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Physical Review A

Single-mode-laser phase dynamics

Eugenio Roldán, Germán J. de Valcárcel, Ramon A. Vilaseca, Paul Mandel

Published Jul 1, 1993• 46 citations
Publisher

Abstract

We study the phase dynamics of a single-mode ring laser described by the complex Maxwell-Bloch equations. We identify three reference-frame frequencies and determine the properties of the field dynamics observed in these frames. In one of these reference frames, the phase jumps are always equal to \ensuremath{\pi}, irrespective of the detuning, while in another reference frame quasiperiodic field portraits reduce to periodic field portraits. We also apply the recent theory of Ning and Haken [Phys. Rev. Lett. 68, 2109 (1992)] to prove that the laser phase can be decomposed into a geometrical component that is frame invariant and a dynamical component that is frame dependent.

DOI
10.1103/physreva.48.591

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Le Journal de Physique Colloques•Jun 1, 1988•4 citations
We study the phase dynamics of a single-mode ring laser described by the complex Maxwell-Bloch equations. We identify three reference-frame frequencies and determine the properties of the field dynamics observed in these frames. In one of these reference frames, the phase jumps are always equal to \ensuremath{\pi}, irrespective of the detuning, while in another reference frame quasiperiodic field portraits reduce to periodic field portraits. We also apply the recent theory of Ning and Haken [Phys. Rev. Lett. 68, 2109 (1992)] to prove that the laser phase can be decomposed into a geometrical component that is frame invariant and a dynamical component that is frame dependent.
In the previous chapters we considered the Maxwell–Bloch equations with the boundary condition appropriate for a unidirectional ring cavity and calculated the stationary solutions for laser and optical bistability, respectively. Next, we showed that the exact stationary solutions simplify appreciably in the low-transmission limit. In this chapter we want, first of all, to apply the low-transmission approximation on the dynamical equations directly, beyond the stationary case. We start from the Maxwell–Bloch equations (4.35)–(4.37) and the boundary condition (8.36). The latter is complex and includes the retardation Δ t and some parameters such as the transmissivity and reflectivity coefficients T and R , the input field y and the cavity detuning δ 0 . We introduce a transformation of the coordinates z and t and a transformation of the variables F and P [121–123]. As a result, the boundary condition reduces to a simple periodic boundary condition, and the electric-field equation incorporates the parameters which previously appeared in the boundary condition. The virtue of the periodic boundary condition is that it allows one to expand the electric field in terms of modal amplitudes and to recast the field equation in the form of a set of time-evolution equations for the modal amplitudes. All of this is shown in Section 12.1. In Section 12.2 we apply the low-transmission approximation and the Maxwell–Bloch equations assume a form that is a generalization of a set of equations commonly used in the literature to describe the ring laser [124]. Together with the Bloch equations for the atomic variables, the modal equations provide a model that lends itself ideally to a numerical resolution of the equations. In Section 12.3 we show that, if we assume that only the resonant cavity mode is active, this model reduces to a simple single-mode model consisting in three coupled ordinary differential equations, which will be analyzed in Part II of this book. Next, in the same section we derive the single-mode model in a more direct way, without passing through the multimodal equations.
The dynamic behavior of a coherently pumped single-mode unidirectional ring laser with a homogeneously broadened three-level active medium is studied. Our formulation is based on a set often real equations of the plane-wave, mean-field Maxwell–Bloch type. The instability domain in the main control parameters space is determined. Our numerical study of these equations for a parameter range of the type explored in the recent experiments by Weiss Brock [ Phys. Rev. Lett.57, 2804 ( 1986)] reveals some similarities, but striking differences between our theoretical predictions and their experimental observations are also noted.
Homogeneously broadened bidirectional ring lasers show alternation of the lasing direction induced dynamically by a long-lived population grating and slight detuning of the laser cavity from the atomic resonance1. We show that the mode switching is accompanied by abrupt jumps by pi in the total relative phase of the system2. Our analysis reveals that the carrier frequencies of the two modes are not equal even though the two unidirectional steady state solutions have the same frequency2. When one mode dominates and the other is suppressed, the dominant mode has the steady state frequency. However, a mode growing in strength has a frequency closer to the atomic resonance while a mode falling in strength has a frequency further from the atomic resonance. The dynamically induced mode-pulling and mode-pushing can be as much as 25 % of the steady state frequency detuning. The shifting frequencies of the modes indicate that the standing wave interference pattern rotates in the laser ring cavity, alternately moving clockwise and counterclockwise with nearly equal and constant speeds. The longitudinal spatial hole-burning pattern consequently also rotates in the ring tending to wash out the population grating and keeping it from developing an amplitude of much more than one percent of the spatially averaged population inversion.
It is well known that transverse intensity profiles of lasers may show a nontrivial spatial structure. Usually, depending on the degree of symmetry, the spatial patterns have been associated with modes of a passive resonator or combinations of them which cause oscillations in the laser intensity (mode beating).1 It was recently shown2 that the fluctuations may be due, instead, to the nonlinear coupling between atoms and the field. Here we show that a spontaneous symmetry-breaking process is also at the origin of the formation of complex spatiotemporal dynamics in which oscillations at a single frequency, quasiperiodicity, and chaos may coexist within in the same spatial structure. We compare our experimental results with those obtained from bifurcation theory and numerical integration of the Maxwell–Bloch equations. The transition from a simple (Gaussian) profile to a complex (turbulent) profile is studied by means of the average intensity patterns, frequencies, and phases of the local intensity and cross-correlation functions.
This paper contains a description of the behavior of a multimode unidirectional ring laser with a homogeneously broadened active medium. Our formulation is based on the conventional Maxwell-Bloch (MB) equations, but is distinguished from other treatments by the inclusion of a finite mirror reflectivity and an arbitrary value of the gain parameter. We review the steady-state behavior of the system and analyze the longitudinal profile of the field and of the atomic variables. With an appropriate transformation of variables, we transform the boundary conditions of the ring cavity into standard periodicity type, even in the presence of a finite reflectivity, and derive an infinite hierarchy of coupled mode equations. We analyze exactly the linear stability of the system, and investigate the dependence of the instability domain on the reflectivity and gain parameters. A numerical study of the full MB equations for a parameter range of the type explored in the recent experiments by Hillman et al. [Phys. Rev. Lett. 52, 1605 (1984)] reveals similarities, but also considerable differences between the results of the theory and the main experimental signatures of their instability. However, the injection of numerical noise shows the presence of numerous coexisting basins of attraction which are likely to play a significant role in the dynamics of a noisy laser.
We study the dynamic behaviour of a coherently pumped single-mode unidirectional ring laser with an active medium of Doppler broadened three-level atoms. Our formulation is based on a set of 9N+1 real equations of the plane-wave mean-field Maxwell-Bloch type (N is the number of velocity groups of molecules considered). This model predicts, for a parameter range of the type explored in the experiments by Weiss and Brock (Phys. Rev. Lett. 57 (1986) 2804), a behaviour that reproduces most of their main experimental features. These include the instability pump threshold and the sequence of instabilities ocurring on resonator tuning and on increasing pump strength.