  We extend the powerful frequency-domain analysis of femtosecond two-photon
absorption to the intermediate-field regime, which involves both two- and
four-photon transitions. Consequently, we find a broad family of shaped pulses
that enhance the absorption over the transform-limited pulse. It includes any
spectral phase that is anti-symmetric around half the transition frequency. The
spectrum is asymmetric around it. The theoretical framework and results for Na
are verified experimentally. This work opens the door for rational femtosecond
coherent control in a regime of considerable absorption yields.
