k[xi]/f = p, k[eta]/f = q, (1),
we have for the general expression (S 11) of the intensity
[lambda]^2f^2I^2 = S^2 + C^2 (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 [sqrt](p^2 + q^2). It is thus
sufficient to determine the intensity along the axis of p. Putting q =
0, we get
_ _ _+R
/ / / /
C = | | cos px dx dy = 2 | cos px \/(R^2 - x^2) dx,
_/_/ _/-R
R being the radius of the aperture. This integral is the Bessel's
function of order unity, defined by
_[pi]
z /
J1(z) = ---- | cos(z cos [phi]) sin^2 [phi] d[phi] (6).
[pi] _/0
Thus, if x = R cos [phi],
2J1(pR)
C = [pi]^2R ------- (7);
pR
and the illumination at distance r from the focal point is
/ 2[pi]Rr \
4J1^2( --------- )
[pi]^2R^4 \f[lambda]/
I^2 = ----------- . ----------------- (8).
[lambda]^2f^2 / 2[pi]Rr \^2
( --------- )
\f[lambda]/
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
z z^3 z^5 z^7
J1(z) = - - ----- + --------- - ------------- + ... (9).
2 2^2.4 2^2.4^2.6 2^2.4^2.6^2.8
When z is great, we may employ the semi-convergent series
_
/ / 2 \ | 3.5.1 /1\^2
J1(z) = / ( ----- ) sin (z - 1/4[pi]) |1 + ------ ( - )
\/ \[pi]z/ |_ 8.16 \z/
_
3.5.7.9.1.3.5 /1\^4 |
- ------------- ( - ) + ... |
8.16.24.32 \z/ _|
_
/ / 2 \ | 3 1 3.5.7.1.3 /1\ ^3
+ / ( ----- ) cos (z - 1/4[pi]) | - . - - --------- ( - )
\/ \[pi]z/ |_8 z 8.16.24 \z/
_
3.5.7.9.11.1.3.5.7 /1\^5 |
+ ------------------ ( - ) - ... | ... (10).
8.16.24.32.40 \z/ _|
A table of the values of 2z^-1J1(z) has been given by E. C. J. Lommel
(_Schlomilch_, 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
f[lambda]z1 f[lambda]z2
-----------, -----------, ...
2[pi]R 2[pi]R
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 [phi] so as to obtain the
result for an infinitely thin annular aperture. Thus, if
x = [rho] cos [phi], y = [rho] sin [phi],
_ _ _R _2[pi]
/ / / /
C = | | cos px dx dy = | | cos (p[rho] cos [theta]) [rho]d[rho] d[theta].
_/_/ _/0 _/0