## Abstract

The interaction of two closely spaced resonances in degenerate four-wave-mixing spectra is investigated theoretically. Degenerate four-wave-mixing spectra for the phase-conjugate geometry (counterpropagating pump beams) are calculated by integrating the time-dependent density-matrix equations at numerous grid points along the phase-matching axis and summing the polarization contribution from each of these grid points. Both purely homogeneously broadened resonances and resonances that are both collision and Doppler broadened are considered. The homogeneously broadened results are found to agree with the results of the widely used Abrams–Lind expression for the case of stationary absorbers exposed to a steady-state excitation. We examine the effects of power broadening, pressure broadening, transition dipole moment, and population on the spectra of two neighboring homogeneously broadened resonances. As the laser power is increased, the real parts of the resonance susceptibilities destructively interfere with each other in the region between the lines, resulting in a significant and persistent dip in reflectivity. When Doppler broadening is included in the calculations, similar interference effects are observed, although at first the interference dip begins to disappear until the homogeneous part of the line shapes is broadened significantly by saturation. Using this numerical approach, we obtain good agreement with experimental results of primarily Doppler-broadened neighboring resonances.

© 1997 Optical Society of America

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### Equations (14)

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(1)
$$\frac{\partial {\rho}_{11,a}(z,t)}{\partial t}=-\frac{i}{\hslash}({V}_{12,a}{\rho}_{21,a}-{\rho}_{12,a}{V}_{21,a})+{\mathrm{\Gamma}}_{a}{\rho}_{22,a},$$
(2)
$$\frac{\partial {\rho}_{21,a}(z,t)}{\partial t}=-{\rho}_{21,a}(i{\omega}_{a}+{\gamma}_{a})-\frac{i}{\hslash}{V}_{21,a}({\rho}_{11,a}-{\rho}_{22,a}),$$
(3)
$$\frac{\partial {\rho}_{11,b}(z,t)}{\partial t}=-\frac{i}{\hslash}({V}_{12,b}{\rho}_{21,b}-{\rho}_{12,b}{V}_{21,b})+{\mathrm{\Gamma}}_{b}{\rho}_{22,b},$$
(4)
$$\frac{\partial {\rho}_{21,b}(z,t)}{\partial t}=-{\rho}_{21,b}(i{\omega}_{b}+{\gamma}_{b})-\frac{i}{\hslash}{V}_{21,b}({\rho}_{11,b}-{\rho}_{22,b}),$$
(5)
$$\beta ={\alpha}_{0}\frac{i+\mathrm{\Delta}x}{1+(\mathrm{\Delta}x{)}^{2}}\frac{2{I}_{\mathrm{pump}}/{I}_{\mathrm{sat}}}{(1+4{I}_{\mathrm{pump}}/{I}_{\mathrm{sat}}{)}^{3/2}}.$$
(6)
$${I}_{\mathrm{sat}}={I}_{\mathrm{sat}}^{0}[1+(\mathrm{\Delta}x{)}^{2}],$$
(7)
$${I}_{\mathrm{sat}}^{0}=\frac{{\u220a}_{0}c{\hslash}^{2}\gamma \mathrm{\Gamma}}{2{\mu}_{21}^{2}}=\frac{2{\pi}^{2}c\hslash \gamma \mathrm{\Delta}{\omega}_{C}}{{A}_{21}{\mathrm{\lambda}}^{3}}.$$
(8)
$$R=|\beta L{|}^{2}={L}^{2}[({\beta}_{\mathrm{real},a}+{\beta}_{\mathrm{real},b}{)}^{2}+({\beta}_{\mathrm{imag},a}+{\beta}_{\mathrm{imag},b}{)}^{2}],$$
(9)
$${\beta}_{\mathrm{real},a}={\alpha}_{0,a}\frac{\mathrm{\Delta}{x}_{a}}{1+(\mathrm{\Delta}{x}_{a}{)}^{2}}\frac{2{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},a}}{(1+4{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},a}{)}^{3/2}},$$
(10)
$${\beta}_{\mathrm{real},b}={\alpha}_{0,b}\frac{\mathrm{\Delta}{x}_{b}}{1+(\mathrm{\Delta}{x}_{b}{)}^{2}}\frac{2{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},b}}{(1+4{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},b}{)}^{3/2}},$$
(11)
$${\beta}_{\mathrm{imag},a}={\alpha}_{0,a}\frac{1}{1+(\mathrm{\Delta}{x}_{a}{)}^{2}}\frac{2{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},a}}{(1+4{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},a}{)}^{3/2}},$$
(12)
$${\beta}_{\mathrm{imag},b}={\alpha}_{0,b}\frac{1}{1+(\mathrm{\Delta}{x}_{b}{)}^{2}}\frac{2{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},b}}{(1+4{I}_{\mathrm{pump}}/{I}_{\mathrm{sat},b}{)}^{3/2}}.$$
(13)
$${\mathrm{\Omega}}_{\mathrm{pump}}=\frac{{\mu}_{21}{A}_{\mathrm{pump}}}{2\hslash}.$$
(14)
$${\mathrm{\Omega}}_{\mathrm{sat}}=\frac{(\gamma \mathrm{\Gamma}{)}^{1/2}}{2},$$