Kerker Transform: Expanding Fields in a Discrete Basis of Directional Harmonics

We present a linear coordinate transform to expand the solution of scattering and emission problems into a basis of forward and backward directional vector harmonics. The transform provides intuitive algebraic and geometric interpretations of systems with directional scattering/emission across a broad range of wavelength-to-size ratios. The Kerker, generalized Kerker, and transverse Kerker effect as well as other forms of highly directional scattering/emission are easily understood through open and closed loop contours in the complex plane. Furthermore, the theoretical maximum directivity of any scattering/emissive system is easily defined. The transformed far field harmonics have coordinates that are polar-angle invariant, interference between forward and backward harmonics weakly interact, and interference of same type harmonics alters directivity. Examples of highly directional scattering are presented including a Kerker scattering magnetic sphere, a directional scattering photonic nanojet, both under plane wave illumination, as well as generalized backward Kerker and transverse Kerker emission from sub-wavelength spheres that are near-field coupled to emitters. Solutions of scattering/emission under the Kerker transform are contrasted to the traditional Mie expansion for comparison.