The Refractive Index in Electron Optics

The application of methods of geometrical optics for the investigation of electron trajectories requires the knowledge of the refractive index $\mu$ of the equivalent optical problem. A general formula for $\mu$ in terms of the electrostatic potential $V$ and of the magnetic vector potential A is known. If $\mathbf{A}\ne 0$ this formula contains a unit vector s parallel to the electron velocity, which is not known in general. It is therefore important that s be eliminated. This can be done only if a suitable integral of the equations of motion of the electron is known. If $V$ and A are symmetric about an axis, s may be eliminated from $\mu$ by means of a momentum integral (Glaser). This elimination is generalized in the present paper to include cylindrical fields (not necessarily symmetrical) as well as some more general fields with a transversal field component. The result is reached by interpreting the magnetic scalar potential $U$ as a velocity potential of an ideal fluid and by making use of the corresponding stream function $w$ to obtain an expression for $\mu$.