§ 6. Achromatic Interference Bands.—We have already seen that in the ordinary arrangement, where the source is of white light entering through a narrow slit, the heterogeneity of the light forbids the visibility of more than a few bands. The scale of the various band-systems is proportional to λ. But this condition of things, as we recognize from (2) (see § 5), depends upon the constancy of b, i.e. upon the supposition that the various kinds of light all come from the same place. Now there is no reason why such a limitation need be imposed. If we regard b as variable, we see that we have only to take b proportional to λ, in order to render the band-interval Λ independent of colour. In such a case the system of bands is achromatic, and the heterogeneity of the light is no obstacle to the formation of visible bands of high order.
These requirements are very easily met by the use of Lloyd’s mirrors, and of a diffraction grating (see [Diffraction]) with which to form a spectrum. White light enters the dark room through a slit in the window-shutter, and falls in succession upon a grating and an achromatic lens, so as to form a real diffraction spectrum, or rather a series of such, in the focal plane. The central image and all the lateral coloured images except one are intercepted by a screen. The spectrum which is allowed to pass is the proximate source of light in the interference experiment, and since the deviation of any colour from the central white image is proportional to λ, it is only necessary to arrange the mirror so that its plane passes through the white image in order to realize the conditions for the formation of achromatic bands.
When a suitable grating is at hand, the experiment in this form succeeds very well. If we are satisfied with a less perfect fulfilment of the achromatic conditions, the diffraction spectrum may be replaced by a prismatic one, so arranged that d(λ/b) = 0 for the most luminous rays. The bands are then achromatic in the sense that the ordinary telescope is so. In this case there is no objection to a merely virtual spectrum, and the experiment may be very simply executed with Lloyd’s mirror and a prism of (say) 20° held just in front of it.
The number of black and white bands shown by the prism is not so great as might be expected. The lack of contrast that soon supervenes can only be due to imperfect superposition of the various component systems. That the fact is so is at once proved by observing according to the method of Fizeau; for the spectrum from a slit at a very moderate distance out is seen to be traversed by bands. If the adjustment has been properly made, a certain region in the yellow-green is uninterrupted, while the closeness of the bands increases towards the other end of the spectrum. So far as regards the red and blue rays, the original bands may be considered to be already obliterated, but so far as regards the central rays, to be still fairly defined. Under these circumstances it is remarkable that so little colour should be apparent on direct inspection of the bands. It would seem that the eye is but little sensitive to colours thus presented, perhaps on account of its own want of achromatism.
§ 7. Airy’s Theory of the White Centre.—If a system of Fresnel’s bands be examined through a prism, the central white band undergoes an abnormal displacement, which has been supposed to be inconsistent with theory. The explanation has been shown by Airy (Phil. Mag., 1833, 2, p. 161) to depend upon the peculiar manner in which the white band is in general formed.
“Any one of the kinds of homogeneous light composing the incident heterogeneous light will produce a series of bright and dark bars, unlimited in number as far as the mixture of light from the two pencils extends, and undistinguishable in quality. The consideration, therefore, of homogeneous light will never enable us to determine which is the point that the eye immediately turns to as the centre of the fringes. What then is the physical circumstance that determines the centre of the fringes?
“The answer is very easy. For different colours the bars have different breadths. If then the bars of all colours coincide at one part of the mixture of light, they will not coincide at any other part; but at equal distances on both sides from that place of coincidence they will be equally far from a state of coincidence. If then we can find where the bars of all colours coincide, that point is the centre of the fringes.
“It appears then that the centre of the fringes is not necessarily the point where the two pencils of light have described equal paths, but is determined by considerations of a perfectly different kind.... The distinction is important in this and in other experiments.”
The effect in question depends upon the dispersive power of the prism. If v be the linear shifting due to the prism of the originally central band, v must be regarded as a function of λ. Measured from the original centre, the position of the nth bar is now
v + nλD / b.