Page 83 - Introduction to Information Optics

68 2. Signal Processing with Optics

ui(t)

Fig. 2.1. A wavefront propagates in space.

from several sources, we cannot obtain an exact result unless the degrees of
coherence among the separate sources are taken into account. In such a
situation, it is desirable to obtain a statistical ensemble average for the most
likely result; for example, from any combination of sources. It is therefore more
useful to provide a statistical description than to follow the dynamic behavior
of a wave field in detail.
Let us assume an electromagnetic wave field propagating in space, as
depicted in Fig. 2.1, where u^t) and u 2(t) denote the instantaneous wave
disturbances at positions 1 and 2, respectively. The mutual coherence function
(i.e., cross-correlation function) between these two disturbances can be written as

r)u* 2(t) dt, (2.1)

where the superasterisk denotes the complex conjugate, the < > represents the
time ensemble average, and T is a time delay.
The complex degree of coherence between u^t) and u 2(x) can be defined as

(2.2)

where r u (t) and r 22(r) are the self-coherence functions of u^t) and M 2(f),
respectively.