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International Journal of the Physical Sciences

Multiphoton angular distribution: Occurrence of an isotropic term and effect of light polarization

Mamadou Faye, S T Wane, Samba Ndiaye

Published Sep 4, 2010• 1 citation
Publisher

Abstract

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.

DOI
10.5897/ijps.9000388

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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.
Total and differential two-photon ionization cross sections are derived in the framework of perturbation theory, assuming a simple one-electron atomic model. It is shown that for an unpolarized target the angular distribution of the ejected electrons has the general form $\mathrm{a}+b{cos}^{2}\ensuremath{\theta}+c{cos}^{4}\ensuremath{\theta}$, whatever the quantum numbers nnn, lll, of the initial energy level, and the state of polarization (linear or circular) of the incident radiation. The total transition rates for circularly and for linearly polarized light are quite different from each other. For an arbitrary nl\mathrm{nl}nl state, the theoretically allowed maximum value of their ratio is 32\frac{3}{2}23​. Further analytical and numerical results are presented for hydrogen. The Appendix contains several useful summation formulas for spherical harmonics.
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.
We study in some detail the photoelectron angular distributions in two-colour three-photon ionization of the ground state hydrogen atom in its dependence on the polarizations of the two electromagnetic fields. We treat the particular case of photons propagating in the same direction and having elliptic polarizations with the same axes. The azimuthal distributions presented refer to the electrons emitted with momenta orthogonal to the photons direction. The calculations are done in nonrelativistic dipole approximation, in velocity gauge, using third-order perturbation theory. We show that the photoelectron azimuthal angular distributions have a fourfold symmetry only for identical linear or circular polarizations. In all the other cases the distribution has a twofold symmetry.
The infinite summations over the complete set of unperturbed atomic states appearing in the NNNth-order perturbation theory for multiphoton ionization are performed using the Dalgarno-Lewis method. The relevant transition matrix elements are written in a closed integral form exhibiting all the analytic properties of the amplitude as a function of incident photon energy. The cross sections for two- and three-photon ionization for atomic hydrogen are calculated for both linearly and circularly polarized light with a wide range of photon energy spectrum including the near resonance, and numerical comparison is made with the values obtained by different methods.
This paper illustrates the study on asymmetry of the angular distribution of the ejected electrons in a multiphoton process from the ground state of a hydrogen atom. General expressions for the multiphoton ionization rates have been derived based on a variation of the Dalgarno–Lewis method for the evaluation of matrix elements in higher order perturbation theory. In addition, these analytical expressions for the various multiphoton transition rates and their variation as a function of polarization, phase, frequency and intensity of the incident radiation have been studied extensively. The general expression obtained can be used to analyse any number of incident beams with arbitrary polarization, phase and intensity.