenuation of the reflected wave.

It turns out that Snell's law and Fresnel's equations are applicable to total reflection if disregard the fact that sinqt > 1.

Note the negative sign before the square root. You should be able to verify that this equation begins with the well-known relation
.

Note also the factor of i pulled out front. Since

this factor compensates the factor of -1 that was multiplied inside the square root sign. If we now return to the coordinate system used for the study of reflection and refraction we can write the waves as:

The reflected wave is exactly the same for total reflection as for any other reflection. However, the transmitted wave can be expressed in terms of the condition for total reflection and the above massaged expression for cosqt.

Note that

Also according to the coordinate system, the transmitted wave (in medium 2) has negative values of z. Therefore, the wave consists of a traveling component in the x direction:

and an exponentially decaying component in the z direction (across the interface in medium 2)

.

Example: Calculate the critical angle for radiation in BK-7 glass (n1 = 1.47) that is in contact with a buffered solution (n2 = 1.33).

Sample problem solution

Attenuation

The reduction of light intensity as it traverses a dielectric medium is called attenuation. Attenuation is an exponential function of the thickness of the dielectric medium.

where g is the absorption coefficient of the medium. An evanescent wave penetrates the dielectric medium across the interface as shown in the diagram below.

The evanescent wave leads to interesting kinds of spectroscopy. For example, a wave guide can be used to obtain multiple internal reflections of an infrared beam. Attentuated total reflection (ATR) Fourier-transform infrared is an important technique for the study of surfaces. ATR fluoresence and Raman are also valuable techniques.