Optical resolution question

rogerinnh · Jul 22, 2008

If you have a telescope set up so that it filters out all but one wavelength of light and so that the image it produces is at the optical limit of its resolution you should see, for each point source of light in the field of view (such as stars) the corresponding Airy disk. But what happens when you continue to increase the magnification of the telescope? Do you just see a bigger and bigger Airy disk? Or does the Airy disk disappear, replaced by just a blurring of the image?

Carrickagh · Jul 22, 2008

<pre style="tab-stops:.5in">It may have less to do with magnification than the focal length and diameter of the optics on your scope. </pre><pre style="tab-stops:.5in">Assuming the optics are diffraction-limited, I don’t believe magnification will have that big of an effect. </pre><pre style="tab-stops:.5in">The formula for r, the radius or ½ size of the Airy-disk is given as…</pre><pre style="tab-stops:.5in">r   = 1.22 lambda f/D</pre><pre style="tab-stops:.5in">r is dependent on f and D</pre><pre style="tab-stops:.5in">lambda is your key wavelength.</pre><pre style="tab-stops:.5in">So if you let f stay the same, the greater the D value then the smaller the r and if you let D constant: </pre><pre style="tab-stops:.5in">the longer the f the bigger the r. </pre><pre style="tab-stops:.5in"> Consider if you have two amateurs with 2 very different scopes. </pre><pre style="tab-stops:.5in">If you have a really, really big telescope (1m f3m) the airy disk would have the same diameter as with </pre><pre style="tab-stops:.5in">a telescope 10cm f 30cm. Again, based around the f and D function.</pre><pre style="tab-stops:.5in"> Also, I assume when you are looking atmospheric turbulence is low, thus you would have an ideal state and the Airy function should hold. </pre><pre style="tab-stops:.5in"> You may also want to consider the Dawes criterion, whereby we can determine the observable gap between the </pre><pre style="tab-stops:.5in">Airy disc rings, or more to the practical, the ability to resolve two close stars given certain seeing conditions a</pre><pre style="tab-stops:.5in">nd the ability of our telescope.</pre><pre style="tab-stops:.5in"> </pre><pre style="tab-stops:.5in">If you provide your scope’s aperture I may be able to calculate the theoretical resolution at some given wavelength.</pre><pre style="tab-stops:.5in"> Both Dawes as well as Airy.</pre><pre style="tab-stops:.5in"> **** </pre> <div class="Discussion_UserSignature"> </div>

rogerinnh · Jul 26, 2008

I am aware of the Dawes limit and what the Airy disk is and how they are calculated. My question is about what you would see in the image plane when a scope is set up perfectly and you get the absolutely best possible image from it, with each point-source of light resulting in its set of diffraction rings )the central ring effectively being the Airy disk). In such a situaiton if you increase the magnification, so that the image is simply larger, do you get a larger, better-resoultion image of the diffraction rings, or does it just get blurry? I was considering these issues and came up with an algorithm that could take the image, with its diffraction rings, and generate an "ideal" image from it, in which each set of diffraction rings is effectively reduced to a single, ideal point of light in the image. So, I'm wondering if my algorithm will actually work on a real diffraction-limited image. I've implemented my algorithm and it does seem to work, at least on simulated test images. If someone can tell me how to include images in a forum posting I'll be glad to show what I've done.Does this kind of algorithm already exist?

Carrickagh · Jul 26, 2008

[Does this kind of algorithm already exist? If your algorithm contains some variation of Besel function then I believe AiP currently has a seeing-limited/diffraction limited application or toolset. Primarily it is a tool for resolution adjustment rather than magnification. Also, RITM-soft has a similar ability, however that is primarily a tool for stellar spectroscopy. The other way of minimizing seeing is to observe at longer wavelengths, since the size of the seeing disc decreases slowly towards the IR while the Airy disc grows larger. Large scopes become diffration-limited at wavelengths longer than a few microns. This will not improve angular resolution (the diffraction-limited resolution is worse than in the visual wavebands) but it does not help to remove the time-dependent distorting effects of the atmosphere.I believe you can post here as jpg. If you are processing stellar images as .fit files that will be more problematic. Transferring them to jpg (rffectively "dumbing" them down) tends to dump a lot of data, including the nuances that some imaging s/w allow.*** <div class="Discussion_UserSignature"> </div>

rogerinnh · Jul 28, 2008

No, my algorithm does not use a Bessel function. <img src="http://sitelife.space.com/ver1.0/Content/images/store/6/13/167462e5-0464-458a-bd40-30f71a24a142.Medium.jpg" alt="" /> {I"m hoping that the above image shows up in full size when displayed on the forum.}The above shows the images I'm processing. I generate the first image o the left by randomly setting indiidual lixel values. The idea of this picture is that it shows what a star field image would look like if there were no ineterference, i.e. if each point-source of light resulted in an individual point-dot on the image (no interference rings, no Airy disk). I then process that first image into the sceond image, effectively taking each pixel and generating the interference pattern that would actually appear in the image. Thus, the energy from the individual pixel is distributed amongst the neary pixels. This is what you would get from a resolution-limited telescope (ideal seeing) with high magnification. The third image is then generated using my image-enhancment algorithm. It analysizes the second image (the one with the Airy disks) and attempts to generate an image in which each Airy disk is reduced to a single pixel, taking all o fthe energy from the Airy disk and placing it into a single pixel. You will notice that it is quite similar to the original image, which is what I'm aiming for. Ideally my algorithm would produce an exact copy of the original image. The fourth image gives a representation of the accuracy of my algorithm. It is basically a display of the differences between the first image and the third image (i.e. my enhanced image). So, at least in my tests where I artificially create a starfield image that includes the diffraction patterns of the "star", it seems to work quite well. Meaning that it potentially provides greater resolution than a diffraction-limited image.

rogerinnh · Jul 28, 2008

Well, my picture didn't show up full size in my prior post. Here are the four pictures... <img src="http://sitelife.space.com/ver1.0/Content/images/store/12/2/5c445769-107a-4136-8214-455187ab7e67.Medium.jpg" alt="" />An "ideal" image of stars, no diffraction.  <img src="http://sitelife.space.com/ver1.0/Content/images/store/1/5/f1aab147-7970-4ab3-95f3-22c2d22e4dbf.Medium.jpg" alt="" /> The same image as the first but with each ideal point shown as it would in a real image, i.e. with diffraction, Airy disks, the energy of each pinpoint of light distributed amongst adjacent pixels. <img src="http://sitelife.space.com/ver1.0/Content/images/store/12/6/dc520e82-4c07-4f4e-86fe-701cb296b143.Medium.jpg" alt="" /> My generated image, using my enhancement algorithm. Pretty close to the original "Ideal" image. http://sitelife.space.com/ver1.0/Forums/# <img src="http://sitelife.space.com/ver1.0/Content/images/store/6/2/36044385-436a-4da6-8207-e10c6160db1a.Medium.jpg" alt="" /> The differences between the original ideal image and the results of my enhancement algroithm.

Carrickagh · Jul 28, 2008

The images are rather interesting. Is the interference that you are generating is a randomized spread of the PSD?Also, are you calculating around a given centroid for a particular "splotch" of light? This is typically how some algorithms identify a star from a given background as the stars and inherent diffraction may be spread over many pixels on a given CCD. The stars are not typically considered to be imaged as a point-source and those algorithms typically take that into account.I'd be curious to see this system tested on an optical bench. Instead of stars you might create an artifical star field on a glass plate. If you had access to a microlithographic system (or simply purchase such services) you could make a plate that holds a pattern of "stars." These would be of known diameters (representative arc-sec) at known distances from one another. Projecting this onto a telescope and then a CCD of known dimension (z" x y" or plate size) you could actually generate a plot of which "stars" fall onto which pixels on the CCD. Then you could play with resolution and utilize your algorithm accordingly. The above technique has been done with some stellar photographic and spectrographic systems. It might be applicable to your system. If you can't access microlithographic services, you could stick pins of known gage size into a cardboard or foam-core blank. If it had some known cartesian coordinate grid then you would know "star" size and relative distance from pinhole to pinhole. This could then serve as a known pattern as you use your algorithm at differing resolutions. It would be less accurate but might at least let you test your software. Once you calibrate against known standard diameters give it a try on the real sky.*** <div class="Discussion_UserSignature"> </div>

rogerinnh · Jul 29, 2008

No, I don't generate a randomized spread of the point-source of light. I generate an approximation to the diffraction pattern (rings of light around the central point) that you would expect to get in an image from a point-source of light. That spread is indeed spread out over a group of adjacent pixels. For my initial tests I'm using a 5 X 5 array of pixels as the simulated diffraction pattern. But it's set up to accommodate any size pattern. And, yes, the algorigthm basically goes through the entire image and "looks for" the expected diffraction pattern. My algorithm "should" work even when there are close stars or even when the entire image consists of multiple overlapping diffraction patterns. Can you point me to those existing algorithms that you mention? I certainly don't want to continue pursuingthis if it's already been done.   I don't have access to any of the equipment that you mention, so at present I'm working with just my simuated star field diffraction patterns.

Carrickagh · Jul 29, 2008

.Can you point me to those existing algorithms that you mention? AiP: http://www.aips.nrao.edu/RITm-soft is currently proprietary. I used to use it at the observatory I worked at on a very specific piece of equipment. If it ever goes on the market it would doubtless be 2-4 years.You don't need special equipment as I initially suggested. A home-made pattern should work nicely.Good luck with your project.*** <div class="Discussion_UserSignature"> </div>

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