SIGEST

Abstract

Previous article Next article Full AccessSIGESThttps://doi.org/10.1137/SIREAD000045000003000521000001BibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThe SIAM Journal on Applied Mathematics (SIAP) has a long history of publishing important papers that use mathematical models and numerical studies to enhance understanding of applications from the physical and biological sciences and engineering. For this issue's SIGEST paper, "Hopf Bifurcation Subject to a Large Delay in a Laser System," by D. Pieroux, T. Erneux, A. Gavrielides, and V. Kovanis, which originally appeared in 2000 (volume 61 of SIAP), the motivating physical problem is the sensitivity to optical feedback of semiconductor lasers.Lasers are ubiquitous today, mostly in the form of semiconductor lasers (also known as laser diodes), which are characterized by smallness in size, weight, cost of production, and power requirements. Optical networks depend on semiconductor lasers for generation, amplification, and distribution of the light that transmits voice, video, and data. The 2004 worldwide market for semiconductor lasers is expected to be more than $20 billion in communication applications, and more than $2 billion for other applications such as DVD video players and DVD-ROM drives.An unfortunate property of these remarkable devices is their high susceptibility to unavoidable optical feedback, such as reflection from any optical element of the system surrounding the laser. Even tiny amounts of optical feedback (less than 0.01%) can cause the laser to enter a state of erratic pulsating instabilities and irregular chaotic transitions, revealing highly complex structures of nonlinear dynamics and bifurcations.Fourier spectra measurements show a gradual increase of oscillatory instabilities, but do not reveal the bifurcation mechanisms, and other experimental studies are difficult because of the small (10-12 seconds, i.e., picosecond) time scale of the intensity pulsations. Most of the progress in understanding these instabilities has come from comparisons between experimentally obtained Fourier spectra and numerical studies of (relatively) simple mathematical models, such as the one explored in this paper.Beginning with the 1980 Lang--Kobayashi (LK) nonlinear delay-differential equations, Pieroux et al. motivate the importance to laser instabilities of Hopf bifurcation subject to a strong delay. LK solutions with "two-time" behavior---the first time is the photon lifetime, the second the round-trip time from laser to mirror and back---lead to a Hopf formulation containing two parameters, one large and one small. The authors' analysis exploits the small parameter, leading to a problem that is mathematically simpler than the fully general LK formulation. In addition to a lucid presentation, clear even to a nonexpert, of the mathematical model, the paper demonstrates the importance of Hopf local theory in guiding the numerical studies. Furthermore, the authors' results are relevant to future physical experiments because they show the possibility of different forms of bistable response near the first instability of the laser.We are pleased to present a paper that perfectly represents SIAP's distinguished tradition by combining an important application, analytical mathematics, and numerical experiments. Previous article Next article FiguresRelatedReferencesCited ByDetails Volume 45, Issue 3| 2003SIAM Review History Published online:04 August 2006 Article & Publication DataArticle DOI:10.1137/SIREAD000045000003000521000001Article page range:pp. 521-521ISSN (print):0036-1445ISSN (online):1095-7200Publisher:Society for Industrial and Applied Mathematics