• What is Brewster’s angle for this case?
• What is the angle between the reflected and transmitted light if the light is incident at Brewster’s angle?
• What is the degree of polarization P of the reflected and transmitted light if the light is incident at Brewster’s angle?
• Make a plot of P for reflected and transmitted light as a function of the angle of incidence. Comment upon the physical significance of your graph. What is the direction of the polarization of the reflected light seen by an observer above the surface of the water, such as a person in a boat? What does that tell you about wearing polarized sunglasses when you go boating? What does your graph tell you about the likelihood of the evolution of polarization sensitivity in the vision of fish (i.e. is the ability to sense polarization in light likely to be of any advantage to a fish)?
• How will your calculations differ for blue light compared to red light?
• Extra credit for the astronomically-minded: At what time on July 4, 2000 will the light from the Sun strike the surface of University Lake at Brewster’s angle?

 
2.  Conservation of energy
It may have come to your attention that although the energy incident on an interface must equal the reflected plus the transmitted energy, it is not true that R + T = 1 (this is most obvious at Brewster’s angle, where R_|| = 0 and |T_||| is not equal to 1). However, you can use conservation of energy to prove that the intensity transmittance and reflectance coefficients at an interface are related by

 Solution to problem #2

3.  The Fresnel rhomb, a circular polarizer

As we saw in class, when light is totally internally reflected, a phase difference is introduced between the ^ and || components of the reflected wave. This can be used to transform linearly-polarized light, in which the two components are of equal amplitude and have zero phase difference, into circularly-polarized light, in which the two components differ in phase by p /2. (We will learn more about circularly-polarized light in the next week or so.) Your task in this exercise is to design a Fresnel rhomb, which is a device based on this principle. It consists of a rhombohedron made of some transparent substance, upon which linearly-polarized light is incident normal to the front face. Inside the rhomb the light undergoes two total internal reflections before emerging from the back face. (See the drawing on page 448 of your textbook.) By appropriate choice of the rhomb angle (the acute angle in the corner of the rhomb) , the two reflections can produce a net phase difference of p /2 between the ^ and || components.

• Assuming that the rhomb is made of glass (refractive index m = 1.5), what rhomb angle (and thus what angle of incidence for the total internal reflections) is needed to accomplish this?
• Could you make a device, perhaps out of some other material, in which the transformation was accomplished with one total internal reflection rather than two? If so, how? If not, why not?

 
4.  The evanescent wave
As we discussed in class, when total internal reflection occurs, an evanescent wave is produced in the second medium.

• What is the direction of propagation of this wave?
• What is its speed (in terms of the indices of refraction, the angle of incidence, and the speed of light in vacuum)? For light incident upon a glass-air interface, make a plot of the wave speed as a function of the angle of incidence.
• What is the amplitude of the electric field as a function of distance from the interface?