Abstract

The response of a Class B laser to a rapid change in one of its parameters is known to be accompanied by delay and ringing. It has been theoretically and numerically shown that the transition can be modified by using adequate functional shapes for the control parameter (e.g., the laser pump) in order to steer the laser from one point of operation to another. Here we experimentally show the implementation of these ideas in a commercial device: a semiconductor laser. We establish a procedure for optimizing a controlled switch-on and switch-off and obtain a clean, fast, and reliable square pulse, either in a single shot or in a repetitive sequence. The generality of this procedure promises a wide field of application for a variety of laser systems.

© 2005 Optical Society of America

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  68. For simplicity, in this discussion we assume that the laser is switched on from below threshold. One can easily introduce a generalization to consider the transition between states where the laser is active with different output power levels. (This has been discussed for telecommunication semiconductor lasers.60)
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  70. The step that we describe in detail in Section 4 is necessary for laser systems, i.e., devices that package together both the laser and some controls (e.g., electronic, optical, or others). If one desires to optimize a Class B laser with no additional elements, the general considerations of Refs. 55, 60, and 64 apply directly. The procedure that we propose in this paper can still be applied and is likely to provide faster results, but the tests described in detail in Section 4 become unnecessary.
  71. Although some degree of steering is possible in an oscillator, one can show that the time necessary to attain the new equilibrium state cannot be shortened, contrary to what happens in Class B lasers, and in the more complex device that we analyze here.
  72. Embedding techniques amount to generating N-dimensional spaces starting from a one-dimensional data sequence by constructing vectors with elements shifted by an arbitrary number of elements. For a discussion see Refs. 74 and 75.
  73. Compare Fig. 14 in Ref. 60 with Fig. 1.55 From the latter the presence of the saddle point is much more readily recognized.
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  78. On the basis of general principles63 one prefers the opposite approach, fixing V1 and V2 and varying t1 and t2. If the instrumentation allows it, this is certainly preferred. However, we see that even the opposite one, imposed by our generator, produces successful results.
  79. In general, it is not meaningful to attempt a number of levels much larger than 100, no matter what system is taken into consideration. Maximum estimates of the number of trials can therefore be based on this worst-case assumption if other information is missing.
  80. If the generator’s time resolution is small compared with the time interval to be explored, we come back to the considerations already made79 and consider that 100 trials are more than enough for a scan of the parameter space. In such a case, if no other restrictions are applied, the amount of data to be analyzed may rapidly become too large to be practicable. A way of reducing the portion of parameter space to analyze is presented in Subsection 5.B.
  81. The time estimate for one loop can be obtained on the basis of the following considerations. For 1 kbyte the typical access time for a hard disk is nowadays ≈15 μs, while a standard GIPB interface is currently capable of transferring the same amount of data in ≈150 μs. Choosing a large safety factor in the duty cycle, to make sure that the measurement is not affected by external factors influencing the laser (e.g., heating, memory effects), we can set the signal frequency to ≈10 kHz; thus the waiting time to synchronize the cycles is at most 100 μs. In addition we have to take into account the time Labview takes to update the parameters and send them to the generator, and to activate it, and for the oscilloscope to trigger and store the data in memory. Given the speed of computer clocks, even if the program is not written in an efficient way, the bottleneck of the operation is the arbitrary waveform generator’s reaction time in responding and sending out the signal. In our measurements this time was particularly long (3 s) because of the very old technology of the apparatus. With sufficient error margin we can generically assume modern generators to be ~50 times faster; therefore we arrive at an estimated cycle time of ≈0.06 s.
  82. Although we have not used this option in our measurements, most modern oscilloscopes offer a window discrimination or smart trigger on the data acquired. This option can be used for prefiltering the data. Alternatively this filtering can be done on the computer, once the data are transferred from a lower-class oscilloscope or from an older model, before the information is stored on the hard disk.
  83. This statement is valid with the assumption that the range of values over which the steering parameter (voltage in our system) can be varied above and below threshold covers a comparable range. In our device the laser response above threshold grows considerably more for 1.8 V ≤ V≲ 3.5 V than for 3.5 V ≲ V≤ 5 V. Thus we can consider that the range of below-threshold voltages ΔVbt≈1.8 V is of the same order as the main contribution in the above-threshold interval ΔVat≈ 1.7 V. This response can be tested a priori in each system, and a weighted version of our statement can be used as an educated guess for determining a reasonable interval of ratios between t1 and t2 to be tested.
  84. In all cases in which the time resolution of the arbitrary function generator is sufficiently fine, one would effectively invert the roles of the scans on the time values, t1 and t2, and voltage values, V1 and V2. In this latter case the number of voltage values to be chosen could also be rather small, since one would immediately start by considering in a preliminary run only those that are sufficiently close to Vmax for V1 and to Vmin for V2.
  85. This situation is represented graphically by the trajectory labeled A in Fig. 7 in Ref. 62 when the target point in the phase space is approached from below.
  86. This corresponds to the part of the composite trajectory that approaches the saddle point in phase space,55 starting from the initial operating point (with the laser switched off). It is in the neighborhood of this point that the laser field grows rapidly out of the intrinsic noise (not shown in the figures in Ref. 55).
  87. In the phase-space picture55,60 this phase corresponds to aiming at the fixed point, thereby removing the oscillations.

2004 (2)

N. Dokhane, G. L. Lippi, “Faster modulation of single-mode semiconductor lasers through patterned current switching: numerical investigation,” IEE Proc.-J Optoelectron. 151, 61–68 (2004).
[CrossRef]

H-J. R. Kim, G. L. Lippi, H. Maurer, “Minimizing the transition time in lasers by optimal control methods. Single-mode semiconductor laser with homogeneous transverse profile,” Physica D 191, 238–260 (2004).
[CrossRef]

2003 (1)

G. J. de Valcàrcel, K. Staliunas, “Excitation of phase patterns and spatial solitons via two-frequency forcing of a 1:1 resonance,” Phys. Rev. E 67, 026604 (2003).
[CrossRef]

2002 (4)

B. Ségard, S. Matton, P. Glorieux, “Targeting steady states in a laser,” Phys. Rev. A 66, 053819 (1–5) (2002).
[CrossRef]

J. B. Khurgin, F. Jin, G. Solyar, C.-C. Wang, S. Trivedi, “Cost-effective low timing jitter passively Q-switched diode-pumped solid-state laser with composite pumping pulses,” Appl. Opt. 41, 1095–1097 (2002).
[CrossRef] [PubMed]

N. Dokhane, G. L. Lippi, “Improved direct modulation technique for faster switching of diode lasers,” IEE Proc.-J Optoelectron. 149, 7–16 (2002).
[CrossRef]

J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, M. Motzkus, “Quantum control of energy flow in light harvesting,” Nature 417, 533–535 (2002).
[CrossRef] [PubMed]

2001 (3)

R. J. Levis, G. M. Menkir, H. Rabitz, “Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses,” Science 292, 709–713 (2001).
[CrossRef] [PubMed]

T. Brixner, N. H. Damrauer, P. Niklaus, G. Gerber, “Photoselective adaptive femtosecond quantum control in the liquid phase,” Nature 414, 57–60 (2001).
[CrossRef] [PubMed]

N. Dokhane, G. L. Lippi, “Chirp reduction in semiconductor lasers through injection current patterning,” Appl. Phys. Lett. 78, 3938–3940 (2001).
[CrossRef]

2000 (5)

P. A. Porta, L. M. Hoffer, H. Grassi, G. L. Lippi, “Analysis of nonfeedback technique for transient steering in Class B lasers,” Phys. Rev. A 61, 033801 (1–10) (2000).
[CrossRef]

G. L. Lippi, S. Barland, F. Monsieur, “Invariant integral and the transition to steady states in separable dynamical systems,” Phys. Rev. Lett. 85, 62–65 (2000).
[CrossRef] [PubMed]

E. Benkler, M. Kreuzer, R. Neubecker, T. Tschudi, “Experimental control of unstable patterns and elimination of spatiotemporal disorder in nonlinear optics,” Phys. Rev. Lett. 84, 879–882 (2000).
[CrossRef] [PubMed]

G. L. Lippi, S. Barland, N. Dokhane, F. Monsieur, P. A. Porta, H. Grassi, L. M. Hoffer, “Phase space techniques for steering laser transients,” J. Opt. B: Quantum Semiclassical Opt. 2, 375–381 (2000).
[CrossRef]

P.-Y. Wang, P. Xie, “Eliminating spatiotemporal chaos and spiral waves by weak spatial perturbations,” Phys. Rev. E 61, 5120–5123 (2000).
[CrossRef]

1999 (11)

T. C. Weinacht, J. Ahn, P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, T. Tchudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos Solitons Fractals 10, 701–707 (1999).
[CrossRef]

V. Raab, A. Heuer, J. Schultheiss, N. Hodbson, J. Kurths, R. Menzel, “Transverse effects in phase conjugate laser mirrors based on stimulated Brillouin scattering,” Chaos Solitons Fractals 10, 831–838 (1999).
[CrossRef]

C. Simmendinger, M. Münkler, O. Hess, “Controlling complex temporal and spatiotemporal dynamics in semiconductor lasers,” Chaos Solitons Fractals 10, 851–864 (1999).

G.-L. Oppo, R. Martin, A. J. Scroggie, G. K. Harkness, A. Lord, W. J. Firth, “Control of spatiotemporal complexity in nonlinear optics,” Chaos Solitons Fractals 10, 865–874 (1999).

F. T. Arecchi, S. Boccaletti, P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

G. D. VanWiggeren, R. Roy, “Chaotic communication using time-delayed optical systems,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, 2129–2156 (1999).
[CrossRef]

G. L. Lippi, P. A. Porta, L. M. Hoffer, H. Grassi, “Control of transients in ‘lethargic’ systems,” Phys. Rev. E 59, R32–R35 (1999).
[CrossRef]

T. Ackemann, B. Giese, B. Schäpers, W. Lange, “Investigations of pattern forming mechanisms by Fourier filtering: properties of hexagons and the transition to stripes in an anisotropic system,” J. Opt. B: Quantum Semiclassical Opt. 1, 70–76 (1999).
[CrossRef]

G. K. Harkness, G.-L. Oppo, E. Benkler, M. Kreuzer, R. Neubecker, T. Tschudi, “Fourier space control in an LCLV feedback system,” J. Opt. B: Quantum Semiclassical Opt. 1, 177–182 (1999).
[CrossRef]

M. Bruensteiner, G. C. Papen, “Extraction of VCSEL rate-equation parameters for low-bias system simulation,” IEEE J. Sel. Top. Quantum Electron. 5, 487–494 (1999).
[CrossRef]

1998 (12)

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2586 (1998).
[CrossRef]

A. V. Mamaev, M. Saffman, “Selection of unstable patterns and control of optical turbulence by Fourier plane filtering,” Phys. Rev. Lett. 80, 3499–3502 (1998).
[CrossRef]

S. Juul Jensen, M. Schwab, C. Denz, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering,” Phys. Rev. Lett. 81, 1614–1617 (1998).
[CrossRef]

P. A. Porta, L. M. Hoffer, H. Grassi, G. L. Lippi, “Control of turn-on in Class B lasers,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 8, 1811–1819 (1998).
[CrossRef]

M. Brambilla, A. Gatti, L. A. Lugiato, “Optical pattern formation,” Adv. At. Mol. Opt. Phys. 40, 229–306 (1998).

G. D. VanWiggeren, R. Roy, “Optical communication with chaotic waveforms,” Phys. Rev. Lett. 81, 3547–3550 (1998).
[CrossRef]

G. D. VanWiggeren, R. Roy, “Communicating with chaotic lasers,” Science 279, 1198–1200 (1998).
[CrossRef] [PubMed]

J.-P. Goedgebuer, L. Larger, H. Laporte, “Optical crypto-system based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

L. Larger, J.-P. Goedgebuer, F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57, 6618–6624 (1998).
[CrossRef]

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Swyfried, M. Strehle, G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

D. Meshlach, Y. Silberberg, “Coherent quantum control of two-photon transitions by a femtosecond laser pulse,” Nature 396, 239–242 (1998).
[CrossRef]

M. Möller, B. Forsmann, W. Lange, “Instabilities in coupled Nd:YVO4 microchip lasers,” J. Opt. B: Quantum Semiclassical Opt. 10, 839–848 (1998).

1997 (7)

D. C. DeMott, D. J. Ulness, A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

D. Hochheiser, J. V. Moloney, J. Lega, “Controlling optical turbulence,” Phys. Rev. A 55, R4011–R4014 (1997).
[CrossRef]

M. E. Bleich, D. Hochheiser, J. V. Moloney, J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

V. N. Chizevsky, E. V. Grigorieva, S. A. Kashchenko, “Optimal timing for targeting periodic orbits in a loss-driven CO2 laser,” Opt. Commun. 133, 189–195 (1997).
[CrossRef]

L. A. Kotomtseva, A. V. Naumenko, A. M. Samson, S. I. Turovets, “Targeting unstable orbits and steady states in Class B lasers by using simple off/on manipulations,” Opt. Commun. 136, 335–348 (1997).
[CrossRef]

K. S. Thornburg, M. Möller, R. Roy, T. W. Carr, R.-D. Li, T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

1996 (1)

R. Martin, A. J. Scroggie, G.-L. Oppo, W. J. Firth, “Stabilization, selection, and tracking of unstable patterns by Fourier space techniques,” Phys. Rev. Lett. 77, 4007–4010 (1996).
[CrossRef] [PubMed]

1995 (1)

V. N. Chizhevsky, P. Glorieux, “Targeting unstable periodic orbits,” Phys. Rev. E 51, R2701–R2704 (1995).
[CrossRef]

1994 (2)

R. Roy, K. S. Thornburg, “Experimental synchronization of chaotic lasers,” Phys. Rev. Lett. 72, 2009–2012 (1994).
[CrossRef] [PubMed]

T. Sugawara, M. Tachikawa, T. Tsukamoto, T. Shimizu, “Observation of synchronization in laser chaos,” Phys. Rev. Lett. 72, 3502–3505 (1994).
[CrossRef] [PubMed]

1993 (2)

S. Bielawski, D. Derozier, P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

P. Colet, C. Mirasso, M. San Miguel, “Memory diagram of single-mode semiconductor lasers,” IEEE J. Quantum Electron. 29, 1624–1630 (1993).
[CrossRef]

1990 (1)

L. M. Pecora, T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824 (1990).
[CrossRef] [PubMed]

1986 (1)

L. Bickers, L. D. Westbrook, “Reduction of transient laser chirp in 1.5 μm DFB lasers by shaping the modulation pulse,” IEE Proc. J 133, 155–162 (1986).

1985 (3)

K. Petermann, “Analysis of reduced chirping of semiconductor lasers for improved single-mode-fiber transmission capacity,” Electron. Lett. 21, 1143–1145 (1985).
[CrossRef]

J.-P. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617–656 (1985).
[CrossRef]

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
[CrossRef]

1984 (1)

R. Olshansky, D. Fye, “Reduction of dynamic linewidth in single-frequency semiconductor lasers,” Electron. Lett. 20, 928–929 (1984).
[CrossRef]

1979 (1)

P. Torphammar, R. Tell, H. Eklund, A. R. Johnston, “Minimizing pattern effects in semiconductor lasers at high rate pulse modulation,” IEEE J. Quantum Electron. QE-15, 1271–1276 (1979).
[CrossRef]

1976 (1)

M. Danielsen, “A theoretical analysis for gigabit/second pulse code modulation of a semiconductor laser,” IEEE J. Quantum Electron. QE-12, 657–660 (1976).
[CrossRef]

Ackemann, T.

T. Ackemann, B. Giese, B. Schäpers, W. Lange, “Investigations of pattern forming mechanisms by Fourier filtering: properties of hexagons and the transition to stripes in an anisotropic system,” J. Opt. B: Quantum Semiclassical Opt. 1, 70–76 (1999).
[CrossRef]

Ahn, J.

T. C. Weinacht, J. Ahn, P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Albrecht, A. C.

D. C. DeMott, D. J. Ulness, A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

Arecchi, F. T.

F. T. Arecchi, S. Boccaletti, P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

J. R. Tredicce, F. T. Arecchi, G. L. Lippi, G. P. Puccioni, “Instabilities in lasers with an injected signal,” J. Opt. Soc. Am. B 2, 173–183 (1985).
[CrossRef]

Assion, A.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Swyfried, M. Strehle, G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Barland, S.

G. L. Lippi, S. Barland, F. Monsieur, “Invariant integral and the transition to steady states in separable dynamical systems,” Phys. Rev. Lett. 85, 62–65 (2000).
[CrossRef] [PubMed]

G. L. Lippi, S. Barland, N. Dokhane, F. Monsieur, P. A. Porta, H. Grassi, L. M. Hoffer, “Phase space techniques for steering laser transients,” J. Opt. B: Quantum Semiclassical Opt. 2, 375–381 (2000).
[CrossRef]

G. L. Lippi, N. Dokhane, X. Hachair, S. Barland, J. R. Tredicce, “High speed direct modulation of semiconductor lasers,” in Semiconductor Lasers and Optical Amplifiers for Lightwave Communication Systems, R. P. Mirin, C. S. Menoni, eds., Proc. SPIE4871, 103–114 (2002).
[CrossRef]

Baumert, T.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Swyfried, M. Strehle, G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Benkler, E.

E. Benkler, M. Kreuzer, R. Neubecker, T. Tschudi, “Experimental control of unstable patterns and elimination of spatiotemporal disorder in nonlinear optics,” Phys. Rev. Lett. 84, 879–882 (2000).
[CrossRef] [PubMed]

G. K. Harkness, G.-L. Oppo, E. Benkler, M. Kreuzer, R. Neubecker, T. Tschudi, “Fourier space control in an LCLV feedback system,” J. Opt. B: Quantum Semiclassical Opt. 1, 177–182 (1999).
[CrossRef]

Bergt, M.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Swyfried, M. Strehle, G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Bickers, L.

L. Bickers, L. D. Westbrook, “Reduction of transient laser chirp in 1.5 μm DFB lasers by shaping the modulation pulse,” IEE Proc. J 133, 155–162 (1986).

Bielawski, S.

S. Bielawski, D. Derozier, P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Bleich, M. E.

M. E. Bleich, D. Hochheiser, J. V. Moloney, J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, P. L. Ramazza, “Pattern formation and competition in nonlinear optics,” Phys. Rep. 318, 1–83 (1999).
[CrossRef]

Brambilla, M.

M. Brambilla, A. Gatti, L. A. Lugiato, “Optical pattern formation,” Adv. At. Mol. Opt. Phys. 40, 229–306 (1998).

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

Brixner, T.

T. Brixner, N. H. Damrauer, P. Niklaus, G. Gerber, “Photoselective adaptive femtosecond quantum control in the liquid phase,” Nature 414, 57–60 (2001).
[CrossRef] [PubMed]

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Swyfried, M. Strehle, G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Bruensteiner, M.

M. Bruensteiner, G. C. Papen, “Extraction of VCSEL rate-equation parameters for low-bias system simulation,” IEEE J. Sel. Top. Quantum Electron. 5, 487–494 (1999).
[CrossRef]

Bucksbaum, P. H.

T. C. Weinacht, J. Ahn, P. H. Bucksbaum, “Controlling the shape of a quantum wavefunction,” Nature 397, 233–235 (1999).
[CrossRef]

Carr, T. W.

K. S. Thornburg, M. Möller, R. Roy, T. W. Carr, R.-D. Li, T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

Carroll, T. L.

L. M. Pecora, T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. 64, 821–824 (1990).
[CrossRef] [PubMed]

Chizevsky, V. N.

V. N. Chizevsky, E. V. Grigorieva, S. A. Kashchenko, “Optimal timing for targeting periodic orbits in a loss-driven CO2 laser,” Opt. Commun. 133, 189–195 (1997).
[CrossRef]

Chizhevsky, V. N.

V. N. Chizhevsky, P. Glorieux, “Targeting unstable periodic orbits,” Phys. Rev. E 51, R2701–R2704 (1995).
[CrossRef]

Cogdell, R. J.

J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, M. Motzkus, “Quantum control of energy flow in light harvesting,” Nature 417, 533–535 (2002).
[CrossRef] [PubMed]

Colet, P.

P. Colet, C. Mirasso, M. San Miguel, “Memory diagram of single-mode semiconductor lasers,” IEEE J. Quantum Electron. 29, 1624–1630 (1993).
[CrossRef]

Damrauer, N. H.

T. Brixner, N. H. Damrauer, P. Niklaus, G. Gerber, “Photoselective adaptive femtosecond quantum control in the liquid phase,” Nature 414, 57–60 (2001).
[CrossRef] [PubMed]

Danielsen, M.

M. Danielsen, “A theoretical analysis for gigabit/second pulse code modulation of a semiconductor laser,” IEEE J. Quantum Electron. QE-12, 657–660 (1976).
[CrossRef]

de Valcàrcel, G. J.

G. J. de Valcàrcel, K. Staliunas, “Excitation of phase patterns and spatial solitons via two-frequency forcing of a 1:1 resonance,” Phys. Rev. E 67, 026604 (2003).
[CrossRef]

Delorme, F.

L. Larger, J.-P. Goedgebuer, F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57, 6618–6624 (1998).
[CrossRef]

DeMott, D. C.

D. C. DeMott, D. J. Ulness, A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

Denz, C.

M. Schwab, M. Sedlatschek, B. Thüring, C. Denz, T. Tchudi, “Origin and control of dynamics of hexagonal patterns in a photorefractive feedback system,” Chaos Solitons Fractals 10, 701–707 (1999).
[CrossRef]

S. Juul Jensen, M. Schwab, C. Denz, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering,” Phys. Rev. Lett. 81, 1614–1617 (1998).
[CrossRef]

Derozier, D.

S. Bielawski, D. Derozier, P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Dokhane, N.

N. Dokhane, G. L. Lippi, “Faster modulation of single-mode semiconductor lasers through patterned current switching: numerical investigation,” IEE Proc.-J Optoelectron. 151, 61–68 (2004).
[CrossRef]

N. Dokhane, G. L. Lippi, “Improved direct modulation technique for faster switching of diode lasers,” IEE Proc.-J Optoelectron. 149, 7–16 (2002).
[CrossRef]

N. Dokhane, G. L. Lippi, “Chirp reduction in semiconductor lasers through injection current patterning,” Appl. Phys. Lett. 78, 3938–3940 (2001).
[CrossRef]

G. L. Lippi, S. Barland, N. Dokhane, F. Monsieur, P. A. Porta, H. Grassi, L. M. Hoffer, “Phase space techniques for steering laser transients,” J. Opt. B: Quantum Semiclassical Opt. 2, 375–381 (2000).
[CrossRef]

G. L. Lippi, N. Dokhane, X. Hachair, S. Barland, J. R. Tredicce, “High speed direct modulation of semiconductor lasers,” in Semiconductor Lasers and Optical Amplifiers for Lightwave Communication Systems, R. P. Mirin, C. S. Menoni, eds., Proc. SPIE4871, 103–114 (2002).
[CrossRef]

N. Dokhane, “Amélioration de la modulation logique directe des diodes laser par la technique de l’espace des phases,” Ph.D. dissertation (in French) (Université de Nice-Sophia Antipolis, Valbonne, France, 2000).

Eckmann, J.-P.

J.-P. Eckmann, D. Ruelle, “Ergodic theory of chaos and strange attractors,” Rev. Mod. Phys. 57, 617–656 (1985).
[CrossRef]

Eklund, H.

P. Torphammar, R. Tell, H. Eklund, A. R. Johnston, “Minimizing pattern effects in semiconductor lasers at high rate pulse modulation,” IEEE J. Quantum Electron. QE-15, 1271–1276 (1979).
[CrossRef]

Erneux, T.

K. S. Thornburg, M. Möller, R. Roy, T. W. Carr, R.-D. Li, T. Erneux, “Chaos and coherence in coupled lasers,” Phys. Rev. E 55, 3865–3869 (1997).
[CrossRef]

Firth, W. J.

G.-L. Oppo, R. Martin, A. J. Scroggie, G. K. Harkness, A. Lord, W. J. Firth, “Control of spatiotemporal complexity in nonlinear optics,” Chaos Solitons Fractals 10, 865–874 (1999).

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2586 (1998).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, W. J. Firth, “Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. 79, 2042–2045 (1997).
[CrossRef]

R. Martin, A. J. Scroggie, G.-L. Oppo, W. J. Firth, “Stabilization, selection, and tracking of unstable patterns by Fourier space techniques,” Phys. Rev. Lett. 77, 4007–4010 (1996).
[CrossRef] [PubMed]

Forsmann, B.

M. Möller, B. Forsmann, W. Lange, “Instabilities in coupled Nd:YVO4 microchip lasers,” J. Opt. B: Quantum Semiclassical Opt. 10, 839–848 (1998).

Fye, D.

R. Olshansky, D. Fye, “Reduction of dynamic linewidth in single-frequency semiconductor lasers,” Electron. Lett. 20, 928–929 (1984).
[CrossRef]

Gatti, A.

M. Brambilla, A. Gatti, L. A. Lugiato, “Optical pattern formation,” Adv. At. Mol. Opt. Phys. 40, 229–306 (1998).

Gerber, G.

T. Brixner, N. H. Damrauer, P. Niklaus, G. Gerber, “Photoselective adaptive femtosecond quantum control in the liquid phase,” Nature 414, 57–60 (2001).
[CrossRef] [PubMed]

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Swyfried, M. Strehle, G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Giese, B.

T. Ackemann, B. Giese, B. Schäpers, W. Lange, “Investigations of pattern forming mechanisms by Fourier filtering: properties of hexagons and the transition to stripes in an anisotropic system,” J. Opt. B: Quantum Semiclassical Opt. 1, 70–76 (1999).
[CrossRef]

Glorieux, P.

B. Ségard, S. Matton, P. Glorieux, “Targeting steady states in a laser,” Phys. Rev. A 66, 053819 (1–5) (2002).
[CrossRef]

V. N. Chizhevsky, P. Glorieux, “Targeting unstable periodic orbits,” Phys. Rev. E 51, R2701–R2704 (1995).
[CrossRef]

S. Bielawski, D. Derozier, P. Glorieux, “Experimental characterization of unstable periodic orbits by controlling chaos,” Phys. Rev. A 47, R2492–R2495 (1993).
[CrossRef] [PubMed]

Goedgebuer, J.-P.

J.-P. Goedgebuer, L. Larger, H. Laporte, “Optical crypto-system based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

L. Larger, J.-P. Goedgebuer, F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57, 6618–6624 (1998).
[CrossRef]

Grassi, H.

G. L. Lippi, S. Barland, N. Dokhane, F. Monsieur, P. A. Porta, H. Grassi, L. M. Hoffer, “Phase space techniques for steering laser transients,” J. Opt. B: Quantum Semiclassical Opt. 2, 375–381 (2000).
[CrossRef]

P. A. Porta, L. M. Hoffer, H. Grassi, G. L. Lippi, “Analysis of nonfeedback technique for transient steering in Class B lasers,” Phys. Rev. A 61, 033801 (1–10) (2000).
[CrossRef]

G. L. Lippi, P. A. Porta, L. M. Hoffer, H. Grassi, “Control of transients in ‘lethargic’ systems,” Phys. Rev. E 59, R32–R35 (1999).
[CrossRef]

P. A. Porta, L. M. Hoffer, H. Grassi, G. L. Lippi, “Control of turn-on in Class B lasers,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 8, 1811–1819 (1998).
[CrossRef]

Grigorieva, E. V.

V. N. Chizevsky, E. V. Grigorieva, S. A. Kashchenko, “Optimal timing for targeting periodic orbits in a loss-driven CO2 laser,” Opt. Commun. 133, 189–195 (1997).
[CrossRef]

Hachair, X.

G. L. Lippi, N. Dokhane, X. Hachair, S. Barland, J. R. Tredicce, “High speed direct modulation of semiconductor lasers,” in Semiconductor Lasers and Optical Amplifiers for Lightwave Communication Systems, R. P. Mirin, C. S. Menoni, eds., Proc. SPIE4871, 103–114 (2002).
[CrossRef]

Harkness, G. K.

G. K. Harkness, G.-L. Oppo, E. Benkler, M. Kreuzer, R. Neubecker, T. Tschudi, “Fourier space control in an LCLV feedback system,” J. Opt. B: Quantum Semiclassical Opt. 1, 177–182 (1999).
[CrossRef]

G.-L. Oppo, R. Martin, A. J. Scroggie, G. K. Harkness, A. Lord, W. J. Firth, “Control of spatiotemporal complexity in nonlinear optics,” Chaos Solitons Fractals 10, 865–874 (1999).

G. K. Harkness, G.-L. Oppo, R. Martin, A. J. Scroggie, W. J. Firth, “Elimination of spatiotemporal disorder by Fourier space techniques,” Phys. Rev. A 58, 2577–2586 (1998).
[CrossRef]

Herek, J. L.

J. L. Herek, W. Wohlleben, R. J. Cogdell, D. Zeidler, M. Motzkus, “Quantum control of energy flow in light harvesting,” Nature 417, 533–535 (2002).
[CrossRef] [PubMed]

Hess, O.

C. Simmendinger, M. Münkler, O. Hess, “Controlling complex temporal and spatiotemporal dynamics in semiconductor lasers,” Chaos Solitons Fractals 10, 851–864 (1999).

Heuer, A.

V. Raab, A. Heuer, J. Schultheiss, N. Hodbson, J. Kurths, R. Menzel, “Transverse effects in phase conjugate laser mirrors based on stimulated Brillouin scattering,” Chaos Solitons Fractals 10, 831–838 (1999).
[CrossRef]

Hochheiser, D.

D. Hochheiser, J. V. Moloney, J. Lega, “Controlling optical turbulence,” Phys. Rev. A 55, R4011–R4014 (1997).
[CrossRef]

M. E. Bleich, D. Hochheiser, J. V. Moloney, J. E. S. Socolar, “Controlling extended systems with spatially filtered, time-delayed feedback,” Phys. Rev. E 55, 2119–2126 (1997).
[CrossRef]

Hodbson, N.

V. Raab, A. Heuer, J. Schultheiss, N. Hodbson, J. Kurths, R. Menzel, “Transverse effects in phase conjugate laser mirrors based on stimulated Brillouin scattering,” Chaos Solitons Fractals 10, 831–838 (1999).
[CrossRef]

Hoffer, L. M.

G. L. Lippi, S. Barland, N. Dokhane, F. Monsieur, P. A. Porta, H. Grassi, L. M. Hoffer, “Phase space techniques for steering laser transients,” J. Opt. B: Quantum Semiclassical Opt. 2, 375–381 (2000).
[CrossRef]

P. A. Porta, L. M. Hoffer, H. Grassi, G. L. Lippi, “Analysis of nonfeedback technique for transient steering in Class B lasers,” Phys. Rev. A 61, 033801 (1–10) (2000).
[CrossRef]

G. L. Lippi, P. A. Porta, L. M. Hoffer, H. Grassi, “Control of transients in ‘lethargic’ systems,” Phys. Rev. E 59, R32–R35 (1999).
[CrossRef]

P. A. Porta, L. M. Hoffer, H. Grassi, G. L. Lippi, “Control of turn-on in Class B lasers,” Int. J. Bifurcation Chaos Appl. Sci. Eng. 8, 1811–1819 (1998).
[CrossRef]

Jin, F.

Johnston, A. R.

P. Torphammar, R. Tell, H. Eklund, A. R. Johnston, “Minimizing pattern effects in semiconductor lasers at high rate pulse modulation,” IEEE J. Quantum Electron. QE-15, 1271–1276 (1979).
[CrossRef]

Juul Jensen, S.

S. Juul Jensen, M. Schwab, C. Denz, “Manipulation, stabilization, and control of pattern formation using Fourier space filtering,” Phys. Rev. Lett. 81, 1614–1617 (1998).
[CrossRef]

Kashchenko, S. A.

V. N. Chizevsky, E. V. Grigorieva, S. A. Kashchenko, “Optimal timing for targeting periodic orbits in a loss-driven CO2 laser,” Opt. Commun. 133, 189–195 (1997).
[CrossRef]

Khurgin, J. B.

Kiefer, B.

A. Assion, T. Baumert, M. Bergt, T. Brixner, B. Kiefer, V. Swyfried, M. Strehle, G. Gerber, “Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses,” Science 282, 919–922 (1998).
[CrossRef] [PubMed]

Kim, H-J. R.

H-J. R. Kim, G. L. Lippi, H. Maurer, “Minimizing the transition time in lasers by optimal control methods. Single-mode semiconductor laser with homogeneous transverse profile,” Physica D 191, 238–260 (2004).
[CrossRef]

Kotomtseva, L. A.

L. A. Kotomtseva, A. V. Naumenko, A. M. Samson, S. I. Turovets, “Targeting unstable orbits and steady states in Class B lasers by using simple off/on manipulations,” Opt. Commun. 136, 335–348 (1997).
[CrossRef]

Kreuzer, M.

E. Benkler, M. Kreuzer, R. Neubecker, T. Tschudi, “Experimental control of unstable patterns and elimination of spatiotemporal disorder in nonlinear optics,” Phys. Rev. Lett. 84, 879–882 (2000).
[CrossRef] [PubMed]

G. K. Harkness, G.-L. Oppo, E. Benkler, M. Kreuzer, R. Neubecker, T. Tschudi, “Fourier space control in an LCLV feedback system,” J. Opt. B: Quantum Semiclassical Opt. 1, 177–182 (1999).
[CrossRef]

Kurths, J.

V. Raab, A. Heuer, J. Schultheiss, N. Hodbson, J. Kurths, R. Menzel, “Transverse effects in phase conjugate laser mirrors based on stimulated Brillouin scattering,” Chaos Solitons Fractals 10, 831–838 (1999).
[CrossRef]

Lange, W.

T. Ackemann, B. Giese, B. Schäpers, W. Lange, “Investigations of pattern forming mechanisms by Fourier filtering: properties of hexagons and the transition to stripes in an anisotropic system,” J. Opt. B: Quantum Semiclassical Opt. 1, 70–76 (1999).
[CrossRef]

M. Möller, B. Forsmann, W. Lange, “Instabilities in coupled Nd:YVO4 microchip lasers,” J. Opt. B: Quantum Semiclassical Opt. 10, 839–848 (1998).

Laporte, H.

J.-P. Goedgebuer, L. Larger, H. Laporte, “Optical crypto-system based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Larger, L.

L. Larger, J.-P. Goedgebuer, F. Delorme, “Optical encryption system using hyperchaos generated by an optoelectronic wavelength oscillator,” Phys. Rev. E 57, 6618–6624 (1998).
[CrossRef]

J.-P. Goedgebuer, L. Larger, H. Laporte, “Optical crypto-system based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. 80, 2249–2252 (1998).
[CrossRef]

Lega, J.

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On the basis of general principles63 one prefers the opposite approach, fixing V1 and V2 and varying t1 and t2. If the instrumentation allows it, this is certainly preferred. However, we see that even the opposite one, imposed by our generator, produces successful results.

In general, it is not meaningful to attempt a number of levels much larger than 100, no matter what system is taken into consideration. Maximum estimates of the number of trials can therefore be based on this worst-case assumption if other information is missing.

If the generator’s time resolution is small compared with the time interval to be explored, we come back to the considerations already made79 and consider that 100 trials are more than enough for a scan of the parameter space. In such a case, if no other restrictions are applied, the amount of data to be analyzed may rapidly become too large to be practicable. A way of reducing the portion of parameter space to analyze is presented in Subsection 5.B.

The time estimate for one loop can be obtained on the basis of the following considerations. For 1 kbyte the typical access time for a hard disk is nowadays ≈15 μs, while a standard GIPB interface is currently capable of transferring the same amount of data in ≈150 μs. Choosing a large safety factor in the duty cycle, to make sure that the measurement is not affected by external factors influencing the laser (e.g., heating, memory effects), we can set the signal frequency to ≈10 kHz; thus the waiting time to synchronize the cycles is at most 100 μs. In addition we have to take into account the time Labview takes to update the parameters and send them to the generator, and to activate it, and for the oscilloscope to trigger and store the data in memory. Given the speed of computer clocks, even if the program is not written in an efficient way, the bottleneck of the operation is the arbitrary waveform generator’s reaction time in responding and sending out the signal. In our measurements this time was particularly long (3 s) because of the very old technology of the apparatus. With sufficient error margin we can generically assume modern generators to be ~50 times faster; therefore we arrive at an estimated cycle time of ≈0.06 s.

Although we have not used this option in our measurements, most modern oscilloscopes offer a window discrimination or smart trigger on the data acquired. This option can be used for prefiltering the data. Alternatively this filtering can be done on the computer, once the data are transferred from a lower-class oscilloscope or from an older model, before the information is stored on the hard disk.

This statement is valid with the assumption that the range of values over which the steering parameter (voltage in our system) can be varied above and below threshold covers a comparable range. In our device the laser response above threshold grows considerably more for 1.8 V ≤ V≲ 3.5 V than for 3.5 V ≲ V≤ 5 V. Thus we can consider that the range of below-threshold voltages ΔVbt≈1.8 V is of the same order as the main contribution in the above-threshold interval ΔVat≈ 1.7 V. This response can be tested a priori in each system, and a weighted version of our statement can be used as an educated guess for determining a reasonable interval of ratios between t1 and t2 to be tested.

In all cases in which the time resolution of the arbitrary function generator is sufficiently fine, one would effectively invert the roles of the scans on the time values, t1 and t2, and voltage values, V1 and V2. In this latter case the number of voltage values to be chosen could also be rather small, since one would immediately start by considering in a preliminary run only those that are sufficiently close to Vmax for V1 and to Vmin for V2.

This situation is represented graphically by the trajectory labeled A in Fig. 7 in Ref. 62 when the target point in the phase space is approached from below.

This corresponds to the part of the composite trajectory that approaches the saddle point in phase space,55 starting from the initial operating point (with the laser switched off). It is in the neighborhood of this point that the laser field grows rapidly out of the intrinsic noise (not shown in the figures in Ref. 55).

In the phase-space picture55,60 this phase corresponds to aiming at the fixed point, thereby removing the oscillations.

For simplicity, in this discussion we assume that the laser is switched on from below threshold. One can easily introduce a generalization to consider the transition between states where the laser is active with different output power levels. (This has been discussed for telecommunication semiconductor lasers.60)

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).

The step that we describe in detail in Section 4 is necessary for laser systems, i.e., devices that package together both the laser and some controls (e.g., electronic, optical, or others). If one desires to optimize a Class B laser with no additional elements, the general considerations of Refs. 55, 60, and 64 apply directly. The procedure that we propose in this paper can still be applied and is likely to provide faster results, but the tests described in detail in Section 4 become unnecessary.

Although some degree of steering is possible in an oscillator, one can show that the time necessary to attain the new equilibrium state cannot be shortened, contrary to what happens in Class B lasers, and in the more complex device that we analyze here.

Embedding techniques amount to generating N-dimensional spaces starting from a one-dimensional data sequence by constructing vectors with elements shifted by an arbitrary number of elements. For a discussion see Refs. 74 and 75.

Compare Fig. 14 in Ref. 60 with Fig. 1.55 From the latter the presence of the saddle point is much more readily recognized.

F. Takens, “Detecting strange attractors in turbulence,” in Dynamical Systems and Turbulence, Vol. 898 of Springer Lecture Notes in Mathematics, D. A. Rand, L.-S. Young, eds. (Springer-Verlag, 1981), pp. 366–381.

Y. Fainman, K. Kitayama, T. W. Mossberg, eds., Innovative Physical Approaches to the Temporal or Spectral Control of Optical Signals, J. Opt. Soc. Am. B19, 2740–2823 (2002).

A. E. Siegman, Lasers (University Science, 1986).

G. L. Lippi, N. Dokhane, X. Hachair, S. Barland, J. R. Tredicce, “High speed direct modulation of semiconductor lasers,” in Semiconductor Lasers and Optical Amplifiers for Lightwave Communication Systems, R. P. Mirin, C. S. Menoni, eds., Proc. SPIE4871, 103–114 (2002).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Numerical integration of the direct current step (dashed curve) and of the optimally steered transition (solid curve): (a) laser intensity, (b) population inversion. Parameters: ∊ = 2.5 × 10−2, Poff = 0.9, Pon = 1.2; for the steered transition (see Fig. 6 below for definitions) P1 = 1.8, P2 = 0.9, t1 = 1.46 × 10−6 s, t2 × 3.61 × 10−6 s.

Fig. 2
Fig. 2

Numerically reconstructed phase space for the switch-on. Parameters are as in Fig. 1.

Fig. 3
Fig. 3

Experimental setup: DC, stabilized laser power supply (MVP driver by Thorlabs); Laser System, Thorlabs modulatable semiconductor laser; bias, electronic adder (built into the laser system); D, Si p-i-n detector (Thorlabs DET 210); L, focusing lens; Oscill., HP 54616B with the GPIB; PFG, programmable function generator (LeCroy 9100); PC, personal computer running LabVIEW.

Fig. 4
Fig. 4

Laser turn-on subject to a stepwise current switch (as schematically shown in the inset). Laser threshold: Vth ≈ 1.78 V; Voff = 0.941; Von = 2.549 V. These values hold throughout the paper. All symbols concerning the laser pumping levels (and durations of steering steps, when applicable) are defined in Fig. 6 below. The switch-on step is applied at t = 30 ns.

Fig. 5
Fig. 5

Reconstructed phase space for direct switch-on (Fig. 4). Variables are as in Eq. (3). The inset shows an equivalent numerical picture obtained from Eqs. (1) and (2); compared with Fig. 2, a faster relaxation (∊ = 0.05) has been used, and the larger amount of stimulated emission (typical of a semiconductor laser) has been simulated by considering a larger initial value (by nearly 4 orders of magnitude) for laser intensity I; c = 0.6 is for the numerical trajectory.

Fig. 6
Fig. 6

Shaped current function for optimal switch-on. The voltage values for the added plateaus and their respective durations are identified by labels 1 and 2, respectively (see text for discussion).

Fig. 7
Fig. 7

Laser switch-on. Solid curve, optimal steering; dashed curve, stepwise switch. The inset shows the shape of the steering function experimentally applied to the laser. The switch is applied at t = 30 ns. The two steering levels are V1,opt = 4.706 V and V2,opt = 0.235 V with durations t1,opt = 40 ns and t2,opt = 20 ns, respectively.

Fig. 8
Fig. 8

Response of the system to variations in amplitude: bold curve, nearly optimal transition (V2 = 0.1 V, all other parameter values are the optimal ones specified in the caption of Fig. 7). All other curves are obtained with V2 = 0.1 V and 3 V < V1 < 5 V varied in 0.2 V steps. Here and in Fig. 9 the dashed curve corresponds to the switch obtained with a stepwise function (as in Fig. 4).

Fig. 9
Fig. 9

Sensitivity of the system: bold curve, optimal transition. All other curves are obtained with V1 = 4.7 V and 0.2 V < V2 < 2.6 V varied in 0.2 V steps.

Fig. 10
Fig. 10

Shaped current function for optimal switch-off. The voltage values for the added plateaus and their respective durations are identified by labels 3 and 4, respectively (see text for discussion).

Fig. 11
Fig. 11

Experimental switch-off: solid curve, stepwise; dashed curve, with level 3; dotted curve, with both levels 3 and 4. Schematics of the electrical signal is shown in the corresponding insets. The addition of level 3 noticeably speeds up the switching and removes the residual modulation visible in the bottom trace. Level 4 does not contribute visibly to speeding up the laser intensity switch-off, as expected from the theory.60 Parameter values: V3 = 0.039 V, V4 = 1.568 V, t3 = 40 ns, and t4 = 20 ns.

Fig. 12
Fig. 12

Improvement introduced by the steered DMPCF: solid curve, steered off–on–off sequence with optimal electrical steering as obtained from Figs. 7 and Fig. 11: dotted curve, same sequence but with simple square switching.

Fig. 13
Fig. 13

Laser optical response for optimal steering before (dotted curve) and after (solid curve) the abrupt change in steering values (probably) due to an electrical spike that modified the laser’s characteristics. The response to steering is practically unchanged, but the new optimal values are V1,opt′ = 4.237 V, V2,opt′ = 0.686 V, t1,opt′ = 40 ns, and t2,opt′ = 30 ns.

Tables (1)

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Table 1 Tested Durationsa

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

I ˙ = ( D - 1 ) I ,
D ˙ = - [ D ( 1 + I ) - P ] ,
s = 1 + c ( I ˙ / I ) ,
T j = t j , max - t j . min + Δ t Δ t             ( j = 1 , 2 ) ,
N j = V j , max - V j , min + Δ V Δ V             ( j = 1 , 2 ) .

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