Helmholtz-Gauss waves.

A detailed study of the propagation of an arbitrary nondiffracting beam whose disturbance in the plane z = 0 is modulated by a Gaussian envelope is presented. We call such a field a Helmholtz-Gauss (HzG) beam. A simple closed-form expression for the paraxial propagation of the HzG beams is written as the product of three factors: a complex amplitude depending on the z coordinate only, a Gaussian beam, and a complex scaled version of the transverse shape of the nondiffracting beam. The general expression for the angular spectrum of the HzG beams is also derived. We introduce for the first time closed-form expressions for the Mathieu-Gauss beams in elliptic coordinates and for the parabolic Gauss beams in parabolic coordinates. The properties of the considered beams are studied both analytically and numerically.