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Journal de physique

Parallel three-wave interaction

François Henri Julien, Jean–Michel Lourtioz, Thomas A. DeTemple

Published Jan 1, 1986• 2 citations
Publisher

Abstract

A semiclassical density matrix treatment is developed for the parallel near-resonant interaction of three waves in a four-level atomic or molecular system. Explicit solutions are developed and contain implicit AC Stark-shift and splitting information. The role of splitting and AC Stark-shifts is discussed for a system of two competing emitted waves starting from a common optically pumped level; a situation encountered in many optically pumped lasers. Application of the results to a Raman-Raman competition situation in ammonia is illustrated.

DOI
10.1051/jphys:01986004705078100

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A semiclassical density matrix treatment is developed for the parallel near-resonant interaction of three waves in a four-level atomic or molecular system. Explicit solutions are developed and contain implicit AC Stark-shift and splitting information. The role of splitting and AC Stark-shifts is discussed for a system of two competing emitted waves starting from a common optically pumped level; a situation encountered in many optically pumped lasers. Application of the results to a Raman-Raman competition situation in ammonia is illustrated.
A semiclassical treatment is given for the serial, near-resonant interaction of three waves in seven equivalent four-level atomic or molecular systems. Explicit equations are presented for the complex susceptibilties for the three waves, valid for $t\ensuremath{\gg}{T}_{2}$, which contain implicit ac Stark-shift and -splitting information, and can be separated into one-, two-, and three-photon contributions. For the case of a weak third or terminal wave, equations for the degree of ac Stark shifting and splitting are found and indicate that the three-photon or triple-resonance contribution is immune to the optical Stark effect if the remaining two waves are tuned to at least two-photon resonance conditions. The conditions for Doppler-free triple resonance are outlined, and it is shown that an equivalent ac Stark broadening may exist but may also be minimized by an appropriate choice of field intensities.
The extraction of ac Stark shifts and splittings from a semiclassical treatment of multiple wave-atom interactions is considered. Within the rotating-wave and quasistatic approximations, we show that the information can be extracted from the complex Beer's coefficient sensed by a weak tunable probe wave, whose lower level is fully populated and unconnected to the strong fields. This situation leads to a great simplification of the density-matrix equations of motion, from which algorithms are derived for the algebraic construction of the Beer's coefficients appropriate to other cases not treated. Both level-shifted resonance and line-strength information are contained in the Beer's coefficient, many examples of which are given, some for the first time. Generalized sum rules for the transitions are derived along with a connection of this approach to other approaches including dressing transformations.
A signal-flow-graph representation of Schr\"odinger's equation appropriate to the analysis of multiphoton interactions in the semiclassical limit is described. Within the rotating-wave and quasistatic approximations the equations of motion for the density-matrix elements can be recast into a system of algebraic equations on the basis of which the signal flow graphs are constructed. By introducing selected graph manipulations, the graph can be simplified and direct extraction of information is facilitated by using standard graph algebra. Illustrations are given for typical cases involving three waves and three- or four-level systems for serial, parallel, and parametric interactions. Also, the construction of an ac Stark graph, which is specifically used for the determination of ac Stark shifts, is illustrated. We show that perturbation results for virtual transitions as well as for real transitions can be obtained directly from the graph.
The density-matrix approach, in the quasistatic or adiabatic limit, is applied to the case of resonant four-wave interactions in four-level systems resulting in explicit analytical expressions for the nonlinear optical parametric and nonparametric polarizations, including ac Stark effects. The results are applicable to any closed-linkage diagram of this type but are applied only to the specific case of resonant frequency tripling. The results revealed the presence of up to six Stark-shifted nonparametric resonances and the presence of saturation and interference effects on the parametric susceptibility. In particular, the third-order susceptibility is shown to saturate to a very small value under high-field conditions which restricts the range of useful pump intensities. Under near-resonant to resonant conditions, these imply that because of either nonparametric loss present at the pump or harmonic frequencies or power-dependent dispersive and saturation effects, conversion efficiencies higher than 25% may be difficult to realize in practice.
Atomic mirrors and beam splitters based on adiabatic following by delayed laser pulses in a three-level Raman configuration are analyzed. We show that in a pure three-level system of two ground states and one excited state, laser pulses tuned precisely on resonance with the excited state do not produce any accumulated phase shifts due to ac Stark shifts even if the process is nonadiabatic. A numerical simulation suggests that multiple delayed laser pulses are attractive for precision interferometry since high transfer efficiencies (up to 98.7% per pulse for the cesium D1 line) and low ac Stark shifts are expected. A possible application to optical two-photon spectroscopy for an optical clock is discussed.
The semiclassical density matrix treatment is applied to serial four-level systems for the steady state. Phase and energy relaxations as well as Doppler effects andMdegeneracy are accounted for. The calculations are restricted to parallel polarized fields, avoiding problems of quantum interference. The formalism is used to calculate the gain spectra of different four-level far-infrared (FIR) laser schemes: double resonance pumped laser, Raman laser, two-photon pumped laser, and hyper-Raman-laser.