Theory of spatial coherence in near-field Raman scattering

A theoretical study describing the coherence properties of near-field Raman scattering in two- and one-dimensional systems is presented. The model is applied to the Raman modes of pristine graphene and graphene edges. Our analysis is based on the tip-enhanced Raman scheme, in which a sharp metal tip located near the sample surface acts as a broadband optical antenna that transfers the information contained in the spatially-correlated (but non-propagating) near-field to the far-field. The dependence of the scattered signal on the tip-sample separation is explored, and the theory predicts that the signal enhancement depends on the particular symmetry of a vibrational mode. The model can be applied to extract of the correlation length $L_{\rm c}$ of optical phonons from experimentally recorded near-field Raman measurements. Although the coherence properties of optical phonons have been broadly explored in the time and frequency domains, the spatially-resolved approach presented here provides an alternative probe for the study of local material properties at the nanoscale.