## Rotating and Fugitive Cavity Solitons in semiconductor microresonators

Optics Express, Vol. 11, Issue 26, pp. 3612-3621 (2003)

http://dx.doi.org/10.1364/OE.11.003612

Acrobat PDF (190 KB)

### Abstract

We describe two different methods that exploit the intrinsic mobility properties of cavity solitons to realize periodic motion, suitable in principle to provide soliton-based, all-optical clocking or synchronization. The first method relies on the drift of solitons in phase gradients: when the holding beam corresponds to a doughnut mode (instead of a Gaussian as usually) cavity solitons undergo a rotational motion along the annulus of the doughnut. The second makes additional use of the recently discovered spontaneous motion of cavity solitons induced by the thermal dynamics, it demonstrates that it can be controlled by introducing phase or amplitude modulations in the holding beam. Finally, we show that in presence of a weak 2D phase modulation, the cavity soliton, under the thermally induced motion, performs a random walk from one maximum of the phase profile to another, always escaping from the temperature minimum generated by the soliton itself (Fugitive Soliton).

© 2003 Optical Society of America

## 1. Introduction

1. D. W. McLaughlin, J. V. Moloney, and A. C. Newell, “Solitary Waves as Fixed Points of Infinite-Dimensional Maps in an Optical Bistable Ring Cavity,” Phys. Rev. Lett. **51**, 75–78 (1983). [CrossRef]

3. G. S. McDonald and W. J. Firth “Spatial solitary wave optical memory,” J. Opt. Soc. Am. B **7**, 1328–1335 (1990). [CrossRef]

4. M. Tlidi, P. Mandel, and R. Lefever, “Localized structures and localized patterns in optical bistability,” Phys. Rev. Lett. **73**, 640–643 (1994). [CrossRef] [PubMed]

5. For review, see L. A. Lugiato, “Introduction to the Special Issue on Cavity Solitons,” IEEE J. Quant. Electron. **39**, 193 (2003) [CrossRef]

7. 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]

8. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth,“Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. **79**, 2042 (1997). [CrossRef]

7. 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]

8. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth,“Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. **79**, 2042 (1997). [CrossRef]

8. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth,“Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. **79**, 2042 (1997). [CrossRef]

9. D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A **56**, R3366–R3369 (1997). [CrossRef]

10. 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]

11. G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitons in passive bulk semiconductor microcavities. I. Microscopic model and modulational instabilities,” J. Opt. Soc. Am. B **16**, 2083 (1999). [CrossRef]

12. 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). [CrossRef]

13. S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knoedl, M. Miller, and R. Jaeger,“Cavity Solitons as pixels in semiconductor microcavities,” Nature **419**, 699–702 (2002). [CrossRef] [PubMed]

- Motion of CSs induced by the presence of phase/amplitude gradients in the holding beam.
- Spontaneous motion of CSs induced by the slow thermal dynamics. This phenomenon was predicted in [14] for the case of a driven VCSEL with population inversion, extended to the 2D case in [15
14. 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]] and [1615. A. J. Scroggie, J. M. McSloy, and W. J. Firth, “Self-propelled cavity solitons in semiconductor microcavities,” Phys. Rev. E

**66**, 036607 (2002). [CrossRef]], and extended to a configuration without population inversion in [1516. G. Tissoni, L. Spinelli, and L. A. Lugiato,“Spatio-temporal dynamics in semiconductor microresonators with thermal effects,” Opt. Ex.

**10**, 1009 (2002). [CrossRef], 17]. It is caused by the circumstance that the temperature field evolves with a time scale much larger than that of the carrier field and of the electric field. In an appropriate parameter range, this gives rise to a pitchfork bifurcation [15**66**, 036607 (2002). [CrossRef]**66**, 036607 (2002). [CrossRef]] which induces a spontaneous motion of patterns and of CSs. When the CS is switched on, the optical spot remains stationarily in the location where it was created for times shorter than the time scale (microsecond) which characterizes the thermal dynamics, then it starts moving in a random direction; after an initial transient, the velocity becomes constant when the holding beam (HB) is flat. When several CSs are present, they move, in general, in different directions. The destabilization process arises in the following way: after excitation of the CS, a minimum appears in the temperature profile, at the spatial location of the optical spot. This is a consequence of the fact that a maximum of intensity corresponds to less carriers and less heating (in active systems). When the temperature in the minimum reaches a certain critical value, the optical spot starts moving towards larger values of temperature [1618. D. V. Skryabin, A. Yulin, D. Michaelis, W. J. Firth, G-L. Oppo, U. Peschel, and F. Lederer, “Perturbation Theory for Domain Walls in the Parametric Ginzburg-Landau Equation,” Phys. Rev. E

**64**, 56618-1-9 (2001). [CrossRef]]. The minimum of the temperature profile gradually disappears and a dynamical equilibrium is reached, in which the optical spot and the temperature front move together (see Fig. 4 of [16**10**, 1009 (2002). [CrossRef]**10**, 1009 (2002). [CrossRef]

- The first part concerns only point 1) and is based on a model which does not include the temperature. Up to now, the effects of gradients in the HB have been mainly studied by introducing a phase modulation in a plane-wave HB [7], or an amplitude modulation which converts the plane-wave configuration of the HB into a Gaussian one [10
**76**, 1623–1626 (1996). [CrossRef] [PubMed]]. Here we consider the case of a doughnut-shaped holding beam, which is both amplitude and phase-modulated, and show that - as one can easily understand - one has a uniform rotational motion of CSs along the annulus of the doughnut. This is the paradigmatic configuration of a HB which gives rise to a rotatory motion. In addition, this is a visual demonstration of the orbital angular momentum carried by the doughnut mode [1910. 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], 20].19. L. Allen, M.W. Beijersbergen, R. J. C. Spreuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A

**45**, 8185–8189 (1992). [CrossRef] [PubMed] - The second part is based on a model which includes the temperature dynamics and we analyze how the presence of phase/amplitude gradients affects the spontaneous motion described above in point 2). Such a motion represents in principle a problem but, as we show, can be controlled by gradients and this feature may open new opportunities for application. In [14] our analysis was limited to the case of one transverse dimension and we showed that in 1D the spontaneous motion can be confined or even suppressed by introducing a phase modulation. In 2D there is a far richer scenario of possibilities for example, Ref. [15
**66**, 023817 (2002). [CrossRef]**66**, 036607 (2002). [CrossRef]

## 2. The model

### 2.1. a) Without thermal effects

**58**, 2542–2559 (1998) and references quoted therein. [CrossRef]

*E, N*are the normalized electric field and the carrier density normalized to the transparency value, respectively,

*κ*is the cavity damping constant,

*γ*

_{‖}is the carrier nonradiative recombination rate,

*θ*=(

*ω*

_{c}-

*ω*

_{0})/

*κ*is the cavity detuning parameter, with

*ω*

_{0}being the frequency of the holding field and

*ω*

_{c}the longitudinal cavity frequency closest to

*ω*

_{0}. The transverse Laplacian, defined as usual as

*∂*

^{2}/

*∂x*

^{2}+

*∂*

^{2}/

*∂y*

^{2}, represents diffraction (in Eq. 1), and carrier diffusion (in Eq. 2 through the diffusion parameter

*d*),

*η*is proportional to the linear absorption coefficient per unit length due to the material in the region between the QWs and the reflectors,

*β*=

*BN*

_{0}/

*γ*

_{‖}where

*B*is the coefficient of radiative recombination involving two carriers,

*N*

_{0}is the carrier density at transparency. The transverse coordinates

*x*and

*y*are scaled to the diffraction length. The parameter

*E*

_{I}is the normalized injected field (taken real and positive for definiteness),

*I*is the normalized injected current,

*C*is the bistability parameter.

*i*)/(1+Δ

^{2}) and Δ=(

*ω*

_{e}-

*ω*

_{0})/

*γ*

_{e}, where

*ω*

_{e}is the central frequency of the excitonic absorption line, approximated by a Lorentzian curve, and

*γ*

_{e}is the half-width of the excitonic line. In the active configuration Θ=

*α*+

*i*, where

*α*is the line-width enhancement factor typical of semiconductor lasers.

21. B. Sfez, J. L. Oudar, J. C. Michel, R. Kuszelewicz, and R. Azoulay, “High contrast multiple quantum well optical bistable device with integrated Bragg reflectors,” Appl. Phys. Lett. **57**, 324 (1990). [CrossRef]

22. B. Sfez, J. L. Oudar, J. C. Michel, R. Kuszelewicz, and R. Azoulay, “External-beam switching in monolithic bistable GaAs quantum well etalons,” Appl. Phys. Lett. **57**, 1849 (1990). [CrossRef]

**58**, 2542–2559 (1998) and references quoted therein. [CrossRef]

### 2.2. b) Including thermal effects

**a**), but here we also take into account the thermal dynamics.

*T*, that is coupled to the field and carrier equations through the temperature dependence of the nonlinear susceptibility and of the cavity detuning parameter (see [14

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

**66**, 036607 (2002). [CrossRef]

*T*is the lattice temperature normalized to the room temperature

*T*

_{0},

*γ*

_{th}is its decay rate towards the environmental temperature.

*Z*and

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

_{0}is the cavity detuning at room temperature,

*n*is the background refractive index,

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

*χ*

_{nl}, as in [14

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

12. 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). [CrossRef]

## 3. Numerical Analysis

*a*) and

*b*), when the input filed

*E*

_{I}is a plane-wave (i.e. it does not depend on the transverse variables

*x*and

*y*) the dynamical equations admit homogeneous (i.e.

*x*-and

*y*-independent) stationary solutions. In all cases considered in this paper, the steady-state curve of |

*E*

_{S}|, where

*E*

_{S}is the stationary value of the field

*E*, as a function of

*E*

_{I}is

*S*-shaped (see e.g. Fig. 2 and 4 in the following) and its lower branch is stable. On the contrary, the negative-slope branch and part of the upper branch are unstable against the growth of spatially modulated perturbations.

### 3.1. a) Without thermal effects

*I*=0,

*η*=0.25,

*β*=1.6,

*d*=0.2 and Δ=-1. The values are derived from Ref. [10

**58**, 2542–2559 (1998) and references quoted therein. [CrossRef]

*θ*=-3 and

*C*=40.

*TEM**

_{10}or

*TEM**

_{01}), as shown in Fig. 1(a).

*I*larger than the transparency value

*I*

_{0}, in such a way that it becomes an amplifier, slightly below the threshold for laser emission, with an injected field

*E*

_{I}.

**58**, 2542–2559 (1998) and references quoted therein. [CrossRef]

*I*

_{th}=2.11, and we set

*I*=2. We considered the parameter

*α*=5,

*θ*=-2,

*C*=0.45,

*η*=

*β*=0 and

*d*=0.052. In Fig. 2 we show the steady-state curve, the dotted segment is unstable because of a modulational instability. We consider values of

*E*

_{I}just below the righthand turning point of the

**S**-shaped curve, so that the presence of CSs is observed. We create a pair of cavity solitons, then we change the plane-wave HB into a doughnut-mode as before. Fig. 3 shows a movie in which the two CSs rotate in the same direction.

*iφ*) and we got

*ν*=2.02

*µm/nsec*, while by changing exp(±

*iφ*) into exp(±2

*iφ*) we got ν=2.2

*µm/nsec*. Hence, the cavity soliton velocity does not seem to be significantly affected. This is only apparently in contrast with the predicted linear dependence of the drift speed on the field gradient [23, 24

24. G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitons in passive bulk semiconductor microcavities. II. Dynamical proprties and control,” J. Opt. Soc. Am. B **16**, 2095 (1999). [CrossRef]

25. T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity Solitons in Semiconductor Microresonators: Existence, Stability and Dynamical Properties,” Phys Rev E **62**, 8726–8739 (2000). [CrossRef]

26. G-L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Optical Parametric Oscillators,” Phys Rev E **63** 066209-1/15 (2001). [CrossRef]

### 3.2. b) Including thermal effects

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

*Z*≃1.2×10

^{-4},

*P*≃8.1×10

^{-8}, Σ=80, Δ=3,

*θ*

_{0}=-18.5,

*I*=1.43,

*d*=0.1 and

*D*

_{T}=1. The homogeneous stationary solution is shown in Fig. 4, where the dotted segment is unstable because of a dynamical modulational instability, which indicates the drift of CSs.

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

*E*

_{I}=2.55, where the lower branch of the steady-state curve is stable. We remove the writing pulse and the optical spot persists in the position where it has been excited for an initial interval time on the order of

*ρ*

_{1}and

*ρ*

_{2}, respectively, and out of phase by

**58**, 2542–2559 (1998) and references quoted therein. [CrossRef]

*ε*

_{i}=2

*ρ*

_{i}/

*ε*

_{i}is sufficiently small,

*E*

_{I}acquires essentially a pure phase modulation, becoming

*φ*(

*x,y*)=

*ε*cos(

*Kx*) by setting

*ε*

_{2}=0,

*ε*

_{1}=

*ε*=0.2, as it is displayed in Fig. 5(a). In Fig. 5(b) we show the motion of two cavity solitons in presence of this holding beam. It is well known [7

**76**, 1623–1626 (1996). [CrossRef] [PubMed]

*ν*≅47

*µm/µs*, while in presence the phase modulation we get

*ν*≅48.4

*µm/µs*. Hence the drift velocity does not seem to be significantly affected by the presence of the modulation, whose effect is thus mainly of steering the CS in the initial phase of the drift.

*σ*

_{1}, the other with phase equal to

*π*and width equal to

*σ*

_{2}, with

*σ*

_{1}>

*σ*

_{2}. The amplitudes of the two Gaussian beams have been taken equal.

*ε*

_{1}=

*ε*

_{2}=

*ε*in Eq. 9 (see Fig. 7(a)). The cavity soliton moves towards the nearest maximum of the phase landscape and remains trapped there for a while. When the CS is trapped, a minimum of temperature develops in the location of the optical spot and tends to destabilize it. In Ref. [14

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

*ε*=0.05. The optical spot starts moving and is momentarily captured by one of the nearest phase maxima where, however, the temperature field starts digging a dip which, in this case, is capable of expelling the CS. As a consequence, the CS is captured by another phase maximum for a while, then it is again expelled, and the process repeats again and again, generating random walk in the phase landscape, in a sense similar to a slow pinball game.

## 4. Conclusions and discussion

*TEM**

_{10}or a

*TEM**

_{01}mode as holding beam, both in the case of passive and active configurations, in absence of thermal effects.

## Acknowledgments

*Formazione e controllo di solitoni di cavità in microrisonatori a semiconduttore*of the Italian MIUR, and the European Network VISTA (

*VCSELs for Information Society Technology Applications*).

## References and links

1. | D. W. McLaughlin, J. V. Moloney, and A. C. Newell, “Solitary Waves as Fixed Points of Infinite-Dimensional Maps in an Optical Bistable Ring Cavity,” Phys. Rev. Lett. |

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

3. | G. S. McDonald and W. J. Firth “Spatial solitary wave optical memory,” J. Opt. Soc. Am. B |

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

5. | For review, see L. A. Lugiato, “Introduction to the Special Issue on Cavity Solitons,” IEEE J. Quant. Electron. |

6. | For review, see W. J. Firth and G. K. Harkness, “Existence, Stability and Properties of Cavity Solitons,” in “Spatial Solitons,” Springer Series in Optical Sciences Vol. 82, eds. S. TrilloW. Torruellas, pp. 343–358 (Springer Velag, 2002). |

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

8. | M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth,“Spatial soliton pixels in semiconductor devices,” Phys. Rev. Lett. |

9. | D. Michaelis, U. Peschel, and F. Lederer, “Multistable localized structures and superlattices in semiconductor optical resonators,” Phys. Rev. A |

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

11. | G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitons in passive bulk semiconductor microcavities. I. Microscopic model and modulational instabilities,” J. Opt. Soc. Am. B |

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

13. | S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knoedl, M. Miller, and R. Jaeger,“Cavity Solitons as pixels in semiconductor microcavities,” Nature |

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

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

16. | G. Tissoni, L. Spinelli, and L. A. Lugiato,“Spatio-temporal dynamics in semiconductor microresonators with thermal effects,” Opt. Ex. |

17. | I. M. Perrini, G. Tissoni, T. Maggipinto, and M. Brambilla, “Thermal effects and cavity solitons in passive semiconductor microresonators,” submitted to J. Opt. B (2003). |

18. | D. V. Skryabin, A. Yulin, D. Michaelis, W. J. Firth, G-L. Oppo, U. Peschel, and F. Lederer, “Perturbation Theory for Domain Walls in the Parametric Ginzburg-Landau Equation,” Phys. Rev. E |

19. | L. Allen, M.W. Beijersbergen, R. J. C. Spreuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

20. | L. Allen, S. M. Barnett, and M. J. Padgett, “Optical angular Momentum,” Institute of Physics Publishing, Bristol, (2003). |

21. | B. Sfez, J. L. Oudar, J. C. Michel, R. Kuszelewicz, and R. Azoulay, “High contrast multiple quantum well optical bistable device with integrated Bragg reflectors,” Appl. Phys. Lett. |

22. | B. Sfez, J. L. Oudar, J. C. Michel, R. Kuszelewicz, and R. Azoulay, “External-beam switching in monolithic bistable GaAs quantum well etalons,” Appl. Phys. Lett. |

23. | W. J. Firth and G. Harkness, “Cavity Solitons,” Asian J. Phys. |

24. | G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini, and L. A. Lugiato, “Cavity solitons in passive bulk semiconductor microcavities. II. Dynamical proprties and control,” J. Opt. Soc. Am. B |

25. | T. Maggipinto, M. Brambilla, G. K. Harkness, and W. J. Firth, “Cavity Solitons in Semiconductor Microresonators: Existence, Stability and Dynamical Properties,” Phys Rev E |

26. | G-L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Optical Parametric Oscillators,” Phys Rev E |

**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: October 31, 2003

Revised Manuscript: December 18, 2003

Published: December 29, 2003

**Citation**

R. Kheradmand, L. Lugiato, G. Tissoni, M. Brambilla, and H. Tajalli, "Rotating and Fugitive Cavity Solitons in semiconductor microresonators," Opt. Express **11**, 3612-3621 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-26-3612

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### References

- D. W. McLaughlin, J. V. Moloney and A. C. Newell, �??Solitary Waves as Fixed Points of Infinite-Dimensional Maps in an Optical Bistable Ring Cavity,�?? Phys. Rev. Lett. 51, 75-78 (1983). [CrossRef]
- N. N. Rosanov and G. V. Khodova, �??Autosolitons in bistable interferometers,�?? Opt. Spectrosc. 65, 449-450 (1988).
- G. S. McDonald, andW. J. Firth �??Spatial solitary wave optical memory,�?? J. Opt. Soc. Am. B 7, 1328-1335 (1990). [CrossRef]
- For review, see L. A. Lugiato, �??Introduction to the Special Issue on Cavity Solitons,�?? IEEE J. Quant. Electron.39, 193 (2003) [CrossRef]
- For review, see W. J. Firth and G. K. Harkness, �??Existence, Stability and Properties of Cavity Solitons,�?? in �??Spatial Solitons,�?? Springer Series in Optical Sciences Vol. 82, eds. S. Trillo and W. Torruellas, pp. 343-358 (Springer Velag, 2002).
- 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, F. Prati, L. Spinelli andW. J. Firth,�??Spatial soliton pixels in semiconductor devices,�?? Phys. Rev. Lett. 79, 2042 (1997). [CrossRef]
- D. Michaelis, U. Peschel and F. Lederer, �??Multistable localized structures and superlattices in semiconductor optical resonators,�?? Phys. Rev. A 56, R3366-R3369 (1997). [CrossRef]
- 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]
- G. Tissoni, L. Spinelli, M. Brambilla, T. Maggipinto, I. Perrini and L. A. Lugiato, �??Cavity solitons in passive bulk semiconductor microcavities. I. Microscopic model and modulational instabilities,�?? J. Opt. Soc. Am. B 16, 2083 (1999). [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]
- S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knoedl, M. Miller and R. Jaeger,�??Cavity Solitons as pixels in semiconductor microcavities,�?? Nature 419, 699-702 (2002). [CrossRef] [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]
- G. Tissoni, L. Spinelli and L. A. Lugiato,�??Spatio-temporal dynamics in semiconductor microresonators with thermal effects,�?? Opt. Ex. 10, 1009 (2002). [CrossRef]
- I. M. Perrini, G. Tissoni, T. Maggipinto, and M. Brambilla, �??Thermal effects and cavity solitons in passive semiconductor microresonators,�?? submitted to J. Opt. B (2003).
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