Randomness-Enhanced Chaotic Source With Dual-Path ... - IEEE Xplore

IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 24, NO. 19, OCTOBER 1, 2012

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Randomness-Enhanced Chaotic Source With Dual-Path Injection from a Single Master Laser Shui Ying Xiang, Wei Pan, Nian Qiang Li, Bin Luo, Lian Shan Yan, Xi Hua Zou, Liyue Zhang, and Penghua Mu

Abstract— The randomness and bandwidth enhancement of chaotic signals in optically injected semiconductor lasers (SLs) with dual-path injection from a single master laser are investigated experimentally and numerically. It is shown that both the randomness and bandwidth of chaotic signals generated in a slave SL (SSL) can be enhanced significantly, especially for positive frequency detuning, by simply adding an additional injection path in the conventional master–slave configuration. The region of injection parameter space leading to wideband randomnessenhanced chaos in SSL can be greatly broadened compared to the SSL with single-path injection. This low-cost and simple configuration for wideband randomness-enhanced chaotic source is highly desirable for high-speed random number generators based on chaotic SLs. Index Terms— Bandwidth, chaos, randomness, semiconductor lasers.

dual-path

injection,

Fig. 1. Experimental setup of an SSL subject to DPI from one MSL. M-DFB: master distributed-feedback laser. S-DFB, slave distributed-feedback laser. PC: polarization controller. VA: variable attenuator. EDFA: erbiumdoped optical amplifier.

I. I NTRODUCTION

C

HAOTIC semiconductor lasers (SLs) are promising candidates for many potential applications including, for example, secure optical communications [1]–[4] and high speed random number generators (RNGs) [5]–[8]. It has been shown that, the bandwidth of the chaotic signal can be enhanced roughly three times by using external optical injection [9]–[12], which is extremely useful to increase the bit rate of RNGs. Besides, enhancing the randomness of physical chaos is helpful to further improve the randomness quality of high speed RNGs with reduced post-processing. For conventional master-slave configuration with single-path-injection (SPI), the synchronization of the bandwidth-enhanced chaos in semiconductor lasers has been demonstrated theoretically and experimentally [13], [14]. However, for the purpose of wideband randomness-enhanced chaotic source, which is highly desirable for high-speed randomness-enhanced RNGs, the injection-locking chaos synchronization observed in masterslave configuration with SPI would narrow the parameter range contributing to wideband randomness-enhanced chaos.

Manuscript received June 11, 2012; revised July 25, 2012; accepted August 15, 2012. Date of current version September 12, 2012. This work was supported in part by the National Natural Science Foundation of China under Grant 60976039 and Grant 61274042, in part by the Fundamental Research Foundation of Sichuan Province under Grant 2011JY0030, and in part by the Funds for the Excellent Ph.D. Dissertation of Southwest Jiaotong University, 2010. The authors are with the School of Information Science and Technology, Southwest Jiaotong University, Chengdu 610031, China (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this letter are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LPT.2012.2214208

Hence, it is valuable to explore a simple and low-cost technique to obtain wider range of parameters corresponding to wideband randomness-enhanced chaos. In this letter, we experimentally demonstrate the randomness and bandwidth enhancement of chaotic signals generated by slave SL (SSL) subject to dual-path-injection (DPI) from single master SL (MSL), to obtain low-cost and simple configuration for generation of wideband randomness-enhanced chaos in SLs. The DPI is realized by simply adding an additional path in the conventional master-slave configuration, hence, is quite simple and low-cost compared to SSL subject to optical injections from several MSLs [15]–[16]. Numerical results are further presented to provide more insight into the randomness enhancement due to DPI. The all-fiber experimental setup is shown in Fig. 1. Master distributed-feedback laser (M-DFB) renders chaotic dynamic by means of the fiber ring optical feedback loop. The feedback power P f can be adjusted by variable attenuator (VA). The chaotic output of M-DFB is amplified by erbium-doped optical amplifier (EDFA) and then injected into slave DFB (S-DFB) via two different paths. Different injection delays can be achieved by simply adding a fiber jumper. Note that, when Path 2 is disconnected, the configuration is simplified to be SPI. The injection power Pinj1 and Pinj2 can be adjusted freely by EDFA and VA. Both lasers are driven by the ultralow-noise current sources (ILX-Lightwave, LDX-3620B) and are temperature stabilized by precise temperature controllers (ILX-Lightwave, LDT-5412). The frequency detuning between two lasers, defined as f = f m − f s , was achieved by adjusting the wavelength of S-DFB through temperature

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controllers. Optical spectrum analyzer is used to observe the center wavelength of DFBs. The output of S-DFB is captured by high-speed photo-detectors (PD) with a bandwidth of 10 GHz. The electrical signals are then sent to a 13GHzbandwidth oscilloscope with a sampling rate of 40Gs/s and a power spectrum analyzer. In the experiments, we fix the bias current at 20 mA for both DFBs (with threshold being 9.6mA for M-DFB and 9.8mA for S-DFB), the feedback power for M-DFB, is P f = 100μw. Note that, when the path lengths are set to be identical, the experimental system is very unstable due to interference [17], the total injection power Pinj1 + Pinj2 is varied time to time. Hence, two different path lengths are desired to obtain stable system for the case of DPI. The experimental results in the time and frequency domains for different injection conditions are shown in Fig. 2. The gray lines denote the noise floor, where the sharp change is due to the spectrum analyzer and PD, and the RF spectra for output of S-DFB shown here are obtained by subtracting the noise level. It can be seen that, for f = −7.5GHz, the outputs of S-DFB for SPI with Pinj1 = 300μw shown in Figs. 2(a1) and (a2) are similar to those of M-DFB which are not shown here, due to the injection locking effect. While for DPI with Pinj1 = Pinj2 = 150 μw, the fluctuations of time series in Fig. 2(b1) seem to be more complex, and the spectrum shown in Fig. 2(b2) is much flatter. That is to say, due to the DPI, the chaos synchronization cannot be realized, even the injection locking effect is favored by negative frequency detuning [13], [14], and hence is highly desired for the purpose of bandwidth-enhanced chaotic source. On the other hand, for f = 15 GHz, it can be seen from Fig. 2(c2) that, the spectrum becomes much flatter, and another peak near 15GHz is observed resulting from the beating between the injected light and S-DFB field [10]–[16], and the bandwidth is enhanced obviously compared to that for both SPI and DPI under the case of f = −7.5 GHz. That is to say, positive frequency detuning is preferred to obtain bandwidthenhanced chaos. Besides, we also consider the case of DPI with f = 15 GHz and Pinj1 = Pinj2 = 100 μw, the experimental results are similar to those shown in Figs. 2(c1)–(c2), and are not shown here. However, we believe that the bandwidth for S-DFB with DPI is enhanced more significantly than those for S-DFB with SPI, similar to the case of negative frequency detuning, but the bandwidth is too broad to be accurately observed due to the bandwidth limitation of the PD in experiments. To quantify the bandwidth of chaotic signals, we adopt the bandwidth definition of chaotic waveform as the span between the dc and the frequency component where 80% of the energy is contained within it [12], [18]. The chaotic bandwidth is denoted as BWm and BWs for M-DFB and S-DFB, respectively. Considering the advantages of simplicity, fast calculation, robustness, we adopt the normalized permutation entropy (PE) to evaluate quantitatively the randomness, or unpredictability degree of chaotic signals [19]. The definition and parameter selection for PE has been detailed in Refs. [19]–[22], and we choose embedding dimension d = 6, embedding delay τe = 1. The randomness for the chaotic


Fig. 2. Output time series (left column) and RF spectra (right column). (a1) and (a2) SPI with f = −7.5 GHz and Pinj1 = 300 μW. (b1) and (b2) DPI with f = −7.5 GHz and Pinj1 = Pinj2 = 150 μW. (c1) and (c2) SPI with f = 15 GHz and Pinj1 = 200 μW. The gray lines in the spectrum correspond to the noise floor.

signal generated by M-DFB (S-DFB) is denoted as h m (h s ). We have done calculations for the experimental collected data shown in Fig. 2. It is verified that, when f = −7.5 GHz, the values of BWs and h s for the DPI are higher than those for SPI. While for f = 15 GHz, the values are higher than those for the negative frequency detuning case. Due to the limited bandwidth of the PD, we are only able to show the slower dynamics of the system, and it is difficult to capture the bandwidth and randomness enhancement in S-DFB with DPI. Hence, we further numerically investigate the bandwidth and randomness properties, to identify the parameter region leading to wideband randomness-enhanced chaos. Our rate equation models are based on the well-known Lang-Kobayashi (LK) equations [23]. The equations for the slowly varying amplitude of the electric field E(t) and the carrier number N(t), assuming single-mode operation and low to moderate feedback strengths, read [13], [14], [23]: (1 + j α) 1 d E m (t) = E m (t) + kd E m (t − τd ) G m (t) − dt 2 τp (1) × exp( j ωm τd ) d E s (t) (1 + i α) 1 = G s (t) − E s (t) dt 2 τp +k S P E m (t − τr1 ) exp( j ωm τr1 ) × exp[− j (ωm − ωs )t] kri E m (t − τri ) exp( j ωm τri ) + i=1,2

×exp[− j (ωm − ωs )t] I Nm,s (t) d Nm,s (t) = − − G m,s (t)|E m,s (t)|2 dt q τn G m,s (t) = g[Nm,s (t) − N0 ]/[1 + ε|E m,s (t)|2 ]

(2) (3) (4)

where subscripts m and s stand for MSL and SSL. The last term in Eq. (1) represents optical feedback, kd and τd are

XIANG et al.: RANDOMNESS-ENHANCED CHAOTIC SOURCE

feedback strength and feedback delay. The second (last) term in Eq. (2) denotes the injection lights from MSL to SSL for the case of SPI (DPI). For convenience, we introduce k S P = 2 × kr1 (k D P = kr1 + kr2 ) as the effective total injection strength for SPI (DPI), where kri (i = 1, 2) and τri are the injection strength and injection delay correspond to two injection paths. ωm is the angular frequency of MSL, ωs is the angular frequency of SSL, the frequency detuning between the lasers is defined as f = (ωm − ωs ) 2π. I is bias current. The other typical parameters are [16], [22]: linewidth enhancement factor α = 5, differential gain is g = 1.5 × 104 s −1 , carrier number at transparency N0 = 1.5 × 108, gain saturation coefficient ε = 5 × 10−7 , photon lifetime τ p = 2 ps, carrier lifetime τn = 2 ns, electron charge q = 1.602 × 10−19 C. With these typical parameters, the threshold current is It h = 14.7 mA [16], [22]. For simplicity, we have neglected noise effects and assume that the internal parameters are identical. We numerically solve the equations by the fourth order Runge-Kutta method with an integration step of 0.5ps. At first, to compare with the experimental results, we have simulated the outputs in the time and frequency domains for some experimentally controllable parameters corresponding to Fig. 2, and similar numerical results are obtained. Next, to obtain general results, we consider two cases of frequency detuning f = −7.5 GHz and f = 15 GHz, and present the values of BWs and h s as functions of kr1 for both SPI and DPI in Fig. 3. When numerically calculating PE values, the sampling time is chosen as s = 10ps [16]. Here, the bias current is chosen at I = 1.5It h and the feedback parameters are fixed as kd = 10ns−1 and τd = 3 ns, the bandwidth and randomness values are BWm = 9.6 GHz and h m = 0.43. Note that, when kr1 = 0, the SSL operates continuous wave output and h s = 0, with the increase of kr1 , the SSL also rendered chaotic outputs due to the external chaotic injection. It can be seen that, for both f , with the increase of kr1 , BWs and h s increase firstly due to the new generated frequency components by beating effect, and then decrease and eventually saturate for sufficiently large kr1 , originating from the injection locking effect. Besides, the values of BWs and h s for DPI are much higher than those for SPI. Moreover, for SPI, the range of kr1 leading to high BWs and h s is much narrow, because the easy realization of injection-locking chaos synchronization. However, due to DPI, the injection locking effect is not obvious. In addition, the maximum BWs and h s for positive f are much higher than those for negative f , indicating that positive f is preferred to obtain wideband randomness-enhanced chaos. More specifically, as can be seen from Figs. 3(c) and (d), the maxima for BWs and h s are 33GHz (26GHz) and 0.82 (0.72) for DPI (SPI), besides, BWs > 18 GHz and h s > 0.6 are achieved when 18ns−1 < kr1 < 190ns−1 (30ns−1 < kr1 < 57ns−1 ) for DPI (SPI). Interestingly, even with this simple and low cost configuration, the enhancement is similar to the finding obtained for wideband randomness- enhanced chaotic SLs subject to dual chaotic optical injections from two MLs with identical frequency detuning [16], which is highly desirable for practical implementation.

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Fig. 3. Values of (a) and (c) B Ws and (b) and (d) h s as functions of kr1 , with τr1 = 3 ns for SPI, and τr1 = 3 ns, τr2 = 4 ns, and kr2 = kr1 for DPI. f = −7.5 and 15 GHz for the left and right columns, respectively.

Fig. 4. 2-D maps of (a) B Ws and (b) h s for SPI with τr1 = 3 ns, and (c) B Ws and (d) h s for DPI with τr1 = 3 ns, τr2 = 4 ns, in the parameter space of injection strength and frequency detuning.

Following, two dimensional maps of BWs and h s are shown in Fig. 4 in the parameter space of f and kr1 . It is verified that, higher BWs and h s benefit from positive f , and moreover, the parameter region leading to wideband randomnessenhanced chaos for DPI is much wider than those for SPI. At last, without loss of generality, we also consider different combinations of injection strength and time delay in our system. The h s as a function of kr2 for given kr1 for DPI is shown in Fig. 5. It can be seen that, when kr1 = 100ns−1 , the trend is similar to that for kr1 = kr2 . However, when kr1 = 200ns−1 , the value of h s is relative low compared to the case of kr1 = kr2 , which can be attributed to the strong injection of Path 1 (i.e., large kr1 ) induced injection locking effect. The h s as a function of kr1 for different time delays is shown in Fig. 6. Interestingly, we find that, the randomness enhancement for the case of τr2 −τr1 = τd is not as obvious as those for τr2 −τr1 = τd , but are more significant than those for

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Fig. 5. h s as a function of kr2 for specific values of kr1 , f = 20 GHz, τr1 = 3 ns, and τr2 = 4 ns.

Fig. 6. h s as a function of kr1 (kr2 = kr1 ) for different combinations of the time delay. (a) τd = 3 ns and τr1 = 5 ns. (b) τd = 2 ns and τr1 = 3 ns. f = 20 GHz.

SPI. To verify it, we have done lots of calculations for different combinations of time delay. For example, it can be seen from Fig. 6(b) that similar results are obtained when τd = 2ns. Note that, the relative low PE values for τr2 − τr1 = τd can be explained by the external-cavity modes caused periodicity in the chaotic signals in MSL. II. C ONCLUSION In summary, we have demonstrated experimentally and numerically that, by simply adding an additional path in the conventional master-slave configuration, the bandwidth and randomness can be enhanced significantly in wider injection parameter region compared to SSL with conventional SPI, and positive frequency detuning is preferred. In addition, for DPI, when the injection delay difference is chosen at the feedback delay of MSL, the enhancement may not so obvious as other cases. This low-cost simple configuration for generation of wideband randomness-enhanced chaos in SSL with DPI from single MSL is extremely useful to increase the bit rate and randomness quality of high speed RNGs based on chaotic SLs. R EFERENCES [1] A. Argyris, et al., “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature, vol. 437, pp. 343–346, Nov. 2005. [2] J. Ohtsubo, “Chaos synchronization and chaotic signal masking in semiconductor lasers with optical feedback,” IEEE J. Quantum Electron., vol. 38, no. 9, pp. 1141–1154, Sep. 2002.

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