A procedure is proposed for generating rigorous closed-form orthonormal bases for the expansion of strongly focused (high-numerical-aperture), monochromatic, electromagnetic fields. The performance of three such bases is tested in terms of a parameter that determines their directional spread, for several truncation orders. Simple example fields corresponding to beams with differing polarizations focused by a thin lens are expanded in terms of these bases.
A closed form formula is found for the Mie scattering coefficients of incident complex focus beams (which are a nonparaxial generalization of Gaussian beams) with any numerical aperture. This formula takes the compact form of multipoles evaluated at a single complex point. Included are the cases of incident scalar fields as well as electromagnetic fields with many polarizations, such as linear, circular, azimuthal and radial. Examples of incident radially and azimuthally polarized beams are presented.
Two bases (one biorthogonal and one orthonormal) are proposed for the expansion of strongly focused (high numerical aperture) scalar monochromatic fields. The performance of these bases is tested and compared, both with each other and with a similar basis proposed by Alonso et al. [Opt. Express14, 6894 (2006)]. It is found that the orthonormal basis proposed herein exhibits the lowest truncation error of these three bases for the same truncation order for the examples considered. Additionally, this basis is advantageous because it allows for the expansion of fields without rotational symmetry.
The Lorenz-Mie scattering of a wide class of focused electromagnetic fields off spherical particles is studied. The focused fields in question are constructed through complex focal displacements, leading to closed-form expressions that can exhibit several interesting physical properties, such as orbital and/or spin angular momentum, spatially-varying polarization, and a controllable degree of focusing. These fields constitute complete bases that can be considered as nonparaxial extensions of the standard Laguerre-Gauss beams and the recently proposed polynomials-of-Gaussians beams. Their analytic form turns out to lead also to closed-form expressions for their multipolar expansion. Such expansion can be used to compute the field scattered by a spherical particle and the resulting forces and torques exerted on it, for any relative position between the field's focus and the particle.
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