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Journal of Physics B Atomic and Molecular Physics

A quantised field treatment of two-photon resonant third-harmonic generation

J N Elgin

Published Feb 14, 1980• 1 citation
Publisher

Abstract

The problem of two-photon resonant third-harmonic generation is analysed using a quantised field formalism. Photons are generated at the harmonic frequency as a result of transitions between sets of doublet states of the Autler-Townes type and hence, in the general case, necessarily have more than one spectral component.
DOI
10.1088/0022-3700/13/3/025

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The analysis of two-photon resonant third-harmonic generation is generalised to include, in particular, the effects of imperfect index matching and two-photon tuning. Furthermore, whereas previously the authors considered only the temporal properties of the generated harmonic signal, here is spectral characteristics are also considered. The authors comment on a discrepancy between the predictions of semiclassical and quantised field formalisms. In the semiclassical analysis, it is shown that for large energy input pulses the harmonic spectrum is double peaked even near exact resonance and that this structure can be interpreted in terms of transitions between sets of doublet states of the Autler-Townes (1955) type. In a quantised field formalism, however, the picture is radically altered near exact resonance where a three-peaked spectrum is predicted.
A numerical investigation is made of third harmonic generation (THG) under conditions of two-photon resonance between the fundamental frequency and a selected transition in the atoms of a medium. It is shown that under conditions of coherent interaction between the pump pulse and the medium, a THG efficiency of ∼20% or higher may be achieved. In this case, THG in the form of high-power ultrashort pulses may be obtained. It is shown that experimental observation of similar effects in specific media should not present particular difficulties.
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.
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.
We establish a connection between the simple harmonic oscillator and a two-level atom interacting with resonant, quantized cavity and strong driving fields, which suggests an experiment to measure the harmonic-oscillator's probability distribution function. To achieve this, we calculate the Autler-Townes spectrum by coupling the system to a third level. We find that there are two different regions of the atomic dynamics depending on the ratio of the Rabi frequency ${\ensuremath{\Omega}}_{c}$ of the cavity field to that of the Rabi frequency $\ensuremath{\Omega}$ of the driving field. For ${\ensuremath{\Omega}}_{c}<\ensuremath{\Omega}$ and moderate coupling of the transition to the cavity mode the spectral peaks are composed of multiplets. A quantized dressed-atom approach provides a simple explanation of the spectral features and shows that the oscillations in the spectral components arise from the oscillations of the population distribution in the dressed states. The observation of these features would provide evidence for the quantum nature of the cavity field. The distribution is an analog of the harmonic-oscillator probability distribution function, and should be experimentally observable. For ${\ensuremath{\Omega}}_{c}>~\ensuremath{\Omega}$ there is no Autler-Townes splitting and the spectrum is composed of a single peak located at the frequency of the probe transition. We show that this effect results from the collapse of the atom to the ground state, which has been predicted by Alsing, Cardimona, and Carmichael [Phys. Rev. A 45, 1793 (1992)] for a two-level atom in a lossless cavity.