Try drawing two small black circles on a piece of paper, a couple of millmetres apart, without showing anyone what you have drawn. Stand across the room from your mates and ask what they see - they'll probably report that they see just one dot. Reducing the distance between you will reveal the truth.
Resolution is the separation of images of objects by an optical system. We can calculate the limiting factors.
Key Concepts
Two sources of light may be easily resolved, just resolved or unresolved.
There are two factors that influence the extent to which two sources of light can be resolved:
- The width of the slit
- The wavelength of the light
When light passes through a circular aperture, it diffracts in the same way as when light passes through a narrow slit except that the diffraction is in 2 dimensions. The result is a circle in the centre with rings around the outside.
\(\theta=1.22{\lambda\over b}\)
- \(\theta\) is the angle subtended by the centre and the first minimum (rad)
- \(\lambda\) is the wavelength of light (m)
- \(b\) is the diameter of the aperture (m)
The Rayleigh criterion gives the condition for two point sources to be resolved:
Two points are resolved if the principal maximum of one diffraction pattern coincides with the first minimum of the other. Look for this point of overlap in this intensity graph below.
The angle is the same as that subtented by the two sources of light when the Rayleigh criterion applies. For a fixed separation of sources, we can calculate the minimum distance at which the sources can be resolved:
\(\theta = {\text{separation}\over D}\)
- \(D\) is the perpendicular distance from the centre of the two objects to the slit (m)
Diffraction gratings can be used to resolve two wavelengths of light.
\(R={\lambda\over \Delta \lambda}=mN\)
- \(R\) is the resolvance of the diffraction grating
- \(\lambda\) is the average wavelength (m)
- \(\Delta \lambda\) is the smallest resolvable wavelength difference (m)
- \(m\) is the order of the diffraction
- \(N\) is the total number of slits illuminated
Find out more about diffraction gratings and interference.
Use quizzes to practise application of theory.
START QUIZ!
Two points can just be resolved by the camera in the image below.
Which of the lengths indicated would result in better resolution if increased?
Increasing A will increase the angle subtended by the objects.
Increasing B will reduce the diffracting angle.
Increasing D will increase the detail collected within the aperture.
Two points can just be resolved by the camera in the image below.
Which of the lengths indicated would not change the resolution?
As C is increased both the image separation and diffraction angle increase.
Two points can just be resolved by the camera in the image below.
If the angles are small, then the angle between the principal maximum and first minimum is given by:
If just resolved then the Rayligh criterion applies. The diffracting angle is equal to the angle subtended by the objects.
Two points can just be resolved by the camera in the image below.
If D is reduced the points would no longer be resolved. The points could be resolved by:
Increasing the separation of the objects would mean that the angle subtended was larger. Unfortunately it's impossible for us to move distant stars further apart in the universe!
Two points can just be resolved by the camera in the image below.
A = 5 mm
B = 100 m
D = 1 cm
The wavelength of light used is:
\(θ = {\text{separation}\over \text{distance to aperture}}= {5 \times 10^{-3} \over 100} = 5 \times 10^{-5} \text{ rad}\)
Since the Rayleigh criterion applies, \(θ = 1.22 {λ\over \text {diameter}}\). Equating the two expressions:
\(λ = {θ\times \text{diameter}\over 1.22} = 4.1 \times 10^{-7} \text{ m}\)
Light of wavelength 600 nm passes through the 0.5 cm diameter aperture of an optical instrument. What is the minimum angle that two point objects can subtend at the aperture if they are to be resolved?
\(θ =1.22 {\lambda\over \text{diameter}}=1.22 \times {600 \times 10^{-9}\over 0.5 \times 10^{-2}}\)
\(\theta\approx1\times 10^3\times 10^{-9}\times 10^2\approx 10^{-4}\)
NB: You need to be prepared to perform these calculations without your calculator!
The image shows light diffracted by the lens in the eye.
Two points will be resolved if they subtend an angle:
The graph of intensity and distance indicates that the diagram shown is at the Rayleigh criterion; \(\theta\) is the angle between the principal maximum and the first minimum and the point at which resolution just takes place.
How much of Resolution have you understood?
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