Transfer-matrix and Green-function quantum-mechanical theory of electronic field emission applied to the simulation of diffraction by a carbon fiber in the Fresnel projection microscope

Field emission is described in the framework of both transfer-matrix and Green-function formalisms. The transfer-matrix methodology is used to compute the scattering in the diffuser and the Green-function formalism to propagate the scattered wave function. The computation procedure takes advantage of the presence of an n-fold symmetry axis. The method is applied to simulate electronic field emission from a nanotip and the beam diffraction by a carbon fiber in the Fresnel projection microscope. The influence of the voltage and tip-sample distance on the diffraction is studied. The simulations confirm the existence of Fraunhofer and Fresnel diffraction, that appear in good agreement with the criteria derived from two-dimensional models. A result reveals the fact that the interaction between the parts of the beam traveling, respectively, through and beside the fiber can shape the diffraction figure in a way not accounted for by simplified models.