Page 243 - Phase Space Optics Fundamentals and Applications

224 Chapter Seven

observation, the squared modulus of the field is given by

sin u 2 sx

I (Q) = I 1 (Q) + I 2 (Q) + 2 I 1 (Q)I 2 (Q) cos (7.20)
u d
In this expression, d is the distance of the plane of observation from
the two-slit plane, and x is the coordinate of the point Q above the z
axis. The fringe spatial frequency is f = s/( d). In Eq. (7.20), I i (Q),
for i = 1 or 2, is the squared modulus of the field at Q contributed
by either of the two slits, individually. For simplicity we let I 1 (Q) =
I 2 (Q) = I 0 and

sin u 2 sx
I (Q) = 2I 0 1 + cos (7.21)
u d
In this form, it is clear that the normalized spatial coherence function

sin u
(s, ) = = V (7.22)
u
plays the role of visibility of the fringes. In an experiment, it may be
better to leave the slit separation s and the distance d fixed, so that
the fringe period is kept fixed. The spatial coherence function may
be varied by changing the distance d 0 of the source plane from the
two-slit plane.

7.5 Stationary Phase Approximation
The diffracted field as described by the Rayleigh-Sommerfeld diffrac-
tion theory is given by

1 z exp(ikR)
(x, y, z) = (x s ,y s , 0) (1 − ikR) 2 dx s dy s
2 R R
A
(7.23)
In this expression, (x s ,y s , 0) is the amplitude distribution of the
field in the diffracting aperture centered at the origin, and R is the
distance between the aperture point and the point of observation R =

2
2
2
r
r
| − s |= (x − x s ) + (y − y s ) + z .
Consider the two-dimensional spatial Fourier transform of the
diffracted field

2
(x, y, z) = ( p, q, 0) exp i ( px + qy + mz) dp dq (7.24)