Page 89 - Introduction to Information Optics

74 2. Signal Processing with Optics

is the spatial coherence function, also known as mutual intensity function, at the
input plane (x, >').
By choosing two arbitrary points <2 t and Q 2 at the input plane, and if r l
and r 2 are the respective distances from Q^ and Q 2 to dZ, the complex light
disturbances at Q l and Q 2 due to dZ can be written as

u, (x, y) =
r
\
and
1 2
, , , C/OUXT ' ,., ,
M 2(x , v ) = exp(//cr 2)
r
\
where I(£, n,) is the intensity distribution of the light source. By substituting
preceding equations in Eq. (2.9), we have

TV // \ i I - visJ 'I/ r-/ / \TJV ^ /^ < r»\
1 (x, v;xv) = expl ik(r, — r 2 )\ d^L. (2.10)
I ; Y Y
/I 1' 2
In the paraxial case, rj — r 2 may be approximated by

where r is the separation between the source plane and the signal plane. Then
Eq. (2.10) can be reduced to
1 ff f k 1
F(x, y; x' /) = -5 /(& i/) exp ^ i - K(x - x') + ifiy - y')~] \ d^ dn (2.11)
r l
JJs ( )
which is known as the Van Cittert-Zernike theorem. Notice that Eq. (2.11)
forms an inverse Fourier transformation of the source intensity distribution.
One of the two extreme cases is by letting the light source become infinitely
large; for example, /(£, fj) « K; then Eq. (2.11) becomes

T(x, y; x'y') = K 16(x - x', y - /), (2. i 2)

which describes a completely incoherent illumination, where Kj is an appropri-
ate constant.
On the other hand, if the light source is vanishingly small; i.e., /(£, rj) %
K, (5(c, rj), Eq. (2.11) becomes

r(x,y;*y) = K 2 , (2.13)