Diffraction Patterns Resulting From Cell Induced Mirror Deformation

How good are Plop Cells?

A mirror supported on a Plop computed cell still has some surface deformation. That deformation must degrade the images formed by the mirror. How much degradation is there? The RMS and P-V deformation values listed by Plop give one way to estimate the degree of image degradation, but do not show the magnitude of the degradation on an image. Calculating diffraction patterns allows one to reliably predict the effect on one type of image, a single star. Knowledgeable interpretation allows one to extend the prediction to other types of images. There are more sophisticated methods (Fourier convolution) to directly calculate the effect on more complex images, but I am not yet that sophisticated in my ability to use them.

The following diffraction plots show the effects of support induced mirror deformation on image formation at the center of the field of an otherwise perfect parabolic mirror with no central obstruction or spider vanes.

The mirror is the same for all cases: Diameter = 317.5 mm (12.5 in). Focal Length = 1587.5 mm (62.5 in). f/ Number = 5. Thickness = 30.48 mm (1.2 in). Material = Pyrex 7740. I chose these dimensions, because they are the dimensions of a mirror an ATM asked me for advice about. This ATM was sceptical of the quality of images that would be had from 5 and 6 point Plop optimized cells. I didn't have a quantitative answer for him, so I decided to research one.

Deformations listed in the table are surface deformations as reported by Plop. Wavefront deformations are about twice as large. (Reference wavelength is 500 nM.) Click the links in the tables to see the diffraction patterns resulting from the Plop derived support systems.



Support TypeP-V DeformationRMS DeformationStrehl Ratio
Perfect001.000
3-Point
Plop
1/12 wave1/56 wave0.947
4-Point
Plop
1/25 wave1/121 wave0.9891
5-Point
Plop
1/41 wave1/227 wave0.9968
6-Point
Plop
1/60 wave1/334 wave0.9984

Interpretation

My interpretation of the data is that, with a very good mirror, under very good conditions, an experienced observer would probably be able to tell the difference between the mirror mounted on the 3-point support and one mounted on a more perfect support. Someone making careful photometric measurements of the diffraction pattern in a lab would probably pick up the difference. This amount of scattering would cause a small, but potentially observable loss of detail contrast at high power.

Very careful examination of the diffraction pattern of a bright star, under essentially perfect conditions might allow one to detect the light scattered by the 4-point cell. One would have to look carefully. There would be a very faint brightening between the first and second diffraction rings. The only visual observing situations that might be affected would be trying to separate very closely spaced, and very unequal (in brightness) stars, or observing very faint lunar detail very close to a very bright feature. Honestly, I doubt that even the most observant astronomer would notice the scattered light unless she specifically set out to observe the diffraction pattern of a bright star under the absolute best conditions.

I seriously doubt that even the most observant astronomer, working under the best conditions, could detect the imperfections resulting from the 5 or 6 point cells. The amount of light involved is just too small. I think that even laboratory measurements would have trouble detecting these small differences.

Method

I took mirror deformations from the color deformation plots produced by Plop. This process was automated so that the deformations used in my calculation are a very close approximation of the deformations reported by Plop.

Diffraction patterns were calculated by calculating the path lengths for a grid of points spaced 1 mm across the face of the mirror. The central "ray" path length was subtracted from each path length to get the path length difference. The path length differences were scaled to wavelengths (reference wavelength = 500 nM) and converted to angular units. The sines and cosines of the angular path length differences were summed. The intensity at each image point is the sum of the squares of the mean cosine and sine values. This was repeated for each point in the image plane. I used 1 micron spacing of points on the image plane. For the images presented here, over 1.5 million path lengths were calculated for each diffraction pattern. All path length calculations were done with double precision floating point arithmetic. The resolution of double precision arithmetic should be easily good enough for the job.

Most diffraction pattern calculations make use of an approximation called, I believe, the Fraunhoffer approximation. This allows one to calculate the path length differences for only one image point, then use a Fourier transform to derive the diffraction pattern. I understand that the Fraunhoffer approximation is considered a very good one for this type of simulation. When I started this work, I did not know how to use that technique. (Jim Burrows has enlightened me, but I still haven't managed to put the new knowledge into working code.) My method takes a lot more calculation, but is based very soundly on straightforward wave theory and, except for the use of 1mm spaced sample points, does not involve any approximations. My PC's CPU only had to grind for 5 or 10 minutes on each diffraction pattern. That was fast enough for my purpose. Since I did these patterns, I have learned more about the use of Fourier transforms to do this type of calculation. An interesting result is that the usual Fourier transform method ends up using a substantially coarser grid of sample points across the mirror.

Strehl Ratios

Strehl ratio is defined as the the peak intensity of the abberated diffraction pattern divided by the peak intensity of the perfect diffraction pattern. It is possible to approximate Strehl ratio from the RMS deviation of the mirror surface, however that is not the method I used. The Strehl ratios presented in the table above are calculated directly from the diffraction patterns. In this way, any approximation involved in the RMS Strehl calculation is avoided.