Phase variations in microwave cavities for atomic clocks

We analyse the phase variations of the microwave field in a TE(011) microwave cavity and how these variations affect the frequency of an atomic clock. We analytically solve for the microwave fields in a TE(011) cavity. These 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. The cavities used in atomic clocks have holes in the endcaps, and we use finite element methods to study the large fields and phase shifts near these holes. The effects of the phase variations on atoms traversing a cavity are analysed 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 within +/-0.1 murad and produce no systematic frequency error for a nearly homogeneous and expanding cloud of atoms.