Page:EB1911 - Volume 08.djvu/260

We may compare this with the corresponding result for a rectangular aperture of width a,

ξ/ƒ =λ/a;

and it appears that in consequence of the preponderance of the central parts, the compensation in the case of the circle does not set in at so small an obliquity as when the circle is replaced by a rectangular aperture, whose side is equal to the diameter of the circle.

Again, if we compare the complete circle with a narrow annular aperture of the same radius, we see that in the latter case the first dark ring occurs at a much smaller obliquity, viz.

r/ƒ = ·7655 × λ/2R.

It has been found by Sir William Herschel and others that the definition of a telescope is often improved by stopping off a part of the central area of the object-glass; but the advantage to be obtained in this way is in no case great, and anything like a reduction of the aperture to a narrow annulus is attended by a development of the external luminous rings sufficient to outweigh any improvement due to the diminished diameter of the central area.^[1]

The maximum brightnesses and the places at which they occur are easily determined with the aid of certain properties of the Bessel’s functions. It is known (see Spherical Harmonics) that

J0′(z)＝−J1(z),

(16);

J2(z)＝(1/z) J1(z) − J1′(z)

(17);

J0(z) + J2(z) = (2/z) J1(z)

(18).

The maxima of C occur when

${\frac {d}{dz}}\left({\frac {\mathrm {J} _{1}(z)}{z}}\right)={\frac {\mathrm {J} _{1}{}^{\prime }(z)}{z}}-{\frac {\mathrm {J} _{1}(z)}{z^{2}}}=0;$

or by 17 when J₂(z) = 0. When z has one of the values thus determined,

${\frac {2}{z}}\mathrm {J} _{1}(z)=\mathrm {J} _{0}(z).$

The accompanying table is given by Lommel, in which the first column gives the roots of J₂(z) = 0, and the second and third columns the corresponding values of the functions specified. If appears that the maximum brightness in the first ring is only about 1/57 of the brightness at the centre.

z	2z−1J1(z)	4z−2J1²(z)
·000000	+1·000000	1·000000
5·135630	− ·132279	·017498
8·417236	+ ·064482	·004158
11·619857	− ·040008	·001601
14·795938	+ ·027919	·000779
17·959820	− ·020905	·000437

We will now investigate the total illumination distributed over the area of the circle of radius r. We have

$\mathrm {I} ^{2}={\frac {\pi ^{2}\mathrm {R} ^{4}}{\lambda ^{2}f^{2}}}\cdot {\frac {4\mathrm {J} _{1}{}^{2}(z)}{z^{2}}}$

(19),

where

z = 2πRr/λƒ

(20).

Thus

$2\pi \int \mathrm {I} ^{2}rdr={\frac {\lambda ^{2}f^{2}}{2\pi \mathrm {R} ^{2}}}\int \mathrm {I} ^{2}zdz=\pi \mathrm {R} ^{2}\cdot 2\int z^{-1}\mathrm {J} _{1}{}^{2}(z)dz.$

Now by (17), (18)

z⁻¹J₁(z) = J₀(z) − J₁′(z);

so that

$z^{-1}\mathrm {J} _{1}{}^{2}(z)=-{\tfrac {1}{2}}{\frac {d}{dz}}\mathrm {J} _{0}{}^{2}-{\tfrac {1}{2}}{\frac {d}{dz}}\mathrm {J} _{1}{}^{2}(z),$

and

$2\int _{0}^{z}z^{-1}\mathrm {J} _{1}{}^{2}(z)dz=1-\mathrm {J} _{0}{}^{2}(z)-\mathrm {J} _{1}{}^{2}(z)$

(21).

If r, or z, be infinite, J₀(z), J₁(z) vanish, and the whole illumination is expressed by πR², in accordance with the general principle. In any case the proportion of the whole illumination to be found outside the circle of radius r is given by

J₀²(z) + J₁²(z).

For the dark rings J₁(z) = 0; so that the fraction of illumination outside any dark ring is simply J₀²(z). Thus for the first, second, third and fourth dark rings we get respectively .161, .090, .062, .047, showing that more than ⁹⁄₁₀ths of the whole light is concentrated within the area of the second dark ring (Phil. Mag., 1881).

When z is great, the descending series (10) gives

${\frac {2\mathrm {J} _{1}(z)}{z}}={\frac {2}{z}}{\sqrt {\big .}}\left({\frac {2}{\pi z}}\right)\sin(z-{\tfrac {1}{4}}\pi )$

(22);

so that the places of maxima and minima occur at equal intervals.

The mean brightness varies as z^-3 (or as r^-3), and the integral found by multiplying it by zdz and integrating between 0 and ∞ converges.

It may be instructive to contrast this with the case of an infinitely narrow annular aperture, where the brightness is proportional to J₀²(z). When z is great,

$\mathrm {J} _{0}(z)={\sqrt {\big .}}\left({\frac {2}{\pi z}}\right)\cos(z-{\tfrac {1}{4}}\pi ).$

The mean brightness varies as z⁻¹; and the integral ∫∞0 J₀²(z)z dz is not convergent.

5. Resolving Power of Telescopes.—The efficiency of a telescope is of course intimately connected with the size of the disk by which it represents a mathematical point. In estimating theoretically the resolving power on a double star we have to consider the illumination of the field due to the superposition of the two independent images. If the angular interval between the components of a double star were equal to twice that expressed in equation (15) above, the central disks of the diffraction patterns would be just in contact. Under these conditions there is no doubt that the star would appear to be fairly resolved, since the brightness of its external ring system is too small to produce any material confusion, unless indeed the components are of very unequal magnitude. The diminution of the star disks with increasing aperture was observed by Sir William Herschel, and in 1823 Fraunhofer formulated the law of inverse proportionality. In investigations extending over a long series of years, the advantage of a large aperture in separating the components of close double stars was fully examined by W. R. Dawes.

The resolving power of telescopes was investigated also by J. B. L. Foucault, who employed a scale of equal bright and dark alternate parts; it was found to be proportional to the aperture and independent of the focal length. In telescopes of the best construction and of moderate aperture the performance is not sensibly prejudiced by outstanding aberration, and the limit imposed by the finiteness of the waves of light is practically reached. M. E. Verdet has compared Foucault’s results with theory, and has drawn the conclusion that the radius of the visible part of the image of a luminous point was equal to half the radius of the first dark ring.

The application, unaccountably long delayed, of this principle to the microscope by H. L. F. Helmholtz in 1871 is the foundation of the important doctrine of the microscopic limit. It is true that in 1823 Fraunhofer, inspired by his observations upon gratings, had very nearly hit the mark.^[2] And a little before Helmholtz, E. Abbe published a somewhat more complete investigation, also founded upon the phenomena presented by gratings. But although the argument from gratings is instructive and convenient in some respects, its use has tended to obscure the essential unity of the principle of the limit of resolution whether applied to telescopes or microscopes.

Fig. 4.

In fig. 4, AB represents the axis of an optical instrument (telescope or microscope), A being a point of the object and B a point of the image. By the operation of the object-glass LL′ all the rays issuing from A arrive in the same phase at B. Thus if A be self-luminous, the illumination is a maximum at B, where all the secondary waves agree in phase. B is in fact the centre of the diffraction disk which constitutes the image of A. At neighbouring points the illumination is less, in consequence of the discrepancies of phase which there enter. In like manner if we take a neighbouring point P, also self-luminous, in the plane of the object, the waves which issue from it will arrive at B with phases no longer absolutely concordant, and the discrepancy of phase will increase as the interval AP

1. Airy, loc. cit. “Thus the magnitude of the central spot is diminished, and the brightness of the rings increased, by covering the central parts of the object-glass.”
2. “Man kann daraus schliessen, was moglicher Weise durch Mikroskope noch zu sehen ist. Ein mikroskopischer Gegenstand z. B, dessen Durchmesser = (λ) ist, und der aus zwei Theilen besteht, kann nicht mehr als aus zwei Theilen bestehend erkannt werden. Dieses zeigt uns eine Grenze des Sehvermögens durch Mikroskope” (Gilbert’s Ann. 74, 337). Lord Rayleigh has recorded that he was himself convinced by Fraunhofer’s reasoning at a date antecedent to the writings of Helmholtz and Abbe.