Resolution 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
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?
Our Eyes 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.
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
Light from the source S1 enters the eye and is diffracted by thepupil such that the central maximum of the diffraction pattern isformed on the retina at P1. Similarly, light from S2 produces amaximum at P2. If the two central maxima are well separated thenthere is a fair chance that we will see the two sources as separatesources. If they overlap then we will not be able to distinguish onesource from another. From the diagram we see as the sources aremoved close to the eye then the angle increases and so does theseparation of the central maxima.
THE RAYLEIGH CRITERION 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.
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
From m = d sin , if m =1, Sin = for small angles i.e. = /d where d is the width of the slit through which the light from the sources passes.
However, from the diagram of the two sources above this is just the angle that the two sources subtend at the slit. Hence we conclude that two sources will be resolved by a slit if the angle that they subtend at the slit is greater than or equal to /d
We started this discussion on optical resolution with the eye as the "slit". But the eye is not a slit but a circular aperture. So to find the "resolving power" of the eye we have to know the half-angular width of the central maximum of the diffraction formed by a circular aperture.
Tricky Calculations It can be shown that = a sin Where a 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
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.
Example 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.
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.
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.
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.
Microscopes 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.
Significance of Resolution in the Development of Devices CDs – the information on the drive is read by lasers reflecting from the surface. A laser with a higher resolution can read more information from the surface of a disk.
Significance of Resolution in the Development of Devices Electron Microscope – electrons are propagated with a very short wavelength through a species. Hence, the image produced is very clear due to the microscope’s high resolution power
Significance of Resolution in the Development of Devices Radio Telescopes – the wavelength of radio waves is large, so the telescope needs to be large so that the resolution is good