OPTIMAL BEAM FORMING FOR LASER BEAM PROPAGATION ...

wherew(z) = w 0[1 +( zz 0) 2] 1/2, (3.21)R(z) = z[1 +( z0) ] 2, (3.22)zζ(z) = tan −1 z z 0, (3.23)w 0 =( ) 1/2 λz0. (3.24)πThe Gaussian beam is the lowest order solution in an infinite solutionfamily of the free space paraxial wave equation. There are higher order solutionssets for the paraxial Helmhotz equation, such as the Hermite-Gaussianbeams in rectangular coordinates and the Laguerre-Gaussian beam in cylindricalcoordinates.3.2.2 Hermite-Gaussian beamsThe Hermite-Gaussian beams are the most widely used solution set of theparaxial wave equation in rectangular coordinates. The complex amplitudefor Hermite-Gaussian beams can be expressed as [67]( ) ( )w 0 √2f l,m (x, y, z) = a l,mw(z) H x √2 ylH m (3.25)w(z) w(z)]× exp[− r2w 2 (z) − jkr22R(z) − jkz + j(l + m + 1)ζ(z) ,where r = √ x 2 + y 2 , H m is the Hermite polynomial of order m:H m+1 (x) = 2xH m (x) − 2mH m−1 (x), (3.26)21