Homework 4

4/22/02

Physics 110B

1. A reflector is made out of plastic with an index of refraction of $1.5$. Light is incident on it as shown below. The reflector is very thick compared to the wavelength of light so that interference effects need not be considered. The bottom zig zags at 45 degrees. You can also ignore edge effects at the ridges. Take the index of refraction of air to be $1$.

(a)(10 points) First just consider the reflection off the upper surface. What fraction of light is transmitted, and what fraction is reflected?

(b)(10 points) Now consider the part of the light that is transmitted into the plastic. How much of that light is reflected off the lower face after the first bounce?

(c)(10 points) Taking into consideration all bounces of light, up, down, in and out of the plastic, what is the total fraction of incident light that is reflected off the reflector?

2. An electromagnetic plane wave of frequency $\omega$ propagates in a material with complex dielectric constant $\epsilon$, which can be written as $\epsilon = \vert\epsilon \vert\exp (i\phi)$. As usual, take $\mu  = \mu_0$.

(a)

(10 points) For a linearly polarized wave, what is the angle between the electric and magnetic fields?

(b)

(20 points) Do the same for a circularly polarized wave

Recall that a circularly polarized wave traveling along the $x$ direction can be written as

$${\bf E}  =  Re\big( E_0 ({\hat k} + i{\hat j}) e^{i(kx-\omega t)}\big) .$$

Also you may want to use $\sin (\alpha +\beta)  =  \sin(\alpha)\cos(\beta)+\sin(\beta)\cos(\alpha)$.

3. An underwater beam of light is incident on the water's surface at an angle of 60 degrees with respect to the normal. Calculate the transmission and reflection coefficients at the surface given that the polarization is parallel to the plane of incidence. Assume the index of refraction of water is 4/3, and that the other side of the interface has an index of refraction of 1.

4. A beam of light starts in air ($n_1 = 1$), and is incident upon a medium made up of slabs with indexes of refraction $n_2, ..., n_i, ..., n_N$ (see figure). Show that the angle at which the transmitted light ray penetrates into the ith slab, $\theta_i$, depends only on the initial angle of incidence, $\theta_1$ and the index of refraction of the ith layer. What is this dependence?

5. Imagine a giant metallic sail in space, facing the sun, with a surface density of $\sigma kg/m^2$. The sun acting on the sail will impart momentum to it. If the average power per unit area of sunlight is $S$, calculate the acceleration of the sail. Hint: the momentum density stored in the fields is $S/c^2$. In a time $\Delta t$, a length $c \Delta t$ of wave, passes through an area A. From this calculate the momentum per unit area per unit time carried by the wave, and then the acceleration of the sail. Assume it is a perfect reflector.

6. Assume that some light is reflected from a plane glass surface at an angle of reflection of 57 degrees and is completely polarized. The polarized light will be vibrating principally in a plane making an angle $\theta$ with the reflecting surface. Calculate $\theta$.

7. Two light rays from separate incandescent sources each illuminates a surface with an intensity of illumination $I$. If the two light rays are combined, they will illuminate the surface with an intensity of illumination

(a) $I$

(b) $\sqrt {2} I$

(c) $2I$

(d) $4I$

(e) $8I$

8. Two plane-parallel coherent beams of light each of which illuminates a surface with an intensity of illumination $I$, are combined so that they are in phase with each other. The combination will illuminate the surface with an intensity

(a) $I$

(b) $\sqrt {2} I$

(c) $2I$

(d) $4I$

(e) $8I$

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