The Optics Laboratory

GFR, Classical Derivation

In the classical one-dimensional derivation of the Raman effect, the polarization P of the material depends on the polarizability a of the material and on the incident electric field E, which is provided by the incident light at frequency n _o:

The material itself may oscillate at some frequency n _m about its equilibrium position:

This oscillation may induce a change in the polarizability of the material, which can be approximated by a Taylor expansion of a :

Combining these and simplifying gives, to first order in q:

The first of these terms is the Rayleigh scattering. The second is the Raman scattering, which depends on the Raman activity ¶ a /¶ q. The - or + in the cosines refer to the Stokes and anti-Stokes Raman lines. The two relate to the creation or annihilation of a phonon (vibration quantum) and requisite energy transfer from or to the light.

The above derivation assumes that the only change in the electric field is due to the wave nature of the incident light, and that the wavelength of the light is large compared to the induced oscillation. Thus there is essentially no spatial dependence to the electric field, only time dependence. In NSOM-Raman, these assumptions are not valid. In particular, the close proximity to the aperture gives rise to a localized field gradient, which decays on a length scale comparable to that of the induced oscillation.

To investigate the effect of this field gradient, take a Taylor expansion of the electric field:

Adding this into the previous derivation gives:

The first two terms of (6) are the same as before. The third describes a scattering with the same frequency shifts as the Raman lines, but no dependence on Raman activity. Instead, this scattering term is dependent on the field gradient and the polarizability. The GFR term also exists for both Stokes and anti-Stokes modes.

In summary, the GFR term results from the acknowledgement that the electric field is not a constant, so the product rule for differentiation must be used rather than removing it from the derivative as a constant.

Selection rules for GFR derive from the coefficient of the term in (6). The derivative must be nonzero, which implies that the electric field gradient must be along, or at least have a component along, the vibrating bond. Since the electric field gradient will usually be perpendicular to a metal surface, either supporting molecules or as part of an NSOM probe tip, the vibrations should be normal to this surface.

The strength of the peaks also depends upon the polarizability. A large polarizability will give a large GFR peak. Strong IR modes tend to have large polarizabilities, so these will tend to be strong in GFR also (if they are in the proper direction). Both Raman active and IR active vibration modes can be GFR active.