Page 85 - Introduction to Information Optics

70 2. Signal Processing with Optics

where c\ and c 2 are the appropriate complex constants. The corresponding
irradiance at P is written by

— i _i_r _i_ o RA / /> i/ I* * I /**..* I t
]
l
M
C
— \ + 2 + 2K C \ C 1 1 1 * 7 I 2"2 ( f —
where / j and / 2 are proportional to the squares of the magnitudes of M 5 (r) and
u 2(t). By letting
r i r 2 j
?!=—-, r, = — and t = f-, — t ?,
c c

the preceding equation can be written as

l J
IP = i + 2 + 2c,c! Re< Ml (f + rM(r)>.
In view of the mutual coherence and self-coherence function, we show that

1 2
/ p = /!-f 7 2 + 2(/ 1/ 2) / Re[y l2(T)].
Thus, we see that for / = /j = / 2, the preceding equation reduces to

in which we see that

K = |y 12(r)|, (2.4)

the visibility measure is equal to the degree of coherence.
Let us now proceed with the Young's experiment further, by letting d
increase. We see that the visibility drops rapidly to zero, then reappears, and
so on, as shown in Fig. 2.3. There is also a variation in fringe frequency as d
varies. In other words, as d increases the spatial frequency of the fringes also
increases. Since the visibility is equal to the degree of coherence, it is, in fact,
the degree of spatial coherence between points Q v and Q 2- If we let the source
size £ deduce to very small, as illustrated in Fig. 2.3, we see that the degree of
(spatial) coherence becomes unity (100%), over the diffraction screen. In this
point of view, we see that the degree of spatial coherence is, in fact, governed
by the source size.
As we investigate further, when the observation point P moves away from
the center of the observing screen, visibility decreases as the path difference
Ar = r 2 — r, increases, until it eventually becomes zero. The effect also depends