FIG. 25.
The action of the grating can be made clear by means of Fig. 25. Let A, B, C, D represent the equidistant slits in a grating, and let the straight lines to the left of the grating represent at any instant the crests of some simple plane waves coming up to the grating. The small fractions of the original waves emerging from the slits A, B, C, D will spread out from the slits so that the crests of the small wavelets may at any instant be represented by a series of concentric circles, starting from each slit as centre. The series of crests from each slit are represented in the figure.
Now notice that a line PQ parallel to the original waves lies on one of the crests from each slit, and therefore the wavelets will make up a plane wave parallel to the original wave. This may therefore be brought to a focus by means of a convex lens just as if the grating were removed, except that the intensity of the wave is less. But a line, LM, also lies on a series of crests, the crest from A being one wave-length behind that from B, the one from B a wave-length behind that from C, and so on. The wavelets will therefore form a plane wave LM, which will move in the direction perpendicular to itself (i.e. the direction DK) and may be brought to a focus in that direction by means of a lens.
Draw CH and DK perpendicular to LM, and draw CE perpendicular to DK, i.e. parallel to LM. The difference between CH and DK is evidently one wave-length, i.e. DE is one wave-length. If α is the angle between the direction of PQ and LM, DE is evidently equal to CD sin α and therefore one wave-length=CD sin α.
From the ruling of the grating we know the value of CD, and therefore by measuring α we can calculate the wave-length.
We find that a third line RS also lies on a series of crests, and therefore a plane wave sets out in the direction perpendicular to RS. We notice here that the crest from A is two wave-lengths behind that from B, and so on, and therefore if β is the angle between RS and PQ, CD sin β is equal to two wave-lengths.
Similarly we get another plane wave for a three wave-lengths difference, and so on. The intensity of the wavelets falls off fairly rapidly as they become more oblique to their original direction, and therefore the intensity of these plane waves also falls off rather rapidly as they become more oblique to the direction in which PQ goes.
We see that the essential condition for the plane wave to set out in any direction, is that the difference in the distances of the plane wave from two successive slits shall be exactly a whole number of wave-lengths. Should it depart ever so little from this condition we should see, on drawing the line, that there lie on the line an equal number of crests and troughs, and therefore, if a lens focus waves in this direction, the resulting effect is zero. The directions of the waves PQ, LM, RS, &c., will therefore be very sharply defined and will admit of very accurate determination.
Dispersion by Grating.—Evidently the deviations α, β will be greater the greater is DE, i.e. the greater the wave-length, and therefore the light or heat will be "dispersed" into its different wave-lengths as in the prism; but in this case the dispersion is opposite to that in the normal prism, the long waves being dispersed most and the short waves least.