1.  Resolution

2.  The astronomers tell us that many of the stars that we observe with the naked eye are in fact binary stars  That is, what we see as a single star actually consists of two stars in orbit about a common centre  Furthermore the astronomers tell us that if we use a "good" telescope then we will actually see the two stars  we will resolve the single point source into its two component parts

3.  So what is it that determines whether or not we see the two stars as a single point source i.e. what determines whether or not two sources can be resolved?

4.  In each of our eyes there is an aperture, the pupil, through which the light enters.  This light is then focussed by the eye lens onto the retina.  But we have seen that when light passes through an aperture it is diffracted  and so if we look at a point source a diffraction pattern will be formed on the retina.

5.  If we look at two point sources then two diffraction patterns will be formed on the retina and these patterns will overlap.  The width of our pupil and the wavelength of the light emitted by the sources will determine the amount that they overlap.  But the degree of overlap will also depend on the angular separation of the two point sources.  We can see this from the next diagram

6. Light from the source S 1 enters the eye and is diffracted by the pupil such that the central maximum of the diffraction pattern is formed on the retina at P 1. Similarly, light from S 2 produces a maximum at P 2. If the two central maxima are well separated then there is a fair chance that we will see the two sources as separate sources. If they overlap then we will not be able to distinguish one source from another. From the diagram we see as the sources are moved close to the eye then the angle  increases and so does the separation of the central maxima.

7.  Rayleigh suggested by how much they should be separated in order for the two sources to be just resolved.  If the central maximum of one diffraction pattern coincides with the first minima of the other diffraction pattern then the two sources will just be resolved.  This is known as the Rayleigh Criterion.

12.  In diagram 3 the two sources will just be resolved  Since this is when the peak of the central maximum of one diffraction pattern coincides with the first minimum of the other diffraction pattern.  This means that the angular separation of the peaks of the two central maxima formed by each source is just the half angular width of one central maximum

13.  It can be shown that = b sin   Where b is the aperture of a rectangular objective  It can also be shown that  = 1.22 /b for a circular apperture and a small angle  Where b is the diameter of the aperture

14.   A circular aperture will resolve two sources if the angle that they subtend at the aperture is greater than or equal to   = 1.22 / b  As mentioned above the angle  is sometimes called the resolving power but should more accurately be called the minimum angle of resolution.  Clearly the smaller  the greater the resolving power.

15.  If we take  the average wavelength of white light to be 500 nm  & the average diameter of the human eye to be 3 mm  Using  = 1.22 / b  the resolving power of the eye is about 2 x 10 ‑ 4 rad.

16.  So suppose that you are looking at car headlights on a dark night and the car is a distance D away.  If the separation of the headlight is say 1.5 m  Then the headlights will subtend an angle 1.5/D at your eye.  Your eye will resolve the headlights into two separate sources if this angle equals 2 x 10 ‑ 4 rad.  i.e. 1.5/D = 2 x 10 ‑ 4  This gives D = 7.5 km.

17.  In other words if the car is approaching you on a straight road then you will be able to distinguish the two headlights as separate sources when the car is 7.5 km away from you.

18.  Actually because of the structure of the retina and optical defects  the resolving power of the average eye is about 3 x 10 ‑ 4 rad.  This means that the car is more likely to be 5 km away before your resolve the headlights.

19.  For a microscope, the actual distance when two point objects are just barely resolvable is known as the resolving power.  It is given by  R.P. = 1.22 D Where D is the diameter of the aperture  As a general rule we can say that it is impossible to resolve details of objects smaller than a wavelength of the radiation being used.  Microscopes that use visible light are not useful to resolve small objects of size in the order of nm.

20.  Electrons have wave properties. The electron microscope uses the wave properties of electrons to produce magnified images of very small objects.  An EM has greater resolving power than a light microscope and can reveal the structure of smaller objects because electrons have wavelengths about 100,000 times shorter than visible light photons. (courtesy to Wikipedia)  The EM can achieve better than 50 pm resolution and magnifications of up to about 10,000,000x whereas ordinary light microscopes are limited by diffraction to about 200 nm resolution and useful magnifications below 2000x. (courtesy to Wikipedia)

22.  One thing astronomers like to have is images that can resolve fine detail in structure.  In optical wavelengths, the resolution of images is limited by the atmosphere, which blurs optical light.  This is why Hubble is such an amazing telescope and why there is a lot of work on adaptive optics to correct for atmospheric blurring for ground telescopes. http://alfalfasurvey.wordpress.com/2009/02/13/radio-telescope-arrays/

23.  However, the limitation that comes from diffraction is much worse for radio telescopes than for visible-light telescopes.  Since the wavelengths of radio waves are thousands of times longer, diffraction causes the resolution of a radio telescope to be thousands of times worse, all other things being equal http://alfalfasurvey.wordpress.com/2009/02/13/radio-telescope-arrays/

24.  Typical resolution limits achieved for optical telescopes are on the order of one arcsecond.  There are sixty arcseconds in one arcminute and sixty arcminutes in one degree. The moon and sun are both about half a degree across  Arecibo, the largest single dish radio telescope, has a resolution of about 3.5 arcminutes at a wavelength of 21 cm  This is almost 200 times worse than the resolution one would have with a small optical telescope.  In order to achieve comparable resolution at 21 cm, you would need a radio telescope with a diameter of over 40 km.  building such a telescope is not feasible. http://alfalfasurvey.wordpress.com/2009/02/13/radio-telescope-arrays/

25.  Radio astronomers overcame this limitation in the following way: they came up with the idea of using multiple smaller antennas to synthesize a larger dish.  This is the motivation behind the VLA and other radio telescope arrays.  Some number of telescopes work together, observing the same source. http://alfalfasurvey.wordpress.com/2009/02/13/radio-telescope-arrays/

26.  The signals from the separate telescopes are then combined and processed so that you can produce an image with a resolution set by the largest baseline (separation between any two dishes) rather than the size of the radio dishes.  This means that you don’t need one giant dish; rather, you can have lots of smaller (and cheaper) telescopes spread over a large area. http://alfalfasurvey.wordpress.com/2009/02/13/radio-telescope-arrays/