OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

from finite surfaces with small diffraction angle, the mutual coherence andthe cross spectral density in the far field can be derived asW (⃗v 1 , ⃗v 2 ; ν)Γ(⃗v 1 , ⃗v 2 ; τ)≈≈∫∫W (⃗u 1 , ⃗u 2 ; ν)h ∗ (⃗v 1 , ⃗u 1 ; ν)h(⃗v 2 , ⃗u 2 ; ν)d⃗u 1 d⃗u 2 ,(3.48)∫∫A 2 Γ(⃗v 1 , ⃗v 2 ; τ)h ∗ (⃗v 1 , ⃗u 1 )h(⃗v 2 , ⃗u 2 )d⃗u 1 d⃗u 2 . (3.49)A 2where h(⃗v, ⃗u) is the impulse response function of wave propagation and τ =(d 1 − d 2 )/c. When τ = 0, Zernike’s propagation law [50] for the mutualintensity is derived as∫∫J(⃗v 1 , ⃗v 2 ) = J(⃗u 1 , ⃗u 2 )h ∗ (⃗v 1 , ⃗u 1 )h(⃗v 2 , ⃗u 2 )d⃗u 1 d⃗u 2 .A 2 (3.50)For a planar and spatially incoherent source at A, assuming that Fresnelapproximation is valid, the mutual intensity in the far field can be derivedasJ(⃗v 1 , ⃗v 2 ) =( ) 2 ∫1e −j π λd (|⃗v 2| 2 −|⃗v 1 | 2 )i(⃗u)e −jk(⃗v 2−⃗v 1 )·⃗u d⃗u. (3.51)λdAThis is the so called van Citter-Zernike theorem [30].3.3.3 Coherent mode representationWe know that for a stationary optical field f(⃗r, t) in some finite closed domainD in free space, its mutual coherence function Γ(⃗r 1 , ⃗r 2 , τ) and the crossspectral density function W (⃗r 1 , ⃗r 2 , ν) form Fourier transform pairsW (⃗r 1 , ⃗r 2 , ν) =∫ ∞−∞Γ(⃗r 1 , ⃗r 2 , τ)e 2πντ dτ. (3.52)29