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The relationship between the length gauge (LG) and the velocity gauge (VG)exact forms of the photoionization probability amplitude is considered. Ourmotivation for this paper comes from applications of the Keldysh-Faisal-Reiss(KFR) theory, which describes atoms (or ions) in a strong laser field (in thenonrelativistic approach, in the dipole approximation). On the faith of acertain widely-accepted assumption, we present a simple proof that thewell-known LG form of the exact photoionization (or photodetachment)probability amplitude is indeed the gauge-invariant result. In contrast, toobtain the VG form of this probability amplitude, one has to either (i) neglectthe well-known Goeppert-Mayer exponential factor (which assures gaugeinvariance) during all the time evolution of the ionized electron or (ii) putsome conditions on the vector potential of the laser field.

Simple proof of gauge invariance for the S-matrix element of strong-field photoionization