OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

When the propagation distance d is much greater than the maximumlinear dimension of both the aperture A and the observe region R, and onlya finite region around the z axis is of interest, the following approximationscan be usedcos θ ≈ 1, (3.10)1r≈ 1 d , (3.11)exp(jkr) ≈ exp [ jk(d + ‖⃗v − ⃗u‖ 2 /(2d)) ] . (3.12)Then we get the Green’s function of Fresnel diffraction as3.2 Optical Beamsh(⃗v, ⃗u) ≃ ejkdjλd ej π λd ‖⃗v−⃗u‖2 . (3.13)3.2.1 Paraxial waves and Gaussian beamWe are interested primarily in the paraxial waves because they form beamlikeoptical waves. The complex amplitude of a paraxial wave can be expressedasf(⃗u, z) = a(⃗u, z)e jkz , (3.14)where a(⃗u, z) is a slowly varying function and ⃗u = (x, y) is the positionvector in plane of constant z. Substituting the paraxial wave function intothe Helmholtz equation, and making the assumption that∂a∂z ≪ ka,∂ 2 a∂ 2 z ≪ k2 a, (3.15)19