## Random optical pulse generation with bistable semiconductor ring lasers |

Optics Express, Vol. 19, Issue 8, pp. 7439-7450 (2011)

http://dx.doi.org/10.1364/OE.19.007439

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### Abstract

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 OSA

## 1. Introduction

1. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. **80**, 3051–3053 (2002). [CrossRef]

7. A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional ring lasers,” Phys. Rev. A **81**, 043817 (2010). [CrossRef]

8. M. T. Hill, H. J. S. Dorren, T. Vries, X. J. M. Leijtens, J. H. Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature **432**, 206–209 (2004). [CrossRef] [PubMed]

11. A. Trita, M. J. Latorre Vidal, M. Zanola, G. Mezosi, J. Javaloyes, M. Sorel, F. Bragheri, I. Cristiani, A. Scirè, S. Balle, and G. Giuliani, “All-Optical Set-Reset Flip-Flop based on Semiconductor Ring Laser: Ultrafast Response and Error-Free Bit-Error-Rate Operation,” in *International Conference on Photonics in Switching* (IEEE, 2009), pp. 1–2. [CrossRef]

8. M. T. Hill, H. J. S. Dorren, T. Vries, X. J. M. Leijtens, J. H. Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature **432**, 206–209 (2004). [CrossRef] [PubMed]

11. A. Trita, M. J. Latorre Vidal, M. Zanola, G. Mezosi, J. Javaloyes, M. Sorel, F. Bragheri, I. Cristiani, A. Scirè, S. Balle, and G. Giuliani, “All-Optical Set-Reset Flip-Flop based on Semiconductor Ring Laser: Ultrafast Response and Error-Free Bit-Error-Rate Operation,” in *International Conference on Photonics in Switching* (IEEE, 2009), pp. 1–2. [CrossRef]

12. B. Qi, Y-M. Chi, H-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. **35**, 312–314 (2010). [CrossRef] [PubMed]

14. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Karashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical bit generation with chaotic semiconductor lasers,” Nat. Photonics **2**, 728–732 (2008). [CrossRef]

14. A. Uchida, K. Amano, M. Inoue, K. Hirano, S. Naito, H. Someya, I. Oowada, T. Karashige, M. Shiki, S. Yoshimori, K. Yoshimura, and P. Davis, “Fast physical bit generation with chaotic semiconductor lasers,” Nat. Photonics **2**, 728–732 (2008). [CrossRef]

18. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express **18**, 18763–18768 (2010). [CrossRef] [PubMed]

19. S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Scirè, G. Mezosi, M. Sorel, and J. Danckaert, “Topological Insight into the Non-Arrhenius Mode Hopping of Semiconductor Ring Lasers,” Phys. Rev. Lett. **101**, 093903 (2008). [CrossRef] [PubMed]

12. B. Qi, Y-M. Chi, H-K. Lo, and L. Qian, “High-speed quantum random number generation by measuring phase noise of a single-mode laser,” Opt. Lett. **35**, 312–314 (2010). [CrossRef] [PubMed]

18. A. Argyris, S. Deligiannidis, E. Pikasis, A. Bogris, and D. Syvridis, “Implementation of 140 Gb/s true random bit generator based on a chaotic photonic integrated circuit,” Opt. Express **18**, 18763–18768 (2010). [CrossRef] [PubMed]

## 2. Method of random optical pulse generation

### 2.1. Model for SRLs

1. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. **80**, 3051–3053 (2002). [CrossRef]

7. A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional ring lasers,” Phys. Rev. A **81**, 043817 (2010). [CrossRef]

2. M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillation in semiconductor ring lasers,” Opt. Lett. **27**, 1992–1994 (2002). [CrossRef]

3. M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers,” IEEE J. Quantum Electron. **39**, 1187–1195 (2003). [CrossRef]

20. T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express **15**, 12941–12948 (2007). [CrossRef] [PubMed]

*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/τ*→

_{p}*t*. In the above,

*τ*is the photon lifetime,

_{p}*α*is the linewidth enhancement factor,

*s*and

*c*are respectively the (dimension-less) self and cross saturation coefficients.

*k*and

_{d}*k*represent the dissipative and conservative components of the backscattered field, respectively. For sake of convenience, without loss of generality, it is assumed that

_{c}*k*is a positive value. The last term of Eq. (1) represents the effect of spontaneous emission noise coupled to the CW(CCW) mode:

_{d}*D*represents the noise strength expressed as follow,

*D*=

*C*(

_{s}*N*+

*G*

_{0}

*τ*

_{p}N_{0}), where

*C*is the spontaneous emission factor,

_{s}*G*

_{0}is the differential gain,

*N*

_{0}is the transparent carrier density. ξ

_{1(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.

*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

*k*and

_{d}*k*[3

_{c}3. M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers,” IEEE J. Quantum Electron. **39**, 1187–1195 (2003). [CrossRef]

21. L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E **79**, 016213 (2009). [CrossRef]

### 2.2. Random pulse generation by SRLs

*N*

_{0}= (1

*–*2

*k*)

_{d}*/*(1

*–*(

*s*+

*c*)

*|E*

_{0}

*|*

^{2}), and relative phase between the CW and CCW modes

*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

2. M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillation in semiconductor ring lasers,” Opt. Lett. **27**, 1992–1994 (2002). [CrossRef]

*μ*is increased larger than a certain critical value

*μ*

_{1}, which is obtained by solving the following equation, where

*K*= 1

*/*2

*N*

_{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).

*μ*is much larger than a certain value

*μ*

_{2}(

*> μ*

_{1}), which mainly depends on the value of the backscattering terms (

*k*and

_{d}*k*) and gain saturation terms

_{c}*s*and

*c*[3

3. M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers,” IEEE J. Quantum Electron. **39**, 1187–1195 (2003). [CrossRef]

*|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

21. L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E **79**, 016213 (2009). [CrossRef]

*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),

*U*

_{CW}_{(}

_{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

*U*and

_{CW}*U*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

_{CCW}*U*

_{CW}_{(}

_{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

*U*

_{CCW}_{(}

_{CW}_{)}[21

21. L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E **79**, 016213 (2009). [CrossRef]

22. L. Gelens, S. Beri, G. Van der Sande, J. Danckaert, N. Calabretta, H. J. S. Dorren, R. Nötzel, E. A. J. M. Bente, and M. K. Smit, “Optical injection in semiconductor ring lasers: backfire dynamics,” Opt. Express **16**, 10968–10974 (2008). [CrossRef] [PubMed]

20. T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express **15**, 12941–12948 (2007). [CrossRef] [PubMed]

23. B. Li, M. I. Memon, G. Mezosi, G. Yuan, Z. Wang, M. Sorel, and S. Yu, “All-Optical Response of Semiconductor Ring Laser to Dual-Optical Injections,” IEEE Photon. Technol. Lett. **20**, 770–772 (2008). [CrossRef]

*S*, and they relax toward different final states

*U*or

_{CW}*U*.

_{CCW}*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

*U*or

_{CW}*U*. The resetting of the final lasing state can be achieved by again decreasing the injection to the bidirectional regime and relaxing to stable point

_{CCW}*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

5. M. F. Booth, A. Schremer, and J. M. Ballantyne, “Spatial beam switching and bistability in a diode ring laser,” Appl. Phys. Lett. **76**, 1095–1097 (2000). [CrossRef]

*B*1 and

*B*2. The noises emitted from

*B*1 and

*B*2 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

*B*1 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

1. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguide ring lasers,” Appl. Phys. Lett. **80**, 3051–3053 (2002). [CrossRef]

## 3. SRL device: design and fabrication

*μ*m 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.

*B*1 and

*B*2 are 50

*μ*m-long SOAs used to control the amounts of the spontaneous emission noises coupled to the CCW and CW modes, respectively. Applying forward bias to

*B*1 (

*B*2), the amplified spontaneous emission noise is injected to the ring laser part in the CCW (CW) direction.

*B*1 and

*B*2 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.

6. G. Mezösi, M. J. Strain, S. Fürst, Z. Wang, S. Yu, and M. Sorel, “Unidirectional Bistability in AlGaInAs Microring and Microdisk Semiconductor Lasers,” IEEE Photon. Technol. Lett. **21**, 88–90 (2009). [CrossRef]

*B*1, and

*B*2. Then, the unnecessary part was etched off using a SiO

_{2}mask, and the passive layer and the intrinsic InP cladding layer were made by butt-joint selective growth. After the SiO

_{2}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 SiO

_{2}mask. The contact areas and the metals for the positive electrodes were separated for SOA,

*B*1 and

*B*2. To reduce the facet reflectivity, the cleaved facet of the output waveguide was anti-reflection coated at wavelength 1.55

*μ*m.

*±*0.01 °C, and the output signals from the

*B*1 and

*B*2 are respectively extracted via microstrip lines with 50-Ω chip resistors for impedance matching.

## 4. Experimental results

### 4.1. L-I characteristics

*B*1 and

*B*2 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

19. S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Scirè, G. Mezosi, M. Sorel, and J. Danckaert, “Topological Insight into the Non-Arrhenius Mode Hopping of Semiconductor Ring Lasers,” Phys. Rev. Lett. **101**, 093903 (2008). [CrossRef] [PubMed]

*μ*s, 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

24. M. Choi, T. Tanaka, S. Sunada, and T. Harayama, “Linewidth properties of active-passive coupled monolithic InGaAs semiconductor ring lasers,” Appl. Phys. Lett. **94**, 231110 (2009). [CrossRef]

### 4.2. Generation of random optical pulse train

*B*1 and

*B*2 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

*μ*s). 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.

## 5. Dependence on noise bias current

*B*1 and

*B*2 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

*B*1 and

*B*2 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

*B*2, 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

*B*2. In this experiment,

*B*1 was used as photodiode to detect the output signals in the CW direction. The current value applied to

*B*2 was maintained with accuracy

*±*0.01 mA. The statistical frequency ratio was calculated for 10

^{5}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

*B*2, 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.

## 6. Statistical randomness property

*B*1 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*= 10

^{6}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.

25. G. Marsaglia, DIEHARD: A battery of tests of randomness (1996). http://stat.fsu.edu/geo.

*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.

## 7. Conclusion

## Acknowledgments

## References and links

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2. | M. Sorel, P. J. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillation in semiconductor ring lasers,” Opt. Lett. |

3. | M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, S. Donati, and P. J. R. Laybourn, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers,” IEEE J. Quantum Electron. |

4. | W. Coomans, S. Beri, G. Van der Sande, L. Gelens, and J. Danckaert, “Optical injection in semiconductor ring lasers,” Phys. Rev. A |

5. | M. F. Booth, A. Schremer, and J. M. Ballantyne, “Spatial beam switching and bistability in a diode ring laser,” Appl. Phys. Lett. |

6. | G. Mezösi, M. J. Strain, S. Fürst, Z. Wang, S. Yu, and M. Sorel, “Unidirectional Bistability in AlGaInAs Microring and Microdisk Semiconductor Lasers,” IEEE Photon. Technol. Lett. |

7. | A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional ring lasers,” Phys. Rev. A |

8. | M. T. Hill, H. J. S. Dorren, T. Vries, X. J. M. Leijtens, J. H. Besten, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature |

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19. | S. Beri, L. Gelens, M. Mestre, G. Van der Sande, G. Verschaffelt, A. Scirè, G. Mezosi, M. Sorel, and J. Danckaert, “Topological Insight into the Non-Arrhenius Mode Hopping of Semiconductor Ring Lasers,” Phys. Rev. Lett. |

20. | T. Pérez, A. Scirè, G. Van der Sande, P. Colet, and C. R. Mirasso, “Bistability and all-optical switching in semiconductor ring lasers,” Opt. Express |

21. | L. Gelens, G. Van der Sande, S. Beri, and J. Danckaert, “Phase-space approach to directional switching in semiconductor ring lasers,” Phys. Rev. E |

22. | L. Gelens, S. Beri, G. Van der Sande, J. Danckaert, N. Calabretta, H. J. S. Dorren, R. Nötzel, E. A. J. M. Bente, and M. K. Smit, “Optical injection in semiconductor ring lasers: backfire dynamics,” Opt. Express |

23. | B. Li, M. I. Memon, G. Mezosi, G. Yuan, Z. Wang, M. Sorel, and S. Yu, “All-Optical Response of Semiconductor Ring Laser to Dual-Optical Injections,” IEEE Photon. Technol. Lett. |

24. | M. Choi, T. Tanaka, S. Sunada, and T. Harayama, “Linewidth properties of active-passive coupled monolithic InGaAs semiconductor ring lasers,” Appl. Phys. Lett. |

25. | G. Marsaglia, DIEHARD: A battery of tests of randomness (1996). http://stat.fsu.edu/geo. |

**OCIS Codes**

(140.3560) Lasers and laser optics : Lasers, ring

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.1450) Nonlinear optics : Bistability

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: December 23, 2010

Revised Manuscript: February 28, 2011

Manuscript Accepted: March 9, 2011

Published: April 4, 2011

**Citation**

Satoshi Sunada, Takahisa Harayama, Kenichi Arai, Kazuyuki Yoshimura, Ken Tsuzuki, Atsushi Uchida, and Peter Davis, "Random optical pulse generation with bistable semiconductor ring lasers," Opt. Express **19**, 7439-7450 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7439

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### References

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