## Bistability of time-periodic polarization dynamics in a free-running VCSEL |

Optics Express, Vol. 22, Issue 6, pp. 6772-6777 (2014)

http://dx.doi.org/10.1364/OE.22.006772

Acrobat PDF (1624 KB)

### Abstract

We report experimentally a bistability between two limit cycles (i.e. time-periodic dynamics) in a free-running vertical-cavity surface-emitting laser. The two limit cycles originate from a bifurcation on two elliptically polarized states which exhibit a small frequency difference and whose main axes are symmetrical with respect to the linear polarization eigenaxes at threshold. We demonstrate theoretically that this peculiar behavior can be explained in the framework of the spin-flip model model by taking into account a small misalignment between the phase and amplitude anisotropies.

© 2014 Optical Society of America

## 1. Introduction

1. K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical cavity surface emitting laser polarization,” Appl. Phys. Lett. **64**, 2062–2064 (1994). [CrossRef]

7. L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers.” Opt. Express **19**, 2476–2484 (2011). [CrossRef] [PubMed]

8. M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics **7**, 60–65 (2012). [CrossRef]

10. K. Panajotov, B. Ryvkin, J. Danckaert, M. Peeters, H. Thienpont, and I. Veretennicoff, “Polarization switching in VCSEL’s due to thermal lensing,” IEEE Photon. Technol. Lett. **10**, 6–8 (1998). [CrossRef]

2. K. D. Choquette, R. P. Schneider, K. L. Lear, and R. E. Leibenguth, “Gain-dependent polarization properties of vertical-cavity lasers,” IEEE J. Sel. Top. Quantum Electron. **1**, 661–666 (1995). [CrossRef]

11. M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A **52**, 1728–1739 (1995). [CrossRef] [PubMed]

13. M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A **87**, 013834 (2013). [CrossRef]

5. M. Sondermann, T. Ackemann, S. Balle, J. Mulet, and K. Panajotov, “Experimental and theoretical investigations on elliptically polarized dynamical transition states in the polarization switching of vertical-cavity surface-emitting lasers,” Opt. Commun. **235**, 421–434 (2004). [CrossRef]

14. F. Prati, P. Caccia, M. Bache, and F. Castelli, “Analysis of elliptically polarized states in vertical-cavity-surface-emitting lasers,” Phys. Rev. A **69**, 033810 (2004). [CrossRef]

6. L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. **21**, 1008–1010 (2009). [CrossRef]

8. M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics **7**, 60–65 (2012). [CrossRef]

13. M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A **87**, 013834 (2013). [CrossRef]

7. L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers.” Opt. Express **19**, 2476–2484 (2011). [CrossRef] [PubMed]

12. J. Martin-Regalado, F. Prati, M. San Miguel, and N. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quant. Electron. **33**, 765–783 (1997). [CrossRef]

15. M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers.” Phys. Rev. A **54**, 1647–1660 (1996). [CrossRef] [PubMed]

16. M. Travagnin, “Linear anisotropies and polarization properties of vertical-cavity surface-emitting semiconductor lasers,” Phys. Rev. A **56**, 4094–4105 (1997). [CrossRef]

## 2. Experimental observation: bistability between two limit cycles

7. L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers.” Opt. Express **19**, 2476–2484 (2011). [CrossRef] [PubMed]

*C*and, in this case, the laser turns into chaos at high injection current [8

8. M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics **7**, 60–65 (2012). [CrossRef]

17. F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. **89**, 141106 (2006). [CrossRef]

*GHz*, see the time-series in Fig. 1(b) and the frequency evolution in Fig. 1(e). The amplitude of the LC increases along with the current but, around 1.95 mA, it is destabilized and a new limit cycle appears. The latter exhibits a slightly larger frequency around 6.55

*GHz*but, even more importantly, the cycle oscillates around the second, symmetrical EP with respect to the polarization of the laser at threshold, see the inversion of the +45° and −45° curves shown in Fig. 1(a). At this point, decreasing the current unveils the region of bistability between the two LCs - in Fig. 1(a), 1(d) and 1(f) - until the laser settles back on the first LC for currents below 1.77 mA. Moreover, before the switching, we observe that the cycle fades out to an EP steady-state as can be seen very clearly in Fig. 1(f) where the FFT peak vanishes before the switching. Due to a high sensitivity of the device to experimental conditions, e.g. stress from the probes, the range of current at which the bistability appear can change between different measurements. In particular the current value at which polarization switchings occur in Fig. 1(a) are slightly different from what can be concluded from panels 1(c)–1(d) or 1(e)–1(f).

## 3. Asymmetric SFM model: misaligned phase and amplitude anisotropies

11. M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A **52**, 1728–1739 (1995). [CrossRef] [PubMed]

12. J. Martin-Regalado, F. Prati, M. San Miguel, and N. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quant. Electron. **33**, 765–783 (1997). [CrossRef]

16. M. Travagnin, “Linear anisotropies and polarization properties of vertical-cavity surface-emitting semiconductor lasers,” Phys. Rev. A **56**, 4094–4105 (1997). [CrossRef]

*E*

_{±}the electrical field for the right (+) and left (−) circular polarization,

*N*the normalized total carrier population and

*n*the normalized carrier population difference between the two reservoirs.

*κ*is the electric field decay rate in the cavity and

*γ*the spin-flip relaxation rate that accounts for the spin homogenization of the spin up and spin down carrier populations.

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

*μ*is the normalized injection current. Finally, the phase anisotropy or birefringence is

*γ*whereas

_{p}*γ*is the amplitude anisotropy.

_{a}*θ*is defined as the angle between the axis of maximum frequency and the axis of maximum losses. All parameters and variables are dimensionless and the time is normalized by the carrier lifetime.

13. M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A **87**, 013834 (2013). [CrossRef]

18. T. Erneux, J. Danckaert, K. Panajotov, and I. Veretennicoff, “Two-variable reduction of the San MiguelFeng-Moloney model for vertical-cavity surface-emitting lasers,” Phys. Rev. A **59**, 4660–4667 (1999). [CrossRef]

*R*

_{+}and

*R*

_{−}and their phase difference

*ϕ*: The last two equations remain unchanged. In the next sections, unless stated otherwise, we use the following parameters:

*κ*= 600,

*α*= 3,

*γ*= 35,

_{p}*γ*= −10 and

_{a}*γ*= 100.

_{s}## 4. Simulation results: asymmetry impact on laser dynamics

*θ*= 0, we obtain the dynamical evolution given in Fig. 2. At threshold the laser emits LP light - see Fig. 2(a) - but an increase of the injection leads to a pitchfork bifurcation at

*μ*∼ 3.4 which creates two symmetrical EP states - see Fig. 2(b). Because the system is completely symmetrical, the two EPs are strictly equivalent, the polarization selection is therefore only made at random depending on the noise and initial conditions. With a further increase of the current, these steady-states are both destabilized by two identical Hopf bifurcations at

*μ*∼ 3.5, hence creating two symmetrical LCs oscillating around the now unstable EP states - see Fig. 2(c). Finally, for

*μ*> 5, the system experiences a cascade of period doubling bifurcation and enters a large region of chaotic dynamics [8

**7**, 60–65 (2012). [CrossRef]

**87**, 013834 (2013). [CrossRef]

*θ*= −0.023; this very specific angle between the two anisotropies has been chosen to provide a good match with the features observed in the experiments. For simplicity, we call

*EP*

_{+}(

*EP*

_{−}) the EP state exhibiting a dominant emission at +45° (−45°). We also designate the two limit cycles oscillating around these states

*LC*

_{+}and

*LC*

_{−}.

*EP*

_{+}elliptical. For levels of current where the pitchfork bifurcation would appear when no asymmetry is considered, we observe a smooth transition towards an elliptical

*EP*

_{+}emission with increasing ellipticity, see Fig. 3(a). This observation clearly confirms that the two EP states are no longer equivalent as the asymmetry obviously strengthens the

*EP*

_{+}emission.

*LC*

_{+}and

*LC*

_{−}, as can be seen from Fig. 3(a)–3(b) and 3(d)–3(e). Despite the misalignment induced asymmetry, a Hopf bifurcation still occurs around

*μ*= 3.5 hence creating

*LC*

_{+}. The amplitude of the cycle largely increases along with the current until it is destabilized by a cascade of period doubling bifurcations at

*μ*∼ 4.9 (Fig. 3(b)). Without asymmetry the system would then enter in a region of polarization chaos [8

**7**, 60–65 (2012). [CrossRef]

*LC*

_{−}. The FFT of the output power time-series confirms a slight frequency shift, about 0.07 between the two cycles - i.e. about 70 MHz for a carrier lifetime of 1 ns. From this point, decreasing the injection current unveils a large bistability region between the two time-periodic solutions oscillating with different polarization and delimited by the two vertical dashed lines of Fig. 3(b)–3(e). The amplitude decreases along with the current until it reaches a steady

*EP*

_{−}emission around

*μ*= 3.5 (see inset of Fig. 3(e)). This transition is also visible on the FFT spectrum, Fig. 3(f), as the frequency peak vanishes close to the switching point. Indeed the

*EP*

_{−}is stable only in a small region and the system quickly settles back on

*LC*

_{+}. The frequency shift we observe here is slightly larger around 0.1, i.e. about 100 MHz for a carrier lifetime of 1 ns, i.e. the same order of magnitude than in the experiment.

*θ*= −0.023 - which makes the

## 5. Conclusion

## Acknowledgments

## References and links

1. | K. D. Choquette, D. A. Richie, and R. E. Leibenguth, “Temperature dependence of gain-guided vertical cavity surface emitting laser polarization,” Appl. Phys. Lett. |

2. | K. D. Choquette, R. P. Schneider, K. L. Lear, and R. E. Leibenguth, “Gain-dependent polarization properties of vertical-cavity lasers,” IEEE J. Sel. Top. Quantum Electron. |

3. | M. van Exter, M. Willemsen, and J. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A |

4. | T. Ackemann and M. Sondermann, “Characteristics of polarization switching from the low to the high frequency mode in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. |

5. | M. Sondermann, T. Ackemann, S. Balle, J. Mulet, and K. Panajotov, “Experimental and theoretical investigations on elliptically polarized dynamical transition states in the polarization switching of vertical-cavity surface-emitting lasers,” Opt. Commun. |

6. | L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. |

7. | L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, and D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers.” Opt. Express |

8. | M. Virte, K. Panajotov, H. Thienpont, and M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics |

9. | K. Panajotov and F. Prati, |

10. | K. Panajotov, B. Ryvkin, J. Danckaert, M. Peeters, H. Thienpont, and I. Veretennicoff, “Polarization switching in VCSEL’s due to thermal lensing,” IEEE Photon. Technol. Lett. |

11. | M. San Miguel, Q. Feng, and J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A |

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

13. | M. Virte, K. Panajotov, and M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A |

14. | F. Prati, P. Caccia, M. Bache, and F. Castelli, “Analysis of elliptically polarized states in vertical-cavity-surface-emitting lasers,” Phys. Rev. A |

15. | M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, and J. P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers.” Phys. Rev. A |

16. | M. Travagnin, “Linear anisotropies and polarization properties of vertical-cavity surface-emitting semiconductor lasers,” Phys. Rev. A |

17. | F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, and P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. |

18. | T. Erneux, J. Danckaert, K. Panajotov, and I. Veretennicoff, “Two-variable reduction of the San MiguelFeng-Moloney model for vertical-cavity surface-emitting lasers,” Phys. Rev. A |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(190.3100) Nonlinear optics : Instabilities and chaos

(140.7260) Lasers and laser optics : Vertical cavity surface emitting lasers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 9, 2014

Revised Manuscript: February 28, 2014

Manuscript Accepted: March 3, 2014

Published: March 17, 2014

**Virtual Issues**

Physics and Applications of Laser Dynamics (2014) *Optics Express*

**Citation**

M. Virte, M. Sciamanna, E. Mercier, and K. Panajotov, "Bistability of time-periodic polarization dynamics in a free-running VCSEL," Opt. Express **22**, 6772-6777 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6772

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### References

- K. D. Choquette, D. A. Richie, R. E. Leibenguth, “Temperature dependence of gain-guided vertical cavity surface emitting laser polarization,” Appl. Phys. Lett. 64, 2062–2064 (1994). [CrossRef]
- K. D. Choquette, R. P. Schneider, K. L. Lear, R. E. Leibenguth, “Gain-dependent polarization properties of vertical-cavity lasers,” IEEE J. Sel. Top. Quantum Electron. 1, 661–666 (1995). [CrossRef]
- M. van Exter, M. Willemsen, J. Woerdman, “Polarization fluctuations in vertical-cavity semiconductor lasers,” Phys. Rev. A 58, 4191–4205 (1998). [CrossRef]
- T. Ackemann, M. Sondermann, “Characteristics of polarization switching from the low to the high frequency mode in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 78, 3574–3576 (2001). [CrossRef]
- M. Sondermann, T. Ackemann, S. Balle, J. Mulet, K. Panajotov, “Experimental and theoretical investigations on elliptically polarized dynamical transition states in the polarization switching of vertical-cavity surface-emitting lasers,” Opt. Commun. 235, 421–434 (2004). [CrossRef]
- L. Olejniczak, M. Sciamanna, H. Thienpont, K. Panajotov, A. Mutig, F. Hopfer, D. Bimberg, “Polarization switching in quantum-dot vertical-cavity surface-emitting lasers,” IEEE Photon. Technol. Lett. 21, 1008–1010 (2009). [CrossRef]
- L. Olejniczak, K. Panajotov, H. Thienpont, M. Sciamanna, A. Mutig, F. Hopfer, D. Bimberg, “Polarization switching and polarization mode hopping in quantum dot vertical-cavity surface-emitting lasers.” Opt. Express 19, 2476–2484 (2011). [CrossRef] [PubMed]
- M. Virte, K. Panajotov, H. Thienpont, M. Sciamanna, “Deterministic polarization chaos from a laser diode,” Nat. Photonics 7, 60–65 (2012). [CrossRef]
- K. Panajotov, F. Prati, Polarization dynamics of vcsels, in VCSELs (Springer, 2013), 181–231.
- K. Panajotov, B. Ryvkin, J. Danckaert, M. Peeters, H. Thienpont, I. Veretennicoff, “Polarization switching in VCSEL’s due to thermal lensing,” IEEE Photon. Technol. Lett. 10, 6–8 (1998). [CrossRef]
- M. San Miguel, Q. Feng, J. Moloney, “Light-polarization dynamics in surface-emitting semiconductor lasers,” Phys. Rev. A 52, 1728–1739 (1995). [CrossRef] [PubMed]
- J. Martin-Regalado, F. Prati, M. San Miguel, N. Abraham, “Polarization properties of vertical-cavity surface-emitting lasers,” IEEE J. Quant. Electron. 33, 765–783 (1997). [CrossRef]
- M. Virte, K. Panajotov, M. Sciamanna, “Bifurcation to nonlinear polarization dynamics and chaos in vertical-cavity surface-emitting lasers,” Phys. Rev. A 87, 013834 (2013). [CrossRef]
- F. Prati, P. Caccia, M. Bache, F. Castelli, “Analysis of elliptically polarized states in vertical-cavity-surface-emitting lasers,” Phys. Rev. A 69, 033810 (2004). [CrossRef]
- M. Travagnin, M. P. van Exter, A. K. Jansen van Doorn, J. P. Woerdman, “Role of optical anisotropies in the polarization properties of surface-emitting semiconductor lasers.” Phys. Rev. A 54, 1647–1660 (1996). [CrossRef] [PubMed]
- M. Travagnin, “Linear anisotropies and polarization properties of vertical-cavity surface-emitting semiconductor lasers,” Phys. Rev. A 56, 4094–4105 (1997). [CrossRef]
- F. Hopfer, A. Mutig, M. Kuntz, G. Fiol, D. Bimberg, N. N. Ledentsov, V. a. Shchukin, S. S. Mikhrin, D. L. Livshits, I. L. Krestnikov, a. R. Kovsh, N. D. Zakharov, P. Werner, “Single-mode submonolayer quantum-dot vertical-cavity surface-emitting lasers with high modulation bandwidth,” Appl. Phys. Lett. 89, 141106 (2006). [CrossRef]
- T. Erneux, J. Danckaert, K. Panajotov, I. Veretennicoff, “Two-variable reduction of the San MiguelFeng-Moloney model for vertical-cavity surface-emitting lasers,” Phys. Rev. A 59, 4660–4667 (1999). [CrossRef]

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