Page 241 - Phase Space Optics Fundamentals and Applications

222 Chapter Seven

i.e., independent of the time origin. In Eq. (7.11) the angular brackets
represent the time average. It is useful to define a normalized MCF.
( 1 , r 2 , )
r
12 ( ) ≡ √ (7.12)
r
( 1 , 1 , 0) ( 2 , 2 , 0)
r
r
r
4
By the Cauchy-Schwarz inequality, it can be shown to have the prop-
erty
0 ≤ | 12 ( )| ≤ 1 (7.13)
The Fourier transform of the MCF is called the cross-spectral density
function or mutual spectral density (MSD) and may be defined with an
ensemble average of the transformed fields.

r
∗
( 1 , r 2 , ) = ( r 1 , ) ( r 2 , ) (7.14)
The same symbol is used for the transformed function provided they
are identified by the arguments or as the case may be. In Eq. (7.14),
the angular brackets represent the ensemble average. A normalized
MSD function is defined by,

r
( 1 , r 2 , )
12 ( ) ≡ ( r 1 , r 2 , ) = √ (7.15)
r
r
r
r
( 1 , 1 , ) ( 2 , 2 , )
Again by the Cauchy-Schwarz inequality we have
r
0 ≤| ( 1 , r 2 , )|≤ 1 (7.16)
The MCF is a convenient theoretical quantity to describe the quality
of the fringes in an interference experiment. A term such as contrast
of the fringes is used, but a better term is the visibility of fringes and
is denoted by V, defined by
I max − I min
V = (7.17)
I max + I min
In this expression, the symbol I is used for the time average of the
2
squared modulus of the optical field (r, t), that is, I = | (r, t)| .
Clearly when I max = I min , the visibility is zero; and when I min = 0,
the visibility is unity.

0 ≤ V ≤ 1 (7.18)

It is a good estimate of how dark the minimum of the fringes is.
A typical setup of a simple interference experiment is shown in
5
Fig. 7.2. The source plane shows a noncoherent source in the shape
of a slit of width 2a. The source slit is placed at a distance d 0 from
the double-slit plane with the slits labeled s 1 and s 2 separated by a