## Spatio-temporal dynamics in semiconductor microresonators with thermal effects

Optics Express, Vol. 10, Issue 19, pp. 1009-1017 (2002)

http://dx.doi.org/10.1364/OE.10.001009

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### Abstract

In this paper we study the dynamics of the intracavity field, carriers and lattice temperature in externally driven semiconductor microcavities. The combination/competition of the different time-scales of the dynamical variables together with diffraction and carrier/thermal diffusions are responsible for new dynamical behaviors. We report here the occurrence of a spatio-temporal instability of the Hopf type giving rise to Regenerative Oscillations and travelling patterns and cavity solitons.

© 2002 Optical Society of America

## 1 Introduction

*e.g.*[2, 3

3. M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. **73**, 640ߝ643 (1994). [CrossRef] [PubMed]

4. W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. **76**, 1623ߝ1626 (1996). [CrossRef] [PubMed]

5. M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. **34**, 109ߝ114 (1996). [CrossRef]

*e.g.*[6

6. M. Saffman, D. Montgomery, and D. Z. Anderson, “Collapse of a transverse-mode continuum in a self-imaging photorefractively pumped ring resonator,” Opt. Lett. **19**, 518ߝ520 (1994). [CrossRef] [PubMed]

7. V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,” Phys. Rev. A **56**, 1582ߝ1591 (1997). [CrossRef]

*e.g.*[8

8. B. Schaepers, M. Feldmann, T. Ackemannand, and W. Lange, “Interaction of Localized Structures in an Optical Pattern-Forming System,” Phys. Rev. Lett. **85**, 748ߝ751 (2000). [CrossRef]

10. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A **58**, 2542ߝ2559 (1998) and references quoted therein. [CrossRef]

11. L. Spinelli, G. Tissoni, M. Tarenghi, and M. Brambilla, “First principle theory for cavity solitons in semiconductor microresonators,” Eur. Phys. J. D **15**, 257ߝ266 (2001) and references quoted therein. [CrossRef]

*i.e.*with population inversion, Multi-Quantum-Well structure.

12. E. Abraham, “Modelling of regenerative pulsations in an InSb etalon,” Opt. Comm. **61**, 282ߝ286 (1987) and references quoted therein. [CrossRef]

16. L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, “Thermal effects and transverse structures in semiconductor microcavities with population inversion,” Phys. Rev. A **66**, 023817 (2002). [CrossRef]

16. L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, “Thermal effects and transverse structures in semiconductor microcavities with population inversion,” Phys. Rev. A **66**, 023817 (2002). [CrossRef]

## 2 The model

16. L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, “Thermal effects and transverse structures in semiconductor microcavities with population inversion,” Phys. Rev. A **66**, 023817 (2002). [CrossRef]

18. T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, “Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A **58**, 3279ߝ3292 (1998). [CrossRef]

**66**, 023817 (2002). [CrossRef]

*E*is the adimensional slowly varying envelope of the intracavity electric field;

*N*and

*T*are the carrier density normalized to the transparency value

*N*

_{0}and the temperature normalized to the room temperature

*T*

_{0}, respectively.

*κ, γ*and

*γ*

_{th}are the decay rates of

*E, N*and

*T*, respectively. We assumed for the ratio between

*γ*

_{th}and

*γ*a typical value of 10

^{-3},

*i.e.*we consider a temperature dynamics on a microsecond scale, and a carrier dynamics on nanosecond scale. As for the electric field dynamics, it is 2 or 3 order of magnitude faster than carrier dynamics for such a kind of microresonators.

*∂*

^{2}/

*∂x*

^{2}+

*∂*

^{2}/

*∂y*

^{2}; it represents diffraction (in Eq. (1)), and carrier and thermal diffusion (in Eqs. (2) and (3) through the diffusion parameters

*d*and

*D*

_{T}, respectively) in the paraxial approximation. The transverse coordinates

*x*and

*y*are scaled to the diffraction length.

*ω*

_{0}is the reference frequency;

*E*

_{I}is the adimensional slowly varying envelope of the injected field;

*I*is the adimensional injected current; Σ is the bistability parameter. On the other hand, the coefficients

*Z*and

*P*describe the heating of the device due to carriers and to Joule effect, respectively.

*χ*

_{nl}, we adopt here the microscopic description of the radiation-matter interaction already described in [11

11. L. Spinelli, G. Tissoni, M. Tarenghi, and M. Brambilla, “First principle theory for cavity solitons in semiconductor microresonators,” Eur. Phys. J. D **15**, 257ߝ266 (2001) and references quoted therein. [CrossRef]

*T*

_{0}, was assumed. Here, instead, the temperature dependence in the material susceptibility

*χ*

_{nl}is taken into account. It mainly consists in a red shift of the band gap

*∊*

_{gap}of the semiconductor upon an increase of temperature [16

**66**, 023817 (2002). [CrossRef]

*∊*

_{gap}(

*T*

_{0})/

*ħ*-

*ω*

_{0})/

*γ*

_{p}, with

*γ*

_{p}being the polarization decay rate.

**66**, 023817 (2002). [CrossRef]

## 3 Numerical results

*χ*

_{nl}, it has been calculated and tabulated in a file as a function of

*N*and

*T*, and then interpolated during the time integration [16

**66**, 023817 (2002). [CrossRef]

*K*= 0. Correspondingly, global regenerative oscillations [12

12. E. Abraham, “Modelling of regenerative pulsations in an InSb etalon,” Opt. Comm. **61**, 282ߝ286 (1987) and references quoted therein. [CrossRef]

*K*≠ 0. In this case the steady state curve is monostable and no cavity solitons are present. By means of numerical simulations in two transverse dimensions we showed the existence of a travelling pattern in the unstable region. In Fig. 3 we show a movie that demonstrates this result (travelling honeycomb pattern).

**66**, 023817 (2002). [CrossRef]

## 4 Conclusions and discussion

*i. e.*a regime where the system displays an oscillatory homogeneous output intensity, for a constant homogeneous input intensity. Moreover the numerical simulations performed here led us to concluding that regenerative oscillations appear close to conditions of nascent hysteresis, when an interval of values of the driving field exists where no stationary solution is stable. Finally, a very small parameter range was found, where the Hopf instability was dominating over the Turing instability for

*k*≠ 0. In this case the steady state curve is monostable and no cavity solitons are present. In the unstable branch, numerical simulations showed the existence of a travelling honeycomb pattern, showed here by a 2-D movie. This behavior of travelling spatial pattern is just due to the presence of thermal effects.

## 5 Acknowledgments

*Processing of Information by Arrays of Nonlinear Optical Solitons*and of the PRIN project

*Formazione e controllo di solitoni di cavità in microrisonatori a semiconduttore*of the Italian Ministry for University and Research.

## References and links

1. | L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical Pattern Formation,” in Advances in Atomic, Molecular and Optical Physics, Vol. 40, edited by B. Bederson and H. Walther , Academic Press, 1998, pp. 229ߝ306, and references quoted therein. |

2. | N. N. Rosanov and G. V. Khodova, “Autosolitons in bistable interferometers,” Opt. Spectrosc. |

3. | M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. |

4. | W. J. Firth and A. J. Scroggie, “Optical bullet holes: robust controllable localized states of a nonlinear cavity,” Phys. Rev. Lett. |

5. | M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and control of optical localized structures,” Europhys. Lett. |

6. | M. Saffman, D. Montgomery, and D. Z. Anderson, “Collapse of a transverse-mode continuum in a self-imaging photorefractively pumped ring resonator,” Opt. Lett. |

7. | V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,” Phys. Rev. A |

8. | B. Schaepers, M. Feldmann, T. Ackemannand, and W. Lange, “Interaction of Localized Structures in an Optical Pattern-Forming System,” Phys. Rev. Lett. |

9. | S. Barland, J.R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Koedl, M. Miller, and R. Jaeger, “Cavity solitons work as pixels in semiconductors,” Nature, to appear. See also references quoted therein. |

10. | L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. A. Lugiato, “Spatial solitons in semiconductor microcavities,” Phys. Rev. A |

11. | L. Spinelli, G. Tissoni, M. Tarenghi, and M. Brambilla, “First principle theory for cavity solitons in semiconductor microresonators,” Eur. Phys. J. D |

12. | E. Abraham, “Modelling of regenerative pulsations in an InSb etalon,” Opt. Comm. |

13. | S. Barland, O. Piro, S. Balle, M. Giudici, and J. Tredicce, “Thermo-optical pulsation in semiconductor lasers with injected signal: Relaxation oscillations, excitability, phase-locking and coherence resonance,” preprint. |

14. | R. Kuszelewicz et al., 2 |

15. | L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, “Thermal instabilites in semiconductor amplifiers,” submitted to J. Mod. Opt., special issue for the Proceedings of the Physics of Quantum Electronics Conference (Snowbird USA January6ߝ10, 2002) edited by R. W. Boyd and M. O. Scully. |

16. | L. Spinelli, G. Tissoni, L. A. Lugiato, and M. Brambilla, “Thermal effects and transverse structures in semiconductor microcavities with population inversion,” Phys. Rev. A |

17. | A. J. Scroggie, J. M. McSloy, and W. J. Firth, “Self-Propelled Cavity Solitons in Semiconductor Microcavities,” submitted to Phys. Rev. E. |

18. | T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney, and C. Z. Ning, “Modeling the interplay of thermal effects and transverse mode behavior in native-oxide-confined vertical-cavity surface-emitting lasers,” Phys. Rev. A |

**OCIS Codes**

(160.6000) Materials : Semiconductor materials

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.4870) Nonlinear optics : Photothermal effects

**ToC Category:**

Research Papers

**History**

Original Manuscript: August 1, 2002

Revised Manuscript: September 12, 2002

Published: September 23, 2002

**Citation**

Giovanna Tissoni, Lorenzo Spinelli, Luigi Lugiato, Massimo Brambilla, Ida Perrini, and Tommaso Maggipinto, "Spatio-temporal dynamics in semiconductor microresonators with thermal effects," Opt. Express **10**, 1009-1017 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-19-1009

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### References

- L. A. Lugiato, M. Brambilla and A. Gatti, �??Optical Pattern Formation,�?? in Advances in Atomic, Molecular and Optical Physics, Vol. 40, edited by B. Bederson and H. Walther, Academic Press, 1998, pp. 229-306, and references quoted therein.
- N. N. Rosanov and G. V. Khodova, �??Autosolitons in bistable interferometers,�?? Opt. Spectrosc. 65, 449-450 (1988).
- M. Tlidi, P. Mandel and R. Lefever, �??Localized structures and localized patterns in optical bistability,�?? Phys. Rev. Lett. 73, 640-643 (1994). [CrossRef] [PubMed]
- W. J. Firth and A. J. Scroggie, �??Optical bullet holes: robust controllable localized states of a nonlinear cavity,�?? Phys. Rev. Lett. 76, 1623-1626 (1996). [CrossRef] [PubMed]
- M. Brambilla, L. A. Lugiato and M. Stefani, �??Interaction and control of optical localized structures,�?? Europhys. Lett. 34, 109-114 (1996). [CrossRef]
- M. Saffman, D. Montgomery and D. Z. Anderson, �??Collapse of a transverse-mode continuum in a self-imaging photorefractively pumped ring resonator,�?? Opt. Lett. 19, 518-520 (1994). [CrossRef] [PubMed]
- V. B. Taranenko, K. Staliunas and C. O. Weiss, �??Spatial soliton laser: localized structures in a laser with a saturable absorber in a self-imaging resonator,�?? Phys. Rev. A 56, 1582-1591 (1997). [CrossRef]
- B. Schaepers, M. Feldmann, T. Ackemannand, W. Lange, �??Interaction of Localized Structures in an Optical Pattern-Forming System,�?? Phys. Rev. Lett. 85, 748-751 (2000). [CrossRef]
- S. Barland, J.R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Koedl, M. Miller and R. Jaeger, �??Cavity solitons work as pixels in semiconductors,�?? Nature, to appear. See also references quoted therein.
- L. Spinelli, G. Tissoni, M. Brambilla, F. Prati and L. A. Lugiato, �??Spatial solitons in semiconductor microcavities,�?? Phys. Rev. A 58, 2542-2559 (1998) and references quoted therein. [CrossRef]
- L. Spinelli, G. Tissoni, M. Tarenghi and M. Brambilla, �??First principle theory for cavity solitons in semiconductor microresonators,�?? Eur. Phys. J. D 15, 257-266 (2001) and references quoted therein. [CrossRef]
- E. Abraham, �??Modelling of regenerative pulsations in an InSb etalon,�?? Opt. Commun. 61, 282-286 (1987) and references quoted therein. [CrossRef]
- S. Barland, O. Piro, S. Balle, M. Giudici and J. Tredicce, �??Thermo-optical pulsation in semiconductor lasers with injected signal: Relaxation oscillations, excitability, phase-locking and coherence resonance,�?? preprint.
- R. Kuszelewicz et al., 2nd yearly report of the PIANOS Project (2000). I. Ganne, Ph. D. Thesis (2000).
- L. Spinelli, G. Tissoni, L. A. Lugiato and M. Brambilla, �??Thermal instabilites in semiconductor amplifiers,�?? submitted to J. Mod. Opt., special issue for the Proceedings of the Physics of Quantum Electronics Conference (Snowbird USA January 6-10, 2002) edited by R. W. Boyd and M. O. Scully.
- L. Spinelli, G. Tissoni, L. A. Lugiato and M. Brambilla, �??Thermal e.ects and transverse structures in semiconductor microcavities with population inversion,�?? Phys. Rev. A 66, 023817(2002). [CrossRef]
- A. J. Scroggie, J. M. McSloy and W. J. Firth, �??Self-Propelled Cavity Solitons in Semiconductor Microcavities,�?? submitted to Phys. Rev. E.
- T. Rossler, R. A. Indik, G. K. Harkness, J. V. Moloney and C. Z. Ning, �??Modeling the interplay of thermal e.ects and transverse mode behavior in native-oxide-confined vertical-cavity surfaceemitting lasers,�?? Phys. Rev. A 58, 3279-3292 (1998). [CrossRef]

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