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Physical Review A

"Multiple-" versus "multi-" photon absorption in the theory of resonant ionization

Burke Ritchie

Published Oct 1, 1979• 3 citations
Publisher

Abstract

The author has identified his result for resonant ionization by the simultaneous absorption of two photons by a ground-state electron [Phys. Rev. A 17, 659 (1978)] with the high-laser-power large-intermediate-state ionization rate result derived by Eberly and co-workers using a steady-state rate theory. This result gives $R{\ensuremath{\sigma}}_{22}$ for all power levels considered, where $R$ is the one-photon ionization rate for the intermediate state and ${\ensuremath{\sigma}}_{22}$ is its steady-state population factor. The result can be cast in this form, but it is found that ${\ensuremath{\sigma}}_{22}$ must be much less than ${\ensuremath{\sigma}}_{11}$, the ground-state population factor. Thus the resonant process can be described by a rate limit for conserving energy by simultaneous absorption only when the ground-state population factor remains unity to order $\frac{{\ensuremath{\Omega}}^{2}}{{R}^{2}}$, where $\ensuremath{\Omega}$ is the Rabi rate for the bound-bound transition. In other words, the simultaneous and sequential processes are physically equivalent in this limit.

DOI
10.1103/physreva.20.1734

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The author has identified his result for resonant ionization by the simultaneous absorption of two photons by a ground-state electron [Phys. Rev. A 17, 659 (1978)] with the high-laser-power large-intermediate-state ionization rate result derived by Eberly and co-workers using a steady-state rate theory. This result gives R\ensuremathσ22R{\ensuremath{\sigma}}_{22}R\ensuremathσ22​ for all power levels considered, where RRR is the one-photon ionization rate for the intermediate state and \ensuremathσ22{\ensuremath{\sigma}}_{22}\ensuremathσ22​ is its steady-state population factor. The result can be cast in this form, but it is found that \ensuremathσ22{\ensuremath{\sigma}}_{22}\ensuremathσ22​ must be much less than \ensuremathσ11{\ensuremath{\sigma}}_{11}\ensuremathσ11​, the ground-state population factor. Thus the resonant process can be described by a rate limit for conserving energy by simultaneous absorption only when the ground-state population factor remains unity to order \ensuremathΩ2R2\frac{{\ensuremath{\Omega}}^{2}}{{R}^{2}}R2\ensuremathΩ2​, where \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ is the Rabi rate for the bound-bound transition. In other words, the simultaneous and sequential processes are physically equivalent in this limit.
The author presents a theory for NNN-photon ionization in which two intermediate states can be in resonance with the ground state over a narrow range of photon frequencies. The theory is applied to the three-photon 6F6F6F resonance-state spin-orbit doublet in the four-photon ionization of Cs. The amplitude for the process is the sum of two components, each containing a complex line-shape function. In the limit of zero splitting, the previously studied isolated resonance result, in which the line shape is Lorentzian is recovered. For the splitting in Cs (6F6F6F), the absorption profile is studied as a function of the splitting parameters of the state, the $6S\ensuremath{\rightarrow}6F$ three-photon Rabi rate, and the field-induced shift and width for the $6S\ensuremath{\rightarrow}6F$ transition. This profile depends in a complicated way on the relative magnitudes of the shift and the spacing between the fine-structure levels.
Two-photon-resonant three-photon ionization of atomic phosphorus is studied in the 298--306-nm wavelength range. P atoms are produced by Ar+{\mathrm{Ar}}^{+}Ar+ ion sputtering of an InP solid sample and the photoions are detected by a quadrupole mass spectrometer. The laser power dependence of the 3p4{\mathit{p}}^{4}p4S3/20{\mathit{S}}_{3/2}^{0}S3/20​--4 4S3/20^{4}\mathrm{S}_{3/2}^{0}4S3/20​ two-photon excitation and one-photon ionization of 4p4{\mathit{p}}^{4}p4S3/20{\mathit{S}}_{3/2}^{0}S3/20​ phosphorus is determined in the 10--200 MW/cm2{\mathrm{cm}}^{2}cm2 range. The cross section for the ionization step is estimated in the frame of the quantum-defect theory: \ensuremath{\sigma}\ensuremath{\sim}3\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}18}$ cm2{\mathrm{cm}}^{2}cm2. The two-photon Rabi frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{R}}$ is deduced from the comparison of our experimental results with the extended two-level model developed by Eberly [Phys. Rev. Lett. 42, 1049 (1979)] to describe (m+n) resonant multiphoton ionization processes and with the rate equation approximation analysis. The best fit gives ${\mathrm{\ensuremath{\Omega}}}_{\mathit{R}}$(${\mathrm{s}}^{\mathrm{\ensuremath{-}}1}$)=8IL{\mathit{I}}_{\mathit{L}}IL​(W/cm2{\mathrm{cm}}^{2}cm2). This value compares relatively well with the theoretical estimate ${\mathrm{\ensuremath{\Omega}}}_{\mathit{R}}$=5.5IL{\mathit{I}}_{\mathit{L}}IL​ obtained by limiting the perturbation-theory summation to the dominant intermediate states. The combination of well-characterized Ar+{\mathrm{Ar}}^{+}Ar+ ion sputtering of a solid sample with resonant photoionization is used to perform actual trace analysis of materials. For example, 0.5 ppm of phosphorus in a silicium sample was measured by this method with a lateral resolution of \ensuremath{\sim}300 \ensuremath{\mu}m.
A perturbation theory of the ionization of atoms by simultaneous absorption of several photons, each of whose energy is less than the ionization potential, is developed from the evolution-operator formalism. A precise computation is made for the hydrogen atom, giving transition rates as a function of photon energy for two- through twelve-photon photoionization. The eighth-order ionization rate (in cgs units) at the 1.78-eV ruby-laser line is found to be \ensuremath{\sim}${10}^{\ensuremath{-}244}$\ifmmode\times\else\texttimes\fi{}(photonflux)8{(\mathrm{photon}\mathrm{flux})}^{8}(photonflux)8 and should be observable using available techniques. Good agreement is obtained with Zernik's exact calculation of the two-photon ionization rate of metastable 2S2S2S hydrogen. Approximate calculations are made for the rare gases. Assuming "typical" experimental conditions of a gas density of \ensuremath{\sim}1020{10}^{20}1020 atoms ${\mathrm{cm}}^{\ensuremath{-}3}$ and a ruby laser focused into a volume of \ensuremath{\sim}${10}^{\ensuremath{-}8}$ cm3{\mathrm{cm}}^{3}cm3, we find that the flux required to liberate one electron during a 10-nsec pulse is \ensuremath{\sim}1029{10}^{29}1029 ${\mathrm{cm}}^{\ensuremath{-}2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for Xe, Kr, and Ar and \ensuremath{\sim}5\ifmmode\times\else\texttimes\fi{}1030{10}^{30}1030 photons ${\mathrm{cm}}^{\ensuremath{-}2}$ ${\mathrm{sec}}^{\ensuremath{-}1}$ for Ne and He. These gases ionize with the simultaneous absorption of 7, 8, 9, 13, and 14 photons, respectively. The predicted rate for Xe is found to be in excellent agreement with the recent direct measurements of Voronov and Delone. We conclude that multiphoton ionization provides the initial electrons required for the optical breakdown of gases, though it does not account for the over-all growth of the discharge except possibly at very low pressures. Impurity atoms (particularly heavy rare gases) may be the source of "initiating" electrons in Ne and He.
The two-photon resonance ionization probability of atoms in strong extreme-ultraviolet free-electron laser (EUV FEL) pulses has been investigated by the model of time-dependent wave packet propagation of a light-coupled multilevel atom. Under the simulation within the model assuming single-mode FEL pulses, the ionization probability Pion{P}_{\mathrm{ion}}Pion​ has shown characteristic dependences on the scaled coupling parameter Ugi{U}_{gi}Ugi​ between two levels of the ground (ggg) and intermediate (iii) resonance states, namely, ${P}_{\mathrm{ion}}\ensuremath{\propto}{({U}_{gi})}^{n}$, with nnn being equal to \ensuremath{\sim}2, less than 1, and \ensuremath{\sim}1 for the small, medium, and large Ugi{U}_{gi}Ugi​ regimes, respectively. This power dependence of the ionization probability has been interpreted due to Rabi oscillations between ggg and iii states. To compare with recent experimental results on the same condition, the multimode nature of self-amplitude spontaneous emission (SASE) FEL pulses has been managed in the simulation. Then, the recent experimental laser-power dependence of the two-photon resonance ionization of He [Sato et al., J. Phys. B 44, 161001 (2011)] has been well described by that for the large Ugi{U}_{gi}Ugi​ regime of the simulation, i.e., nnn \ensuremath{\sim} 1. Thus, the observed linear laser-power dependence has been rationalized as being caused by the strong Rabi oscillations between the (2ppp)--(1sss) states.
The advent of the laser has brought about a revolution in the kinds of experimental phenomena that can be investigated. In particular, ultra-high power monochromatic, collimated sources have intensities of sufficient magnitude to enable studies to be carried out in the regime where 'non-linear' effects are important, even dominant. Important among these are processes which are characterised by the simultaneous absorption of two or more optical photons. While multiple photon processes are not completely new, those in the optical part of the spectrum, like Raman scattering, have, until the advent of the laser, been thought of as linear in the incident intensity, or, like the two-quantum decay of the metastable 2s states of hydrogen-like atoms and ions, been confined to theory. Of the multiple photon, non-linear optical processes, multiquantum ionization is one of the more fruitful for detailed, quantitative measurements, and for comparison with theory, since (1) one does not require precise coincidence between the laser light and an absorption line (2) the ionizations can be detected unambiguously with 100% efficiency (3) atomic wave functions are simple, and sufficiently accurate to enable one to make reasonable precise calculations.
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{}.