  We describe the structure of the time-harmonic electromagnetic field of a
vertical Hertzian electric dipole source radiating over an infinite,
translation invariant two-dimensional electron system. Our model for the
electron flow takes into account the effects of shear and Hall viscosities as
well as an external static magnetic field perpendicular to the sheet. We
identify two wave modes, namely, a surface plasmon and a diffusive mode. In the
presence of an external static magnetic field, the diffusive mode combines the
features of both the conventional and Hall diffusion and may exhibit a negative
group velocity. In our analysis, we solve exactly a boundary value problem for
the time-harmonic Maxwell equations coupled with linearized hydrodynamic
equations for the flat, two-dimensional material. By numerically evaluating the
integrals for the electromagnetic field on the sheet, we find that the plasmon
contribution dominates in the intermediate-field region of the dipole source.
In contrast, the amplitude of the diffusive mode reaches its maximum value in
the near-field region, and quickly decays with the distance from the source. We
demonstrate that the diffusive mode can be distinguished from the plasmon in
the presence of the static magnetic field, when the highly oscillatory plasmon
is gapped and tends to disappear.
