We present an ab initio account of the paraxial complex geometrical optics (CGO) in application to scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of the Riccati type. This substantially simplifies the desc...

We "nd conditions for the existence of an elliptic periodic solution of a singular equation that governs the motion of a magnetically focused axially symmetric electron beam with Brillouin #ow by using a monotone iterative scheme starting from a couple of upper and lower solutions on the reversed order. Also, a uniqueness result is proved by using Brouwer degree and index of solutions. Copyrigh...