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Optics Letters

Exact solution for dispersive optical bistability with inhomogeneous broadening

M. Gronchi, Luigi A. Lugiato

Published Mar 1, 1980• 23 citations
Publisher

Abstract

We solve exactly at steady state. fully including propagation effects, the Maxwell–Bloch equations for dispersive optical bistability from an inhomogeneously broadened atomic system placed in a ring cavity. In order to obtain an analytical solution, a Lorentzian atomic line shape is assumed. In the so-called mean-field limit, the exact solution approaches the previously derived mean-field-state equation. For large values of the bistability parameter, one finds situations with multistability.

DOI
10.1364/ol.5.000108

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We solve exactly at steady state. fully including propagation effects, the Maxwell–Bloch equations for dispersive optical bistability from an inhomogeneously broadened atomic system placed in a ring cavity. In order to obtain an analytical solution, a Lorentzian atomic line shape is assumed. In the so-called mean-field limit, the exact solution approaches the previously derived mean-field-state equation. For large values of the bistability parameter, one finds situations with multistability.
The problem of dispersive optical bistability has so far been treated only in the mean-field approximation. A rigorous justification of the mean-field theory can only be obtained from exact solutions of the steadystate Maxwell-Bloch equations which retain the spatial dependence of the field. In this paper exact analytic solutions are presented for these equations. The authors demonstrate that the mean-field equation connecting the input and the output fields follows naturally from these solutions in the limits $T\ensuremath{\rightarrow}0$, ${\ensuremath{\delta}}_{F}\ensuremath{\rightarrow}0$, and $\ensuremath{\alpha}L\ensuremath{\rightarrow}0$ for the mirror transmission coefficient, the detuning of the field from the cavity resonance, and the linear absorption, respectively, with $\frac{\ensuremath{\alpha}L}{2T}$ and $\frac{{\ensuremath{\delta}}_{F}L}{2cT}$ remaining finite. The results are illustrated with the help of graphs showing the output versus input intensity for different values of the relevant parameters. The effect of these parameters on the phase shift of the output field is also displayed.
The steady state behavior of a bistable system of a homogeneously or inhomogeneously broadened two sorts of two-level atoms placed in a ring cavity and driven by a coherent field is analyzed within the mean field limit where the transverse effect of the radiation field is taken into account within the single-transverse-mode model. Effects on the input–output relation, due to the transverse parameter and the Lorentzian and the Gaussian line widths which measure the inhomogeneous broadening of the atoms, are examined. The enlargement of the optical bistable (tristable) region according to the previously mentioned parameters is explained.
The steady-state bistability and the evolution of the field envelope of a ring-cavity optical-bistable system including the effect of the transverse radiation field and the effect of an inhomogeneously broadened atomic medium are studied by means of the mean-field approximation and dressed-mode theory in this paper. Our study shows that the bistable region enlarges with an increase of the ratio r0{\mathit{r}}_{0}r0​/w0{\mathit{w}}_{0}w0​ (r0{\mathit{r}}_{0}r0​ is the radius of the cylindrical atomic sample and w0{\mathit{w}}_{0}w0​ the beam waist of the Gaussian beam), and decreases with increasing width of the inhomogeneously broadened line. The transverse-field effect has more impact on the magnitude of the bistable region than the inhomogeneous broadening under certain conditions. In the case of two unstable modes, the evolution of the field envelope before reaching the stable oscillation regime differs remarkably from that in the case of the plane-wave approximation with a homogeneously broadened atomic medium.
Optical bistable behavior for a system of inhomogeneously broadened two sorts of two-level atoms in a ring cavity is investigated outside the rotating wave approximation (RWA). The model Maxwell–Bloch equations are treated with Fourier decomposition up to first harmonic. The first harmonic output field component exhibits reversed or mushroom bistability simultaneously with bi- and double-bistability in the fundamental field component. Inhomogeneous broadening and transverse field effects are also considered.
The steady state behaviour of a bistable system of a homogeneously/inhomogeneously broadened two-level atomic medium in contact with a squeezed vacuum field and placed in a ring cavity driven by a coherent field is analysed within the mean field limit where the transverse effect of the radiation field is taken into account within the single-transverse-mode model. Effects on the input–output relation and the critical value of the cooperative parameter C crit are examined and compared with these results in the plane wave approximation. The 'cooperative switching effect' between the output states of the system due to variation of the cooperative parameter C is also investigated. Linear stability analysis for the model Maxwell–Bloch equations is presented in the homogeneously broadened case.
We analyze instabilities in mixed absorptive and dispersive optical bistability in the rate-equation approximation and the mean-field and good-cavity limits. Our starting point is a set of multimode equations derived from the Maxwell-Bloch equations for ring-cavity boundary conditions. We obtain analytic expressions for the instability conditions. In a plane-wave analysis, we find that a portion of the lower transmission branch can be unstable in addition to the upper-branch instability found in purely absorptive bistability. Also, a new disconnected region of instability can exist on the upper branch. Our analysis becomes particularly simple for equal and opposite cavity detunings and we explore this case in detail. We extend our treatment to include a Gaussian transverse intensity profile and show that the instabilities remain in the presence of Gaussian averaging. We also show that many of the results obtained in the rate-equation approximation hold when the atomic linewidth and atomic decay rate are of the same order.