Structure of a nonparaxial Gaussian beam near the focus: III. Stability, eigenmodes, and vortices

Exact analytical structurally stable solutions of the Maxwell equations for singular mode beams propagating in free space or a uniform isotropic mediu m are obtained. Approximate boundary conditions are chosen in the form of t he requirement that in the paraxial approximation the fields of nonparaxial mode beams in the waist plane are transformed into the fields of eigenmode s and vortices of weakly guiding optical fibers with the axial symmetry of refractive index. It is shown that optical vortices, in spite of a rather c omplex structure of field distribution, do not experience substantial chang es in the beam form and reproduce, in general features, the field of paraxi al vortices. Linear perturbations of the characteristic parameters of mode beams do not change the structure of their electromagnetic field. Nonparaxi al singular beams have one more important property, in addition to the fact that the structure of these beams in the paraxial approximation is similar to the structure of the fields of eigenmodes in a fiber. The propagation c onstants of eigenmodes of a fiber exactly coincide (in the first approximat ion of perturbation theory) with the projection of the wave vector of a mod e beam on the optical axis (an analog of the propagation constant). The pos sibility of the paraxial transition for nonparaxial mode beams with arbitra ry values of azimuthal and radial indices is shown. The properties of nonpa raxial modes are illustrated by numerous examples. The solutions obtained a nd the results of their analysis can be used for exact matching optical fib ers and laser beams in various applications. (C) 2001 MAIK "Nauka/Interperi odica".