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Journal of Physics B Atomic Molecular and Optical Physics

Study and characterization of the four-wave mixing signal in the frequency space

José Luis Paz, Alfredo E. Cárdenas, A. J. Hernández

Published Oct 28, 1994• 1 citation
Publisher

Abstract

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.

DOI
10.1088/0953-4075/27/20/023

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Journal of Physics B Atomic Molecular and Optical Physics•Dec 28, 1995•9 citations
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.
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.
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.
An analytical expression for the degenerate four-wave mixing signal produced in a medium composed of two-level atoms by pump and probe lasers of arbitrary intensities and parallel linear polarization, suffering negligible absorption and depletion, has been derived. The Abrams–Lind approach [Opt. Lett.2, 94 and 3, 205 (1978)] for deriving the phase-matched polarization has been extended to find all the other higher-order polarization terms contributing to the signal. The signal field amplitude is expressed as an infinite series whose convergence rate depends on the intensity of the input fields and their detuning from resonance. The solution is successfully verified by comparison with a full numerical nonperturbative calculation and is found to be equivalent to the Abrams–Lind result in the limit of weak probe intensity. The result is used to calculate the saturation effects on spectral line shape that are relevant to simulation of molecular spectra obtained by degenerate four-wave mixing with saturating pump and probe fields. The result is extended to treat atomic motion in the case of forward geometry of the input beams.
We have derived an analytic expression for the intensity distribution of two counterpropagating waves of arbitrary input intensity in a saturable absorber. We use this distribution to extend the theory of degenerate four-wave mixing to include the effects of pump wave absorption. In particular, the spatial variation of the nonlinear absorption and coupling constants that appear in the coupled-amplitude equations for the probe and the signal (i.e., conjugate) waves are determined. These coupled-amplitude equations are solved numerically in a noniterative manner, leading to a prediction for the phase-conjugate reflectivity. The nature of the predictions differs qualitatively from those1,2 derived from theories that ignore pumpwave absorption, especially for the case of off-line-center excitation. At large values of the input pump intensities, the phase conjugate reflectivity can be larger than that predicted by theories that ignore pump-wave absorption.
A theoretical description for a forward degenerate four-wave mixing process in homogeneously broadened two-level absorbers with arbitrary intensities of all four beams is presented. Assuming no pump absorption and depletion, we obtain an analytical solution of forward degenerate four-wave mixing signal intensity in the limit of the strong pumps and weak probe and signal. In the case of low absorption or large detuning, this solution is shown to become equivalent to that of the phase-conjugate degenerate four-wave mixing derived by Abrams and Lind [Opt. Lett.2, 94 (1978)]. For arbitrary beam intensities and absorption parameters, the signal intensity is numerically calculated by solution of the coupled equations of complex wave amplitudes. Comparing the results of the numerical calculation with those of the analytical solution, we discuss the validity of using the analytical solution in practical experiments and show the saturation behavior that is due to the strong probe beam. The line shapes under the various ratios among the input beams and absorption parameters are also obtained and discussed.
Properties of the resonance peaks associated with the non-degenerate four-wave mixing signal in a two-level molecular system with non-zero permanent dipole moments are studied in the space defined by the pump ( omega 1 ) and probe ( omega 2 ) frequencies. An analytical expression for the Fourier components of the nonlinear homogeneous macroscopic polarization are obtained, where the terms responsible for the oscillation (at frequency omega 3 =2 omega 1 - omega 2 ) of the populations and coherences are also identified. Topological properties of the resulting nonlinear spectra are studied in frequency space using three-dimensional curves and contour plots. We have additionally identified the dominant contributions of the coherence and/or population that determine the intensity and the spectral linewidth. The critical quantities for the present analysis are the transition and permanent dipole moments when the rotating-wave approximation (RWA) is not considered.