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We experimentally show that a random optical pulse train can be generated by modulating a bistable semiconductor ring laser. When the ring laser is switched from the monostable to the bistable regime, it randomly selects one of two different stable unidirectional lasing modes, clockwise or counterclockwise modes. Non-deterministic random pulse sequences are generated by driving the switch parameter, the injection current, with a periodic pulse signal. The origin of the nondeterministic randomness is the amplified spontaneous emission noise coupled to the counter-propagating lasing modes. The statistical randomness properties are optimized by adjusting the relative strength of amplified spontaneous emission noise sources for the two lasing modes. It is also shown that it is possible to generate optical pulse sequences which pass a standard suite of statistical randomness tests.
©2011 Optical Society of America
1. Introduction
Semiconductor ring lasers (SRLs) exhibit a rich variety of operating regimes, such as bidirectional operation, alternative oscillation, chaotic behavior, and unidirectional bistability, due to highly nonlinear interaction and backscattering coupling between the clockwise (CW) and counter-clockwise (CCW) propagating waves [1–7]. In particular, unidirectional bistability, i.e. the coexistence of two stable unidirectional lasing states corresponding to the CW and CCW propagating modes, has attracted much attention because of potential applications in optical logic systems [8–11]. In the bistable operation regime, the SRLs can be switched from one direction to another by means of an external optical injection. Various potentially useful functions using this property have been demonstrated, such as optical flip-flop and gate functions [8–11].
Recently, dynamics of semiconductor lasers have been used for physical random bit generation [12–14]. Using dynamical phenomena can realize efficient methods to transform microscopic noises such as spontaneous emission noises to macroscopic random signals. In particular, chaotic lasers with delayed optical feedback have been used to achieve fast bit generation at gigabits per second. [14–18]. The non-deterministic behavior of the chaotic lasers can be theoretically explained on the basis of nonlinear amplification of spontaneous emission noises and the mixing property of chaotic dynamics.
On the other hand, in the case of the SRLs operating in the bistable regime, spontaneous emission noises can directly cause non-deterministic switching between the two stable lasing states. As demonstrated in Ref. [19], stochastic mode-hopping induced by spontaneous emission noise occurs in a particular regime of SRL injection current, where the switching time intervals are non-deterministic. However, according to Ref. [13], a complicated post-processing is needed to produce high-quality random bit sequences from the switching time intervals.
In this paper, we show that non-deterministic random binary signal generation with high-quality statistical randomness can be achieved by using the bistable regime of the SRLs if the state of the SRL is repeatedly re-set to an unstable initial state in which the light intensities and spontaneous emission noise of CW and CCW modes are balanced. The final lasing state in the bistable regime is then strongly affected even by the small perturbations due to the spontaneous emission noises, and becomes unpredictable. The setting of the initial state to the unstable state can be easily achieved by controlling the SRL injection current, and the noise can be balanced by controlling the spontaneous emission noise injected into each mode. The method can be theoretically modeled using semiconductor ring laser equations including spontaneous emission. Moreover, in this paper, we show an experimental demonstration of a random binary optical pulse train generation on the basis of the above method by pulse-modulating the injection current to the SRL toward the bistable regime. The generation rate of the random pulses is also easily controlled by the modulation rate of the injection current. It is also shown that the probability of selection of each mode is controlled by controlling the strength of an amplified spontaneous emission source. It is confirmed that binary optical pulse sequences obtained from this experiment pass a standard statistical test suite for randomness, known as the Diehard test.This random pulse signal generator is more compact than the existing random bit generators using other optical phenomena [12–18], and is more suitable for generating random optical pulses.
2. Method of random optical pulse generation
2.1. Model for SRLs
First, let us review the dynamical properties of SRLs before we explain the method of random optical pulse generation. As reported in many papers [1–7], SRLs exhibit various types of dynamical behaviors due to the linear and nonlinear couplings between the counter-propagating waves induced by the backscattering of the ring waveguide and the gain saturation, respectively. The behavior of the SRL is described by a standard semiconductor ring laser model [2,3,20], which consists of equations for two slowly varying complex amplitudes of the counter-propagating waves E 1 (CW mode) and E 2 (CCW mode) taking into account the effect of spontaneous emission noise,
and the rate equation for normalized carrier density N, where time t is made dimensionless by the scale transformation t/τp → t. In the above, τp is the photon lifetime, α is the linewidth enhancement factor, s and c are respectively the (dimension-less) self and cross saturation coefficients. kd and kc represent the dissipative and conservative components of the backscattered field, respectively. For sake of convenience, without loss of generality, it is assumed that kd is a positive value. The last term of Eq. (1) represents the effect of spontaneous emission noise coupled to the CW(CCW) mode: D represents the noise strength expressed as follow, D = Cs(N + G 0 τpN 0), where Cs is the spontaneous emission factor, G 0 is the differential gain, N 0 is the transparent carrier density. ξ1(2) are two independent complex white Gaussian noises with zero mean and unitary variance. In Eq. (2), γ is the ratio of the photon lifetime to the carrier lifetime, and μ is the normalized pumping power.At the normalized pumping power, μ ≈ 1, laser action starts. When the pumping power μ is increased but close to the threshold, i.e., for weak nonlinear coupling, the linear coupling due to the backscattering is a relatively dominant factor characterizing the dynamical behavior of the SRL. In this case, a bidirectional operation is induced, where the CW and CCW modes oscillate with the same amplitudes, |E 1 | = |E 2 |. For large pumping power μ ≫ 1, the nonlinear coupling due to the gain saturation is enhanced and the mode-competition is caused between the CW and CCW modes. Therefore, the SRL operates at either of the CW or CCW lasing states, and a bistability is exhibited between the two stable lasing states. For intermediate values of μ, an oscillatory behavior (called alternative oscillation) and multi-stable operations are observed depending on the values of the backscattering terms kd and kc [3, 21].
2.2. Random pulse generation by SRLs
The bidirectional and bistable regimes are used for random pulse generation. The two operation regimes are characterized by the stationary solutions of Eqs. (1) and (2) without spontaneous emission noise terms. In the bidirectional regime, there exists one stable stationary solution of Eqs. (1) and (2), in which the two modes have the same field amplitude , the carrier density N 0 = (1 – 2kd)/(1 – (s + c)|E 0 | 2), and relative phase between the CW and CCW modes . We call the stable solution B and plot it in a two-dimensional phase space of the system for the CW and CCW intensities, (|E 1 | 2 ,|E 2 | 2) [see Fig. 1 (a)], where O is the unstable fixed point corresponding to non-lasing state (|E 1 | = |E 2 | = 0). In this regime, any arbitrary initial states always converge to the stable state B. However, according to the stability analysis for the stable solution B [2], it loses stability when the pumping power μ is increased larger than a certain critical value μ 1, which is obtained by solving the following equation,
where K = 1/2N 0 |E 0 | 2(c – s). For μ > μ 1, the solution B can be described as an unstable saddle solution in the sense that it has unstable manifolds in a space where the total power is conserved (|E 1 | 2 +|E 2 | 2 = μ – 1) and a stable manifold along symmetry line |E 1 | 2 = |E 2 | 2 with the relative phase Ψ = π owing to the symmetry of Eqs. (1) and (2).
Fig. 1 (a) The phase space in a bidirectional regime. O; unstable fixed point corresponding to the non-lasing state, B; stable fixed point corresponding to the bidirectional lasing state. The line |E 1 | 2 = |E 2 | 2 is shown as a blue dotted line. In this regime, evolution from any initial state converges to the stable state B. (b) The phase space in a bistable regime. S; saddle fixed point corresponding to the bidirectional lasing state, UCW ( CCW ); stable fixed point corresponding to the CW (CCW) lasing state. Blue dotted line (line |E 1 | 2 = |E 2 | 2) represents the stable manifold of the saddle point S. Red and green curves represent two trajectories starting from an initial state (indicated by open circle) on the stable manifold with different noise instances. The trajectories are numerically calculated by using Eqs. (1) and (2) with the parameter values for our SRL devices, α = 3.5, 2s = c = 0.01, kd = 3.27 × 10−4, kc = 2.5 × 10−3, γ = 0.02, μ = 4.921, Cs = 5 × 10−6, τp = 40 ps, G 0 = 10−12 m3s−1, N 0 = 1.4 × 1024m−3.
The bistable regime is realized when the pumping power μ is much larger than a certain value μ 2 (> μ 1), which mainly depends on the value of the backscattering terms (kd and kc) and gain saturation terms s and c [3]. In this regime, there exsit two stable solutions corresponding to lasing states dominated by either CW or CCW propagation (i.e. where |E 1 | 2 > |E 2 | 2, or vice versa). These two solutions emerge either from a pitch-fork bifurcation of the saddle solution (bidirectional solution) or a saddle node bifurcation [21]. In particular, for large pumping power μ ≫ μ 2, the backscattering coupling is negligible compared to the nonlinear coupling. Therefore, the dynamical behavior of this regime does not depend on the relative phase Ψ and can be essentially described by projection onto a two-dimensional phase space (|E 1 | 2 , |E 2 | 2). Figure 1 (b) shows the phase space (|E 1 | 2 , |E 2 | 2), where O is the unstable fixed point corresponding to non-lasing state (|E 1 | = |E 2 | = 0), UCW ( CCW ) is the stable fixed point corresponding to the CW (CCW) stable solutions, and S is the saddle point corresponding to the bidirectional solution. The blue dotted line denotes the stable manifold of the saddle point S along the line |E 1 | 2 = |E 2 | 2. The manifold separates the basins of attraction of the stable fixed points UCW and UCCW and plays an important role in switching due to external perturbations. For instance, switching from the CW (CCW) mode to the CCW (CW) mode can be realized with an optical injection pulse causing a transition of the state of the SRL from the stable fixed point UCW ( CCW ) to a point in the basin of CCW (CW) mode on the other side of the the stable manifold of the saddle point, so that it subsequently evolves autonomously to the other stable fixed point UCCW ( CW ) [21, 22]. This switching operation is very robust with respect to microscopic noises and so the switching is deterministic [20, 23].
If the initial state is set on the stable manifold of the saddle point, i.e., the boundary of the two basins, the dynamics will be sensitive to noise, so that the final lasing state of the system will be unpredictable. In real lasers, there always exist spontaneous emission noise. If the noise is equally coupled to the CW and CCW lasing modes, the selection process of the lasing modes is expected to be statistically random. In the example, two trajectories starting from the same initial state on the stable manifold with different noise instances are shown in Fig. 1 (b). These trajectories were numerically calculated by using Eqs. (1) and (2) with the spontaneous emission noise terms. During the transient relaxation oscillation, the trajectories separate due to the noise perturbing them to different sides of the unstable manifold of the saddle point S, and they relax toward different final states UCW or UCCW.
The setting of the initial state on the stable manifold and the random optical pulse generation can be carried out by changing the pumping power and switching the dynamics of the SRL between the bidirectional and bistable regimes: First, the injection current to the SRL is adjusted so that the SRL operates in the bidirectional regime. In this case, the state of the system always relaxes to the stable point B in the phase space of the bidirectional regime (see Fig. 1 (a)). It is important to note that the stable point B corresponds to a point on the stable manifold of the saddle point S in the bistable regime. This means that when the injection current is suddenly increased so that the SRL operates in the bistable regime, the state is lain on the stable manifold of the saddle point S in a bistable regime, as indicated by open circle in Fig. 1 (b). However, since the spontaneous emission noises are always coupled to the counter-propagating modes, the fluctuation of the state of the system due to the noises is amplified by the unstable manifold of the saddle point S. Consequently, the state of the system relaxes to either of UCW or UCCW. The resetting of the final lasing state can be achieved by again decreasing the injection to the bidirectional regime and relaxing to stable point B. Accordingly, the stochastic mode-selection is repeated by the modulation of the injection current between the bidirectional and bistable regimes, so that a random optical pulse train can be emitted in the CW and CCW directions.
2.3. Control of spontaneous emission noises
However, in the actual SRL devices, the spontaneous emission will not be isotropic due to material non-uniformities, and they will not be equally coupled to the CW and CCW modes. Thus, actual SRL devices have a preferred direction, and the dominant output direction is reproducible [5]. For achieving the random operation with the equal probability of the appearance of the CW or CCW lasing state, the amounts of the spontaneous emission noises coupled to the CW and CCW modes should be controlled so that the asymmetry of the coupling is reduced. We show that this is achieved by using two spontaneous emission noise sources. Figure 2 shows a schematic of a SRL device with two spontaneous emission noises sources B1 and B2. The noises emitted from B1 and B2 are injected into a ring laser part in the CCW and CW direction via a weakly coupled waveguide used as a directional coupler. For example, when B1 is active, the amount of the spontaneous emission coupled to the CCW mode can be enhanced. A similar method for the control of the amounts of the spontaneous emission has been used for achieving the switching operation from CW (CCW) mode to CCW (CW) mode [1].
3. SRL device: design and fabrication
In order to implement the random optical pulse generation scheme mentioned in the previous section, a SRL device was designed and fabricated in a InP/InGaAsP material system with an active-passive integration. Figures 3 (a) and (b) show the schematic and picture of the fabricated device. The ring laser cavity consists of a 1.5 mm-long semiconductor optical amplifier (SOA) as a gain medium and a passive ring waveguide of 2 μm width to form a ring cavity. The racetrack-shaped ring has a 1.25 mm radius of curvature and 2.65 mm-long straight sections. The total cavity length is 12.3 mm.
Fig. 3 (a) A schematic and (b) a photograph of the fabricated SRL device. The ring laser part consists of a semiconductor optical amplifier (SOA) and a passive waveguide. B1 and B2 are 50 μm-long SOAs used to enhance the amounts of spontaneous emission noises coupled to the CCW and CW modes, respectively. They are also used as the photodiodes to detect the CW and CCW output signals, when they are not used as the ASE noise sources.
B1 and B2 are 50 μm-long SOAs used to control the amounts of the spontaneous emission noises coupled to the CCW and CW modes, respectively. Applying forward bias to B1 (B2), the amplified spontaneous emission noise is injected to the ring laser part in the CCW (CW) direction. B1 and B2 can be also used as photodiodes to detect the output signals of the CW and CCW modes, when forward biases are not applied and they are not used as the noise sources.
This active-passive integrated SRL device is designed so that the gain of the SRL is localized in one section of the ring cavity and the asymmetry of amplified spontaneous emission noise coupling induced by the gain section is as small as possible. Then, the device size is decided so that the effect of the directional coupler (the curved waveguide in Fig. 3) is reduced as far as possible: This is because as the device size is larger, the backscattering effect due to the directional coupler is relatively reduced and the injection current needed for the achievement of the bistable operation is decreased [6]. The control of the spontaneous emission noise coupling and the device size are important points for achieving random operation.
This SRL device was fabricated as follows: First, a strained multi-quantum well (MQW) layer, the non-doped InGaAsP separate-confinement-heterostructure layer, and p-InP cladding layer were grown by using metal-organic vapor phase epitaxy on the n-InP substrate in order to make SOA, B1, and B2. Then, the unnecessary part was etched off using a SiO2 mask, and the passive layer and the intrinsic InP cladding layer were made by butt-joint selective growth. After the SiO2 was etched off, a p-InP cladding layer and InGaAsP contact layers were made over the whole region. Then the waveguide patterns were fabricated using a SiO2 mask. The contact areas and the metals for the positive electrodes were separated for SOA, B1 and B2. To reduce the facet reflectivity, the cleaved facet of the output waveguide was anti-reflection coated at wavelength 1.55 μm.
The fabricated SRL device is soldered on a temperature-controlled mount with stability ± 0.01 °C, and the output signals from the B1 and B2 are respectively extracted via microstrip lines with 50-Ω chip resistors for impedance matching.
4. Experimental results
4.1. L-I characteristics
The static property of the fabricated SRL device is evaluated by dc light-injection current (LI) characteristics. The typical L-I curves are shown in Fig. 4. These curves are plotted with photocurrent response taken simultaneously from B1 and B2 used as photodiodes. The bidirectional regime is observed when the injection current J is larger than the threshold current 52.6 mA and smaller than about 70 mA. For a regime of relatively weak injection current just above the threshold, the two counter-propagating modes have almost the same intensities. For 70 mA < J < 81.3 mA, a mode-hopping phenomenon is observed with hopping between the CW and CCW modes. This can be explained as a noise-induced hopping between two attractors corresponding to the CW and CCW modes [19]. In our device, the residence time is very long, typically the order of μs, and it is strongly dependent on the injection current. so that the L-I characteristics are strongly affected by the output averaging time. We note that the mode-hopping dynamics has a long correlation time unsuitable for fast random signal generation. For J > 81.3 mA, the laser device operates stably in either of the unidirectional modes (in the CW direction in Fig. 4) i.e., the operation is bistable. In this regime, a single frequency operation, with a high mode-extinction ratio between the CW mode and the CCW mode, is obtained due to the spatial hole burning and the mode competition [24].
Fig. 4 L-I curves of the fabricated SRL device. The bidirectional and bistable regimes are observed for 52.6 mA < J < 70 mA and J > 81.3 mA, respectively. The mode-extinction ratio of the CW mode to the CCW mode is more than 16 dB for J > 117 mA.
4.2. Generation of random optical pulse train
A random pulse train was generated by modulating the injection current of the SRL device. The typical example of the generated pulse train is shown in Fig. 5 (a). The injection current is modulated with a square wave modulation. In this case, B1 and B2 are both used as photodiodes without applying forward biases to them. It is seen that the SRL switches between CW and CCW lasing states in a random sequence. Here, the low level J 0 of the pulse current is set to be 55mA to realize the bidirectional operation, while the high level J 1 is 117mA to realize the unidirectional bistable operation (see Fig. 4). The repetition rate is 3 MHz. In order to avoid the influence of the mode-hopping fluctuation, the rise and fall times are set to be 2.5 ns, which is much faster than the characteristic time of the mode-hopping fluctuation (∼ 1 μs). A transient process of the operation is shown in Fig. 5 (b) (the enlargement in Fig. 5 (a)): When the injection current is increased from J 0, initially the light intensities of both the two modes both increase. After the injection current reaches J 1 the mode competition starts, and after short time the SRL starts to operate in just one direction, in this instance, the CCW direction. When the injection current is decreased toward the bidirectional regime, the intensity of the lasing modes suddenly decreases (the fall time ∼ 2.5 ns) and the SRL operates in the bidirectional mode again, with small equal intensity in each direction. The memory of the previous lasing state vanishes.
Fig. 5 (a) Time traces of the CW (red) and CCW (green) light intensities under the pulse-modulation of the injection current to the SRL. The upper panel represents the pulse current. J 0 and J 1 are respectively 55 mA and 117 mA. (b) The enlargement of the left figure (a).
Similar random pulse sequences can be obtained for current modulation rates up to 10 MHz. The limit is decided by the rise time of the optical random pulse i.e., the time required by the laser to switch between the bidirectional mode and a bistable unidirectional mode. In order to achieve faster generation rates of the random pulse signals in the present scheme, it would be needed to optimize the parameters of the device structure so that the mode-competition interaction is made stronger. Investigations along this line will be reported elsewhere.
5. Dependence on noise bias current
Next, let us explain the dependence of the random pulse generation on the bias current of the spontaneous emission noise source. As mentioned in section 2.3, if the noise sources B1 and B2 are not used, the statistical frequency ratio of the appearance of CW (CCW) lasing modes will never be equal to 50 % due to material non-uniformities and the asymmetry of the amounts of the noise coupled into the counter-propagating modes from sources inside the cavity. Actually, the statistical frequency for the CW lasing state was about 16.6 % in our SRL when noise sources B1 and B2 are not used. In order to increase the amount of the amplified spontaneous emission noise coupled to the CW mode and make the statistical frequency of the appearance of the CW lasing mode close to 50 %, a forward bias is applied to B2, and the amount of the noise coupled to the CW lasing modes is enhanced. Figure 6 shows the dependence of the statistical frequency for the CW mode lasing on the bias current applied to B2. In this experiment, B1 was used as photodiode to detect the output signals in the CW direction. The current value applied to B2 was maintained with accuracy ± 0.01 mA. The statistical frequency ratio was calculated for 105 samples obtained from random pulse signals generated at rate 10 MHz. The error bars represent three standard deviations. It is clearly seen that with increase of the bias current, the frequency monotonically increases. In particular, for the bias current 20.11mA applied to B2, the ratio of CW pulses becomes close to 50 % with three standard deviation of about 0.3 %. The value of the statistical frequency ratio is stably attained over many hours of continual operation.
Fig. 6 (a) The bias current dependence of the statistical frequency of the appearance of the CW lasing mode. Dots and error bars respectively represent the mean value and three standard deviations. (b) An enlargement of the left figure (a).
6. Statistical randomness property
Finally, we evaluate the statistical randomness of the generated optical pulse signals by converting them to binary bit sequences. In the conversion, the pulse signals (in the CW direction) obtained from the photodiode B1 were assigned to bits 0 and 1 by comparing them with a threshold. Figure 7 shows the absolute values of the autocorrelation function for the bit sequences of length N = 106 obtained for the bias current 20.11mA. It can be clearly confirmed that the correlation is statistically insignificant i.e., no larger than could be expected from a truly random sequence.
Fig. 7 The absolute values of the autocorrelation function of the random optical bit sequences of length N = 106 generated from the experiment.
We evaluated the statistical randomness by using a standard statistical test suite for randomness, the so-called ”Diehard test” [25]. As shown in Table 1, the Diehard test consists of 18 tests and is performed using 92 Mbit data obtained from the experiment for modulation rates up to 10 MHz and statistical significance level, α = 0.01, which means that the p-value of each test should be in the range of [0.01,0.99]. We confirmed that the bit sequences passed all of the Diehard tests at this significance level. A typical result is shown in Table 1. These results confirm that the generated optical pulse train is statistically random to a strict level of statistical significance.
Table 1. Typical Results of “Diehard” Statistical Test Suitea
7. Conclusion
We experimentally demonstrated that random selection of two counter-propagating CW and CCW modes can be achieved by switching to directional bistability in a SRL with amplified spontaneous emission noises. It was shown that a random optical pulse train can be generated by pulse-modulating the injection current of the SRL so that a directional mode is selected randomly during each injection current pulse. We found that although asymmetries in actual SRL devices degrade the statistical quality of the generated random signals, it can be improved by controlling the amounts of the amplified spontaneous emission noises coupled to the counter-propagating modes. It was checked that with the noise control technique, the generated optical bit sequences pass a standard statistical test suite for randomness, the Diehard test suite. Hence we have shown that the bistable SRL device could be used not only as optical logic element as previously proposed, but also as compact random optical signal generators with applications in ranging, sensing and random bit generation.
Acknowledgments
The authors would like to thank the members of NTT Communication Science Laboratories for their support.
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