Calculation of resonant cross sections for multiphoton ionization using very-narrow-bandwidth sources

We consider the infinite-order set of coupled, integral equations for the states of an atom in a very-narrow-bandwidth classical radiation field. In the vicinity of a one-photon resonance for two-photon ionization, this set is reduced to a pair of coupled equations describing the response of the atom to the radiation field. This set is derived by dropping terms of the original set whose lowest-order contributions to the radiation-induced shift and width of the resonance are quadratic in the intensity. In the absence of simultaneous absorption and emission, impossible except for two-photon processes from excited states, this pair can be decoupled. The remaining uncoupled integral equation describes the ac Stark effect based on contributions to the shift and width linear in the intensity. These contributions derive from radiative corrections to the intermediate atomic state and depend on the virtual processes of ionization from and recombination into, as well as emission from and reabsorption into, this state. A method for obtaining the exact numerical solution is discussed. An approximate solution is obtained by use of the single-eigenfunction approximation to the Green's function belonging to the response function. This Green's function has a pole at the static (unshifted) position of the resonance; thus, this approximation is quite accurate very near the pole. The order of nonlinearity is calculated as a function of the static detuning through the resonance and is shown to undergo rapid fluctuations about the nonresonance value of 2. The error inherent in the single-eigenfunction approximation to the Green's function is estimated and found to range from about 1% to 27% corresponding to static detunings ranging from about 1 to 25 A, respectively