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Topical Meeting on Laser Techniques in the Extreme Ultraviolet

Two-photon Excitation of Argon

Michael Stuart Pindzola

Published Jan 1, 1982• 0 citations
Publisher

Abstract

Multiphoton excitation is describable in terms of rate equations1,2 when the generalized n-photon Rabi frequency and the excited-state photo-ionization rate are much less than the laser bandwidth. This is generally the case when powerful excimer lasers are used to excite atomic levels in rare gases. The two photon excitation rate may be written as where ω1 and ω2 are the frequencies of lasers 1 and 2 respectively, I1 and I2 are the laser intensities, G(ω1, ω2) is the bandwidth-dependent line-shape function and α is the two-photon excitation parameter defined by Eq. (1).

DOI
10.1364/euvs.1982.me2

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Multiphoton excitation is describable in terms of rate equations1,2 when the generalized n-photon Rabi frequency and the excited-state photo-ionization rate are much less than the laser bandwidth. This is generally the case when powerful excimer lasers are used to excite atomic levels in rare gases. The two photon excitation rate may be written as where ω1 and ω2 are the frequencies of lasers 1 and 2 respectively, I1 and I2 are the laser intensities, G(ω1, ω2) is the bandwidth-dependent line-shape function and α is the two-photon excitation parameter defined by Eq. (1).
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.
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.
Laser excitation of a two-level atom is analytically studied for arbitrary pulse shape, area, and energy. The ionization mechanism is modeled as a homogeneous width $\ensuremath{\gamma}$ of the atomic excited state. A transformation of the optical Bloch equations allows the dynamics to be completely described by a single equation for the area of the pulse at time ttt, $\ensuremath{\Phi}(t)$. Analytic solutions for the total bound population for a wide variety of pulse shapes can be found in two regions: $\ensuremath{\Phi}(t)\ensuremath{\approx}sin\ensuremath{\Phi}(t)$ and $\ensuremath{\Omega}(t)>\ensuremath{\gamma}$, where $\ensuremath{\Omega}(t)$ is the resonance Rabi frequency. The generalization of the two-level atom results to the NNN-level atom is discussed.
A time-independent single-rate theory for multiphoton ionization processes is developed to cover cases with a partially coherent (finite-bandwidth) driving laser. We use the Ornstein-Uhlenbeck fluctuating-phase model to describe the driving field, which has therefore a non-Lorentzian spectrum, and the effective two-level model to represent the atomic system. We present analytic and numerical methods that permit discussion of the effects of non-Lorentzian band-shape laser light, and we give analytically the bandwidth dependence of the rate in the wings of the ionization curve. Our model contains a simple scaling relationship that permits the rate for NNN-photon-resonant (N+1N+1N+1)-photon ionization to be obtained directly from the one-photon-resonant two-photon ionization rate. We treat explicitly the cases N=1N=1N=1 and N=2N=2N=2.
Compared to single-photon rates, multiphoton excitation rates of atoms and molecules are generally quite small, even at high laser intensities, at least in a 2-level system. Small multiphoton coupling strengths, large ac Stark shifts, and ionization all inhibit real population transfer between bound states. However, it has recently been shown that a 3-level system consisting of a ground state and a pair of nearly degenerate strongly coupled upper states greatly enhances the multiphoton coupling with the ground state, while also greatly reducing the ac Stark shift of the ground state. In this paper I will derive an analytic expression for the n-photon Rabi frequency, for this system, in the case of degenerate upper states, as well as an expression for the level shifts induced in the case of nondegenerate upper states. Numerical calculations based on the 3-level system are presented to verify the analytic results and to show that high-order {pi} pulses can be driven at moderate field strengths. To demonstrate the feasibility of this process in a real physical system, I will present fully correlated 2-electron calculations in a model one-dimensional molecular potential, including ionization that show a 12-photon {pi} pulse driven with near-infrared photons. In other words, amore » single excited state 18.6 eV above the ground state can be populated with over 90% efficiency with little ionization while the ground state is almost completely depopulated. Besides the possibility of producing an amplifying medium in the vacuum ultraviolet, this 3-level configuration may open the door to other strong field effects previously restricted to single-photon interactions, such as adiabatic passage and inner-shell ionization.« less