A calculation of Airy disks for various telescopes. Ref: Jackson, _Classical Electrodynamics_ second edition (1975) pp. 443-446.

By David Whysong 

After reading all the discussion about how a central obstruction changes the Airy disk, I decided to work it out for myself. So I opened up my Jackson (the standard physics book on electromagnetic fields), turned to the treatment of diffraction, and spent a couple hours trying to figure it all out. In case you're wondering, yes there IS black magic involved. Really. :-)

Some of you may have seen these graphs before if you have a telescope optics book; I don't, so I did it myself. If you haven't seen these graphs, they are very instructive. This is all theory (that's all I'm good for, right now).

I'll spare you the math (if you want the details, email me). I plotted the intensity of an Airy disk versus angle theta from the optical axis for five different "ideal" telescopes: an 8" refractor, a 4" refractor, a 5" refractor, an 8" SCT, and a 9.25" SCT. I know the 9.25" isn't a common aperture, but I was interested because I'm planning on ordering one soon.

My graphs have one important drawback: I haven't bothered to integrate the total power in the Airy disks, and so the y-axis scales (the intensities) for different 'scopes are arbitrary, and they are all different. I can try to do the more detailed calculations to get the intensities all scaled correctly, but it will take time.

The equations and graphs are available as a Mathematica notebook or in html/gif format. I can send them in a zip file by email. If you want a plot like this for a different instrument, I can probably do the calculation pretty easily.

The results:

An ideal 8" SCT with a 34% central obstruction should always outperform an ideal 4" refractor. The airy disk of the SCT is more compact by a factor of 2 (obvious from Rayleigh's criterion) and the first diffraction ring is still within the outer edge of the central maximum of the refractor. However, a 5" refractor has a very slightly more compact outer edge to its Airy disk than the 8" SCT. The 8" SCT does appear to have a tighter Airy disk overall though.

By the way, about that 99% obstruction -- it would narrow the central peak of the Airy disk, but the first central maximum has about 20% the intensity of the peak. After that, the intensity of the diffraction patterns falls off as roughtly 1/theta. Not a useful situation for most purposes, unless maybe you can do a lot of complicated math with the resultant image and filter out the "bad" spatial frequencies. This might be nice for splitting binaries. An 8" SCT with a 99.9% central obstruction has a central maximum of about 1.8 microradians, whereas with a 34% obstruction the central maximum is around 2.6-2.7 microradians. You gain some resolution with the huge obstruction, but not a tremendous amount.

Notes: I had to adjust the power for each graph by hand. Mathematica had trouble integrating the Bessel functions to get the total intensity. As a result of this, each different instrument's graph has a different, arbitrary scaling factor in intensity. I have roughly normalized each graph so that the intensity at the center of the Airy disk is one. Think of it this way: the following graphs assume that the peak intensity of the Airy disk is always the same, regardless of the aperture of the instrument. Each graph describes the intensity of the diffracted light as a function of angle, from zero (the optic axis) to 2 arc seconds.

For all graphs, the incoming light is assumed to have wavelength of 500 nm. The y-axis is intensity (with dimensions of power per unit solid angle), and has arbitrary scale as mentioned above. The x-axis is angle in radians from the optic axis.

For an ideal 102 mm refractor with incoming plane waves, this is the transmitted intensity I as a function of theta:

[Graphics:diffractiongr2.gif][Graphics:diffractiongr6.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr7.gif]
Here is a 130 mm refractor:

[Graphics:diffractiongr2.gif][Graphics:diffractiongr10.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr11.gif]
Here is a 203.2 mm refractor:

[Graphics:diffractiongr2.gif][Graphics:diffractiongr14.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr15.gif]
Here is a 203.2 mm Schmidt-Cassegrain with a 69.85 mm obstruction (34%):

[Graphics:diffractiongr2.gif][Graphics:diffractiongr18.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr19.gif]
Here is the 203.2 mm SCT and the 102 mm refractor together:

[Graphics:diffractiongr2.gif][Graphics:diffractiongr22.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr23.gif]
Here is the 203.2 mm SCT and the 130 mm refractor together:

[Graphics:diffractiongr2.gif][Graphics:diffractiongr26.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr27.gif]
Here is 230 mm Schmidt-Cassegrain with an 86.06 mm obstruction (Celestron 9.25", 37% obstruction)

[Graphics:diffractiongr2.gif][Graphics:diffractiongr30.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr31.gif]
Here is the 230 mm SCT with the 130 mm refractor:

[Graphics:diffractiongr2.gif][Graphics:diffractiongr34.gif]

[Graphics:diffractiongr2.gif][Graphics:diffractiongr35.gif]
Here is an 8" telescope with a 99.9% central obstruction. Note the 1/theta falloff.

[Graphics:diffractiongr2.gif][Graphics:diffractiongr38.gif] 


Mike McJimsey has calculated the OTF's (MTF's) for these same telescopes. See OTF.html.

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