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

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