The fundamental solution to Maxwell's equations may appear in different forms when formulated in different coordinate systems. To solve the scattering at a spherical object, it is more efficient to contemplate the problem in spherical coordinates, so that the vectorial spherical harmonics (VSH) can then be found as the solution to Maxwell's equations. Based on this conclusion, a rigorous regional solver for the Mie scattering problem has been implemented. Using this solver, the scattering properties at spherical particles with varying sizes can be analyzed in a rigorous manner (Fig. 1).

Fig. 1: Scattering at spheres made from fused silica with different radii. From left to right, the radii of the spheres are 0.1 µm, 1 µm, 10 µm respectively. The angular distribution of the scattered light depends strongly on the size of the spheres. (rights: IAP) |

For rigorous multiple-sphere Mie scattering, non-sequential propagation between spheres should be taken into account. In our solver, the VSH propagation between spheres is analytically calculated by the addition theory of VSH, based on which the scattering coefficients of each particle can be solved by the generalized minimal residual method (GMRES); thus, the non-sequential propagation between spheres is considered. Having solved for the scattering coefficients, the scattered field from multiple spheres can be simulated by the generalized Mie theory.

Fig. 2: Scattering field simulation of three spheres with different sphere distance, (a) 200nm, (b) 300nm, (c) 500nm, and (d) 700nm. The input field is a plane wave linearly polarized along x. The spheres are located along the y axis. The scattered field from each sphere becomes gradually more independent from the others as the distance between the spheres increases. (rights: IAP) |