Hopf Bifurcation Subject to a Large Delay in a Laser System

Hopf bifurcation theory for an oscillator subject to a weak feedback but a large delay is investigated for a specific laser system. The problem is motivated by semiconductor laser instabilities which are initiated by undesirable optical feedbacks. Most of these instabilities are starting from a single Hopf bifurcation. Because of the large delay, a delayed amplitude appears in the slow time bifurcation equation which generates new bifurcations to periodic and quasi-periodic states. We determine analytical expressions for all branches of periodic solutions and show the emergence of secondary bifurcation points from double Hopf bifurcation points. We study numerically different cases of bistability between steady, periodic, and quasi-periodic regimes. Finally, the validity of the Hopf bifurcation approximation is investigated numerically by comparing the bifurcation diagrams of the original laser equations and the slow time amplitude equation.

[1]  Jack K. Hale,et al.  Nonlinear Oscillations in Equations with Delays. , 1978 .

[2]  Takuya T. Sano.  Antimode dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback , 1993, Optics & Photonics.

[3]  Hermann Haken,et al.  Theory of quasiperiodicity in model of lasers with delayed optoelectronic feedback , 1999 .

[4]  J. Cole,et al.  Multiple Scale and Singular Perturbation Methods , 1996 .

[5]  Laurent Larger,et al.  Chaotic oscillator in wavelength: a new setup for investigating differential difference equations describing nonlinear dynamics , 1998 .

[6]  Antonio Politi,et al.  Multiple scale analysis of delayed dynamical systems , 1998 .

[7]  Athanasios Gavrielides,et al.  Bifurcation Cascade in a Semiconductor Laser Subject to Optical Feedback , 1999 .

[8]  Daan Lenstra,et al.  Sisyphus effect in semiconductor lasers with optical feedback , 1995 .

[9]  I. Epstein,et al.  An Introduction to Nonlinear Chemical Dynamics , 1998 .

[10]  S. Orszag,et al.  Advanced Mathematical Methods For Scientists And Engineers , 1979 .

[11]  Jesper Mørk,et al.  Chaos in semiconductor lasers with optical feedback: theory and experiment , 1992 .

[12]  J. Mork,et al.  The mechanism of mode selection for an external cavity laser , 1990, IEEE Photonics Technology Letters.

[13]  Shui-Nee Chow,et al.  Singularly Perturbed Delay-Differential Equations , 1983 .

[14]  Jesper Mørk,et al.  Bistability and low-frequency fluctuations in semiconductor lasers with optical feedback: a theoretical analysis , 1988 .

[15]  Gavrielides,et al.  Lang and Kobayashi phase equation. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[16]  Otsuka,et al.  Minimal model of a class-B laser with delayed feedback: Cascading branching of periodic solutions and period-doubling bifurcation. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[17]  Hartmut Haug,et al.  Theory of the bistable limit cycle behavior of laser diodes induced by weak optical feedback , 1993 .

[18]  R. Lang,et al.  External optical feedback effects on semiconductor injection laser properties , 1980 .

[19]  Andrew C. Fowler,et al.  Mathematical Models in the Applied Sciences , 1997 .

[20]  Jack K. Hale,et al.  Period Doubling in Singularly Perturbed Delay Equations , 1994 .

[21]  Klaus Petermann,et al.  Numerical analysis of the feedback regimes for a single-mode semiconductor laser with external feedback , 1988 .

[22]  P. Saboureau,et al.  Injection-locked semiconductor lasers with delayed optoelectronic feedback , 1997 .

[23]  Thomas Erneux,et al.  Lang and Kobayashi phase equation and its validity for low pump , 1996, Photonics West.

[24]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[25]  池田 研介,et al.  Successive Higher-harmonic Bifurcations in Systems with Delayed Feedback(カオスとその周辺,研究会報告) , 1982 .

[26]  Masoller.  Coexistence of attractors in a laser diode with optical feedback from a large external cavity. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[27]  Mork,et al.  Route to chaos and competition between relaxation oscillations for a semiconductor laser with optical feedback. , 1990, Physical review letters.

[28]  Erneux,et al.  Strongly pulsating lasers with delay. , 1996, Physical review. A, Atomic, molecular, and optical physics.

[29]  Athanasios Gavrielides,et al.  Bifurcations in a semiconductor laser subject to delayed incoherent feedback , 2002 .