Maximal Raman enhancement factor (EF) calculations for hot spots at two metallic spheres

The interaction between two metallic spheres with radius R with external electromagnetic (EM) field polarized in the symmetric z direction is described. Solutions of Laplace equation with bi-spherical coordinates are developed. Hot spots are obtained under the condition that the shortest distance between the two spheres surfaces is very small relative to their radius. Boundary conditions are applied which assume very large real negative value for the dielectric constant of the metallic spheres which is valid for atoms like that of AG or Au at certain frequencies. Under these conditions the EM field is amplified by many orders of magnitudes relative to the incident EM field. Analytical results for maximal Raman enhancement factor (EF ) are obtained as function of various parameters. The present study can be applied to surface-enhanced Raman spectroscopy (SERS) and two-photon induced illumination (TPI-PL) in which the amplification is proportional to the fourth power of the incident EM field.