## Resolvance of a Diffraction Grating

Illuminating a diffraction grating with monochromatic light from a He/Ne laser shows a typical pattern, out in the photograph to m=3 on both sides. The spots are equally spaced and we notice that the m=2 spot is hidden under the first single slit diffraction minimum – a “missing order”.

The geometry is identical to that for a double slit, d being the distance between the centre of one slit and the next. For a bright maximum:

Unlike two-slit interference, only at very particular angles do the contributions from each slit add constructively. Everywhere else, the contribution from one slit has a partner somewhere else down the grating which cancels its contribution out, hence the very bright spots and a lot of empty space.

You are strongly encouraged to go to the Wolfram Demonstrations Project, download the CDF player and experiment with this demonstration. 1, 2 or many slits -the choice is yours. With 15 slits the pattern is almost indistinguishable from a diffraction grating – screenshot below – the single slit diffraction envelope is clearly shown. Light intensity (y-axis) is proportional to amplitude squared.

A flame test for sodium displays a very bright yellow emission. This emission is due to the sodium D-lines – two lines very close together.

The diagram shows the absorption spectrum of the Sun by FraunhĂ¶fer who labelled the lines. The sodium doublet is seen at wavelengths of about 589.0 nm and 589.6nm.

How could these be resolved using a diffraction grating? We recall that a diffraction grating gives sharp, clear orders.

More accurately, the D lines have wavelength1 = 589.592nm and wavelength2 = 588.995nm

We can find the resolvance or the resolving power required for the doublet to be resolved.

For N lines of the diffraction grating, we can write (without derivation) for the mth order:

So, in this case, for a required resolvance of about 1000, viewing the second order would need N=500 grating lines to be illuminated – even the coarsest of gratings manages this easily – a grating with 1800 lines per mm is quite common, if rather expensive. The larger N the better the resolution. If third, fourth or greater orders are visible, a coarser hence cheaper grating will do.

## 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. Blue, green or red dots work well – try to make them both small and the same size and separation. 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. BTW, the red one blurs closer than the green (or blue) one. Any idea why?

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 criterionÂ 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.