OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

The wave number k(⃗r) can be expressed as a scalar function of the refractiveindex n(⃗r)k(⃗r) = k 0 n(⃗r), (3.68)where k 0 represents the average wave number of the medium and n(⃗r) is therefractive index of the medium given by√µ(⃗r)ɛ(⃗r)n(⃗r) = , (3.69)µ 0 ɛ 0where µ and ɛ are the magnetic permeability and dielectric constant of themedium, and µ 0 and ɛ 0 are corresponding average values.The refractiveindex n(⃗r) is a random variable and can be written in the average and fluctuationterms asn(⃗r) = 〈n〉 + n s (⃗r). (3.70)In general cases such as propagation in the atmosphere, we can assume that〈n〉 = 1 . Then the wave equation becomes∇ 2 f(⃗r) + k 2 0 [1 + δ n (⃗r)]f(⃗r) = 0, (3.71)whereδ n (⃗r) = 2n s (⃗r) + n 2 s(⃗r). (3.72)It is difficult to solve this inhomogeneous wave equation analytically. Severalapproximations have been developed for certain turbulence cases andsolutions for these approximations can be derived.35