Preprint of: T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop “Propagation of arbitrary non-paraxial beams by expansion in spherical functions” in D. Neilson (ed) Proceedings of the Australian Institute of Physics 15th Biennial Congress 2002 CD-ROM, Causal Productions (2002) Presented at: Fifteenth AIP Biennial Congress 2002/AOS2002 Sydney, Australia (2002) Propagation of arbitrary non-paraxial beams by expansion in spherical functions T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop Centre for Biophotonics and Laser Science, Department of Physics, The University of Queensland, Brisbane QLD 4072, Australia Phone +617-3365 2422, fax +617-3365 1242 timo@physics.uq.edu.au Abstract While the paraxial approximation (kw ≫ 1, where w is the transverse width of the beam or system) is applicable to many, even most, optical systems, the highly non-paraxial regime, where the paraxial approximation and simple corrections to it fail, is becoming increasingly important with the development of intrinsi- cally non-paraxial optical devices and structures such as nano/micro-cavities, photonic crystals, VCSELs, and others of sizes comparable to, or smaller than, the optical wavelength λ. We calculate the propagation of a highly non-paraxial beam by expansion into spherical functions, which can be considered as fundamental non-paraxial modes. This method is applicable to arbitrary beams. 1 Propagation of beams The propagation of arbitrary monochromatic electromagnetic radiation fields in isotropic homoge- neous media is described by the vector Helmholtz equation: ∇×∇× E + k 2 E = 0 (1) where E is the complex amplitude of the electric field, and k = 2π n/λ is the wavenumber. In prin- ciple, the propagation of such fields can be calculated by solving this differential equation, with suitable boundary conditions. In practice, analytical solutions are elusive, and direct numerical so- lution (such as by using finite-difference methods) are impractical for computational domains large compared to the wavelength. Therefore, when considering the propagation of beams, that is, radiation fields with a general direc- tion of propagation, and originating from a finite angular region, it is natural to consider theoretical simplifications. The usual choice is the use of the scalar paraxial wave equation, which is a close approximation as long as the wavevector at all points is directed almost parallel to the propagation direction of the beam. Analytical solutions can be given in terms of beam modes—eigenfunctions of the scalar paraxial wave equation—or propagation can be calculated using scalar diffraction theory. 1 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by University of Queensland eSpace