# Rayleigh’s Criterion.

Here’s an exercise for a physics class. At eye level, make two tiny dots on a whiteboard as close together as possible. Then walk backwards looking at the dots with one eye. Eventually the two dots cannot be resolved as separate.

When the light from either one of the dots reaches our pupil, it will be diffracted through a circular aperture and a diffraction pattern is formed on our retina. When light from both dots reaches our eye, the diffraction patterns overlap.

As a reminder, you will have seen single slit diffraction with a laser, the light passing through a very narrow slit and displayed on a distant screen. The angle in the diagram below is exaggerated for clarity. Notice the central bright maximum is twice as wide as the other secondary maxima on either side of it.

Each tiny element down the length of the slit ( width a) behaves like a point source which can be thought of as producing a circular ripple, like on a pond. These superpose at the screen. When the path difference between contributions at the top and bottom of the slit is one wavelength, (m=1) each contribution has a partner halfway down the slit which has a path difference of half a wavelength. So, every point source has a partner exactly out of phase. At the screen, all these contributions superpose and we get a dark first minimum. So, we see the familiar pattern of a wide central bright maximum and minima on each side, fainter maxima, minima and so on.

Just while we’re here – as a is decreased, pattern smears out (y increased). Narrower slit means broader diffraction pattern in other words.

If we decrease the wavelength (use blue light), y decreases.

Lord Rayleigh ( who told us why the sky is blue and discovered argon) gave us the accepted standard for the measurement of angular resolution. Rayleigh’s criterion is the generally accepted criterion for the minimum resolvable detail – the imaging process is said to be diffraction-limited when the first diffraction minimum of the image of one source point coincides with the maximum of another. This is the definition an examiner might want to see. This image of two circular apertures shows what it means; the middle picture shows two images which are JUST resolved.

More rigorously, see the middle diagram below. First minimum of one diffraction is exactly underneath central maximum of the other.

In exams, they sometimes ask you to either draw this or calculate it. Clearly, it’s wavelength-dependent and also dependent on the width of the slit or aperture diameter (a).

Calculation:

As an example, how far away from two point sources of green paint of wavelength 400nm separated by a distance of 2 mm would you have to stand so they could no longer be resolved as separate?

Solution:

For a circular aperture (our own pupil), we have to invoke the factor 1.22

If the paint were blue we could walk further away and still resolve them, theoretically.

NB, in reality – this is an upper limit for people with perfect vision – most people can’t do as well as this. Most people would only be able to resolve the dots as separate at about 4m, which makes this a good little exercise for a class.