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Physical Review A

Theory of the effect of third-harmonic generation on three-photon resonantly enhanced multiphoton ionization in focused beams

M. G. Payne, W. R. Garrett

Published Dec 1, 1983• 72 citations
Publisher

Abstract

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 $n$-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.

DOI
10.1103/physreva.28.3409

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We consider effects which occur when four-wave sum frequency generation and multiphoton ionization are induced by lasers tuned near a three-photon resonance and simultaneously near or at a dipole allowed four-photon resonance. In studies with unfocused laser beams, if the phase mismatch of the generated four-wave-mixing field is large and the related two-photon resonance for the absorption of a four-wave-mixing photon and a laser photon results in strong absorption of the four-wave-mixing field, a coherent cancellation occurs between the pumping of the resonance by two- and four-photon processes. This interference effect occurs when the first laser is tuned on either side of the three-photon resonance and \ensuremath{\Vert}\ensuremath{\Delta}${\mathit{k}}_{\mathit{r}}$L\ensuremath{\Vert}\ensuremath{\gg}1, where \ensuremath{\Delta}${\mathit{k}}_{\mathit{r}}$ is the mismatch and L is the length of the path of the laser beams in the gas. With focused laser beams large differences occur between ionization with unidirectional beams and with counterpropagating laser beams when \ensuremath{\Vert}\ensuremath{\Delta}${\mathit{k}}_{\mathit{r}}$b\ensuremath{\Vert}\ensuremath{\gg}1, where b is the confocal parameter of the focused laser beams. Strong absorption of the four-wave-mixing field is shown not to be necessary for strong destructive interference with focused laser beams when the phase mismatch is large. This work also suggests an explanation for earlier experiments where the presence of a four-photon resonance enabled the generation of third-harmonic light in a positively dispersive wavelength region. We argue that this process can occur when the laser used to achieve the four-photon resonance is focused on the small z (z is the coordinate in the direction of propagation) side of the focal point of the laser responsible for the third-harmonic generation. In this way the absorption of the third-harmonic light on one side of the focal point is made much stronger than on the other side. Correspondingly, contributions to the third-harmonic field from the two sides of the focal point are unequal and cannot undergo complete destructive interference as would otherwise occur in a positively dispersive region.

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Theory of the cancellation of four-photon resonances by an off-resonance three-photon cancellation

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We present a complete account of our recent work [Phys. Rev. A 44, 31 (1991)] in which we investigate the theory of cancellation by interference between the absorption of three fundamental laser photons and one third-harmonic photon. The theory is formulated in terms of the density matrix so as to take detunings, dephasing, and laser bandwidth into account. The result is a theory of cancellation for finite detuning that explains how four-photon resonances can be canceled by a three-photon mechanism if there is an atomic level at near-three-photon resonance. The treatment is extended to focused beams and the interplay between phase matching and cancellation is investigated. We obtain explicit conditions for cancellation to occur, and perform calculations pertaining to a recent experiment where cancellation of 4+1 resonantly enhanced multiphoton ionization has been observed.

Physical Review A•Jul 1, 1992•8 citations
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.
Several extensions are made in theoretical studies of multiphoton ionization associated with third-harmonic fields that are generated near three-photon resonances in gaseous media. The treatment includes the strong coherent cancellation effect which occurs between three-photon pumping of a resonant level and pumping by the third-harmonic field that is generated in the same medium. The incomplete spoiling of the cancellation effect with counterpropagating beams is quantitatively described. Also, treatment of the multiphoton-ionization problem involving phase-matched, third-harmonic fields is generalized to include quantitative effects of index changes due to a buffer gas, absorption due to dimers, and collisional dephasing of coherent excitations. Finally, we show how measurements of linewidths, phase-matching positions, and pressure dependences of ionization yields associated with third-harmonic generation and subsequent absorption can be used to determine oscillator strengths of a resonant level, absorption coefficients for a resonant species and a buffer-gas species, and the refractive index of the buffer gas at the third-harmonic frequency.
We consider effects which occur when four-wave sum frequency generation and multiphoton ionization are induced by lasers tuned near a three-photon resonance and simultaneously near or at a dipole allowed four-photon resonance. In studies with unfocused laser beams, if the phase mismatch of the generated four-wave-mixing field is large and the related two-photon resonance for the absorption of a four-wave-mixing photon and a laser photon results in strong absorption of the four-wave-mixing field, a coherent cancellation occurs between the pumping of the resonance by two- and four-photon processes. This interference effect occurs when the first laser is tuned on either side of the three-photon resonance and \ensuremath{\Vert}\ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​L\ensuremath{\Vert}\ensuremath{\gg}1, where \ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​ is the mismatch and L is the length of the path of the laser beams in the gas. With focused laser beams large differences occur between ionization with unidirectional beams and with counterpropagating laser beams when \ensuremath{\Vert}\ensuremath{\Delta}kr{\mathit{k}}_{\mathit{r}}kr​b\ensuremath{\Vert}\ensuremath{\gg}1, where b is the confocal parameter of the focused laser beams. Strong absorption of the four-wave-mixing field is shown not to be necessary for strong destructive interference with focused laser beams when the phase mismatch is large. This work also suggests an explanation for earlier experiments where the presence of a four-photon resonance enabled the generation of third-harmonic light in a positively dispersive wavelength region. We argue that this process can occur when the laser used to achieve the four-photon resonance is focused on the small z (z is the coordinate in the direction of propagation) side of the focal point of the laser responsible for the third-harmonic generation. In this way the absorption of the third-harmonic light on one side of the focal point is made much stronger than on the other side. Correspondingly, contributions to the third-harmonic field from the two sides of the focal point are unequal and cannot undergo complete destructive interference as would otherwise occur in a positively dispersive region.
Three-photon resonance enhancement of third-harmonic generation and multiphoton ionization have been studied in xenon under excitation by the Bessel and the focused annular laser beams. It has been shown that the transformation of a Gaussian beam into a Bessel or an annular one leads to significant changes of third-harmonic excitation profiles and ionization spectra. These changes result from a principal difference of the beams, where either collinear (Gaussian beams) or noncollinear (Bessel and focused annular beams) excitation geometries are realized. It has been shown that the noncollinear ring-type phase matching shifts the maximum of the generated third harmonic to the longer wavelength toward the atomic resonance. Numerical calculations of third-harmonic excitation spectra have been carried out and compared with experimental results. The efficiency of frequency tripling in collinear and noncollinear excitation geometries have been compared. Cancellation of the three-photon 6s6s6s resonance and the influence of harmonic photons on the excitation of the four-photon 4f4f4f resonance in the Bessel beams are discussed.
The destructive interference between the three-photon and the third-harmonic one-photon absorption in the vicinity of the 6s[3/2]1{]}_{1}]1​ state of Xe has been studied in the resonantly enhanced multiphoton ionization via nf states, through the cancellation of the nf, J=2 resonances of the spectrum, accessible by four-photon absorption. The gas-pressure, i.e., phase-matching, dependence of the three-photon detuning range from the 6s[3/2]1{]}_{1}]1​ state for which cancellation occurs has been investigated in one- and two-color experiments, including positive detuning from this state in positively dispersive regions. For obtaining cancellation, the theoretically predicted requirement of optically thick medium at the third-harmonic frequency is corroborated by present observations for negative detunings. The different cancellation behavior of the nf[3/2]2{]}_{2}]2​ and nf[5/2]2{]}_{2}]2​ states observed indicates the dependence of the effect on the atomic parameters of the coupled four-photon resonance. The experimental results are found to be in partial agreement with existing theoretical models.
Three-photon resonance enhancement of third-harmonic generation and multiphoton ionization is treated for a slab geometry configuration. Striking pressure effects which become important at concentrations $n\ensuremath{\gtrsim}{10}^{12}$ /cm3{/\mathrm{c}\mathrm{m}}^{3}/cm3 for narrow-bandwidth short-pulse lasers are predicted for both the third-harmonic signal and the multiphoton ionization. For example, part of the third-harmonic signal exists the slab simultaneously with the laser pulse while another part is time delayed for small detunings from resonance. At intermediate detuning from the three-photon resonance the two parts can interfere. Multiphoton ionization yields near three-photon resonance are strongly suppressed under certain conditions, and peaks may occur for proper detunings on both sides of the three-photon resonance. A detailed experimental verification is suggested for Xe.
We present a complete account of our recent work [Phys. Rev. A 44, 31 (1991)] in which we investigate the theory of cancellation by interference between the absorption of three fundamental laser photons and one third-harmonic photon. The theory is formulated in terms of the density matrix so as to take detunings, dephasing, and laser bandwidth into account. The result is a theory of cancellation for finite detuning that explains how four-photon resonances can be canceled by a three-photon mechanism if there is an atomic level at near-three-photon resonance. The treatment is extended to focused beams and the interplay between phase matching and cancellation is investigated. We obtain explicit conditions for cancellation to occur, and perform calculations pertaining to a recent experiment where cancellation of 4+1 resonantly enhanced multiphoton ionization has been observed.