Theoretical and numerical investigation of the interaction between phase‐shaped electron probes and plasmonic modes

Abstract

Optical vortices ‐ i.e. electromagnetic fields with phase singularity ‐ are well known objects and have been used for a decade in a wide range of applications in e.g. optics or astrophysics. Recently, it has been demonstrated that such vortices can be created in an electron microscope by tailoring the phase of the beam [1] and these so‐called electron‐vortex‐beams have already proven their efficiency detecting magnetic state in a material or to probe chirality in a crystal [2]. Simultaneously, EEL spectroscopy in the low‐loss region has attracted a tremendous interest due to its efficiency in resolving plasmonic resonance at the nanometer scale [3] and the underlying formalism is now firmly established [4]. However, because of the invariance of the electron probe along the propagation axis, low‐loss EELS remained unable to detect plasmonic optical activity. However, electron‐vortex‐beams constitute a perfect candidate to overcome this limitation and measure the dichroic behavior of plasmons in an electron microscope ‐ as recently pointed out through simulations by Asenjo‐Garcia and García de Abajo [5]. In the present work, we developed a semiclassical formalism describing the interaction between an electron probe with an arbitrary phase profile and a plasmonic mode. Following [6], we used a quasi‐static and classical description of the plasmon resonances while the electron probe is described in a fully quantum way. We showed that the equation ruling this interaction takes the elegant form of a transition matrix ‐ between two electron states mediated by the eigenpotentials of the plasmon modes. Starting from this formalism, we built an analytical model describing the interaction between point charges and a vortex beam, which gave us a good insight into the physics of plasmonic dichroism. Important experimental inputs, such as convergence and collection angles, were considered. We also implemented a Matlab script within MNPBEM [7] in order to compute our equation and investigate the dichroic behavior of arbitrary plasmonic nano‐structures (see Figure 1). In the conference, we will present the theoretical formalism and a wide variety of numerical studies of interactions between different nano‐structures (e.g. helix, rod) and phase shaped electron probes (e.g. vortex beams, HG‐like beams…), with a special emphasis on the experimental feasibility of the proposed geometries.