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How Does the Wavelength of Light Influence Resolving Power?

In the 1600s, glass was cloudy, greenish and uneven. Through a telescope made of such lenses, Galileo discovered four of Jupiter's moons, appearing as tiny dots moving in the neighborhood of the spot of light that was Jupiter. Telling one tiny dot from another was not easy. As the centuries passed, the quality of glass and the workmanship of optics have improved the situation. Still, any imaging system must answer one question: How well can it distinguish two points of light? That is resolving power.

The Point-Spread Function
- A star is what is called a "point source." A point source is a beam of light that comes from so far away that all the light seems to be coming from one specific location. The sun and planets are not point sources. When light from a point source goes through an optical system, you'd like the image to also be a point. That never happens, partly because of limitations of the optical system and partly because of the laws of physics. The intensity pattern of the image of a point source is called the point-spread function (PSF).
The Rayleigh Criterion
- If you look at the point spread function for a good optical system, you will see that most of the light gets put into a central circular disk. Then there are circular "halos" of light and dark outlining the central blob. So how far apart do two point sources have to be to tell that they are two sources and not one brighter source? The Rayleigh criterion states that two point sources can be distinguished if the center of one is imaged at the first dark halo of its neighbor.
Other Options
- The clarity of a telescope is a function of its resolution---an image like this has some resolved stars and some that merge into a glowing cloud.
  
  Another possible metric is the Dawes criterion, based on the observation that an astronomer can separate two stars when they're separated by only 82 percent of the distance between the peak and the first halo. Another criterion is the Sparrow limit: two point sources so close that there was no lessening of intensity between them cannot be distinguished but at any larger separation there is a little darkening between the two spots. Sparrow's limit ends up being about three-quarters as far as the Rayleigh criterion, but it depends upon the shape of the PSF.
Wavelength
- It would be great if perfect optics would make perfect points, but that's not the way the world works. The physics of light results in a phenomenon called diffraction. Diffraction is a result of the fact that light doesn't exactly travel in straight lines. For precision telescopes and microscopes, diffraction forces a perfect point source to be shaped like a circular spot surrounded by perfect rings, each successive ring lessening in intensity. The central spot has an angular width of 1.22*wavelength/diameter. So as the wavelength goes down, so does the spot size.
The Equations
- If the optics are really good, they are called "diffraction-limited," and the PSF is a central bright spot surrounded by rings called an Airy pattern. The central bright spot is called an Airy disk. Because all the resolution criteria are expressed in terms of the characteristics of the PSF---the Airy disk---and the Airy disk depends on wavelength, the angular resolution can be written as a function of wavelength. The Rayleigh criterion is 1.22*wavelength/diameter. Dawes is 1.01*wavelength/diameter; and Sparrow's Limit is 0.94*wavelength/diameter.

Astronomy