By "filling factor" we mean the square of the atomaveraged z component of the oscillatory field divided by the square of the total rf field amplitude averaged over the cavity volume. , Phys. Rev. 141, 55 (1966).This relaxation rate is given in Refs. 3 and 4 as y& = TH = (T1) s' e -no Vr, where n is the hydrogen density, 0 is the spin-exchange cross section, and Vr is the mean relative velocity.

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In this paper we show that although in one dimension multiplicity of solutions to the phase reconstruction problem presents a serious problem, in two or more dimensions multiplicity is pathologically rare. We derive from a given solution pair (g,G) necessary conditions for the existence of alternative solution pairs (h,H), and a characterization of their form. The mathematical tools employed are from the theory of functions of two complex variables.

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