Application of optical beams to electrons in graphene

The technique of beam optics is applied to the description of the wave function of Dirac electrons. This approach is illustrated by considering electron transmission through simple nonhomogeneous structures, such as flat and bent p-n junctions and superlattices. We found that a convex p-n junction compresses the beam waist, while a concave interface widens it without loosing its focusing properties. At a flat p-n junction the waist of the transmitted Gaussian beam can be narrowed or widened, depending on the angle of incidence. A general condition is derived for the occurrence of beam collimation in a superlattice which is less stringent than previous discussed.

Figure 1

Illustration of a Gaussian beam with its important parameters.

Figure 2

Potential barrier and the band structure of graphene.

Figure 3

Beam penetration into the flat potential step at $x=0$ for $E<V$.

Figure 4

Diverging beam when penetrating the flat potential step at $x=0$ with $E>V$.

Figure 5

Reflection and transmission at an interface.

Figure 6

Superlattice and an electron beam propagating through it.