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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 4705–4713
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Experimental and numerical study of the symbolic dynamics of a modulated external-cavity semiconductor laser

Andrés Aragoneses, Taciano Sorrentino, Sandro Perrone, Daniel J. Gauthier, M. C. Torrent, and Cristina Masoller  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 4705-4713 (2014)
http://dx.doi.org/10.1364/OE.22.004705


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Abstract

We study the symbolic dynamics of a stochastic excitable optical system with periodic forcing. Specifically, we consider a directly modulated semiconductor laser with optical feedback in the low frequency fluctuations (LFF) regime. We use a method of symbolic time-series analysis that allows us to uncover serial correlations in the sequence of intensity dropouts. By transforming the sequence of inter-dropout intervals into a sequence of symbolic patterns and analyzing the statistics of the patterns, we unveil correlations among several consecutive dropouts and we identify clear changes in the dynamics as the modulation amplitude increases. To confirm the robustness of the observations, the experiments were performed using two lasers under different feedback conditions. Simulations of the Lang-Kobayashi (LK) model, including spontaneous emission noise, are found to be in good agreement with the observations, providing an interpretation of the correlations present in the dropout sequence as due to the interplay of the underlying attractor topology, the external forcing, and the noise that sustains the dropout events.

© 2014 Optical Society of America

1. Introduction

Inferring signatures of determinism in stochastic high-dimensional complex systems is a challenging task, and much effort is focused on developing efficient and computationally fast methods of time-series analysis that are useful even in the presence of high levels of noise [1

1. C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88, 174102 (2002). [CrossRef] [PubMed]

6

6. A. Aragoneses, N. Rubido, J. Tiana-alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013). [CrossRef] [PubMed]

]. In optics, a long standing discussion about the roles of stochastic and deterministic nonlinear processes comes from the dynamics of semiconductor lasers with optical feedback. Their dynamical behavior has been studied for decades and is still the object of intense research, allowing for the observation of a great variety of phenomena [7

7. D. Lenstra, B. H. Verbeek, and A. J. Den Boef, “Coherence collapse in single-mode semiconductor-lasers due to optical feedback,” IEEE J. Quantum. Electron. 21, 674–679 (1985). [CrossRef]

10

10. S. Donati and R-H Horng, “The diagram of feedback regimes revisited,” IEEE J. Sel. Top. Quantum Electron. 19, 1500309 (2013). [CrossRef]

], including excitability [11

11. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414 (1997). [CrossRef]

, 12

12. A. M. Yacomotti, M. C. Eguia, J. Aliaga, O. E. Martinez, and G. B. Mindlin, “Interspike time distribution in noise driven excitable systems,” Phys. Rev. Lett. 83, 292 (1999). [CrossRef]

], regular pulses [13

13. T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,” Phys. Rev. Lett. 87, 243901 (2001). [CrossRef] [PubMed]

, 14

14. A. Tabaka, K. Panajotov, I. Veretennicoff, and M. Sciamanna, “Bifurcation study of regular pulse packages in laser diodes subject to optical feedback,” Phys. Rev E 70, 036211 (2004). [CrossRef]

], extreme pulses and intermittency [15

15. J. A. Reinoso, J. Zamora-Munt, and C. Masoller, “Extreme intensity pulses in a semiconductor laser with a short external cavity,” Phys. Rev. E 87, 062913 (2013). [CrossRef]

], quasiperiodicity [16

16. S. D. Cohen, A. Aragoneses, D. Rontani, M. C. Torrent, C. Masoller, and D. J. Gauthier, “Multidimensional subwavelength position sensing using a semiconductor laser with optical feedback,” Opt. Lett. 38, 4331 (2013). [CrossRef] [PubMed]

] and chaos [17

17. L. Junges, T. Pöschel, and J. A. C. Gallas, “Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang–Kobayashi model,” Eur. Phys. J. D. 67, 149 (2013). [CrossRef]

].

A particular dynamical behavior occurs for moderate feedback near the solitary laser threshold, and is referred to as low-frequency fluctuations (LFFs) [18

18. D. W. Sukow, J. R. Gardner, and D. J. Gauthier, “Statistics of power-dropout events in semiconductor lasers with time-delayed optical feedback,” Phys. Rev. A. 56, R3370 (1997). [CrossRef]

25

25. K. Hicke, X. Porte, and I. Fischer, “Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback,” Phys. Rev. E 88, 052904 (2013). [CrossRef]

]. In the LFF regime, the laser output intensity displays irregular, apparently random and sudden, dropouts. In particular, the LFF dynamics has been studied in detail when the laser current is periodically modulated [26

26. D. Baums, W. Elsässer, and E. O. Göbel, “Farey tree and Devil’s staircase of a modulated external-cavity semiconductor laser,” Phys. Rev. Lett. 63, 155 (1989). [CrossRef] [PubMed]

, 27

27. J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18, 1695516972 (2010). [CrossRef]

], not only because the LFFs can be controlled via current modulation [28

28. Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995). [CrossRef]

], but also, from a complex systems perspective, because the interplay of nonlinearity, noise, periodic forcing and delayed feedback leads to entrainment and synchronization [29

29. D. W. Sukow and D. J. Gauthier, “Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current,” IEEE J. Quantum Electron. 36, 175 (2000). [CrossRef]

, 30

30. W-S Lam, N. Parvez, and R. Roy, “Effect of spontaneous emission noise and modulation on semiconductor lasers near threshold with optical feedback,” Int. J. of Modern Phys. B 17, 4123–4138 (2003). [CrossRef]

], providing a controllable experimental setup for studying these phenomena. In addition, because the LFF dynamics is excitable, the influence of external forcing has also attracted attention from the point of view of improving our understanding of how excitable systems respond to external signals to encode information [31

31. J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, “Dynamics of periodically forced semiconductor laser with optical feedback,” Phys. Rev. E 63, 066218 (2001). [CrossRef]

35

35. T. Schwalger, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and B. Lindner, “Interspike-interval correlations induced by two-state switching in an excitable system,” Europhys. Lett. 99, 10004 (2012). [CrossRef]

].

By using ordinal analysis applied to experimentally recorded sequences of inter-dropout intervals (IDIs), we identify clear changes in the symbolic dynamics as the modulation amplitude increases. Specifically, our analysis uncovers the presence of serial correlations in the sequence of dropouts, and reveals how they are modified by the amplitude of the external forcing. To demonstrate the robustness and the generality of the observations, the experiments are performed with two lasers under different feedback conditions.

We also show that simulations of the Lang-Kobayashi (LK) model [36

36. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347 (1980). [CrossRef]

] are in good qualitative agreement with the experimental observations. While the LK model has been shown to adequately reproduce the main statistical features of the LFF dynamics, such as the IDI distribution with and without modulation [19

19. J. Mulet and C. R. Mirasso, “Numerical statistics of power dropouts based on the Lang–Kobayashi mod,” Phys. Rev. E 59, 5400 (1999). [CrossRef]

, 29

29. D. W. Sukow and D. J. Gauthier, “Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current,” IEEE J. Quantum Electron. 36, 175 (2000). [CrossRef]

, 30

30. W-S Lam, N. Parvez, and R. Roy, “Effect of spontaneous emission noise and modulation on semiconductor lasers near threshold with optical feedback,” Int. J. of Modern Phys. B 17, 4123–4138 (2003). [CrossRef]

], it has also been shown that the LFFs are noise sustained in the LK model [23

23. A. Torcini, S. Barland, G. Giacomelli, and F. Marin, “Low-frequency fluctuations in vertical cavity lasers: experiments versus Lang–Kobayashi dynamics,” Phys. Rev. A 74, 063801 (2006). [CrossRef]

, 24

24. J. Zamora-Munt, C. Masoller, and J. Garcia-Ojalvo, “Transient low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 81, 033820 (2010). [CrossRef]

] (i.e., the LFFs are a transient dynamics that dies out when the trajectory finds a stable cavity mode, in deterministic simulations of the LK model). Therefore, it is remarkable that, in spite of the fact that the inclusion of noise is required for simulating sustained LFFs, the model adequately reproduces the symbolic dynamics, and in particular, the correlations present in the sequence of dropouts, and how they vary with the modulation amplitude.

2. Experimental setup and LFF dynamics with current modulation

The experimental setup is shown in Fig. 1: we perform the experiments with a laser emitting at 650 nm with free-space feedback provided by a mirror, and with a laser emitting at 1550 nm, with feedback provided by an optical fiber.

Fig. 1 Experimental setup for (a) 650 nm laser (Hitachi HL6714G) and (b) 1550 nm laser (Mitsubishi ML925B45F). LD stands for laser diode, BS for beam-splitter and OSC for oscilloscope.

For the 650 nm laser, the external cavity is 70 cm (giving a feedback time delay of 4.7 ns) and the feedback threshold reduction is 8%. A 50/50 beam-splitter sends light to a photo-detector (Thorlabs DET210) connected with a 1 GHz oscilloscope (Agilent DSO 6104A). The solitary threshold is 38 mA and the current and temperature (17 C) are stabilized with an accuracy of 0.01 mA and 0.01 C, respectively, using a controller (Thorlabs ITC501). Through a bias-tee in the laser head, a sinusoidal RF component from a leveled waveform generator (HP Agilent 3325A) is combined with a constant dc current of 39 mA. The modulation frequency is fmod = 17 MHz and the modulation amplitude varies from 0 mV to 78 mV in steps of 7.8 mV (from 0% to 4% of the dc current in steps of 0.4%). For each modulation amplitude, five measurements of 3.2 ms were recorded. The time series contains between 74,000 and 207,000 dropouts, at low and high modulation amplitude, respectively.

For the 1550 nm laser, the time delay is 25 ns and feedback threshold reduction is 10.7%. The solitary threshold is 11.20 mA, the dc value of the pump current is 12.50 mA, the modulation frequency is fmod = 2 MHz and the modulation amplitude varies from 0 mV to 150 mV in steps of 10 mV (from 0% to 24% of the dc current in steps of 1.6%). The time series contain between 8,000 and 19,000 dropouts, at low and high modulation amplitude, respectively. While, for the 1550 nm laser, the modulation frequency is about one order of magnitude smaller than for the 650 nm laser, the relation with the characteristic time-scale of the LFF dynamics, given by the average inter-dropout interval 〈ΔT〉 is about the same: for the 650 nm laser, 〈ΔT〉 = 365 ns and thus 〈ΔT〉 × fmod = 6.2. For the 1550 nm laser, 〈ΔT〉 = 2.55 μs and 〈ΔT〉 × fmod = 5.1.

Fig. 2 Time traces of the laser intensity (650 nm laser), probability distribution functions (PDFs) of the inter-dropout intervals, ΔTi (IDIs), and return maps (ΔTi+1 vs. ΔTi) in units of the modulation period (Tmod) for increasing modulation amplitude: from top to bottom, No modulation, 23.4 mV (1.2%), 31.2 mV (1.6%), and 39.0 mV (2%). In panel (a) the words ’10’ (D = 2) and ’210’ (D = 3) are depicted as examples; in panel (c) the transition for ’10 → 01’ is depicted as example (see text for details).

3. Lang and Kobayashi model and method of symbolic time-series analysis

The Lang and Kobayashi (LK) rate equations for the slowly varying complex electric field amplitude E and the carrier density N are given by [36

36. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347 (1980). [CrossRef]

]
dEdt=12τp(1+α)(G1)E+ηE(tτ)eiω0τ+2βspξ,
(1)
dNdt=1τN(μNG|E|2),
(2)
where τp and τN are the photon and carrier lifetimes respectively, α is the line-width enhancement factor, G is the optical gain, G = N/(1 + ε|E|2) (with ε being a saturation coefficient), μ is the pump current parameter, η is the feedback strength, τ is the feedback delay time, ω0τ is the feedback phase, and βsp is the noise strength, representing spontaneous emission.

For simulating the dynamics with current modulation, the pump current parameter is μ = μ0 + a sin(2πfmodt), where a is the modulation amplitude, fmod is the modulation frequency, and μ0 is the dc current. Simulations of 2 ms were performed. The intensity time-series were averaged over a moving window of 1 ns to simulate the bandwidth of the experimental detection system. The averaged time series contained between 12,000 and 30,000 dropouts for low and high modulation amplitude, respectively. The best agreement with the dynamics found in the experimental data was for μ0 = 1.01, fmod = 21 MHz, ε = 0.01, k = 300 ns−1, τ = 5 ns, γ = 1 ns−1, βsp = 10−4 ns−1, η = 10 ns−1, and α = 4. For these parameters 〈ΔT〉 = 127 ns and 〈ΔT〉 × fmod = 2.7.

In order to select the optimal length of the words for the analysis, we need to consider the length of the correlations present in the time-series: if D is much longer than the correlation length, most words will appear in the sequence with similar probabilities. In addition, we need to consider the length of the time-series, because the number of possible words increases with D as D!, and for large D values, long time series will be needed for computing the word probabilities with robust statistics. Here, we recorded long time series of dropouts and the main limitation for the value of D comes from the large level of stochasticity of the LFF dynamics, which results in correlations among only few consecutive dropouts. Thus, we limit the ordinal analysis to D = 2 and D = 3 words. We will show that the LFF symbolic dynamics is such that the analysis with words of D = 3 allows us to uncover correlations which are not seen with D = 2 words.

From the sequence of words, additional information can be extracted by computing the transition probabilities (TPs) [3

3. N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: experiments and model comparison,” Phys. Rev. E 84, 026202 (2011). [CrossRef]

] from one word to the next. In Fig. 2(d), the transition 10 → 01 is depicted as example. With D = 2 words, the TP analysis can uncover correlations among five consecutive dropouts, and thus allows us to extract information about the memory of the system in a longer time scale. The TPs can be normalized in two different ways: normalized for all transitions, such that the sum of all possible TPs is one, ∑i,j TPij = 1, or normalized for each word i, such that ∑j TPij = 1 ∀ i. We compute these two sets of TPs corresponding to D = 2 words. For the first set, the TPs are normalized such that TP(01→01)+TP(01→10) = 1 and TP(10→01)+TP(10→10) = 1. For the second set: TP(0101)*+TP(0110)*+TP(1001)*+TP(1010)*=1, and TP(0110)*=TP(1001)*.

4. Results

Figure 3 shows the probabilities of words of D = 2 (a, b, c) and D = 3 (d, e, f), vs. the modulation amplitude, for the 650 nm laser (a, d), for the 1550 nm laser (b, e), and for the simulated time series (c, f). The gray region indicates probability values consistent with the null hypothesis (NH) that the words are equally probable, and thus, that there are no correlations among the dropouts. In other words, probability values outside the gray regions are not consistent with a uniform distribution of word probabilities and reveal serial correlations in the IDI sequence. It can be noticed that the gray region is narrower in (a, d) than in (b, e) and (c, f). This is due to the fact that the number of dropouts recorded for the 650 nm laser is much larger than for the 1550 nm laser (the corresponding delay times being 4.7 ns and 25 ns respectively), and is also larger than the number of dropouts in the simulated data.

Fig. 3 Probabilities of the words of D = 2 (a, b, c) and D = 3 (d, e, f) versus the modulation amplitude for the experiment with the 650 nm laser (a, d), the experiment with the 1550 nm laser (b, e), and the numerical simulations of the LK model (c,f). The gray region (p±3σ, where p = 1/D!, σ=p(1p)/N, and N is the number of words in the symbolic sequence) indicates probability values consistent with 95% confidence level with the null hypothesis that all the words are equally probable (i.e., that there are no correlations present in the sequence of dropouts).

It is observed that the dynamics is consistent with the NH, in the case of D = 2, for small and for high modulation amplitude. However, the analysis with D = 3, reveals that, for high modulation, the probabilities are outside the gray region, revealing correlations among four consecutive IDIs. We note that there are two groups of words, one less probable (‘012’, ‘210’) and one more probable (‘021’,‘102’, ‘120’, ‘201’), resulting, for D = 2, in the same probabilities for ’01’ as for ’10’. With D = 3, the less probable words are those which imply three consecutively increasing or decreasing IDIs and this can be understood in the following terms: strong enough modulation forces a rhythm in the LFF dynamics, and three consecutively increasing or decreasing intervals imply a loss of synchrony with the external rhythm, and thus, are less likely to occur.

By computing the four transition probabilities of D = 2 words, depicted in Fig. 4, we obtain information about correlations among five consecutive dropouts. This analysis is statistically more robust than computing the probabilities of the 24 words of length D = 4.

Fig. 4 Transition probabilities for the 650 nm laser (a, d, g), the 1550 nm laser (b, e, h), and the simulated data (c, f, i). The first row indicates the transition probabilities from word ’01’ to ’01’ and ’10’, such that TP(01→01)+TP(01→10) = 1. The second row indicates the transition probabilities from word ’10’ to ’01’ and ’10’, such that TP(10→01)+TP(10→10) = 1, while the third row consideres all four transitions, such that TP(0101)*+TP(0110)*+TP(1001)*+TP(1010)*=1.

The results in Fig. 4 confirm that, at this time scale, the dynamics is still consistent with the NH for low modulation amplitudes but, as the modulation increases, a transition takes places and the TPs display a deterministic-like behavior. This transitions occur at the same values as in Fig. 3 (at about 1.8% modulation amplitude for the 650 nm case, 16% for the 1550nm case, and 6% for the simulated data). Figure 4 shows that, for high modulation amplitude, the most probable transitions are those which go from one word to the same word (‘01 → 01’ and ‘10 → 10’), because the external forcing imposes a periodicity in the LFF dynamics. The transition in the dynamics, and the qualitative agreement between experiments and simulations, are independent of the type of normalization used to compute the TPs.

In Figs. 3 and 4, there is a good qualitative agreement between experiments and simulations. As discussed in the introduction, within the framework of the LK model, the LFF dynamics is sustained by spontaneous emission noise, and thus, one could expect weak correlations in the sequence of dropouts. While this is indeed the case for no modulation or very weak modulation amplitude, larger modulation induces precise correlations, which are adequately reproduced by the LK model. For strong modulation the reason why some words and transitions are more probable than the others is well understood (as due to the external rhythm imposed by the modulation), but for no modulation and also for moderate modulation amplitude, further investigations are needed in order to understand the symbolic behavior. We note that, without modulation, while in the experimental data the word probabilities are within the NH gray region, in the simulated data they are not. This can be due to a number of model parameters that can be tuned in order to obtain a better fit of the symbolic dynamics without modulation (e.g., the feedback stregth, the linewidth enhancement factor, the dc value of the pump current, etc.). However, our goal here is to analyze the influence of the modulation on the LFF symbolic dynamics, and a detailed comparison experiments-simulations in the absence of modulation will be reported elsewhere.

5. Conclusions

We have studied experimentally and numerically the symbolic dynamics of a semiconductor laser with optical feedback and current modulation in the LFF regime. We have analyzed time series of inter-dropout intervals employing a symbolic transformation that allows us to identify clear changes in the dynamics induced by the modulation. For weak modulation the sequence of dropouts is found to be mainly stochastic, while for increasing modulation it becomes more deterministic, with correlations among several consecutive dropouts. We have identify clear changes in the probabilities of the symbolic words and transitions with increasing modulation amplitude. The LK model has also been tested and we have found a good qualitative agreement with the experimental observations. We speculate that the symbolic behavior uncovered here is a fingerprint of the underlying topology of the phase space, and is due to the interplay of noise-induced escapes from an stable external cavity mode, and the dynamics in the coexisting attractor. It would be interesting for a future study to analyze the influence of varying the modulation frequency and the noise strength. It would also be interesting to use an analytic effective potential [18

18. D. W. Sukow, J. R. Gardner, and D. J. Gauthier, “Statistics of power-dropout events in semiconductor lasers with time-delayed optical feedback,” Phys. Rev. A. 56, R3370 (1997). [CrossRef]

], to further understand the mechanisms underlying the symbolic dynamics of the modulated LFFs.

The methodology proposed here can be a useful tool for identifying signatures of determinism in high-dimensional and stochastic complex systems. It provides a computationally efficient way to unveiling structures and transitions hidden in the time series. As the laser in the LFF regime is an excitable system, our results could be relevant for understanding serial correlations in the spike sequences of other forced excitable systems. Also as a future study, an analysis of simpler models of excitable systems (such as the FitzHughNagumo model or the phenomenological LFF model proposed by Yacomotti et al. [12

12. A. M. Yacomotti, M. C. Eguia, J. Aliaga, O. E. Martinez, and G. B. Mindlin, “Interspike time distribution in noise driven excitable systems,” Phys. Rev. Lett. 83, 292 (1999). [CrossRef]

]), can contribute to improve our understanding of the symbolic behavior induced by periodic modulation.

Acknowledgments

This work was supported in part by grant FA8655-12-1-2140 from EOARD US, grant FIS2012-37655-C02-01 from the Spanish MCI, and grant 2009 SGR 1168 from the Generalitat de Catalunya. C. M. gratefully acknowledges partial support from the ICREA Academia programme. D. J. G. gratefully acknowledges the financial support of the U.S. Army Research Office through Grant W911NF-12-1-0099.

References and links

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C. Bandt and B. Pompe, “Permutation entropy: a natural complexity measure for time series,” Phys. Rev. Lett. 88, 174102 (2002). [CrossRef] [PubMed]

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3.

N. Rubido, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and C. Masoller, “Language organization and temporal correlations in the spiking activity of an excitable laser: experiments and model comparison,” Phys. Rev. E 84, 026202 (2011). [CrossRef]

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L. Zunino, M. C. Soriano, and O. A. Rosso, “Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach,” Phys. Rev. E 86, 046210 (2012). [CrossRef]

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M. C. Soriano, L. Zunino, L. Larger, I. Fischer, and C. R. Mirasso, “Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed opto-electronic oscillator,” Opt. Lett. 36, 2212 (2011). [CrossRef] [PubMed]

6.

A. Aragoneses, N. Rubido, J. Tiana-alsina, M. C. Torrent, and C. Masoller, “Distinguishing signatures of determinism and stochasticity in spiking complex systems,” Sci. Rep. 3, 1778 (2013). [CrossRef] [PubMed]

7.

D. Lenstra, B. H. Verbeek, and A. J. Den Boef, “Coherence collapse in single-mode semiconductor-lasers due to optical feedback,” IEEE J. Quantum. Electron. 21, 674–679 (1985). [CrossRef]

8.

K. Lüdge, editor, Nonlineal Laser Dynamics. From Quantum Dots to Cryptography (Wiley-VCH, 2011). [CrossRef]

9.

D. M. Kane and K. A. Shore, eds. Unlocking Dynamical Diversity (John Wiley & Sons, 2005). [CrossRef]

10.

S. Donati and R-H Horng, “The diagram of feedback regimes revisited,” IEEE J. Sel. Top. Quantum Electron. 19, 1500309 (2013). [CrossRef]

11.

M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor lasers with optical feedback,” Phys. Rev. E 55, 6414 (1997). [CrossRef]

12.

A. M. Yacomotti, M. C. Eguia, J. Aliaga, O. E. Martinez, and G. B. Mindlin, “Interspike time distribution in noise driven excitable systems,” Phys. Rev. Lett. 83, 292 (1999). [CrossRef]

13.

T. Heil, I. Fischer, W. Elsäßer, and A. Gavrielides, “Dynamics of semiconductor lasers subject to delayed optical feedback: the short cavity regime,” Phys. Rev. Lett. 87, 243901 (2001). [CrossRef] [PubMed]

14.

A. Tabaka, K. Panajotov, I. Veretennicoff, and M. Sciamanna, “Bifurcation study of regular pulse packages in laser diodes subject to optical feedback,” Phys. Rev E 70, 036211 (2004). [CrossRef]

15.

J. A. Reinoso, J. Zamora-Munt, and C. Masoller, “Extreme intensity pulses in a semiconductor laser with a short external cavity,” Phys. Rev. E 87, 062913 (2013). [CrossRef]

16.

S. D. Cohen, A. Aragoneses, D. Rontani, M. C. Torrent, C. Masoller, and D. J. Gauthier, “Multidimensional subwavelength position sensing using a semiconductor laser with optical feedback,” Opt. Lett. 38, 4331 (2013). [CrossRef] [PubMed]

17.

L. Junges, T. Pöschel, and J. A. C. Gallas, “Characterization of the stability of semiconductor lasers with delayed feedback according to the Lang–Kobayashi model,” Eur. Phys. J. D. 67, 149 (2013). [CrossRef]

18.

D. W. Sukow, J. R. Gardner, and D. J. Gauthier, “Statistics of power-dropout events in semiconductor lasers with time-delayed optical feedback,” Phys. Rev. A. 56, R3370 (1997). [CrossRef]

19.

J. Mulet and C. R. Mirasso, “Numerical statistics of power dropouts based on the Lang–Kobayashi mod,” Phys. Rev. E 59, 5400 (1999). [CrossRef]

20.

M. Sciamanna, C. Masoller, N. B. Abraham, F. Rogister, P. Mégret, and M. Blondel, “Different regimes of low-frequency fluctuations in vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 20, 37 (2003). [CrossRef]

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Y. Hong and K. A. Shore, “Statistical measures of the power dropout ratio in semiconductor lasers subject to optical feedback,” Opt. Lett. 30, 3332 (2005). [CrossRef]

23.

A. Torcini, S. Barland, G. Giacomelli, and F. Marin, “Low-frequency fluctuations in vertical cavity lasers: experiments versus Lang–Kobayashi dynamics,” Phys. Rev. A 74, 063801 (2006). [CrossRef]

24.

J. Zamora-Munt, C. Masoller, and J. Garcia-Ojalvo, “Transient low-frequency fluctuations in semiconductor lasers with optical feedback,” Phys. Rev. A 81, 033820 (2010). [CrossRef]

25.

K. Hicke, X. Porte, and I. Fischer, “Characterizing the deterministic nature of individual power dropouts in semiconductor lasers subject to delayed feedback,” Phys. Rev. E 88, 052904 (2013). [CrossRef]

26.

D. Baums, W. Elsässer, and E. O. Göbel, “Farey tree and Devil’s staircase of a modulated external-cavity semiconductor laser,” Phys. Rev. Lett. 63, 155 (1989). [CrossRef] [PubMed]

27.

J. P. Toomey, D. M. Kane, M. W. Lee, and K. A. Shore, “Nonlinear dynamics of semiconductor lasers with feedback and modulation,” Opt. Express 18, 1695516972 (2010). [CrossRef]

28.

Y. Liu, N. Kikuchi, and J. Ohtsubo, “Controlling dynamical behavior of a semiconductor laser with external optical feedback,” Phys. Rev. E 51, R2697–R2700 (1995). [CrossRef]

29.

D. W. Sukow and D. J. Gauthier, “Entraining power-dropout events in an external-cavity semiconductor laser using weak modulation of the injection current,” IEEE J. Quantum Electron. 36, 175 (2000). [CrossRef]

30.

W-S Lam, N. Parvez, and R. Roy, “Effect of spontaneous emission noise and modulation on semiconductor lasers near threshold with optical feedback,” Int. J. of Modern Phys. B 17, 4123–4138 (2003). [CrossRef]

31.

J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, “Dynamics of periodically forced semiconductor laser with optical feedback,” Phys. Rev. E 63, 066218 (2001). [CrossRef]

32.

F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88, 040601 (2002). [CrossRef] [PubMed]

33.

J. M. Buldú, J. Garcia-Ojalvo, C. R. Mirasso, and M. C. Torrent, “Stochastic entrainment of optical power dropouts,” Phys. Rev. E 66, 021106 (2002). [CrossRef]

34.

J. M. Buldú, D. R. Chialvo, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Ghost resonance in a semiconductor laser with optical feedback,” Europhys. Lett. 64, 178 (2003). [CrossRef]

35.

T. Schwalger, J. Tiana-Alsina, M. C. Torrent, J. Garcia-Ojalvo, and B. Lindner, “Interspike-interval correlations induced by two-state switching in an excitable system,” Europhys. Lett. 99, 10004 (2012). [CrossRef]

36.

R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. 16, 347 (1980). [CrossRef]

OCIS Codes
(140.1540) Lasers and laser optics : Chaos
(140.2020) Lasers and laser optics : Diode lasers
(140.5960) Lasers and laser optics : Semiconductor lasers
(190.3100) Nonlinear optics : Instabilities and chaos

ToC Category:
Nonlinear Optics

History
Original Manuscript: January 21, 2014
Revised Manuscript: February 6, 2014
Manuscript Accepted: February 7, 2014
Published: February 21, 2014

Virtual Issues
Physics and Applications of Laser Dynamics (2014) Optics Express

Citation
Andrés Aragoneses, Taciano Sorrentino, Sandro Perrone, Daniel J. Gauthier, M. C. Torrent, and Cristina Masoller, "Experimental and numerical study of the symbolic dynamics of a modulated external-cavity semiconductor laser," Opt. Express 22, 4705-4713 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4705


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