## Physical model for the incoherent writing/erasure of cavity solitons in semiconductor optical amplifiers

Optics Express, Vol. 15, Issue 19, pp. 12457-12463 (2007)

http://dx.doi.org/10.1364/OE.15.012457

Acrobat PDF (578 KB)

### Abstract

We present a physical mechanism that explains the recent observations of incoherent writing and erasure of Cavity Solitons in a semiconductor optical amplifier [S. Barbay *et al*, Opt. Lett. **31**, 1504–1506 (2006)]. This mechanism allows to understand the main observations of the experiment. In particular it perfectly explains why writing and erasure are possible as a result of a local perturbation in the carrier density, and why a delay is observed along with the writing process. Numerical simulations in 1D are performed and show very good qualitative agreement with the experimental observations.

© 2007 Optical Society of America

## 1. Introduction

2. S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödel, M. Miller, and R. Jäger, “Cavity solitons work as pixels in semiconductors,” Nature **419**, 699–702 (2002). [CrossRef] [PubMed]

1. S. Barbay, Y. Ménesguen, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, “Incoherent and coherent writing and erasure of cavity solitons in an optically pumped semiconductor amplifier,” Opt. Lett. **31**, 1504–1506 (2006). [CrossRef] [PubMed]

4. Y. Menesguen, S. Barbay, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, “Optical self-organization and cavity solitons in optically pumped semiconductor microresonators,” Phys. Rev. A **74**, 023818 (2006). [CrossRef]

5. Y. Tanguy, T. Ackemann, and R. Jager, “Characteristics of bistable localized emission states in broad-area vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A **74**, 053824 (2006). [CrossRef]

6. I. Ganne, G. Slekys, I. Sagnes, and R. Kuszelewicz, “Precursor forms of cavity solitons in nonlinear semiconductor microresonators,” Phys. Rev. E **66**, 066613 (2002). [CrossRef]

7. F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. **89**, 221111 (2006). [CrossRef]

8. X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle, and J. Tredicce, “Cavity-solitons switching in semiconductor microcavities,” Phys. Rev. A **72**, 013815 (2005). [CrossRef]

1. S. Barbay, Y. Ménesguen, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, “Incoherent and coherent writing and erasure of cavity solitons in an optically pumped semiconductor amplifier,” Opt. Lett. **31**, 1504–1506 (2006). [CrossRef] [PubMed]

9. M. Brambilla, L.A. Lugiato, F. Prati, L. Spinelli, and W. Firth, “Spatial Soliton Pixels in Semiconductor Devices,” Phys. Rev. Lett. **79**, 2042–2045 (1997). [CrossRef]

4. Y. Menesguen, S. Barbay, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, “Optical self-organization and cavity solitons in optically pumped semiconductor microresonators,” Phys. Rev. A **74**, 023818 (2006). [CrossRef]

## 2. Model

10. 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]

9. M. Brambilla, L.A. Lugiato, F. Prati, L. Spinelli, and W. Firth, “Spatial Soliton Pixels in Semiconductor Devices,” Phys. Rev. Lett. **79**, 2042–2045 (1997). [CrossRef]

11. G. Tissoni, L. Spinelli, L. A. Lugiato, M. Brambilla, I. M. Perrini, and T. Maggipinto, “Spatiotemporal dynamics in semiconductor microresonators with thermal effects,” Opt. Express **10**, 1009–1017 (2002). [PubMed]

12. 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]

13. A. J. Scroggie, J. M. McSloy, and W. J. Firth, “Self-propelled cavity solitons in semiconductor microcavities,” Phys. Rev. E **66**, 036607 (2002). [CrossRef]

*E*, the carrier density

*N*and the detuning

*θ*:

*E*is the holding beam field amplitude,

_{I}*C*the bistability parameter,

*α*is the Henry enhancement factor. The carrier recombination rate is

*γ*and the thermal relaxation rate towards the steady state value

*θ*

_{0}is

*γ*. The pumping term for the carriers Λ also provides a source term in the detuning equation through a linear function

_{T}*f*. Diffusion and diffraction processes are also included through second derivative terms. A small noise term

*ξ*in the intracavity field equation has been added to mimic spontaneous emission such that <

*ξ(t)ξ(t*>=2

^{′})*D*(

_{E}δ*t-t*) where

^{′}*D*=10

_{E}^{-6}. Local carrier injection by an external address beam occurs on a very fast time scale (~100fs) such that we consider that it amounts to a perturbation at

*x*of duration

_{i}*τ*on the pump term Λ=Λ

_{0}[1+δΛΠ(

*t-t*)Π(

_{0},τ*x-x*)], where Π(

_{i},δ x*x,δ x*) is a boxcar function centered on x of width

*δ x*. The corresponding source term f is such that

*f*(Λ

_{0})=0 and is given by

*f*(Λ)=(Λ-3.2)/0.2. We choose standard parameters for semiconductor materials that give bistability and a modulational instability :

*C*=0.2,

*θ*

_{0}=-2, Λ

_{0}=3.2,

*E*=0.75,

_{I}*γ*=0.01,

*γ*=10

_{T}^{-4}. The laser threshold is given by Λ

_{th}=1+1/2

*C*while transparency is at Λ

_{0}=1, hence we are in an amplifying medium. The diffusion of carriers and of detuning are taken to be equal. Although this may not be true in general, this is a reasonable starting point since DT may vary a lot according to the design details of the microresonator, and in particular how thermal management is taken care of [14]. The same holds for the relaxation time of the detuning

*γ*, and the ratio

_{T}*γ/γ*=100 is chosen to be large as expected in the experiment. Larger ratios may be required to fit the actual parameters in the experiments at the expense of much longer computation times but without significant qualitative impact on the dynamics.

_{T}## 3. Results

15. R. L. Honeycutt, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A **45**, 600–603 (1992). [CrossRef] [PubMed]

*τ*=20.

*θ*shows a decrease of the detuning after the pump pulse followed by a slow relaxation towards the steady-state value. Once the writing pulse energy has been released, the system is brought to an unstable state and a slow dynamics governed by the detuning takes place. CS switch-on is triggered by this slow dynamics. On the contrary, CS switch-off takes place immediately after the beginning of the injection of carriers and the following detuning has little effect if not a small but visible relaxation of the system to the steady state.

16. B. Segard, J. Zemmouri, and B. Macke, “Noncritical slowing down in optical bistability,” Opt. Commun. **63**, 339–343 (1987). [CrossRef]

17. F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. **71**, 385–392 (1989). [CrossRef]

*A*(product of the perturbation time and perturbation amplitude) is larger than a critical value

*A*, the system switches to the other state with a delay (non-critical slowing down) which has a logarithmic scaling law

_{c}*δτ*~-ln(

*A-A*). When the perturbation area is close to its critical value, infinite delay can be observed whereas, on the other side, the larger the perturbation, the smaller the delay. This behavior is reproduced here in our 1D system as shown on Fig. 4.

_{c}*δ*Λ

_{c}≃1.0225. Below this value the switch-off fails and above this value the system evolves towards the homogeneous background (see Fig. 5). On this figure, we have prepared the system with a cavity soliton and after a time

*t*=100 launched an erasing pulse of a given amplitude. When the switch-off is effective, for an erasure power larger than the critical value, the erasure of the cavity soliton is very fast. For erasing powers slightly below the critical value, the dynamics is slowed down around the unstable point and the system returns back to its on-state. If the erasing power is high enough however, the switch down is followed by a switch-on if the local heat-induced detuning is large. We note that precise rules for the writing/erasing conditions can in principle be found, at least numerically, e.g. using the numerical method introduced in [10

10. 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]

*E*) over a range of input fields

_{I}*E*and writing/erasing powers.

_{I}## 4. Conclusion

1. S. Barbay, Y. Ménesguen, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, “Incoherent and coherent writing and erasure of cavity solitons in an optically pumped semiconductor amplifier,” Opt. Lett. **31**, 1504–1506 (2006). [CrossRef] [PubMed]

18. D.N. Maywar, G.P. Agrawal, and Y. Nakano, “All-optical hysteresis control by means of cross-phase modulation in semiconductor optical amplifiers,” J. Opt. Soc. Am. B **18**, 1003–1013 (2001). [CrossRef]

**31**, 1504–1506 (2006). [CrossRef] [PubMed]

## References and links

1. | S. Barbay, Y. Ménesguen, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, “Incoherent and coherent writing and erasure of cavity solitons in an optically pumped semiconductor amplifier,” Opt. Lett. |

2. | S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödel, M. Miller, and R. Jäger, “Cavity solitons work as pixels in semiconductors,” Nature |

3. | X. Hachair, S. Barland, L. Furfaro, M. Giudici, S. Balle, J. R. Tredicce, M. Brambilla, T. Maggipinto, I. M. Perrini, G. Tissoni, and L. Lugiato, “Cavity solitons in broad-area vertical-cavity surface-emitting lasers below threshold,” Phys. Rev. |

4. | Y. Menesguen, S. Barbay, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, “Optical self-organization and cavity solitons in optically pumped semiconductor microresonators,” Phys. Rev. A |

5. | Y. Tanguy, T. Ackemann, and R. Jager, “Characteristics of bistable localized emission states in broad-area vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A |

6. | I. Ganne, G. Slekys, I. Sagnes, and R. Kuszelewicz, “Precursor forms of cavity solitons in nonlinear semiconductor microresonators,” Phys. Rev. E |

7. | F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Positioning cavity solitons with a phase mask,” Appl. Phys. Lett. |

8. | X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle, and J. Tredicce, “Cavity-solitons switching in semiconductor microcavities,” Phys. Rev. A |

9. | M. Brambilla, L.A. Lugiato, F. Prati, L. Spinelli, and W. Firth, “Spatial Soliton Pixels in Semiconductor Devices,” Phys. Rev. Lett. |

10. | W. J. Firth and A. J. Scroggie, “Optical Bullet Holes: Robust Controllable Localized States of a Nonlinear Cavity,” Phys. Rev. Lett. |

11. | G. Tissoni, L. Spinelli, L. A. Lugiato, M. Brambilla, I. M. Perrini, and T. Maggipinto, “Spatiotemporal dynamics in semiconductor microresonators with thermal effects,” Opt. Express |

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

13. | A. J. Scroggie, J. M. McSloy, and W. J. Firth, “Self-propelled cavity solitons in semiconductor microcavities,” Phys. Rev. E |

14. | Y. Ménesguen and R. Kuszelewicz, unpublished. |

15. | R. L. Honeycutt, “Stochastic Runge-Kutta algorithms. I. White noise,” Phys. Rev. A |

16. | B. Segard, J. Zemmouri, and B. Macke, “Noncritical slowing down in optical bistability,” Opt. Commun. |

17. | F. Mitschke, C. Boden, W. Lange, and P. Mandel, “Exploring the dynamics of the unstable branch of bistable systems,” Opt. Commun. |

18. | D.N. Maywar, G.P. Agrawal, and Y. Nakano, “All-optical hysteresis control by means of cross-phase modulation in semiconductor optical amplifiers,” J. Opt. Soc. Am. B |

**OCIS Codes**

(190.1450) Nonlinear optics : Bistability

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

(250.5980) Optoelectronics : Semiconductor optical amplifiers

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: May 11, 2007

Revised Manuscript: July 31, 2007

Manuscript Accepted: July 31, 2007

Published: September 14, 2007

**Citation**

S. Barbay and R. Kuszelewicz, "Physical model for the incoherent writing/erasure of cavity solitons in semiconductor optical amplifiers," Opt. Express **15**, 12457-12463 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-19-12457

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### References

- S. Barbay, Y. Ménesguen, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, "Incoherent and coherent writing and erasure of cavity solitons in an optically pumped semiconductor amplifier," Opt. Lett. 31, 1504-1506 (2006). [CrossRef] [PubMed]
- S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knödel, M. Miller, and R. Jäger, "Cavity solitons work as pixels in semiconductors," Nature 419, 699-702 (2002). [CrossRef] [PubMed]
- X. Hachair, S. Barland, L. Furfaro, M. Giudici, S. Balle, J. R. Tredicce, M. Brambilla, T. Maggipinto, I. M. Perrini, G. Tissoni, and L. Lugiato, "Cavity solitons in broad-area vertical-cavity surface-emitting lasers below threshold," Phys. Rev. A 69, 043817 (2004).
- Y. Menesguen, S. Barbay, X. Hachair, L. Leroy, I. Sagnes, and R. Kuszelewicz, "Optical self-organization and cavity solitons in optically pumped semiconductor microresonators," Phys. Rev. A 74, 023818 (2006). [CrossRef]
- Y. Tanguy, T. Ackemann, and R. Jager, "Characteristics of bistable localized emission states in broad-area vertical-cavity surface-emitting lasers with frequency-selective feedback," Phys. Rev. A 74, 053824 (2006). [CrossRef]
- I. Ganne and G. Slekys and I. Sagnes and R. Kuszelewicz, "Precursor forms of cavity solitons in nonlinear semiconductor microresonators," Phys. Rev. E 66, 066613 (2002). [CrossRef]
- F. Pedaci, P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, "Positioning cavity solitons with a phase mask," Appl. Phys. Lett. 89, 221111 (2006). [CrossRef]
- X. Hachair, L. Furfaro, J. Javaloyes, M. Giudici, S. Balle, and J. Tredicce, "Cavity-solitons switching in semiconductor microcavities," Phys. Rev. A 72, 013815 (2005). [CrossRef]
- M. Brambilla, L.A. Lugiato, F. Prati, L. Spinelli, andW. Firth, "Spatial Soliton Pixels in Semiconductor Devices," Phys. Rev. Lett. 79, 2042-2045 (1997). [CrossRef]
- 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]
- G. Tissoni and L. Spinelli and L. A. Lugiato and M. Brambilla and I. M. Perrini and T. Maggipinto, "Spatiotemporal dynamics in semiconductor microresonators with thermal effects," Opt. Express 10, 1009-1017 (2002). [PubMed]
- 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]
- A. J. Scroggie, J. M. McSloy, and W. J. Firth, "Self-propelled cavity solitons in semiconductor microcavities," Phys. Rev. E 66, 036607 (2002). [CrossRef]
- Y. Ménesguen and R. Kuszelewicz, unpublished.
- R. L. Honeycutt, "Stochastic Runge-Kutta algorithms. I. White noise," Phys. Rev. A 45, 600-603 (1992). [CrossRef] [PubMed]
- B. Segard, J. Zemmouri, and B. Macke, "Noncritical slowing down in optical bistability," Opt. Commun. 63, 339-343 (1987). [CrossRef]
- F. Mitschke, C. Boden, W. Lange, and P. Mandel, "Exploring the dynamics of the unstable branch of bistable systems," Opt. Commun. 71, 385-392 (1989). [CrossRef]
- D.N. Maywar, G.P. Agrawal, and Y. Nakano, "All-optical hysteresis control by means of cross-phase modulation in semiconductor optical amplifiers," J. Opt. Soc. Am. B 18, 1003-1013 (2001). [CrossRef]

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