# Reflection and Refraction in an Elliptical Imaging Mirror

Mr_Allod
Homework Statement:
An elliptical imaging mirror is constructed as shown in the diagram where light originating at point A should be reflected from the off of the surface to the point B.

a. With ##n_1 = 1##, ##n_2 = \sqrt 3##, ##d = \frac h {\sqrt 2}## find the maximum and minimum power reflected for both polarizations assuming the light reflects off the mirror between ##-b \leq x \leq b## and the critical angle occurs when light is reflected from the origin.

b. Calculate a new refractive index ##n_2## which would result in all light reflected between ##-b \leq x \leq b## to be reflected at an angle greater than or equal to the critical angle.

c. For the design of b. estimate the proportion of light leaving A which exists at B.
Relevant Equations:
Snells Law: ##n_1\sin(\theta_i) = n_2\sin(\theta_t)##
Fresnel Equations:
$$r_ {\perp}= \frac {n_1\cos(\theta_i) - n_2\cos(\theta_t)}{n_1\cos(\theta_i) + n_2\cos(\theta_t)}$$
$$r_{\parallel} = \frac {n_2\cos(\theta_i) - n_1\cos(\theta_t)}{n_2\cos(\theta_i) + n_1\cos(\theta_t)}$$

My thoughts so far:

a. Since the critical angle occurs at the origin for the given parameters I would imagine that the maximum power reflected would be 100% since at the critical angle ##\theta_t = \frac \pi 2## and ##r_ {\perp} = r_{\parallel} = 1##. I do not know how I might go about finding the minimum power reflected though.

b. Approximate the curve between ##-b \leq x \leq b## as a circle of radius ##R = h##. I thought that if I can ensure that the critical angle is present at the end points (##x = -d## and ##x = d##) then any angle in between would be greater than the critical angle (##\theta_c##). I sketched a ray going directly down from A to the the mirror and towards B. Under the circle approximation the the sine of the angle of incidence is:
$$\sin(\theta_i) = \frac {2d} h$$
Then using Snell's law:
$$\sin(\theta_c) = \frac {n_1} {n_2} = \frac {1} {n_2} = \frac {2d} h$$
$$n_2 = \frac h {2d}$$
However I'm not sure whether I was right to approximate the curve as a circle in this case though and I get the feeling the question is expecting an actual value for ##n_2## as opposed to an expression

c. I presume this is similar to part a. but instead using the new parameters.

So the most trouble I'm having is with part a. and b. If someone could help me out I'd really appreciate it, thank you!