  Double Resonance is a powerful method spectroscopic method that provides an
unambiguous assignment of the rigorous quantum numbers of one state of a
transition. However, there is often ambiguity as to the branch ($\Delta J$) of
the transition. The dependence of the intensity of the double resonance signal
on the relative polarization of pump and probe radiation can be used to resolve
this ambiguity and has been used for this in the past. However, the published
theoretical predictions for this ratio are based upon a weak (i.e.
non-saturating) field approximation. In this paper, we present theoretical
predictions for these intensity ratios for cases where the pump field is
strongly saturating, in the two limits of transitions dominated by homogeneous
and inhomogeneous broadening. While saturation, as can be expected, reduces the
magnitude of the polarization effect (driving the intensity ratio closer to
unity), polarization anisotropy remains even with a strongly saturating probe
field in most cases. For the case of an inhomogeneously broadened line, as when
Doppler broadening linewidth dominates over even the power broadened
homogeneous line width, a large fraction of the low pump power anisotropy
remains. Results are presented for both the case of linear and circular pump
and probe field polarizations. The present predictions are compared with
experimental measurements on CH$_4$ ground state $\rightarrow \nu_3 \rightarrow
3\nu_3$ transitions recently reported by de Oliveira et al and found to be in
better agreement than the weak field predictions.
