1911 Encyclopædia Britannica/Diffraction of Light/4

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).

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.^[2]

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.

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.
2. 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.”