## All-optical fast random number generator |

Optics Express, Vol. 18, Issue 19, pp. 20360-20369 (2010)

http://dx.doi.org/10.1364/OE.18.020360

Acrobat PDF (2460 KB)

### Abstract

We propose a scheme of all-optical random number generator (RNG), which consists of an ultra-wide bandwidth (UWB) chaotic laser, an all-optical sampler and an all-optical comparator. Free from the electric-device bandwidth, it can generate 10Gbit/s random numbers in our simulation. The high-speed bit sequences can pass standard statistical tests for randomness after all-optical exclusive-or (XOR) operation.

© 2010 OSA

## 1. Introduction

1. C. S. Petrie and J. A. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. **47**(5), 615–621 (2000). [CrossRef]

2. J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, post-processing free, quantum random number generator,” Appl. Phys. Lett. **93**(3), 031109 (2008). [CrossRef]

3. J. Walker, “HotBits: Genuine Random Numbers, Generated by Radioactive Decay,” http://www.fourmilab.ch/hotbits/*.*

4. M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. **52**(4), 403–409 (2003). [CrossRef]

5. D. S. Ornstein, “Ergodic theory, randomness, and “chaos”,” Science **243**(4888), 182–187 (1989). [CrossRef] [PubMed]

8. T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators - Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. **48**(3), 382–385 (2001). [CrossRef]

*et al*. [9

9. 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**(12), 728–732 (2008). [CrossRef]

10. I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. **103**(2), 024102 (2009). [CrossRef] [PubMed]

11. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics **4**(1), 58–61 (2010). [CrossRef]

*et al*.demonstrated another fast random bit generation using bandwidth-enhanced chaotic laser [12

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**(6), 5512–5524 (2010). [CrossRef] [PubMed]

## 2. Principle of all-optical RNG

### 2.1 UWB chaotic laser

*E*and carrier density

*N*, respectively, as expressed in the following equations: where κ

_{f}and κ

_{j}denote the feedback and injection strength, the amplitude of injection laser |

*E*

_{j}| is equal to that of the solitary slave laser, and Δν = ν

_{inj}– ν

_{s}is the frequency detuning between the injection and the slave lasers. The following parameters were used in simulations: transparency carrier density

*N*

_{0}= 0.455 × 10

^{6}µm

^{−3}, threshold current

*I*

_{th}= 22 mA, differential gain

*g*= 1.414 × 10

^{−3}µm

^{−3}ns

^{−1}, carrier lifetime

*τ*

_{N}= 2.5 ns, photon lifetime

*τ*

_{p}= 1.17 ps, round-trip time in laser intra-cavity

*τ*

_{in}= 7.38 ps, line-width enhancement factor

*α*= 5.0, gain saturation parameter

*ε*= 5 × 10

^{−5}µm

^{3}, active laser volume

*V*= 324 µm

^{3}, the frequency detuning Δν = 0 GHz, and the optical injection strength

*κ*

_{j}= 0.15. The slave laser was biased at 1.7

*I*

_{th}. More details see [13

13. A. B. Wang, Y. C. Wang, and H. C. He, “Enhancing the Bandwidth of the Optical Chaotic Signal Generated by a Semiconductor Laser With Optical Feedback,” IEEE Photon. Technol. Lett. **20**(19), 1633–1635 (2008). [CrossRef]

14. A. B. Wang, Y. C. Wang, and J. F. Wang, “Route to broadband chaos in a chaotic laser diode subject to optical injection,” Opt. Lett. **34**(8), 1144–1146 (2009). [CrossRef] [PubMed]

### 2.2 All-optical sampler

15. H. Sotobayashi, C. Sawaguchi, Y. Koyamada, and W. Chujo, “Ultrafast walk-off-free nonlinear optical loop mirror by a simplified configuration for 320-Gbit / s time-division multiplexing signal demultiplexing,” Opt. Lett. **27**(17), 1555–1557 (2002). [CrossRef] [PubMed]

_{2}, are launched as the control light into the Sagnac loop via the WDM coupler, while the UWB chaotic light operating at λ

_{1}, is injected as the probe light into the loop via the 3dB coupler. The interaction between the UWB chaotic light and clock pulses in HNLF is described in terms of the well-known nonlinear Schrödinger equations [18,19

19. J. Z. Zhang, A. B. Wang, J. F. Wang, and Y. C. Wang, “Wavelength division multiplexing of chaotic secure and fiber-optic communications,” Opt. Express **17**(8), 6357–6367 (2009). [CrossRef] [PubMed]

*j*,

*k*is chosen to be 1, or 2.

*E*

_{1}and

*E*

_{2}represent the slowly varying complex electrical field amplitude of the UWB chaotic light and clock pulses train, respectively.

*z*is the propagation distance, and

*T*is the time measured in a reference frame moving at the group velocity.

*α*,

*β*

_{2},

*γ*are the fiber attenuation coefficient, the second-order dispersion parameter and the nonlinear coefficient, respectively. The two terms on the right-hand side of Eq. (3) are due to self-phase modulation (SPM) and cross-phase modulation (XPM), respectively. The factor of 2 shows that XPM is twice as effective as SPM for the same intensity. In our numerical simulations, the typical parameter values of HNLF is set to be

*α*= 0.2 dB/km,

*β*

_{2}= 5.1 ps

^{2}/km and

*γ*= 20 W

^{−1}km

^{−1}.

*P*

_{out}satisfies the following formula:where

*P*

_{in}= |

*E*

_{in}|

^{2}is the power of the injected chaotic light and

*φ*

_{CW}-

*φ*

_{CCW}is the phase difference between the clockwise and the counter clockwise traveling chaotic light. The phase shift of the UWB chaotic laser traveling clockwise and counterclockwise,

*φ*

_{CW}and

*φ*

_{CCW}, can be written as

*φ*

_{CW}= 2γ

*P*

_{peak}

*L*, and

*φ*

_{CCW}= 2γ

*P*

_{ave}

*L*, respectively, where,

*P*

_{peak}is the peak power of each clock pulse,

*P*

_{ave}is the average power of the clock pulse and

*L*is the length of HNLF. When the phase difference becomes π or 0 radian through adjusting the peak power and average power of the clock pulses and the length of HNLF, the corresponding chaotic light is entirely transmitted from the output port or reflected back to the Sagnac loop. Specific parameters used in the simulation have been given out in Section 3. Consequently, the sampling of the chaotic laser can be realized.

### 2.3 All-optical comparator

20. K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. **20**(1), 18–20 (2008). [CrossRef]

21. K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express **16**(15), 11405–11410 (2008). [CrossRef] [PubMed]

*et al*. in 2008. This kind of bistability can be described as below: A λ/4-shifted DFB laser (λ/4DFB) with antireflection-coated facets is biased above threshold. When an external light outside the stopband of the grating is injected into the laser, two different stable states where the laser works can be distinguished: one in which the laser is lasing and the other where the laser is switched off.

21. K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express **16**(15), 11405–11410 (2008). [CrossRef] [PubMed]

_{th2}named as the threshold power, the output power of lasing light will jump down to a tiny level of nearly 0 mW. While the injection light power is below P

_{th1}, the output power of lasing light will maintain a higher level around 1 mW. Note that the bistability domain, i.e. the hysteresis curve width expressed as ΔP = P

_{th2}-P

_{th1}, can become narrower by decreasing the bias current of the λ/4DFB laser. This point has been demonstrated by Huybrechts numerically and experimentally [20

20. K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. **20**(1), 18–20 (2008). [CrossRef]

21. K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express **16**(15), 11405–11410 (2008). [CrossRef] [PubMed]

_{th2}, the lasing light output time-train will have a low power level, which represents “0”, and otherwise, represents “1”.

### 2.4 Post-processing of exclusive-or (XOR)

22. Y. Miyoshi, K. Ikeda, H. Tobioka, T. Inoue, S. Namiki, and K. Kitayama, “Ultrafast all-optical logic gate using a nonlinear optical loop mirror based multi-periodic transfer function,” Opt. Express **16**(4), 2570–2577 (2008). [CrossRef] [PubMed]

*P*

^{’}_{in}is the half power of the CW light.

*Φ*

_{arm1}and

*Φ*

_{arm2}are defined as the phase shift induced by control lights, respectively, which are in proportion with the power of the input random number signals (control lights). Similar with the above mentioned all-optical sampler, when the signal is “1” level, the phase of the probe light has a π radian phase-shift. However, when the signal is “0” level, the phase of the probe light keeps invariable. Thus, the all-optical XOR function can be realized and accordingly the single random bit sequence with better randomness can be obtained. Adopted parameters in the simulation can be found in Section 3. The truth table is shown in Table 1 , where signal 1 and signal 2 are corresponding to two random number signals in arm 1 and arm 2, respectively.

## 3. Simulation results and randomness tests

_{1}, 1543nm.

^{−3}W

^{−1}m

^{−1}. The sampling clock pulse sequence generated by the mode-locked laser operating at wavelength λ

_{2}, 1550 nm is shown in Fig. 5(b). The pulse is in the format of Gauss, whose pulse width is 20 ps, peak power is 1 W and frequency is 10 GHz which determine the sampling rate. Figure 5(c) is a sampled chaotic laser pulse train at the output port of Sagnac loop via the filter (BPF1). Its mean power is 1.45 mW by calculating.

_{th2}is 1.7 mW and P

_{th1}is 1.6 mW. Thus the width of the hysteresis is 0.1 mw, which is smaller than the mean power 1.45mW of the sampled chaotic laser. The power of the injected CW light was set to 0.25 mW, with the same wavelength λ

_{1}as the UWB chaotic laser. The wavelength λ

_{1}was outside the stopband of the grating and different from the lasing wavelength of λ/4DFB, 1553nm. In this way, the sum of the CW light power and the mean power of sampled chaotic pulses is equal to the threshold P

_{th2}. Thus, when the sampled chaotic pulse power is above the P

_{th2}, the lasing light output would locate at the “off ” state, and time-train will appear a “hollow”, which represents “0”. Otherwise, it will have no or smaller “hollow”, which represent “1”, as shown in Fig. 6. The extinction ratio between “zeros” and “ones” is as high as 30 dB. Note that every bit occupies a duration time of 100ps. However, the duty ratio of “zeros” has tiny difference that is induced by the existence of the hysteresis curve width.

23. A. Rukhin, *et al*., “NIST Statistical Tests Suite,” http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html*.*

## 4. Discussions

## 5. Conclusions

## Acknowledgements

## References and links

1. | C. S. Petrie and J. A. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. |

2. | J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, post-processing free, quantum random number generator,” Appl. Phys. Lett. |

3. | J. Walker, “HotBits: Genuine Random Numbers, Generated by Radioactive Decay,” http://www.fourmilab.ch/hotbits/ |

4. | M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. |

5. | D. S. Ornstein, “Ergodic theory, randomness, and “chaos”,” Science |

6. | G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circ. Syst. |

7. | T. Stojanovski and L. Kocarev, “Chaos-based random number generators - Part I: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. |

8. | T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators - Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. |

9. | 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 |

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

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. | A. B. Wang, Y. C. Wang, and H. C. He, “Enhancing the Bandwidth of the Optical Chaotic Signal Generated by a Semiconductor Laser With Optical Feedback,” IEEE Photon. Technol. Lett. |

14. | A. B. Wang, Y. C. Wang, and J. F. Wang, “Route to broadband chaos in a chaotic laser diode subject to optical injection,” Opt. Lett. |

15. | H. Sotobayashi, C. Sawaguchi, Y. Koyamada, and W. Chujo, “Ultrafast walk-off-free nonlinear optical loop mirror by a simplified configuration for 320-Gbit / s time-division multiplexing signal demultiplexing,” Opt. Lett. |

16. | K. Ikeda, J. Abdul, S. Namiki, and K. Kitayama, “Optical quantizing and coding for ultrafast A/D conversion using nonlinear fiber-optic switches based on Sagnac interferometer,” Opt. Express |

17. | G. P. Agrawal, “Fiber interferometer,” in Applications of nonlinear fiber optics: Edited by Paul L. Kelley, (Academic press, San Diego, 2001), Chap. 3. |

18. | G. P. Agrawal, |

19. | J. Z. Zhang, A. B. Wang, J. F. Wang, and Y. C. Wang, “Wavelength division multiplexing of chaotic secure and fiber-optic communications,” Opt. Express |

20. | K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. |

21. | K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express |

22. | Y. Miyoshi, K. Ikeda, H. Tobioka, T. Inoue, S. Namiki, and K. Kitayama, “Ultrafast all-optical logic gate using a nonlinear optical loop mirror based multi-periodic transfer function,” Opt. Express |

23. | A. Rukhin, |

24. | K. Huybrechts, A. Ali, T. Tanemura, Y. Nakano, and G. Morthier, “Numerical and experimental study of the switching times and energies of DFB-laser based All-optical flip-flops”, presented at the International Conference on Photonics in Switching, Pisa, Italy, 15–19 Sept. 2009. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(140.1540) Lasers and laser optics : Chaos

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 25, 2010

Revised Manuscript: August 19, 2010

Manuscript Accepted: August 23, 2010

Published: September 9, 2010

**Citation**

Pu Li, Yun-Cai Wang, and Jian-Zhong Zhang, "All-optical fast random number generator," Opt. Express **18**, 20360-20369 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-19-20360

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### References

- C. S. Petrie and J. A. Connelly, “A noise-based IC random number generator for applications in cryptography,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 47(5), 615–621 (2000). [CrossRef]
- J. F. Dynes, Z. L. Yuan, A. W. Sharpe, and A. J. Shields, “A high speed, post-processing free, quantum random number generator,” Appl. Phys. Lett. 93(3), 031109 (2008). [CrossRef]
- J. Walker, “HotBits: Genuine Random Numbers, Generated by Radioactive Decay,” http://www.fourmilab.ch/hotbits/ .
- M. Bucci, L. Germani, R. Luzzi, A. Trifiletti, and M. Varanonuovo, “A high-speed oscillator-based truly random number source for cryptographic applications on a Smart Card IC,” IEEE Trans. Comput. 52(4), 403–409 (2003). [CrossRef]
- D. S. Ornstein, “Ergodic theory, randomness, and “chaos”,” Science 243(4888), 182–187 (1989). [CrossRef] [PubMed]
- G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circ. Syst. 37(9), 1157–1164 (1990). [CrossRef]
- T. Stojanovski and L. Kocarev, “Chaos-based random number generators - Part I: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 281–288 (2001). [CrossRef]
- T. Stojanovski, J. Pihl, and L. Kocarev, “Chaos-based random number generators - Part II: practical realization,” IEEE Trans. Circ. Syst. I Fundam. Theory Appl. 48(3), 382–385 (2001). [CrossRef]
- 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(12), 728–732 (2008). [CrossRef]
- I. Reidler, Y. Aviad, M. Rosenbluh, and I. Kanter, “Ultrahigh-speed random number generation based on a chaotic semiconductor laser,” Phys. Rev. Lett. 103(2), 024102 (2009). [CrossRef] [PubMed]
- I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, “An optical ultrafast random bit generator,” Nat. Photonics 4(1), 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. Express 18(6), 5512–5524 (2010). [CrossRef] [PubMed]
- A. B. Wang, Y. C. Wang, and H. C. He, “Enhancing the Bandwidth of the Optical Chaotic Signal Generated by a Semiconductor Laser With Optical Feedback,” IEEE Photon. Technol. Lett. 20(19), 1633–1635 (2008). [CrossRef]
- A. B. Wang, Y. C. Wang, and J. F. Wang, “Route to broadband chaos in a chaotic laser diode subject to optical injection,” Opt. Lett. 34(8), 1144–1146 (2009). [CrossRef] [PubMed]
- H. Sotobayashi, C. Sawaguchi, Y. Koyamada, and W. Chujo, “Ultrafast walk-off-free nonlinear optical loop mirror by a simplified configuration for 320-Gbit / s time-division multiplexing signal demultiplexing,” Opt. Lett. 27(17), 1555–1557 (2002). [CrossRef] [PubMed]
- K. Ikeda, J. Abdul, S. Namiki, and K. Kitayama, “Optical quantizing and coding for ultrafast A/D conversion using nonlinear fiber-optic switches based on Sagnac interferometer,” Opt. Express 13(11), 4296–4302 (2005). [CrossRef] [PubMed]
- G. P. Agrawal, “Fiber interferometer,” in Applications of nonlinear fiber optics, P. L. Kelley, ed., (Academic press, San Diego, 2001), Chap. 3.
- G. P. Agrawal, Nonlinear fiber optics, 3rd Edition (Academic Press, San Diego, 2001) Chap. 2.
- J. Z. Zhang, A. B. Wang, J. F. Wang, and Y. C. Wang, “Wavelength division multiplexing of chaotic secure and fiber-optic communications,” Opt. Express 17(8), 6357–6367 (2009). [CrossRef] [PubMed]
- K. Huybrechts, W. D'Oosterlinck, G. Morthier, and R. Baets, “Proposal for an All-Optical Flip-Flop Using a Single Distributed Feedback Laser Diode,” IEEE Photon. Technol. Lett. 20(1), 18–20 (2008). [CrossRef]
- K. Huybrechts, G. Morthier, and R. Baets, “Fast all-optical flip-flop based on a single distributed feedback laser diode,” Opt. Express 16(15), 11405–11410 (2008). [CrossRef] [PubMed]
- Y. Miyoshi, K. Ikeda, H. Tobioka, T. Inoue, S. Namiki, and K. Kitayama, “Ultrafast all-optical logic gate using a nonlinear optical loop mirror based multi-periodic transfer function,” Opt. Express 16(4), 2570–2577 (2008). [CrossRef] [PubMed]
- A. Rukhin, et al., “NIST Statistical Tests Suite,” http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html .
- K. Huybrechts, A. Ali, T. Tanemura, Y. Nakano, and G. Morthier, “Numerical and experimental study of the switching times and energies of DFB-laser based All-optical flip-flops”, presented at the International Conference on Photonics in Switching, Pisa, Italy, 15–19 Sept. 2009.

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