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Journal of Physics B Atomic Molecular and Optical Physics

Phase and ellipticity dependence of the photoelectron angular distribution in non-resonant two-photon ionization of atomic hydrogen. I

M Faye, S T Wane

Published Feb 11, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.1088/0953-4075/44/5/055204

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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.
We studied the ellipticity and the dependence on the phase lag (lead) (between the semi major and the semi minor axes of the field components) of the photoelectron angular distribution (PAD) in the non resonant three photon ionization of atomic hydrogen. The exact analytical expressions for azimuthal PAD for 1s, 2s, 3s, 2p, 3p and 3d, initial states, are given.In comparison with dipole-dipole transitions, the number of quantum paths increases from: two to three for s-states; three to six for p-states; four to seven for d-states; while the number of angular coefficients goes from four to six, with two asymmetric terms. It is important that these asymmetric terms giving rise to the elliptic dichroism (ED), are only constituted with the imaginary part of the interference associated to the authorized channels leading to final states. Using the ED expression, we have established the phase shift isolation’s equation for l=0 instead of l=0,1, initial states, previously. Similarly, it is notable that, the submagnetic levels, m=0 for l=1; m=±1, for l=2, initial states, do not contribute to the PAD.Numerical evaluation of the angular coefficients is given for each state. The PAD shapes and the ED signals have been analyzed. It is found that, the maxima or the minima and the directions, of the PAD (1s, 2s, 3s, 3p), depend on the competing angular coefficients, which in return are affected by the interference terms. It is interesting to note that ,the asymmetric terms contribute only when the PAD maxima are shifted from the semi major or semi minor axes(2p,3d);and the isotropic shape(1s,2s,2p,3p,3d) is strongly dependent on the isotropic term. It is also observed that, when the photoelectron has a preference for the left handed or the right handed, the azimuthal detection difference between these two limit angles is reduced of one half than that obtained for dipole-dipole transitions. The highest ED signal (2p, 3p), occurs from the combination of the strong contributing asymmetric terms with the competing four first PAD terms. Besides, for 1s initial state, a nonzero ED signal is observed, at a particular value of the phase lag matching the phase shifts difference, for nearly circularly and nearly linearly polarized light.
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 three-dimensional angular distribution of electrons in two-photon ionization of atoms by an elliptically polarized laser beam, from an initial state with arbitrary angular momentum J i , is expressed in terms of six frequency-dependent invariant atomic parameters. The specific effect of the field polarization on the angular distribution (the elliptic dichroism, ED) is discussed for the general case of elliptic polarization. The necessary conditions for a non-zero ED are established and its physical origin is explained. The general theory is illustrated by a simple analytical calculation for the -potential model. Numerical calculations of the angular distribution parameters for H and Cs atoms are also performed and the dependence of ED magnitude on the atomic quantum numbers and the photon frequency is discussed for different frequencies.
We theoretically study the photoelectron angular distribution (PAD) from the two-photon single ionization of H and He by femtosecond and attosecond extreme-ultraviolet pulses, based on the time-dependent perturbation theory and simulations with the full time-dependent Schrodinger equation. The PAD is formed by the interference of the s and d continuum wave packets, and, thus, contains the information on the relative phase and amplitude ratio between them. We find that, when a spectrally broadened femtosecond pulse is resonant with an excited level, the PAD substantially changes with pulse width, since the competition between resonant and nonresonant ionization paths, leading to distinct from the scattering phase shift difference, changes with it. In contrast, when the Rydberg manifold is excited, and for the case of above-threshold two-photon ionization, and the PAD do not depend much on pulse width, except for the attosecond region. Thus, the Rydberg manifold and the continuum behave similarly in this respect. For a high-harmonic pulse composed of multiple harmonic orders, while the value is different from that for a single-component pulse, the PAD still rapidly varies with pulse width. The present results illustrate a new way to tailor the continuum wave packet.
We study dichroic effects in the two-photon ionization of hydrogen from its ground state due to the interaction with a bichromatic field of commensurate frequencies. The two field components have different polarization: one is linearly polarized and the other is circularly polarized (CP). Circular dichroism (CD) in the angular distribution of the photoelectrons appears if the helicity of the CP field is reversed. Numerical results reveal the influence of the photon frequencies chosen on the CD in the azimuthal angular distribution of the ejected photoelectrons.