d /J1(z)\ J1'(z) J1(z)
-- (-------) = ------ - ----- = 0;
dz \ z / z z^2
or by 17 when J2(z) = 0. When z has one of the values thus determined,
2
- J1(z) = J0(z).
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^-1 J1(z) 4z^-2 J1^2(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
[pi]^2R^4 4J1^2(z)
I^2 = ------------- . ------- (19),
[lambda]^2f^2 z^2
where
z = 2[pi]Rr/[lambda]f (20).
Thus
_ _ _
/ [lambda]^2f^2 / /
2[pi] | I^2rdr = ------------- | I^2zdz = [pi]R^2.2 | z^-1 J1^2(z)dz.
_/ 2[pi]R^2 _/ _/