Alternative angular spectrum derivation of beam-shape coefficients of generalized Lorenz–Mie theory: scattering of light coming from two pinholes

In the scattering theory of light by a spherical particle, the crucial step for solving the scattering problem is to determine the so-called beam-shape coefficients (weighting coefficients of multipolar expansions) for an arbitrary incident electromagnetic field. We present an alternative analytical derivation of such coefficients using the angular spectrum method of field propagation. Different from previous approaches, our derivation requires only recurrence relations and orthogonality properties of associated Legendre functions. We provide closed-form expressions to obtain these coefficients in terms of the distribution of either electric or magnetic field of the secondary source. Our general expressions for obtaining the beam-shape coefficients are applied to the scattering problem in which a sphere is illuminated with light coming from two pinholes. For this problem, the beam-shape coefficients can be analytically obtained and they are merely related to the Cartesian components of the vector spherical harmonics whose arguments are the vector positions from the pinholes to the center of the sphere. Specifically, we simulate the scattering fields when the scatterer is a dielectric sphere (size larger than the wavelength of the incident field) for on- and off-resonance conditions; subwavelength confinement of the electromagnetic field is attained.