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IEEE Journal of Quantum Electronics

Time-dependent pulse amplification in a three level gas

Tsung-Ming Tsai, Theodore F. Morse

Published Jul 1, 1982• 1 citation
Publisher

Abstract

The density matrix equations have been solved in conjunction with the time-dependent field amplitude equations for a three level optically pumped gas. The solutions are valid in the limit in which the pumping pulse is longer than the dephasing time of the system and true coherence effects may be neglected. In order to obtain closed form solutions, it is assumed that the FIR field is smaller than the pump field and saturation effects associated with this field are absent. When the pump pulse is resonant with the absorbing FIR transition, particularly simple solutions for the time-dependent FIR field amplification are obtained. These solutions exhibit the specific dependence of the FIR output pulse on the shape of the input pulse and display certain pulse narrowing features.

DOI
10.1109/jqe.1982.1071664

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The density matrix equations have been solved in conjunction with the time-dependent field amplitude equations for a three level optically pumped gas. The solutions are valid in the limit in which the pumping pulse is longer than the dephasing time of the system and true coherence effects may be neglected. In order to obtain closed form solutions, it is assumed that the FIR field is smaller than the pump field and saturation effects associated with this field are absent. When the pump pulse is resonant with the absorbing FIR transition, particularly simple solutions for the time-dependent FIR field amplification are obtained. These solutions exhibit the specific dependence of the FIR output pulse on the shape of the input pulse and display certain pulse narrowing features.
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 investigate the properties of a homogeneously broadened three-level system interacting with two optical fields, Eω1(x,t) and Eω2(x,t), with central frequencies ω1 and ω2 whose amplitudes are assumed to be slowly varying in time compared with dephasing times of the optical transitions. We calculate Raman susceptibilities allowing each transition to be induced by both fields and without employing the rotating wave approximation (RWA); this enables correct calculation of the optical response of the medium when the fields are on-resonance or arbitrarily far off-resonance with the Raman transition. In some frequency regions the RWA causes substantial errors. A distinction is drawn between two different types of RWA: (a) a neglect of off-resonant components of the polarization for direct optical transitions, and (b) neglect of off-resonant components of off-diagonal density matrix elements of order Eω1(x,t)Eω2(x,t)* in the field amplitudes. The first RWA is known to become inaccurate when the pump frequency is significantly off-resonance with the pump transition. We show that the second RWA becomes inaccurate when ‖ω1−ω2‖ is small or comparable to the active Raman mode frequency (even when the field are close to resonance). Calculations based upon our expressions show that when the pump frequency is close to resonance with the optical transition from the ground state, a nonlinear absorption at the Stokes shifted frequency occurs. This absorption can significantly decrease the Raman gain. Also, we find that Rabi splittings in a three-level system can occur even when the pulse durations of the incident fields are short compared with the T1 times of the system. We solve analytically for the optical response of the system to two fields of arbitrary strength and frequency. We analyze this solution and its perturbative expansion in the field amplitudes. We use the Maxwell–Bloch equations developed here to propagate time-dependent pulses.
We develop an analytical approach to the dynamics of band populations of reverse saturable absorbers modeled by the three-level approximation of the five-level rate equations. We find high-accuracy approximate solutions to these rate equations, taking into account the temporal shape of the incident laser pulse for different regimes of excitation. The results obtained are confirmed by direct numerical integration of the rate equations and are verified by solution of the full system of the rate and the propagation equations. The validity ranges of the approximations are determined. We also prove that for input pulses that are much longer than the lifetime of the first excited state the ground-state depletion obeys the same functional dependence on the input fluence as in the case of rectangular input pulses. The dynamics of the excited states, however, explicitly depends on the pulse shape. We quantitatively estimate the effect of various parameters on the nonlinear absorption coefficient and discuss implementation of the approach by a beam propagation method to reduce the computational time.
Several polar molecules possessing strong far-infrared (FIR) transitions have been optically pumped by a mode-locked CO2TEA laser in a single-pass configuration. All the resulting FIR pulses are shorter than the sub-T2pump pulses; subnanosecond FIR pulses are observed. These short FIR pulses exhibit lethargic gain, stimulated Raman emission, and multiple-pulse superradiance. The experimental results are modeled by numerical solution of the Maxwell-density matrix equations for two fields interacting with a three-level system. Level degeneracy is included, and the pump is described by a multimode power spectrum with a parabolic envelope, resulting in a periodic train of pulses. Variation of the detuning of the pump from resonance allows simulation of the superradiant or Raman cases. This fully coherent model, which is the major contribution of this paper, explains key experimental observations such as the noncoincidence of the Raman peak with the pump modulation minimum and the enhancement of multiple-pulse superradiance due to remanent polarization.