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”.

Screen Shot

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:Screen Shot

Screen Shot 1

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.

Screen Shot

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.

Screen Shot 2.png

The diagram shows the absorption spectrum of the Sun by Fraunhöfer 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.

Screen Shot 3

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

Screen Shot 4

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.

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