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Physical Review A

High-pump-power effects on resonant nearly degenerate four-wave-mixing signal for homogeneously and inhomogeneously broadened two-level systems

L. Márquez, Isaac Reif, Héctor J. Franco, A. Marcano O., José Luis Paz

Published May 1, 1993• 20 citations
Publisher

Abstract

The influence of the high-pump-field intensity saturation and dynamic Stark effects on the resonant nearly degenerate four-wave-mixing signal (${\mathrm{\ensuremath{\omega}}}_{3}$=2${\mathrm{\ensuremath{\omega}}}_{1}$-${\mathrm{\ensuremath{\omega}}}_{2}$) is studied in a frequency space defined by the pump- and probe-field frequencies, ${\mathrm{\ensuremath{\omega}}}_{1}$ and ${\mathrm{\ensuremath{\omega}}}_{2}$, respectively. Variable inhomogeneous widths have been considered ranging from extremely narrow (homogeneous distribution) to extremely wide. Remarkable differences between nearly homogeneously broadened systems and extreme inhomogeneous ones are exhibited. It is shown, for the situation of homogeneous broadening, that the spectra carried on with fixed probe and fixed pump frequency values have substantial differences in shape. These differences disappear in the situation of extreme inhomogeneous broadening. Fixed-pump-frequency resonant and fixed-probe-frequency nonresonant spectra are proposed as very convenient tools to yield useful evidence about the width of the inhomogeneous molecular distribution.

DOI
10.1103/physreva.47.4185

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Four-wave mixing in strongly driven four-level systems

F. A. M. de Oliveira, Cid Bartolomeu de Araujo, J. R. Rios Leite

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.

Physical Review A•Dec 1, 1988•17 citations
The influence of the high-pump-field intensity saturation and dynamic Stark effects on the resonant nearly degenerate four-wave-mixing signal (\ensuremathω3{\mathrm{\ensuremath{\omega}}}_{3}\ensuremathω3​=2\ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​-\ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​) is studied in a frequency space defined by the pump- and probe-field frequencies, \ensuremathω1{\mathrm{\ensuremath{\omega}}}_{1}\ensuremathω1​ and \ensuremathω2{\mathrm{\ensuremath{\omega}}}_{2}\ensuremathω2​, respectively. Variable inhomogeneous widths have been considered ranging from extremely narrow (homogeneous distribution) to extremely wide. Remarkable differences between nearly homogeneously broadened systems and extreme inhomogeneous ones are exhibited. It is shown, for the situation of homogeneous broadening, that the spectra carried on with fixed probe and fixed pump frequency values have substantial differences in shape. These differences disappear in the situation of extreme inhomogeneous broadening. Fixed-pump-frequency resonant and fixed-probe-frequency nonresonant spectra are proposed as very convenient tools to yield useful evidence about the width of the inhomogeneous molecular distribution.
The characteristics and properties of a resonant nearly degenerate four-wave mixing signal are studied in the frequency space defined by the pump- and probe-field frequencies. Simultaneous effects of the intense pump field (perturbative treatment at all orders in the amplitude of the total field), chemical concentration of the solution, inhomogeneous distribution of resonance frequencies, and propagation of the electromagnetic fields are studied. Discussion and interpretation of the results are based on different characteristics of the non-linear signal within the absorptive and saturative regime, and on the behaviour in the frequency space of the component of the propagation-vector mismatch. Results obtained in this model are compared to previous spectral homogeneous-line results and to local cases.
Polarization and Rayleigh type optical mixing spectra of a homogeneously and inhomogeneously broadened two-level system are calculated. We show that the nonlinear signal has contributions from the population pulsation and saturation effect. In the homogeneous system these effects can induce an asymmetric spectrum when the probe frequency is different from the resonant value. In the inhomogeneous system the spectral response is always symmetric independent of the probe frequency value. Line saturation and dynamic stark splitting are substantially reduced in the inhomogeneous case. The mentioned differences can be used to identify the type of broadening of a given spectral line.
The effect of an intense pump field and the chemical concentration of the solution on the forward four-wave mixing signal of molecular systems with an inhomogeneous distribution of resonance frequencies, has been studied. A perturbative treatment to all orders in the pump and to first order in the probe and signal beams, including the propagation of the fields as the variable to be studied, was used. The present results are compared to local regimes of propagation and to the case of a homogeneously distributed spectral line. The behaviour of the non-linear signal, characterized by tridimensional intensity surfaces, is explained in terms of saturation and/or absorption processes taking place under the present study conditions.
We present a theoretical study of phase-conjugate (PC) emission by backward degenerate four-wave mixing (DFWM) processes in a resonant gas medium. In this study, high-intensity (pump) saturation and Doppler broadening are simultaneously taken into account. We develop a general formalism to deal with the saturation effects in the case of a three-level atom (with degenerate frequencies) and obtain analytical expressions of the PC field amplitude (after integration over the velocity distribution) under the following conditions: (1) the probe beam and one of the pump beams have intensities weak enough to be treated at the lowest perturbation order, whereas the other pump intensity is arbitrarily high; (2) the probe crosses the standing pump wave at grazing incidence. It is shown that the saturation induced by the forward pump (copropagating with the probe beam) leads to entirely different behavior of the nonlinear susceptibility from that induced by the backward pump (counterpropagating with the probe beam). This saturation anisotropy is specific to the inhomogeneous broadening of the gas medium, which is also responsible for the frequency splitting experimentally observed in the intensity line shape of the PC emission for large pump intensities. At saturation, this splitting is theoretically shown to be proportional to the Rabi frequency, with a proportionality coefficient critically dependent on the relaxation processes; in the case of a single-relaxation model, the splitting is equal to 1.5 times the Rabi frequency. The PC intensity line shape is demonstrated to be of dispersive origin in the case of backward saturation, whereas for forward saturation, narrow structures are predicted in both real and imaginary parts of the nonlinear susceptibility, also leading to the intensity line-shape splitting. One also finds that the reflectivity of PC mirrors based on resonant DFWM strongly depends on the origin of the saturation. For a saturating forward pump, the PC emission intensity tends toward zero with increasing pump intensities, and it reaches a finite (nonzero) limit in the case of backward saturation. Most of these predictions have been observed experimentally [ BlochD., Phys. Rev. Lett.49, 719 ( 1982)].
The effect of Doppler broadening on degenerate four-wave mixing (DFWM) signal intensities in the regime of high pump and probe laser intensities is investigated theoretically. DFWM reflectivities are calculated by solving the time-dependent density-matrix equations for a two-level system interacting with three laser fields. The density-matrix equations are integrated directly in the time domain on a grid of spatial locations along the phase-matching axis; the DFWM signal level is then calculated by summation of the polarization contribution (with the appropriate phase factor) from each of the spatial grid points. For the case in which the Doppler and the collisional linewidths are comparable, the DFWM reflectivity is found to be inversely proportional to the factor 1 + (bΔωD/ΔωC)2, where ΔωD is the Doppler width, ΔωC is the collisional width, and b is weakly dependent on the pump and the probe laser powers. We developed an analytical expression for the reflectivity of a line that is both collision and Doppler broadened by dividing the widely used Abrams and Lind expression for homogeneous reflectivity Rhom by the factor 1 + (bΔωD/ΔωC)2. This modified reflectivity expression is found to give accurate results for the DFWM reflectivity over a wide range of values for the ratio of Doppler to collisional width. With this modified Abrams–Lind expression, strategies for quantitative DFWM concentration measurements in flames and plasmas are proposed and analyzed. We conclude that, by selection of the appropriate rotational transition, a DFWM reflectivity that is directly proportional to the square of the total species number density can be obtained over a wide range of temperature for constant-laser-intensity spatial profile mapping in flames.
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.