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Physical Review A

Strong-field theory of four-wave mixing. II. Density-matrix treatment of extra resonances

N. Chencinski, W. M. Schreiber, Alfred M. Levine, Yehiam Prior

Published Sep 1, 1990• 16 citations
Publisher

Abstract

A nonperturbative density-matrix formalism is used for the analysis of strong-field four-wave mixing, allowing the treatment of relaxation processes, including lifetime as well as dephasing. The combined role of proper dephasing and level saturation in the generation of extra resonances is studied. It is shown that, whereas in the absence of proper dephasing there is no pressure-induced resonance, there is always a nonzero contribution to the field-induced resonance. For increasing field intensity, a characteristic double saturation is predicted as a function of the field strength. The dependence on the various parameters is investigated, and the conditions under which these effects should be observable are identified.

DOI
10.1103/physreva.42.2839

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A nonperturbative density-matrix formalism is used for the analysis of strong-field four-wave mixing, allowing the treatment of relaxation processes, including lifetime as well as dephasing. The combined role of proper dephasing and level saturation in the generation of extra resonances is studied. It is shown that, whereas in the absence of proper dephasing there is no pressure-induced resonance, there is always a nonzero contribution to the field-induced resonance. For increasing field intensity, a characteristic double saturation is predicted as a function of the field strength. The dependence on the various parameters is investigated, and the conditions under which these effects should be observable are identified.
A nonperturbative theoretical study of four-wave-mixing processes in four-level systems is presented. Using the density-matrix formalism, numerical results are obtained for two arbitrarily intense pump fields with the third field a weak probe. The fourth wave, radiated by the system, is obtained analytically for some special cases. We find, for different frequency detunings, Rabi frequencies, and relaxation rates that the line shapes exhibit: (i) dynamic Stark splitting and shifting, and merging of resonances; (ii) power broadening with two-photon antiresonance behavior resulting from a destructive interference between different contributions to the optical polarization; and (iii) extra resonances induced by field intensities and saturation processes.
Nonlinear perturbative susceptibilities have been used successfully to analyze experiments in nonlinear optics for two decades. By its very nature, a perturbative approach is limited to a small value of the expansion parameter, and nonlinear susceptibilities are not useful for strong saturating fields. Presently, a theory is presented for the treatment of FWM which is applicable to arbitrary strength of the input fields, both on and off resonance. The field harmonic time dependences are removed by a transformation to the proper rotating frame, and the resulting eigenvalue problem is solved numerically. Several new effects are predicted. For strong fields resonance frequencies are shifted and lines are split (similar to the ac Stark effect). When such a split line is acted on by another strong field (a phenomenon not studied so far), a new effect of stirring is observed, which results in the broadening and merging of the two separate components. A new extra resonance appears which is induced by the strong field and is present even in the absence of any collisions or other dephasing processes or of laser phase fluctuation. This new extra resonance is a new member in the family of extra resonances which were discussed in various previous publications.
A nonperturbative semiclassical theory is presented for the treatment of four-wave mixing (FWM) in a four-level system. This treatment is based on a transformation to a generalized rotating frame followed by a numerical diagonalization of the resulting time-independent Hamiltonian. For strong fields, all resonant frequencies are shifted from the atomic (molecular) frequencies. Lines are split up in a way similar to the ac Stark effect for a two-level system, and for even stronger fields, the effect of stirring is observed for the first time in four-wave mixing. In addition, a new extra resonance appears which is induced by the saturating field. This field-induced resonance is the new member in the family of extra resonances (pressure induced, or stochastic fluctuation induced), and it should be observed whenever saturating fields interact with a multilevel atomic or molecular system.
Saturation effects in gas-phase degenerate four-wave mixing (DFWM) are investigated for conditions of interest for diagnostic applications in flames and plasmas. In particular, the case in which Doppler and collisional broadening are comparable, as is often the case for flame species such as NO and OH, is investigated. DFWM line shapes and signal intensities are calculated nonperturbatively and compared with high-resolution laser measurements. In the nonperturbative calculations the time-dependent density-matrix equations for a two-level system interacting with three laser fields are integrated directly on a grid of spatial locations along the phase-matching axis. The electric-field amplitude for the DFWM signal is determined by multiplying the time-varying, laser-induced polarization at each grid point by the appropriate phase factor and then by summing over all grid points. The calculations are in excellent agreement with measurements of DFWM line shapes and signal intensities for NO in a buffer gas of He over a wide range of He pressure. Under saturation conditions the pressure dependence of the DFWM signal is reduced greatly compared with the unsaturated case. The signal level is still dependent on the ratio of pure dephasing to quenching collisions, even for saturation conditions. From the standpoint of minimal dependence on collisional processes it appears that operation with pump-laser intensities approximately equal to the saturation intensity is optimal. The dependence of the DFWM line shape and signal intensity on the ratio of probe to pump-laser intensity is also investigated.
In the present paper, we investigate the saturation mechanism of FDFWM (forward degenerate four-wave-mixing) spectra, observed with different polarizations of the input beams, line coupling, and Doppler effect. Our theoretical approach uses a nonperturbative resolution of the density matrix equations, decomposed on the irreducible tensor basis. This formalism greatly simplifies analytical expressions and allows for a correct treatment of magnetic sublevels structure in a three-J level system. Systematic calculations of the FDFWM signal for specific NO lines are performed to characterize intensities and line shapes as a function of saturation. Relative intensities exhibit strong changes in the intermediate regime of saturation further supporting the necessity for using such a complete model. Under saturating conditions, line coupling is shown to distort the spectra, according to the type of transition and the polarization configuration. This effect is observed in the following paper [V. Kr\"uger et al., paper II, Phys. Rev. A 2001 (to be published)], where saturated experimental spectra of NO are presented and interpreted for two polarization configurations of interest for diagnostic purposes.