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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 15, Iss. 19 — Sep. 17, 2007
  • pp: 12457–12463
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Physical model for the incoherent writing/erasure of cavity solitons in semiconductor optical amplifiers

S. Barbay and R. Kuszelewicz  »View Author Affiliations


Optics Express, Vol. 15, Issue 19, pp. 12457-12463 (2007)
http://dx.doi.org/10.1364/OE.15.012457


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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

Cavity solitons (CS) are localized structures whose existence is now experimentally well established in semiconductor systems [2

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]

, 3

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. A 69, 043817 (2004).

, 1

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

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

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]

]. They appear in spatially extended systems in the presence of bistability and of a modulational instability. These conditions can be obtained in semiconductor microcavities with an injected signal (holding beam, HB). Although CS can appear in passive or active systems, active systems have proved more favorable from an experimental point of view, especially to get rid of a global, thermally driven instability [6

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]

]. CS currently attract much interest because of the possible applications to all-optical processing of information. In particular, CS can be written or erased at any transverse location of the microresonator, and can be manipulated by phase gradients [7

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]

].

2. Model

Fig. 1. Plane-wave, bistability curves (Eqs.1 without spatial terms and without the equation for θ) with the parameters α=5,C=0.2, θ=-2 and Λ=3.2 except blue curve (Λ=3.3) and red curve (θ=-2.2). The portions of the curves with negative slopes are unstable.
Et=(1+iθ)E+EI2iC(α+i)(N1)E+2Ex2+ξ
Nt=γ[NΛ+(N1)E2D2Nx2]
θt=γT[(θθ0)+f(Λ)DT2θx2]
(1)

The equations are in rescaled form. Time is expressed in cavity lifetime units, and the spatial dimension is in units of the diffraction length. EI is the holding beam field amplitude, 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 γT. The pumping term for the carriers Λ also provides a source term in the detuning equation through a linear function 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)>=2DEδ (t-t) where DE=10-6. 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 xi of duration τ on the pump term Λ=Λ0[1+δΛΠ(t-t0)Π(x-xi,δ x)], where Π(x,δ x) is a boxcar function centered on x of width δ x. The corresponding source term f is such that f0)=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, Λ0=3.2, EI=0.75, γ=0.01, γT=10-4. The laser threshold is given by Λth=1+1/2C while transparency is at Λ0=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

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

]. The same holds for the relaxation time of the detuning γT, and the ratio γ/γT=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.

3. Results

Results of the simulation are shown in Fig. 2. We use a numerical split-step method with a stochastic, second order Runge-Kutta scheme [15

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

] for the temporal part and a Fast Fourier Transform scheme for the Laplacian operators part. The boundary conditions are periodic. The carrier injection time is taken to be τ=20.

As can be seen on Fig. 2, the qualitative behavior observed in experiments is well reproduced here featuring a delay in the switch-on process and a fast switch-off. Moreover, the transition itself is fast in both cases. Inspection of the variable 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.

The steady-state response of the system when ramping the pump intensity is shown on Fig. 3. It is obtained by monitoring the steady-state response of the system at a CS location while ramping adiabatically the pump. The maximum of the field is plotted in each branch. The lower branch corresponds to the homogeneous steady state (the dashed line and full line curves are superimposed) and is obtained by starting the system with a uniform state. When the system jumps to the higher branch, a pattern solution is selected. The noise visible on the curve when further increasing the pump is due to the finite size of the simulation box that constrains the available transverse wavevectors. When ramping down the pump, the solution evolves from the patterned solution to a solution when several CS can coexist. In the latter region, we can check that the single CS solution is stable by starting the simulation on the upper branch near the upper turning point with a single CS as initial condition. This single CS solution becomes unstable when leaving the bistable region and transforms into an extended pattern. The sense of the hysteresis is now in accordance with the one observed in experiments.

Fig. 2. Time traces of the intensity at a CS peak |E|2 vs time (blue line) and local injection of carrier (Λ, red line). Upper figure : experimental results obtained in [1]. Lower figure : incoherent switch-on and off of a CS, with respectively EI=0.76, δΛ=3.6, δx=20 and EI=0.75, δΛ=1.37 and δx=20 (in inset is a zoom of the initial part of the switch-off process).
Fig. 3. Adiabatic response |E|2when ramping the pump Λ: 1D numerical simulation (full line, the maximum of the field is plotted) and plane wave hysteresis (dashed line). The injected field amplitude is EI=0.76. The arrows indicate the sense followed by the system on the hysteresis cycle.
Fig. 4. Scaling of the switch-on delay versus writing power for a fixed duration of the writing pulse τ=20. The critical writing power is δΛc≃3.50.

Let us now focus on the switching process itself. Switching in bistable systems has already been the subject of many theoretical and experimental studies, especially in zero-dimensional systems [16

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

, 17

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]

]. When a bistable system is perturbed by a pulsed excitation and brought into an unstable state and if the perturbation area A(product of the perturbation time and perturbation amplitude) is larger than a critical value Ac, the system switches to the other state with a delay (non-critical slowing down) which has a logarithmic scaling law δτ~-ln(A-Ac). 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.

The critical writing intensity perturbation has been obtained by a fitting procedure. The time-jitter on each delay point was negligible at the noise level used in the simulation, hence only one realization of the switch-on has been done for each writing power. The logarithmic scaling seems to hold very well, especially at writing powers close to the critical value. At much higher writing levels, a small deviation is observed. This may be due to the fact that at high perturbation levels, the lasing threshold is transitorily crossed thus changing the subsequent scaling law.

The switch-off process is also characterized by a critical erasing power δΛc≃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]

] to find the bistability curves for the CS, but are out of the scope of this paper. With the particular parameter values used in the simulations, switch-on and off are possible for the same parameters (in particular same HB amplitude EI) over a range of input fields EI and writing/erasing powers.

Fig. 5. Switch-off dynamics with succeeded (blue) and failed (red) switch-offs. From bottom to top, the erasing powers are δΛc=1.10,1.03,1.0225,1.022,1.02. The erasing pulse in launched at t=100 and has a duration δτ=20.

4. Conclusion

In conclusion we have proposed a model to describe the incoherent writing and erasure process of CS in a semiconductor optical amplifier. This model includes a thermally-induced detuning making it possible to write and erase CS incoherently. It is worth pointing out that this detuning is always present when injecting carriers, because it is impossible not to produce some heat in the system when doing so (be it by releasing energy through the quantum defect or by the non-radiative recombination of carriers). However, dissipation of this energy can be somehow controlled by a proper thermal management of the device. Indeed in [1

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]

], heat dissipation has been carefully taken care of in order to obtain a sample with a small thermal resistance and a fast relaxation time, which is all the more critical when dealing with a broad area system. Despite this, and because the system is very sensitive to small detunings, there remains a small effect which is responsible for the incoherent processes presented here. In addition, the thermal timescale involved is rather fast (some hundreds of nanoseconds) since carrier injection for writing a CS is made over a rather small area. In the case where the thermally driven detuning would become too small to be able to induce the CS switch-on, then other methods could be employed like the one presented in [18

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]

] in a zero-dimensional system, using two control beams with different wavelengths. Our model is in qualitative agreement with what has been observed in experiments [1

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]

] : it explains why it is possible to write and erase CS with a local addition of carriers, reconciles the observed sense of the hysteresis cycle with the predicted one and explains the observed delay in the CS switch-on. Although this delay represents a disadvantage for high bit-rate applications, it could prove interesting in applications requiring a tunable delay if the noise in the system is not too high.

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

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]

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. A 69, 043817 (2004).

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]

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]

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]

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]

14.

Y. Ménesguen and R. Kuszelewicz, unpublished.

15.

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

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]

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]

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

  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]
  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]
  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. A 69, 043817 (2004).
  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 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]
  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]
  9. 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]
  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]
  11. 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]
  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]
  14. Y. Ménesguen and R. Kuszelewicz, unpublished.
  15. R. L. Honeycutt, "Stochastic Runge-Kutta algorithms. I. White noise," Phys. Rev. A 45, 600-603 (1992). [CrossRef] [PubMed]
  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]
  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]

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