Keldysh theory at arbituary photon energies and a multiphoton perspective on tunnel ionization

We have generalized the Keldysh amplitude for strong field ionization in atoms from the long wavelength tunneling regime to the short wavelength regime (down to soft x-rays) to provide a nearly analytic ionization model (for short range potentials). Comparison with numerical results based on simulations of the time-dependent Schrodinger equation show that the model is valid at arbitrary photon energies. In the model we discard the saddle point approximation and evaluate the time-integrals analytically. We will discuss how this model is both useful to predict results in an experiment and give a multiphoton perspective on the transition from multiphoton ionization to tunnel ionization. We include Coulomb corrections in limiting cases and describe difficulties encountered in the multiphoton regime.