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:

- One is star-collimating with an artificial star at finite distance. Some spherical aberration is acceptable, but if it is too large, the increased intensity in the first diffraction ring may overwhelm the asymmetry caused by coma
- Another is mirror testing. With common Foucault testing (including several variations of this basic theme), the "spherical aberration" is considerable for large, fast mirrors. With the light source inside the paraxial COC and the KE outside it, you could come considerably closer to "null" testing - the KE position, as well as the visual pattern, will be less influenced by the deviation from spherical.

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 + (Y ^{2}/2)(1/A - 1/R) + s)** and

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 = y ^{4}(1/(2AR^{2) }- 1/(4A^{2}R));**

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 = y ^{4}/(4R^{3});**

Let us calculate this term for some values of A:

A | (s + t)*R^{3}/Y^{4} |
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 = (D ^{4}/16)(1/(8AF^{2}) - 1/(8A^{2}F))**;

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=D ^{4}/128AF^{2} or A~14D^{4}/F^{2}
**;

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:

- The focuser out.travel is limited (you may of course put an extension tube between focuser and eyepiece, or fasten the eyepiece as far out of the holder as possible)
- If you move the focus outwards, somewhere you will reach a point where the
secondary will vignette the primary mirror's edge - this lowers the
sensitivity of star collimation. Put a peephole cap flush with the drawtube,
and move it outwards till you find the position where the primary appears to
fill the whole secondary. If the focal plane of the high-power eyepiece you
use is flush with the focuser drawtube (as is often the case), you have the
outer limit of travel. If you know the position of the focuser for a star, you
can measure the outward travel
**d**. Then the minimum distance**A**is approx.**A~F**.^{2}/d

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