Two related basic transition operators, T1 and T2, are found that transform arbitrary solutions of the parabolic equation of the paraxial approximation into exact monochromatic solutions of the scalar wave equation or of the corresponding Helmholtz equation. The operators realize different boundary conditions. The operator T1 preserves the transverse field distribution of the paraxial approximation in the plane z = 0 for the obtained exact solution. The method is applied to calculate the complete corrections to the paraxial approximation of the fundamental Gaussian beam solutions of the n-dimensional wave equation. The lowest-order correction to the paraxial approximation in the three-dimensional case is found to be in agreement with the result of Agrawal and Pattanayak [ J. Opt. Soc. Am. 69, 575 ( 1979)]. The complete series of corrections on the axis is summed up to a transcendental function and discussed for the three-dimensional case.
© 1992 Optical Society of America
Original Manuscript: October 17, 1990
Revised Manuscript: October 21, 1991
Manuscript Accepted: November 1, 1991
Published: May 1, 1992
A. Wünsche, "Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams," J. Opt. Soc. Am. A 9, 765-774 (1992)