OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

3.3 Optical CoherenceThe optical coherence theory is used to describe the statistical fluctuationsof an optical field by using coherence functions, and is very important inthe study of optical beam propagation through turbulence. Basic conceptsand methods of coherent optics are introduced in this section. We follow thenotations in [50].3.3.1 Mutual coherence functionPerhaps the most important concept in statistical optics is the mutual coherence,which is defined as the cross correlation function of two optical fieldsΓ(⃗r 1 , ⃗r 2 ; t 1 , t 2 ) = 〈f ∗ (⃗r 1 , t 1 )f(⃗r 2 , t 2 )〉 , (3.30)where ∗ denotes the complex conjugate and 〈·〉 denotes ensemble average.For stationary fields, the mutual coherence becomesΓ(⃗r 1 , ⃗r 2 ; τ) = 〈f ∗ (⃗r 1 , t)f(⃗r 2 , t + τ)〉 . (3.31)There are two types of coherence, temporal coherence characterized by Γ(⃗r, ⃗r, τ)and spatial coherence characterized by Γ(⃗r 1 , ⃗r 2 , 0). The average field intensityis expressed in terms of the coherence asi(⃗r, t) = Γ(⃗r, ⃗r, 0) = 〈f ∗ (⃗r, t)f(⃗r, t)〉 . (3.32)From the relationship between correlation functions of complex analytic randomprocesses, the cross correlation function Γ(⃗r 1 , ⃗r 2 ; τ) of an analytic func-25