## Thursday, May 24, 2018

### Selected Questions-Answers From All Experts Astronomy Forum (Diffraction and Optical Resolution)

Question: Could you explain in detail how diffraction applies to stellar sources and how one computes the optical resolution, say of a telescope  - using this?

Answer:  Diffraction refers to the bending of light waves as they encounter edges or in this specific case, lenses (say in an astronomical refracting telescope) which cause refraction and the production of a diffraction disk or image.

The diagram below illustrates the problem for two sources and a single rectangular aperture. (For simplicity) On the far right we see the detecting screen with the two separate  (diffraction) patterns superposed, the peaks of each clearly visible.  We basically have two lenses on either side of the aperture.  Two light sources are detected which are very close together in the line of sight. These lenses produce two diffraction patterns on the screen depicted as two overlapping intensity patterns. Note that the central maxima of the two patterns are separated by an angle a which is the same as the angle (q) subtended by the two sources at the slit center. Thus, for each pattern, the principal maximum just falls on the second minimum.

The resolving power or resolution of an optical instrument just means its ability to distinguish the images of two light sources very near to one another. In terms of application, it is the diffraction pattern of the sources via the aperture that sets the upper theoretical limit to the resolving power.
For a rectangular slit the resolving power may be expressed as:

q1 = nl/ d

Where n is the number of minima from the center, l is the wavelength of the light used and d is the slit width.  Hence, in this case we find:  q1 = 2l/ d

For visible light, l = 5.5 x 10 -7 m. In performing the calculation, care must be taken to ensure that l and d are expressed in the same units. For the case depicted the two sources are clearly resolved with the angular separation:

a =  2q1 = 2l/ d

It shouldn’t be difficult to see that the degree of resolution deteriorates as one reduces the angular separation. Of particular interest is the case where a = p, corresponding to the condition a =  q1, for which the images are just barely resolved. This is known as the Rayleigh criterion and occurs when the central maximum of one diffraction pattern just falls on the first minimum of the other, or:

q1  = l/ d

Having considered resolution in the context of the single slit Fraunhofer diffraction pattern we’re now in position to discuss resolving power as it applies to circular apertures.  Such a pattern is arrived at by rotating the single slit about its rotation axis leading to the result shown below: Note we again have secondary maxima and minima but these are now observed as concentric rings around a bright central disc. The latter is called Airy’s disc  after Sir George Airy, and is what the observer actually sees when he focuses his telescope on a distant star. (We do not observe a star as it actually is, it's simply too distant, we can only capture its diffraction disk. The smaller that disk, the better the resolution).

The expression for resolving power is analogous to that for the single rectangular aperture already considered. The chief difference is that n is no longer a whole number. In particular for the Rayleigh criterion:

q 1 = 1.22  l/ D

Where D is now the diameter of the circular aperture. One can see from this that the separation of two light sources will depend very directly on the objective diameter D of the telescope. The larger D the smaller  q1 and hence the better the resolution.

To fix ideas consider the diagram below which shows two stars and their corresponding diffraction patterns for the Rayleigh criterion. Here, one of the stars is on the principal axis and the other is off it. The diagram shows the intensity distributions for sources S1, S2 and what might actually be observed, say through a telescope. Note at this critical threshold the two stars are just barely separated.   As an exercise we can compute the resolving power of the Harry Bayley Observatory Celestron 14 telescope (the one which I am shown using in the accompanying blog image and profile "About Me").
Then: q1 =  1.22  l/ D

q1 = 1.22  l/ D = 1.22 (5.5 x 10 -7 m) / 0.35 m

q1 = 1.91 X 10 -6 rad = 0.40 arc sec

Thus, the C-14 telescope will be able to (theoretically)  resolve double stars separated by as little as 0.40 seconds of arc.

While the apparent angular diameter of a celestial objects (a discrete object, say like a planet) does increase with higher magnification, one does not increase the resolving power at the same time. In effect, one cannot extract any more detail than the diffraction pattern already allows for a star. Further, if one recklessly increases the telescope magnification without regard to the aperture, then one only succeeds in producing a blurred image.

In general, as one increases D, the telescope aperture, the diameter of the Airy disc produced by a distant star is reduced in scale. Thus, we observe star images as smaller and smaller points of light the larger the telescope.