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Optics Letters

Probe-beam intensity-dependent reflectivity in degenerate four-wave mixing in a saturable absorber

Jin Hyuk Kwon, Sang Soo Lee

Published Aug 1, 1983• 7 citations
Publisher

Abstract

A degenerate four-wave mixing experiment is performed in the saturable absorber of cryptocyanine. When the pump-beam intensity is below the saturation intensity, the reflectivity of the probe beam does not change, even though the probe-beam intensity is increased; however, the reflectivity begins to decrease slowly for moderate pump-beam intensity and finally decreases as l/q (q is the ratio of probe-to-forward pump-beam intensity) for pump-beam intensity far above the saturation intensity. The results are explained by extending Kogelnik's coupled-wave theory of volume grating to the four-wave case.

DOI
10.1364/ol.8.000428

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The Journal of Chemical Physics•Dec 8, 1999•22 citations
A degenerate four-wave mixing experiment is performed in the saturable absorber of cryptocyanine. When the pump-beam intensity is below the saturation intensity, the reflectivity of the probe beam does not change, even though the probe-beam intensity is increased; however, the reflectivity begins to decrease slowly for moderate pump-beam intensity and finally decreases as l/q (q is the ratio of probe-to-forward pump-beam intensity) for pump-beam intensity far above the saturation intensity. The results are explained by extending Kogelnik's coupled-wave theory of volume grating to the four-wave case.
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.
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.
Transient phase conjugation by degenerate four-wave mixing in saturable dyes is investigated theoretically and experimentally. Two special cases are examined: (1) the case in which two pump waves and a probe wave are simultaneously turned on and (2) the case in which the pump waves are turned on well before the incidence of the probe wave. For case (1) a peak appears in the transient reflectivity for high pump intensity, while for case (2) the reflectivity increases monotonically with time up to the steady-state value irrespective of the value of the pump intensity.
An analytic solution for the intensity distribution of two counterpropagating pump waves within a saturable absorber is derived. From this distribution, 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 results of the exact theory are compared with those of previously published theories. It is found that at large values of the input-pump intensities, the predicted phase-conjugate reflectivity is larger when pump-absorption effects are included in the theory.
We have derived the polarization dependent coupling constants for transmission and reflection gratings in degenerate four-wave mixing, measured phase-conjugate signals for various polarization conditions, and determined the relative contribution of the two gratings to the signal. The results obtained with cryptocyanine dissolved in methanol as the nonlinear medium show that the signal from transmission grating is 1.53–2.14 times larger than that from a reflection grating depending on the angle between the wave vectors of the forward pump beam and the probe beam.
The physics of the degenerate four-wave mixing process for resonant transitions between two degenerate energy levels is investigated by direct numerical integration of the time-dependent density matrix equations. The Zeeman structure of the upper and lower energy levels is included in a multistate formulation of the density matrix equations. The inclusion of the Zeeman structure enables the investigation of the degenerate four-wave mixing process for different polarization configurations of the forward pump, backward pump, and probe beams. Saturation curves and lineshapes are calculated for different polarization configurations and for numerous low-J transitions. At low laser intensity, the results of our calculations are in excellent agreement with perturbation theory in terms of the relative intensities of the degenerate four-wave mixing signal for linear polarization configurations. As the laser intensity increases and the resonance starts to saturate, we find in general that the relative degenerate four-wave mixing reflectivity increases for the crossed polarization configurations compared to the parallel polarization configuration because the saturation intensity is higher. However, for some resonance transitions, some of the crossed polarization configurations saturate at lower laser intensities than the parallel polarization configuration, even though the reflectivity for these crossed polarization configurations is much lower than for the parallel polarization configuration in the perturbative intensity limit. This result is explained in terms of the coupling of the various Zeeman states during the degenerate four-wave mixing interaction for specific polarization configurations. The effect of saturation on the resonance line shapes for the different polarization configurations is also investigated. Finally, a limited number of calculations are performed for resonances that are Doppler broadened as well as collision broadened. The effect of saturation on the reflectivity of the crossed polarization configurations compared to the parallel polarization configuration is even more significant for resonances with comparable Doppler and collisional broadening.