## Numerical characterization of transient polarization square-wave switching in two orthogonally coupled VCSELs |

Optics Express, Vol. 19, Issue 21, pp. 20269-20278 (2011)

http://dx.doi.org/10.1364/OE.19.020269

Acrobat PDF (2624 KB)

### Abstract

We study the dynamics of two vertical-cavity surface-emitting lasers (VCSELs) mutually coupled such that the natural lasing polarization of each laser is rotated by 90 degrees and then is injected into the other laser. Simulations based on the spin-flip model show transient square-wave polarization switchings before a stationary state is reached. The influence of various model parameters on the duration of the stochastic transient time and on the lasers’ dynamics in the stationary state is investigated.

© 2011 OSA

## 1. Introduction

2. A. Pikovsky, M. Rosenblum, and J. Kurths, *Synchronization: A Universal Concept in Nonlinear Science* (Cambridge University Press, Cambridge, 2001). [CrossRef]

3. A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. **104**114102 (2010). [CrossRef] [PubMed]

4. M. W. Lee, J. Paul, C. Masoller, and K. A. Shore, “Observation of cascade complete chaos synchronisation with zero time lag in laser diodes,” J. Opt. Soc. Am. B **23**846 (2006). [CrossRef]

5. I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-Lag Long-Range Synchronization via Dynamical Relaying,” *Phys. Rev. Lett.*97123902 (2006). [CrossRef] [PubMed]

6. N. Jiang, W. Pan, L. Yan, B. Luo, S. Xiang, L. Yang, and D. Zheng, “Isochronal chaos synchronization of semiconductor lasers with multiple time-delayed couplings,” J. Opt. Soc. Am. B **28**1139 (2011). [CrossRef]

8. J. F. Martinez Avila and J. R. Rios Leite, “Time delays in the synchronization of chaotic coupled lasers with feedback,” Opt. Express **17**21442 (2009). [CrossRef]

9. M. Y. Kim, R. Roy, J. L. Aron, T. W. Carr, and I. B. Schwartz, “Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment,” Phys. Rev. Lett. **94**088101 (2005). [CrossRef] [PubMed]

10. M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback”, IEEE J. Quantum Electron. **41**1333 (2005). [CrossRef]

11. D. W. Sukow, K. L. Blackburn, A. R. Spain, K. J. Babcock, J. V. Bennett, and A. Gavrielides, “Experimental synchronization of chaos in diode lasers with polarization-rotated feedback and injection”, Opt. Lett. **29**2393 (2004). [CrossRef] [PubMed]

12. R. Ju, P. S. Spencer, and K. Alan Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **41**1461 (2005). [CrossRef]

13. D. W. Sukow, A. Gavrielides, T. McLachlan, G. Burner, J. Amonette, and J. Miller, “Identity synchronization in diode lasers with unidirectional feedback and injection of rotated optical fields,” Phys. Rev. A **74**023812 (2006). [CrossRef]

14. N. Shibasaki, A. Uchida, S. Yoshimori, and P. Davis, “Characteristics of chaos synchronization in semiconductor lasers subject to polarization-rotated optical feedback,” IEEE J. Quantum Electron. **42**, 342 (2006). [CrossRef]

15. D. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E **81**025206R (2010). [CrossRef]

15. D. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E **81**025206R (2010). [CrossRef]

16. C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: theory and experiment,” Phys. Rev. A **84**023838 (2011). [CrossRef]

17. F. Koyama, “Recent advances of VCSEL photonics,” J. Lightwave Technol. **24**, 4502 (2006). [CrossRef]

18. K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5 mu m wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. **20**779 (2008). [CrossRef]

19. W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, “Influence of polarization-selected mutual injection on the polarization-switching dynamics of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B **24**2472 (2007). [CrossRef]

20. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **33**, 765 (1997). [CrossRef]

15. D. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E **81**025206R (2010). [CrossRef]

*x*and

*y*, and while one laser emits a

*x*-polarized output, the other laser emits a

*y*-polarized output), or they can both emit on the two polarizations. In addition, the polarization intensities that can be either constant in time in the two lasers, or time-dependent in the two lasers, or one laser can emit a cw intensity while the other, a time-dependent one.

## 2. Model

### 2.1. Rate equations

**81**025206R (2010). [CrossRef]

20. J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **33**, 765 (1997). [CrossRef]

*τ*, and the coupling strength,

*η*: when the

*x*polarization of one laser is injected into the

*y*polarization of the other laser (

*x*→

*y*),

*η*=

_{x}*η*and

*η*= 0, and when

_{y}*y*→

*x*,

*η*= 0 and

_{x}*η*=

_{y}*η*. Because of the interplay of birefringence (represented by the parameter

*γ*) and the phase-amplitude coupling (represented by the

_{p}*α*factor), these two coupling schemes are not symmetric, and the frequency difference between the two polarizations plays a role similar to that of the detuning between an injected and a master laser.

### 2.2. Steady state solutions

**81**025206R (2010). [CrossRef]

16. C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: theory and experiment,” Phys. Rev. A **84**023838 (2011). [CrossRef]

*x*→

*y*coupling we obtain the equation: with Δ = 2

*γ*–

_{p}*αγ*,

_{a}*I*= (

_{x}*μ*– 1 –

*γ*/

_{a}*k*)/(1 +

*γ*/

_{a}*k*) and

*N*=

*μ*/(1+

*I*). This implicit equation for

_{y}*I*=

_{y}*f*(

*η*

^{2}) can be solved for

*η*

^{2}considering

*I*a parameter. A very similar equation can be obtained in the case of

_{y}*y*→

*x*coupling.

*I*

_{1x}=

*I*

_{2x};

*I*

_{1y}=

*I*

_{2y}; however, their equations are too complicated to solve analytically. They can be found numerically for low coupling strengths, as will be discussed in the next section.

## 3. Results

*k*= 300 ns

^{−1},

*γ*= 2 ns

_{n}^{−1},

*γ*= 50 ns

_{s}^{−1},

*γ*= 0.4 ns

_{a}^{−1},

*α*= 3,

*β*= 10

_{sp}^{−5}ns

^{−1}, and

*τ*= 3 ns. The pump current parameter and the birefringence parameter are chosen such that the solitary lasers display polarization mono-stability (the stability region of the two linear polarizations is displayed Fig. 1). For a pump current parameter about twice the threshold,

*μ*≃ 2, the

*x*polarization is stable if the birefringence is large enough (

*γ*≥ 50 rad/ns), while the

_{p}*y*polarization is stable if the birefringence is low enough (

*γ*≤ 8 rad/ns). The simulations start with initial conditions are such that both lasers are off.

_{p}*x*intensity in one laser in red, and the extreme values of the oscillations of the

*y*intensity in the other laser in blue), as a function of the coupling strength. In Figs. 2(a), (b) the coupling is

*x*→

*y*while in Figs. 2(c), (d) the coupling is

*y*→

*x*. One can notice that in both cases, for strong coupling the pure mode is dominant, while for low coupling, the mixed state dominates. Analytically we find that, depending on the coupling strength, there are either one or three pure mode solutions. The stability analysis of these solutions can not be done analytically but numerically we find that the middle and lower solutions are unstable, while the upper solution is stable as soon as it exists [as in Fig. 2(b)] or becomes stable after a series of bifurcations [as in Fig. 2(c)].

*x*-polarization in red and the intensity of the

*y*-polarization in blue; the intensities were averaged to simulate the finite experimental detection bandwidth). During the transient there are square-wave polarization switchings, and it can be noticed that the lasers emit the same polarization and switch simultaneously to the orthogonal one. The sum of two consecutive switching intervals is slightly larger than 2

*τ*. This dynamical switching is metastable and eventually evolves as shown in the right panels: the intervals that one laser emits one polarization grow, while the intervals that emits the orthogonal one decrease, until a steady-state is reached where the lasers emit polarization-orthogonal cw outputs. In this “pure-mode” one laser emits its natural polarization and acts as a “master laser” injecting polarization-rotated cw light into the other laser (referred to as “injected laser”). The injection is strong enough to turn off the natural polarization of the injected laser and turn on the orthogonal one. Since the lasers are coupled via their natural polarizations, the “master laser” does not receive light from the “injected laser” and therefore, their coupling is in fact unidirectional.

- Both lasers emit the two polarizations with cw intensities (“mix-mode solution”).
- There are irregular, chaotic-like oscillations in the two polarizations in the two lasers.
- The lasers emit orthogonal polarizations: the master laser emits a cw output at the natural polarization while the injected laser displays sustained oscillations at the laser relaxation frequency after a Hopf bifurcation. This occurs, for example, in the region of coupling strengths 20 ns
^{−1}≤*η*≤ 80 ns^{−1}in the bifurcation diagrams presented in Figs. 2(c), (d), where the intensity of the master laser is constant in time, Fig. 2(d), while the intensity of the injected laser is time-dependent, Fig. 2(c). Typical time-traces of the intensities of the two lasers are presented in the right panels of Fig. 4. The injected laser can display the rich variety of dynamical behaviors that are typical of orthogonal optical injection (see, e.g., Ref. [21, 2221. A. Hurtado, I. D. Henning, and M. J. Adams, “Different forms of wavelength polarization switching and bistability in a 1.55 m vertical-cavity surface-emitting laser under orthogonally polarized optical injection,” Opt. Lett.

**34**365 (2009). [CrossRef] [PubMed]] and references therein), with the frequency difference between the two polarizations playing a role similar to that of the detuning between the injected and master lasers. It is worth noting that when the injection is22. M. S. Torre, A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams “Polarization switching in long-wavelength VCSELs subject to orthogonal optical injection,” IEEE J. Quantum Electron.

**47**92 (2011). [CrossRef]*y*→*x*higher coupling strengths are need for injection locking, as compared with*x*→*y*.

*x*polarization of the solitary laser loses stability (see the green region in Fig. 1) and the dominant

*x*polarization acquires a small component in the orthogonal

*y*polarization. For even larger pump current the intensities

*I*and

_{x}*I*of the solitary laser both display regular oscillations. In this situation, when the lasers are mutually coupled, the “master” laser exhibits oscillations in both polarizations (the “solitary laser” solution) while the “injected” laser emits only the

_{y}*y*polarization and shows the same oscillations as the master laser, as shown in the left panels of Fig. 4.

*μ*s (i.e., this is the longest transient time that we could compute). We limited the study to parameter regions where, for the solitary lasers, only one polarization is stable (regions red and blue in Fig. 1). We found that the transient time is a stochastic quantity that can display large deviations from its mean value. As shown in Fig. 5(a), the average duration of the transient time is not affected by the noise strength, if this is not too strong. The injection current parameter also does not seem to significantly affect the duration of the transient, Fig. 5(b). The average transient increases with the delay time, Fig. 6(a), and with the coupling strength, Fig. 6(b). The linear anisotropy parameter,

*γ*, and the spin-flip rate,

_{a}*γ*, do not significantly affect the duration of the transient, as shown in Figs. 7 (a), (b).

_{s}*x*→

*y*injection; for

*y*→

*x*injection we found qualitatively similar results, except on the influence of the birefringence parameter,

*γ*, that is displayed in Fig. 8. For

_{p}*y*→

*x*injection there is no significant influence while for

*x*→

*y*there are three distinct regions: for

*γ*≤ 55 rad/ns (but larger than ≈50 rad/ns, such that for the solitary lasers only the

_{p}*x*polarization is stable), the duration of the transient decreases with

*γ*. For intermediate values of

_{p}*γ*(55 rad/ns ≤

_{p}*γ*≤ 65 rad/ns) the “pure-mode” solution is stable and the transient time decreases abruptly. For larger

_{p}*γ*the transient time diverges.

_{p}*x*low-frequency mode, or in the

*y*high-frequency mode), square-wave switchings were found to be a metastable dynamics, towards one of the various possible stationary states described above.

## 4. Discussion and conclusion

*x*-polarization, for high birefringence, or the high-frequency,

*y*-polarization, for low birefringence.

*x*to the

*y*polarization (

*x → y*) and vice-versa (

*y*→

*x*). Because of the phase-amplitude coupling, represented by the

*α*factor, these two coupling schemes are not symmetric; however, for strong enough coupling, in both cases we found a transient dynamics towards a stationary state characterized by square-wave-like polarization switchings of periodicity about twice the coupling delay time.

16. C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: theory and experiment,” Phys. Rev. A **84**023838 (2011). [CrossRef]

**84**023838 (2011). [CrossRef]

*x*→

*y*and

*y*→

*x*coupling schemes results in different dynamical regimes at intermediate coupling strengths (this difference was not studied in [16

**84**023838 (2011). [CrossRef]

**81**025206R (2010). [CrossRef]

**84**023838 (2011). [CrossRef]

## Acknowledgments

## References and links

1. | I. Kanter, M. Butkovski, Y. Peleg, M. Zigzag, Y. Aviad, I. Reidler, M. Rosenbluh, and W. Kinzel, “Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography,” Opt. Express |

2. | A. Pikovsky, M. Rosenblum, and J. Kurths, |

3. | A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. |

4. | M. W. Lee, J. Paul, C. Masoller, and K. A. Shore, “Observation of cascade complete chaos synchronisation with zero time lag in laser diodes,” J. Opt. Soc. Am. B |

5. | I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-Lag Long-Range Synchronization via Dynamical Relaying,” |

6. | N. Jiang, W. Pan, L. Yan, B. Luo, S. Xiang, L. Yang, and D. Zheng, “Isochronal chaos synchronization of semiconductor lasers with multiple time-delayed couplings,” J. Opt. Soc. Am. B |

7. | M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. A |

8. | J. F. Martinez Avila and J. R. Rios Leite, “Time delays in the synchronization of chaotic coupled lasers with feedback,” Opt. Express |

9. | M. Y. Kim, R. Roy, J. L. Aron, T. W. Carr, and I. B. Schwartz, “Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment,” Phys. Rev. Lett. |

10. | M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback”, IEEE J. Quantum Electron. |

11. | D. W. Sukow, K. L. Blackburn, A. R. Spain, K. J. Babcock, J. V. Bennett, and A. Gavrielides, “Experimental synchronization of chaos in diode lasers with polarization-rotated feedback and injection”, Opt. Lett. |

12. | R. Ju, P. S. Spencer, and K. Alan Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. |

13. | D. W. Sukow, A. Gavrielides, T. McLachlan, G. Burner, J. Amonette, and J. Miller, “Identity synchronization in diode lasers with unidirectional feedback and injection of rotated optical fields,” Phys. Rev. A |

14. | N. Shibasaki, A. Uchida, S. Yoshimori, and P. Davis, “Characteristics of chaos synchronization in semiconductor lasers subject to polarization-rotated optical feedback,” IEEE J. Quantum Electron. |

15. | D. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E |

16. | C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: theory and experiment,” Phys. Rev. A |

17. | F. Koyama, “Recent advances of VCSEL photonics,” J. Lightwave Technol. |

18. | K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5 mu m wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. |

19. | W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, “Influence of polarization-selected mutual injection on the polarization-switching dynamics of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B |

20. | J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. |

21. | A. Hurtado, I. D. Henning, and M. J. Adams, “Different forms of wavelength polarization switching and bistability in a 1.55 m vertical-cavity surface-emitting laser under orthogonally polarized optical injection,” Opt. Lett. |

22. | M. S. Torre, A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams “Polarization switching in long-wavelength VCSELs subject to orthogonal optical injection,” IEEE J. Quantum Electron. |

**OCIS Codes**

(140.1540) Lasers and laser optics : Chaos

(140.2020) Lasers and laser optics : Diode lasers

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

(250.7260) Optoelectronics : Vertical cavity surface emitting lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: August 23, 2011

Revised Manuscript: September 16, 2011

Manuscript Accepted: September 16, 2011

Published: September 30, 2011

**Citation**

Maria S. Torre, Athanasios Gavrielides, and Cristina Masoller, "Numerical characterization of transient polarization square-wave switching in two orthogonally coupled VCSELs," Opt. Express **19**, 20269-20278 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-21-20269

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### References

- I. Kanter, M. Butkovski, Y. Peleg, M. Zigzag, Y. Aviad, I. Reidler, M. Rosenbluh, and W. Kinzel, “Synchronization of random bit generators based on coupled chaotic lasers and application to cryptography,” Opt. Express 18, 18292 (2010).
- A. Pikovsky, M. Rosenblum, and J. Kurths, Synchronization: A Universal Concept in Nonlinear Science (Cambridge University Press, Cambridge, 2001). [CrossRef]
- A. Englert, W. Kinzel, Y. Aviad, M. Butkovski, I. Reidler, M. Zigzag, I. Kanter, and M. Rosenbluh, “Zero lag synchronization of chaotic systems with time delayed couplings,” Phys. Rev. Lett. 104114102 (2010). [CrossRef] [PubMed]
- M. W. Lee, J. Paul, C. Masoller, and K. A. Shore, “Observation of cascade complete chaos synchronisation with zero time lag in laser diodes,” J. Opt. Soc. Am. B 23846 (2006). [CrossRef]
- I. Fischer, R. Vicente, J. M. Buldu, M. Peil, C. R. Mirasso, M. C. Torrent, and J. Garcia-Ojalvo, “Zero-Lag Long-Range Synchronization via Dynamical Relaying,” Phys. Rev. Lett.97123902 (2006). [CrossRef] [PubMed]
- N. Jiang, W. Pan, L. Yan, B. Luo, S. Xiang, L. Yang, and D. Zheng, “Isochronal chaos synchronization of semiconductor lasers with multiple time-delayed couplings,” J. Opt. Soc. Am. B 281139 (2011). [CrossRef]
- M. Ozaki, H. Someya, T. Mihara, A. Uchida, S. Yoshimori, K. Panajotov, and M. Sciamanna, “Leader-laggard relationship of chaos synchronization in mutually coupled vertical-cavity surface-emitting lasers with time delay,” Phys. Rev. A 79026210 (2009).
- J. F. Martinez Avila and J. R. Rios Leite, “Time delays in the synchronization of chaotic coupled lasers with feedback,” Opt. Express 1721442 (2009). [CrossRef]
- M. Y. Kim, R. Roy, J. L. Aron, T. W. Carr, and I. B. Schwartz, “Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment,” Phys. Rev. Lett. 94088101 (2005). [CrossRef] [PubMed]
- M. C. Chiang, H. F. Chen, and J. M. Liu, “Experimental synchronization of mutually coupled semiconductor lasers with optoelectronic feedback”, IEEE J. Quantum Electron. 411333 (2005). [CrossRef]
- D. W. Sukow, K. L. Blackburn, A. R. Spain, K. J. Babcock, J. V. Bennett, and A. Gavrielides, “Experimental synchronization of chaos in diode lasers with polarization-rotated feedback and injection”, Opt. Lett. 292393 (2004). [CrossRef] [PubMed]
- R. Ju, P. S. Spencer, and K. Alan Shore, “Polarization-preserved and polarization-rotated synchronization of chaotic vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 411461 (2005). [CrossRef]
- D. W. Sukow, A. Gavrielides, T. McLachlan, G. Burner, J. Amonette, and J. Miller, “Identity synchronization in diode lasers with unidirectional feedback and injection of rotated optical fields,” Phys. Rev. A 74023812 (2006). [CrossRef]
- N. Shibasaki, A. Uchida, S. Yoshimori, and P. Davis, “Characteristics of chaos synchronization in semiconductor lasers subject to polarization-rotated optical feedback,” IEEE J. Quantum Electron. 42, 342 (2006). [CrossRef]
- D. Sukow, A. Gavrielides, T. Erneux, B. Mooneyham, K. Lee, J. McKay, and J. Davis, “Asymmetric square waves in mutually coupled semiconductor lasers with orthogonal optical injection,” Phys. Rev. E 81025206R (2010). [CrossRef]
- C. Masoller, D. Sukow, A. Gavrielides, and M. Sciamanna, “Bifurcation to square-wave switching in orthogonally delay-coupled semiconductor lasers: theory and experiment,” Phys. Rev. A 84023838 (2011). [CrossRef]
- F. Koyama, “Recent advances of VCSEL photonics,” J. Lightwave Technol. 24, 4502 (2006). [CrossRef]
- K. H. Jeong, K. H. Kim, S. H. Lee, M. H. Lee, B. S. Yoo, and K. A. Shore, “Optical injection-induced polarization switching dynamics in 1.5 mu m wavelength single-mode vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 20779 (2008). [CrossRef]
- W. L. Zhang, W. Pan, B. Luo, X. F. Li, X. H. Zou, and M. Y. Wang, “Influence of polarization-selected mutual injection on the polarization-switching dynamics of vertical-cavity surface-emitting lasers,” J. Opt. Soc. Am. B 242472 (2007). [CrossRef]
- J. Martin-Regalado, F. Prati, M. San Miguel, and N. B. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 33, 765 (1997). [CrossRef]
- A. Hurtado, I. D. Henning, and M. J. Adams, “Different forms of wavelength polarization switching and bistability in a 1.55 m vertical-cavity surface-emitting laser under orthogonally polarized optical injection,” Opt. Lett. 34365 (2009). [CrossRef] [PubMed]
- M. S. Torre, A. Hurtado, A. Quirce, A. Valle, L. Pesquera, and M. J. Adams “Polarization switching in long-wavelength VCSELs subject to orthogonal optical injection,” IEEE J. Quantum Electron. 4792 (2011). [CrossRef]

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