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I classify multiphoton absorption into separable, linked, and simultaneousprocesses. The first and second types can be distinguished when therate-equation approximation is valid whereas the third type refers to the casewhen the full description of multiphoton absorption is essential. For thispurpose, rate equations are solved analytically without decay processes whichshows that even if many photons are absorbed the interaction with the lightfield is linear and one has the case of separable multiphoton absorption. Nexta short-pulse approximation is investigated in which I first solve the rateequations without decay processes and then solve only rate equations for theensuing decay. Finally, the full rate equations are examined and a successiveapproximation of the underlying Volterra integral equation of the second kindis derived leading to linked multiphoton absorption by the involved decaywidths. The three methods are applied to a nitrogen atom in intense andultrafast x rays from free-electron lasers (FELs). The linearity theorem barelyapproximates the results in the presence of decay processes which is also notsatisfactorily corrected for by the short-pulse approximation. The successiveapproximation gives excellent agreement with the numerically-exact solution ofthe rate equations.

Nonlinearity in the sequential absorption of multiple photons