## Fast random bits generation based on a single chaotic semiconductor ring laser |

Optics Express, Vol. 20, Issue 27, pp. 28603-28613 (2012)

http://dx.doi.org/10.1364/OE.20.028603

Acrobat PDF (950 KB)

### Abstract

The use of the postprocessing method consisting of bitwise Exclusive-OR and least significant bits extraction to generate random bit sequences typically requires two distinct chaotic outputs. While the two signals are, in general, generated using two separated devices, e.g. two Fabry-Perot lasers, a single semiconductor ring laser can be used as an alternative due to its circular symmetry which facilitates lasing in two counterpropagating mode directions. We consider a chaotic semiconductor ring laser and investigate both numerically and experimentally its characteristics for fast random bit generation. In particular, we show that by sampling each directional mode’s output signal using a 8-bit analog-digital converter and through Exclusive-OR operation applied to the two resulting signals (after throwing away 4 most significant bits), we can achieve fast random bit-streams with a bit rate 4 × 10 = 40 Gbit/s, passing the statistical randomness tests. To optimize the system performance, we also study the dependence of randomness on the main system parameters and on noise.

© 2012 OSA

## 1. Introduction

2. W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I , **44**, 521–528 (1997). [CrossRef]

3. T. Stojanovski and L. Kocarev, “Chaos-based random number generators part I: Analysis [cryptography],” IEEE Trans. Circuits Syst. I: Fundam. Theory Applicat. **48**, 281–288 (2001). [CrossRef]

4. R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. **107**, 034103/1–4 (2011). [CrossRef]

5. R. M. Nguimdo and P. Colet, “Electro-optic phase chaos systems with an internal variable and a digital key,” Opt. Express **20**, 25333–25344 (2012). [CrossRef] [PubMed]

*in real time*by directly sampling the output of two chaotic semiconductor lasers with one-bit analog-digital converters (ADCs) [6

6. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics **2**, 728–732 (2008). [CrossRef]

8. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. **45**, 1367–1379 (2009). [CrossRef]

12. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express **18**, 5512–5524 (2010). [CrossRef] [PubMed]

9. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultra high-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. **103**, 024102 (2009). [CrossRef] [PubMed]

13. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. **36**, 4632 (2011). [CrossRef] [PubMed]

12. K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express **18**, 5512–5524 (2010). [CrossRef] [PubMed]

14. C. R. S. Williams, J. C. Salevan, X.-W. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express **18**, 23584–23597 (2010). [CrossRef] [PubMed]

15. R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Commun. **17**, 539–550 (1999) [CrossRef]

11. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics **4**, 58–61 (2010). [CrossRef]

16. M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAsAl-GaAs semiconductor ring lasers: Experiment and model,” IEEE J. Quantum Electron. **39**, 1187–1195 (2003). [CrossRef]

17. J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Elect. **45**, 431–438 (2009). [CrossRef]

18. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, K. Tsuzuki, A. Uchida, and P. Davis, “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Express **19**, 7439–7450 (2011) [CrossRef] [PubMed]

## 2. Dynamics characterization

19. I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability”, Commun. Nonlinear Sci. Numer. Simul. **17**, 4767–4779 (2012). [CrossRef]

20. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. **37**, 2541–2544 (2012). [CrossRef] [PubMed]

16. M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAsAl-GaAs semiconductor ring lasers: Experiment and model,” IEEE J. Quantum Electron. **39**, 1187–1195 (2003). [CrossRef]

19. I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability”, Commun. Nonlinear Sci. Numer. Simul. **17**, 4767–4779 (2012). [CrossRef]

20. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. **37**, 2541–2544 (2012). [CrossRef] [PubMed]

*E*and

_{cw}*E*, and the carrier number

_{ccw}*N*as where the parameters are the linewidth enhancement factor

*α*, renormalized bias current

*μ*, field decay rate

*κ*, carrier inversion decay rate

*γ*, solitary laser frequency

*ω*

_{0}, feedback rate

*η*, delay time

*T*, feedback phase

*ω*

_{0}

*T*, backscattering coefficients

*k*+

_{d}*ik*where

_{c}*k*and

_{c}*k*are the conservative and the dissipative couplings, respectively. The relationship between the theoretical parameters and real-world devices are detailed in [16

_{d}16. M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAsAl-GaAs semiconductor ring lasers: Experiment and model,” IEEE J. Quantum Electron. **39**, 1187–1195 (2003). [CrossRef]

*= 1 −*

_{cw}*s*|

*E*|

_{cw}^{2}−

*c*|

*E*|

_{ccw}^{2}and 𝒢

*= 1 −*

_{ccw}*s*|

*E*|

_{ccw}^{2}−

*c*|

*E*|

_{cw}^{2}where

*s*and

*c*account for the phenomenological self- and cross-saturations, respectively. All the parameters in Eqs. (1)–(3) are needed to reproduce the dynamics encountered in experiments on SRLs [16

**39**, 1187–1195 (2003). [CrossRef]

*c*and

*s*are necessary to get unidirectional emission (i.e emission in only one of the directional modes),

*k*and

_{d}*k*model different reflections on the end-facets of the device, facilitating therefore the emergence of bidirectional emission (i.e emission in both directional modes). The last terms in Eqs. (1) and (2) represent the effect of spontaneous emission noise coupled to the CW/CCW modes [18

_{c}18. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, K. Tsuzuki, A. Uchida, and P. Davis, “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Express **19**, 7439–7450 (2011) [CrossRef] [PubMed]

*D*represents the noise strength expressed as

*D*=

*D*(

_{m}*N*+

*G*

_{0}

*N*

_{0}/

*κ*), where

*D*is the spontaneous emission factor,

_{m}*G*

_{0}is the gain parameter,

*N*

_{0}is the transparent carrier density.

*ξ*(

_{i}*t*) (

*i*=

*cw*,

*ccw*) are two independent complex Gaussian white noises with zero mean and correlation

*D*= 0 (no noise) unless stated otherwise. We consider the following values for the key parameters which are chosen within the range of experimentally accessible values [20

_{m}20. R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. **37**, 2541–2544 (2012). [CrossRef] [PubMed]

*α*= 3.5,

*μ*= 1.75,

*s*= 0.005,

*c*= 0.01,

*κ*= 100 ns

^{−1},

*γ*= 0.2 ns

^{−1},

*ω*

_{0}

*T*= 0,

*k*= 0.033 ns

_{d}^{−1},

*k*= 0.44 ns

_{cw}^{−1},

*T*= 50 ns,

*η*= 2.5 ns

^{−1}. With our parameters, the relaxation period of the free-running SRL is

*τ*

_{R}_{0}determines how fast the intrinsic dynamics of the system changes.

25. R. Vicente, J. Dauden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. **41**, 541–548 (2005). [CrossRef]

*T*≠

*τ*/2 and its near multiples. To illustrate, Fig. 3 shows open windows close to

_{RO}*T*≈

*τ*/2 and its near multiples, evidencing that the dynamics of the system is either periodic or multi-periodic. Thus the system is more stable for

_{RO}*T*≈

*τ*/2 meaning that the interplay between the intrinsic dynamics and the external delay is rather destructive so that the system is not destabilized enough to enter in a chaotic regime.

_{RO}## 3. Random bit generation

*τ*and confirmed in Fig. 2(b) that the bandwidth of the chaos is small for direct random bits extraction at 10 GSamples/s. The reason is that, if the sampling interval is shorter than

_{RO}*τ*, consecutively extracted points lead to the same value most of the time. As a result, some tests fail.

_{RO}8. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. **45**, 1367–1379 (2009). [CrossRef]

*D*= 5 × 10

_{m}^{6}ns

^{−1},

*G*

_{0}= 10

^{−12}m

^{3}s

^{−1}and

*N*

_{0}= 1.4 × 10

^{24}m

^{−3}[18

18. S. Sunada, T. Harayama, K. Arai, K. Yoshimura, K. Tsuzuki, A. Uchida, and P. Davis, “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Express **19**, 7439–7450 (2011) [CrossRef] [PubMed]

## 4. Influence of parameters on random bit generation

*μ*, the feedback rate

*η*, the delay time

*T*and the linewidth enhancement factor

*α*. This can be well understood because each of them directly affects the characteristics of the chaotic signals generated by SRLs. While the delay time signatures can be overcome through the digitization and LSB extraction as discussed in section 3, other parameters have to be within a suitable range.

*μ*,

*η*and

*α*for which sequences with acceptable randomness can be generated. As already mentioned, the increase (decrease) of

*μ*leads to the increase (decrease) of the generated chaos bandwidth. For our parameters, we have found that random bit sequences pass all the NIST test when

*μ*is in the range of 1.7 ≲

*μ*≲ 4. For

*μ*≲ 1.7, the bandwidth is not large enough to generate random numbers at the current bit rate whereas for

*μ*≳ 4 we have found that the system is not chaotic enough. In fact, as

*μ*is increased, the relaxation period decreases, rendering the system more stable. Therefore the current feedback rate is not enough to bring the system into a strongly chaotic regime. Nonetheless, this deteriorating effect for

*μ*≳ 4 can be compensated by increasing the feedback rate so that the system again gets more chaotic. It is worth noting that for two arbitrary values of

*μ*, if the feedback rate is set so that the system operates with the same complexity, better results will be achieved for higher

*μ*because it corresponds to shorter relaxation period, i.e faster intrinsic dynamics. Thus, for random bit generations, it is preferable to operate the SRL with a

*μ*value as high as possible (to get a large bandwidth) and adjust the feedback rate to optimize the system performance.

**37**, 2541–2544 (2012). [CrossRef] [PubMed]

23. D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: A dynamical point of view,” IEEE J. Quantum Electron. **45**, 879–891 (2009). [CrossRef]

^{−1}≲

*η*≲ 2.8ns

^{−1}. Note that for

*η*≲ 2.2ns

^{−1}, the system is not chaotic enough while for ≳ 2.8ns

^{−1}the delay time signature is not completely suppressed by disregarding 4 MSBs.

*α*can also play an important role for the randomness of the sequences as increasing

*α*leads to a larger amplitude-phase coupling rendering the system more chaotic. As a consequence the delay signatures are reduced [20

**37**, 2541–2544 (2012). [CrossRef] [PubMed]

*c*,

*s*and

*k*only slightly influence the randomness of the sequences.

_{c}## 5. Experiments

31. X. Leijtens, “JePPIX: the platform for InP-based photonics,” IET Optoelectronics **5**, 202–206 (2011). [CrossRef]

32. I. V. Ermakov, S. Beri, M. Ashour, J. Danckaert, B. Docter, J. Bolk, X. Leijtens, and G. Verschaffelt, “Semiconductor ring laser with On-Chip Filtered Optical Feedback for discrete wavelength tuning,” IEEE J. Quantum Electron. **48**, 129–136 (2012). [CrossRef]

*I*= 600 mA leads to strong peaks in the autocorrelation and therefore fails the NIST tests. The NIST tests also fail for small feedback strength, e.g

_{OSA}*I*= 200 mA because the system is not chaotic enough. We also recorded the time series at higher injection currents (keeping

_{OSA}*I*= 295 mA as before). The NIST tests failed for this data as the complexity of the chaos is lowered compared to the previously discussed injection current of 127 mA. As the laser noise, detector’s noise and ADC noise are similar for both tested values of the injection current. We can thus conclude that this noise sources are not sufficient to generate random bits, and the chaotic nature of the signal is needed in our system.

_{OSA}## 6. Concluding remarks

8. K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. **45**, 1367–1379 (2009). [CrossRef]

11. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics **4**, 58–61 (2010). [CrossRef]

**45**, 1367–1379 (2009). [CrossRef]

10. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express **18**, 18763–18768 (2010). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | N. Ferguson, B. Schneier, and T. Kohno, |

2. | W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I , |

3. | T. Stojanovski and L. Kocarev, “Chaos-based random number generators part I: Analysis [cryptography],” IEEE Trans. Circuits Syst. I: Fundam. Theory Applicat. |

4. | R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett. |

5. | R. M. Nguimdo and P. Colet, “Electro-optic phase chaos systems with an internal variable and a digital key,” Opt. Express |

6. | A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics |

7. | T. E. Murphy and R. Roy, “The worlds fastest dice, Nat. Photonics |

8. | K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron. |

9. | I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultra high-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. |

10. | A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express |

11. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics |

12. | K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express |

13. | N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. |

14. | C. R. S. Williams, J. C. Salevan, X.-W. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express |

15. | R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Commun. |

16. | M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAsAl-GaAs semiconductor ring lasers: Experiment and model,” IEEE J. Quantum Electron. |

17. | J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Elect. |

18. | S. Sunada, T. Harayama, K. Arai, K. Yoshimura, K. Tsuzuki, A. Uchida, and P. Davis, “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Express |

19. | I. V. Ermakov, G. Van der Sande, and J. Danckaert, “Semiconductor ring laser subject to delayed optical feedback: bifurcations and stability”, Commun. Nonlinear Sci. Numer. Simul. |

20. | R. M. Nguimdo, G. Verschaffelt, J. Danckaert, and G. Van der Sande, “Loss of time-delay signature in chaotic semiconductor ring lasers,” Opt. Lett. |

21. | L. Gelens, S. Beri, G. Van der Sande, G. Mezosi, M. Sorel, J. Danckaert, and G. Verschaffelt, “Exploring multi-stability in semiconductor ring lasers: theory and experiment,” Phys. Rev. Lett. |

22. | N. Jiang, W. Pan, B. Luo, L. Yan, S. Xiang, L. Yang, D. Zheng, and N. Li, “Influence of injection current on the synchronization and communication performance of closed-loop chaotic semiconductor lasers,” Opt. Lett. |

23. | D. Rontani, A. Locquet, M. Sciamanna, D. S. Citrin, and S. Ortin, “Time-delay identification in a chaotic semiconductor laser with optical feedback: A dynamical point of view,” IEEE J. Quantum Electron. |

24. | R. M. Nguimdo, M. C. Soriano, and P. Colet, “Role of the phase in the identification of delay time in semiconductor lasers with optical feedback,” Opt. Lett. |

25. | R. Vicente, J. Dauden, P. Colet, and R. Toral, “Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop,” IEEE J. Quantum Electron. |

26. | M. F. Booth, A. Schremer, and J. M. Ballantyne, “Spatial beam switching and bistability in a diode ring laser,” Appl. Phys. Lett. |

27. | I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “Towards the generation of random bits at terahertz rates based on a chaotic semiconductor laser,” Int. Workshop on Statistical-Mechanical Informatics 1–8 (2010), 5861 (2010). |

28. | A. Argyris, E. Pikasis, S. Deligiannidis, and Dimitris Syvridis, “Sub-Tb/s physical random bit generators based on direct detection of amplified spontaneous emission signals,” J. Lightwave Technol. |

29. | A. Rukhin, J. Soto, J. Nechvatal, M. Smid, E. Barker, S. Leigh, and M. Levenson, “A statistical test suite for random and pseudorandom number generators for cryptographic applications,” Nat. Inst. Standards and Technology, Special Publication 800-22, 2001, Revision 1, 2008 [Online]. Available: http://csrc.nist.gov/publications/nist-pubs/800-22-rev1/SP800-22rev1.pdf. |

30. | S. J. Kim, K. Umeno, and A. Hasegawa, “Corrections of the NIST statistical test suite for randomness,” 2004, arXiv:nlin.CD/0401040v1. |

31. | X. Leijtens, “JePPIX: the platform for InP-based photonics,” IET Optoelectronics |

32. | I. V. Ermakov, S. Beri, M. Ashour, J. Danckaert, B. Docter, J. Bolk, X. Leijtens, and G. Verschaffelt, “Semiconductor ring laser with On-Chip Filtered Optical Feedback for discrete wavelength tuning,” IEEE J. Quantum Electron. |

**OCIS Codes**

(030.6600) Coherence and statistical optics : Statistical optics

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(140.1540) Lasers and laser optics : Chaos

(140.3560) Lasers and laser optics : Lasers, ring

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: October 5, 2012

Revised Manuscript: November 16, 2012

Manuscript Accepted: November 16, 2012

Published: December 10, 2012

**Citation**

Romain Modeste Nguimdo, Guy Verschaffelt, Jan Danckaert, Xaveer Leijtens, Jeroen Bolk, and Guy Van der Sande, "Fast random bits generation based on a single chaotic semiconductor ring laser," Opt. Express **20**, 28603-28613 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-28603

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### References

- N. Ferguson, B. Schneier, and T. Kohno, Cryptography Engineering: Design Principles and Practical Applications (Wiley, 2010).
- W. T. Holman, J. A. Connelly, and A. B. Dowlatabadi, “An integrated analog/digital random noise source,” IEEE Trans. Circuits Syst. I, 44, 521–528 (1997). [CrossRef]
- T. Stojanovski and L. Kocarev, “Chaos-based random number generators part I: Analysis [cryptography],” IEEE Trans. Circuits Syst. I: Fundam. Theory Applicat.48, 281–288 (2001). [CrossRef]
- R. M. Nguimdo, P. Colet, L. Larger, and L. Pesquera, “Digital key for chaos communication performing time delay concealment,” Phys. Rev. Lett.107, 034103/1–4 (2011). [CrossRef]
- R. M. Nguimdo and P. Colet, “Electro-optic phase chaos systems with an internal variable and a digital key,” Opt. Express20, 25333–25344 (2012). [CrossRef] [PubMed]
- A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Kurashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical random bit generation with chaotic semiconductor lasers,” Nat. Photonics2, 728–732 (2008). [CrossRef]
- T. E. Murphy and R. Roy, “The worlds fastest dice, Nat. Photonics2, 714–715 (2008). [CrossRef]
- K. Hirano, K. Amano, A. Uchida, S. Naito, M. Inoue, S. Yoshimori, K. Yoshimura, and P. Davis, “Characteristics of fast physical random bit generation using chaotic semiconductor lasers,” IEEE J. Quantum Electron.45, 1367–1379 (2009). [CrossRef]
- I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultra high-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett.103, 024102 (2009). [CrossRef] [PubMed]
- A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express18, 18763–18768 (2010). [CrossRef] [PubMed]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics4, 58–61 (2010). [CrossRef]
- K. Hirano, T. Yamazaki, S. Morikatsu, H. Okumura, H. Aida, A. Uchida, S. Yoshimori, K. Yoshimura, T. Harayama, and P. Davis, “Fast random bit generation with bandwidth-enhanced chaos in semiconductor lasers,” Opt. Express18, 5512–5524 (2010). [CrossRef] [PubMed]
- N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett.36, 4632 (2011). [CrossRef] [PubMed]
- C. R. S. Williams, J. C. Salevan, X.-W. Li, R. Roy, and T. E. Murphy, “Fast physical random number generator using amplified spontaneous emission,” Opt. Express18, 23584–23597 (2010). [CrossRef] [PubMed]
- R. H. Walden, “Analog-to-digital converter survey and analysis,” IEEE J. Sel. Areas Commun.17, 539–550 (1999) [CrossRef]
- M. Sorel, G. Giuliani, A. Sciré, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAsAl-GaAs semiconductor ring lasers: Experiment and model,” IEEE J. Quantum Electron.39, 1187–1195 (2003). [CrossRef]
- J. Javaloyes and S. Balle, “Emission directionality of semiconductor ring lasers: A traveling-wave description,” IEEE J. Quantum Elect.45, 431–438 (2009). [CrossRef]
- S. Sunada, T. Harayama, K. Arai, K. Yoshimura, K. Tsuzuki, A. Uchida, and P. Davis, “Random optical pulse generation with bistable semiconductor ring lasers,” Opt. Express19, 7439–7450 (2011) [CrossRef] [PubMed]
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