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Physical Review A

Theory of resonant multiphoton processes

C. S. Chang

Published May 1, 1974• 14 citations
Publisher

Abstract

The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order ${10}^{8}$ W/${\mathrm{cm}}^{2}$) is determined by the intensity-dependent level width ($6f$ configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order ${10}^{12}$ W/${\mathrm{cm}}^{2}$) are determined by both the intensity-dependent widths and shifts a$\stackrel{\mathrm{`}}{\mathrm{s}}$ well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.

DOI
10.1103/physreva.9.1769

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Experimental and theoretical results are presented for resonantly enhanced three-photon ionization of cesium atoms in the wavelength range from 830 to 950 nm. The accidental near resonance of the $6p^{2}P_{\frac{3}{2},\frac{1}{2}}$ levels at the one-photon energy and the near resonance at the two-photon level with the $6d^{2}D_{\frac{3}{2},\frac{5}{2}}$ states gives rise to large ionization signals and interesting anomalies in the relative intensities of the fine-structure levels. "Hybrid resonances" are observed which involve the production of $6p ^{2}P_{\frac{1}{2}}$ atoms from molecular cesium. One hybrid resonance involves ionization of the $6p ^{2}P_{\frac{1}{2}}$ state via an electric quadrupole transition ($6p ^{2}P_{\frac{1}{2}}\ensuremath{-}7p ^{2}P_{\frac{3}{2}}$). Both experimental and theoretical photoelectron angular distributions are compared for all of the observed resonances except for the $6p ^{2}P_{\frac{3}{2}}$ state where the signals were too weak for such a measurement. Agreement between experiment and theory is fair for three-photon ionization via the $6p^{2}P_{\frac{1}{2}}$ state. Comparison of experiment and theory for the case of the $6d^{2}D_{\frac{3}{2},\frac{5}{2}}$ states indicates that hyperfine coupling during the laser-pulse duration may be important. The angular distributions for the hybrid resonances show the general behavior predicted by theory but the features appear broadened.

Physical Review A•Oct 1, 1984•11 citations
The present paper applies the method of high-intensity quantum electrodynamics developed earlier to several problems of current experimental interest. In this paper the intensitydependent shifts and widths of atomic levels are calculated and applied to a given transition near resonance. The technique is demonstrated with three examples: the multiphoton ionization of atomic cesium and hydrogen in atomic beams and third-harmonic production in an atomic vapor. It is found that the multiphoton-ionization rate in a relatively weak laser beam (of order 108{10}^{8}108 W/cm2{\mathrm{cm}}^{2}cm2) is determined by the intensity-dependent level width (6f6f6f configuration) as well as the interference of the fine-structure states, and that the rates for hydrogen, in a much more intense beam (of order 1012{10}^{12}1012 W/cm2{\mathrm{cm}}^{2}cm2) are determined by both the intensity-dependent widths and shifts as‘\stackrel{\mathrm{`}}{\mathrm{s}}s‘ well as the wave-function renormalization constants. The agreement of these two examples of calculations with the experiments done recently is satisfactory. For third-harmonic production the role of two-photon resonance is emphasized, and the dependence of the generated third-harmonic intensity on the intensity and frequency of the fundamental is discussed. Near resonance, the ratio of third-harmonic intensity to fundamental intensity saturates.
This work reports the multiphoton ionization of cesium atoms, via resonance with the 6F6F6F level, by interaction with a single transverse and longitudinal mode neodymium-glass laser beam. At a low rate of ion production, when the laser frequency is tuned through resonance, the laser intensity necessary to create a constant number of ions passes through a minimum, as expected, for a three-photon energy equal to the unperturbed $6S\ensuremath{\rightarrow}6F$ transition energy. But for a higher ionization rate, involving larger laser intensities, a departure from expectation is observed which is interpreted as a consequence of an energy shift of the transition. The level shifts are demonstrated in another way by their strong effect on the experimental order of nonlinearity of the interaction, Kexpt{K}_{\mathrm{expt}}Kexpt​. A sharp variation of Kexpt{K}_{\mathrm{expt}}Kexpt​ is observed when the energy shift of the $6S\ensuremath{\rightarrow}6F$ transition exactly compensates the wavelength detuning of the laser. The latter method gives shift values which are found to be independent of the large variations of the interaction volume under conditions of resonance. Both methods give a linear dependence of the shifts on the laser intensity. Identification of these shifts with energy shifts of the $6S\ensuremath{\rightarrow}6F$ transition is subject to theoretical reservations and needs further experimentation.
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.
We investigate the third order harmonic generation for model atoms interacting with intense short laser pulses. We focus on the cases where the ponderomotive energy is not big enough to generate high-order harmonics, and where resonant transitions between bound states are also as important as multi-photon ionization and low-order harmonic generation. We find that the third order harmonic intensity from a model hydrogen atom can decrease as the laser frequency draws near a resonance between bound states. We explain this result by looking into the ionization and Rabi oscillation from the numerical calculation of the time-dependent 1-D Schrödinger equation. In order to support our explanation we also investigate the populations of bound states that play an important role in this atomic process.
The resonantly enhanced multiphoton ionization of atomic hydrogen is studied for the case of four-photon ionization (three-photon resonant). The time-dependent probabilities, the shape of the resonance profile at different intensities, and the nonlinear index are computed for coherent light at frequencies around the resonance (365 nm). The systematic dependence of the cross section as a function of the principal quantum number of the resonant state (as this tends towards the Rydberg states) is investigated at zero detuning. The probabilities of absorption of one photon more than necessary for the four-photon ionization process (resonantly enhanced above-threshold ionization) is computed and the relevant branching ratios are examined. Then it is shown that the effect of the spatio-temporal structures of the laser pulse is to red shift the maximum of the resonance profile, whereas that of an ideal chaotic light is to blue shift and broaden the resonance peak. The shape, shift, and width of the profile resulting from both these effects are in very fair agreement with the experimental ones.
Multiphoton ionization in the region near a three-photon resonance is treated for focused, plane-polarized Gaussian beams with diffraction-limited beam divergence. In this situation, a third-harmonic field is generated within the laser beam. At, and very near, three-photon resonance the driving rate for the upper-state probability amplitude due to one-photon absorption of third-harmonic light becomes nearly equal to the corresponding three-photon rate due to the laser field, but these effects are 180\ifmmode^\circ\else\textdegree\fi{} out of phase. As a consequence of this cancellation between two pumping terms, the three-photon resonance line essentially disappears at moderate concentrations and the observed ionization has a line shape that is close to the phase-matching curve for third-harmonic generation. The ionization signal, near but not on the resonance, is due almost entirely to absorption of third-harmonic photons plus other laser photons; three-photon resonantly enhanced multiphoton ionization by the laser is much weaker. This is particularly true on the blue side of the three-photon resonance at detunings where phase matching occurs. The problem is treated quite generally with predictions of the full line shape for nnn-photon ionization and third-harmonic light generation near three-photon resonance, including the rather strong influences of positively dispersive buffer gases. We also show that the cancellation between the one-photon and the three-photon process is partially spoiled in the presence of a counterpropagating beam at the same frequency.
Experimental and theoretical results are presented for resonantly enhanced three-photon ionization of cesium atoms in the wavelength range from 830 to 950 nm. The accidental near resonance of the 6p2P32,126p^{2}P_{\frac{3}{2},\frac{1}{2}}6p2P23​,21​​ levels at the one-photon energy and the near resonance at the two-photon level with the 6d2D32,526d^{2}D_{\frac{3}{2},\frac{5}{2}}6d2D23​,25​​ states gives rise to large ionization signals and interesting anomalies in the relative intensities of the fine-structure levels. "Hybrid resonances" are observed which involve the production of 6p2P126p ^{2}P_{\frac{1}{2}}6p2P21​​ atoms from molecular cesium. One hybrid resonance involves ionization of the 6p2P126p ^{2}P_{\frac{1}{2}}6p2P21​​ state via an electric quadrupole transition ($6p ^{2}P_{\frac{1}{2}}\ensuremath{-}7p ^{2}P_{\frac{3}{2}}$). Both experimental and theoretical photoelectron angular distributions are compared for all of the observed resonances except for the 6p2P326p ^{2}P_{\frac{3}{2}}6p2P23​​ state where the signals were too weak for such a measurement. Agreement between experiment and theory is fair for three-photon ionization via the 6p2P126p^{2}P_{\frac{1}{2}}6p2P21​​ state. Comparison of experiment and theory for the case of the 6d2D32,526d^{2}D_{\frac{3}{2},\frac{5}{2}}6d2D23​,25​​ states indicates that hyperfine coupling during the laser-pulse duration may be important. The angular distributions for the hybrid resonances show the general behavior predicted by theory but the features appear broadened.