Analytical approximations of low-frequency scattering from homogeneous spheres

Because of its perfect isotropy, an electrically small homogeneous sphere is an important building block in the design and construction of artificial electromagnetic materials, including metamaterials, over a broadest range of frequencies, from microwave and terahertz to optical. The relative permittivity εr of the sphere may, therefore, broadly vary as its real part can be positive (all natural materials at microwave frequencies) or negative (noble metals at optical frequencies), the absolute value large or small and the imaginary part can correspond to a strongly absorbing (|Im εr| > 1) or an almost lossless (Im εr → 0) material. The relative permeability μr may also deviate from unity if the material is magnetic. Though exact and low-frequency analytical solutions for scattering from homogeneous spheres have been available since more than a century, e.g. (J.A. Stratton, Electromagnetic Theory, 1941, pp. 563–573), a complete description of the low-frequency scattering for the whole variety of possible material parameters is still missing. This work closes the gap by providing a complete set of analytical low-frequency approximations on the complex plane of εr.