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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 9 — Apr. 27, 2009
  • pp: 7592–7608
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Automated correlation dimension analysis of optically injected solid state lasers

J. P. Toomey, D. M. Kane, S. Valling, and A. M. Lindberg  »View Author Affiliations


Optics Express, Vol. 17, Issue 9, pp. 7592-7608 (2009)
http://dx.doi.org/10.1364/OE.17.007592


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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:YVO4 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

Complex instabilities and dynamics are common in many different laser systems. Some examples are listed. CO2 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]

]. Ammonia (NH3) 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]

]. Semiconductor lasers can display complex, nonlinear dynamics when subject to direct current modulation [3

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]

], optical feedback [4

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

], strong optical injection from a master laser [5

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]

], optoelectronic feedback [6

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]

] or a hybrid combination of these [7

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

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]

]. In many instances a perturbation such as feedback or injection, which may cause complex dynamics for some parameter values, can also be used to stabilize the laser with different sets of values [9

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

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

].

Recently, complex laser systems have generated interest, not only for fundamental nonlinear dynamics research, but also for their potential application as chaotic sources for secure communication schemes. This is a concept that was first proposed in nonlinear electronic circuits [11

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

]. The potential to obtain much higher frequency bandwidths in chaos using optical signals rather than electronics, has prompted development of systems that display complex instabilities, such as Erbium doped fiber ring lasers [12

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

], solid state laser microchip lasers (Nd:YVO4) [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]

], and semiconductor lasers [14

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]

]. These have been specifically researched for applications in secure communications. Schemes incorporating chaotically varying wavelength have also been proposed [15

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]

]. Systems employing semiconductor lasers with direct optical feedback and optoelectronic feedback to produce chaotic output power fluctuations have been used successfully to transmit and receive chaotically masked signals over commercial fiber-optic networks [16

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]

].

Nonlinear time-series and chaos analysis techniques can be utilized to gain further insight to these systems and have been used to quantify the observed complex dynamics of real world systems from many areas of research [23

23. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2004).

]. Maps of the largest Lyapunov exponent, a nonlinear time-series measure of the sensitivity of a system to initial conditions [24

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

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]

], have been generated from theoretically modeled optically injected semiconductor lasers for comparison with experimental results [26

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]

], for example.

Another reason chaotic and dynamically unstable lasers are useful is that the experimenter has much more control over system variables than is possible in most other nonlinear systems, such as many biological, physiological, mechanical, hydrodynamical and electronic nonlinear systems. For example, employing nonlinear time-series analysis, and having the ability to systematically vary the signal-to-noise of a photodetected chaotic semiconductor laser signal, has allowed the transition from deterministic chaos, through chaos plus noise, to a noise dominated signal, and allowed the levels of acceptable noise that would not mask the chaotic dynamics to be identified [27

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]

]. In contrast, most biological and physiological nonlinear systems do not permit independent control of signal-to-noise. Study of noise masking effects in laser-based systems can provide insight that helps interpret nonlinear data analysis in these systems.

Nonlinear laser systems are also ideal for developing and testing the nonlinear analysis concepts themselves. This has been suggested in [28

28. 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]

], where they use nonlinear time-series techniques to analyze the chaotic output of a Nd:YAG laser with an intra-cavity second harmonic generating crystal, in order to further understand the impact noise has on the laser output. The high degree of control that one has over time-series length, signal-to-noise level and sampling rate that is experimentally achievable in laser systems make them ideal for refining robust analysis techniques. These procedures can then be implemented for use with other complex systems where short, noisy and/or under sampled data sets are unavoidable.

Despite the vast number of publications on nonlinear time-series analysis, there is still confusion and conflicting reports on whether the results of such analysis are physically meaningful. One of the main problems with implementing such analysis techniques is that the results generally require subjective interpretation by the user. For instance, calculation of the correlation dimension [29

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

], an appropriate analysis measurand for nonlinear time-series, generates a graph of the correlation sum (explained in detail later) from which a region of constant slope needs to be identified. In most cases this identification is done by visual inspection of the graph. Consequently, discrepancies can arise due to varying criteria implemented by the user. Manual interpretation of such results from a complex system over a large parameter range, such as is required to produce a full dynamical map, is impractical. Thus, an automatic computational system able to generate dynamical maps is a tool which will significantly facilitate analysis of the complexity of nonlinear dynamical systems. An important part of any such automated computation is development of appropriate protocols to be incorporated to ensure physically meaningful dynamical measures are calculated.

In this paper, an overview of nonlinear time series analysis techniques, and the correlation dimension in particular, is given in Section 2. The details of the experimental method, automated analysis program and theoretical model, along with main results are given in Section 3. Discussion of the results and comparisons between the experimental and simulated data is given in Section 4. The variety of dynamics identified by the automated analysis program demonstrate how this is a useful tool for investigating and understanding nonlinear systems.

2. Nonlinear time-series analysis

In general, nonlinear systems are best described and analyzed by means of a phase space trajectory or attractor, which represents all the possible states of a particular system. The dimension of the phase space required to completely describe a system needs to be equal to the number of variables describing the system, with each axis representing a single system variable. Each point on the trajectory describes a unique state of the system. Such a trajectory could be constructed from observations made on experimental nonlinear systems if it were possible to take measurements of each variable in time. In most cases this is not possible due to experimental limitations. Typically, measurements of only one system variable in time are available. In many cases even the actual number of system variables for a particular nonlinear system is unknown.

It has been shown that a ‘reconstructed’ attractor can be formed using the embedding method [31

31. 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.

], which has all the same properties as the ‘true’ attractor. The method involves taking a single scalar measurement, in this case the intensity of the laser output power as a function of time 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,

A(t)=[P(t),P(t+T),P(t+2T),,P(t+(m1)T)].
(1)

The value of time delay 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]

]. Constraints on the choice of 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]

]. There have been many other techniques suggested, see for example [34

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

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

], however none have been shown to perform significantly better than others and all give values of similar magnitude [23

23. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2004).

].

The nonlinear time-series measurand used in this work, the correlation dimension, can be calculated using the Grassberger-Procaccia algorithm [29

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

]. This involves calculation of the correlation sum defined as

Cm(r)=limN2N(N1)jNj=i+1NH(ryiyj),
(2)

where ∥yiyj∥ is the distance between pairs of points in the phase space and 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.

In theory the density of the points will scale as a power law with the radius

limNlimr0Cm(r)=rD.
(3)

The correlation dimension is then given by

D=limNlimr0logCm(r)logr.
(4)

So the gradient of a graph of log Cm(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, Dm(r) = log Cm(r)/ log r, as a function of r makes observing this region much clearer. The value of the slope Dm(r) within the scaling region is the correlation dimension.

The graphs in Fig. 1 show the correlation sum, Cm(r), and corresponding slopes, Dm(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]

]. Looking between radius values of 3 and 10.8, the curves of Dm(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.

Fig. 1. The graph in (a) shows correlation sum as a function of radius and (b) shows the gradient of the correlation sum as a function of radius for values of the embedding dimension m between 2 and 12. Time series analyzed is the x-variable data from the chaotic Lorenz equations [36].

The value of the correlation dimension characterizes the shape of a chaotic attractor [29

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

]. A value of 1 is representative of periodic or limit-cycle dynamics and a value of 2 arises from quasi-periodic data that generates a torus in the phase space. A higher fractional value of correlation dimension indicates the phase space trajectory has a higher level of complexity, arising from a chaotic system, where the attractor is bounded within a region of phase space but does not have an exact geometrical shape. A purely stochastic signal will have no structure and will completely fill the phase space, leading to an infinitely increasing value of Dm(r).

When performing this type of analysis on experimental data care must be taken in the interpretation of the results. It has been shown that naïve implementation of the Grassberger and Procaccia algorithm [29

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

] can lead to serious underestimation of dimension value [37

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 47, 2289–2297 (1993). [CrossRef]

]. The two most important factors to take into consideration are (i) the removal of temporally correlated points from the correlation sum calculation [38

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

], and (ii) the requirement that a comparison of results with surrogate data sets must also be performed [39

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 58, 77–94 (1992). [CrossRef]

].

In order to achieve the first of these, the correlation sum (Eq. 2) needs to be slightly modified so that points in the phase space that were close in time are omitted from the calculation by the introduction of a Theiler window [38

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

]. The modified equation is

Cm(r)=limN2(Nw)(Nw1)iNj=i+1+wNH(ryiyj),
(5)

where w is the value of the Theiler window. This omission is especially important when dealing with flow-like data since pairs of points that are close in time are generally close in phase space as well. These temporally correlated pairs introduce a bias in the correlation sum calculation. Different authors suggest different methods of determining the size of w. A mathematically rigorous approach is to use a space-time separation plot which allows the minimum value of w to be determined by looking at the number of pairs of points as a function of both time separation and spatial distance [40

40. 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]

]. In this work, since there are a large number of data points and the dynamics are highly sampled, the size of the Theiler window was set generously at 100 points to virtually eliminate the use of temporally correlated points in the correlation sum. This is likely an over estimate, since the correlation times of all of the time series are less than 10 data points (determined as the number of points for the autocorrelation function to drop to 1/e). The effect of using a value larger than required is a small loss in statistics (about 2w/N) due to using fewer points in the correlation sum [23

23. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2004).

]. This can cause an increase in uncertainty in the value of the correlation dimension. However, in this case the loss of 4% of “point pairs” has no significant impact on the results. Omission of the space time separation plot from our automated system results in a simpler, more computationally efficient process. It was not required in this instance because there was sufficient data density captured from the OISSL system. The “space-time separation” step could be included in the automated computations for time-series data sets where the time series are short and where loss of data points needs to be minimized.

It has also been shown that the correlation dimension analysis can give spuriously low dimension values for linearly filtered random numbers [37

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 47, 2289–2297 (1993). [CrossRef]

]. Thus, a method is required to distinguish between linearly correlated noise and nonlinear structure. The use of surrogate data solves this problem and is regarded as a necessary tool for validating the results of nonlinear time-series analysis techniques [39

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 58, 77–94 (1992). [CrossRef]

]. The idea is that a null hypothesis is specified which then needs to be tested. In this case, the hypothesis would be that the data set being analyzed is from a linearly filtered stochastic process. If this null hypothesis can be rejected then we can have confidence that the data is truly from a deterministic nonlinear process.

The method of discriminating truly nonlinear dynamics involves performing the dynamical analysis, in this case evaluation of the correlation dimension, on the original set of data to obtain a result CDorig. 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, <CDsurr>, is then compared to CDorig to see if they are significantly different.

The surrogates data sets themselves can be formed in many ways [41

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

]. In this work we use random phase surrogates. These are generated by taking the Fourier transform of the original data set, randomizing the phase, and performing the inverse Fourier transform. This results in the surrogates having the same power spectrum and autocorrelation function as the original data. If the results are significantly different then the null hypothesis is rejected and we can have confidence in the correlation dimension value for the original data set. The issue of how many surrogates to use has no simple solution. Statistically, the more surrogates used the better, however the fact that each measurement takes a significant amount of time means one has to balance the accuracy of the analysis with the time required to perform it.

3. Experiment and analysis

The experimental system outlined in Section 3.1 is fully described in [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]

, 42

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]

]. The automated computational system used to analyze this data is described in Section 3.2 and is based on the correlation dimension algorithm [29

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

]. The model for this laser system used to generate simulated data is discussed in Section 3.3 and fully described in [5

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

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

The experimental setup of the optically injected Nd:YVO4 solid state laser system can be seen in Fig. 2. Two 1064 nm solid state Nd:YVO4 lasers were operated in a master-slave configuration. The Nd:YVO4 crystals (CASIX) are 1 mm thick with 1% Nd3+ 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.

Fig. 2. Experimental setup: LD, laser diode; FI, Faraday isolator; TEC, temperature control; IF, interference filter; BS, beam splitter; AOM, acousto-optic modulator; FP, Fabry-Pérot interferometer; PD, photodetectors. Figure reproduced from [21].

The power of the injected beam was modulated using the AOM and the temperature of the master laser was tuned with the Peltier element to control the frequency detuning between the lasers (Δω). 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.

For a fixed detuning, long time-series data of the laser intensity were recorded as the injection strength (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 (Δω).

3.2 Automated Computational Process

An automated computation system was implemented so that the correlation dimension could be mapped at every point in the (K, Δω) plane. The algorithm was coded using a combination of both Matlab and C++ programming languages.

The processing steps involved in the automated computation are shown in Fig. 3. The raw experimental data consists of 315 slave laser intensity time-series, each corresponding to a normalized frequency detuning value Δω 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.

The first step in the automated process was to load a complete time-series for one fixed detuning value. This was then divided up into 176 smaller subsets of 5000 points or 50 μs. Since the sweep time of the injection strength is on a much longer scale (approx. 9 ms), each subset corresponds to practically constant injection strength. The largest change in injection strength for a time-series-subset occurs at maximum detuning. Here the injection strength sweeps from K = 0 to 3.9, resulting in a maximum change of ΔK = 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.

Fig. 3. Structure diagram of the process of the automated computational system.

Each subset was then analyzed individually. First, the time lag 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]

]. Typically, values for this time lag were less than 15 data points. An attractor was then mapped out for embedding dimensions 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, Dm(r), was generated.

Fig. 4. Experimental laser intensity time-series trace and corresponding injection strength sweep (inset) for detuning Δω = −1.0.

Once Dm(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]

]. This involves generating a histogram of how many times each gradient value is obtained, in all embeddings, and locating the peak value. The width of the peak in the histogram is an appropriate way to quantify confidence in the scaling region. A narrow peak indicates a significant plateau whereas a wider peak results from a plateau that is not as well defined, as seen in Fig. 5.

Fig. 5. Plots of Dm(r) and corresponding histograms for examples of (a) ‘good’ and (b) ‘bad’ scaling regions.

If a scaling region was present then the value of Dm(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, Δω) plane. The final dynamical map is shown in Fig. 6.

The final step in the automated system repeats the whole process shown in Fig. 3 several times, each time replacing the 50 μs time-series with a different surrogate data set. The surrogates were created by taking the Fast Fourier Transform (FFT) of the original 50 μs time-series, randomizing the phases and taking the inverse FFT. This gives a surrogate time-series that has the same spectral content as the original time-series, but with the determinism (if any) removed. A total of 10 surrogate data maps were averaged to produce a map for comparison with the original experimental map in Fig. 6. The different dynamical regions identified as I, Locked; II, Periodic; III, ‘Spiky’ Output; IV, Chaotic and V, Noisy, are discussed in Section 4.

3.3 Theoretical Model of Solid State Laser Subject to Optical Injection

A theoretical model of the specific Nd:YVO4 solid-state laser subject to optical injection has been described in [5

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

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

], and is reproduced here:

dadt=[12(1)γcγnγsĴ(n1)12γp(a21)+iΩ]a+κ+Fa
dndt=γs(1n)+γsĴ(1a2)+γna2(1n)+γpγsĴγca2(a21)+Fn.
(6)

Table 1. Parameter values used in the simulations*

table-icon
View This Table

Using the parameter values shown in Table 1 (reproduced from [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]

]), simulated time-series were generated, and the same automated correlation dimension analysis was performed. The correlation dimension map is shown in Fig. 7.

Fig. 6. Correlation dimension map in the (K, Δω) plane for experimental data. Different dynamical region are identified as I: Locked (no data), II: Periodic (CD = 1), III: ‘Spiky’ Output (CD < 1), IV: Chaotic (CD > 2), V: Noisy (CD = ∞).
Fig. 7. Correlation dimension map in the (K, Δω) plane for simulated data without noise.

3.4 Dynamical Maps Using Automated Correlation Dimension Analysis

The automated correlation dimension map method is computationally intensive but does allow maps to be generated that would be practically impossible if they were to be done manually. The actual computation time varies with data set size, maximum embedding dimension and number of time-series analyzed. For example, in this work, using a PC with modest specifications (Pentium 4 3.6 GHz CPU, 3 GB RAM), with data set size of 5000 points, maximum embedding dimension of 10, and 55 440 time-series analyzed, the generation of a complete map took approximately 315 hours to compute. Parallel processing using several PC’s meant the ‘actual’ time to generate a map was only a fraction of this. The same map to be manually constructed would require a person to view 55 440 individual graphs and visually locate plateau regions to record the dimension. Quicker computation could be effected using a more powerful computer and/or carrying out further optimization of the algorithm coding.

Fig. 8. (a) Experimental (K, Δω) map of intensity time-series maxima. Reproduced from [21]. (b) Theoretical bifurcation diagram in the (K, Δω) plane. Showing Hopf (H) bifurcations in red, saddle-node (SN) bifurcations in blue, torus (T) bifurcations in black and period doubling (PD) bifurcations in green. Reproduced from [22].

4. Discussion

The map in Fig. 6 shows the value of the correlation dimension at every point in the (K, Δω) 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.

Figure 9 shows samples of the output power time-series, plots of gradient Dm(r) and the corresponding histograms for each of the different dynamical regions (II-V) in the (K, Δω) 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.

Fig. 9. Typical output power time-series, graphs of Dm(r) versus r and corresponding histograms for regions (a) II, (b) III, (c) IV and (d) V in the (K, Δω) plane.

The automated computation correctly assigned a correlation dimension value of ~1, indicating a limit cycle in the phase space, to the periodic time-series seen in Region II.

In Region III the time-series are characterized by large amplitude spikes, which give rise to values for correlation dimension of 0.1 – 0.3. Within this region, the CD analysis appears to differ depending on the value of the embedding delay T used. When the program detects the first minimum of the mutual information as 1 data point, the Dm(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 Dm(r) graph obtained is similar to that seen in Fig. 10(b).

Clearly, a longer delay results in an incorrect detection of a scaling region, as in Fig. 10(b). This can be detected by looking at the width of the peak which is larger when this erroneous detection occurs. A map showing the value of the width of the scaling region peak is shown in Fig. 11. From this map it can be concluded that many instances of this are occurring within Region III as the scaling region peak width in this region is generally larger than in the other regions identified. In this case we cannot claim a correlation dimension was obtained since no true scaling region is observed. However, even though the values obtained in this region are not a true CD, the automated process does generate a low CD estimate (with a wide peak) reliably which can be identified as a consistent signature to identify the type of ‘spiky’ dynamic that occurs from the OISSL system for parameter values of Region III.

Fig. 10. Typical output power time-series, graphs of Dm(r) versus r and corresponding histograms (showing scaling region peak width) for data from region III with values of embedding delay (a) T = 1 data point, and (b) T = 7 data points.
Fig. 11. Map of the width of scaling region peaks for experimental OISSL data in the (K, Δω) plane.

Correct scaling region detection does occur with delay of 1 data point, as in Fig. 10(a), and in this case low values of CD are still obtained consistently. It is uncertain what such a low dimension estimate means. The output power time-series from this region shows large amplitude spikes that resemble a noisy mode locked laser. In theory an ideal mode-locked laser output would be represented by a zero dimensional point in the phase space. Correspondingly, the correlation dimension analysis may show a small, noise induced, nonzero dimension estimate. Thus, we hypothesize that correlation dimension analysis is potentially a useful tool for analyzing mode-locked lasers where dimension 0 is an ideal mode-locked laser and the magnitude of the fractional CD may be an indicator, and possibly a measure of departure from ideality. This hypothesis will need to be further investigated.

It is of interest to also look at other features of the Region III dynamics, such as the spike amplitude and frequency. This is the subject of further research.

Note that the authors recognize that correlation dimension analysis is not designed to be used with highly correlated, periodic data. However, when dealing with systems that switch between periodic and chaotic nonlinear dynamics with small parameter changes it is unavoidable if a single measurand is to be used to describe the nature of the data. It is possible the very low CD estimates seen in some regions of the map could be aberrative due to the dimensional analysis being used on data for which it is not appropriate. Despite this, the technique still gives consistent results and can identify different regions - this is significant, even if the actual value of the CD for region III may not be physically meaningful.

Regions of higher correlation dimension, Region IV in Fig. 6, generally appear for negative detunings at the transition between Regions II and III, and also between Regions III and V. These higher values of CD, in the range 2 to 2.5, indicate a chaotic output of the laser system. The complexity of this chaos is fairly low dimensional compared to other nonlinear laser systems such as a semiconductor laser subject to optical feedback which typically show correlation dimension values of between 3 and 4 [44

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]

]. This region could be of interest for chaotic secure communication schemes.

The surrogate data map was constructed by averaging 10 maps generated with surrogate time-series data sets. Inspection of each of the 10 individual surrogate maps reveals that no structure, such as that seen for the original data, is present. For the vast majority of parameter values, the automated program does not detect scaling regions in the Dm(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, Δω) 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, Δω) 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.

The correlation dimension map generated from the simulated data, shown in Fig. 7, closely resembles the map generated using experimental data. The model correctly predicts Region I where the output is locked and stable. The low CD, Region III, is present for negative detunings and shows the same behavior as the experimental data when the embedding delay is changed. The high CD, chaotic Region IV is also identified in the same parameter space as in the experimental map. As to discrepancies, the main difference is in the low injection strength region where the simulated data map shows no definition between Regions II and V. This arises due to the fact that this whole region is dominated by periodic oscillations – correctly identified in the simulated map as CD = 1. The reason a distinction is seen in the experimental map is due to the signal-to-noise levels. The map of time-series intensity in Fig. 8(b) shows that the two crescent shapes, corresponding to Region II, have larger amplitude than the surrounding region. In this low amplitude region, the unavoidable noise contained in the experimental time-series data results in a signal-to-noise level too low to allow the scaling region to be identified, therefore making it impossible to determine a value for the CD. Since the theoretical model does not include any noise, the small amplitude periodic oscillations of Region V can be successfully analyzed and assigned a value for CD.

On comparison with the bifurcation diagram from [22

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]

], reproduced in Fig. 8(a), it can be seen that all the identified regions I-V are separated by bifurcations lines. A period-doubling bifurcation surrounds Region II, whilst a torus bifurcation separates Region III from V. Hopf and saddle-node bifurcation lines occur at the boundaries of Regions I and III for negative detuning, and between Regions I and V for positive detunings. The chaotic dynamics seen in Region IV primarily occur at the higher injection end of the period-doubling bifurcation and along the torus bifurcation, both for negative detuning.

5. Conclusion

An automated computation process was developed to generate a dynamic map of the correlation dimension of the output power time series from an optically injected Nd:YVO4 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

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

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]

] and gives further insight in to the level of complexity of the different nonlinear dynamics produced by this system. Maps generated for both experimental and simulated laser outputs show similar structure with discrepancies only arising due to signal-to-noise limitations in the experimental data. The automated process could easily be implemented to analyze other complex systems, from which experimental time-series can be measured, for large ranges of relevant parameter values.

Acknowledgments

This research is supported by an Australian Research Council Discovery Project grant and Macquarie University. J. Toomey is supported by an Australian Postgraduate Award PhD scholarship. Dr Thomas Fordell (formerly of Department of Physics, University of Helsinki, now at Lund Laser Centre, Lund University, Lund, Sweden) is gratefully acknowledged for assistance with generating the simulated data and transferring it to Macquarie University.

References and links

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]

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]

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]

18.

S. Eriksson and A. M. Lindberg, “Observations on the dynamics of semiconductor lasers subjected to external optical injection,” J. Opt. B 4, 149–154 (2002). [CrossRef]

19.

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]

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 416, 1–128 (2005).

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]

23.

H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2004).

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]

28.

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]

29.

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

30.

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

31.

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.

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]

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]

36.

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

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 47, 2289–2297 (1993). [CrossRef]

38.

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

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 58, 77–94 (1992). [CrossRef]

40.

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]

41.

T. Schreiber and A. Schmitz, “Surrogate time series,” Physica D 142, 346–382 (2000). [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]

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]

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]

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


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References

  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]
  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 ?m," 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," Electron. 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]
  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 Semiclassic. Opt. 9, 765-784 (1997). [CrossRef]
  18. 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]
  19. 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]
  20. S. Wieczorek, B. Krauskopf, T. B. Simpson, and D. Lenstra, "The dynamical complexity of optically injected semiconductor lasers," Phys. Rep.  416, 1-128 (2005).
  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]
  23. H. Kantz and T. Schreiber, Nonlinear Time Series Analysis (Cambridge University Press, Cambridge, 2004).
  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]
  28. 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]
  29. P. Grassberger and I. Procaccia, "Measuring the strangeness of strange attractors," Physica D 9, 189-208 (1983). [CrossRef]
  30. T. Fordell and A. M. Lindberg, "Numerical stability maps of an optically injected semiconductor laser," Opt. Commun. 242, 613-622 (2004). [CrossRef]
  31. 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.
  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]
  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]
  36. E. N. Lorenz, "Deterministic Nonperiodic Flow," J. Atmos. Sci. 20, 130-141 (1963). [CrossRef]
  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 47, 2289-2297 (1993). [CrossRef]
  38. J. Theiler, "Spurious dimension from correlation algorithms applied to limited time-series data," Phys. Rev. A 34, 2427-2432 (1986). [CrossRef] [PubMed]
  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 58, 77-94 (1992). [CrossRef]
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