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!

 

[mark726]

I would like to provide some clarification and additional insights on the forum post:

a. The critical angle is the angle of incidence at which the refracted ray travels along the boundary between two media. It is not the angle at which the maximum power is reflected. The maximum power reflected occurs when the angle of incidence is perpendicular to the surface, which is not the case at the critical angle. At the critical angle, the refracted ray travels along the boundary and there is no reflection.

To find the minimum power reflected, you can use the Fresnel equations, which describe the reflection and transmission of light at an interface between two media. These equations take into account the incident angle, the refractive indices of the two media, and the polarization of the incident light. You can use these equations to calculate the power reflected at different angles of incidence and find the minimum value.

b. Approximating the curve as a circle may not be accurate in this case. The critical angle depends on the refractive indices of the two media, not just the distance between them. So, while you can ensure that the critical angle is present at the end points, the angle of incidence at the points in between may not be greater than the critical angle.

To find the refractive index of the second medium, you can use the relationship n2 = c/v, where c is the speed of light in vacuum and v is the speed of light in the second medium. You can also use the Snell's law to find the angle of incidence at the points in between, using the refractive indices of the two media.

c. Yes, part c is similar to part a, but with different parameters. You can use the same approach as in part a to find the minimum power reflected, but with the updated parameters.

In conclusion, it is important to understand the concepts of critical angle and Fresnel equations to accurately calculate the power reflected in this scenario. I hope this helps clarify any confusion and provides a better understanding of the problem.