du / Λ = ndλ / λ
(3).
Now clearly du must be a small fraction of Λ, so that dλ/λ must be many times smaller than 1/n, if the darkest places are to be sensibly black. But the phenomenon will be tolerably well marked if the proportional range of wave-length do not exceed 1/2n, provided, that is, that the distribution of illumination over this range be not concentrated towards the extreme parts.
So far we have supposed the sources at O1, O2 to be mathematically small. In practice, the source is an elongated slit, whose direction requires to be carefully adjusted to parallelism with the reflecting surface or surfaces. By this means an important advantage is gained in respect of brightness without loss of definition, as the various parts of the aperture give rise to coincident systems of bands.
The question of the admissible width of the slit requires consideration. We will suppose that the light issuing from various parts of the aperture is without permanent phase-relations, as when the slit is backed immediately by a flame, or by an incandescent filament. Regular interference can then only take place between light coming from corresponding parts of the two images, and a distinction must be drawn between the two ways in which the images may be situated relatively to one another. In Fresnel’s experiment, whether carried out with the mirrors or with the biprism, the corresponding parts of the images are on the same side; that is, the right of one corresponds to the right of the other, and the left of the one to the left of the other. On the other hand, in Lloyd’s arrangement the reflected image is reversed relatively to the original source; the two outer edges corresponding, as also the two inner. Thus in the first arrangement the bands due to various parts of the slit differ merely by a lateral shift, and the condition of distinctness is simply that the projection of the width of the slit be a small fraction of the width of the bands. From this it follows as a corollary that the limiting width is independent of the order of the bands under examination. It is otherwise in Lloyd’s method. In this case the centres of the systems of bands are the same, whatever part of the slit is supposed to be operative, and it is the distance apart of the images (b) that varies. The bands corresponding to the various parts of the slit are thus upon different scales, and the resulting confusion must increase with the order of the bands. From (1) the corresponding changes in u and b are given by
du = −nλD db/b2;
so that
du/Λ = −n db/b
(4).
If db represents twice the width of the slit, (4) gives a measure of the resulting confusion in the bands. The important point is that the slit must be made narrower as n increases if the bands are to retain the same degree of distinctness.