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Annual Meeting Optical Society of America

Theoretical analysis of the singlefeedback mirror spatiotemporal instabilities

Martine Le Berre, Elisabeth Ressayre, Andree A. Tallet

Published Jan 1, 1989• 0 citations
Publisher

Abstract

The linear stability analysis of the plane wave stationary solution of the dispersive single-feedback mirror device displays a 2τR oscillation only if the phases of the stationary forward and backward field amplitudes are 2π shifted, δφ = 2kπ. The condition provides an instability boundary in the space of the atom field detuning and the input intensity, the shape of which agrees with experimental results. The 2τR oscillation is the signature of an instability which develops in a phase conjugate resonator, constituted of the atomic cell and a totally reflecting ordinary mirror. Thus, a stationary symmetric profile may undergo a 2τR Hopf bifurcation together with a spontaneous breaking of the cylindrical symmetry. The a-c profiles are constituted of spots located on a ring, as observed in the Tucson experiment [G. Giusfredi et al., J. Opt. Soc. Am. B 5, 1181 (1988)]. A mean phase δφp is defined, which results from an average of the grating, weighted with the mode TEM0, ±p. Therefore the system itself determines the number of spots in order that δφp fulfills the requirement δφ= 2kπ.

DOI
10.1364/oam.1989.fw5

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The linear stability analysis of the plane wave stationary solution of the dispersive single-feedback mirror device displays a 2τR oscillation only if the phases of the stationary forward and backward field amplitudes are 2π shifted, δφ = 2kπ. The condition provides an instability boundary in the space of the atom field detuning and the input intensity, the shape of which agrees with experimental results. The 2τR oscillation is the signature of an instability which develops in a phase conjugate resonator, constituted of the atomic cell and a totally reflecting ordinary mirror. Thus, a stationary symmetric profile may undergo a 2τR Hopf bifurcation together with a spontaneous breaking of the cylindrical symmetry. The a-c profiles are constituted of spots located on a ring, as observed in the Tucson experiment [G. Giusfredi et al., J. Opt. Soc. Am. B 5, 1181 (1988)]. A mean phase δφp is defined, which results from an average of the grating, weighted with the mode TEM0, ±p. Therefore the system itself determines the number of spots in order that δφp fulfills the requirement δφ= 2kπ.
Spatiotemporal instabilities of a single-feedback-mirror device are investigated with the help of the linear stability analysis of the stationary regime for a saturable medium that has a complex nonlinear susceptibility; the outcome is compared with experimental results. The first striking result of the analysis is the occurrence of an oscillation with a period nearly twice the round-trip time, regardless of its value with respect to the atomic linewidth for given physical situations, when only longitudinal grating effects are considered. This result cannot be interpreted as a pure phase-conjugate resonator operation, which is expected to occur only when transverse grating effects are included. When such effects are included, the second striking feature is the prediction of the spontaneous breaking of the revolution symmetry; symmetry is broken by rotating light spots, with an instability threshold that depends on the number of spots; this prediction agrees with experiment.
The instability threshold characteristics are investigated for two different optical devices, the ring cavity and the feedback mirror, with passive weakly absorbing two-level atoms cell. The linear stability analysis of the ring cavity predicts emission of several conical shells, depending on the optical arrangement and the buffer gas pressure. In the case of pure vapor, each conical shell has its own frequency. All the cones display the same frequency in the case of large buffer gas pressure. The feedback mirror device displays very different instability characteristics, with only one frequency emitted along a unique cone, in agreement with the experiment. The self-oscillation is shown to be controlled by the reflectivity properties of the cell, which behaves at threshold as a mixture of ordinary mirror and phase conjugate mirror
The threshold characteristics of the static and periodic instability of the single feedback mirror device are investigated for a homogeneously broadened two-level medium as a function of the input intensity, the detuning, the buffer-gas pressure, the atomic concentration, and the round-trip time d, and compared with experimental results. For a large range of parameters that agrees with the experiment, the self-oscillation process is shown to be mainly induced by the longitudinal grating, which ensures a large instability domain, and controlled by the buffer-gas pressure, which must be low enough in order that the threshold period of the oscillation is constant on a doughnutlike transverse profile and nearly equal to 2d. This oscillation, called the 2d oscillation, is neither a result of pure phase-conjugate-mirror operation of the atomic medium nor a result of a Rabi sideband gain process. It is shown to take place through a mixture of phase-conjugate-mirror and ordinary mirror operations, which are both allowed by the specific optical arrangement of the device.
Of the Lorenz-type instabilities expected in optical systems, the single-mode instability in optical bistability has been the subject of detailed theoretical and experimental work by our groups. A treatment employing the plane-wave and mean-field limits of the Maxwell-Bloch equations predicts a rich variety of higher-order bifurcations including period doublings and intermittency.1 An extension of this work to include the transverse dependence of the intracavity field reveals no significant qualitative differences between the plane-wave theory and the Gaussian beam theory as regards the stability boundaries obtained from a linearized analysis.2 The instability persists in the presence of a Gaussian transverse profile and the boundaries are not radically modified compared with those obtained from the plane-wave theory. However, numerical integration of the Maxwell-Bloch equations in the single transverse mode approximation does not show the wealth of higher-order dynamic states present in the plane-wave case. An experimental investigation of optical bistability for two-level atoms in a single transverse, single longitudinal mode cavity exhibits limit cycle oscillation with stability boundaries that agree well with the boundaries obtained from the work in Ref. 2. However, for the range of values C < 300, |Δ| < 5, |θ| < 50, X ~ 102, the experimental results do not show the diversity of higher dynamic states expected from the plane-wave theory, in agreement with our numerical results. In addition the measurements support the existence of a single transverse mode oscillation. The sum of our theoretical and experimental work provides a quantitative statement for the role of the transverse effects in optical instabilities.
Phase conjugate resonators (PCR) are optical resonators in which (at least) one of the mirrors is a phase-conjugate one (PCM), e.g. a four-wave mixing cell. These systems are of considerable experimental and theoretical interest, specially in view of their potential to eliminate phase distorsions within the resonator1. In recent work2, we discussed a PCR consisting of a Fabry-Perot resonator with one end mirror replaced by a lossless Kerr medium of infinitely fast response time pumped by two counterpropagating beams of same frequency. We described the PCM within the Raman-Nath approximation3, but including fully the effects of pump depletion. The normal mirror was positioned such as to couple the first order scattered field back into the PCM. We found that this system can break into self-oscillations corresponding to the half-axial modes predicted by linear theories1,4. For higher pump intensities, this is followed by a sequence of bifurcations to chaos, with in general the coexistence of several basins of attraction. We also showed5 that the first of this sequence of bifurcations is in general already sufficient to induce a quasi-periodic motion of the intracavity field phase, leading in practice to its complete indetermination.
This paper gives a detailed description of the plane-wave Gaussian models of optical bistability and of the experimental investigations of the single-mode instability induced on a beam of sodium atoms in a cavity by an external driving field. Our goal is to provide a careful comparison of the theoretical and experimental results. We focus on the shape of the instability domain, the nature and behavior of the spontaneous output oscillations produced by the instability, and their frequency dependence on the control parameters. We carry out this comparison using both the plane-wave and Gaussian models. Our analysis shows that the latter is in good quantitative agreement with the experimental results. We investigate in detail the main assumption of the Gaussian model, namely, that the internal cavity field retains the same radial profile as the input field.