Two-photon ionisation of atomic hydrogen in the ground state with linearly and circularly polarised light is considered. In the region above the one-photon ionisation threshold expressions for the two-photon ionisation transition matrix elements are obtained using the non-relativistic dipole approximation. The two-photon ionisation rate is calculated above and below the one-photon ionisation threshold. Above threshold the total cross section for two-photon ionisation has no resonances and diminishes more rapidly than the photoionisation cross section as the wavelength decreases.
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E Karule (1978) studied this question.