Vignetting in a Newton telescope

By: Arjan te Marvelde

This article explains how the amount of vignetting at the various apertures found in a telescope can be calculated. The excercise below only takes front aperture and secondary mirror into account, but other candidates would be the focuser drawtube and various baffles.
First a calculation model is given, and then the resultsof the spreadsheet I made for my rich field telescope.

This drawing shows the layout of a Newton telescope, and the various parameters used in the vignetting model. The most important are the diameters of the front aperture (A), the primary mirror (D), the diameter of the secondary (sd) and the required vignetting free field (vff). The diameters of interest are all scaled to the width of the lightcone at the same location. So, the secondary diameter sd is divided by the lightcone diameter at secondary location, resulting in Rs. The front aperture diameter A is divided by the primary mirror diameter D, resulting in Ra. The lightcone diameter itself is normalized to 1 (Rd).

A tight fit (Rd=Ra=Rs) would yield a 100% illumination of the focal plane on the optical axis only. Wider fit (Ra>1 and Rs>1) will result in a wider unvignetted field. The precise drop-off curve can be found by calculating the overlap area of the circles for various offsets (x) from the optical axis. The indefinite integral over a circlesegment is given by: The upper and lower limits are determined by the crossing points of the different circles. The positions of the circles with respect to the light cone are a result of simple trigonometry, deduced from focal length, x and the distance along the optical path of the aperture to the primary mirror. The figure shows the worst case situation, with vignetting from both secondary and front aperture.

The spreadsheet calculating this integral for a given telescope design can be downloaded from the Resources page.
Below the result is shown for my RFT having the following parameters:
D = 150mm
f = 600mm
sd = 40mm
A = 170mm
The vignetting free field I required to be 6mm diameter, and the illumination of the edge of the field not below 70%.