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Physical Review A

Rabi resonance in frequency conversion by four-wave mixing in lasers and its connection with the multimode laser instability

F. Castelli, L. A. Lugiato, R. Pirovano

Published May 1, 1994• 9 citations
Publisher

Abstract

We analyze the behavior of a laser upon injection of a weak-field resonant with a cavity mode adjacent to the lasing mode; the four-wave-mixing coupling generates also a field with the frequency of the symmetrical adjacent mode. Using the Maxwell-Bloch equations, we show that under appropriate conditions there is a spectacular amount of frequency conversion, with a peak when the free spectral range is equal to the Rabi frequency of the laser field multiplied by \ensuremath{\surd}2 . The origin of this effect is identified with the help of the linear stability analysis of the free-running laser, which shows also the deep connections with the classical laser instability, predicted by Risken and Nummedal [J. Appl. Phys. 39, 4662 (1968)] and Graham and Haken [Z. Phys. 213, 420 (1968)]. The Rabi peak is a precursor of the multimode laser instability, but it arises for smaller values of the pump parameter and is only weakly sensitive to the transverse shape of the laser beam.

DOI
10.1103/physreva.49.4031

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We analyze the behavior of a laser upon injection of a weak-field resonant with a cavity mode adjacent to the lasing mode; the four-wave-mixing coupling generates also a field with the frequency of the symmetrical adjacent mode. Using the Maxwell-Bloch equations, we show that under appropriate conditions there is a spectacular amount of frequency conversion, with a peak when the free spectral range is equal to the Rabi frequency of the laser field multiplied by \ensuremath{\surd}2 . The origin of this effect is identified with the help of the linear stability analysis of the free-running laser, which shows also the deep connections with the classical laser instability, predicted by Risken and Nummedal [J. Appl. Phys. 39, 4662 (1968)] and Graham and Haken [Z. Phys. 213, 420 (1968)]. The Rabi peak is a precursor of the multimode laser instability, but it arises for smaller values of the pump parameter and is only weakly sensitive to the transverse shape of the laser beam.
By means of numerical modeling of nonlinear and analytical solution of linearized Maxwell-Bloch equations, we find an intriguing regime of a superradiant laser with a symmetric low-Q Fabry-Perot cavity equipped with a distributed feedback of counter-propagating waves. We find that, in a wide range of parameters, there is lasing of a strongly-asymmetric self-modulated powerful polariton mode accompanied by excitation of several almost symmetric weak polariton or electromagnetic modes. The frequency of the powerful mode is approximately coincides with a center of the laser spectral line, and the frequencies of the weak modes are outside the spectral line of an active medium. We show that such a lasing takes place due to formation of a self-consistent inhomogeneous asymmetric grating of population inversion of active centers that provides undamped Rabi oscillations of the population inversion and dipole moments of centers. The grating makes the powerful ‘central’ mode strongly asymmetric, and the Rabi oscillations of active centers excite the weak ‘sideband’ modes due to the quadratic resonant nonlinearity of the two-level active medium.
An explicit solution is obtained for the four-wave frequency mixing ${\ensuremath{\omega}}_{d}={\ensuremath{\omega}}_{a}\ensuremath{-}{\ensuremath{\omega}}_{b}+{\ensuremath{\omega}}_{c}$ of two strong fields a and c and two weak fields b and d in a four-level system with large Doppler broadening in collinear geometry, where the frequencies of weak fields are nearly equal, ${\ensuremath{\omega}}_{b}\ensuremath{\simeq}{\ensuremath{\omega}}_{d},$ and the medium is optically thin. Without weak fields there are two independent two-level systems. A pair of weak fields probes two other allowed transitions. A peak of the mixing coefficient as a function of intensity is found around an equal Rabi splitting of both two-level systems. The effect is based on a resonance between two closed cycles of four-wave mixing via different dressed states. Three, four, or six peaks are predicted in the dependence of the mixing coefficient on the frequency of the weak field; two of them are a consequence of averaging over velocities. The model allows an interpretation of the dependence of the output wave power on the intensity and detuning in recent experiments on frequency mixing in sodium vapor.
We start from the Maxwell-Bloch equations that govern the dynamics of Fabry-Perot lasers and, considering a doubled cavity, generalize our traveling wave formalism to the case in which the two-level medium does not fill the cavity but occupies only a portion of it. We linearize these equations and focus on the adiabatic elimination of both atomic fluctuations. Next, we restrict our attention to free running lasers under resonant conditions and analyze amplitude and phase instabilities. The examination of the unstable domain leads to the conclusion that the multimode Fabry-Perot instability arises near threshold only when the ratio of the longitudinal to the transverse atomic relaxation rate is substantially smaller than unity. This result agrees with our previous study in the limit of adiabatic elimination of the atomic polarization fluctuations only. We describe the self-pulsing behavior that arises from the multimode instability, and exhibits hysteretic behavior when the pump parameter is swept forward and backward. Finally, we investigate the single-mode instability and show that in the Fabry-Perot case there is no longer the correspondence between single-mode and multimode instabilities that is well known in the case of ring lasers. We confirm that in the case of resonant Fabry-Perot lasers the single-mode amplitude instability arises very far from threshold.
An explicit solution is obtained for the four-wave mixing ω 4 =ω 1 −ω 2 +ω 3 of two strong fields E 1 , E 3 and two weak fields E 2 , E 4 in a four-level system with large Doppler broadening. Resonance of the intensity dependence of the mixing coefficient is found around equal Rabi frequencies, E 1 · d 1 = E 3 · d 3 , where d 1,3 are the dipole moments of the corresponding transitions. The effect is interpreted as a crossing of quasi-energy levels. Up to 6 peaks appear in the dependence of the conversion coefficient on the detuning of the probe field E 2 . The unexpected additional pair of peaks is a consequence of averaging over velocities. The results permit interpretation of the saturation behavior found in recent experiments on mixing in sodium vapor.
We compare theoretical predictions and experimental data for the multimode instability of optical bistability, which arises from the coherent dynamics of a system of two-level molecules contained in an optical cavity and driven by an external stationary laser field. Considerable insight into this phenomenon is provided by two ``rules of thumb'' that govern the behavior of the self-pulsing frequency, and establish its relation with the Rabi frequency of the intracavity field. We describe in detail the experimental apparatus and the main features of the experimental findings. The theoretical results are based on the Maxwell-Bloch equations for a ring cavity in the plane-wave approximation. Despite the crudeness of this model, the numerical data display a satisfactory qualitative agreement with the experimental results. Additional period-doubling and chaotic phenomena, predicted by the theory and not observed in the experiment, are presumably washed out by the longitudinal and radial variation of the electric field in the cavity.
Four-wave mixing (FWM) in a cycling transition Fg =1 ↔ Fe =0 is investigated. Subject to a linearly polarized pump field with frequency ω c and a magnetic field, and simultaneously saturated and probed by the circularly polarized fields (σ ±) with frequencies ω s and ω p , three kinds of FWM fields with frequencies: 2ω s − ω p, ω c+ω s − ω p and ω c − ω s+ω p, emit. The multi-peaked structure of FWM spectra is obtained and explained as the resonant emission between the ‘virtual states’ in the dressed picture. The generated FMW signals have high quantum efficiency for a wide range of drive Rabi frequency and saturation Rabi frequency. Finally, the emitted fields ω c + ω s − ω p and ω c − ω s+ω p have about the same intensity for the well matching among the Zeeman splitting, drive Rabi frequency and saturation Rabi frequency.