In this chapter we turn to the formulation of the theory of the interaction of intense laser fields with atoms in the important case where the laser photon energy is much smaller than the ionization potential of the initial atomic state. When the intensity is sufficiently high and the frequency sufficiently low, ionization proceeds as if the laser electric field were quasi-static. In this regime, it is appropriate to make the “strong-field approximation,” or SFA, in which one assumes that an active electron, after having been ionized, interacts only with the laser field and not with its parent core. Using this approximation, Keldysh [1] showed that analytical expressions for the rate of ionization can be obtained when the electric-field amplitude, the laser frequency and the binding energy of the initial state are such that the Keldysh parameter γK defined by Equation (1.8) is much less than unity and the photoelectron does not escape by over-the-barrier ionization (OBI). However, the applicability of the SFA extends beyond this regime and, more importantly, it can be used to investigate high-order ATI and high-order harmonic generation. The SFA also provides a framework in which the physical origin of these processes, embodied in the semi-classical three-step recollision model introduced in Section 1.3, can be understood.

We begin in Section 6.1 by examining the low-frequency limit of the Floquet theory and showing how the total ionization rate of the atom can be obtained using the adiabatic approximation.