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Physical Review A

Theory of third-harmonic generation in metal vapors under two-photon resonance conditions

A. T. Georges, P. Lambropoulos, John H. Marburger

Published Jan 1, 1977• 81 citations
Publisher

Abstract

The density-matrix formalism is used to develop the theory of two-photon-resonant third-harmonic generation in metal vapors taking into account saturation, ionization, laser-induced Stark shifts, and variations in the wave-vector mismatch as a result of population redistribution. It is shown that for third-harmonic generation in the ultraviolet and the vacuum-ultraviolet regions, where $3\ensuremath{\omega}$ exceeds the ionization limit of the metal vapor, two-photon-resonant three-photon ionization will be the main limiting process. This will also be true for four-wave parametric processes. For efficient energy conversion, the two-photon-resonant metal vapor must be phase matchable to allow large vapor density and cell length. Otherwise, the low incident intensity limits, set by the saturation and the ionization effects, cannot be efficiently converted in only one coherence length. The theory is applied to two-photon-resonant tripling of the ruby laser in cesium vapor.

DOI
10.1103/physreva.15.300

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We have investigated three resonant nonlinear processes in atomic cesium vapor. They are third harmonic generation, two photon absorption, and parametric frequency conversion. Resonant enhancement of each is provided by tuning the ruby laser half the 6S - 9D transition frequency of 28828.90 cm. The parametric process involves, in addition, a resonance with one of the P states. We have studied the variation of each of the three processes with laser intensity and frequency. At exact resonance the THG is found to be proportional to the square of the laser intensity and the TPA is proportional to laser intensity over a large power range. The resonance lineshape for both processes is found to be power dependent. A detailed analysis of these two processes based on steady state density matrix calculations and including A. C. Stark shifts, photoionization, and beam geometry is shown to accurately describe most of our observations.

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The dependence of the third harmonic intensity on the medium length under completely phase-matched and resonant conditions is discussed. It is shown that the conversion efficiency in the coherent transient case is expected to exceed that in the steady state case in a long and/or dense medium when the incident light pulse has sufficient energy. The 3S-5S transition of sodium with a 602.3 nm light is studied experimentally as a model system. The efficiency as a function of the incident light frequency and intensity is in agreement with the predictions. By eliminating the phase-mismatching effect, the dependence on the medium density is also determined. It is demonstrated that the efficiency surely increases even in the higher density of sodium.

Journal of the Physical Society of Japan•Sep 15, 1984•1 citation
The density-matrix formalism is used to develop the theory of two-photon-resonant third-harmonic generation in metal vapors taking into account saturation, ionization, laser-induced Stark shifts, and variations in the wave-vector mismatch as a result of population redistribution. It is shown that for third-harmonic generation in the ultraviolet and the vacuum-ultraviolet regions, where 3\ensuremathω3\ensuremath{\omega}3\ensuremathω exceeds the ionization limit of the metal vapor, two-photon-resonant three-photon ionization will be the main limiting process. This will also be true for four-wave parametric processes. For efficient energy conversion, the two-photon-resonant metal vapor must be phase matchable to allow large vapor density and cell length. Otherwise, the low incident intensity limits, set by the saturation and the ionization effects, cannot be efficiently converted in only one coherence length. The theory is applied to two-photon-resonant tripling of the ruby laser in cesium vapor.
The authors present a detailed theoretical analysis of two-photon resonant third- and fifth-harmonic generation in metal vapors with picosecond laser pulses. These processes are described by self-consistent Maxwell-Bloch equations which take into account the effects of optical Stark shifts, multiphoton ionization, Doppler broadening, two-photon absorption, the reaction of the third-harmonic field on the fundamental, and a nonlinear index of refraction. As an example, a quantitative analysis of third- and fifth-harmonic generation in lithium vapor ($2s\ensuremath{-}4s$ transition) is presented. The limitations on energy conversion efficiency and the temporal as well as spectral characteristics of the harmonic pulses are investigated for two kinds of low-loss propagation fundamental pulses. It is shown that 90\ifmmode^\circ\else\textdegree\fi{} phase-shifted fundamental pulses (zero area) are more efficient than $2\ensuremath{\pi}m$ pulses. Energy conversion efficiencies of 6.5% for third-harmonic (190 nm) and 0.07% for cascade fifth-harmonic (114 nm) pulses are predicted.
Density-matrix formalism is applied to construct a model for third- and fifth-harmonic generation in a four-photon resonant medium. The transition rates and coherent superposition of states associated with the resonant levels are treated rigorously, while the adiabatic following approximation is used to calculate the contribution of all nonresonant levels to the interaction. Expressions for the quadratic and quartic Stark shift and multiphoton ionization emerge naturally from the theory. At the power level required for appreciable four-photon absorption, the intensity-dependent Stark shift and ionization become overwhelming and limit the efficiency of harmonic generation. Because of the Stark shift, an intense pulse suffers a ``self-detuning'' effect and propagates through the resonant medium with minimal absorption. The interference of the phase-dependent Raman scattering with the multiphoton transition is found to be responsible for the vanishing of ionization in the presence of the generated third-harmonic field, an effect which was observed in earlier experiments.
This work examines the use of higher order multiphoton resonances in higher harmonic generation together with judicious exploitation of coherent interaction properties to achieve efficient harmonic generation. A detailed experimental study on third harmonic generation in two photon resonant coherent interaction and a theoretical study on four photon resonant coherent interaction have been conducted. Two photon resonant coheren propagation in lithium vapor (2S-4S and 2S-3D interaction) has been studied in detail as a function of phase and delay of the interacting pulse sequence. Under coherent lossless propagation of 90 phase shifted pulse pair, third harmonic generation is enhanced. A maximum energy conversion efficiency of 1% was measured experimentally. This experiment shows that phase correlated pulse sequence can be used to control multiphoton coherent resonant effects. A larger two photon resonant enhancement does not result in more efficient harmonic generation, in agreement with the theoretical prediction. An accurate (to at least 0.5 A°) measurement of intensity dependent Stark shift has been done with the newly developed "interferometric wavemeter." Stark shifts as big as several pulse bandwidths (of picosecond pulses) result in a poor tuning of multiphoton resonance and become a limiting factor of resonant harmonic generation. A complete theory has been developed for harmonic generation in a four photon resonant coherent interaction. A numerical application of the theory to the Hg atom successfully interprets the experimental observations in terms of the phase dependent stimulated Raman scattering. With the intensity required for four photon resonant transition, the calculation predicts a dramatic Stark shift effect which completely destroys the resonance condition. This model provides a basis for the development of future schemes for efficient higher order coherent upconversion.
We have investigated three resonant nonlinear processes in atomic cesium vapor. They are third harmonic generation, two photon absorption, and parametric frequency conversion. Resonant enhancement of each is provided by tuning the ruby laser half the 6S - 9D transition frequency of 28828.90 cm. The parametric process involves, in addition, a resonance with one of the P states. We have studied the variation of each of the three processes with laser intensity and frequency. At exact resonance the THG is found to be proportional to the square of the laser intensity and the TPA is proportional to laser intensity over a large power range. The resonance lineshape for both processes is found to be power dependent. A detailed analysis of these two processes based on steady state density matrix calculations and including A. C. Stark shifts, photoionization, and beam geometry is shown to accurately describe most of our observations.
The dependence of the third harmonic intensity on the medium length under completely phase-matched and resonant conditions is discussed. It is shown that the conversion efficiency in the coherent transient case is expected to exceed that in the steady state case in a long and/or dense medium when the incident light pulse has sufficient energy. The 3S-5S transition of sodium with a 602.3 nm light is studied experimentally as a model system. The efficiency as a function of the incident light frequency and intensity is in agreement with the predictions. By eliminating the phase-mismatching effect, the dependence on the medium density is also determined. It is demonstrated that the efficiency surely increases even in the higher density of sodium.