Nonlinear polarisation inside metals: A mathematical study of the free-electron model

A mathematical study of the free-electron model proposed by Bloembergen et al. to derive the nonlinear polarisation inside a metal is achieved. Using a local conformal mapping as a tool, we show that this model, which has been declared ambiguous by many authors, is in fact perfectly sound from a mathematical point of view. The values of the tangential and normal components of the surface nonlinear polarization are fully given and the boundary conditions at the metal-vacuum interface are rigorously established. Thus, we do not introduce phenomenological parameters such as the coefficients a and b which appear in the works of Rudnick and Stern, or Sipe and Stegeman. Moreover, we do not need the classical hypothesis which assumes that the nonlinear surface polarization is placed in vacuum.