Calculating diffraction patterns

I've always been fascinated by different telescopes designs and the endless debates about how they offer different degrees of contrast and resolution. Hard figures are however hard to come by.

Tip: In the following paragraphs I describe at a high level how I eventually calculated some figures for myself. You can always trust my workings (ha ha) and skip to the bottom of the page, where I give my results for a few common and a few exotic telescopes.

For example, some manufacturers do quote the angular resolution of their telescopes, typically the Rayleigh limit. However, when you scratch the surface it turns out the manufacturers are simply giving the resolution as calculated for a refractor of matching aperture. The actual resolution of a telescope depends on its optical design. So a Maksutov has higher Rayleigh resolution than a refractor of similar aperture, at the cost of significantly degraded contrast.

When it comes to contrast, there are no figures available and no measures for this important characteristic. Contrast is negatively affected by a central obstruction, and by spider vanes, but by how much? Can we put a number to it, and what will the number mean?

The first challenge is to calculate the Fraunhofer diffraction pattern formed by a given telescope. This usually requires Fourier transforms and a number of mathematical approximations, but - mostly due to my limited mathematical ability and stronger programming skills - I have gone for the alternative approach of simulating the optical system. I take tens of thousands of samples of the inbound wavefront and map these to the focal plane of the telescope.

Normally one thinks of a photon hitting a smooth mirror and bouncing off like a ping-pong ball. Following this model, one would expect a parabolic mirror to perfectly focus all light to a point. We know that does not happen; we get a diffraction pattern instead. A better model for reflection is the Huygens-Fresnel principle, where the light simply ripples off every point of the mirror in every direction. The "reflected" light is simply the interference pattern formed by all the wavelets spraying off the mirror.

As a consequence, every sample of the wavefront must be mapped to every point we sample of the focal plane, and then the interference occurring at each sample point of the focal plane must be calculated, giving the amplitude at that point. That is then squared to give the intensity of the light. This is all rather computationally expensive, but rather than exploiting elaborate maths, we are staying close to the physics.

Doing the calculations

I have written a Java program implementing the above approach. It works similarly to an astrophotographer, in that it "shoots" multiple images of the diffraction pattern and then "stacks" them to form a single, more accurate image. An animation of the program at work is at the top of the page.

The application simulates the shape of the aperture and transform this to a spherical wavefront converging on the focus. I believe this is acceptable for on-axis comparisons, that is comparisons near the center of the focal plane. Off-axis, the different scope designs diverge (e.g. coma, astigmatism) but to analyze that I must do a much more complicated simulation. I have to find a way of computing this in a way which is not computationally prohibitive. I would like to fully simulate refractors and compound telescopes in the future.

Here's a link to my public source code:

https://gist.github.com/anonymous/67ad83b729d140329ffb 

How to calculate resolution?

There is no need to invent a new criterion for resolution, so all one has to do is follow Rayleigh and find the radius of the first minimum of the diffraction pattern. I do this by sampling a row of points from the focal point outward at 0.01 arc second increments.

How to calculate contrast?

At its heart, contrast must be the ratio between the light in the airy disc and the light outside of the Airy disc. But it is not that simple - calculating the extreme reaches of the diffraction pattern is very expensive, and where do you stop anyway? Also, if light is scattered far away from the object being
examined, then it does not contribute to a loss of contrast. For example, spider vanes throw light out so far from the Airy disc that it it does not affect the contrast when viewing the small image of a planet. But these same star spikes will affect contrast significantly when viewing the eyepiece-filling Moon.

So I have broken down contrast into four categories: 

• Light scattered very near the the Airy disk (within two Airy disc radii of the focal point). This will affect contrast for Double stars or very fine high contrast detail on the Moon.
• Light scattered within 8 arc seconds. This will affect contrast on Mars, or show the scatter around bright stars.
• Light scattered within 30 arc seconds. This will affect contrast on Jupiter and Saturn. (The planets are larger than 30 arc seconds, but this radius will fully cover the middle part of the planet which is most observable).
• Light scattered within 30 arc minutes. This will affect contrast on the Moon and the Sun (if viewed with appropriate safety precautions). I do not calculate this last measurement as it is currently too time consuming.

Some results

Here is an entirely random selection of telescopes... I only own the NexStar 4SE and the ST80, so I hope you will agree that I am not suffering from owner's bias. Of course the calculations assume each scope has perfectly figured optics, which may be nearly true of the TMB refractors in this list, but is definitely an unjustified assumption for the ST80.

I am shocked at the results for the NexStar, and will rerun them to higher precision to verify.

Contrast in the following table is quoted as the ratio of power in the Airy disc to the power in the disc of the given radius. The "Fine Contrast" radius is simply double the Airy disc radius, so its size depends on the telescope. The rationale is that different telescopes are used differently, so this is the contrast offered by the given telescope for its highest level of resolution.

All simulations are done with 510nm light, which is approximately the wavelength the human eye is most sensitive to at night.

Telescope	Resolution (arcsec)	Fine Contrast	8 arcsec Contrast	30 arcsec Contrast
NexStar 4SE (102mm Mak)	1.13	0.788	0.660	0.562
MN190 (180mm Mak-Newt)	0.63	0.869	0.716	0.685
APM LZOS 180 (180mm frac)	0.71	0.874	0.794	0.765
C11 (280mm SCT)	0.41	0.856	0.677	0.650
SkyWatcher 200P F6 (200mm Newtonian, 50mm central obstruction)	0.60	0.884	0.741	0.708
GSO CF 200 F4 (200mm Newtonian, 70mm central obstruction)	0.57	0.860	0.661	0.630
ST80 (80mm frac)	1.61	0.773	0.677	0.628
APM TMB 105/650 (105mm frac)	1.22	0.825	0.754	0.704