  This work reports the theoretical investigation into the mechanism
underpinning the anomalous Josephson effect. The prototypical system we study
is a ballistic two-dimensional junction containing a two-dimensional Rashba
spin-orbit interaction. In this paper we demonstrate how this two-dimensional
Rashba interaction mixes the spins of adjacent transverse subbands which leads
to significant spin-asymmetry within the junction. Under an external magnetic
field, applied perpendicular to both the axis of transport and the normal
vector of the junction, the sinusoidal Josephson current can then experience an
anomalous phase shift. The role of this spin mixing in the limit of a single
sub-band is initially explored by deriving an analytical expression for the
resulting anomalous phase shift. The analysis is then extended to systems with
multiple occupied sub-bands; in this later section, starting from a microscopic
model, we derive an analytic formula for the resulting anomalous phase shift
indicating it is linear in both magnetic field and spin-orbit strength. We then
verify and validate all findings by comparing them with numerical results
evaluated by a tight-binding model.
