# Resolution

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:

1. The width of the slit
2. The wavelength of the light

Circular aperture diffraction

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)

Rayleigh criterion

The Rayleigh criterion gives the condition for two point sources to be resolved:

Two points are resolved if the principal maxima of one diffraction pattern coincides with the first minima 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)

Uses

The ability to resolve sources of light has implications in Astrophysics, for example distinguishing two stars as separate. A DVD laser light has a shorter wavelength than for CDs so that more information can be stored for the same separation of 'pits'. Electron diffract with a de Broglie wavelength of ~0.1 nm, enabling us to image on an atomic scale.

Essentials

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.

Test Yourself

Use quizzes to practise application of theory.