Optics of spherical surfaces expressed as functions of y

[razmataz]

Homework Statement

Problem is: A maksutov camera, which is made from a refelector with a spherical surface s and a transparent corrector with two spherical surfaces s1 and s2. The radii of s, s1 and s2 are 2f, r1 and r2, respecitvely. Z=0 is the centre of all these spherical surfaces.

There is a diagram with y-z axis, with the 3 spherical surfaces as shown above centred on z axis.

(a) Express the profiles of the spherical surfaces in Figure 2 as functions of y and then expand these to the 4th power of y.

(b)Suppose the refractive index of the corrector is n and that r1 = f. Derive an expression for r2 that enables the corrector to correct the spherical aberration of the reflector.

Homework Equations

Not sure - hard topic with bad lecturer

The Attempt at a Solution

For (a) i know the equations of the surfaces when expanded to 4th power to be:

S = 2f - y^2/4f - y^4/(64f^3)

S1 = r1 - y^2/2r1 - y^4/(8r1^3)

S2 = r2 - y^2/2r2 - y^4/(8r2^3)


but i need to know the initial function of y before expansion?

Also for part (b)

I need help gettin from

(n-1)((1/r1^3)-(1/r2^3))(y^4/8) = (y^4/32f^3)

to the solution (below)

Question states r1 = f

Supposed to end up with

r2 = ((4n-4)/(4n-5))^(1/3)) * f

Can anyone show me the steps?!

 

[mark726]

Thank you for your question. I will do my best to help you find a solution to this problem. Let's start by defining some terms and equations that will be useful in solving this problem:

- Spherical aberration: This is a type of optical aberration that occurs when light rays from a single point on an object do not converge to a single point after passing through a lens or other optical element. This results in a blurred or distorted image.

- Refractive index (n): This is a measure of how much a material bends light as it passes through it. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material.

- Radius of curvature (r): This is the distance from the center of a sphere to its surface.

Now, let's address part (a) of the problem. You have correctly identified the expanded equations for the three spherical surfaces in the diagram. However, in order to find the initial function of y, we need to use the general equation for a spherical surface, which is given by:

S = r - y^2/2r - y^4/(8r^3)

This equation can be derived using the Pythagorean theorem and the equation for a circle. From this equation, we can see that the initial function of y is simply r. Therefore, the initial functions for the three spherical surfaces in the diagram are:

S = 2f

S1 = r1

S2 = r2

Now, let's move on to part (b) of the problem. We are given that r1 = f and we need to find an expression for r2 that will correct the spherical aberration of the reflector. To do this, we can use the formula for spherical aberration, which is given by:

SA = (n-1)((1/r1^3)-(1/r2^3))(y^4/8)

We know that we want to correct the spherical aberration, which means we want SA to be equal to 0. Therefore, we can set the equation equal to 0 and solve for r2:

0 = (n-1)((1/f^3)-(1/r2^3))(y^4/8)

Solving for r2, we get:

r2 = (4n-4)^(1/3) * f

However, this is not the final solution.