Effect of magnetic flux and of electron momentum on the transmission   amplitude in the Aharonov-Bohm ring
  A characterization of the two-terminal open-ring Aharonov-Bohm interferometer is made by analyzing the phase space plots in the complex transmission amplitude plane. Two types of plots are considered: type I plot which uses the magnetic flux as the variable parameter and type II plot which uses the electron momentum as the variable parameter. In type I plot, the trajectory closes upon itself only when the ratio $R$ of the arm lengths (of the interferometer) is a rational fraction, the shape and the type of the generated flower-like pattern is sensitive to the electron momentum. For momenta corresponding to discrete eigenstates of the perfect ring (i.e. the ring without the leads), the trajectory passes through the origin a certain fixed number of times before closing upon itself, whereas for arbitrary momenta it never passes through the origin. Although the transmission coefficient is periodic in the flux with the elementary flux quantum as the basic period, the phenomenon of electron transmission is shown not to be so when analyzed via the present technique. The periodicity is seen to spread over several flux units whenever $R$ is a rational fraction whereas there is absolutely no periodicity present when $R$ is an irrational number. In type II plot, closed trajectories passing through the origin a number of times are seen for $R$ being a rational fraction. The case R=1 (i.e. a symmetric ring) with zero flux is rather pathological--it presents a closed loop surrounding the origin. For irrational $R$ values, the trajectories never close. 
M.V. Amaresh Kumar and Debendranath Sahoo