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Journal of the Optical Society of America

Collisional contributions to the dynamic Stark effect in coherently driven three-level systems

Gregory R. Osche

Published Oct 1, 1978• 10 citations
Publisher

Abstract

A theoretical investigation of collisional contributions to the gain line shape of coherently driven three-level systems is presented. The theory differs from previous work in that it includes, in addition to the usual thermal-bath mechanisms, direct collisional coupling between optically active states. It is shown that for stronger coupling between the stimulated levels than between the pump and two-quantum levels (i) the small signal gain g obtained with off-resonant pumping is greater than that obtained with resonant pumping, (ii) the unsaturated gain profile becomes asymmetric in proportion to the degree of pump detuning, and (iii) the unsaturated gain asymmetry is translated into the emission profiles gI obtained with nonzero values of stimulated intensities I. These results may be of particular importance in the case of optically pumped submillimeter lasers which exhibit large differences in the thermalization rates of the pump (V-V) and stimulated (R-R) transitions.

DOI
10.1364/josa.68.001293

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A theoretical investigation of collisional contributions to the gain line shape of coherently driven three-level systems is presented. The theory differs from previous work in that it includes, in addition to the usual thermal-bath mechanisms, direct collisional coupling between optically active states. It is shown that for stronger coupling between the stimulated levels than between the pump and two-quantum levels (i) the small signal gain g obtained with off-resonant pumping is greater than that obtained with resonant pumping, (ii) the unsaturated gain profile becomes asymmetric in proportion to the degree of pump detuning, and (iii) the unsaturated gain asymmetry is translated into the emission profiles gI obtained with nonzero values of stimulated intensities I. These results may be of particular importance in the case of optically pumped submillimeter lasers which exhibit large differences in the thermalization rates of the pump (V-V) and stimulated (R-R) transitions.
The Langevin quantum theory has been used to treat the optically pumped three-level laser system. We derive an equation of motion for laser field in the type of Van der Pol equation when the photon number of the laser field is not too high. It includes not only both incoherent and coherent pumping, but also both strong and weak optical pumping. If an average over reservoir is taken, the equation will give all the semi-classical results. Our results show that the laser gain comes from two basic processes, i.e., a two-photon process and a two-step process. When the pumping strength is larger than some value, the dynamic Stark effect will appear. The laser linewidth is mainly contributed by the thermal noise, the vaecum fluctuation, the spontaneous emissions between the laser levels and the spontaneous Raman scattering.
The Langevin quantum theory has been used to treat the optically pumped three-level laser system. We derive an equation of motion for laser field in a type of Van der Pol equation when the photon number of the laser field is not too high. It includes not only both incoherent and coherent pumping, but also both strong and weak optical pumping. If an average over the reservoir is taken, the equation will give all the semi-classical results. Our results show that the laser gain comes from two basic processes, i.e., a two-photon process and a two-step process. When the pumping strength is larger than a certain value, the dynamic Stark effect will appear. The laser linewidth is mainly contributed by the thermal noise, the vacuum fluctuation, the spontaneous emissions between the laser levels and the spontaneous Raman scattering.
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.
A density-matrix calculation of the line shape for an optically pumped coupled three-level system subjected to two traveling wave-laser fields with Gaussian intensity profiles is presented to third-order in perturbation theory. A closed-form expression for the line shape in the extreme transit-time limit is obtained. The line shape of the narrow resonance (Raman-type processes) is found to be an exponential function of the detuning, with half-width at half maximum ${\ensuremath{\Delta}}_{\mathrm{HWHM}}^{(\mathrm{Hz})}\ensuremath{\simeq}0.98(\frac{u}{2\ensuremath{\pi}D})$ (DDD is the intensity 1e\frac{1}{e}e1​ diameter and uuu is the thermal speed). The saturation intensity is also modified by the transit rate ${\ensuremath{\gamma}}_{t}=\frac{\sqrt{2}u}{D}$ (rad/sec).
We present a theoretical analysis of the influence of the collimation of a strong pump beam on a small-signal-gain line shape in a three-level \ensuremath{\Lambda}-type molecular system in the regime of high pump intensity when saturation and Stark (Autler-Townes) splitting of the levels of the pumping transition take place. The cases of homogeneous and Doppler-broadened pumping transitions have been considered. A significant offset, an asymmetric deformation, and a narrowing of the gain line, i.e., a spectrum focusing, have been obtained and explained as a result of the nonlinear (dispersion) and the active (gain) wave guiding of the small signal in the field of the strong pump beam.
An analysis of a three-level system using time-dependent perturbation theory shows that two distinct photon-emission processes take place within such a system when stimulated optical emission occurs: one-and two-photon transitions. The two-photon transition consists of the emission of a stimulated photon followed by the absorption of a coherent pump photon. In three-level systems which possess a narrow pump level, two-photon transitions are dominant at high pump powers while the maser line is split into two lines due to the modulation of the wave function at an angular frequency determined by the rate of pumping. This might lead to the possibility of frequency-modulating the maser line. These results are extended qualitatively to include optical masers which possess broad pump-bands.