Expansion of arbitrary electromagnetic fields in terms of vector spherical wave functions

Abstract

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 ...

Electromagnetic energy in multilayered spherical particles

“…and outside a multilayered sphere and (ii) the total electromagnetic energy stored within its core and each of its shells. Other types of excitation [12,13,15,16,27,83] require substitution of corresponding expressions of expansion coefficients C γL for those in Eq. (10).…”

Toward Waveguide-Based Optical Chromatography

“…which is not complete because there is a spherical Bessel function embedded in the right-hand side that did not cancel explicitly the one on the left side. We solved this problem for an arbitrary electromagnetic field in 2010 [23,24], by taking the Fourier transform, explicitly canceling the spherical Bessel functions, and obtaining the final result:…”

“…However, in 2010 we discovered how to perform this expansion for any type of electromagnetic field in the Fourier space and provided analytical integrals for the Beam Shape Coefficients for general waveguides. Analytical results from the integrals were found in the cases of cylindrical and rectangular metallic waveguides [23,24].…”

“…Our current analysis focuses on cylindrical metallic waveguides, whose modes are largely similar to those in readily available hollow-core fibers waveguides. However, the work can be easily extended to rectangular waveguides, using recently develop analytical expressions for their modes [24], which can be of interest for typical on-chip microfluidic geometries. Recently, Ref.…”

Photonic nanojets in optical tweezers

“…Note that the explicit cancellation of the radial dependence in g p has been the basis of various approximation techniques for the BSC [5,37]. Recently, the analytical solution for any type of Maxwellian beam has been demonstrated [42].…”