  We study the phase diagram of a superconducting ring threaded by an
Aharonov-Bohm flux and an in-plane magnetic Zeeman field. The simultaneous
presence of both the external flux and the in-plane magnetic field leads to the
competition between the Fulde-Ferrell (FF) phase and the Larkin-Ovchinnikov
(LO) phase. Using the Bogoliubov-de Gennes equation, we investigate the spacial
profile of the order parameter. Both the FF phase and the LO phase are found to
exist stably in this system. The phase boundary is determined by comparing the
free energy. The distortion of the phase diagrams due to the mesoscopic effect
is also studied.
