A calculation approach for the virtual source spatial distribution of sub-beams in single-emitter multi-electron-beam...

In this paper, an approach is proposed for computing the spatial distribution characteristics (i.e., positions and sizes) of virtual sources of sub-beams in a single-emitter multi-electron beam system (MEBS). First, the electron trajectories for an arbitrary sub-beam are described by the transfer coefficients up to the fifth order. These coefficients are obtained by tracing only one axial reference ray using the differential algebraic (DA) technique. Then, using these electron trajectories, we derive formulas to describe the virtual source positions of such a sub-beam in the meridional and sagittal planes, respectively. Furthermore, we find that the virtual source position of the off-axis sub-beam differs between these two planes, showing a significant distinction from that of the central sub-beam. Accordingly, for each sub-beam, we define an intermediate virtual source position between its meridional and sagittal virtual source positions. In the plane perpendicular to the optic axis that contains this intermediate virtual source, the virtual image of this sub-beam exhibits a linear magnification or demagnification relationship relative to its emission area. In addition, formulas are derived to describe the curved surface where the defined virtual sources of all sub-beams are located. Moreover, the formulas for calculating the virtual source sizes of sub-beams are also derived using these trajectories described by calculated DA transfer coefficients. Finally, as an example, we have calculated and analyzed a Schottky-emitter MEBS. The spatial distribution characteristics of its virtual sources are obtained using the proposed approach. And the calculation results are cross-checked by employing the ray tracing method.