Simulating diffraction in optical instruments - help settle a dispute

[halfelven]

Consider a Newtonian reflector telescope:

As you can see in the image, the little secondary mirror inside the main tube is supported by an X structure called 'spider'. The four "legs" of the spider, called 'vanes' by astronomers, are actually thin stripes of high-tensile material, usually steel, tensioned for rigidity.

Diffraction at the edges of each vane distorts the image. E.g., the image of a star (a point-like object) will show four spikes, like this (visually it's not that bad, the example image here is a bit exaggerated):

Each vane actually makes two opposite spikes, and the spikes are perpendicular to the vane. So the four vanes will make in reality 4 x 2 = 8 spikes, which are then superimposed two by two.

My question is: Is there a way to simulate numerically the diffraction spikes? Given an instrument, and vanes of a given size and thickness, calculate the intensity of the spikes in any point within the virtual image. I would be satisfied with calculating the spikes produced by even a single vane, or by two opposite vanes forming a full diameter. It's okay to ignore the diffraction produced by the secondary mirror.

Basically, this is a regular telescope (actual type doesn't matter, it could be simulated as either reflector or refractor), with a thin and long wire intersecting the field of view at some distance ahead of it. The distance between wire and primary lens or mirror is almost equal to the focal length of that lens or mirror (at least that's the case with most Newtonian telescopes); let's say the wire-to-telescope distance is about 80% the focal length of the primary lens. The diameter of the primary lens is about 1/6 of its focal length. The telescope is looking at a point-like object at infinity, right in the center of the view field. Now calculate the diffraction effects due to the wire, as seen in the virtual image produced by the telescope, assuming various values for the wire thickness. Producing an intensity map for the diffraction effects would be great.

Slightly modified: I believe (please correct me if I'm wrong) it is enough to calculate the real image formed in the focal plane of the primary mirror or lens. There's no need to simulate a full telescope with a virtual image at infinity. The ocular merely magnifies the real image it sees in the primary focal plane, where diffraction is already spiking the image.

Simulating diffraction from the edge of the primary lens or mirror would be a nice bonus (a.k.a. the Airy rings, for a more realistic simulation), but it's not essential.

The reason I'm asking: It is generally accepted within the community of amateur astronomers that the thicker the vanes, the more intense the spikes, therefore thinner vanes are seen as "better", leading to an arms race towards ever more fragile designs. I don't think that assumption is true. Diffraction is an edge phenomenon, why should it matter what is the distance between the two edges of a vane (thickness)? A thick vane should merely block a little more incoming light, that's all. The two edges should always diffract the same amount of light, no matter what is the distance between them. But this belief is so strong, persistent, and wide spread that only a rigorous numerical simulation may stand a chance to repel it (or confirm it, in case my intuition is wrong).

I assume some kind of numerical analysis software will be needed, such as Matlab or Octave. I'm willing to put some work into this project, but I need a start. Modifying an existing software (something close to this problem, but perhaps not identical) would be perfect. Keep in mind, Optics is not what I do for a living.

 

[wukunlin]

this site appears to have some decent info, have you looked at it yet?
http://www.beugungsbild.de/diffraction/diffraction.html

ooo and I found this one with some codes
http://blog.sigfpe.com/2011/06/simulating-visual-artifacts-with.html

 

[DrDu]

There exist free software to "deconvolute" the image. Once you know the image of a pointlike source (in practice any star), the so-called point spread function (with the four spikes), you can remove the diffraction artifacts from your picture.

 

[mrspeedybob]

This may be a dumb question but why don't they mount the secondary mirror to a piece of glass?

 

[halfelven]

this site appears to have some decent info, have you looked at it yet?
http://www.beugungsbild.de/diffraction/diffraction.html

Fascinating and very informative article. Thank you!

Looks like I was wrong. Thickness does matter, albeit only for the diffraction figures around stars (it doesn't matter for planetary viewing). I just don't understand why that is. If diffraction is basically an edge phenomenon, why are the spikes getting stronger when the vanes get thicker? Hmmm... I guess I'll have to ponder this one for a while.

I've emailed the author of that article, asking what software have they used to simulate the images. Hopefully it's not something proprietary.

This may be a dumb question but why don't they mount the secondary mirror to a piece of glass?

Short answer: That would be worse.

Long answer:

You could install the secondary mirror on a glass flat, of course. In fact, some people have done that. But there are several issues:

If you just steal a chunk of glass from a window and use it to mount the secondary, that would be terrible. The glass would simply not be flat enough, it would distort the image, and performance would take a steep dive.

You need a piece of glass which is optically flat - that is, the deviation from the perfect surface at any given point is less than a fraction of wavelength - usually something like less than λ/4 or so. Additionally, the glass would have to be very thick, so that it doesn't sag under its own weight. Finally, it cannot be regular "green glass", but some kind of optical glass proper. The flat would have to be about the same diameter like the primary mirror, a bit larger in fact.

Basically, whereas before you had only one big optical element with only one high-precision surface (the primary mirror), now you have two elements with 3 high-precision surfaces total (primary mirror and mounting flat), one of them (the flat) made of optical glass. This design is a lot more expensive, which is annoying since a Newtonian in general gives a very high performance / price ratio; the flat would pull that ratio down quite a bit.

Also, whereas before you only had to worry about one large element becoming deformed due to temperature variations, now there are two large pieces of glass that will be distorted until they reach thermal equilibrium.

In fact, the price of the flat would be not prohibitive (but still significant) only for small telescopes, aperture maybe up to 200 mm or so (8"). For larger telescopes, we're talking about a very large flat made of optical glass. That's expensive and heavy. If we're talking about large Newtonians (aperture above 600 mm), just forget it, it's completely impractical.

Please note that there are telescope designs that have a large optical element exactly where that flat would be - except in these cases it's not flat, it's a lens of a certain shape that applies corrections to the whole telescope. The most popular examples are the Schmidt and Maksutov telescopes:

http://en.wikipedia.org/wiki/Schmidt_camera

http://en.wikipedia.org/wiki/Maksutov_telescope

In general, these telescopes are not very large, due to the difficulty of making the large transparent front element. They are typically used when a compact design is one of the primary goals. They are also used a lot for astrophotography.

 

[Drakkith]

Diffraction spikes are not an "edge" phenomenon. They are the result of part of the wavefront actually missing when it is focused down to a spot. Thinner vanes mean more of the wavefront is available and less diffraction.

See the following website for an excellent, if complicated, description of telescope optics: http://www.telescope-optics.net/wave.htm

Edit: Also, while it's not a big deal, I just want to point out that in the picture in the first post the three brightest stars have CCD blooming in addition to the diffraction spikes. The vertical white spikes emerging from the top and bottom of all three stars look like diffraction spikes, but as you can see they don't quite match up with the real diffraction spikes and are pure white, which is the result of the pixels becoming saturated and overflowing into the ones above and below them. The actual diffraction spikes are the dimmer spikes that are almost on top of the blooming effects.

 

[sophiecentaur]

It is common to use offset sub - reflectors and feeds for microwave antennae. Is there a good reason not to use a similar offset arrangement for the mirror in an optical telescope, to avoid the supports? I can appreciate that the reflector would need to be asymmetrical but that need not be a problem these days. And the offset angle need not be very great.

 

[halfelven]

It is common to use offset sub - reflectors and feeds for microwave antennae. Is there a good reason not to use a similar offset arrangement for the mirror in an optical telescope, to avoid the supports? I can appreciate that the reflector would need to be asymmetrical but that need not be a problem these days. And the offset angle need not be very great.

If the primary mirror is parabolic, like in Newtonian telescopes, you need to keep things close to the main axis. Parabolic mirrors are aberration free for objects at infinity only when the object and the image are near the axis. Ofsetting the parabolic mirror would introduce aberrations.

If you still want to offset it, and keep the parabolic shape, you could use a segment of a paraboloid. The "offset" is obtained merely on account of the rest of the mirror missing (you only use an off-axis chunk of the paraboloid). However, such a surface is extremely hard to make, the mirror would be very expensive. It's much easier to make a complete, symmetrical paraboloid.

A primary mirror that is always offsettable is the spherical mirror. A sphere has no preferred axis, so you cannot optically distinguish between an offset sphere and a non-offset. This is what Herschel did for his big telescopes: the primary was spherical and he merely tilted it while shifting it sideways. If you shift the mirror so that its surface always stays on the same sphere, there's no extra aberration introduced.

The problem with spherical mirrors is that they are never free of aberrations, for objects at infinity. Telescopes with spherical primaries, uncorrected, always underperform other designs.

It can be shown that if the focal length of the spherical mirror is big enough compared to the diameter of the mirror (F at least 8x bigger than D), then the aberrations of a sphere become lesser than the diffraction effects, so are negligible. However, F/8 is not a convenient focal ratio for telescopes of large diameter; you want a more tight design, otherwise the total size of the instrument becomes prohibitive. Also, long focal length (compared to diameter) means reduced field of view, which is not always desirable.

Finally, you could tilt the mirrors purposefully, accepting that the tilt would introduce aberrations, but design the whole system so that the aberrations of various elements compensate each other. This is the idea with the Schiefspiegler telescope:

http://www.schiefs.com/schiefs02e.html

In practice, diffraction from support elements in Newtonian telescopes is almost never a problem. In a visual telescope, the diffraction spikes are only visible around a handful of the brightest stars. Most stars don't show visible spikes. It's not a problem, unless you're a purist.

It can be shown that if you curve the support elements such that each one of them becomes a half-circle, the diffracted light, instead of being concentrated in a spike, becomes spread around in a circular cloud, becoming effectively invisible. In theory, this may reduce the contrast around bright sources, in practice it's a pretty effective way of getting rid of spikes. Most people don't bother with that. Even scientific instruments, such as space telescopes, etc, use straight support elements and then they just ignore the spikes during analysis.

Microwave receivers are offset because aberrations don't matter too much for that application.