Gaussian and Bessel Beams and Pulses Beyond the Paraxial Approximation

The Gaussian beam is a fundamental concept in optical beam propagation, but is based upon assumptions of paraxial optics. There are several ways of generalizing the concept of che Gaussian beam to the non-paraxial case, but some of these have limitations if a solution is required that is valid for a complete three-dimensional space. A solution that satisfies che requirements is based on the complex source-sink representation. This can be applied in a full vectorial treatment, in which a lineary polarized (LPO1) mode is generated from transverse and orthogonal electric and magnetic dipoles. The individuai TM and TE modes are represented by electric or magnetic dipoles alone. All three of these beam modes reduce to the ordinary TEMOO mode in the paraxial approximation. Axial dipoles result in radial or azimuthal TM or TE modes. Higher order multipoles generate higher order beams that can be expressed in terms of surns over che Laguerre-Gauss beams of paraxial theory. Another important beam is the Bessel beam, recognized as the fundamental solution of the wave equation in cylindrical coordinates. This too can be generalized to a non-paraxial vectorial form.

Ultra-short pulses can be modeled based on these types of beam. Again there are several ways of constructing solutions corresponding to different assumptions for the spatial distribution of the spectral components. A particulary useful type is the isodiffracting pulse, corresponding to the field of a mode-locked laser.

Beams and pulses can be treated by methods based on Fourier or phase space representations, generalized to the non-paraxial case. It can also be shown that there are relationships between these different representations.

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