Purcell factor and local intensity enhancement in surface-enhanced Raman scattering

We prove that, at any point of any reciprocal structure, the local field intensity enhancement, which is averaged over the incident wave polarizations and the incidence directions, exactly coincides with the radiative part of the Purcell factor, which is averaged over the orientations of the emitter. This result is important for gaining a better understanding of the electromagnetic Raman gain in the surface-enhanced Raman scattering and for the further development of this technique.