## Automated correlation dimension analysis of optically injected solid state lasers

Optics Express, Vol. 17, Issue 9, pp. 7592-7608 (2009)

http://dx.doi.org/10.1364/OE.17.007592

Acrobat PDF (1004 KB)

### Abstract

Nonlinear lasers are excellent systems from which to obtain high signal-to-noise experimental data of nonlinear dynamical variables to be used to develop and demonstrate robust nonlinear dynamics analysis techniques. Here we investigate the dynamical complexity of such a system: an optically injected Nd:YVO_{4} solid state laser. We show that a map of the correlation dimension as a function of the injection strength and frequency detuning, extracted from the laser output power time-series data, is an excellent mirror of the dynamics map generated from a theoretical model of the system. An automated computational protocol has been designed and implemented to achieve this. The correlation dimension map is also contrasted with prior research that mapped the peak intensity of the output power as an experimentally accessible measurand reflecting the dynamical state of the system [Valling et al., Phys. Rev. A **72**, 033810 (2005)].

© 2009 Optical Society of America

## 1. Introduction

_{2}lasers show self pulsing and deterministic chaos when the laser is subject to feedback [1

1. F. T. Arecchi, W. Gadomski, and R. Meucci, “Generation of chaotic dynamics by feedback on a laser,” Phys. Rev. A **34**, 1617–1620 (1986). [CrossRef] [PubMed]

_{3}) ring lasers have been shown to produce similar instabilities and period-doubling routes to chaos have been investigated in this system [2

2. W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser,” Phys. Rev. A **31**, 4049–4051 (1985). [CrossRef] [PubMed]

3. E. Hemery, L. Chusseau, and J. M. Lourtioz, “Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 μm,” IEEE J. Quantum Electron. **26**, 633–641 (1990). [CrossRef]

4. D. M. Kane and K. A. Shore, eds. *Unlocking Dynamical Diversity: Feedback Effects on Semiconductor Lasers* (Wiley, 2005). [CrossRef]

5. T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, “Period-doubling route to chaos in a semiconductor-laser subject to optical-injection,” Appl. Phys. Lett. **64**, 3539–3541 (1994). [CrossRef]

6. S. Tang and J. M. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. **37**, 329–336 (2001). [CrossRef]

7. F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. **221**, 173–180 (2003). [CrossRef]

8. J. S. Lawrence and D. M. Kane, “Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,” IEEE J. Quantum Electron. **38**, 185–192 (2002). [CrossRef]

9. K. R. Preston, K. C. Woollard, and K. H. Cameron, “External cavity controlled single longitudinal mode laser transmitter module,” Electon. Lett. **17**, 931–933 (1981). [CrossRef]

10. H. L. Stover and W. H. Steier, “Locking of laser oscillators by light injection,” Appl. Phys. Lett. **8**, 91–93 (1966). [CrossRef]

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

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

_{4}) [13

13. A. Uchida, H. Shinozuka, T. Ogawa, and F. Kannari, “Experiments on chaos synchronization in two separate microchip lasers,” Opt. Lett. **24**, 890–892 (1999). [CrossRef]

14. S. Donati and C. R. Mirasso “Feature section on optical chaos and applications to cryptography,” IEEE J. Quantum Electron. **38**, 1138–1204 (2002). [CrossRef]

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

16. A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature **438**, 343–346 (2005). [CrossRef] [PubMed]

17. T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclassical Opt. **9**, 765–784 (1997). [CrossRef]

21. S. Valling, T. Fordell, and A. M. Lindberg,“Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A **72**, 033810 (2005). [CrossRef]

22. S. Valling, B. Krauskopf, T. Fordell, and A. M. Lindberg, “Experimental bifurcation diagram of a solid state laser with optical injection,” Opt. Commun. **271**, 532–542 (2007). [CrossRef]

24. H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A **185**, 77–87 (1994). [CrossRef]

25. M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D **65**, 117–134 (1993). [CrossRef]

26. K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. **216**, 405–412 (2003). [CrossRef]

27. J. P. Toomey and D. M. Kane, “Analysis of chaotic semiconductor laser diodes,” in *Proceedings of the Conference on Optoelectronic and Microelectronic Materials and Devices*(IEEE, Perth, Australia, 2006), pp. 164–167. [CrossRef]

29. P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D **9**, 189–208 (1983). [CrossRef]

_{4}solid state laser system [21

21. S. Valling, T. Fordell, and A. M. Lindberg,“Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A **72**, 033810 (2005). [CrossRef]

29. P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D **9**, 189–208 (1983). [CrossRef]

21. S. Valling, T. Fordell, and A. M. Lindberg,“Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A **72**, 033810 (2005). [CrossRef]

22. S. Valling, B. Krauskopf, T. Fordell, and A. M. Lindberg, “Experimental bifurcation diagram of a solid state laser with optical injection,” Opt. Commun. **271**, 532–542 (2007). [CrossRef]

*K*, and the angular frequency detuning between the free-running master laser and free-running slave laser Δ

*ω*. The master/slave OISSL system is described in full detail in [21

**72**, 033810 (2005). [CrossRef]

5. T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, “Period-doubling route to chaos in a semiconductor-laser subject to optical-injection,” Appl. Phys. Lett. **64**, 3539–3541 (1994). [CrossRef]

30. T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. **242**, 613–622 (2004). [CrossRef]

29. P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D **9**, 189–208 (1983). [CrossRef]

## 2. Nonlinear time-series analysis

*P*(

*T*), and plotting it against a time delayed version of itself

*P*(

*t*+

*T*). Since the dimension of the attractor is not known beforehand, we aim to embed the attractor sequentially in a phase space with dimension

*m*, equal to and greater than the dimension of the attractor. This ensures that the orbit of the attractor is completely unfolded and no overlaps due to projection occur. The vectors describing the attractor are given by,

*T*and its effect on the phase space reconstruction has been previously investigated [32

32. M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, “State space reconstruction in the presence of noise,” Physica D **51**, 52–98 (1991). [CrossRef]

*T*are that it must be a multiple of the sampling time and also ensure that data is separated as much as possible without the points becoming completely independent. While there is no rigorous way of determining the optimal value of

*T*, a good approximation, and the method used in this work, is to use the first minimum of the mutual information function [33

33. A. M. Fraser and H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A **33**, 1134–1140 (1986). [CrossRef] [PubMed]

34. M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “Reconstruction expansion as a geometry-based framework for choosing proper delay times,” Physica D **73**, 82–98 (1994). [CrossRef]

35. T. Buzug and G. Pfister, “Comparison of algorithms calculating optimal embedding parameters for delay time coordinates,” Physica D **58**, 127–137 (1992). [CrossRef]

**9**, 189–208 (1983). [CrossRef]

*y*−

_{i}*y*∥ is the distance between pairs of points in the phase space and

_{j}*H*(

*x*) is the Heaviside function (

*H*= 1 if

*x*≥ 0 and

*H*= 0 if

*x*< 0). The value

*r*is the radius of a hypersphere in an

*m*-dimensional space, and

*N*is the number of points in the space.

*C*(

_{m}*r*) versus log

*r*should approach a finite value

*D*in the limit as

*r*→ 0. For data that has finite sampling, the slope of this graph for very small

*r*is very inaccurate so instead a ‘scaling” region of r is taken over which the slope is relatively stable. A plot of the gradient,

*D*(

_{m}*r*) =

*∂*log

*C*(

_{m}*r*)/

*∂*log

*r*, as a function of

*r*makes observing this region much clearer. The value of the slope

*D*(

_{m}*r*) within the scaling region is the correlation dimension.

*C*(

_{m}*r*), and corresponding slopes,

*D*(

_{m}*r*), as a function of radius. The data used is the

*x*-variable time-series (10 000 points) of the chaotic Lorenz equations [36

36. E. N. Lorenz, “Deterministic Nonperiodic Flow,” J. Atmos. Sci. **20**, 130–141 (1963). [CrossRef]

*D*(

_{m}*r*) can be seen to saturate to a fixed level as the data is embedded in higher and higher dimensions

*m*. This ‘plateau’ is the scaling region and, if present, indicates that the data is deterministic. In the case of this Lorenz data we can infer that the minimum embedding dimension required to unfold the attractor is 8, and that the value of the correlation dimension is slightly larger than 2 (the value of the slope in the scaling region). If the time-series was from a purely stochastic process it would give rise to a slope that increases continually for higher embedding dimensions. Thus, the absence of a scaling region indicates that the data could be stochastic in nature, or severely affected by noise.

**9**, 189–208 (1983). [CrossRef]

*D*(

_{m}*r*).

38. J. Theiler, “Spurious dimension from correlation algorithms applied to limited time-series data,” Phys. Rev. A **34**, 2427–2432 (1986). [CrossRef] [PubMed]

*CD*. Surrogate data sets are then created from the original set, and the same dimensional analysis is performed on these. The average value of the correlation dimension obtained from the surrogates, <

_{orig}*CD*>, is then compared to

_{surr}*CD*to see if they are significantly different.

_{orig}41. T. Schreiber and A. Schmitz, “Surrogate time series,” Physica D **142**, 346–382 (2000). [CrossRef]

## 3. Experiment and analysis

**72**, 033810 (2005). [CrossRef]

42. S. Valling, T. Fordell, and A. M. Lindberg,“Experimental and numerical intensity time series of an optically injected solid state laser,” Opt. Commun. **254**, 282–289 (2005). [CrossRef]

**9**, 189–208 (1983). [CrossRef]

5. T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, “Period-doubling route to chaos in a semiconductor-laser subject to optical-injection,” Appl. Phys. Lett. **64**, 3539–3541 (1994). [CrossRef]

30. T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. **242**, 613–622 (2004). [CrossRef]

### 3.1 Laser system

_{4}solid state laser system can be seen in Fig. 2. Two 1064 nm solid state Nd:YVO

_{4}lasers were operated in a master-slave configuration. The Nd:YVO

_{4}crystals (CASIX) are 1 mm thick with 1% Nd

^{3+}atomic doping. The laser cavity was formed by HR coating on the front surface and 5% output coupling at the end facet. The laser crystals were pumped with 150 mW diode lasers (SDL) operating at 809 nm. The pump power was low enough to ensure single mode operation. Both the pump and laser polarization were along the c-axis of the crystal. Faraday isolators were used to block feedback to the pump lasers. The master laser crystal mount could be temperature controlled using a Peltier element. Interference filters removed excess pump light from the 1064 nm beams. Faraday isolators made the coupling between master and slave unidirectional and prevented unwanted feedback. Light from the master laser was injected into the slave via an acousto-optic modulator (AOM) and the relative injection power measured on the detector PD1. The beat frequency between the two lasers was measured on a 400 MHz detector (PD3). The optical input (PD2) of a Tektronix CSA 7404 oscilloscope was used to measure the intensity time-series of the slave laser.

*ω*). The maximum range over which the frequency is shifted is approximately 32 MHz. This is very narrow compared to the mode spacing so no mode hopping occurs.

*K*) was slowly increased from zero until injection locking was achieved. This injection sweep was repeated at a large number of fixed frequency detuning values (Δ

*ω*).

### 3.2 Automated Computational Process

*K*, Δ

*ω*) plane. The algorithm was coded using a combination of both Matlab and C++ programming languages.

*ω*between −4.11 and 3.25 units (a multiplier of the angular relaxation oscillation frequency). Each of these contains 883 000 data points of the laser intensity, sampled at 10 ns intervals, for a complete sweep of the injection strength from zero until the level at which the system reaches frequency locking of the slave to the master laser. The necessary injection strength to achieve frequency locking increases linearly with the frequency detuning between the master laser and the free running slave laser. Each of these complete time-series contains the whole range of dynamics possible for that value of frequency detuning. An example of one of these time-series is shown in Fig. 4.

*K*= 0 to 3.9, resulting in a maximum change of Δ

*K*= 0.03 within the 50 μs subset. The subsets also contain at least a hundred cycles of the prevalent dynamics. This was determined to be enough for robust analysis as very similar results were obtained when analyzing the same data sets but using only the first 2000 data points.

*T*required to reconstruct the phase space trajectory was calculated by locating the first minimum of the mutual information function [33

33. A. M. Fraser and H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A **33**, 1134–1140 (1986). [CrossRef] [PubMed]

*m*= 2–10 by using Eq. (1). Each attractor was analyzed using the modified correlation sum in Eq. (5), from which a graph of the gradient as a function of radius,

*D*(

_{m}*r*), was generated.

*D*(

_{m}*r*) curves for all embeddings were constructed, the scaling region, if present, was located by using a modified ‘most probable dimension value’ method [43

43. A. Corana, G. Bortolan, and A. Casaleggio, “Most probable dimension value and most flat interval methods for automatic estimation of dimension from time series,” Chaos, Solitons Fractals **20**, 779–790 (2004). [CrossRef]

*D*(

_{m}*r*) at the histogram peak was recorded as the correlation dimension. If no plateau was detected then no value was recorded for correlation dimension. The width of this scaling region peak, measured at half the maximum, is a measure of the uncertainty in the correlation dimension. The correlation dimension values were then mapped into the corresponding positions in the (

*K*, Δ

*ω*) plane. The final dynamical map is shown in Fig. 6.

### 3.3 Theoretical Model of Solid State Laser Subject to Optical Injection

_{4}solid-state laser subject to optical injection has been described in [5

**64**, 3539–3541 (1994). [CrossRef]

30. T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. **242**, 613–622 (2004). [CrossRef]

**72**, 033810 (2005). [CrossRef]

### 3.4 Dynamical Maps Using Automated Correlation Dimension Analysis

_{4}solid state laser system have used peak intensity amplitude [21

**72**, 033810 (2005). [CrossRef]

22. S. Valling, B. Krauskopf, T. Fordell, and A. M. Lindberg, “Experimental bifurcation diagram of a solid state laser with optical injection,” Opt. Commun. **271**, 532–542 (2007). [CrossRef]

## 4. Discussion

*K*, Δ

*ω*) plane, as determined by the automated process for the experimental OISSL data. The white triangular region in the middle, identified as Region I in Fig. 6, is where the laser becomes frequency locked. Here the laser output is single frequency and constant in power (with small amplitude relaxation oscillations). Several other distinct regions can be identified within this map, and they have been labeled as Regions II, III, IV and V in Fig. 6.

*D*(

_{m}*r*) and the corresponding histograms for each of the different dynamical regions (II-V) in the (

*K*, Δ

*ω*) plane. Figure 9(a) is a typical periodic signal seen within the light blue (CD = 1) Region II of the experimental CD map. Fig. 9(b) represents the ‘spiky’ output generally seen in Region III, the dark blue (low CD) region of the CD map. Fig. 9(c) shows a ‘chaotic’ output from Region IV, identified by the red (high CD) points in the CD map. Figure 9(d) is representative of Region V, corresponding to noisy time-series that gave no scaling region in the correlation sum and therefore an infinite correlation dimension.

*T*used. When the program detects the first minimum of the mutual information as 1 data point, the

*D*(

_{m}*r*) graph and histogram register a narrow peak as seen in Fig. 10(a). However, it is also observed that when the delay is longer than 1 data point the

*D*(

_{m}*r*) graph obtained is similar to that seen in Fig. 10(b).

44. D. M. Kane, J. P. Toomey, M. W. Lee, and K. A. Shore, “Correlation dimension signature of wideband chaos synchronization of semiconductor lasers,” Opt. Lett. **31**, 20–22 (2006). [CrossRef] [PubMed]

*D*(

_{m}*r*) graphs when using the surrogate data sets. This is as expected since the surrogate construction process should remove any determinism from the time-series, producing stochastic, infinite dimensional attractors. In each individual surrogate map there are spurious CD values recorded, but never at the same point in the (

*K*, Δ

*ω*) parameter space so their impact is averaged out in the final map which shows an infinite CD over the entire parameter range. The map of the original experimental data shows values for CD recorded at 19 369 points in the (

*K*, Δ

*ω*) range (out of 55 440 total points), compared with an average 230 values recorded from the surrogate data sets. Since this surrogate data produces a map that is significantly different to the map generated from the original data we can be confident that the correlation dimension values calculated are a result of deterministic nonlinear dynamics, rather than a linearly filtered stochastic process.

**271**, 532–542 (2007). [CrossRef]

**72**, 033810 (2005). [CrossRef]

## 5. Conclusion

_{4}solid state laser with varying injection strength and frequency detuning. The map is in excellent agreement with prior investigations into the amplitude and bifurcation analysis of this laser system [21

**72**, 033810 (2005). [CrossRef]

**271**, 532–542 (2007). [CrossRef]

## Acknowledgments

## References and links

1. | F. T. Arecchi, W. Gadomski, and R. Meucci, “Generation of chaotic dynamics by feedback on a laser,” Phys. Rev. A |

2. | W. Klische and C. O. Weiss, “Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser,” Phys. Rev. A |

3. | E. Hemery, L. Chusseau, and J. M. Lourtioz, “Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 μm,” IEEE J. Quantum Electron. |

4. | D. M. Kane and K. A. Shore, eds. |

5. | T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, “Period-doubling route to chaos in a semiconductor-laser subject to optical-injection,” Appl. Phys. Lett. |

6. | S. Tang and J. M. Liu, “Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback,” IEEE J. Quantum Electron. |

7. | F. Y. Lin and J. M. Liu, “Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback,” Opt. Commun. |

8. | J. S. Lawrence and D. M. Kane, “Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation,” IEEE J. Quantum Electron. |

9. | K. R. Preston, K. C. Woollard, and K. H. Cameron, “External cavity controlled single longitudinal mode laser transmitter module,” Electon. Lett. |

10. | H. L. Stover and W. H. Steier, “Locking of laser oscillators by light injection,” Appl. Phys. Lett. |

11. | L. M. Pecora and T. L. Carroll, “Synchronization in chaotic systems,” Phys. Rev. Lett. |

12. | G. D. VanWiggeren and R. Roy, “Communication with chaotic lasers,” Science |

13. | A. Uchida, H. Shinozuka, T. Ogawa, and F. Kannari, “Experiments on chaos synchronization in two separate microchip lasers,” Opt. Lett. |

14. | S. Donati and C. R. Mirasso “Feature section on optical chaos and applications to cryptography,” IEEE J. Quantum Electron. |

15. | J. P. Goedgebuer, L. Larger, and H. Porte, “Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode,” Phys. Rev. Lett. |

16. | A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature |

17. | T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, “Nonlinear dynamics induced by external optical injection in semiconductor lasers,” Quantum Semiclassical Opt. |

18. | S. Eriksson and A. M. Lindberg, “Observations on the dynamics of semiconductor lasers subjected to external optical injection,” J. Opt. B |

19. | S. Eriksson, “Dependence of the experimental stability diagram of an optically injected semiconductor laser on the laser current,” Opt. Commun. |

20. | S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, “The dynamical complexity of optically injected semiconductor lasers,” Physics Reports-Review Section of Physics Letters |

21. | S. Valling, T. Fordell, and A. M. Lindberg,“Maps of the dynamics of an optically injected solid-state laser,” Phys. Rev. A |

22. | S. Valling, B. Krauskopf, T. Fordell, and A. M. Lindberg, “Experimental bifurcation diagram of a solid state laser with optical injection,” Opt. Commun. |

23. | H. Kantz and T. Schreiber, |

24. | H. Kantz, “A robust method to estimate the maximal Lyapunov exponent of a time-series,” Phys. Lett. A |

25. | M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “A practical method for calculating largest Lyapunov exponents from small data sets,” Physica D |

26. | K. E. Chlouverakis and M. J. Adams, “Stability maps of injection-locked laser diodes using the largest Lyapunov exponent,” Opt. Commun. |

27. | J. P. Toomey and D. M. Kane, “Analysis of chaotic semiconductor laser diodes,” in |

28. | C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, “Influence of noise on chaotic laser dynamics,” Phys. Rev. E |

29. | P. Grassberger and I. Procaccia, “Measuring the strangeness of strange attractors,” Physica D |

30. | T. Fordell and A. M. Lindberg, “Numerical stability maps of an optically injected semiconductor laser,” Opt. Commun. |

31. | F. Takens, “Dynamical systems and turbulence,” in |

32. | M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, “State space reconstruction in the presence of noise,” Physica D |

33. | A. M. Fraser and H. L. Swinney, “Independent coordinates for strange attractors from mutual information,” Phys. Rev. A |

34. | M. T. Rosenstein, J. J. Collins, and C. J. Deluca, “Reconstruction expansion as a geometry-based framework for choosing proper delay times,” Physica D |

35. | T. Buzug and G. Pfister, “Comparison of algorithms calculating optimal embedding parameters for delay time coordinates,” Physica D |

36. | E. N. Lorenz, “Deterministic Nonperiodic Flow,” J. Atmos. Sci. |

37. | P. E. Rapp, A. M. Albano, T. I. Schmah, and L. A. Farwell, “Filtered noise can mimic low-dimensional chaotic attractors,” Phys. Rev. E |

38. | J. Theiler, “Spurious dimension from correlation algorithms applied to limited time-series data,” Phys. Rev. A |

39. | J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, “Testing for nonlinearity in time-series -The method of surrogate data,” Physica D |

40. | A. Provenzale, L. A. Smith, R. Vio, and G. Murante, “Distinguishing between low-dimensional dynamics and randomness in measured time-series,” Physica D |

41. | T. Schreiber and A. Schmitz, “Surrogate time series,” Physica D |

42. | S. Valling, T. Fordell, and A. M. Lindberg,“Experimental and numerical intensity time series of an optically injected solid state laser,” Opt. Commun. |

43. | A. Corana, G. Bortolan, and A. Casaleggio, “Most probable dimension value and most flat interval methods for automatic estimation of dimension from time series,” Chaos, Solitons Fractals |

44. | D. M. Kane, J. P. Toomey, M. W. Lee, and K. A. Shore, “Correlation dimension signature of wideband chaos synchronization of semiconductor lasers,” Opt. Lett. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(140.1540) Lasers and laser optics : Chaos

(140.3580) Lasers and laser optics : Lasers, solid-state

(190.0190) Nonlinear optics : Nonlinear optics

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 12, 2009

Revised Manuscript: April 3, 2009

Manuscript Accepted: April 4, 2009

Published: April 23, 2009

**Citation**

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg, "Automated correlation dimension analysis of optically injected solid state lasers," Opt. Express **17**, 7592-7608 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-9-7592

Sort: Year | Journal | Reset

### References

- F. T. Arecchi, W. Gadomski, and R. Meucci, "Generation of chaotic dynamics by feedback on a laser," Phys. Rev. A 34, 1617-1620 (1986). [CrossRef] [PubMed]
- W. Klische and C. O. Weiss, "Instabilities and routes to chaos in a homogeneously broadened one- and two-mode ring laser," Phys. Rev. A 31, 4049-4051 (1985). [CrossRef] [PubMed]
- E. Hemery, L. Chusseau, and J. M. Lourtioz, "Dynamic behaviors of semiconductor lasers under strong sinusoidal current modulation: modeling and experiments at 1.3 ?m," IEEE J. Quantum Electron. 26, 633-641 (1990). [CrossRef]
- D. M. Kane, and K. A. Shore, eds., Unlocking Dynamical Diversity: Feedback Effects on Semiconductor Lasers (Wiley, 2005). [CrossRef]
- T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing, "Period-doubling route to chaos in a semiconductor-laser subject to optical-injection," Appl. Phys. Lett. 64, 3539-3541 (1994). [CrossRef]
- S. Tang and J. M. Liu, "Chaotic pulsing and quasi-periodic route to chaos in a semiconductor laser with delayed opto-electronic feedback," IEEE J. Quantum Electron. 37, 329-336 (2001). [CrossRef]
- F. Y. Lin and J. M. Liu, "Nonlinear dynamical characteristics of an optically injected semiconductor laser subject to optoelectronic feedback," Opt. Commun. 221, 173-180 (2003). [CrossRef]
- J. S. Lawrence and D. M. Kane, "Nonlinear dynamics of a laser diode with optical feedback systems subject to modulation," IEEE J. Quantum Electron. 38, 185-192 (2002). [CrossRef]
- K. R. Preston, K. C. Woollard, and K. H. Cameron, "External cavity controlled single longitudinal mode laser transmitter module," Electron. Lett. 17, 931-933 (1981). [CrossRef]
- H. L. Stover and W. H. Steier, "Locking of laser oscillators by light injection," Appl. Phys. Lett. 8, 91-93 (1966). [CrossRef]
- L. M. Pecora and T. L. Carroll, "Synchronization in chaotic systems," Phys. Rev. Lett. 64, 821-824 (1990). [CrossRef] [PubMed]
- G. D. VanWiggeren and R. Roy, "Communication with chaotic lasers," Science 279, 1198-1200 (1998). [CrossRef] [PubMed]
- A. Uchida, H. Shinozuka, T. Ogawa, and F. Kannari, "Experiments on chaos synchronization in two separate microchip lasers," Opt. Lett. 24, 890-892 (1999). [CrossRef]
- S. Donati and C. R. Mirasso, "Feature section on optical chaos and applications to cryptography," IEEE J. Quantum Electron. 38, 1138-1204 (2002). [CrossRef]
- J. P. Goedgebuer, L. Larger, and H. Porte, "Optical cryptosystem based on synchronization of hyperchaos generated by a delayed feedback tunable laser diode," Phys. Rev. Lett. 80, 2249-2252 (1998). [CrossRef]
- A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, "Chaos-based communications at high bit rates using commercial fibre-optic links," Nature 438, 343-346 (2005). [CrossRef] [PubMed]
- T. B. Simpson, J. M. Liu, K. F. Huang, and K. Tai, "Nonlinear dynamics induced by external optical injection in semiconductor lasers," Quantum Semiclassic. Opt. 9, 765-784 (1997). [CrossRef]
- S. Eriksson and A. M. Lindberg, "Observations on the dynamics of semiconductor lasers subjected to external optical injection," J. Opt. B:Quantum Semiclassical Opt. 4, 149-154 (2002). [CrossRef]
- S. Eriksson, "Dependence of the experimental stability diagram of an optically injected semiconductor laser on the laser current," Opt. Commun. 210, 343-353 (2002). [CrossRef]
- S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, "The dynamical complexity of optically injected semiconductor lasers," Phys. Rep. 416, 1-128 (2005).
- S. Valling, T. Fordell, and A. M. Lindberg, "Maps of the dynamics of an optically injected solid-state laser," Phys. Rev. A 72, 033810 (2005). [CrossRef]
- S. Valling, B. Krauskopf, T. Fordell, and A. M. Lindberg, "Experimental bifurcation diagram of a solid state laser with optical injection," Opt. Commun. 271, 532-542 (2007). [CrossRef]
- H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2004).
- H. Kantz, "A robust method to estimate the maximal Lyapunov exponent of a time-series," Phys. Lett. A 185, 77-87 (1994). [CrossRef]
- M. T. Rosenstein, J. J. Collins, and C. J. Deluca, "A practical method for calculating largest Lyapunov exponents from small data sets," Physica D 65, 117-134 (1993). [CrossRef]
- K. E. Chlouverakis and M. J. Adams, "Stability maps of injection-locked laser diodes using the largest Lyapunov exponent," Opt. Commun. 216, 405-412 (2003). [CrossRef]
- J. P. Toomey and D. M. Kane, "Analysis of chaotic semiconductor laser diodes," in Proceedings of the Conference on Optoelectronic and Microelectronic Materials and Devices (IEEE, Perth, Australia, 2006), pp. 164-167. [CrossRef]
- C. Liu, R. Roy, H. D. I. Abarbanel, Z. Gills, and K. Nunes, "Influence of noise on chaotic laser dynamics," Phys. Rev. E 55, 6483-6500 (1997). [CrossRef]
- P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D 9, 189-208 (1983). [CrossRef]
- T. Fordell and A. M. Lindberg, "Numerical stability maps of an optically injected semiconductor laser," Opt. Commun. 242, 613-622 (2004). [CrossRef]
- F. Takens, "Dynamical systems and turbulence," in Springer Lecture Notes in Mathematics, D. A. Rand, and L.-S. Young, eds., (Springer-Verlag, New York, 1980), pp. 366-381.
- M. Casdagli, S. Eubank, J. D. Farmer, and J. Gibson, "State space reconstruction in the presence of noise," Physica D 51, 52-98 (1991). [CrossRef]
- A. M. Fraser and H. L. Swinney, "Independent coordinates for strange attractors from mutual information," Phys. Rev. A 33, 1134-1140 (1986). [CrossRef] [PubMed]
- M. T. Rosenstein, J. J. Collins, and C. J. Deluca, "Reconstruction expansion as a geometry-based framework for choosing proper delay times," Physica D 73, 82-98 (1994). [CrossRef]
- T. Buzug and G. Pfister, "Comparison of algorithms calculating optimal embedding parameters for delay time coordinates," Physica D 58, 127-137 (1992). [CrossRef]
- E. N. Lorenz, "Deterministic Nonperiodic Flow," J. Atmos. Sci. 20, 130-141 (1963). [CrossRef]
- P. E. Rapp, A. M. Albano, T. I. Schmah, and L. A. Farwell, "Filtered noise can mimic low-dimensional chaotic attractors," Phys. Rev. E 47, 2289-2297 (1993). [CrossRef]
- J. Theiler, "Spurious dimension from correlation algorithms applied to limited time-series data," Phys. Rev. A 34, 2427-2432 (1986). [CrossRef] [PubMed]
- J. Theiler, S. Eubank, A. Longtin, B. Galdrikian, and J. D. Farmer, "Testing for nonlinearity in time-series - The method of surrogate data," Physica D 58, 77-94 (1992). [CrossRef]
- A. Provenzale, L. A. Smith, R. Vio, and G. Murante, "Distinguishing between low-dimensional dynamics and randomness in measured time-series," Physica D 58, 31-49 (1992). [CrossRef]
- T. Schreiber and A. Schmitz, "Surrogate time series," Physica D 142, 346-382 (2000). [CrossRef]
- S. Valling, T. Fordell, and A. M. Lindberg, "Experimental and numerical intensity time series of an optically injected solid state laser," Opt. Commun. 254, 282-289 (2005). [CrossRef]
- A. Corana, G. Bortolan, and A. Casaleggio, "Most probable dimension value and most flat interval methods for automatic estimation of dimension from time series," Chaos, Solitons Fractals 20, 779-790 (2004). [CrossRef]
- D. M. Kane, J. P. Toomey, M. W. Lee, and K. A. Shore, "Correlation dimension signature of wideband chaos synchronization of semiconductor lasers," Opt. Lett. 31, 20-22 (2006). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.