OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

where F (⃗r, ν) is the Fourier transform of field f(⃗r, t):f(⃗r, t) =∫ ∞0F ∗ (⃗r, ν)e −j2πνt dν. (3.39)Then the cross spectral density and the mutual coherence function form aFourier transform pairΓ(⃗r 1 , ⃗r 2 , τ) =W (⃗r 1 , ⃗r 2 , ν) =∫ ∞0∫ ∞−∞W (⃗r 1 , ⃗r 2 ; ν)e −j2πντ dν, (3.40)Γ(⃗r 1 , ⃗r 2 ; τ)e j2πντ dτ. (3.41)The spectral density (or power spectrum) S(⃗r, ν) of light is a special case ofthe cross spectral density W (⃗r 1 , ⃗r 2 , ν) at ⃗r 1 = ⃗r 2S(⃗r, ν) = W (⃗r, ⃗r, ν). (3.42)The spectral degree of coherence is defined as the normalized cross spectraldensity functionµ(⃗r 1 , ⃗r 2 ; ν) =W (⃗r 1 , ⃗r 2 ; ν). (3.43)[W (⃗r 1 , ⃗r 1 ; ν)] 1/2 [W (⃗r 2 , ⃗r 2 ; ν)]1/2The degree of coherence can be measured by the value of |γ(⃗r 1 , ⃗r 2 ; τ)|.Fully coherent and fully incoherent are two special cases that occur whenthe value of |γ(⃗r 1 , ⃗r 2 ; τ)| approaches its two extremes: 1 and 0, respectively.Other states of coherence are called partially coherent.27