Bistability of pulsating intensities for double-locked laser diodes M. Nizette and T. Erneux Universite ´ Libre de Bruxelles, Optique Nonline ´aire The ´orique, Campus Plaine, Code Postal 231, 1050 Bruxelles, Belgium A. Gavrielides and V. Kovanis Nonlinear Optics Center, Air Force Research Laboratory, 3550 Aberdeen Avenue SE, Kirtland Air Force Base, New Mexico87117-5776 T. B. Simpson Jaycor, Inc., P. O. Box 85154, San Diego, California 92186-5154 Received 17 August 2001; published 8 May 2002 Rate equations for semiconductor lasers subjected to simultaneous near-resonant optical injection and mi- crowave current modulation are examined by combined analytical-numerical bifurcation techniques. Simple qualitative criteria are given for a bistable response. These results compare well with experimental measure- ments. DOI: 10.1103/PhysRevE.65.056610 PACS numbers: 42.65.Sf, 42.60.Fc, 42.65.Pc, 42.55.Px I. INTRODUCTION Bistability of pulsating regimes for weakly periodically forced oscillators is a well known phenomenon for mechani- cal or electronic systems such as the driven Duffing or van der Pol oscillators 1,2. It is, however, poorly documented for lasers exhibiting limit-cycle intensity oscillations such as semiconductor lasers subject to optical injection. Cases of bistability have been reported for periodically modulated semiconductor lasers but under strong modulation conditions and between different periodic states such as period-1 and period-2 regimes 3–6. Bistability under weak modulation conditions is less well known for lasers in general 7 but this problem was revived by recent experiments using optically injected diode lasers at high injection rates 8. In these ex- periments, cw optical injection induces a limit-cycle instabil- ity in the optical output. When a small periodic current modulation is added to the dc bias current remarkable limit- cycle phase locking performances have been obtained. The laser is doubly locked to the cw optical injection and the microwave current modulation. The output power exhibits a deep, high quality microwave oscillation. A similar perfor- mance has been observed previously by using sideband in- jection locking where a semiconductor laser is stably locked to a modulation sideband of a master laser 9–13. Such devices are promising for a number of applications that re- quire a spectrally pure microwave oscillator with frequencies in the tens of gigahertz. A bistable output means that two distinct amplitudes of the oscillations are available for cer- tain values of the control parameters. As we shall demon- strate analytically and show with experimental measure- ments, bistability can be achieved under strong optical injection with simultaneous weak current modulation, al- though the domain of parameters is not obvious. In this paper, we model such a double-locking experiment using the laser rate equations, which are then analyzed by combined analytical and numerical bifurcation techniques. Our objective is to determine simple conditions for bistabil- ity which will guide our experiments. In our previous analy- sis of the laser equations 14,15, the optical detuning was assumed zero, and we analyzed the laser response as a func- tion of the optical injection rate. We found several branches of periodic and quasiperiodic intensity oscillations but no case of bistability. Our present analysis of the laser equations takes the optical detuning instead of the injection rate as the control parameter and leads to cases of bistability. The paper is organized as follows. In Sec. II, we formu- late the laser rate equations and derive a slow time amplitude equation. The solutions of this equation are then analyzed analytically and numerically and the possible bifurcation dia- grams are shown in Sec. III. We emphasize the domains of bistability in the injection rate versus detuning diagrams. Section IV is devoted to the experiments and the main results are summarized in Sec. V. II. FORMULATION AND AMPLITUDE EQUATION Most of our understanding of the effects of optical injec- tion in a single-mode diode laser comes from numerical and analytical studies of simple rate equations for the complex electric field E and the excess carrier number Z. The formu- lation of dimensionless rate equations is documented in sev- eral places. Introducing the current modulation, these equa- tions are given by 15 d E dt = 1 +i   Z E+i  E+ , 1 T dZ dt = P  1 + cos  t  -Z - 1 +2 Z  | E| 2 . 2 Here time t is measured in units of the photon lifetime.  denotes the amplitude of the injected signal.  and  de- scribe the modulation depth and the frequency of the current modulation, respectively. P is the average value of the di- mensionless pumping current above threshold. The fixed pa- rameters  and T are the linewidth enhancement factor, which measures the degree of amplitude-phase coupling, and the ratio of carrier lifetime to photon lifetime, respectively. T is typically an O (10 3 ) large quantity which allows important PHYSICAL REVIEW E, VOLUME 65, 056610 1063-651X/2002/655/0566105/$20.00 ©2002 The American Physical Society 65 056610-1