It will now be easy to form a distinct idea of the method [p439] which must be followed, in order to calculate the situation and the intensity of the dark and bright stripes, in the different circumstances under which it is proposed to compare the theory with experiment. When the screen is infinitely extended on one side, or is broad enough to allow us to neglect the rays which pass beyond it, we are to determine, for any point P at the distance of the place at which the fringes are to be observed, the result of all the elementary undulations coming from the part AMF only of the incident wave; and comparing the intensities at different collateral points, P, P′, P″, we are to find the situation of the darkest and the brightest points. In this manner we find, for a screen closed on one side, 1st, that the intensity of the light decreases rapidly within the [shadow] beginning from the tangent CAB, and so much the more rapidly as the undulation is smaller; and this in a continuous manner, without any alternations of maxima and minima; 2ndly, that out of the shadow, the intensity of the light, after augmenting considerably to a certain point, which may be called a maximum of the first order, decreases to another point, which is the minimum of the first order: that it increases again to a second maximum, to which succeeds a second minimum, and so forth; 3rdly, that none of these minimums completely vanish, as in the case of fringes produced by the concourse of two luminous pencils of equal intensity, and that the difference between the maxima and minima diminishes in proportion as we go further from the shadow; whence we may understand why the fringes which surround shadows in a homogeneous light, are less marked and less numerous, than those which are obtained by a combination of two mirrors, and those in white light much less brilliant; 4thly, that the intervals been the maxima and minima are unequal, and diminish, as we depart from the shadow, in proportions which remain unaltered, whatever may be the distance from the screen at which we measure them; and 5thly, that the same maxima and minima, calculated for different distances from the screen, are situated in hyperbolas of a sensible curvature, of which the foci are the edge of the screen, and the luminous point. All these consequences of the theory are precisely confirmed by experiment. [p440]
The general formula gives the position of the maxima and minima for any distances whatever of the luminous point from the screen, and from the screen to the micrometer, when the length of the undulation of the light employed is known. In order to submit the theory to a decisive test, instead of determining the length of the undulation by measures of the external fringes, and then employing it in calculations of the same kind, I deduced it from an experiment on diffraction of a very different kind; and after having first verified it by the fringes obtained from two mirrors, of which it represented the breadth within a hundredth part of the truth, I introduced it into the formula which I afterwards compared with 125 measurements of exterior fringes, made under very different circumstances; for the distance of the radiant point from the screen was varied from four inches to six or seven yards, and the distance between the screen and the micrometer was varied from 113th of an inch to more than four yards: and the results of all these comparisons were perfectly satisfactory, as maybe seen in the comparative table published in the XIth volume of the Annales de Chimie et de Physique, p. 339, 343.
When the screen, instead of extending infinitely on one side, is narrow enough to admit some light on that side, not too much weakened by the rapid decrease of intensity produced by obliquity, we must take into the calculation the light on both sides, and find, for each point of the shadow, the general result of all the elementary undulations derived from the points on the right and left. We thus demonstrate that the interior parts of the shadow must be divided by a series of dark and bright stripes, nearly equal in breadth, of which the situations differ very little from those which would be deduced from the approximative formula which has already been given for the same purpose, when they are still separated from the borders of the shadow by an interval of several of their breadths. But when the opaque body is narrow enough, and the micrometer far enough removed for the observed stripes to be very near the exterior stripes, then the results of this more exact calculation, as well as those of experiment, show that the approximation is no longer accurate. The [p441] calculation determines also, with remarkable precision, the singular alterations which the exterior fringes often undergo, when the other series extends beyond the shadow, and mixes its effects with those of the exterior.
I have also verified the theory by examining the fringes derived from a narrow slit of indefinite length; and determining, for the different points enlightened by the luminous pencil, the result of all the elementary undulations derived from the part of the primitive wave comprehended in the breadth of the slit; and I have found a satisfactory agreement between the calculation and the observations, even when the fringes thus obtained afforded the most capricious and apparently irregular appearances.
In this mode of considering the problems relating to diffraction, we have not taken into the calculation the greater or less thickness of the edges of the screen, but merely the extent of the primitive wave which is capable of sending elementary undulations to the points for which we are to find the intensity of illumination; and the opaque substance has no other effect than simply to intercept a part of the wave: for this reason the result is necessarily independent of the nature of the body, of its mass, and of the thickness of its edges. Nevertheless, if the surface of the edges were very extensive, it would be impossible to consider the portion of the wave as quitting the slit without having received some previous modification, and it would be necessary to take into the calculation the small fringes derived from the effect of the remoter parts of the slit. But while the thickness is moderate, or the edges rounded off into a well marked curve, the small fringes derived from this cause may be neglected, and the emerging wave may be considered as of equal intensity throughout, at the moment of its quitting the screen, especially if the intensity of the light is to be calculated for a pretty considerable distance from the screen. We must not, indeed, forget, that according to the reasoning which has been employed, the formulas for diffraction are only sufficiently exact when this distance is very considerable, in comparison with the breadth of an undulation, since it is in this case only that we can neglect the rays that are decidedly oblique, and [p442] can suppose all those, which are essentially concerned in the effect, to be nearly of equal intensity. It is not, however, surprising that the same formulas will give the position of the fringes with sufficient accuracy at small distances from the screen, when its edges are thin, since, the mean breadth of an undulation being but about one fifty thousandth of an inch, a tenth of an inch becomes comparatively a very considerable distance.
These are the three principal kinds of phenomena presented to us by diffraction, when the edges of the screen, or of the opening made in it, are sufficiently extensive to afford fringes independent of any effect from their terminations: and in such cases it is sufficient to make the integral calculation for the plane perpendicular to the edges of the screen only, in order to determine the position of the dark and bright stripes, and their comparative intensities. But when the screen or the opening are of small dimensions in every direction, it becomes necessary to extend the integration to the effects produced in two perpendicular planes: and the results of the calculation agree perfectly with observation, as will appear from two curious instances.