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Amateur Cameras Revealing Airy Patterns
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Amateur Cameras Revealing Airy Patterns
By Steven Bellavia
A point source of light does not image as a point, in any optical system, even if "perfect" (diffraction limited), and precisely focused.
The point becomes an Airy Pattern, named after mathematician and astronomer, George Biddel Airy. John Herschel had previously described the phenomenon, but Airy was the first to explain it theoretically. This is also called the Point Spread Function, or PSF.
Image credit: Resolving some spatial resolution issues - Part 2: When diffraction takes over, Geert J J Verhoeven, Ludwig Boltzmann Institute for Archaeological Prospection and Virtual Archaeology
What is exciting (to me) is that I've consistently been able to show evidence of the Airy pattern, using analysis of individual stars in images, taken with an amateur CMOS astronomy camera. (ZWO ASI 294MM on a 115mm, f/7 triplet refractor)
For this most recent example, I picked a bright non-saturated star from stacked data of my recent Crab Nebula image and took the pixel data to either side (left and right) of the star, and through the center, using the pixel tables in SAO DS9.
By obtaining the individual pixel data, you can see that the intensity does not fall off steadily but has two little "bumps" in the "tails". These bumps are the first diffraction ring (2nd maxima) in the Airy pattern. This is not visible in the image, but it is "visible" in the data.
If you look closely at the pixel values, this happens in all directions. Not just horizontal and vertical, but along the diagonals, and in all the directions that form the circular ring.
(Note, a single 13x13 pixel table in SAO DS9 did not capture the first ring, so I had to use several tables).
It is also just barely evident in the plot of the data. It becomes obvious if you zoom in.
Note: I tried many types of scaling and stretching and was not able to visually see the Airy Pattern. I also do not believe this to be a halo artifact, as it is too thin and distinct, and occurs with and without filters.
The next thing I did was to determine the shape of the curve of the first (central) maxima.
A Gaussian curve with a FWHM of 5.52 (3.29 arc-seconds), and a Moffat curve (Beta=7), both fit the data very well. (But of course, do not capture the rings, as the PSF is not Gaussian, Lorentzian, or even Moffat, but in actuality, a Bessel function).
Summary:
It’s nice to read about Point Spread Functions (Airy patterns) and see images and simulations of them.
But to be able to see them, at least the second maxima, numerically, using modest amateur equipment is very exciting.
- Paulo Gordinho, gundark, glatiak and 3 others like this
57 Comments
James,
That is awesome
Since people are showing general examples of actual Airy patterns and not just the odd long exposure deep sky rings - here is an example using MetaGuide with a 290 camera at f/10 and with about 750nm IR filter on EdgeHD11:
This is a realtime view based on a live, aligned stack of video averaged over about 1 second. The plot at the lower left compares the theoretical diffraction profile with the measured one, with no free parameters except matching the peak height. The location and height of the first ring is in excellent agreement with theory for that wavelength and that OTA.
With modern small pixel cmos video cameras it is very easy to see the Airy pattern directly with fairly large aperture at f/10 and even lower - with no need for a Barlow.
If I didn't align and stack, the pattern would be blurred out and the ring wouldn't be visible. That's why it's particularly odd that sharp rings show up in long exposure images.
I see attempts above to explain it with out of focus patterns and so forth - but that is one of the ways I was trying to model it in the long thread I started on these rings some time ago. I never found an explanation that worked. I was trying to find something that would act tethered to a given pixel - somehow - so the source could move a bit but the image would stay sharp. I still think something about the sensor may make that possible but I don't know what it is.
One thing I meant to correct earlier is the link above talking about how seeing, combined with the central limit theorem, results in Gaussian star spots. That link is just someone's blog and it is extremely incorrect. The Moffat profile that you get is very different from Gaussian in that it is a power law distribution with a lot of energy in the "wings," whereas a Gaussian is much more condensed. You can only apply the central limit theorem if you are summing a set of independently distributed random variables - but that is not what is happening when you accumulate a seeing disk over time. The images are just landing on top of each other and adding up - and that's completely different from the central limit scenario.
If you can read the caption at the top, it says the theoretical fwhm of the Airy pattern should be 0.530" and the aligned fwhm (AFWHM) is 0.616" - i.e. it is slightly bloated relative to theory, but still a very good match. You wouldn't expect it to be perfect since seeing is still affecting the result despite short exposure video and stacking.
Frank
This thread has been an interesting read! Regarding the 2D-slice plot, I propose that the apparent airy-shaped pattern is actually just noise. Notice how the spacing between peaks and valleys is exactly one pixel. Looking through some of the other similar plots on the google drive, most of them don't show any obvious consistent ring pattern (this one for example:)
These plots are also smoothed out and interpolated, which makes them visually look much more like there is some kind of ringing pattern and hides the fact that they are under-sampled. It would be interesting to see a plot of a vertical vs. horizontal slice and see if the peaks are correlated. If I'm right and it's just noise, they won't be.
I'm not sure what caused the 1-pixel-wide ring in the original post, but that seems like a different artifact. The fact that it can't be seen visually with any kind of histogram stretching is suspicious to me, as it appears to be well above the noise floor in the plot. I'm curious if it appears in all radial directions and for different stars in the same image. I would be interested in taking a look at the raw data if possible.
What a minute. An Airy Pattern has nothing to with the air and instead is named after George Biddel Airy?!
Yes, larger aperture = smaller airy disks and worse stability of the airy disk because of seeing conditions. Downside of small sensors with high pixel density (phones) is they do not handle small apertures well, being subject to diffraction effect and blurring.
The linear diameter of the Airy disk - in microns - is a function of focal ratio only - not aperture - a 10cm f/4 scope produces a pattern the same size as a 30cm f/4 one does.
The angular diameter (seconds of arc) yes, is inversely proportional to aperture.
The effects of seeing are worse with increasing aperture.