Dynamics and coherence of a multimode semiconductor laser with optical feedback in an intermediate-length external-cavity regime C. Serrat, S. Prins,* and R. Vilaseca Departament de Fı ´sica i Enginyeria Nuclear, Universitat Polite `cnica de Catalunya, Colom 1, E-08222 Terrassa, Spain Received 26 February 2003; published 4 November 2003 We use a traveling-wave numerical model of the temporal dynamics in semiconductor lasers to investigate the dynamics and coherence properties of a multimode semiconductor laser subject to optical feedback. Con- sidering an external cavity of intermediate length, we have observed features characteristic of the so-called short-cavity regime in a region where the relaxation oscillations frequency is larger than the external-cavity frequency. The time scales of the dynamics are controlled in this case by the strength of the optical feedback. We have examined the coherence properties of the multimode emission and have studied how the coherence time is reduced as the laser becomes more multimode. DOI: 10.1103/PhysRevA.68.053804 PACS numbers: 42.65.Sf, 05.45.Jn, 42.55.Px I. INTRODUCTION Semiconductor lasers play a central role in the growing world of optoelectronic technologies. Intense research activ- ity in recent years has been focused on the effect of optical feedback in the dynamics of these lasers. In this context, several studies have focused their attention on the multimode behavior of semiconductor lasers, since many practical ap- plications normally use semiconductor lasers operating in several longitudinal modes simultaneously 1. On the other hand, delayed dynamical systems have become a broad in- terdisciplinary subject, since this type of systems appear in many different fields of science 2, semiconductor lasers constituting an excellent tool to investigate their behavior. In the context of the optoelectronic industry, relatively short external cavities are relevant for many applications e.g., fiber couplers or compact discs. In a recent study 3, it was shown that the characteristics of the dynamics of a semiconductor laser subject to optical feedback depend strongly on the length of the external cavity. In particular, Heil et al. 3 showed that the dynamical behavior of the system is governed by one of the two basic frequencies, namely, the relaxation oscillations frequency  RO in the case of the so-called long-cavity regime LCR, and the external- cavity frequency  EC in the case of the so-called short-cavity regime SCR. Indeed, for long external cavities (  RO  EC ), the behavior of the system is characterized by ir- regular fast pulses at  RO , while the dynamics show regu- lar pulses at  EC in the case of short external cavities (  RO  EC ). To our knowledge, however, no study of the regime corresponding to cavities of intermediate length has been performed, so that the main features characterizing the laser behavior in such conditions are not known. In the present paper, we analyze the influence of multi- longitudinal-mode emission on the dynamics of a semicon- ductor laser with an external cavity of length of the order of 12 cm, which is intermediate between the LCR and SCR cases. Due to the length of the external cavity that we con- sider, the ratio  RO /  EC is reversed as the injection current is varied, with  RO /  EC 1 near the lasing threshold, and  RO /  EC 1 for larger values of the injection current. Our contribution consists in the study of the region around twice the lasing threshold, which is interesting from an applied point of view for potential applications in chaotic secure communications 5. By thus fixing the injection current at twice the lasing threshold, we obtain, for the case considered in the present study,  RO 2.6 GHz and  EC 1.2 GHz. In such conditions, we will show that the system switches from dynamics dominated by  RO to dynamics that include pulses at  EC by only increasing the feedback strength. Addition- ally, we also analyze in this paper the temporal coherence of the emission, as a function of several laser parameters, by means of the visibility function. As it will be discussed in detail, we observe how the coherence time is degraded as the laser becomes more multimode. In Sec. II we describe the system and the procedure used for numerical integration. In Sec. III our main results are presented and discussed and in Sec. IV the main conclusions are summarized. II. MODEL AND NUMERICAL INTEGRATION We use a traveling-wave model of coupled partial differ- ential equations recently derived by White et al. 4, which *Permanent address: Vrije Universiteit Brussel, Department of Applied Physics and Photonics, Pleinlaan 2, B-1050 Brussels, Belgium. R 1 R 2 R 3 AR E + E - Z=0 Z=1 Z FIG. 1. Illustration of the laser cavity. The mirror effective re- flectivities are R 1 , R 2 , and R 3 . Inside the cavity, the forward and backward propagating fields are E + and E - , respectively. The mir- ror with reflectivity R 2 is antireflection coated AR for beams com- ing from the external mirror R 3 . PHYSICAL REVIEW A 68, 053804 2003 1050-2947/2003/685/0538047/$20.00 ©2003 The American Physical Society 68 053804-1