Page:EB1911 - Volume 08.djvu/258

 where y is a positive quantity which is small when u is large. Substituting this, we find cot y = U − y, whence $y = \frac{1}{\mathrm{U}} \left(1 + \frac{y}{\mathrm{U}} + \frac{y^2}{\mathrm{U}^2} + \ldots\right) - \frac{y^3}{3} - \frac{2y^{5}}{15} - \frac{17y^{7}}{315}.$|undefined This equation is to be solved by successive approximation. It will readily be found that

In the first quadrant there is no root after zero, since tan u &gt; u, and in the second quadrant there is none because the signs of u and tan u are opposite. The first root after zero is thus in the third quadrant, corresponding to m = 1. Even in this case the series converges sufficiently to give the value of the root with considerable accuracy, while for higher values of m it is all that could be desired. The actual values of u/&pi; (calculated in another manner by F. M. Schwerd) are 1·4303, 2·4590, 3·4709, 4·4747, 5·4818, 6·4844, &c.

Since the maxima occur when u = (m + )&pi; nearly, the successive values are not very different from $\frac{4}{9\pi^2}, \frac{4}{25\pi^2}, \frac{4}{49\pi^2},$ &c.

The application of these results to (3) shows that the field is brightest at the centre &xi; = 0, &eta; = 0, viz. at the geometrical image of the radiant point. It is traversed by dark lines whose equations are &xi;＝mf&lambda; / a, &eta;＝mf&lambda; / b. Within the rectangle formed by pairs of consecutive dark lines, and not far from its centre, the brightness rises to a maximum; but these subsequent maxima are in all cases much inferior to the brightness at the centre of the entire pattern (&xi; = 0, &eta; = 0).

By the principle of energy the illumination over the entire focal plane must be equal to that over the diffracting area; and thus, in accordance with the suppositions by which (3) was obtained, its value when integrated from &xi; = &infin; to &xi; = +&infin;, and from &eta; = −&infin; to &eta; = +&infin; should be equal to ab. This integration, employed originally by P. Kelland (Edin. Trans. 15, p. 315) to determine the absolute intensity of a secondary wave, may be at once effected by means of the known formula $\int^{+\infty }_{- \infty }\frac{\sin^2u}{u^2}du = \int^{+\infty }_{- \infty }\frac{\sin u}{u}du = \pi.$ It will be observed that, while the total intensity is proportional to ab, the intensity at the focal point is proportional to a2b2. If the aperture be increased, not only is the total brightness over the focal plane increased with it, but there is also a concentration of the diffraction pattern. The form of (3) shows immediately that, if a and b be altered, the co-ordinates of any characteristic point in the pattern vary as a−1 and b−1.

The contraction of the diffraction pattern with increase of aperture is of fundamental importance in connexion with the resolving power of optical instruments. According to common optics, where images are absolute, the diffraction pattern is supposed to be infinitely small, and two radiant points, however near together, form separated images. This is tantamount to an assumption that &lambda; is infinitely small. The actual finiteness of &lambda; imposes a limit upon the separating or resolving power of an optical instrument.

This indefiniteness of images is sometimes said to be due to diffraction by the edge of the aperture, and proposals have even been made for curing it by causing the transition between the interrupted and transmitted parts of the primary wave to be less abrupt. Such a view of the matter is altogether misleading. What requires explanation is not the imperfection of actual images so much as the possibility of their being as good as we find them.

At the focal point (&xi; = 0, &eta; = 0) all the secondary waves agree in phase, and the intensity is easily expressed, whatever be the form of the aperture. From the general formula (2), if A be the area of aperture,

The formation of a sharp image of the radiant point requires that the illumination become insignificant when &xi;, &eta; attain small values, and this insignificance can only arise as a consequence of discrepancies of phase among the secondary waves from various parts of the aperture. So long as there is no sensible discrepancy of phase there can be no sensible diminution of brightness as compared with that to be found at the focal point itself. We may go further, and lay it down that there can be no considerable loss of brightness until the difference of phase of the waves proceeding from the nearest and farthest parts of the aperture amounts to &lambda;.

When the difference of phase amounts to &lambda;, we may expect the resultant illumination to be very much reduced. In the particular case of a rectangular aperture the course of things can be readily followed, especially if we conceive ƒ to be infinite. In the direction (suppose horizontal) for which &eta; = 0, &xi;/ƒ = sin &theta;, the phases of the secondary waves range over a complete period when sin &theta; = &lambda;/a, and, since all parts of the horizontal aperture are equally effective, there is in this direction a complete compensation and consequent absence of illumination. When sin &theta; = 3⁄2&lambda;/a, the phases range one and a half periods, and there is revival of illumination. We may compare the brightness with that in the direction &theta; = 0. The phase of the resultant amplitude is the same as that due to the central secondary wave, and the discrepancies of phase among the components reduce the amplitude in the proportion $\frac{1}{3\pi}\int^{+\frac{3}{2}\pi}_{-\frac{3}{2}\pi}\cos \phi d\phi : 1$ or -2⁄3&pi; : 1; so that the brightness in this direction is 4⁄9&pi;2 of the maximum at &theta; = 0. In like manner we may find the illumination in any other direction, and it is obvious that it vanishes when sin &theta; is any multiple of &lambda;/a.

The reason of the augmentation of resolving power with aperture will now be evident. The larger the aperture the smaller are the angles through which it is necessary to deviate from the principal direction in order to bring in specified discrepancies of phase—the more concentrated is the image.

In many cases the subject of examination is a luminous line of uniform intensity, the various points of which are to be treated as independent sources of light. If the image of the line be &xi; = 0, the intensity at any point &xi;, &eta; of the diffraction pattern may be represented by

the same law as obtains for a luminous point when horizontal directions are alone considered. The definition of a fine vertical line, and consequently the resolving power for contiguous vertical lines, is thus independent of the vertical aperture of the instrument, a law of great importance in the theory of the spectroscope.

The distribution of illumination in the image of a luminous line is shown by the curve ABC (fig. 3), representing the value of the function sin2u/u2 from u = 0 to u = 2&pi;. The part corresponding to negative values of u is similar, OA being a line of symmetry.

Let us now consider the distribution of brightness in the image of a double line whose components are of equal strength, and at such an angular interval that the central line in the image of one coincides with the first zero of brightness in the image of the other. In fig. 3 the curve of brightness for one component is ABC, and for the other OA′C′; and the curve representing half the combined brightnesses is E′BE. The brightness (corresponding to B) midway between the two central points AA’ is ·8106 of the brightness at the central points themselves. We may consider this to be about the limit of closeness at which there could be any decided appearance of resolution, though doubtless an observer accustomed to his instrument would recognize the duplicity with certainty. The obliquity, corresponding to u = &pi;, is such that the phases of the secondary waves range over a complete period, i.e. such that the projection of the horizontal aperture upon this direction is one wave-length. We conclude that a double line cannot be fairly resolved unless its components subtend an angle exceeding that subtended by the wave-length of light at a distance equal to the horizontal aperture. This rule is convenient on account of its simplicity; and it is sufficiently accurate in view of the necessary uncertainty as to what exactly is meant by resolution.

If the angular interval between the components of a double line be half as great again as that supposed in the figure, the brightness midway between is ·1802 as against 1·0450 at the central lines of each image. Such a falling off in the middle must be more than sufficient for resolution. If the angle subtended by the components of a double line be twice that subtended by the wave-length at a distance equal to the horizontal aperture, the central bands are just clear of one another, and there is a line of absolute blackness in the middle of the combined images.

The resolving power of a telescope with circular or rectangular aperture is easily investigated experimentally. The best object for examination is a grating of fine wires, about fifty to the inch, backed by a sodium flame. The object-glass is provided with diaphragms pierced with round holes or slits. One of these, of width equal, say, to one-tenth of an inch, is inserted in front of the object-glass, and the telescope, carefully focused all the while, is drawn gradually back from the grating until the lines are no longer seen. From a measurement of the maximum distance the least angle between consecutive lines consistent with resolution may be deduced, and a comparison made with the rule stated above.

Merely to show the dependence of resolving power on aperture it is not necessary to use a telescope at all. It is sufficient to look at wire gauze backed by the sky or by a flame, through a piece of blackened cardboard, pierced by a needle and held close to the eye. By varying the distance the point is easily found at which resolution ceases; and the observation is as sharp as with a telescope. The