## Phase-locking phenomenon in a semiconductor laser with external cavities

Optics Express, Vol. 14, Issue 26, pp. 12859-12867 (2006)

http://dx.doi.org/10.1364/OE.14.012859

Acrobat PDF (7169 KB)

### Abstract

Phase-locked solutions are found numerically in a semiconductor laser with one and two external cavities. Different periodic, quasiperiodic, chaotic, and steady-state regimes form Arnold’s tongues in bi-dimensional parameter spaces of the length and feedback strengths of the external cavities and the pump parameter. This rich structure gives additional possibility for controlling complex dynamics and chaos in a semiconductor laser with external cavities by properly adjusting their lengths and feedback strengths.

© 2006 Optical Society of America

## 1. Introduction

2. P. Besnard, B. Meziane, and G. N. Stephan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. **29**, 1271 (1993). [CrossRef]

3. M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor laser with optical feedback,” Phys. Rev. E **55**, 6414 (1997). [CrossRef]

4. R. P. Salathe, “Diode lasers coupled to external resonators,” Appl. Phys. **20**, 1 (1979). [CrossRef]

5. D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback,” IEEE J. Quantum Electron. **21**, 674 (1985). [CrossRef]

6. M. Jesper, T. Bjarne, and M. Jannik, “Chaos in semiconductor laser with optical feedback: theory and experiment,” IEEE J. Quantum Electron. **28**, 93 (1992). [CrossRef]

7. T. K. Sano, “Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback,” Phys. Rev. A **50**, 2719 (1994). [CrossRef] [PubMed]

8. Y. Liu and J. Ohtsubo, “Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer,” IEEE J. Quantum Electron. **33**, 1163 (1997). [CrossRef]

*et al.*[9

9. F. Rogister, P. Mégret, O. Deparis, M. Blondel, and T. Erneux, “Suppression of low frequency fluctuations and stabilization of a semiconductor laser subject to optical feedback from a double cavity: theoretical results,” Opt. Lett. **24**, 1218 (1999). [CrossRef]

10. F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external cavity laser diode,” Opt. Lett. **25**, 808 (2000). [CrossRef]

*Arnold tongues*[11]. Recently Mendez

*et al.*[12

12. J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, “Dynamics of periodically forced semiconductor laser with optical feedback,” Phys. Rev. E **63**, 066218 (2001). [CrossRef]

## 2. Model equations

13. R. Lang and K Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. **16**, 347–355 (1980). [CrossRef]

*et al.*[14

14. S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, “Experimental verification of the synchronization condition for chaotic external cavity diode lasers,” Phys. Rev. E **62**, 7505–7507 (2000). [CrossRef]

15. T.W. Carr, “Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback,” Eur. Phys. J. D **19**, 245–255 (2002). [CrossRef]

*E*

_{0}(

*t*) of the complex electric field

*E*(

*t*)=

*E*

_{0}(

*t*)exp[

*iω*

_{0}

*t*+

*φ*(

*t*)] (

*ω*

_{0}being the angular frequency of the solitary laser), the average carrier density in the active region

*N*(

*t*) and the phase

*φ*(

*t*). The carrier density at threshold

*N*=

_{th}*N*

_{0}+(

*τ*)

_{p}G^{-1}, where

*N*

_{0}is the carrier density at transparency,

*τ*is the photon lifetime and

_{p}*G*is the modal gain coefficient. The initial phases for the first and second external cavities

*ψ*

_{1}=

*ω*0

*τ*

_{1}and

*ψ*

_{2}=

*ω*0

*τ*

_{2}, where

*τ*

_{1}and

*τ*

_{2}are the corresponding round trip times. The other parameters are

*τ*is the carrier lifetime,

_{s}*P*is the pumping term,

*α*is the linewidth enhancement factor, and

*κ*

_{1}and

*κ*

_{2}are the feedback strengths for the first and second external cavities. The last two parameters are defined by the following formula:

*κ*

_{1,2}=(1/

*τ*

_{0})(1-

*r*

_{3,4})(

*r*

_{3,4}/

*r*

_{2})

^{1/2}, where

*τ*

_{0}is the round trip time for the internal cavity. In our simulations we use the following parameter values [16

16. T. Heil, I. Fischer, W. ElsäBer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms,” Phys. Rev. E **67**, 066214 (2003). [CrossRef]

*α*=3.5,

*G*=5×10

^{25}m

^{3}ns

^{-1},

*τ*=1 ns,

_{s}*τ*=1 ps,

_{p}*P*=8×10

^{24}m

^{-3},

*N*=5×10

_{th}^{24}m

^{-3}and

*N*0=3×10

^{24}m

^{-3}. In all simulations performed with two external cavities, the parameters

*τ*

_{1}and κ

_{1}of the first external cavity are fixed.

## 3. Results

*κ*

_{2}=0) and then we show how the second external cavity affects laser dynamics. The analysis is performed with codimensional-one and codimensional-two bifurcation diagrams in the parameter spaces of the lengths and feedback strengths of the external cavities and the pump parameter.

### 3.1 One external cavity

16. T. Heil, I. Fischer, W. ElsäBer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms,” Phys. Rev. E **67**, 066214 (2003). [CrossRef]

*κ*

_{1}=25 ns

^{-1}, the solution of the system Eqs. (1)–(3) is chaotic in the well-known form of LFF shown in Fig. 2.

*l*

_{1}=

*cτ*

_{1}/2 (

*c*being the speed of light) and the feedback strength. The diagram displays a striped structure, the alternation of steady-state, periodic, and chaotic regimes. In Fig. 3 and hereafter, the black dots indicate fix points or cw laser operation, the yellow, blue, and red are, respectively, the period-1, period-2, and period-3 regimes. The white regions indicate the quasiperiodic and chaotic regimes.

*l*

_{1}or

*κ*

_{1}is changed, the laser undergoes different bifurcations: a Hopf bifurcation, a period-doubling bifurcation(s), and a torus bifurcation after which a quasi-periodic regime gives rise. The quasi-periodicity is converted to chaotic LFF terminated in crisis where a new cascade of bifurcations is initiated. In fact, the state diagram shown in Fig. 3 is the codimensional-two bifurcation diagram, where the boundaries between the black and yellow regions represent the Hopf bifurcation lines, the boundaries between the yellow and blue regions are period-doubling bifurcation lines, and the onsets of the white regions are the torus bifurcations. For very weak feedback strengths (

*κ*

_{1}<0.4 ns

^{-1}), the laser works in a cw (steady-state) regime for any length of the external cavity. Another interesting result is that for relatively strong feedback strengths, the width of the chaotic tongues increases with increasing the cavity length. Chaotic oscillations are not observed in the laser with a very short external cavity, i.e. the laser is always stable. Instead, the laser with a very long external cavity is always unstable.

*l*

_{1}<1.7 cm) and a weak feedback strength (

*κ*

_{1}<1 ns

^{-1}) works in a stationary (cw) regime independent on how much the pumping is. However, for a longer external cavity the laser dynamics is very rich. The state diagrams in both graphics form the structure of Arnold tongues which are distributed almost equidistantly along the abscissa axes. One can also see that while

*P*is increased, the chaotic tongues become wider and finally at a very strong pump, chaos dominates over the other regimes.

### 3.2 Two external cavities

*τ*

_{1}=0.22 ns and

*κ*

_{1}=25 ns

^{-1}. One of the parameters of the second external cavity which can be easily controlled in experiments is its length

*l*

_{2}=

*cτ*

_{2}/2. First, we derive the codimensional-one bifurcation diagrams of the laser peak intensity with respect to

*τ*

_{2}/

*τ*

_{1}=

*l*

_{2}/

*l*

_{1}for different fixed values of the feedback strength and the pump parameter. One of such diagrams is shown in Fig. 6. One can see that the laser dynamics is very sophisticated and the diagram is not symmetric with respect to

*l*

_{2}=

*l*

_{1}. By fixing

*τ*

_{1}, we solve Eqs. (1)–(3) for different

*τ*

_{2}. As the delay time of the second external cavity is changed, the laser undergoes different types of bifurcations (Hopf bifurcation, torus bifurcation, period-doubling bifurcation, and crisis). With increasing the delay time, the chaotic attractor, first, transforms to a torus and then the laser undergoes an inverse torus bifurcation resulting in period-1 oscillations which are terminated again in chaos.

16. T. Heil, I. Fischer, W. ElsäBer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms,” Phys. Rev. E **67**, 066214 (2003). [CrossRef]

*τ*

_{2}/

*τ*

_{1}and

*κ*

_{2}/

*κ*

_{1}, and pump parameter

*P*. These codimensional-two bifurcation diagrams are constructed by analyzing the codimensional-one bifurcation diagrams, similar to that shown in Fig. 6, calculated for different fixed values of the feedback strength and the pump parameter. Although the parameters of the first external cavity are maintained to be constants, we plot these diagrams in the coordinates of the ratios of the cavity parameters for better understanding of the physical mechanism underlying the phase-locking phenomenon.

*τ*

_{1}/

*τ*

_{2}=

*p*/

*q*(both

*p*and

*q*being integers), i.e. at

*τ*

_{2}/

*τ*

_{1}=1/1, 1/2, 1/4, 2/1, 3/2, …. It is particularly remarkable that the diagrams around

*τ*

_{2}/

*τ*

_{1}=1 are almost symmetrical. It is also possible for the tongues to overlap. In this case phase-locked solutions coexist at a point in the parameter space leading to multistability. The asymmetry in the diagrams appears just due to multistability. In the numerical simulations the phase-locked solution that is reached depends upon the basin structure for the coexisting attractors and the choice of initial conditions.

12. J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, “Dynamics of periodically forced semiconductor laser with optical feedback,” Phys. Rev. E **63**, 066218 (2001). [CrossRef]

## 4. Conclusions

## References and links

1. | J. Ohtsubo, |

2. | P. Besnard, B. Meziane, and G. N. Stephan, “Feedback phenomena in a semiconductor laser induced by distant reflectors,” IEEE J. Quantum Electron. |

3. | M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, “Andronov bifurcation and excitability in semiconductor laser with optical feedback,” Phys. Rev. E |

4. | R. P. Salathe, “Diode lasers coupled to external resonators,” Appl. Phys. |

5. | D. Lenstra, B. H. Verbeek, and A. J. den Boef, “Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback,” IEEE J. Quantum Electron. |

6. | M. Jesper, T. Bjarne, and M. Jannik, “Chaos in semiconductor laser with optical feedback: theory and experiment,” IEEE J. Quantum Electron. |

7. | T. K. Sano, “Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback,” Phys. Rev. A |

8. | Y. Liu and J. Ohtsubo, “Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer,” IEEE J. Quantum Electron. |

9. | F. Rogister, P. Mégret, O. Deparis, M. Blondel, and T. Erneux, “Suppression of low frequency fluctuations and stabilization of a semiconductor laser subject to optical feedback from a double cavity: theoretical results,” Opt. Lett. |

10. | F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, “Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external cavity laser diode,” Opt. Lett. |

11. | V. I. Arnold, |

12. | J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, “Dynamics of periodically forced semiconductor laser with optical feedback,” Phys. Rev. E |

13. | R. Lang and K Kobayashi, “External Optical Feedback Effects on Semiconductor Injection Laser Properties,” IEEE J. Quantum Electron. |

14. | S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, “Experimental verification of the synchronization condition for chaotic external cavity diode lasers,” Phys. Rev. E |

15. | T.W. Carr, “Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback,” Eur. Phys. J. D |

16. | T. Heil, I. Fischer, W. ElsäBer, B. Krauskopf, K. Green, and A. Gavrielides, “Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms,” Phys. Rev. E |

17. | H. G. Shuster, |

**OCIS Codes**

(140.5960) Lasers and laser optics : Semiconductor lasers

(190.3100) Nonlinear optics : Instabilities and chaos

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: September 28, 2006

Revised Manuscript: November 21, 2006

Manuscript Accepted: November 26, 2006

Published: December 22, 2006

**Citation**

F. R. Ruiz-Oliveras and A. N. Pisarchik, "Phase-locking phenomenon in a semiconductor laser with external cavities," Opt. Express **14**, 12859-12867 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-26-12859

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### References

- J. Ohtsubo, Semiconductor Lasers: Stability, Instability and Chaos, Springer Series in Optical Sciences, vol. 111 (Springer-Verlag, Berlin, 2005).
- P. Besnard, B. Meziane, and G. N. Stephan, "Feedback phenomena in a semiconductor laser induced by distant reflectors," IEEE J. Quantum Electron. 29, 1271 (1993). [CrossRef]
- M. Giudici, C. Green, G. Giacomelli, U. Nespolo, and J. R. Tredicce, "Andronov bifurcation and excitability in semiconductor laser with optical feedback," Phys. Rev. E 55, 6414 (1997). [CrossRef]
- R. P. Salathe, "Diode lasers coupled to external resonators," Appl. Phys. 20, 1 (1979). [CrossRef]
- D. Lenstra, B. H. Verbeek, and A. J. den Boef, "Coherence Collapse in Single-Mode. Semiconductor Lasers Due to Optical Feedback," IEEE J. Quantum Electron. 21, 674 (1985). [CrossRef]
- M. Jesper, T. Bjarne, and M. Jannik, "Chaos in semiconductor laser with optical feedback: theory and experiment," IEEE J. Quantum Electron. 28, 93 (1992). [CrossRef]
- T. K. Sano, "Antimode-dynamics and chaotic itinerancy in the coherence collapse of semiconductor lasers with optical feedback," Phys. Rev. A 50, 2719 (1994). [CrossRef] [PubMed]
- Y. Liu and J. Ohtsubo, "Dynamics and Chaos Stabilization of Semiconductor Lasers with Optical Feedback from an Interferometer," IEEE J. Quantum Electron. 33, 1163 (1997). [CrossRef]
- F. Rogister, P. Mégret, O. Deparis, M. Blondel, and T. Erneux, "Suppression of low frequency fluctuations and stabilization of a semiconductor laser subject to optical feedback from a double cavity: theoretical results," Opt. Lett. 24, 1218 (1999). [CrossRef]
- F. Rogister, D. W. Sukow, A. Gavrielides, P. Mégret, O. Deparis, and M. Blondel, "Experimental demonstration of suppression of low-frequency fluctuations and stabilization of an external cavity laser diode," Opt. Lett. 25, 808 (2000). [CrossRef]
- V. I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, Berlin, 1989).
- J. M. Mendez, R. Laje, M. Giudici, J. Aliaga, and G. B. Mindlin, "Dynamics of periodically forced semiconductor laser with optical feedback," Phys. Rev. E 63, 066218 (2001). [CrossRef]
- R. Lang and K Kobayashi, "External Optical Feedback Effects on Semiconductor Injection Laser Properties," IEEE J. Quantum Electron. 16, 347-355 (1980). [CrossRef]
- S. Sivapakrasam, E. M. Shahverdi, and K. A. Shore, "Experimental verification of the synchronization condition for chaotic external cavity diode lasers," Phys. Rev. E 62, 7505-7507 (2000). [CrossRef]
- T.W. Carr, "Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback," Eur. Phys. J. D 19, 245-255 (2002). [CrossRef]
- T. Heil, I. Fischer, W . ElsäBer, B . Krauskopf, K . Green, and A. Gavrielides, "Delay dynamics of semiconductor lasers with short external cavities: Bifurcation scenarios and mechanisms," Phys. Rev. E 67, 066214 (2003). [CrossRef]
- H. G. Shuster, Deterministic Chaos (VCH Verlag, Weinheim, 1988).

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