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 sys tem characterized by an ABCD ray matrix in two naturally linked comple x planes. In the plane defined by beam transmitter parameters OMEGA0 a nd OMEGA, the propagation path is described by a ray line similar to t he ray line in the yyBAR diagram method, whereas the path in the plane of beam receiver parameters THETA and LAMBDA is described by a circul ar arc. In either plane the amplitude, phase, spot size, and radius of curvature of the Gaussian beam are directly related to the modulus an d argument of the complex number designating a particular transverse p lane along the propagation path. These beam parameters also lead to si mple geometric relations for locating the beam waist, Rayleigh range, focal plane, and sister planes, which share the same radius of curvatu re but have opposite signs. Combined with the paraxial wave propagatio n technique based on a Huygens-Fresnel integral and complex ABCD ray m atrices, this geometric approach provides a new and powerful method fo r the analysis and design of laser systems.