Reflection and transmission coefficients for finite-sized aperiodic aggregates of spheres

Generalized Mie theory is employed to study the reflection and transmission properties of finite-sized spherical arrays of nanoparticles based on aperiodic geometries. To simulate realistic experimental conditions, a circular aperture is used to create an incident field with a finite beamwidth. The diffracted fields from the circular aperture are expanded in terms of vector spherical wave harmonics, which are then employed to derive the scattered fields using the generalized Mie theory. Expansion of diffracted fields in terms of spherical harmonics also led to new analytical expressions for two important integrals involving Bessel, associated Legendre, and trigonometric functions, which arise in electromagnetic diffraction problems. Subsequently, generalized scattering parameters were defined in terms of far-field specular energy fluxes. To verify the results, the method was applied to a truncated periodic array of spherical gold nanoparticles for which the generalized scattering parameters were compared and found to agree with the scattering parameters obtained for an infinite planar structure subject to periodic boundary conditions. The method was then applied to an aperiodic array of gold nanoparticles based on a Penrose geometry, and the lowest-order photonic resonances were observed in the predicted regions. Furthermore, it was shown that by proper scaling, the photonic resonances can be strategically placed in the plasmonic region of the gold, where they are enhanced due to strong coupling between the plasmonic and photonic modes.