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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).
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
Pardon my ignorance, and I don't want to rain on your parade, but what's so special about this? Any decent little telescope can show them visually, and photographing them takes nothing but a smartphone and a high-power eyepiece.
This kind of depends upon the aperture of the scope and the exposures used to record the stars. It can be fairly easy to record the Airy pattern on a small scope and in my experience it actually becomes more difficult with larger scopes. A good 50mm refractor can show very obvious diffraction rings, larger scopes less often.
That said, I've been able to record the first diffraction ring with hints of the second in images that I've taken of double stars and here is a link to a recent example that was taken with two different refractors (one with an 80mm aperture, the other at 100mm):
I do think the OP's graphical plots are interesting and I've seldom tried those measurements on my own images.
Brings back fond memories of my very first experimental confirmation of diffraction theory. I was just a little kid visiting the main Rundel Library in Rochester, New York. My grade-school teacher had sent me over with a letter asking the librarian (friend of hers) to allow me access to the "Adult Science Section". There I found the optics books I wanted to read, and discovered the sections on Physical Optics. I raced home and made various single and double slits, different pinhole sizes etc etc and using colored filters --- was able to see the sinc --- (sin(x)/x)2 --- and Bessel sinc --- (J1(x)/x)2 --- functions with my own eyes. It was one of the most joyous moments of discovery in my entire life... the first of uncountable thousands.
A couple decades later I was doing similar things in the research labs for giant segmented telescopes that we were working for Uncle Sam. And with the same manic euphoria when I could see the customized impulse responses that we theorized and intended. I later found myself working the JWST telescope null testing... and celebrating that observatory's successful deployment, initialization, and release to the astronomers. From pinholes in mom's aluminum foil to giant observatories in a few short decades... I never lost the awe factor in discovering Natural Law, in both its meanings. Tom
a body of unchanging moral principles regarded as a basis for all human conduct.
"an adjudication based on natural law"
an observable law relating to natural phenomena.
I'm surprised to see such sharp peaks for the secondary ring. Have you worked out your imaging scale in arcsec/pixel to verify that those peaks appear in the right place for the first ring of the Airy pattern at the wavelength you are using?
Kind of impressive detecting it at that image scale.
Yes, very impressive.
If we assume green light (0.5micron wavelength), an f/7 scope and the 2.3 native micron pixel size of the ASI294 then the Airy disk diameter is less than 4 pixels by my calculations. So, the radius of the first ring cannot be 17 pixels.
Yes --- to sensibly scrutinize the structure of the Airy (or any other) impulse pattern... you need to massively "oversample" beyond which traditional imagery would need. And back to my kidly experiments... I was astonished how close I had to get the two paired silts to see the fringes and how small the pinhole to see the Airy pattern. This kinda stuff also ingrains the physical optics of what Nature does for us observers, every time we open our eyes --- and look up.
PS: Even the professionals sometimes misjudge the smallness of the Airy pattern. I walked into one of the labs at work, and a young optical engineer was getting completely frustrated (for days) trying to "get fringes" on her complicated concoction on one of the huge Newport tables. Context was white-light interferometry for phasing segmented systems... an admittedly challenging assignment for a fresh from school engineer. I checked things over and offered, "You are lacking spatial coherence by a couple orders of magnitude; you'll never get fringes this way. Blank stare re' how it just had to work, because she designed it. I shrugged and kindly offered (pointing at a relay assembly), "Put a three micron pinhole in here, and you will get fringes." That she did, looked into the far downstream eyepiece, and exclaimed --- "fringes!" I smiled, mission accomplished --- and left.
In her defense, several other senior optical engineers and scientists had looked over her test-set, waived their arms, bloviated meaningless jargon, shrugged --- and slithered out. I have my clueless moments too; no shame in that --- but occasionally exclaiming "Duh!" is perfectly OK. Tom
"I both hate diffraction and am fascinated by it."
I follow the discussion but fail to reach a conclusion, is this or this not an example of the second maxima that Steve has shown, and how does one reconcile the maths that Mark has put out.?
Even better is to see the Airy pattern with your own eyes, which have greater dynamic range than any camera, and ideally through a good reflector (no colour) or maksutov (negligible chromatism).
As a lifelong visual observer keen on high-power I'm more surprised when I don't see it; never occurred to me anyone wants photograph it, particularly.
If you like diffraction, try stopping the scope down with a sub-aperture mask. Then there's the Bhatinov mask; the diffraction theory behind how that works is far from trivial and its a very practical use of diffraction.
If you want an interesting but controversial thread on Airy patterns around stars, there is this one:
It was discussing scopes with central obstructions. The central obstruction "modulates" the Airy pattern in a way that makes some features of it easy to record without Barlows at low-power focal ratios such as f/10.
Central obstruction of the diffraction image is one reason some think SCT's "bloat" star images when they really don't. Just below the eye's threshold for discerning the diffraction disk and rings, a star might "appear" bloated (compared to when viewed in a small refractor) because the star is brighter in the SCT and because the brightness of the first diffraction ring (unresolved at low power) makes it seem like it's just part of the diffraction disk, hence, the star appears larger or bloated. Boosting the power enough (provided the seeing is good) to SEE the diffraction pattern clearly resolves that misconception.
I think the bumps appear sharp due to being undersampled.
I also think the seeing and guiding errors also contribute to further spread out the PSF.
A 5.52 pixel FWHM (3.52 arc-seconds, or 12.8 microns) "seems" reasonable, as does the location of the second maxima "visually".
If only using the diffraction-limited equations, my FWHM should have been 4.7 microns. So that is a factor of 2.7. If this factor is linear, than the second maxima would also be 2.7X further out than the diffraction-limited equation.
You might be correct, but I will check. I've seen these "bumps" a million times in my data, and PSF is the only explanation I can come up with. The bumps also appear in the same exact radius vertically and diagonally, using that same data, so it seems to be a full circle.
But I will check into this further.
Thanks for the comment.
And as I stated in my article, John Herschel also saw them visually a hundred and fifty years ago, as have I more recently.
It was the digital recreation I thought was interesting, as it allows further analysis and discussion.
I have often wondered about the incredible dynamic range of the human eye. Is it static or dynamic? That is, when looking at Jupiter and its moons, is our eye-brain system "flickering" between brightness modes to see both at the same time with great detail? Or is it truly 21 stops of DR, steady state, all the time?
It is indeed 21 stops of DR, steady state, all the time.
I think the novelty described here is that evidence of diffraction can be seen in *long exposure guided* images. People have been imaging the diffraction pattern for years - and it's something I have promoted as the final tweak to collimation for about 15 years. If you can image the diffraction pattern directly, there is no reason to collimate based on more indirect means.
As for the thread Mark/sharkmelley references - the best explanation for these rings appearing readily in recent years is some kind of diffraction - but that explanation still leaves a lot of questions - particularly the fact that these rings show readily in poorly guided or even unguided images. They also show in refractor images and don't require help from a secondary obstruction.
The particular rings in amateur images may be due to a sweet spot of aperture that isn't so large that the diffraction pattern isn't too small on the scale of seeing - and the recent appearance of small pixel cmos cameras suddenly made the effects visible.
But it is still very strange to me how sharp the rings can be even if the image is poorly guided.
At the very least the locations of the rings should be fit to a diffraction model to confirm they are consistent with theory. But even if it matches - it's weird to me that they are so evident *and round* even if the guiding is causing the centroid to drift around.
So I'm still waiting for a really satisfying explanation. I banged on the issue for a while trying to come up with something - but my best guess is that it is somehow diffraction induced - and I assume something else is going on somehow.
The equations for the Airy pattern can be found at this Wiki article
For the example in the original post and using the Wiki article nomenclature we have:
Those parameters generate this plot of the Airy pattern:
As I previously said, the Airy disk diameter is less than 4 pixels. The radius of the peak of the first ring is approx 2.5 pixels.
I've no idea what artifact you are seeing with a radius of 17 pixels.
That is the equation for a "perfect" diffraction limited system, in the absence of an atmosphere.
What I am presenting is real, actual data (which trumps theory every time).
Look at my PSF: the base of the first maxima is literally 10X the size of what you are showing, and models nearly perfectly as a Gaussian (or Moffat) distribution.
So certainly no artifact.
I just randomly picked another 3 images, 3 other stars, from 2 different nights using another telescope (same camera).
Slightly higher sampling (0.87 arc-sec/pixel)
Definitely second and third maxima showing. I can find 100 more examples.
I cannot attach images, so this is a link to the folder. Note I also did the analysis for rows and columns of pixels (horizontal and vertical):
I think you are looking at the Gaussian/Moffat PSF caused by 3 arcsec seeing conditions and that under ideal atmospheric conditions you would actually see the Airy Pattern. After all, isn't this the same model of scope that was tested with a Strehl ratio of 0.978 I'm assuming you are using the TS-Optics Photoline 115mm f/7 triplet APO refractor.
Yes that is the telescope used in the article (a different scope was just used in my previous post)
The PSF includes the diffraction pattern.
No, not perfect, but that IS the diffraction (Airy) pattern, though yes, distorted by several things, mainly the atmosphere.
In the link I just shared, the location of the "bumps" are essentially the same to the left and right and top and bottom of the central maxima. I am sure they are the same diagonally, and all other locations that form a circle.
So that has to be the second (and third) maxima of the Airy pattern, no?
In looking more closely at the profile in your article, showing a broad Gaussian peak surrounded by a sharp ring only one pixel wide - I have seen that over the years in my images - including CCD and not just cmos - and I just don't know what it is. In order to be related to diffraction it needs to be consistent with theory in terms of where the ring appears, and whether it is particularly bright relative to other nearby rings - or if instead those other rings should also show.
It's fine if diffraction theory predicts a particular pattern and you don't see it - and instead you see something else. That is indeed an example where an empirical result doesn't match theory. But in that case you would also need to conclude that this isn't a diffraction phenomenon. It doesn't match general relativity either - but that doesn't mean it *is* due to relativity. Having a sharp ring only one pixel wide at a radius that doesn't match the Airy pattern, and with no similar rings nearby, is very hard to explain - and it's consistent with similar rings that have appeared over the years. Some of the rings do at least roughly match the expected ring locations - but are still perplexingly sharp.
A key thing missing in explaining this stuff is a detailed diagram of the actual pixel and microlens structure. There may also be something funny in terms of pixel crosstalk and electronics or something. I tried many different models that would somehow create a pattern that is anchored to the pixels in one of those "ring" threads - but nothing ever worked.
I think an explanation is lurking out there, though. But I just can't see that lone, single-pixel ring around the broad Gaussian as being part of the Airy pattern.
When I use MetaGuide and stacked video to see the Airy pattern - it matches theory extremely well - both in location of the rings and their width and profile.
Diffraction theory does hold firm and there’s no way round it. The focal ratio would have to be raised to f/20 or f/30 to image it clearly, either by stopping the aperture down (easy) or using a Barlow.
As for perfect optics what you’re using isn’t all that close.
It's certainly a very odd feature. I can't think of anything that might explain it.