A mathematical analysis of spherical aberration
Name: ______Physics Extra Credit
1. (7 points) Suppose that you have a concave mirror that has been shaped like a “generic” parabola. Mathematically prove that its focal point is fixed (and what is that focal point?)
To do the proof, here are some clues:
a. What is the definition of a focal point?
b. Sketch a large generic parabola on the Cartesian plane and write down the formula. You will want to place your parabola at a very convenient location, and you need to find a formula that could give you any parabolic shape (i.e., use a, b, c, etc. for the constants). Where is the principal axis?
c. Sketch an incoming ray that satisfies the definition in part (a) onto your sketch in part (b). Find a “generic” equation that describes the incoming ray.
d.Of course this incoming ray will reflect off the surface of the mirror. All you need to do is to find the equation of the reflected ray (y = mx + b) and this will give you the focal point (why?). Remember, you need to use the law of reflection combined with some simple calculus and trigonometry to find the equation of the reflected ray. What does your solution tell you about the focal point (does it depend on the location of the incoming ray?)?
2. (10 points) Repeat the process above for a spherical mirror. Carefully select the center! What does your solution tell you about the focal point (does it depend on the location of the incoming ray?)?
3. (8 points) Graphically show the difference between an 8” diameter parabolic mirror with a focal length of 56” and a comparable 8” spherical mirror (do this on Excel or some other package that allows you to precisely graph data; set up your formulas and use them to create a table of values that you can then graph). Assuming that light with a wavelength of 580 nm is incident the mirror,
a. Calculate the spherical aberration relative to the parabolic mirror when a ray of light parallel to the principal axis is one inch from the axis. Provide your answer in terms of the number of wavelengths by which the two mirror shapes differ.
b. Repeat for a ray incident 2” from the axis and 4” from the axis.
c. The mirror in Mr. Rhine’s telescope is ground to within one-quarter wavelength of a true parabolic shape. Compared to you results in parts 3(a) and 3(b), is this degree of precision impressive or not? Would it explain why parabolic mirrors cost many time that of spherical? For what applications would you decide to “go cheap” and purchase/manufacture a spherical mirror rather that a more complex parabolic mirror?