## 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 Optical Society of America

## 1. Introduction

Random numbers have a wide range of applications. For example, they are used as ranging signal in radar system, controlling signal in remote control, encryption codes or keys in digital communication, address codes and spread spectrum codes in code division multiple access (CDMA). They are also used in the statistics to solve problems in many fields such as nuclear medicine, finance and computer graphics.

There are two approaches to generate random numbers: software-based and physics-based. The former can produce high-speed pseudorandom numbers with rates of several Gbit/s utilizing deterministic algorithms, but it is vulnerable when such pseudorandom numbers are used as the keys to encryption systems. However, the latter can generate physical random numbers and ensure the confidentiality of secure communication by means of the inherently random or unpredictable processes in the physical world. Stochastic noise [1,2], radioactive decay [3] and frequency jitter of electronic oscillator [4] have gotten used widely as entropy sources in the generation of physical random sequences. Unfortunately, limited by the mechanism of extracting bit sequences, these systems have much slower rates. A more convenient method to generate random numbers is based on chaotic circuits [5–8], but the narrow bandwidths (<1GHz) of electric chaos make the generation rates of this kind of method be not more than 200Mbit/s up to now. An unconditional secure system should be encrypted with truly random number as long as the plaintext, but the generation rates of trusted random numbers achieved with existing physical sources are much slower than the data rates of communications.

In recent years, the chaotic laser has become a more attractive physical source for high-speed random numbers generation, due to its high bandwidth. In 2008, Uchida *et al*. [9] for the fist time realized experimentally a 1.7 Gbit/s random number generator (RNG) by using wide bandwidth chaotic lasers. In their experiment, each of the chaotic lights was firstly converted into an electrical signal by a photodetector, and then to a binary signal using a 1-bit analog-to-digital converter (ADC). The binary bit signals were combined by a logical exclusive-or (XOR) operation to obtain a sequence of random bits. In 2009, Kanter and his colleagues [10,11] achieved RNGs at higher bit rates with a single chaotic laser by extracting more than one bit per sample in off-line processing of experimental chaotic laser time series. Very recently, Uchida *et al*.demonstrated another fast random bit generation using bandwidth-enhanced chaotic laser [12]. However, all these RNGs based on chaotic laser process signals in the electronic domain, so their random number generation rates are limited by the bottleneck of electronic signal processing.

In this paper, we propose and demonstrate an all-optical RNG based on chaotic laser, which does all signal processing only in the optical domain, with no photoelectric conversion. It can provide reliable high-speed random numbers for the cipher, one-time pad. Moreover, the all-optical random numbers can be compatible with the optical signal transmitted in optical communication networks directly, with no need of any external modulators.

## 2. Principle of all-optical RNG

The schematic diagram of the all-optical RNG is shown in Fig. 1 , which includes three parts: an ultra-wide bandwidth (UWB) chaotic laser as random number source, a Sagnac interferometer as all-optical sampler and a λ/4-shifted DFB lasers (λ/4 DFB) as all-optical comparator. The output of UWB chaotic laser, whose intensity varies unpredictably with time, is sampled through the all-optical sampler driven by an optical clock, and then converted into a random number sequence using the all-optical comparator. The specific procedures are given in the following.

#### 2.1 UWB chaotic laser

The implementation of the UWB chaotic laser is shown in Fig. 1(a). A single-mode distributed-feedback (DFB) laser diode, the slave laser, subject to optical feedback with a fiber ring cavity, is referred to as the original chaotic laser. The power of feedback light can be adjusted with a variable attenuator (VA). The polarization state of the feedback light is matched to that of the slave laser by a polarization controller PC2. Another DFB laser diode, the injection laser, is used to enhance the bandwidth of the original chaotic signal by injecting continuous-wave (CW) light into the slave laser through a 30/70 optical fiber coupler. The power and polarization state of the injection light are controlled by an erbium-doped optical fiber amplifier (EDFA) and PC3, respectively. An optical isolator (Isolator) is used to prevent unwanted optical feedback into the injection laser. The injection laser is wavelength adjusted to an appropriate optical frequency detuning to the free-running slave laser. Thus, a UWB chaotic laser can be generated.

The system can be modeled by a set of rate equations for slave laser electric complex amplitude *E* and carrier density *N*, respectively, as expressed in the following equations:

_{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] and [14].

#### 2.2 All-optical sampler

As shown in Fig. 1(b), the all-optical sampler consists of an optical switch based on nonlinear Sagnac interferometer [15–17]. This optical switch is comprised of a 3dB coupler (50/50), a WDM coupler, a mode-locked laser, a length of highly nonlinear fiber (HNLF), two polarization controllers (PC4 and PC5), an isolator and an optical bandpass filter (BPF1). The PC4 placed in the Sagnac loop adjusts the polarization to maximize the on-off ratio of the interferometer. This switch has three ports: one control light input, one probe light input, and one switched output port of the probe light. The isolator prevents the back flow of the reflected probe light and the control light. The BPF1 removes the control light and transmits only the probe light. The output level is determined by the amount of the cross-phase modulation (XPM) exerted by the control light onto the probe light.

In order for this device to work as an all-optical sampler, a train of clock pulses, generated by the mode-locked laser operating at wavelength λ_{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].

Here, *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}.

Transmitted optical power of UWB chaotic light *P*
_{out} satisfies the following formula:

*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

The realization of the all-optical comparator is based on the bistability [20,21] proposed by Huybrechts *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.

Under injection of only continuous-wave (CW) light, a curve of the bistability about the lasing light of the λ/4DFB is shown in Fig. 2
, which is Huybrechts’ experimental result in [21]. From Fig. 2, we can see clearly that only when injection light power is above P_{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 [20,21].

Based on the above, we have designed a comparator as shown in Fig. 1(c), where the sampled chaotic pulses and a CW light with the same wavelength as the sampled chaotic pulses are simultaneously injected into a λ/4DFB. The bandpass filter (BPF2) removes the amplified injection light and transmits only the lasing light of the λ/4DFB. When the sampled chaotic pulse power is above the P_{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)

After the above operations, a random number sequence with a rate corresponding to the frequency of the control clock in all-optical sampler, has been generated. But it contains an unwanted bias: nonuniform ratio of 0 and 1. To eliminate the bias, the random number sequence need to be exclusive-or processed by an all-optical XOR gate, as shown in Fig. 3 .

The configuration of the all-optical XOR gate is based on a Mach-Zehnder interferometer with an identical HNLF in each arm [22]. A beam of continuous-wave (CW) light is split by a 50/50 coupler into two arms as probe light, and the output of two independent RNGs are respectively amplified by two EDFAs and launched into arm 1 and arm 2 as control light, through WDM couplers (WDM). The control light induces the phase shift of the probe light due to the XPM in HNLF. After that, the probe light is finally exported via another 50/50 coupler. A BPF is utilized to remove the control light and only transmit the probe light. The interaction between the random number sequence and the CW light in HNLF is numerically simulated according to Eq. (3).

The output power of probe light can be expressed as below:

*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

Figure 4
is the power spectrum of the generated UWB chaotic laser. The dash line is corresponding to the bandwidth of UWB chaotic laser about 10.5GHz. A time sequence of the UWB chaotic laser, from 0 to 2 ns, is given in Fig. 5(a)
. The UWB chaotic laser is operating at wavelength λ_{1}, 1543nm.

In the simulation on the all-optical sampler, the length of the HNLF was 50 m and nonlinear coefficient was 20 × 10^{−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.

Figure 6
is the output waveform of the all-optical comparator. In simulations, the λ/4DFB was adjusted so that it worked under the condition shown in Fig. 2. From the Fig. 2, we can see that the P_{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.

Figure 7
is an output waveform of the all-optical XOR gate. In the simulation, the CW light power was about 1 mW, and two uncorrelated random signals from different all-optical RNGS were amplified by different EDFAs respectively so that their “1” level was around 1 W. And both of the HNLF were 50 m long, whose nonlinear coefficient was 20 × 10^{−3} W^{−1}m^{−1}. From Fig. 7, we can see clearly that the random bit sequence is in the format of return-to-zero (RZ), with a rate of 10 Gbit/s corresponding to the frequency 10 GHz of the sampling optical clock pulses in all-optical sampler.

To evaluate the statistical randomness of the single random bit sequence, we used the standard statistical test suite for random number generators provided by the National Institute of Standard Technology (NIST) [23]. The tests were performed using 1,000 instances of 1Mbit sequences. And the test results show that bit sequences obtained by our method pass all of the NIST tests. Typical results of the NIST tests are shown in Table 2 .

## 4. Discussions

The generation rate of the proposed all-optical RNG is only limited by the bandwidth of UWB chaotic laser. The all-optical sampler and all-optical XOR gate are both based on the cross phase modulation of the HNLF. The Kerr nonlinearity of silica is as fast as a few femto seconds [17], which translate into a few hundred terahertz bandwidth. That indicates the all-optical sampler and XOR gate could operate potentially at much higher rates than 10 Gbit/s. The all-optical comparator is based on the flip-flop proposed by Huybrechts. The rising time of the flip-flop can reach 40 ps in experiment and even 10 ps in theory [24], which corresponds to a bandwidth larger than 30 GHz. Thus, these three components will not limit the maximum generation rate of random number generator. With the 10.5 GHz chaotic laser, the obtained maximum rate of the random numbers generation was 10 Gbit/s in our scheme. Sequences generated at higher rates did not pass the NIST statistical tests. We confirm that faster generation rates can be achieved through further enhancing the bandwidth of chaotic laser.

The bias of random numbers is induced mainly by the weak periodicity of UWB chaotic laser and the threshold of all-optical comparator. Firstly, the weak periodicity is caused by the external cavity feedback of the slave laser, which can get improved by some methods, such as increasing the cavity length to extremely long feedback time, reducing the feedback strength, and introducing feedback from multiple external cavities or a single external cavity with a changing length of cavity. Secondly, the selection principle of an appropriate threshold of all-optical comparator is to get a high quality output of random numbers with slight bias between “0” and “1”. The appropriate threshold is located around the mean power of the chaotic pulses train. The number of “0” in the random bits will greatly increase when the threshold value is set to a bigger one, and vice versa. Thus, to set an appropriate threshold of all-optical comparator, we must calculate the very accurate average power of the sampled pulses. However, it is very difficult, because this calculation needs a large number of data to get an absolutely exact value. We can only adjust the power of holding CW light or the output power of the UWB chaotic laser to calibrate it in some degree. These are also the reason why a post-processing of XOR is applied to wash out the bias.

In addition, the finally obtained random bit sequence exist tiny jitters as shown in Fig. 7, which essentially is induced by two factors. One is that the rising times of the chaotic pulses in Fig. 5(c) have small fluctuation in time. The other is that the bistability curve has a width.

## 5. Conclusions

In conclusion, we have proposed and demonstrated a 10Gbit/s all-optical RNG. It can overcome the bottleneck of electronic signal processing and be compatible with the data in optical communications directly. We confirm the increasing maturity of all-optical components must provide tremendous convenience for the actual implementation of the all-optical RNG in the near future.

## Acknowledgements

We gratefully acknowledge Koen Huybrechts from Ghent University for providing the data about the bistability. This work is supported partially by the Key Program of National Natural Science Foundation of China under grant 60927007 and in part by the open subject of the State Key Laboratory of Quantum Optics and Quantum Optics devices of China under grant 200903.

## 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. **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]

**6. **G. M. Bernstein and M. A. Lieberman, “Secure random number generation using chaotic circuits,” IEEE Trans. Circ. Syst. **37**(9), 1157–1164 (1990). [CrossRef]

**7. **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]

**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]

**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]

**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]

**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]

**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]

**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 **13**(11), 4296–4302 (2005). [CrossRef] [PubMed]

**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, *Nonlinear fiber optics, 3rd Edition* (Academic Press, San Diego, 2001) Chap. 2.

**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]

**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]

**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]

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

**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.