Two mathematical methods are utilized (one a form of adiabatic approximation, and the other closely related to the Zener method from collision theory) in order to calculate the probability of three-photon ionization when strong counter propagating pulses are tuned very near a two-photon resonant state. In this case the inverted populations predicted by Grischkowsky and Loy for smooth laser pulses lead to larger ionization probabilities than would be obtained for a square pulse of equal peak power and energy per pulse. The line shape of the ionization probability is also quite unusual in this problem. A sharp onset in the ionization probability occurs as the lasers are tuned through the exact unperturbed two-photon resonance. Under proper conditions, the change can be from a very small value to one near unity. It occurs in a very small frequency range determined by the larger of the residual Doppler effect and the reciprocal duration of the pulse. Thus, the line shape retains a Doppler-free aspect even at power levels such that power broadening would dwarf even the full Doppler effect in the case of a square pulse of equal energy and peak power. The same mathematical methods have been used to calculate line …