We consider the interaction of hydrogen-like atoms with a strong laser field and show that the strong field approximation and all its variants may be grouped into a set of families of approximation schemes. This is done by introducing an ansatz describing the electron wave packet as the sum of the initial state wave function times a phase factor and a function which is the perturbative solution in the Coulomb potential of an inhomogeneous time-dependent Schrodinger equation. It is the phase factor that characterizes a given family. In each of these families, the velocity and length gauge version of the approximation scheme lead to the same results at each order in the Coulomb potential. By contrast, irrespective of the gauge, approximation schemes belonging to different families give different results. Furthermore, this new formulation of the strong field approximations allows us to gain deeper insight into the validity of the strong field approximation schemes. In particular, we address two important questions: the role of the Coulomb potential in the output channel and the convergence of the perturbative series in the Coulomb potential. In all the physical situations we consider here, our results are compared to those obtained by solving numerically the time-dependent Schrodinger equation.
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