Time-Harmonic Gaussian Beams: Exact Solutions of the Helmhotz Equation in Free Space

Abstract

An exact solution of the Helmholtz equation u_xx + u_yy + u_zz + k²u = 0 is presented, which describes propagation of monochromatic waves in the free space. The solution has the form of a superposition of plane waves with a specific weight function dependent on a certain free parameter a. If ka→∞, the solution is localized in the Gaussian manner in a vicinity of a certain straight line and asymptotically coincides with the famous approximate solution known as the fundamental mode of a paraxial Gaussian beam. The asymptotics of the aforementioned exact solution does not include a backward wave.