×

zbMATH — the first resource for mathematics

Minimizing the transition time in lasers by optimal control methods: single-mode semiconductor laser with homogeneous transverse profile. (English) Zbl 1050.78008
Summary: The introduction of patterned current fronts that steer the transition between directly modulated optical bits allows for a strong modification in the shape of the trajectory evolving from the initial to the final state in semiconductor lasers. The problem of minimal transition time between two stationary points of the system will be formulated in the framework of optimal control theory. In this approach, the patterned current fronts serve as time-dependent control functions for which upper and lower bounds are imposed. We first calculate the optimal control on a Lotka-Volterra model, which qualitatively well describes the dynamic response of several types of lasers. The optimal control is of “bang-bang” type and switches from the upper to the lower value of the control bounds. Then, we consider a more specific semiconductor laser model and show again that the optimal control is bang-bang, and calculate specific trajectories for the transition between the below-threshold state and one where the laser is emitting. The sensitivity of the trajectory to external perturbations is also studied.
MSC:
78A60 Lasers, masers, optical bistability, nonlinear optics
49N90 Applications of optimal control and differential games
82D37 Statistical mechanical studies of semiconductors
Software:
BNDSCO
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lippi, G.L.; Porta, P.A.; Hoffer, L.M.; Grassi, H., Control of transients in “lethargic” systems, Phys. rev. E, 59, R32-R35, (1999)
[2] Lippi, G.L.; Barland, S.; Dokhane, N.; Monsieur, F.; Porta, P.A.; Grassi, H.; Hoffer, L.M., Phase space techniques for steering laser transients, J. opt. B: quant. semiclass. opt., 2, 1-7, (2000)
[3] Lippi, G.L.; Barland, S.; Monsieur, F., Invariant integral and the transition to steady states in separable dynamical systems, Phys. rev. lett., 85, 1, 62-65, (2000)
[4] Lau, K.Y.; Yariv, A., Ultra-high speed semiconductor lasers, IEEE J. quant. electron., QE-21, 121-138, (1985)
[5] Meland, E.; Holmstrom, R.; Schlafer, J.; Lauer, R.B.; Powazinik, W., Extremely high efficiency (24ghz) ingaasp diode lasers with excellent modulation efficiency, Electron. lett., 26, 1827-1829, (1990)
[6] Kobayashi, K.; Lang, R.; Minemura, K., Novel optical methods for high speed direct modulation of semiconductor lasers, Jpn. J. appl. phys. suppl., 15, 281-287, (1976)
[7] Suematsu, Y.; Hong, T.-H., Suppression of relaxation oscillations in light output of injection lasers by electrical resonance circuit, IEEE J. quant. electron., QE-13, 756-762, (1977)
[8] Hong, T.-H.; Suematsu, Y., Suppression of resonance-like phenomena in the light output of directly modulated injection lasers by π-type suppressor circuit, Trans. IECE jpn. E, 61, 121-124, (1978)
[9] Dellunde, J.; Torrent, M.C.; Sancho, J.M.; San Miguel, M., Frequency dynamics of gain-switched injection-locked semiconductor lasers, IEEE J. quant. electron., QE-33, 1537-1542, (1997)
[10] Mirasso, C.R.; Hernàndez-Garcı́a, E.; Dellunde, J.; Torrent, M.C.; Sancho, J.M., Current modulation and transient dynamics of single-mode semiconductor lasers under different feedback conditions, IEE proc. J., 142, 17-22, (1995)
[11] Bickers, L.; Westbrook, L.D., Reduction of transient laser chirp in 1.5\( μm\) DFB lasers by shaping the modulation pulse, IEE proc., 133, 155-162, (1986)
[12] Dokhane, N.; Lippi, G.L., Chirp reduction in semiconductor lasers through injection current patterning, Appl. phys. lett., 78, 25, 3938-3940, (2001)
[13] Dokhane, N.; Lippi, G.L., Improved direct modulation technique for faster switching of diode lasers, IEE proc. optoelectron., 149, 1, 7-16, (2002)
[14] G.L. Lippi, N. Dokhane, X. Hachair, S. Barland, J.R. Tredicce, High speed direct modulation of semiconductor lasers, in: R.P. Mirin, C.S. Menoni (Eds.), Proceedings of the SPIE, vol. 4871, 2002, pp. 103-114.
[15] N. Dokhane, G.L. Lippi, High quality 10Gb/s data transmission with single mode diode lasers through improved direct current modulation, IEE Proc. Optoelectron., submitted for publication.
[16] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Miscĕnko, The Mathematical Theory of Optimal Processes, Fizmatgiz, Moscow, 1961 (English Translation: Pergamon Press, New York, 1961).
[17] M. Hestenes, Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966. · Zbl 0173.35703
[18] A.E. Bryson, Y.C. Ho, Applied Optimal Control, Revised Printing, Hemisphere Publishing Corporation, New York, 1975.
[19] Hartl, R.F.; Sethi, S.P.; Vickson, R.G., A survey of the maximum principles for optimal control problems with state constraints, SIAM rev., 37, 181-218, (1995) · Zbl 0832.49013
[20] Betts, J.T., Survey of numerical methods for trajectory optimization, J. guidance contr. dyn., 21, 193-207, (1998) · Zbl 1158.49303
[21] J.T. Betts, Practical methods for optimal control using nonlinear programming, in: Advances in Design and Control, SIAM, Philadelphia, 2001. · Zbl 0995.49017
[22] Ch. Büskens, Optimierungsmethoden und Sensitivitätsanalyse für optimale Steuerprozesse mit Steuer- und Zustandsbeschränkungen, Dissertation, Institut für Numerische Mathematik der Universität Münster, Münster, Germany, 1998 (in German).
[23] Büskens, Ch.; Maurer, H., SQP-methods for solving optimal control problems with control and state constraints: adjoint variables, sensitivity analysis and real-time control, J. comp. appl. math., 120, 85-108, (2000) · Zbl 0963.65070
[24] Pesch, H.J., A practical guide to the solution of real-life optimal control problems, Contr. cybernet., 23, 1-2, 7-60, (1994) · Zbl 0811.49029
[25] S.P. Sethi, G.L. Thompson, Optimal Control Theory, Applications to Management Science and Economics, 2nd ed., Kluwer Academic Publishers, Boston, 2000. · Zbl 0998.49002
[26] Kim, J.-H.R.; Maurer, H.; Astrov, Yu.A.; Bode, M.; Purwins, H.-G., High-speed switch-on of a semiconductor gas discharge image converter using optimal control methods, J. comput. phys., 170, 395-414, (2001) · Zbl 1053.82521
[27] Tredicce, J.R.; Arecchi, F.T.; Lippi, G.L.; Puccioni, G.P., Instabilities in lasers with an injected signal, J. opt. soc. am. B, 2, 173-183, (1985)
[28] H. Maurer, N. Osmolovskii, Second order sufficient conditions for time-optimal bang-bang control problems, SIAM J. Contr. Opt., in press. · Zbl 1068.49015
[29] A.E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986.
[30] E. Schoell, Non-equilibrium Phase Transitions in Semiconductors—Self-organization Induced by Generation and Recombination Processes, Springer-Verlag, Berlin, 1987.
[31] K. Petermann, Laser Diode Modulation and Noise, Kluwer Academic Publishers, Dordrecht, 1988.
[32] H.J. Oberle, W. Grimm, BNDSCO—a program for the numerical solution of optimal control problems, Internal Report No. 515-89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany, 1989.
[33] Byrne, D.M., Accurate simulation of multifrequency semiconductor laser dynamics under gigabit-per-second modulation, J. lightwave technol., LT-10, 1086-1096, (1992)
[34] Danielsen, M., A theoretical analysis for gigabit/second pulse code modulation of a semiconductor laser, IEEE J. quant. electron., QE-12, 657-660, (1976)
[35] Torphammar, P.; Tell, R.; Eklund, H.; Johnston, R., Minimizing pattern effects in semiconductor lasers at high rate pulse modulation, IEEE J. quant. electron., QE-15, 1271-1276, (1979)
[36] Colet, P.; Mirasso, C.; San Miguel, M., Memory diagram of single-mode semiconductor lasers, IEEE J. quant. electron., QE-29, 1624-1630, (1993)
[37] Maurer, H., Numerical solution of singular control problems using multiple shooting techniques, J. opt. theory appl., 18, 235-257, (1976) · Zbl 0302.65063
[38] Ch. Büskens, H.J. Pesch, S. Winderl, Real-time solutions of bang-bang and singular optimal control problems, in: M. Grötschel, et al. (Eds.), Online Optimization of Large Scale Systems, Springer-Verlag, Berlin, 2001, pp. 129-142. · Zbl 0984.49015
[39] Fraser-Andrews, G., Finding candidate singular optimal controls: a state of the art survey, J. opt. theory appl., 60, 173-190, (1989) · Zbl 0633.49015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.