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Physical Review

Quantitative Theory of the Two-Photon Ionization of the Alkali Atoms

H. Barry Bebb

Published Sep 9, 1966• 139 citations
Publisher

Abstract

The theory of two-photon ionization of the alkali atoms is developed from the well-known perturbation formula for the second-order transition rate. Quantum defect Coulomb functions are used for both the negative and positive energy states. Spin-orbit splitting of the intermediate $p$ states is taken into account. The results are presented as dispersion curves of the transition rate for photon energies between the two-photon and one-photon thresholds. Cesium is of particular interest because the photon energy for the second harmonic of ruby laser light falls very near the $10{p}^{2}{P}_{\frac{3}{2}}$ level. For a photon energy of 3.57 eV, the transition rate (in cgs units) is found to be 2.6\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}49}$\ifmmode\times\else\texttimes\fi{}${(\mathrm{photon}\mathrm{flux})}^{2}$.

DOI
10.1103/physrev.149.25

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Three-photon ionization transition rates are calculated for the alkali atoms (except Li) using quantum defect Coulomb functions to evaluate the well-known perturbation formula. Several corrections of previous work are mentioned. The often-used approximation of keeping only a small number of intermediate states is investigated. The predicted three-photon ionization rate (in cgs units) of the alkalis for the ruby photon energy of 1.78 eV is of the order of ${10}^{\ensuremath{-}77}$ \ifmmode\times\else\texttimes\fi{} ${(\mathrm{photon}\mathrm{flux})}^{3}$. Detailed results are presented as dispersion curves of the transition rate for photon energies between the three-photon and two-photon thresholds.

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I present a perturbation theory of multiphoton ionization based on the resolvent operator. The theory is applied to a detailed study of two-photon ionization under near-resonance conditions. It is shown that a distinction must be made between linear level shifts and widths, and resonance shifts and widths which appear in the transition probability. The resonance shifts and widths exhibit saturation effects and their values tend to be much smaller than those of the linear level shifts and widths. Using numerical examples corresponding to tunable dye lasers and alkali-metal atoms, we show that the above shifts can have a dramatic effect on the ionization rate even when the laser is not in exact resonance with the intermediate level. The question of "one-step" vs "two-step" two-photon ionization is also discussed. It is shown that exact resonance does not necessarily imply a two-step process. In fact, in most cases it is quite the opposite. A quantitative criterion for this distinction is established. Finally, a distinction is made between instrumental saturation and saturation of the transition probability. The two are shown to be of different nature. Further extensions of the work are also discussed.

Physical Review A•May 1, 1974•104 citations
The theory of two-photon ionization of the alkali atoms is developed from the well-known perturbation formula for the second-order transition rate. Quantum defect Coulomb functions are used for both the negative and positive energy states. Spin-orbit splitting of the intermediate ppp states is taken into account. The results are presented as dispersion curves of the transition rate for photon energies between the two-photon and one-photon thresholds. Cesium is of particular interest because the photon energy for the second harmonic of ruby laser light falls very near the 10p2P3210{p}^{2}{P}_{\frac{3}{2}}10p2P23​​ level. For a photon energy of 3.57 eV, the transition rate (in cgs units) is found to be 2.6\ifmmode\times\else\texttimes\fi{}10\ensuremath−49{10}^{\ensuremath{-}49}10\ensuremath−49\ifmmode\times\else\texttimes\fi{}(photonflux)2{(\mathrm{photon}\mathrm{flux})}^{2}(photonflux)2.
Three-photon ionization transition rates are calculated for the alkali atoms (except Li) using quantum defect Coulomb functions to evaluate the well-known perturbation formula. Several corrections of previous work are mentioned. The often-used approximation of keeping only a small number of intermediate states is investigated. The predicted three-photon ionization rate (in cgs units) of the alkalis for the ruby photon energy of 1.78 eV is of the order of ${10}^{\ensuremath{-}77}$ \ifmmode\times\else\texttimes\fi{} (photonflux)3{(\mathrm{photon}\mathrm{flux})}^{3}(photonflux)3. Detailed results are presented as dispersion curves of the transition rate for photon energies between the three-photon and two-photon thresholds.
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A theoretical method previously developed for the two-photon ionization of alkali-metal atoms is extended to the spin-orbit coupling scheme. We have obtained the Green's function of this problem in the framework of quantum-defect theory and so we have been able to perform the infinite summations related to perturbation theory. The generalized transition probability rates are calculated in the off-resonant and resonant regions, respectively, for polarized incident radiation. Destructive interference effects between the unperturbed atomic states in the infinite summations producing a finite deep minimum in the off-resonant region, are predicted by our theory, both for linearly and circularly polarized incident radiation. A detailed calculation of electron-spin polarization is given. Comparisons of values predicted by the present work with experiment and other theoretical calculations are made for the case of atomic cesium initially in the ground 6S6S6S state.
This paper will review briefly a recent experiment which measured the photodetachment probability for I-ions due to simultaneous absorption of two ruby quanta. We have calculated the two-quantum photoionization rate for atomic cesium as a function of wavelength. The "transition" from the initial state to the bound intermediate state is expressed in terms of oscillator strengths. The second "transition," from these excited valence states into the coulomb continuum is expressed as a cross section. We have observed what appear to be genuine two-photon ionization signals in three cases: 1) near resonance in the10P_{3/2}state using the second-harmonic ofQ-switched ruby, 2) away from resonances, using the 5300 Å second-harmonic ofQ-switched neodymium laser, and 3) away from resonances using the first anti-Stokes line at 5750 Å from ruby-pumped D2gas. Present experiments aim at scanning the interesting range near resonance, e.g., by thermally tuning the ruby.
This paper discusses the general theory of two-photon ionization of a hydrogenic state in the nonrelativistic dipole approximation. An application is made to the photoionization of the metastable 2S2S2S state of atomic hydrogen by optical radiation, for example, by the light from a ruby laser. The sum over intermediate states is carried out exactly by means of an implicit technique. The results are given for a range of wave-lengths from the threshold at 7290 down to 4556 \AA{}. At 6943 \AA{}, the total cross section per unit intensity is found to be 5.35\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}29}$ cm4{\mathrm{cm}}^{4}cm4 ${\mathrm{W}}^{\ensuremath{-}1}$. The lowest order damping corrections to the perturbation-theory results are worked out.
I present a perturbation theory of multiphoton ionization based on the resolvent operator. The theory is applied to a detailed study of two-photon ionization under near-resonance conditions. It is shown that a distinction must be made between linear level shifts and widths, and resonance shifts and widths which appear in the transition probability. The resonance shifts and widths exhibit saturation effects and their values tend to be much smaller than those of the linear level shifts and widths. Using numerical examples corresponding to tunable dye lasers and alkali-metal atoms, we show that the above shifts can have a dramatic effect on the ionization rate even when the laser is not in exact resonance with the intermediate level. The question of "one-step" vs "two-step" two-photon ionization is also discussed. It is shown that exact resonance does not necessarily imply a two-step process. In fact, in most cases it is quite the opposite. A quantitative criterion for this distinction is established. Finally, a distinction is made between instrumental saturation and saturation of the transition probability. The two are shown to be of different nature. Further extensions of the work are also discussed.