Matching Pixels to the Point Spread Function

The point-spread function of a telescope (ideally, the Airy disk) defines a characteristic dimension for the smallest details in a telescope image. To reproduce all of the detail present in the image, the sample size must be small enough to define the bright central core of the diffraction disk reliably. The Nyquist sampling theorem in communication theory states that in sampling a wave, the sampling frequency must be two times the highest frequency present in the original. Music recorded on CDs is therefore sampled at 44 kHz, a bit more than twice the highest frequency (20 kHz) most people can hear.

Applied to image sampling, the Nyquist theorem suggests that the size of a pixel must be half the diameter of the diffraction disk as defined by its full-width half-maximum dimension. Images sampled at this rate are called "critically sampled," because the image has been broken into just enough pixels to capture all detail in the image.

Images sampled with pixels larger than half the full-width half-maximum of the diffraction disk are undersampled, because some of the fine structure in the telescope image will be lost. Undersampling is not necessarily a bad thing, since it may be a trade-off necessary to cover a large field of view.

Images sampled with more than two pixels across the core of the diffraction disk are oversampled. Oversampling with three to five pixels across the diameter of the full-width half-maximum diffraction disk insures that none of the information present in the continuous telescopic image is lost because the image is broken into discrete samples.

Atmospheric turbulence, telescope shake, poor guiding, and slightly out-of-

focus images often enlarge the diffraction disk to many times the size of the Airy disk. A practical observer matches the pixel size not to the Airy disk, but to the size of the best seeing encountered at the observing site.

To match pixel size and diffraction disk, recall the formula for ¿/FWHM »the diameter of the core region of the diffraction disk:

where A is the aperture of the telescope, F is the telescope focal length, and N is its focal ratio. Because the real point-spread function dPSF is always equal to or larger than the Airy disk of a perfect telescope, the following holds:

Since the condition for critical sampling is that two pixels of dimension ¿/pixel must equal the linear dimension of the diffraction disk at the focus of the telescope,

This implies that the minimum focal length, Fmin, required for critically sampling a telescopic image is:

and that the minimum focal ratio, Nmin, necessary for critically sampling a telescopic image is:

To better understand the implications, consider an example: you have a telescope with an aperture of 200 mm, a focal length of 2,000 mm, a CCD camera with 9 micrometer photosites, and you want to take diffraction-limited images in red light at a wavelength of 630 nanometers. What is the optimum focal length? Converting these units to meters and substituting into the above:

r 200x10"3 x 9xl0-6 ,ocx

0.51 x 630x10

works out to Fmin = 5.6 meters, or 5,600 mm. Under conditions of perfect seeing, using a 3x Barlow lens to raise the telescope's focal length of 2,000 mm to 6,000 mm would give you a slightly longer focal length than the minimum required for critical sampling. However, if a combination of seeing, drive errors, and telescope shake were to triple the effective diameter of the point-spread function, the images would be critically sampled at the/710 focus of the telescope.

Here is another example: you want to take diffraction-limited images of Jupiter in green light (550 nanometers) with a camera that has a CCD with 12-mi-

Figure 1.19 To match the focal length and seeing conditions at large modern telescopes, professional astronomers use CCDs that have large pixels and large pixel counts. This CCD is 50 mm square, and has 24 micron pixels. On an amateur-size telescope, it would grossly undersample the images.

cron pixels using the excellent optics of your 16-inch f/6 Newtonian. What is the best focal ratio to use?

Set up the solution by converting to meter units and substituting:

0.51 x 550x10

The result is N > 43 for critical sampling. To capture all the detail present in the image on a night of exquisite seeing, you will need to enlarge the image of the planet 7 times using eyepiece projection from the//6 focus.

On nights of poor seeing, of course, you would use a lower focal ratio because the blurry image of a star would be much larger than a perfect diffraction disk. By dropping to//30 or even/720, you could use shorter integration times, thereby raising your chances of "freezing" moments of steady air. Today's webcam and video observers acquire hundreds, or even thousands, of images at a critical-sampling focal ratio; scan them to find moments of best seeing; and then combine those best moments into a diffraction-limited image.

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