| C = π²R | 2J1(pR) | (7); |
| pR |
and the illumination at distance r from the focal point is
| · | 4J1² ( | 2πRr | ) | (8). | ||
| I² = | π²R4 | ƒλ | ||||
| λ²f² | ( | 2πRr | ) ² | |||
| ƒλ | ||||||
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
| J1(z) = | z | − | z³ | + | z5 | − | z7 | + ... (9). |
| 2 | 2²·4 | 2²·4²·6 | 2²·4²·6²·8 |
When z is great, we may employ the semi-convergent series
| J1(z) = √( | 2 | ) sin (z − ¼π) { 1 + | 3·5·1 | ( | 1 | ) | ² | − | 3·5·7·9·1·3·5 | ( | 1 | ) | 4 | + ... } |
| πz | 8·16 | z | 8·16·24·32 | z |
| + √( | 2 | ) cos (z − ¼π) {3/8 · 1/z − | 3·5·7·1·3 | ( | 1 | ) | ³ | + | 3·5·7·9·11·1·3·5·7 | ( | 1 | ) | 5 | − ... } (10). |
| πz | 8·16·24 | z | 8·16·24·32·40 | z |
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