Items 1-8 are best done after watching the lecture on reflection and refraction; the remaining items practice the transfer-matrix method developed in the lecture on optical multilayers.

Unless stated otherwise, assume there is a single interface lying in the \(xy\) plane between two semi-infinite regions with indices of refraction \(n\) and \(n'\) and magnetic permeabilities \(\mu=\mu'=\mu_0\). Assume that the incident and reflected beams have a finite cross-section and do not overlap far away from the interface. (This means we can neglect interference between them far from the interface.)

1. (3 pts) Verify that the Fresnel equations for two different polarizations become identical in the case of normal incidence (as they should). Reconcile any inconsistencies in the signs.

2. (3 pts) For \(\mathbf{E}\) parallel to the plane of incidence, show there is a certain incidence angle for which the reflection amplitude vanishes. Show there is no such angle for the perpendicular orientation.

3. (5 pts) Using the expression for the time-averaged Poynting vector \(\mathbf{S}=\frac12\mathbf{E}\times\mathbf{H}^*\), find the energy flux \(\mathbf{S}\cdot\hat{\mathbf{z}}\) carried away from the interface by the transmitted and reflected beams, expressing them, respectively, though the amplitudes \(E'\) and \(E''\) that appear in the Fresnel equations. Do it for an arbitrary angle of incidence for both polarizations (\(\mathbf{E}\) perpendicular or parallel to the plane of incidence).

4. (3 pts) Consider the case \(n>n'\) and the angle of incidence (from the \(n\) side) in the region of total internal reflection. Review the arguments showing that \(k^\prime_z\) is purely imaginary. Using this fact, show that (a) the transmitted energy flux is zero, and (b) the reflected energy flux is equal to the incident energy flux.

5. (3 pts) For the situation when total internal reflection does not occur, verify that the sum of the transmitted and reflected energy fluxes is equal to the incident energy flux.

6. (3 pts) Under the conditions of item #3, prove that the transmission coefficient (ratio of the transmitted flux to the incident flux) is the same no matter which side the beam is incident from (assuming that the incidence angles are related as in Snell’s law).

7. (10 pts) Consider a beam of light incident from air on the flat surface of water (index of refraction \(n'=1.33\)). For each of the two polarizations, calculate and plot (in the same figure) the reflection coefficient (ratio of the reflected energy flux to the incident energy flux) as a function of the angle of incidence (from 0 to 90°).

8. (8 pts) Assume an *unpolarized* beam of light is incident from air on the flat surface of water. Calculate and plot the degree of polarization of the reflected beam as a function of the angle of incidence. What are the Stokes parameters for the incident and reflected beams?

9. (4 pts) Fill in the steps leading from the boundary conditions at an interface to the explicit expression for the transfer-matrix for that interface.

10. (15 pts) It is desired to make a single-layer coating for a lens to minimize the reflection of light at normal incidence. To this end, carry out the following steps:

(a) Introduce an optical trilayer with refraction indices \(n_1\) (air), \(n_2\) (coating layer of thickness \(d\)), and \(n_3\) (glass).

(b) Follow the prescription from the lecture to find the transfer matrix of the trilayer at normal incidence. This will take a bit of algebra.

(c) Using the transfer matrix, find the ratio of the reflected amplitude to the incident amplitude.

(d) Show that the reflected amplitude vanishes if the thickness \(d\) is equal to one quarter of a wavelength in the coating layer while \(n_2\) is equal to the geometric mean of \(n_1\) and \(n_3\).

11. (15 pts) Solve Jackson problem 7.3 using the transfer-matrix method.

Total: 72 points.