Since 1908, when Mie reported analytical expressions for the fields scattered by a spherical particle upon incidence of an electromagnetic plane-wave, generalizing his analysis to the case of an arbitrary incident wave has proved elusive. This is due to the presence of certain radially-dependent terms in the equation for the beam-shape coefficients of the expansion of the electromagnetic fields in terms of vector spherical wave functions. Here we show for the first time how these terms can be canceled out, allowing analytical expressions for the beam shape coefficients to be found for a completely arbitrary incident field. We give several examples of how this new method, which is well suited to numerical calculation, can be used. Analytical expressions are found for Bessel beams and the modes of rectangular and cylindrical metallic waveguides. The results are highly relevant for speeding up calculation of the radiation forces acting on spherical particles placed in an arbitrary electromagnetic field, such as in optical tweezers.PACS numbers: 03.50. De,42.25.Bs Gustav Mie, in his celebrated 1908 paper [1], used the vector spherical wave function (VSWF), or partial wave expansion (PWE), of a linear polarized plane-wave to generalize scattering theories to spherical particles of any size, from geometrical optics to the Rayleigh regime, and thus was able to clarify many phenomena, for example in atmospheric physics. He obtained analytical expressions for the expansion coefficients based on special mathematical identities related to a plane-wave. This beam expansion was necessary for applying boundary conditions at a spherical interface. Since then, with the arrival of lasers and optical waveguides, the diversity and complexity of possible incident fields has become enormous so that the restriction to an incident plane-wave has become unrealistic.Different experiments, ranging from particle levitation and trapping [2,3], to the ultrahigh-Q microcavities used in cavity QED experiments [4,5], use different beams. For example, very high numerical aperture beams are used in optical tweezers and confocal microscopy [6-8], evanescent fields in near-field microscopy [9,10], and the waveguide modes of a fiber taper are employed to couple light to the whispering gallery modes of spherical microcavities [11]. Optical forces, absorption, Raman scattering and fluorescence can be greatly enhanced inside spherical microcavities at Mie resonances [12][13][14][15]. Laguerre-Gaussian, Hermite-Gaussian and Bessel beams [16,17], and the internal electromagnetic field of hollow core photonic crystal fibers [18,19], are used to trap and transport particles. The understanding of all these phenomena requires a precise knowledge of the VSWF coefficients of the incident beams. A generalized Lorenz-Mie theory was developed to handle the many variants of beams beyond classical planewaves, and the expansion coefficients in these cases are known as beam shape coefficients (BSC) [20,21]. Moreover, because the VSWFs form an orthogonal complete ...

We used a double tweezers setup to perform ultrasensitive force spectroscopy and observe the forces due to light scattering in a single isolated particle. We demonstrate how to selectively couple the light to the transverse electric ͑TE͒, transverse magnetic ͑TM͒, or both TE and TM microsphere modes by means of the beam polarization and positioning, and to observe correspondent morphology-dependent resonances ͑MDR͒. The results show how the usually assumed azimuthal symmetry in the horizontal plane no longer holds because of the symmetry break caused by the beam polarization. Also, the MDR resonances can change the force values by more than 30-50%.

Optical tweezers are tools made of light that enable contactless pushing, trapping, and manipulation of objects ranging from atoms to space light sails. Since the pioneering work by Arthur Ashkin in the 1970s, optical tweezers have evolved into sophisticated instruments and have been employed in a broad range of applications in life sciences, physics, and engineering. These include accurate force and torque measurement at the femtonewton level, microrheology of complex fluids, single micro- and nanoparticle spectroscopy, single-cell analysis, and statistical-physics experiments. This roadmap provides insights into current investigations involving optical forces and optical tweezers from their theoretical foundations to designs and setups. It also offers perspectives for applications to a wide range of research fields, from biophysics to space exploration.

Using an analytical expression for an integral involving Bessel and Legendre functions, we succeed in obtaining the partial wave decomposition of a general optical beam at an arbitrary location relative to the origin. We also showed that solid angle integration will eliminate the radial dependence of the expansion coefficients. The beam shape coefficients obtained are given by an exact expression in terms of single or double integrals. These integrals can be evaluated numerically on a short time scale. We present the results for the case of a linear-polarized Gaussian beam.

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