  We study in detail coherent phase control of femtosecond resonance-mediated
(2+1) three-photon absorption and its dependence on the spectral bandwidth of
the excitation pulse. The regime is the weak-field regime of third perturbative
order. The corresponding interference mechanism involves a group of
three-photon excitation pathways that are on resonance with the intermediate
state and a group of three-photon excitation pathways that are near resonant
with it. The model system of the study is atomic sodium (Na), for which
experimental and numerical-theoretical results are obtained. Prominent among
the results is our finding that with simple proper pulse shaping an increase in
the excitation bandwidth leads to a corresponding increase in the enhancement
of the three-photon absorption over the absorption induced by the (unshaped)
transform-limited pulse. For example, here, a 40-nm bandwidth leads to an
order-of-magnitude enhancement over the transform-limited absorption.
