Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.
Within the framework of second order time dependent perturbation theory and non relativistic dipole approximation, the exact expressions of the radial transition amplitude from 3d, 3p and 3s excited states, were obtained from the integral representation of the Green Coulomb function. Below threshold and above one photon ionization threshold, it can be expressed as an analytic combination of three Appell hypergeometric functions of first order, one of which is degenerated; while at threshold, it can be obtained in terms of an expansion of Gauss confluent hypergeometric functions. The angular coefficients are numerically evaluated and analysed for some available laser wavelengths.
Two-photon ionization of atomic hydrogen with an excess photon is revisited. The non-relativistic dipole approximation and Coulomb Green function (CGF) formalism are applied. Using the CGF Sturmian expansion straightforwardly, one gets the radial transition amplitude in the form of an infinite sum over Gauss hypergeometric functions which are polynomials. It is convergent if all intermediate states are in the discrete spectrum. In the case of two-photon ionization with an excess photon, when photoionization is also possible, intermediate states are in the continuum. We performed the explicit summation over intermediate states and got a simple general expression for the radial transition amplitude in the form of a finite sum over Appell hypergeometric functions, which are not polynomials. An Appell function may be expressed as an infinite sum over Gauss functions. In the case of ionization by an excess photon, Gauss functions are transformed to give a convergent radial transition amplitude for the whole region. The generalized cross sections for two-photon above-threshold ionization of atomic hydrogen in the ground state and excited states calculated by us agree very well with results of previous calculations. Generalized cross sections for two-photon ionization of positronium in the ground state are obtained by scaling those for atomic hydrogen.
The angular distribution of the emitted electrons, following the two-photon ionization of hydrogen-like ions, is studied within the framework of second order perturbation theory and the Dirac equation. Using a density matrix approach, we have investigated the effects which arise from the polarization of the incoming light as well as from the higher multipoles in the expansion of the electron–photon interaction. For medium- and high- Z ions, in particular, the non-dipole contributions give rise to a significant change in the angular distribution of the emitted electrons, if compared with the electric dipole approximation. This includes a strong forward emission while, in the dipole approximation, the electron emission always occurs symmetrically with respect to the plane which is perpendicular to the photon beam. Detailed computations for the dependence of the photoelectron angular distributions on the polarization of the incident light are carried out for the ionization of H, and Xe 53+ and U 91+ (hydrogen-like) ions.
Within the framework of perturbation theory and of dipole approximation, the angular distribution for third order processes can be expressed for any state of quantum numbers (n, l) of the hydrogen atom as θθθ 642 coscoscos dcba +++; whatever the polarisation state (linear or circular). The explicit expressions of the corresponding angular coefficients are given for ns and np initial states. It is shown for the particular case of linear polarization that, the isotropic term a occurs only for initial states with orbital quantum number l greater than zero, and contains exactly twenty one contributing terms for np states; in the case of circularly polarized light, the distribution follows a sin6θ behaviour for ns states, while for np states, it is expressed as a combination of sin4θ and sin6θ. Using the implicit summation technique, nine radial transitions matrix elements are determined and a quantitative analysis of these coefficients is made for 1s, 2s, 3s, 2p and 3p initial states at various wavelengths, including the three photons ionization threshold; for linear polarization, the shapes are strongly dependent on both quantum numbers (n, l); while for circular polarization the shapes remain the same. Furthermore, the corresponding total cross sections obtained indirectly from these coefficients, are globally in agreement with that obtained directly from other methods.
We study the ellipticity and the dependence on the phase lag (lead) (between the semimajor and the semiminor axes of the field components) of the photoelectron angular distribution (PAD) in the non-resonant two-photon ionization of atomic hydrogen. We establish exact analytical expressions for azimuthal PAD for 3s, 3p and 3d excited initial states, marked by the occurrence of an asymmetric term. This term gives rise to elliptic dichroism (ED), which can be obtained in two ways: either with the left (versus right) ellipticity, or with the phase lag (versus lead); for 3s and 3p initial states, it is shown that the quantum phase of continua is directly related to the phase lag, one-photon below-threshold ionization, and indirectly one photon above. Another important result is that the magnetic sublevels, m = 0, for 3p and m = ±1, for 3d, do not contribute to the azimuthal PAD. Our numerical results show, for 3s and 3d, and near-threshold ionization, that the PAD has maxima either along the semimajor or the semiminor axis, while for above-threshold ionization, they are always shifted from these axes. However, the maxima of the corresponding ED coincide with the PAD maxima, while for 3p, they are shifted from the PAD minima. A strong dependence of the ED sign is noted, regardless of the state or the process. However, strong ED signals are obtained for the 3s initial state and below-threshold ionization.
The infinite summations over the complete set of unperturbed atomic states appearing in the Nth-order time-dependent perturbation theory are performed using the Green's function formalism. In order to test the method extended calculations of the cross section for two- and three-photon ionization for atomic hydrogen are made and numerical comparison is shown with the values obtained by different methods. The resonant character of the process is made evident since the poles of the radial part of the Green's function represent the actual energy spectrum of the atom. The relevant transition matrix elements are written in closed form and then computed exactly without averaging over the intermediate states. The cross sections vanish for circularly polarized light near some energies which explains the form of the cross sections for circularly and linearly polarized light.
The general theory of multiphoton ionization of hydrogen atoms by simultaneous absorption of several photons is discussed. In the nonrelativistic dipole approximation, the calculations lead to a recursion relationship concerning angular functions and involve radial matrix elements containing many summations over intermediate states. A general method for exactly evaluating several sums is described in detail and applied to the photo-ionization of atomic hydrogen in the ground state. Numerical results are presented as dispersion curves of the transition rate $\frac{{\ensuremath{\sigma}}_{n}}{{I}^{N\ensuremath{-}1}}$ for a range of wavelengths from 1000 up to 7200 \AA{}.