Letter

Vol. 40, No. 22 / November 15 2015 / Optics Letters

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Truly random bit generation based on a novel random Brillouin fiber laser DAO XIANG,1 PING LU,1,2 YANPING XU,1 SONG GAO,1 LIANG CHEN,1

AND

XIAOYI BAO1,3

1

Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada e-mail: [email protected] 3 e-mail: [email protected] 2

Received 23 September 2015; revised 20 October 2015; accepted 22 October 2015; posted 22 October 2015 (Doc. ID 250683); published 13 November 2015

We propose a novel dual-emission random Brillouin fiber laser (RBFL) with bidirectional pumping operation. Numerical simulations and experimental verification of the chaotic temporal and statistical properties of the RBFL are conducted, revealing intrinsic unpredictable intensity fluctuations and two completely uncorrelated laser outputs. A random bit generator based on quantum noise sources in the random Fabry–Perot resonator of the RBFL is realized at a bit rate of 5 Mbps with verified randomness. © 2015 Optical Society of America OCIS codes: (060.3510) Lasers, fiber; (190.3100) Instabilities and chaos; (290.5900) Scattering, stimulated Brillouin; (290.5870) Scattering, Rayleigh. http://dx.doi.org/10.1364/OL.40.005415

Reflection due to multiple light scattering in random lasers replacing conventional optical mirrors enhances the optical path length inside and outside the gain medium to provide sufficient feedback to reach the random lasing threshold. Ambartsumyan et al. proposed the first laser with the non-resonant feedback in 1966 [1], where the narrowing amplified spontaneous emission was observed. Further studies were concentrated on the statistical properties of random lasers [2–6]. The statistics of decay rates of quasi-modes was used to predict intensity fluctuations in the random laser system with dynamic disorder [7]. The resonant spikes stochastically vary and appear even in the random laser with static disorder [8]. Interest in random fiber lasers has been further sparked since the pioneering work of the random distributed feedback fiber laser combining the Raman gain and Rayleigh scattering feedback which operates in continuous-wave regime with strongly suppressed power fluctuations [9–11]. In this paper we want to explore the random fiber laser based on Brillouin gain and Rayleigh scattering for random bit generation, which can enable applications in secure quantum communication, cryptography, Monte Carlo simulation, and stochastic modeling [12–16]. We propose a novel random bit generator (RBG) employing a bi-directional pumped random Brillouin fiber laser (RBFL) that can simultaneously and independently produce two 0146-9592/15/225415-04$15/0$15.00 © 2015 Optical Society of America

chaotic laser outputs. Origins of the chaotic temporal behavior of the RBFL are analyzed, and experimental results confirm that the time-domain statistical characteristics of the output signals of the RBFL meet the requirements of a true RBG with a bit rate of 5 Mbps. The random fiber laser consists of a single pumping source split and launched into two ends of a long standard single-mode fiber (SMF) as the Brillouin gain medium and randomly distributed feedback, as illustrated in Fig. 1. The resonant feedback in the bidirectional pumped RBFL is achieved based on the inhomogeneous distribution of the refractive index in the fiber. The intuitive model of nonlinear coupling equations with the steady-state analysis of the RBFL in the previous study [17] indicated that the pump depletion effects of the stimulated Brillouin scattering (SBS) at both ends of the fiber could create two effective Rayleigh mirrors randomly distributed within two 250 m long fiber regions, forming a random fiber Fabry–Perot (FP) resonator. The resonant lasing components appear at stochastic frequencies and keep drifting on top of the SBS gain in the form of narrow spikes owing to the thermally induced instability in Rayleigh scattering. The amplitude of the spectral components within the SBS gain profile fluctuates with time, which is further enhanced by strong relative intensity noise by Rayleigh-scattering-induced random mode changes on the Stokes wave [18], resulting in chaotic intensity fluctuations in the RBFL output. Numerical simulation of the SBS in the RBFL configuration is performed based on a set of temporal-spatial (t − z) equations for complex amplitudes of the forward-propagating (+) and

Fig. 1. Configuration of the RBFL based on the random FP resonator.

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Letter

backward-propagating (−) pump waves E  P z; t, the Stokes  waves E  S z; t, and the hypersound waves ρ z; t: 8 n ∂E  ∂E  > P P  α  >   iκ 1 E  > S ρ − EP > ∂z c ∂t 2 > > < n ∂E  ∂E  α    S S ; 1   iκ 1 E P ρ  η E  S − ES > ∂t ∂z c 2 > > >  > > : ∂ρ  1 Γ ρ  iκ E  E   f  2 P S ∂t 2 B where c is the velocity of light in a vacuum, n is the fiber refractive index, α is the linear fiber loss coefficient, and ΓB is the phonon decay rate. κ1 and κ 2 are Brillouin coupling constants defined by κ 1  γ e ωP ∕2ncρ0 and κ 2  γ e ωP ∕2c 2 υAeff , where γ e and ρ0 are, respectively, the electrostrictive coupling constant and density of silica; ωp is the pump wave frequency; υ is the sound velocity in the fiber; and Aeff is the effective mode area [19]. The Rayleigh backscattering coefficients η z; t and the Langevin noise source f  z; t both follow a spatially and temporally δ-correlated Gaussian random process with zero mean [20,21]: hη z 0 ; t 0 η z 0 0 ; t 0 0 i  rδz 0 − z 0 0 δt 0 − t 0 0  and hf  z 0 ; t 0 f  z 0 0 ; t 0 0 i  Qδz 0 − z 0 0 δt 0 − t 0 0 , where r is the Rayleigh scattering back-reflection coefficient, and Q is the noise intensity [22] with an expression of Q  2kB T ρ0 ΓB ∕υ2 Aeff , where kB is the Boltzmann constant and T is the temperature. The fiber length of the SMF is assumed to be 10 km, and the pump power is set to be 35 mW for both fiber ends in simulation. The simulated results of the laser dynamics of RBFL-1 and RBFL-2 are, respectively, shown in Figs. 2(a) and 2(b). Although a fully quantum-mechanical manipulation of the RBFL for which the noise sources of stochastic nature are Langevin operators with the Gaussian character is not taken into account here, the straightforward classical equations containing quantumnoise sources in the random FP resonator still correctly represent the laser-output properties. The Stokes waves with dynamic phase shifts are backscattered from fluctuations of thermodynamic quantities, proceeding to the SBS process in the counter-propagating direction. In addition, the temporal dynamics of chaotic behavior in the Rayleigh mirrors establishes an intrinsic independent relationship between RBFL-1 and RBFL-2, although they adopt the same random FP resonator shape. As illustrated in Fig. 2(c), the output power distribution exhibits an approximately Gaussian distribution as predicted. Thus the RBFL exhibits a chaotic behavior in the form of random output power spikes with unpredictable amplitude and timing, which means that the time series of power output is independent of both the recent history and the current one. The power spectral density of the thermal noise is controlled by the bandwidth of the SBS gain, so the upper bound of the chaotic oscillation period of the Stokes power is phonon decay time τphonon ∼ 10 ns. These unique properties make the RBFL an ideal candidate as a physical entropy source to create truly random bit sequences in the megahertz-frequency range. The scheme to generate a random binary sequence of N bits is shown in Fig. 2. The chaotic intensity oscillations are independently generated from the two outputs of RBFL-1 and RBFL-2 with the random FP resonator. Such a power fluctuation from either output port of each individual RBFL is first converted to an AC electrical signal by using an AC-coupled photodetector (PD) or a DC-coupled PD that any nonzero

Fig. 2. Schematic diagram of the random bit sequence generation using the RBFL. The simulated chaotic power fluctuations of the two independent outputs: (a) RBFL-1, (b) RBFL-2, and (c) statistical distribution.

average of the signal is suppressed. The electrical signal is then converted to a binary signal using a one-bit analog-to-digital converter (1-bit A/D) by comparing it to a threshold voltage V th with a sampling period τs controlled by a sample clock. The nondeterministic signal source can thus be produced by any arbitrary output port considering the strongly chaotic dynamics in the RBFL. The combination of dual-port laser outputs via a logical exclusive disjunction (XOR) operation gives a completely unpredictable random bit sequence due to the incommensurate chaos in RBFL-1 and RBFL-2. A schematic diagram of the RBFL-based RBG configuration is shown in Fig. 3. A fiber laser with 3.5 kHz linewidth (Rock Module, NP Photonics) was amplified to 70 mW by a C-band EDFA (APEDFA-C-10-B-FA, Amonics) and then split into two channels with equal power by a 50/50 fiber coupler. The bidirectional pumping scheme was realized by launching pump light into a 10 km single-mode fiber (SMF-28, Corning) from its two ends (in-1 and in-2) via Port1-to-Port2 of two fiber circulators (circulator-1 and circulator-2). The SMF was placed inside a sound-proof box isolating from undesired external spatial and temporal disturbances in the local refractive index of the SMF that may cause some form of regularity in laser oscillations. The output light from the two RBFLs (RBFL-1 and RBFL-2) traveled through Port2-to-Port3 of the fiber circulators and two C-band tunable filters (TFC-1 and TFC-2) with 3 GHz bandwidth, respectively. The two narrow filters were used to remove the pump light from the RBFL output (out-1 and out-2). Then, the two RBFL light beams were sent to an oscilloscope (Scope; WaveRunner 64Xi-A, Teledyne LeCroy) for analysis of their temporal properties as well as demonstrating random bit generation based on random laser dynamics. Two variable optical attenuators (VOA-1 and VOA-2) and two photodetectors (PD-1 and PD-2) with 4.5 GHz bandwidth (1592, New Focus) were used in two channels, respectively.

Letter

Vol. 40, No. 22 / November 15 2015 / Optics Letters

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Fig. 3. Experimental setup of the RBG based on the RBFL.

The RBFL-1 output power monitoring of the DC component obtained a mean value of 1.53 V, and the AC-component signal is presented in Fig. 4(a), exhibiting a temporal chaotic behavior of the RBFL due to the significant mode-hopping caused by the random distributed feedback using the Rayleigh mirrors. Figure 4(b) shows the calculated power spectra of RBFL-1 and the PD dark current noise, and the latter exhibits a distinct plateau occurring below around tens of megahertz. This falling edge of the RBFL spectrum is associated with the bandwidth of the Brillouin gain spectrum in the SMF, which limits the rate of chaotic oscillations in the laser output power. Figure 4(c) displays the histograms of the output power distributions of RBFL-1 and RBFL-2 calculated from the 1 ms digitized scope trace with a sampling rate of 500 MS/s. Two fitting curves to the histograms both exhibit the Gaussian distributions, revealing that the generated random Boolean sequences via sampling traces are statistically unbiased (i.e., equal subsequent probabilities of 0 and 1), which is essential for a true RBG. The autocorrelation function (ACF) of the output power signals can provide additional statistical properties of the RBFL radiation and its definition is Cτ  hPtPt τi∕hPt2 i. The ACFs of the chaotic waveforms of the two RBFLs are calculated and shown in Fig. 5. Initially, the correlation peaks for

Fig. 4. (a) Observed time-domain output signals of RBFL-1. (b) Power spectra of RBFL-1 and the PD dark current noise. (c) Statistics for the power Gaussian distributions and the corresponding fitting curves for RBFL-1 and RBFL-2.

Fig. 5. Autocorrelation functions of the chaotic waveforms of RBFL1 and RBFL-2. Inset: enlargement of the short-time autocorrelation.

RBFL-1 and RBFL-2 monotonically decay to zero due to the strongly chaotic dynamics with respective timescales of τc1  0.134 μs and τc2  0.140 μs. The correlation time τc1 and τc2 are defined by the chaos bandwidth corresponding to the high frequency edge (∼7 MHz) of the chaotic plateau of power as indicated in Fig. 4(b), and their drifts Δτc1c2 ) are on the scale of several tens of nanoseconds under transient SBS conditions. Then, the correlation peaks further fluctuate above and below zero over some ripple periods, and no obvious chaotic oscillation can be observed in the relative flat power spectrum of the RBFL as shown in Fig. 4(b). It is noted that the ripples experience relative high amplitude around τ1 and τ2 between 95 and 100 μs, which is associated with the relaxation oscillation of Stokes power in the two RBFLs [23] determined by the effective random cavity length of 9.5–10 km and its corresponding round-trip time. Thus, the sampling time τs was chosen for reliable random bit sequence generation if the following conditions are satisfied: (1) τc1c2  Δτc1c2 < τs , the overall ripples on the ACF curves with a relatively small amplitude level of 0.008 rapidly become decorrelated at about τc1c2 ) and (2) τs < τ12 , the increases in the ACFs occurring at about τ12 may lead to insignificant recurrent features in the timedomain traces, and these ACFs have much smaller values than those in other chaotic semiconductor laser-based RBGs [16]. An example of the random bit generation at a bit rate of 5 Mbps was demonstrated when the sampling rate was set to be 5 MHz, which meets the criteria for selecting the sampling time. The temporal waveforms of the chaotic outputs from RBFL-1 and RBFL-2 are, respectively, displayed in Figs. 6(a) and 6(b) where the solid lines illustrate the threshold values of zero and the solid dots mark the sampling points every 0.2 μs. The corresponding random bit sequence output in a non-return-to-zero format from the XOR operation is shown in Fig. 6(c). 1000 random bit sequences of 1 Mbit per sequence passed the National Institute of Standards and Technology (NIST) statistical test for randomness [24], while the extracted sequences generated at a bit rate higher than 5 Mbps failed the stringent NIST tests, which is consistent with the bandwidth of the chaotic oscillations in the RBFL.

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Fig. 6. Temporal waveforms of output signals from (a) RBFL-1 and (b) RBFL-2. (c) The corresponding random bit sequence.

Compared to the ACFs of either RBFL-1 or RBFL-2 in Fig. 5, the ripples on the ACF curves have been further suppressed by the XOR operation as shown in Fig. 7(a), effectively avoiding the recurrent features of the random bit sequences. Figure 7(b) shows the statistics of 0 and 1 in random bits and the 0/1 ratio of the generated random sequence is 500050/499950. At the 5 MHz sampling rate, the probability in any bin of the histogram is independent of recent history, which specifies that the two bins of 0 and 1 are of nearly equal size. The current method of generating random bit sequences is based on two independent RBFL outputs and requires only a single 1-bit A/D. In order to further minimize the possible bias under the same bit rate, broadband random phase modulation could be induced to the SBS process to extend the bandwidth range; taking postprocessing schemes could also enhance the bit rate with verified randomness. A single bit sequence as a 500 ×

Fig. 7. (a) Autocorrelation function of random bit sequence. One bit corresponds to τs  0.2 μs. (b) Statistics of 0 and 1 in a random bit stream. (c) Random bit patterns in a 2D plane. Bits 0 and 1 are converted into red and green dots, respectively, and placed from left to right and from top to bottom; 500 × 500 bits are shown.

Letter 500 pattern of red (RBFL-1) and green (RBFL-2) dots is plotted in Fig. 7(c) to visualize the randomness of the bits. In conclusion, we have successfully achieved a unique random bit generator based on a novel random Brillouin fiber laser with the SBS gain and Rayleigh feedback. The chaotic temporal behavior in the RBFL has been numerically simulated and statistically investigated, and the quantum noise sources with the Gaussian property in the random FP resonator guarantee the uncorrelated double laser outputs with no distinct oscillation periods or specific cavity features. The inherent chaotic power evolution is thus employed to generate random bit sequences at a bit rate of 5 Mbps, passing the NIST statistical test. The performance of the proposed RBG could be further improved by taking various postprocessing schemes. It presents great potential for truly fast random bit generation in various applications in quantum communication and other areas. Funding. Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant; Canada Research Chairs Program (CRC). REFERENCES 1. R. V. Ambartsumyan, N. G. Basov, P. G. Kryukov, and V. S. Letokhov, IEEE J. Quantum Electron. 2, 442 (1966). 2. H. Cao, Y. G. Zhao, S. T. Ho, E. W. Seelig, Q. H. Wang, and R. P. H. Chang, Phys. Rev. Lett. 82, 2278 (1999). 3. L. I. Deych, Phys. Rev. Lett. 95, 043902 (2005). 4. A. Yamilov, X. Wu, H. Cao, and A. Burin, Opt. Lett. 30, 2430 (2005). 5. L. Angelani, C. Conti, G. Ruocco, and F. Zamponi, Phys. Rev. Lett. 96, 065702 (2006). 6. D. Sharma, H. Ramachandran, and N. Kumar, Opt. Lett. 31, 1806 (2006). 7. K. L. van der Molen, A. P. Mosk, and A. Lagendijk, Phys. Rev. A 74, 053808 (2006). 8. S. Mujumdar, V. Türck, R. Torre, and D. S. Wiersma, Phys. Rev. A 76, 033807 (2007). 9. S. Turitsyn, S. Babin, A. E. El-Taher, P. Harper, D. V. Churkin, S. I. Kablukov, J. D. Ania-Castañón, V. Karalekas, and E. V. Podivilov, Nat. Photonics 4, 231 (2010). 10. M. Bravo, M. Fernandez-Vallejo, and M. Lopez-Amo, Opt. Lett. 38, 1542 (2013). 11. O. Gorbunov, S. Sugavanam, and D. Churkin, Opt. Lett. 40, 1783 (2015). 12. 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, Nat. Photonics 2, 728 (2008). 13. I. Kanter, Y. Aviad, I. Reidler, E. Cohen, and M. Rosenbluh, Nat. Photonics 4, 58 (2009). 14. C. Gabriel, C. Wittmann, D. Sych, R. Dong, W. Mauerer, U. L. Andersen, C. Marquardt, and G. Leuchs, Nat. Photonics 4, 711 (2010). 15. M. Sciamanna and K. A. Shore, Nat. Photonics 9, 151 (2015). 16. X.-Z. Li, S.-S. Li, J.-P. Zhuang, and S.-C. Chan, Opt. Lett. 40, 3970 (2015). 17. Y. Xu, D. Xiang, Z. Ou, P. Lu, and X. Bao, Opt. Lett. 40, 1920 (2015). 18. B. Saxena, X. Bao, and L. Chen, Opt. Lett. 39, 1038 (2014). 19. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012). 20. A. A. Fotiadi, R. Kiyan, O. Deparis, P. Megret, and M. Blondel, Opt. Lett. 27, 83 (2002). 21. A. A. Fotiadi, P. Mégret, and M. Blondel, Opt. Lett. 29, 1078 (2004). 22. R. W. Boyd, K. Rzaewski, and P. Narum, Phys. Rev. A 42, 5514 (1990). 23. A. A. Fotiadi and R. V. Kiyan, Opt. Lett. 23, 1805 (1998). 24. National Institute of Standards and Technology Statistical Tests Suite, http://csrc.nist.gov/groups/ST/toolkit/rng/documentation_software.html.

Truly random bit generation based on a novel random Brillouin fiber laser.

We propose a novel dual-emission random Brillouin fiber laser (RBFL) with bidirectional pumping operation. Numerical simulations and experimental veri...
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