Numerically stable formulation of Mie theory for an emitter close to a sphere

Numerical implementations of Mie theory make extensive use of spherical Bessel functions. These functions are, however, known to overflow/underflow (grow too large/small for floating point precision) for orders much larger than the argument. This is not a problem in applications such as plane wave excitation, as the Mie series converge before these numerical problems arise. However, for an emitter close to the surface of a sphere, the scattered field in the vicinity of the sphere is expressed as slowly converging series, with multipoles up to order 1000 required in some cases. These series may be used to calculate experimentally relevant quantities such as the decay rate of an emitter near a sphere. In these cases, overflow/underflow prevents any calculation in double precision using Mie theory, and alternatives are either computationally intensive (e.g., arbitrary precision calculations) or not accurate enough (e.g., the electrostatics approximation). We present here a formulation of Mie theory that overcomes these limitations. Using normalized Bessel functions where the large growth/decay is extracted as a prefactor, we re-express the Mie coefficients for scattering by spheres in a normalized form. These normalized expressions are used to accurately compute the series for the electric field and decay rate of a dipole emitter near a spherical surface, in cases where the Mie coefficients would normally overflow before any degree of accuracy can be obtained.