VOLUME 70, NUMBER 8 P H YSICAL REVI EW LETTERS Hesitation Phenomenon in Dynamical Hysteresis 22 FEBRUARY 1993 J. Zemmouri, B. Segard, W. Sergent, and B. Macke I.aboratoire de Spectroscopic Hertzienne, Universite de Lille I, 59655 Villeneul e d'Ascq, France (Received 11 March 1992) The effect of the sweeping velocity on the hysteresis loop of a bistable system is examined in the case where the velocity is so large that the system is prevented from undergoing any transition during the for- ward sweep. The experiments, made on optical and electronic devices, evidence a dramatic instability ("hesitation") of the return path of the hysteresis loop for a critical value of the sweeping velocity. The main features of this generic phenomenon are well covered by a one-dimensional analytic theory which provides scaling laws in good agreement with the observations. PACS numbers: 64.60. Ht, 42. 65.Pc The phenomenon of hysteresis is common to various systems in physics, mechanics, chemistry, etc. We report in this Letter on new effects incidentally uncovered in the course of an experimental study of dynamical hysteresis in optical bistability. Optical bistability [1,2] provides an example of purely deterministic hysteresis, observed in the absence of fluctuations. On the other hand, hysteresis is also commonly associated with first-order phase transi- tions, primarily governed by fluctuations. In fact the dis- tinction between these two types of hysteresis is not clear cut. Intrinsic fluctuations and technical noise are indeed unavoidable in bistability experiments and deterministic effects play an important role in the dynamics of first- order phase transitions. We notice in particular the current interest in the search of quasicritical phenomena near the absolute boundaries of metastability in such transitions [3]. The phenomena described hereafter, gen- eric to deterministic bistable systems, are then expected to occur in a wider class of hysteretic systems. To be definite we consider a bistable system whose steady-state characteristic x vs p is an S-shaped curve as given in Fig. 1. x is the output variable and p is one of B' I I Np Pg NA VM i CONTROL PARAMETER ( p ) p FIG. 1. Steady-state characteristic of a bistable system and standard sweeping scheme. the external parameters controlling the system. In the absence of fluctuations, the states corresponding to the upper and lower branches of the S are strictly stable, whereas those belonging to the intermediate branch are unstable. If the control parameter p is adiabatically swept forth and back through the bistability domain (ptt & p & p~), the system describes an hysteresis cycle as shown in Fig. 1. In any real experiment the sweep duration r is obviously finite and dynamical effects occur [4] even at low sweep rate because of the divergence of the evolution times in the vicinity of the turning points 8 and B (critical slowing down) [1,2]. The hysteresis loop actually observed depends on the sweeping velocity v=dp/dt. For moderate velocities, the clear-cut transi- tions AA' and BB' are generally smoothed and delayed, and the hysteresis loop widens. The first optical study of these phenomena has been made on a CO2 laser with an intracavity saturable absorber [5], and quantitative re- sults, including scaling laws, have been recently obtained on a bistable semiconductor laser [6] and on a passive bistable device [7]. Qualitatively different phenomena occur when the sweeping velocity becomes so large that the system may be prevented from switching up during the forward sweep although the control parameter p goes beyond the critical value p~ [4,8]. In this regime, so- called frustrated switching [9], we evidenced — for a par- ticular value of the velocity — a dramatic sensitivity of the return path of the hysteresis loop to very small changes of the experimental parameters. The return path seems then to "hesitate" between quite different trajectories, a phenomenon largely overlooked in previous works. Our first demonstration of hesitation was made in ab- sorptive all-optical bistability. The experiments were realized at a millimetric wavelength (X =3.5 mm) and the experimental setup, adapted from that extensively de- scribed in Ref. [10], consisted of a 23-m-long waveguide Fabry-Perot cavity filled with HC' N gas at low pressure. The source and the cavity were tuned to the frequency of the (I) 0-1 rotational line of HC' N which behaves then as a saturable absorber. The output variable and the con- trol parameter were, respectively, the power transmitted by the cavity and the voltage applied to the modulator 1993 The American Physical Society 1135