  Electron flow through a quantum point contact in presence of spin-orbit
coupling is investigated theoretically in the context of the scanning gate
microscopy (SGM) conductance mapping. Although in the absence of the floating
gate the spin-orbit coupling does not significantly alter the conductance, we
find that the angular dependence of the SGM images of the electron flow at the
conductance plateaux is substantially altered as the spin-orbit interaction
mixes the orbital modes that enter the quantum point contact. The radial
interference fringes that are obtained in the SGM maps at conductance steps are
essentially preserved by the spin-orbit interaction as backscattering by the
tip preserves the electron spin although the effects of the mode mixing are
visible.
