Theoretical Study of Surface Plasmon Frequencies in a System of Two Coupled Spheres and Comparison with Experimental Data

We consider the problem of surface plasmon (SP) oscillations in a pair of coupled spherical metallic nanoparticles (MNP) analytically and compare the results with those obtained experimentally as well as by numerical methods: discrete dipole approximation (DDA), boundary integral method and T-matrix. The calculation of SP frequencies of the pairs of spherical MNPs with size less than SP wavelength is reduced to the electrostatic boundary problem and is solved analytically. Such reduction becomes impossible when the system size is comparable with SP wavelength and the retardation effects in this case must be accounted for. Since this problem does not allow exact solution we develop an approximate analytical approach in which we account for retardation effects within each of the spheres, neglecting it in the electromagnetic interaction between the spheres. We prove that this approximation is accurate for interparticle gaps down to 0.1 of sphere diameter. To check the validity of the approximation we performed also numerical calculations based on DDA method for the system of two dielectrically coated small spheres, and the pair of larger spheres for which the retardation effects are essential. Good agreement demonstrated in both cases indicates the applicability of presented analytical approach allowing quick calculation of SP frequencies of coupled spheres. The theoretical results are compared with known experimental data for the pairs of 42 nm and 87 nm particles. In the valuable for biological applications gap range 5 divided by 50 nm there is a good agreement between experimental data and the results of our calculations.