## Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback

Optics Express, Vol. 17, Issue 12, pp. 10025-10034 (2009)

http://dx.doi.org/10.1364/OE.17.010025

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### Abstract

We investigate the dynamics of two semiconductor lasers with separate optical feedback when they are driven by a common signal injected from a chaotic laser under the condition of non-identical drive and response. We experimentally and numerically show conditions under which the outputs of the two lasers can be highly correlated with each other even though the correlation with the drive signal is low. In particular, the effects of the phase of the feedback light on the correlation characteristics are described. The maximum correlation between the two response lasers is obtained when the phase of the feedback light is matched between the two response lasers, while the minimum correlation is observed when the difference in the optical phase is π. On the other hand, the correlation between the drive and response is not sensitive to the phase of the feedback light, unlike the previously studied case of identical drive and response. We numerically examine the difference between the maximum and minimum cross correlations over a wide range of parameters, and show that it is largest when there is a balance between the injection strength and the feedback strength.

© 2009 Optical Society of America

## 1. Introduction

1. S. Sivaprakasam and K. A. Shore, “Demonstration of optical synchronization of chaotic external-cavity laser diodes,” Opt. Lett. **24**, 466–468 (1999).
[CrossRef]

6. S. Tang and J. M. Liu, “Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers,” Phys. Rev. Lett. **90**, 194101 (2003).
[CrossRef] [PubMed]

7. T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. Goto, and P. Davis, “Common-chaotic-signal induced synchronization in semiconductor lasers, ” Opt. Express **15**, 3974–3980 (2007).
[CrossRef] [PubMed]

## 2. Experimental setup

7. T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. Goto, and P. Davis, “Common-chaotic-signal induced synchronization in semiconductor lasers, ” Opt. Express **15**, 3974–3980 (2007).
[CrossRef] [PubMed]

*I*were 8.7 mA (Drive), 7.6 mA (Response 1), and 9.2 mA (Response 2), respectively.

_{th}*I*) for Response 1, and 11.90 mA (1.29

_{th}*I*) for Response 2, respectively. We set the optical wavelength to 1547.333 nm for the solitary Drive laser and 1547.308 nm for both the solitary Response 1 and 2 lasers, corresponding to wavelength detuning between the Drive and the Response lasers of -0.025 nm (-3.125 GHz), so that optical injection locking could be achieved. The injection strengths from the Drive to the two Response lasers are adjusted to achieve injection locking. Under conditions for injection locking, the wavelengths of the Response lasers were pulled to that of the Drive so all three optical wavelengths were matched at 1547.333 nm.

_{th}## 3. Experimental results

### 3.1 Temporal waveforms

*Δ*

_{ϕ}_{r1,r2}=

*ϕ*

_{r1}-

*ϕ*

_{r2}, and the difference between optical feedback phases in Drive and Response 1 as

*Δ*

_{ϕ}_{d,r1}=

*ϕ*

_{d}-

*ϕ*

_{r1}. Figures 2(a) and 2(b) show the temporal waveforms of Response 1 and 2 and their correlation plot when

*Δϕ*

_{r1,r2}=0, and Figs. 2(c) and 2(d) show the temporal waveforms of Response 1 and 2 and their correlation plot when

*Δϕ*

_{r1,r2}=π. Accurate synchronization is achieved between Response 1 and 2 when the optical feedback phases are matched,

*Δϕ*

_{r1,r2}=0. On the other hand, when the optical feedback phase difference is maximum,

*Δϕ*

_{r1,r2}=π, synchronization is completely destroyed and the temporal waveforms are dissimilar as shown in Figs. 2(c) and 2(d). This shows that synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback is sensitive with respective to the optical phase of the feedback light in the two Response lasers, as in the case of identical synchronization between drive and response lasers shown in [12

12. M. Peil, T. Heil, I. Fischer, and W. Elsäßer, “Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario,” Phys. Rev. Lett. **88**, 1741011-1–1741011-4 (2002).
[CrossRef]

*I*

_{1,2}are the total intensities of the two temporal waveforms,

*I*Ī

_{1,2}are their mean values, and

*σ*

_{1,2}are their standard deviations. The angle brackets denote time averaging. The best synchronization corresponds to cross-correlation value C=1. The cross correlation values corresponding to Figs. 2(b) and 2(d) are 0.903 and 0.015, respectively.

*Δϕ*

_{d,r1}=0, and Figs. 3(c) and 3(d) show the temporal waveforms of Drive and Response 1 and their correlation for

*Δϕ*

_{d,r1}=π. The cross correlation values of Figs. 3(b) and 3(d) are 0.601 and 0.599, respectively. These results confirm that good synchronization is not achieved between Drive and Response even when optical feedback phases are matched. It can be understood that the Drive and Response waveforms do not synchronize because different external cavity lengths and different relaxation oscillation frequencies were used for Drive and Response lasers.

### 3.2 Cross correlation characteristics

*Δϕ*

_{r1,r2}=0), whereas the minimum value is observed at

*Δϕ*

_{r1,r2}=π. The maximum value of cross correlation stays almost constant at 0.95, while the minimum value decreases as the feedback strength is increased. Therefore, the difference between maximum and minimum values of cross correlation (

*ΔC*=

*C*-

_{max}*C*) increases as the feedback strength is increased, and the maximum difference of

_{min}*ΔC*=0.880 is obtained at the maximum feedback strength which can be achieved in the experimental setup. We could not further increase the feedback strength due to limitations of our experimental setup. However, we can anticipate that at some larger value of feedback strength, the injection locking will breakdown. So we should expect the maximum correlation value, and hence the difference between maximum and minimum correlation values, to decrease again at larger feedback strength.

*ΔC*is observed at different optical feedback strengths and the maximum value of

*ΔC*is 0.105. This shows that the cross correlation between Drive and Response 1 is not sensitive to the optical feedback phase even at strong feedback strengths.

## 4. Numerical results

*ΔC*between the maximum and minimum cross correlation values (

*Δ*=

_{C}*C*-

_{max}*C*). The difference

_{min}*ΔC*characterizes the influence of feedback phase on synchronization. We used the Lang-Kobayashi equations for semiconductor lasers with optical feedback, assuming the configuration for synchronization by injection of common chaotic signal. The equations for Drive are given by

*E*and

*N*are the complex electric field and the carrier density, respectively. The equations for Response

*j*(

*j*=1, 2) are given as follows:

*=ω*

_{j}_{0}-ω

*is the detuning of optical angular frequency between Drive and Response*

_{j}*j*and the last term in the first equation represents the optical injection from Drive. In what follows, we also use the optical frequency detuning

*Δf*defined by

*Δf*=Δω

*/*

_{1}*2π*. The optical phase of the feedback light is represented by

*ϕ*. Parameter values used in our simulations are as follows: α=3,

_{j}*G*=8.4×10

_{N}^{-13}m

^{3}s

^{-1},

*N*=1.4×10

_{0}^{24}m

^{-3},

*N*

_{th}=2.018×10

^{24}m

^{-3},

*τ*

_{in}=8.0 ps,

*τ*=2.04 ns,

_{s}*κ*=0.00745,

_{D}*τ*=9.333 ns,

_{D}*J*=1.3

_{D}*J*

_{th},

*τ*=4.0 ns,

*J*=1.19

*J*

_{th}, where

*J*

_{th}=

*N*

_{th}/

*τ*is the threshold of the injection current. Drive and Response have different relaxation oscillation frequencies of 2.5 and 2.0 GHz, respectively, which coincide with the values in our experiments. The relaxation oscillation frequency for Drive is detuned from that for Response 1 and 2. For these parameter values, Drive is in a chaotic regime. Response 1 and 2 are assumed to have a small mismatch in their optical frequencies (ω

_{s}_{1}-ω

_{2})/

*2π*=0.25 GHz. As for the optical phases of the feedback light,

*ϕ*

_{1}is fixed as

*ϕ*

_{1}=0 while

*ϕ*is varied. We use

_{2}*ϕ*=0 to obtain

_{2}*C*and

_{max}*ϕ*

_{2}=

*π*to obtain

*C*. Using these

_{min}*C*and

_{max}*C*, we calculate

_{min}*ΔC*(=

*C*-

_{max}*C*).

_{min}*ΔC*as a function of the optical frequency detuning between Drive and Response 1 (

*Δf*) and the injection strength from Drive to Response (κ

_{inj}). In the calculation, the feedback strengths of Response 1 and 2 are fixed as κ

_{r}=0.08946. The synchronization region, in which the maximum cross correlation

*C*between Response 1 and 2 is close to unity, is shown by black dashed line. There exists a wide parameter region for synchronization. The synchronization region is asymmetric with respect to the line

_{max}*Δf*=0: the region is shifted to the negative frequency detuning side due to the α-parameter of semiconductor lasers. It should be noted that

*ΔC*strongly depends on the parameters

*Δf*and κ

_{inj}in the synchronization region. Large values of

*ΔC*, say

*ΔC*>0.75, are achieved for relatively small κ

_{inj}while

*ΔC*is small for large κ

_{inj}. In the region of large κ

_{inj},

*ΔC*decreases with increasing κ

_{inj}. This is a natural behavior because the injection light comes to dominate the response laser dynamics and thus the influence of the optical feedback becomes relatively weak. Our experimental observation in Fig. 2 corresponds to the conditions

*Δf*≈-3.125 GHz and κ

_{inj}≈0.1~0.2. The injection strength inside the laser cannot be measured directly, so the correspondence with the simulation value is roughly estimated by the following procedure. The injection strength is measured as a ratio of the threshold power required for the onset of injection locking in the absence of external feedback, and then the correspondence between the threshold powers in experiment and simulation is used to estimate the corresponding value of κ

_{inj}.

*ΔC*as a function of the feedback strength of Response (κ

_{r}) and the injection strength from Drive to Response (κ

_{inj}), where the detuning

*Δf*is fixed as

*Δf*=-3.125GHz, which coincides with the experimental value. The boundary curve, above which synchronization between Response 1 and 2 occurs, is also shown by black dashed line. It is clearly observed that

*ΔC*strongly depends on the parameters κ

_{r}and κ

_{inj}. The difference

*ΔC*takes large values (

*ΔC*>0.9) in a region, which is roughly located as 0.10<κ

_{r}<0.16 and 0.18<κ

_{inj}<0.33. This region of large

*ΔC*is an elongated region parallel to the boundary curve of the synchronization region. This figure shows that

*ΔC*takes a large value when κ

_{inj}and κ

_{r}balance with each other: as κ

_{r}increases, κ

_{inj}also has to increase for

*ΔC*to remain large.

*ΔC*increases monotonically up to a value of

*ΔC*=0.88 as the feedback strength is increased up to the maximum feedback strength possible in the experimental setup. In Fig. 7 this corresponds to values of κ

_{inj}and κ

_{r}in the ranges κ

_{inj}≈0.1~0.2 and κ

_{r}≈0.0~0.1. The value of κ

_{r}corresponding to the experiment is estimated from the estimate of the injection strength (as explained above) and the power ratio between the injection and feedback lights. We note that the numerical results confirm the possibility of this scenario. Further, the numerical results show additional features in parameter regions which could not be accessed in the experiment due to technical setup limitations. In particular, they confirm the prediction that the maximum correlation difference should eventually decrease again as the injection-locking boundary is approached. Further, the numerical simulations show that the effect of the feedback phase can be large (

*ΔC*>0.9) over a significantly wider range of feedback strength and injection strength than could be achieved in the experiment. The reason for this dependence of ΔC on injection strength and feedback strength is as follows. When the injection effect is dominant both the maximum and minimum correlations are high. When the feedback effect is dominant, both the maximum and minimum correlations are low. The difference between maximum and minimum correlation

*ΔC*is largest when there is a balance between these two effects.

## 5. Conclusion

## Acknowledgments

## References and links

1. | S. Sivaprakasam and K. A. Shore, “Demonstration of optical synchronization of chaotic external-cavity laser diodes,” Opt. Lett. |

2. | I. Fischer, Y. Liu, and P. Davis, “Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication,” Phys. Rev. A , |

3. | H. Fujino and J. Ohtsubo, “Experimental synchronization of chaotic oscillations in external-cavity semiconductor lasers,” Opt. Lett. |

4. | A. Locquet, C. Masoller, and C. R. Mirasso, “Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers,” Phys. Rev. E |

5. | Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, “Experimental observation of complete chaos synchronization in semiconductor lasers,” Appl. Phys. Lett. |

6. | S. Tang and J. M. Liu, “Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers,” Phys. Rev. Lett. |

7. | T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. Goto, and P. Davis, “Common-chaotic-signal induced synchronization in semiconductor lasers, ” Opt. Express |

8. | D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, “Observation of generalized synchronization of chaos in a driven chaotic system,” Phys. Rev. E |

9. | A. Uchida, K. Higa, T. Shiba, S. Yoshimori, F. Kuwashima, and H. Iwasawa, “Generalized synchronization of chaos in He-Ne lasers,” Phys. Rev. E |

10. | A. Uchida, R. McAllister, R. Meucci, and R. Roy, “Generalized synchronization of chaos in identical systems with hidden degrees of freedom,” Phys. Rev. Lett. |

11. | R. Vicente, T. Perez, and C. R. Mirasso, “Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers,” IEEE J. Quantum Electron. |

12. | M. Peil, T. Heil, I. Fischer, and W. Elsäßer, “Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario,” Phys. Rev. Lett. |

13. | T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, “On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers,” IEEE J. Quantum Electron. |

14. | A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, “Feedback phase in optically generated chaos: a secret key for cryptographic applications,” IEEE J. Quantum Electron. |

**OCIS Codes**

(060.4510) Fiber optics and optical communications : Optical communications

(140.1540) Lasers and laser optics : Chaos

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: April 23, 2009

Revised Manuscript: May 24, 2009

Manuscript Accepted: May 24, 2009

Published: May 29, 2009

**Citation**

Isao Oowada, Hiroki Ariizumi, Mao Li, Shigeru Yoshimori, Atsushi Uchida, Kazuyuki Yoshimura, and Peter Davis, "Synchronization by injection of common chaotic signal in semiconductor lasers with optical feedback," Opt. Express **17**, 10025-10034 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-12-10025

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### References

- S. Sivaprakasam and K. A. Shore, "Demonstration of optical synchronization of chaotic external-cavity laser diodes," Opt. Lett. 24, 466-468 (1999). [CrossRef]
- I. Fischer, Y. Liu, and P. Davis, "Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication," Phys. Rev. A, 62, 011801(R)-1--011801(R)-4 (2000).
- H. Fujino and J. Ohtsubo, "Experimental synchronization of chaotic oscillations in external-cavity semiconductor lasers," Opt. Lett. 25, 625-627 (2000). [CrossRef]
- A. Locquet, C. Masoller, and C. R. Mirasso, "Synchronization regimes of optical-feedback-induced chaos in unidirectionally coupled semiconductor lasers," Phys. Rev. E 65, 056205-1--056205-12 (2002).
- Y. Liu, Y. Takiguchi, P. Davis, T. Aida, S. Saito, and J. M. Liu, "Experimental observation of complete chaos synchronization in semiconductor lasers," Appl. Phys. Lett. 80, 4306-4308 (2002). [CrossRef]
- S. Tang and J. M. Liu, "Experimental verification of anticipated and retarded synchronization in chaotic semiconductor lasers," Phys. Rev. Lett. 90,194101 (2003). [CrossRef]
- T. Yamamoto, I. Oowada, H. Yip, A. Uchida, S. Yoshimori, K. Yoshimura, J. Muramatsu, S. Goto, and P. Davis, "Common-chaotic-signal induced synchronization in semiconductor lasers, " Opt. Express 15, 3974-3980 (2007). [CrossRef]
- D. Y. Tang, R. Dykstra, M. W. Hamilton, and N. R. Heckenberg, "Observation of generalized synchronization of chaos in a driven chaotic system," Phys. Rev. E 57, 5247-5251 (1998). [CrossRef]
- A. Uchida, K. Higa, T. Shiba, S. Yoshimori, F. Kuwashima, and H. Iwasawa, "Generalized synchronization of chaos in He-Ne lasers," Phys. Rev. E 68, 016215-1—016215-7 (2003).
- A. Uchida, R. McAllister, R. Meucci, and R. Roy, "Generalized synchronization of chaos in identical systems with hidden degrees of freedom," Phys. Rev. Lett. 91, 174101-1—174101-4 (2003).
- R. Vicente, T. Perez, and C. R. Mirasso, "Open- versus closed-loop performance of synchronized chaotic external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1197-1204 (2002). [CrossRef]
- M. Peil, T. Heil, I. Fischer, and W. Elsäßer, "Synchronization of chaotic semiconductor laser systems: a vectorial coupling-dependent scenario," Phys. Rev. Lett. 88, 1741011-1—1741011-4 (2002).
- T. Heil, J. Mulet, I. Fischer, C. R. Mirasso, M. Peil, P. Colet, and W. Elsäßer, "On/off phase shift keying for chaos-encrypted communication using external-cavity semiconductor lasers," IEEE J. Quantum Electron. 38, 1162-1170 (2002). [CrossRef]
- A. Bogris, P. Rizomiliotis, K. E. Chlouverakis, A. Argyris, and D. Syvridis, "Feedback phase in optically generated chaos: a secret key for cryptographic applications," IEEE J. Quantum Electron. 44, 119-124 (2008). [CrossRef]

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