## Relaxation and square-wave oscillations in a semiconductor laser with polarization rotated optical feedback |

Optics Express, Vol. 22, Issue 6, pp. 6905-6918 (2014)

http://dx.doi.org/10.1364/OE.22.006905

Acrobat PDF (1169 KB)

### Abstract

The rate equations for a laser with a polarization rotated optical feedback are investigated both numerically and analytically. The frequency detuning between the polarization modes is now taken into account and we review all earlier studies in order to motivate the range of values of the fixed parameters. We find that two basic Hopf bifurcations leading to either stable sustained relaxation or square-wave oscillations appear in the detuning versus feedback rate diagram. We also identify two key parameters describing the differences between the total gains of the two polarization modes and discuss their effects on the periodic square-waves.

© 2014 Optical Society of America

## 1. Introduction

1. K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. **16**, 1759–1761 (1991). [CrossRef] [PubMed]

2. D.-L. Cheng, T.-C. Yen, J.-W. Chang, and J.-K. Tsai, “Generation of high-speed single-wavelength optical pulses in semiconductor lasers with orthogonal-polarization optical feedback,” Opt. Commun. **222**, 363–369 (2003). [CrossRef]

3. W. H. Loh, Y. Ozeki, and C. L. Tang, “High-frequency polarization self-modulation and chaotic phenomena in external cavity semiconductor lasers,” Appl. Phys. Lett. **56**, 2613–2615 (1990). [CrossRef]

4. A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave self-modulation in diode lasers with polarization-rotated optical feedback,” Opt. Lett. **31**, 2006–2008 (2006). [CrossRef] [PubMed]

6. A. Gavrielides, T. Erneux, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave oscillations in edge-emitting diode lasers with polarization rotated optical feedback,” in *Semiconductor Laser and Laser Dynamics II*, D. Lenstra, M. Pessa, and I. H. White, eds., Proc. SPIE6184, 255–261 (2006).

7. N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. **36**, 4632–4634 (2011). [CrossRef] [PubMed]

8. J. Mulet, M. Giudici, J. Javaloyes, and S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A **76**,043801 (2007). [CrossRef]

10. D. W. Sukow, T. Gilfillan, B. Pope, M. S. Torre, A. Gavrielides, and C. Masoller, “Square-wave switching in vertical-cavity surface-emitting lasers with polarization-rotated optical feedback: experiments and simulations,” Phys. Rev. A **86**,033818 (2012). [CrossRef]

11. S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, “High-frequency polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. **63**, 3545–3547, (1993). [CrossRef]

12. H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. **72**, 2355–2357, (1998). [CrossRef]

12. H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. **72**, 2355–2357, (1998). [CrossRef]

13. M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Megret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A **65**,041801 (2002). [CrossRef]

13. M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Megret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A **65**,041801 (2002). [CrossRef]

14. L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express **20**, 22503–22516 (2012). [CrossRef] [PubMed]

19. M. Peil, M. Jacquot, Y. Kouomou Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic-oscillators,” Phys. Rev. E **79**,026208 (2009). [CrossRef]

21. L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slow-fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A **371**,20120459 (2013). [CrossRef] [PubMed]

*a*and

*τ*represent the gain and the delay of the feedback. By analyzing the stability of the zero solution [22, 23

23. F. M. Atay, “Oscillation control in delayed feedback systems,” in Dynamics, Bifurcations, and Control, F. Colonius and L. Grne, eds. (Lect. Notes in Control and Information Sciences 273, 2002), pp. 103–116. [CrossRef]

*a*,

*τ*) parameter space. Figure 1 shows the two first Hopf bifurcation lines H

_{1}and H

_{2}.

_{1}admits a frequency inversely proportional to the delay while H

_{2}exhibits the frequency of the free harmonic oscillator and is independent of the feedback strength.

*T*∼ 10

^{3},

*τ*∼ 10

^{3}, and

*P*∼ 1,

*f*<<

_{D}*f*. As we shall demonstrate in this paper, both frequencies are associated with two distinct Hopf bifurcations.

_{RO}24. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaotic dynamics in semiconductor lasers subjected to polarization-rotated optical feedback,” Appl. Phys. Lett. **93**,181105 (2008). [CrossRef]

25. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. **17**, 144–151 (2010). [CrossRef]

## 2. Formulation

1. K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. **16**, 1759–1761 (1991). [CrossRef] [PubMed]

*E*), the TM electric field (

_{TE}*E*) and the carrier density (

_{TM}*n*) are given by (equations formulated in [26

26. T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A **68**,033811 (2003). [CrossRef]

25. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. **17**, 144–151 (2010). [CrossRef]

*G*

_{1}and

*G*

_{2}are the gain coefficients for the TE and TM mode, respectively and

*n*

_{0}is the carrier density at transparency.

*γ*is the feedback amplitude,

*γ*

_{1}and

*γ*

_{2}are the cavity decay rates for the TE and TM modes, respectively,

*γ*is the inverse of the carrier lifetime,

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

*J*is the injection current,

*ω*

_{1}and

*ω*

_{2}are defined as the angular frequencies of the TE and TM modes, respectively.

*C*≡

*ω*

_{1}

*τ*is the feedback phase which will be removed later and Δ ≡

*ω*

_{1}−

*ω*

_{2}is defined as the frequency detuning. In the appendix, we list the values of the fixed parameters considered in [26

26. T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A **68**,033811 (2003). [CrossRef]

*Y*

_{1},

*Y*

_{2}

*,*and

*N*are the new TE field, TM field, and carrier density, respectively. The new time is

*s*=

*γ*

_{1}

*t*, Ω = (

*ω*

_{1}−

*ω*

_{2})/

*γ*

_{1}, and

*θ*=

*γ*

_{1}

*τ. η*is the feedback strength,

*T*is the ratio of carrier to cavity lifetimes, and

*P*is the pump parameter above threshold. Two important parameters measure the differences between the total gains of the TE and TM modes. They are defined as

*k*is the ratio of the gain coefficients of the TM and TE modes.

*β*measures the losses of the TM mode compared to the TE mode. It depends on both the ratio of the gains coefficients and the ratio of the cavity rates for the two modes. The TM mode has greater inherent losses than the natural TE mode as expressed by the inequality

*β*> 0. The rotated optical feedback—delayed by one cavity round-trip

*θ*— appears in Eq. (9) through the term

*Y*

_{1}(

*s*−

*θ*). The TM mode does not influence the TE mode directly, but instead is mediated through the carrier equation (10).

*k*∼ 0.8 and

*β*∼ 0.1. For all references,

*α*= 2 − 3 and

*P*= 0.5 − 0.6. The delay

*θ*= 10

^{3}− 6 × 10

^{3}and the ratio of carrier to cavity lifetimes

*T*= 10

^{2}− 10

^{3}are large. The feedback amplitude

*η*= 3 × 10

^{−2}− 10

^{−1}.

*Y*

_{1}=

*E*

_{1}and

*Y*

_{2}=

*E*

_{2}exp(−

*iC*+

*i*Ω

*s*) allows us to remove the feedback phase

*C*and the exponential in Eq. (9). Equations (8)–(10) become We next introduce the decomposition

*E*=

_{j}*A*exp(

_{j}*iϕ*) (

_{j}*j*= 1, 2) and obtain five equations for

*A*

_{1},

*ϕ*

_{1},

*A*

_{2},

*ϕ*

_{2}, and

*N*. By formulating an equation for Φ ≡

*ϕ*

_{1}(

*s*−

*θ*) −

*ϕ*

_{2}, it is possible to reduce the problem to the following four equations

*A*

_{1}=

*A*

_{2}= 0 is always unstable if

*P*> 0. We have verified that there exists no pure mode solution with

*A*

_{1}= 0 and

*A*

_{2}≠ 0. There exists a mixed mode solution

*A*

_{1}≠ 0 and

*A*

_{2}≠ 0 if

*N*= 0. We find that the intensities of the two polarization modes are given by and the phase Φ satisfies

## 3. Hopf stability boundaries

*λ*. Assuming

*λ*=

*O*(

*θ*

^{−1}), we find as

*θ*→ ∞. Inserting

*λ*=

*iω*into Eq. (23) and separating the real and imaginary parts, we find that the first Hopf bifurcation satisfies the conditions Using (21), the location of this Hopf bifurcation in the Ω vs

*η*diagram is given by

*θ*large, we expect 2

*θ*–periodic square-waves with two constant plateaus connected by fast transition layers. The mathematical analysis is similar to the one documented in [4

4. A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave self-modulation in diode lasers with polarization-rotated optical feedback,” Opt. Lett. **31**, 2006–2008 (2006). [CrossRef] [PubMed]

*θ*is sufficiently large and provided Φ remains bounded, we may neglect all the time derivatives in Eqs. (16)–(19). Equation (16) then requires that Equation (26) motivates to seek a 2

*θ*–periodic square wave exhibiting (1)

*N*= 0 during the time interval 0 <

*s*<

*θ*and (2)

*A*

_{1}= 0 during the time interval

*θ*<

*s*< 2

*θ*. But because of the periodicity condition,

*A*

_{1}(

*s*−

*θ*) = 0 in part (1) and

*N*(

*s*−

*θ*) = 0 in part (2). From the remaining equations we then determine the values of the other variables. We obtain In part (2), the solutions for

*A*

_{2}and

*N*are given by (in parametric form -

*N*is the parameter) The stability of the two plateaus of the square-wave oscillations can be analyzed by linearizing Eqs. (16)–(19) along each constant plateau [28]. From Eq. (16), the linearized equation is for part (2).

*A*

_{1}=

*a*

_{1}(

*s*) is the small perturbation from

*A*

_{1}= 0 and

*N*is the constant value obtained from (31). Stability clearly requires

*N*< 0. The stability boundary corresponding to

*N*= 0 is determined from Eq. (31) and is exactly the same as the Hopf bifurcation approximation (25). This suggests that a nearly vertical branch of periodic solutions emerges from the unstable steady state, stabilizes at a fixed amplitude, and leads to the stable square-waves described in (27)–(31). Figure 2 represents (25) in the Ω versus

*η*diagram (line H

_{SW}).

_{RO}) which leads to sustained ROs. In order to determine this Hopf bifurcation condition, we insert

*λ*=

*iω*into the full characteristic equation and separate the real and imaginary parts. An approximation based on the large value of

*θ*is delicate because of the presence of fast changing trigonometric functions of

*ωθ*(

*ωθ*=

*O*(

*θ*

^{1/2}) if

*T*=

*O*(

*θ*)), in the Hopf bifurcation conditions. However, it is possible to reduce the two Hopf conditions to a single transcendental equation which we solved numerically. The analysis is long and tedious and we omit all details [28]. The Hopf line is denoted by H

*in Fig. 2. The fast periodic pulsating oscillations were observed experimentally [Fig. 11(a) in [29*

_{RO}29. A. Gavrielides, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Simple and complex square waves in an edge-emitting diode laser with polarization-rotated optical feedback,” Phys. Rev. E **81**,056209 (2010). [CrossRef]

*from the steady state. The Hopf bifurcation now admits a frequency close to*

_{SW}*π/θ*. The simulations shown in Fig. 3 illustrate the transition from the RO oscillations to the square-waves. Figures 3(c)–3(e) correspond to the points labeled by c, d, and e in Fig. 2 left. As the feedback rate increases, the oscillations are first harmonic, then grow in amplitude and become square-waves as anticipated from the stability diagram.

*critical point. Moreover, solutions combining pulsating and square-wave forms have been found numerically. They were observed experimentally and were called ”complex oscillations” [Figs. 8 and 10 in [29*

_{SW}29. A. Gavrielides, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Simple and complex square waves in an edge-emitting diode laser with polarization-rotated optical feedback,” Phys. Rev. E **81**,056209 (2010). [CrossRef]

*. Figure 2 right indicates that near Ω = −0.15, the second Hopf line H*

_{RO}*appears before the first Hopf line H*

_{SW}*. Consequently, the bifurcation to square-waves appears first as we increase the feedback rate and if the bifurcation is supercritical, we may expect a gradual change from nearly sinusoidal to square-waves close to H*

_{RO}*. This is exactly what we observe numerically in Fig. 4. In Fig. 4(a), the oscillations are of small amplitude and nearly sinusoidal but the period is already close to 2 (two delays). They have been obtained numerically very close to H*

_{SW}*[red triangle labelled by a in Fig. 2]. If we slightly increases the feedback rate, square-wave oscillations clearly appear [Fig. 4(b) and red triangle labelled by b in Fig. 2]. Note the damped RO oscillations on the upper plateaus.*

_{SW}## 4. Discussion

*α*factor and the gain/loss difference. We found that two basic Hopf bifurcations leading to stable solutions appear in parameter space. Except for a small range of detuning, the bifurcation to sustained ROs always appears before the bifurcation to the square-wave oscillations as we increase the feedback rate from zero. Close to the bifurcation to the square-waves, regimes involving both RO and square-wave oscillations have been found numerically. For other ranges of the fixed parameters, square-waves with rapidly sustained ROs on the top of one of the two plateaus have been found numerically but not observed experimentally yet.

*k*, the ratio of the differential gain coefficients and

*β*, the dimensionless cavity loss parameter of the passive TM mode with respect to the TE mode. Assuming equal cavity decay rate for both the TE and TM modes, only

*k*controls the differences between the total gains. We have investigated the limit

*k*→ 1 (

*β*= (1 −

*k*)/(2

*k*) → 0) both numerically and analytically [28]. We found that the 2

*θ*–periodic square-wave progressively degrades both in form and stability. Specifically, the 2

*θ*-periodic square-wave becomes asymmetric and exhibit more than 2 plateaus. Figure 5 gives a typical example of this behaviour. The total period becomes larger than 2

*θ*and the periodic regime is highly sensitive to noise. Our results suggest that the total gains of the two polarization modes cannot be too close for a successful generation of stable and robust square-waves. This is in agreement with recent work on VCSELs with the PROF setup [9

9. M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A **87**,013827 (2013). [CrossRef]

## A. Appendix: Dimensionless equations

26. T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A **68**,033811 (2003). [CrossRef]

25. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. **17**, 144–151 (2010). [CrossRef]

27. K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. **19**,1501610 (2013). [CrossRef]

## 4.1. Equations by Heil et al [2626. T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A **68**,033811 (2003). [CrossRef]

]

**68**,033811 (2003). [CrossRef]

**68**,033811 (2003). [CrossRef]

*ℰ*(

_{TE,TM}*t*) =

*E*(

_{TE,TM}*t*) exp(

*iω*). The rate equations are given by where

_{TE,TM}t*C*=

*ω*. The values of the parameters are listed in Table 2.

_{TE}τ## 4.2. Takeuchi et al equations [2525. Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. **17**, 144–151 (2010). [CrossRef]

]

**17**, 144–151 (2010). [CrossRef]

**17**, 144–151 (2010). [CrossRef]

*ℰ*(

_{TE,TM}*t*) =

*E*(

_{TE,TM}*t*) exp(−

*iω*). The rate equations are now formulated as where Δ =

_{TE,TM}t*ω*−

_{TE}*ω*and

_{TM}*C*=

*ω*. The values of the parameters are listed in Table 3.

_{TE}τ## 4.3. Fischer et al equations [2727. K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. **19**,1501610 (2013). [CrossRef]

]

27. K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. **19**,1501610 (2013). [CrossRef]

**19**,1501610 (2013). [CrossRef]

## Acknowledgments

## References and links

1. | K. Otsuka and J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. |

2. | D.-L. Cheng, T.-C. Yen, J.-W. Chang, and J.-K. Tsai, “Generation of high-speed single-wavelength optical pulses in semiconductor lasers with orthogonal-polarization optical feedback,” Opt. Commun. |

3. | W. H. Loh, Y. Ozeki, and C. L. Tang, “High-frequency polarization self-modulation and chaotic phenomena in external cavity semiconductor lasers,” Appl. Phys. Lett. |

4. | A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave self-modulation in diode lasers with polarization-rotated optical feedback,” Opt. Lett. |

5. | A. Gavrielides, T. Erneux, D.W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-waveforms in edge-emitting diode lasers subject to polarization rotated optical feedback,” in |

6. | A. Gavrielides, T. Erneux, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Square-wave oscillations in edge-emitting diode lasers with polarization rotated optical feedback,” in |

7. | N. Oliver, M. C. Soriano, D. W. Sukow, and I. Fischer, “Dynamics of a semiconductor laser with polarization-rotated feedback and its utilization for random bit generation,” Opt. Lett. |

8. | J. Mulet, M. Giudici, J. Javaloyes, and S. Balle, “Square-wave switching by crossed-polarization gain modulation in vertical-cavity semiconductor lasers,” Phys. Rev. A |

9. | M. Marconi, J. Javaloyes, S. Barland, M. Giudici, and S. Balle, “Robust square-wave polarization switching in vertical-cavity surface-emitting lasers,” Phys. Rev. A |

10. | D. W. Sukow, T. Gilfillan, B. Pope, M. S. Torre, A. Gavrielides, and C. Masoller, “Square-wave switching in vertical-cavity surface-emitting lasers with polarization-rotated optical feedback: experiments and simulations,” Phys. Rev. A |

11. | S. Jiang, Z. Pan, M. Dagenais, R. A. Morgan, and K. Kojima, “High-frequency polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. |

12. | H. Li, A. Hohl, A. Gavrielides, H. Hou, and K. D. Choquette, “Stable polarization self-modulation in vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. |

13. | M. Sciamanna, T. Erneux, F. Rogister, O. Deparis, P. Megret, and M. Blondel, “Bifurcation bridges between external-cavity modes lead to polarization self-modulation in vertical-cavity surface-emitting lasers,” Phys. Rev. A |

14. | L. Mashal, G. Van der Sande, L. Gelens, J. Danckaert, and G. Verschaffelt, “Square-wave oscillations in semiconductor ring lasers with delayed optical feedback,” Opt. Express |

15. | D. W. 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. | M. Sciamanna, M. Virte, C. Masoller, and A. Gavrielides, “Hopf bifurcation to square-wave switching in mutually coupled semiconductor lasers,” Phys. Rev. E |

18. | C. Masoller, M. Sciamanna, and A. Gavrielides, “Two-parameter study of square-wave switching dynamics in orthogonally delay-coupled semiconductor lasers,” Phil. Transac. Roy. Soc. A |

19. | M. Peil, M. Jacquot, Y. Kouomou Chembo, L. Larger, and T. Erneux, “Routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro-optic-oscillators,” Phys. Rev. E |

20. | L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Strongly asymmetric square-waves of time delayed systems,” Phys. Rev. E |

21. | L. Weicker, T. Erneux, O. d’Huys, J. Danckaert, M. Jacquot, Y. Chembo, and L. Larger, “Slow-fast dynamics of a time-delayed electro-optic oscillator,” Phil. Trans. R. Soc. A |

22. | S.A. Campbell, “Stability and bifurcation in the harmonic oscillator with multiple, delayed feedback loops,” Dynamics of Continuous, Discrete and Impulsive Systems |

23. | F. M. Atay, “Oscillation control in delayed feedback systems,” in Dynamics, Bifurcations, and Control, F. Colonius and L. Grne, eds. (Lect. Notes in Control and Information Sciences 273, 2002), pp. 103–116. [CrossRef] |

24. | Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaotic dynamics in semiconductor lasers subjected to polarization-rotated optical feedback,” Appl. Phys. Lett. |

25. | Y. Takeuchi, R. Shogenji, and J. Ohtsubo, “Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback,” Opt. Rev. |

26. | T. Heil, A. Uchida, P. Davis, and T. Aida, “TE-TM dynamics in a semiconductor laser subject to polarization-rotated optical feedback,” Phys. Rev. A |

27. | K. Hicke, M. Escalona-Moran, D. Brunner, M. C. Soriano, I. Fischer, and C. R. Mirasso, “Information processing using transient dynamics of semiconductor lasers subject to delayed feedback,” IEEE J. Sel. Top. Quantum Electron. |

28. | G. Friart, “Routes to square-wave oscillations in a semiconductor laser subject to polarization rotated feedback”, MS Thesis in Physics, Université Libre de Bruxelles (2013). |

29. | A. Gavrielides, D. W. Sukow, G. Burner, T. McLachlan, J. Miller, and J. Amonette, “Simple and complex square waves in an edge-emitting diode laser with polarization-rotated optical feedback,” Phys. Rev. E |

**OCIS Codes**

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 17, 2014

Revised Manuscript: March 2, 2014

Manuscript Accepted: March 2, 2014

Published: March 18, 2014

**Virtual Issues**

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

**Citation**

Gaetan Friart, Lionel Weicker, Jan Danckaert, and Thomas Erneux, "Relaxation and square-wave oscillations in a semiconductor laser with polarization rotated optical feedback," Opt. Express **22**, 6905-6918 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-6905

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### References

- K. Otsuka, J.-L. Chern, “High-speed picosecond pulse generation in semiconductor lasers with incoherent optical feedback,” Opt. Lett. 16, 1759–1761 (1991). [CrossRef] [PubMed]
- D.-L. Cheng, T.-C. Yen, J.-W. Chang, J.-K. Tsai, “Generation of high-speed single-wavelength optical pulses in semiconductor lasers with orthogonal-polarization optical feedback,” Opt. Commun. 222, 363–369 (2003). [CrossRef]
- W. H. Loh, Y. Ozeki, C. L. Tang, “High-frequency polarization self-modulation and chaotic phenomena in external cavity semiconductor lasers,” Appl. Phys. Lett. 56, 2613–2615 (1990). [CrossRef]
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