| We analyze the phase variations of the microwave field in a TE011 microwave cavity and how these variations affect the frequency of an atomic clock. We analytically solve for the microwave fields in TE011 cavities which are used in atomic fountain clocks. The analytic solutions show significant new terms that are not present in previous two dimensional treatments. The new terms show that cavities with small radii, near 2.1 cm for a 9.2 GHz cavity, have smaller phase shifts than cavities with larger radii. We also show that the three dimensional phase variations near the axis of the cavity can be efficiently calculated with a rapidly converging series of two dimensional finite element calculations. We use finite clement methods to study the large fields and phase shifts associated with the holes in the cavity endcaps. The effects of the phase variations on atoms traversing a cavity are analyzed using the sensitivity function and we present a cavity design that has small phase shifts for all atomic trajectories. For two pi/2 pulses, the proposed cavity has transverse variations of the effective phase that are +/-0.1 murad and produce no systematic frequency offset for a nearly homogeneous and expanding cloud of atoms. |
