Amplitude modulation in a pair of time-delay coupled external-cavity semiconductor lasers

The phenomenon of amplitude death in coupled nonlinear oscillators has been a topic of recent interest. We demonstrate that a similar phenomenon can occur in a pair of time-delay coupled, external-cavity semiconductor lasers. In particular, with coupling chaotic oscillations of the laser field can be converted into quasiperiodic motion and low-frequency fluctuations in laser power can be suppressed.  2003 Elsevier B.V. All rights reserved. PACS: 42.65.Sf; 42.55.Px; 05.45.-a There has been an interest in the phenomenon of amplitude death [1–3] in the context of time-delay coupled limit-cycle oscillators [4,5]. The phenomenon was first observed in coupled chemical oscillators [1]. It was then established theoretically that, if the coupling is sufficiently strong and the spread in the natural frequencies of the oscillators is sufficiently broad, the amplitudes of the oscillations can reach zero [2,3]. The issue of time-delayed coupling, which is physically important, was recently addressed both theoretically [4] and experimentally [5]. An interesting result is that in the presence of a time delay, amplitude death can occur even if the natural frequencies of the limit* Corresponding author. E-mail address: [email protected] (Y.-C. Lai). 0375-9601/$ – see front matter  2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2003.08.072 cycle oscillators are all identical [4], in sharp contrast to the situation of zero time delay, where a broad distribution in the natural frequencies of the oscillators is necessary for the amplitude death [2,3]. Since time delay is present in many physical applications, amplitude death may be pervasive in coupled oscillators. The aim of this Letter is to present evidence that a similar phenomenon can occur in time-delay coupled, externalcavity semiconductor lasers. The observable phenomenon is that the laser oscillation can be modulated significantly. We call the phenomenon amplitude modulation in coupled semiconductor lasers. In many applications of semiconductor lasers, optical feedbacks are deliberately introduced to improve the performances of the laser such as the enhancement of the single longitudinal mode operation, spectral line narrowing, improved frequency stability, and 72 A. Prasad et al. / Physics Letters A 318 (2003) 71–77 Fig. 1. Our coupling scheme generating amplitude modulation in external-cavity semiconductor lasers. wavelength tunability, etc. [6]. However, at moderate feedback levels, which can be anticipated in most applications, the laser power can exhibit sudden, downto-zero dropouts at random times, followed by a slow and gradual recovery after each dropout. The phenomenon is most serious when the pumping current is close to the solitary threshold. The average frequency of the dropouts is typically at the MHz-level, which is several orders of magnitude smaller than that of the solitary laser relaxation oscillation, hence the term lowfrequency fluctuations (LFFs). In most applications, LFFs are undesirable. As we will show, our amplitudemodulation phenomenon can be used to potentially suppress the LFFs. We consider a pair of coupled semiconductor lasers, as schematically illustrated in Fig. 1. Two laser diodes, denoted by L1 and L2, are mutually coupled through a coupling parameter η1 < 1, i.e., a η1 fraction of the laser field of L1 is injected into L2 and vice versa. Laser L2 is subject to an additional optical feedback, characterized by the external mirror M with reflection coefficient η2. The physical distance between L1 and L2 is l1 and the length of the external cavity of L2 is l2. We shall demonstrate that with the coupling scheme, which can be implemented in laboratory experiments, the optical output from the laser L1 can be modulated so that its maximum values are decreased but its minimum values, which is zero when it is uncoupled with L2 and subject to optical feedbacks from M , is bounded away from zero. The average power of the output from L1 maintains at approximately the same level as that in the absence of coupling. Such sustained operation of the laser without complete power dropouts occurs in parameter regions of positive measure. The fundamental equations modeling a single external-cavity semiconductor laser is a set of delaydifferential equations, known as the Lang–Kobayashi (LK) equations [7], which describe the time evolutions of the complex electrical field E(t) of a single longitudinal mode and the carrier density n(t) averaged spatially over the laser medium. The equations can be written in a standard normalized form [8]