Multipulse Excitability in Optically Injected Semiconductor Laser.

The talk was given as a part of the evaluation of Nonlinear Dynamics course of the fourth semester of my course-work at IMSc.

Transcript

1. Multipulse Excitability in Optically Injected Semiconductor Lasers Vinay Vaibhav The Institute of Mathematical Sciences June 22, 2017
2. Resources • S. Wieczorek, B. Krauskopf and D. Lenstra, Multipulse Excitability in a Semiconductor Laser with Optical Injection, Phys. Rev. Lett. 88 063901 (2002). • S. Wieczorek, B. Krauskopf and D. Lenstra, A unifying view of bifurcations in a semiconductor laser subject to optical injection,Optics Communications. 4172 279 (1999). • S. Wieczorek, B. Krauskopf, T.B. Simpson and D. Lenstra The dynamical complexity of optically injected semiconductor lasers, Physics Reports. 416.1 (2005) 1-128. • Yuri A. Kuznetsov, Elements of applied bifurcation theory, Springer New York. (2013). • Thomas S. Parker and Leon Chua, Practical numerical algorithms for chaotic systems, Springer-Verlag. (1989). 2 of 30
3. L Amplification by timulated S mission LASER ight E adiation R of • Coherent • Highly Monochromatic • Almost Directional Lasing Requirements: • Direct Bandgap Semiconductor • Population Inversion • Optical Resonant Cavity 3 of 30
4. Laser As Dynamical System • First quantum theory of laser was developed in 1960 • Semiclassical Theory: Large average number of photons such that quantum fluctuations are not important • Rate equations: Dynamics of a single-mode laser is fully described by the four equations so called Maxwell-Bloch equations, along with one equation which determines lasing frequency • Four dynamical variables: Electric field amplitude E, Complex polarization amplitude P and Population-inversion density N 4 of 30
5. Free-running Semiconductor Laser • Semiconductor lasers fall under the category of class-B laser MB equations are reduced to two dimensional system Dynamics more complicated than periodic oscillations cannot appear here and hence cannot produce chaos 5 of 30
6. Free-running Semiconductor Laser • Transitions happen between energy bands of a certain bandwidth • Detailed study of the system in pure form is very complicated • With a number of approximations, the rate equations can be modified to study the dynamics of a semiconductor laser Laser pumped at a rate above a threshold has steady output at a characteristic frequency ⌦th 6 of 30
7. Semiconductor Laser Under Perturbation Perturbation: Pump modulation, Optical injection, Optical feedback etc • For small perturbation about the stationary point, Intensity oscillates about Is with frequency ⌦R Relaxation oscillation: Amplitude of these oscillations decrease exponentially • Apart from relaxation oscillations, they can show a wide spectrum of complex dynamics and chaos 7 of 30
8. Semiconductor Laser with Optical Injection 8 of 30
9. Rate Equations In the presence of externally injected signal of constant amplitude and optical frequency (!i ), the rate equations are modified as: ˙ E = K + [ 1 2 (1 + ◆↵)n ◆!]E ˙ n = 2 n (1 + 2Bn)(|E|2 1) • E = Ex + ◆Ey : Complex electric field • K: Injected field strength • n: Polulation inversion • !: Detuning between injected field frequency and free-running laser frequency i.e., (!i ⌦th ) • : Damping rate of relaxation oscillation • B: Cavity lifetime of photons • ↵ (linewidth enhancement factor): quantifies the change in index of refraction of laser medium with n at a given frequency 9 of 30
10. • Di erent parameters in the rate equations have been rescaled in order to make equations dimensionless • B, and ↵ are the material properties of laser used. • The values of B and have been kept fixed to physically realistic values of 0.015 and 0.035 respectively • B and don’t influence the dynamics very much but changing ↵ has very large e ect on dynamics • The value of ↵ strongly depend on the material. Here we take ↵ = 2 10 of 30
11. Di erent Dynamics in Phase Space Qualitatively di erent phase portraits which correspond to di erent physical processes taking place in laser 1. Stationary point dynamics: Injection locking (K = 0.27, ! = 0.28) Laser operates at constant power and the injected light frequency 11 of 30
12. 2. Bounded-phase limit cycle dynamics (K = 0.27, ! = 0.20) 12 of 30
13. 3. Unbounded-phase limit cycle dynamics (K = 0.10, ! = 1.2) 13 of 30
14. 4. Quasiperiodic dynamics on an invariant torus (K = 0.18, ! = 1.2) 14 of 30
15. 5. Chaotic laser dynamics (K = 0.18, ! = 0.6) 15 of 30
16. K-! Plane Still we don’t have any analytical theory which relates all these qualitatively di erents behaviors 16 of 30
17. Bifurcation • Bifurcation: Phase portrait changes qualitatively when parameter(s) (here K and !) is/are varied • Bifurcation point: Parameter values at which transition occurs • Codimension of a bifurcation: The minimum number of parameters necessary to describe a bifurcation • Local bifurcations: Change in parameter causes the stability of a fixed point to change Example- Saddle-node, Hopf, Saddle-node of limit cycle, Period-doubling, Torus (Neimark-Sacker) • Global bifurcations: Cannot be detected by looking at the stability of a single stationary point or limit cycle Example- Homoclinic, Heteroclinic • Bifurcation diagrams can be obtained by using the method of Numerical Continuation 17 of 30
18. Bifurcations Curves of Stationary Points Figure: SN:Saddle-node bifurcation, H:Hopf bifurcation, a:attractor, r:repellor, s:saddle point. Shaded region is locking region • Whenever there is an attractor the laser locks to the input 18 of 30
19. SN bifurcation: Two stationary points are created Hopf bifurcation: A stationary point (stable spiral) bifurcates to a unstable spiral surrounded by a limit cycle 19 of 30
20. • Adler’s locking mechanism: SN bifurcation occurs on a periodic orbit. • This doesn’t happen along whole of the curve SN • Locking region: Confined between curves SN and H • h1: Homoclinic bifurcation curve • A1, A2, A3, ... : Codimension-two saddle-node homoclinic points 20 of 30
21. Figure: (a)Outside homoclinic tooth, (b)On h1 (c1) Simple Shil’nikov bifurcation and (c2) Chaotic Shil’nikov bifurcation • ns: Neutral saddle line (saddle focus is as attracting as repelling) • B1, B2 : Codimension-two Belyakov bifurcation point 21 of 30
22. Homoclinic bifurcation structure associated with Belyakov points 22 of 30
23. Mechanism of Multipulse Excitability • Multipulse excitability occurs for the values of K and ! inside the tongues bounded by the curves hn and below ns • The upper branch of the unstable manifold Wu of the saddle makes n loops above the stable manifold Ws and then dives under Ws to end up at the stable equilibrium 23 of 30
24. Computing 1-D Unstable Manifold • Find fixed points and Jacobian of the system at the fixed points • Find eigenvalues and eigenvectors of the Jacobian Here one eigenvalue is positive and other two are complex with negative real parts: Unstable manifold is one-dimensional • Find a point close to the fixed point P lying on unstable manifold Wu: P±✏V1 ✏ is very small number V1 is eigenvector corresponding to positive eigenvalue 24 of 30
25. One pulse: (K=0.45, ! = 0.93); Two pulses: (K=0.472, ! = 0.98) 25 of 30
26. Three pulses: (K=0.48, ! = 0.97); Four pulses: (K=0.455, ! = 0.95725) 26 of 30
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28. Figure: The dynamics and bifurcation diagrams become more complicated with increasing ↵ 28 of 30
29. Summary • Semicalssical approach to lasers gives Maxwell-Bloch equations. • Free running semiconducting lasers pumped at a rate above a threshold has steady output at a characteristic frequency ⌦th • Perturbed semiconductor laser can show relaxation oscillation and even some more complex dynamics including chaos • Optically injected semiconductor lasers are modeled using four dynamical equations called rate equations • It can show di erent bifurcations depending on parameters K and ! in rate equations. • It can show multipulse excitation if K and ! lie inside the tongues bounded by the curves hn and below ns • Also, the upper branch of the unstable manifold Wu of the saddle makes n loops above the stable manifold Ws and then dives under Ws to end up at the stable equilibrium 29 of 30
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