Geometrical representation of Gaussian beams propagating through complex paraxial optical systems

Geometric relations are used to study the propagation environment of a Gaussian beam wave propagating through a complex paraxial optical system characterized by an ABCD ray matrix in two naturally linked complex planes. In the plane defined by beam transmitter parameters Ω(o) and Ω, the propagation path is described by a ray line similar to the ray line in the y? diagram method, whereas the path in the plane of beam receiver parameters θ and Λ is described by a circular arc. In either plane the amplitude, phase, spot size, and radius of curvature of the Gaussian beam are directly related to the modulus and argument of the complex number designating a particular transverse plane along the propagation path. These beam parameters also lead to simple geometric relations for locating the beam waist, Rayleigh range, focal plane, and sister planes, which share the same radius of curvature but have opposite signs. Combined with the paraxial wave propagation technique based on a Huygens-Fresnel integral and complex ABCD ay matrices, this geometric approach provides a new and powerful method for the analysis and design of laser systems.