1. Resolution refers to the ability to distinguish between two point sources or details of an object. The resolving power depends on the wavelength of light and the diameter of the aperture. 2. According to Rayleigh's criterion, two point sources are just resolved when the central maximum of one diffraction pattern coincides with the first minimum of the other. 3. For the human eye, the resolving power is about 2x10^-4 rad, meaning two sources can be distinguished when separated by an angle of at least this amount.

1. WAVE PHENOMENA
9.4 Resolution

2. 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

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. 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.

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 S1 enters the eye and is diffracted by the
pupil such that the central maximum of the diffraction pattern is
formed on the retina at P1. Similarly, light from S2 produces a
maximum at P2. 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. 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.

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. 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.

14. 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

15. 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.

16. 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

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. 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.

23. SIGNIFICANCE OF RESOLUTION INTHE
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.

24. 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
RadioTelescopes – the wavelength of radio waves is
large, so the telescope needs to be large so that the
resolution is good