Page 244 - Phase Space Optics Fundamentals and Applications

Radiometry, Wave Optics, and Spatial Coherence 225

The stationary phase approximation is carried out on the phase
function

2
2

( p, q) = px + qy + mz = px + qy + z 1 − p − q . (7.25)
The direction cosines ( p, q) are chosen to make the first partial
derivatives of
( p, q) equal to zero. These values are substituted in
the second partial derivatives of the phase function, and the higher-
order terms are neglected. This approximated phase function is used
in the evaluation of the double integral of Eq. (7.24). Alternatively, it
is simpler to substitute the values of ( p, q) in the formula given in
2
Born and Wolf, Eq. (20) of Section 3 on double integrals, contained in
Appendix III, entitled “Asymptotic Approximations to Integrals.”
This procedure leads to the following expression for the diffracted
field. 6
−i z exp (ikr)
(x, y, z) =
r r

2 xx s + yy s
× (x s ,y s , 0) exp −i dx s dy s
r
A
(7.26)

The form of this expression suggests that we can rewrite it in the
form

( r, ) = (r, p, q, )
−i exp (ikr)
= m
r

2
× (x s ,y s , 0) exp −i ( px s + qy s ) dx s dy s (7.27)

A
Here, we have used the direction cosines p = x/r, q = y/r, and
m = z/r.
The diffracted field on a hemisphere is simply the spatial Fourier
transform of the field distribution in the aperture as long as the dis-
tance r to the observation point satisfies the far-field condition
2 2
2m a
r >> (7.28)

In this expression m = z/r = cos which is the third direction co-
sine. The symbol a is the radius of the aperture. The z axis is perpen-
dicular to the aperture plane and is measured from the z axis.