The spherical aberration of a paraboloid mirror for objects of finite distance

A perfect paraboloid mirror imaging an object on-axis at an infinite distance will have no spherical aberration, i.e. the optical pathlengths object to focus will be equal, no matter where the light strikes the mirror.

When imaging at a finite distance, this is not so, and there will be some spherical aberration present. How large is this? I can think of two situations where the solution to this problem is of some interest:

I do not believe the latter case has been exploited much - but I think it would be well worth investigating. I also confidently believe SIXTESTS, the very versatile evaluation program by Jim Burrows, will handle this test situation with no complications.

We consider the situation illustrated in the figure. The paraxial ROC is R, the source is at a distance B from the vertex, and the KE is at A from it:

1/A + 1/B = 2/R;

Consider a point on the mirror, at a radius Y from the optical axis. The distances from this point to the KE and source, respectively, are taken to be:

(A + (Y2/2)(1/A - 1/R) + s) and (B + (Y2/2)(1/B - 1/R) + t);

where the sum s + t is the deviation of the optical path length as a function of Y. Both s and t are small compared to A, B and R.

Using the theorem of Pythagoras and a lot of gruelling algebra (I should expect a better mathematician to find a simpler solution!), and ignoring the small terms, I get an expression for the total path length change:

 s + t = y4(1/(2AR2) - 1/(4A2R));

For A=R/2, as well as A=infinity, this expression is zero, as it should be. For A=R it is, also as expected,

s + t = y4/(4R3);

Let us calculate this term for some values of A:

A (s + t)*R3/Y4  B
R 0.25 R
2R 0.1875 0.6667R
4R 0.1094 0.5714R
8R 0.0586 0.5331R

Let us consider collimation with an artificial star. Let the diameter of the mirror be D, and maximum y = D/2 Also, let the focal length F = R/2.Thus,

s + t = (D4/16)(1/(8AF2) - 1/(8A2F));

However, at best focus, the P/V deviation is only 1/4 of this. What distance to the artificial star do we need in order to have a P-V wavefront error of 1/4 wave, or RMS of 1/14 wave?  When A>>R, we can ignore the second term and get, using mm and a wavelength of 560 nm:

0.00056=D4/128AF2 or A~14D4/F2 ;

First, take a 6" f/6 mirror: D=152 mm, R=912 mm. This gives A~9000mm, quite close. Next a 16" f/4.5 mirror: D=406mm, F=1827mm. Here, A~114000 mm. When doing a star collimation there are also two other, independent constraints:

Let u assume the maximum useful focuser outward travel for the first mirror is d = 50 mm. Then the minimum distance is A=16640 mm - this is the limiting condition here. Let us take the second mirror, assuming a maximum useful outward travel also of 50 mm. Here, the minimum A=66800mm, and at this distance, the spherical aberration is closer to 1/2 wavelength P-V.

Nils Olof Carlin, March 26, 2003