Page:EB1911 - Volume 08.djvu/259

function of the telescope is in fact to allow the use of a wider, and therefore more easily measurable, aperture. An interesting modification of the experiment may be made by using light of various wave-lengths.

Since the limitation of the width of the central band in the image of a luminous line depends upon discrepancies of phase among the secondary waves, and since the discrepancy is greatest for the waves which come from the edges of the aperture, the question arises how far the operation of the central parts of the aperture is advantageous. If we imagine the aperture reduced to two equal narrow slits bordering its edges, compensation will evidently be complete when the projection on an oblique direction is equal to ½λ, instead of λ as for the complete aperture. By this procedure the width of the central band in the diffraction pattern is halved, and so far an advantage is attained. But, as will be evident, the bright bands bordering the central band are now not inferior to it in brightness; in fact, a band similar to the central band is reproduced an indefinite number of times, so long as there is no sensible discrepancy of phase in the secondary waves proceeding from the various parts of the same slit. Under these circumstances the narrowing of the band is paid for at a ruinous price, and the arrangement must be condemned altogether.

A more moderate suppression of the central parts is, however, sometimes advantageous. Theory and experiment alike prove that a double line, of which the components are equally strong, is better resolved when, for example, one-sixth of the horizontal aperture is blocked off by a central screen; or the rays quite at the centre may be allowed to pass, while others a little farther removed are blocked off. Stops, each occupying one-eighth of the width, and with centres situated at the points of trisection, answer well the required purpose.

It has already been suggested that the principle of energy requires that the general expression for I² in (2) when integrated over the whole of the plane ξ, η should be equal to A, where A is the area of the aperture. A general analytical verification has been given by Sir G. G. Stokes (Edin. Trans., 1853, 20, p. 317). Analytically expressed—

$\iint _{-\infty }^{+\infty }\mathrm {I} ^{2}d\xi d\eta =\iint dxdy=\mathrm {A}$

(9).

We have seen that I₀² (the intensity at the focal point) was equal to A²/λ²f². If A′ be the area over which the intensity must be I₀² in order to give the actual total intensity in accordance with

$\mathrm {A} ^{\prime }\mathrm {I} _{0}^{2}=\iint _{-\infty }^{+\infty }\mathrm {I} ^{2}d\xi d\eta ,$

the relation between A and A′ is AA′ = λ²f². Since A′ is in some sense the area of the diffraction pattern, it may be considered to be a rough criterion of the definition, and we infer that the definition of a point depends principally upon the area of the aperture, and only in a very secondary degree upon the shape when the area is maintained constant.

4. Theory of Circular Aperture.—We will now consider the important case where the form of the aperture is circular.

Writing for brevity

kξ/f＝p, kη/f＝q,

(1),

we have for the general expression (§ 11) of the intensity

λ²f²I²＝S² + C²

(2),

where

S＝∫∫ sin(px + qy)dx dy,

(3),

C＝∫∫ cos(px + qy)dx dy,

(4).

When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = 0, and C reduces to

C＝∫∫ cos px cos qy dx dy,

(5).

In the case of the circular aperture the distribution of light is of course symmetrical with respect to the focal point p = 0, q = 0; and C is a function of p and q only through √(p² + q²). It is thus sufficient to determine the intensity along the axis of p. Putting q = 0, we get

$\mathrm {C} =\iint \cos pxdxdy=2\int _{-\mathrm {R} }^{+\mathrm {R} }\cos px{\sqrt {(}}\mathrm {R} ^{2}-x^{2})dx,$

R being the radius of the aperture. This integral is the Bessel’s function of order unity, defined by

$\mathrm {J} _{1}(z)={\frac {z}{\pi }}\int _{0}^{\pi }\cos(z\cos \phi )\sin ^{2}\phi d\phi$

(6).

Thus, if x = R cos φ,

$\mathrm {C} =\pi ^{2}\mathrm {R} {\frac {2\mathrm {J} _{1}(p\mathrm {R} )}{p\mathrm {R} }}$

(7);

and the illumination at distance r from the focal point is

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

(8).

The ascending series for J₁(z), used by Sir G. B. Airy (Camb. Trans., 1834) in his original investigation of the diffraction of a circular object-glass, and readily obtained from (6), is

$\mathrm {J} _{1}(z)={\frac {z}{2}}-{\frac {z^{3}}{2^{2}\cdot 4}}+{\frac {z^{5}}{2^{2}\cdot 4^{2}\cdot 6}}-{\frac {z^{7}}{2^{2}\cdot 4^{2}\cdot 6^{2}\cdot 8}}+\ldots$

(9).

When z is great, we may employ the semi-convergent series

$\mathrm {J} _{1}(z)={\sqrt {\bigg .}}\left({\frac {2}{\pi z}}\right)\sin(z-{\tfrac {1}{4}}\pi )\left\{1+{\frac {3\cdot 5\cdot 1}{8\cdot 16}}\left({\frac {1}{z}}\right)^{2}\right.$

$\left.-{\frac {3\cdot 5\cdot 7\cdot 9\cdot 1\cdot 3\cdot 5}{8\cdot 16\cdot 24\cdot 32}}\left({\frac {1}{z}}\right)^{4}+\ldots \right\}$

$+{\sqrt {\bigg .}}\left({\frac {2}{\pi z}}\right)\cos(z-{\tfrac {1}{4}}\pi )\left\{{\frac {3}{8}}\cdot {\frac {1}{z}}-{\frac {3\cdot 5\cdot 7\cdot 1\cdot 3}{8\cdot 16\cdot 24}}\left({\frac {1}{z}}\right)^{3}\right.$

$\left.+{\frac {3\cdot 5\cdot 7\cdot 9\cdot 11\cdot 1\cdot 3\cdot 5\cdot 7}{8\cdot 16\cdot 24\cdot 32\cdot 40}}\left({\frac {1}{z}}\right)^{5}-\ldots \right\}.$

(10).

A table of the values of 2z^-1J₁(z) has been given by E. C. J. Lommel (Schlömilch, 1870, 15, p. 166), to whom is due the first systematic application of Bessel’s functions to the diffraction integrals.

The illumination vanishes in correspondence with the roots of the equation J₁(z) = 0. If these be called z₁ z₂, z₃, ... the radii of the dark rings in the diffraction pattern are

${\frac {f\lambda z_{1}}{2\pi \mathrm {R} }},{\frac {f\lambda z_{2}}{2\pi \mathrm {R} }},\ldots$

being thus inversely proportional to R.

The integrations may also be effected by means of polar co-ordinates, taking first the integration with respect to φ so as to obtain the result for an infinitely thin annular aperture. Thus, if

x＝ρ cos φ, y＝ρ sin φ,

$\mathrm {C} =\iint \cos pxdxdy=\int _{0}^{\mathrm {R} }\int _{0}^{2\pi }\cos(p\rho \cos \theta )\rho d\rho d\theta .$

Now by definition

$\mathrm {J} _{0}(z)={\frac {2}{\pi }}\int _{0}^{{\frac {1}{2}}\pi }\cos(z\cos \theta )d\theta =1-{\frac {z^{2}}{2^{2}}}+{\frac {z^{4}}{2^{2}\cdot 4^{2}}}-{\frac {z^{6}}{2^{2}\cdot 4^{2}\cdot 6^{2}}}+\ldots$

(11).

The value of C for an annular aperture of radius r and width dr is thus

dC＝2 π J0 (pρ) ρ dρ,

(12).

For the complete circle,

$\mathrm {C}$	$={\frac {2\pi }{p^{2}}}\int _{0}^{p\mathrm {R} }\mathrm {J} _{0}(z)zdz$
	$={\frac {2\pi }{p^{2}}}\left\{{\frac {p^{2}\mathrm {R} ^{2}}{2}}-{\frac {p^{4}\mathrm {R} ^{4}}{2^{2}\cdot 4^{2}}}+{\frac {p^{6}\mathrm {R} ^{6}}{2^{2}\cdot 4^{2}\cdot 6^{2}}}-\ldots \right\}$
	$=\pi \mathrm {R} ^{2}\cdot {\frac {2\mathrm {J} _{1}(p\mathrm {R} )}{p\mathrm {R} }}$ as before.

In these expressions we are to replace p by kξ/f, or rather, since the diffraction pattern is symmetrical, by kr/f, where r is the distance of any point in the focal plane from the centre of the system.

The roots of J₀(z) after the first may be found from

${\frac {z}{\pi }}=i-.25+{\frac {.050561}{4i-1}}-{\frac {.053041}{(4i-1)^{3}}}+{\frac {.262051}{(4i-1)^{5}}}$

(13),

and those of J₁(z) from

${\frac {z}{\pi }}=i+.25-{\frac {.151982}{4i+1}}+{\frac {.015399}{(4i+1)^{3}}}-{\frac {.245835}{(4i+1)^{5}}}$

(14),

formulae derived by Stokes (Camb. Trans., 1850, vol. ix.) from the descending series.^[1] The following table gives the actual values:—

i	z/π for J0(z) = 0	z/π for J1(z) = 0
1	·7655	1·2197
2	1·7571	2·2330
3	2·7546	3·2383
4	3·7534	4·2411
5	4·7527	5·2428
6	5·7522	6·2439
7	6·7519	7·2448
8	7·7516	8·2454
9	8·7514	9·2459
10	9·7513	10·2463

In both cases the image of a mathematical point is thus a symmetrical ring system. The greatest brightness is at the centre, where

dC＝2πρ dρ, C＝π R².

For a certain distance outwards this remains sensibly unimpaired and then gradually diminishes to zero, as the secondary waves become discrepant in phase. The subsequent revivals of brightness forming the bright rings are necessarily of inferior brilliancy as compared with the central disk.

The first dark ring in the diffraction pattern of the complete circular aperture occurs when

r/f＝1·2197 × λ/2R

(15).

1. The descending series for J₀(z) appears to have been first given by Sir W. Hamilton in a memoir on “Fluctuating Functions,” Roy. Irish Trans., 1840.