## Bistability and all-optical switching in semiconductor ring lasers

Optics Express, Vol. 15, Issue 20, pp. 12941-12948 (2007)

http://dx.doi.org/10.1364/OE.15.012941

Acrobat PDF (223 KB)

### Abstract

Semiconductor ring lasers display a variety of dynamical regimes originating from the nonlinear competition between the clockwise and counter-clockwise propagating modes. In particular, for large pumping the system has a bistable regime in which two stationary quasi-unidirectional counter-propagating modes coexist. Bistability is induced by cross-gain saturation of the two counter-propagating modes being stronger than the self-saturation and can be used for data storage when the semiconductor ring laser is addressed with an optical pulse. In this work we study the response time when an optical pulse is injected in order to make the system switch from one mode to the counter-propagating one. We also determine the optimal pulse energy to induce switching.

© 2007 Optical Society of America

## 1. Introduction

2. Q. L. Williams and R. Roy, “Fast polarization dynamics of an erbium-doped fiber ring laser,” Opt. Lett , **21**, 1478 (1996). [CrossRef] [PubMed]

3. H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett , **50**, 109 (1983) [CrossRef]

4. E. J. D’Angelo, E. Izaguirre, G. B. Mindlin, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett , **68**, 3702 (1992). [CrossRef] [PubMed]

5. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. **57**61 (1985). [CrossRef]

6. T. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation of semiconductor ring lasers,” Electron. Lett. **26**, 2095 (1990). [CrossRef]

7. M. Sorel, J. P. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. **27**, 1992 (2002). [CrossRef]

8. M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. **39**, 1187 (2003). [CrossRef]

8. M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. **39**, 1187 (2003). [CrossRef]

8. M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. **39**, 1187 (2003). [CrossRef]

9. J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Applied Phys. Lett. **70**, 1192 (1997). [CrossRef]

10. S. Zhang, Y. Liu, D. Lenstra, M.T. Hill, H. Ju, G.D. Khoe, and H.J.S. Dorre, “Ring-laser optical flip-flop memory with single active element,” J. Sel. Top. Q. Electron. **10**, 1093 (2004). [CrossRef]

11. V. R. Almeida, C. A. Barrios, R. P. Panepucci, M. Lipson, M. A. Foster, D. G. Ouzounov, and A. L. Gaeta, “All-optical switching on a silicon chip,” Opt. Lett. **29**, 2867 (2004). [CrossRef]

12. V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. **29**, 2387 (2004). [CrossRef] [PubMed]

13. M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature **432**, 206 (2004). [CrossRef] [PubMed]

## 2. Model

*E*

_{1}Clockwise (CW mode) and

*E*

_{2}Counter Clockwise (CCW mode), which has provided a good quantitative description of the two-mode dynamics in SRLs [7

7. M. Sorel, J. P. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. **27**, 1992 (2002). [CrossRef]

**39**, 1187 (2003). [CrossRef]

*α*accounts for phase-amplitude coupling and the self and cross saturation coefficients are given by

*s*and

*c*, respectively; the parameters

*k*and

_{d}*k*represent the dissipative and conservative components of the backscattered field, respectively. The term

_{c}*F*

_{1,2}(

*t*) represents resonant (zero-detuning) optical injection in the two modes, and it will be used to trigger the switching. Formally, this term is introduced according to the standard theory of injection in semiconductor laser [14

14. G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semi-class. Opt. **7**, 87 (1995). [CrossRef]

15. V. Annovazzi-Lodi, A. Scirè, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. **34**, 2350 (1998). [CrossRef]

*τ*is the photon lifetime and

_{p}*τ*the flight time in the ring cavity. The practical implementation of the optical injection is sketched in Fig. 1. The last term of Eq. (1) represents spontaneous emission noise;

_{in}*β*= 5 ∙ 10

^{-3}ns

^{-1}represents the fraction of the spontaneously emitted photons coupled to mode 1 or 2. ζ

_{1,2}are two independent complex Gaussian random numbers, with zero mean 〈ζ

_{i}(

*t*)〉 = 0 and correlation 〈ζ

_{i}(

*t*)ζ

_{j}

^{*}(

*t*′)〉 =

*2δ*(

_{ij}δ*t*-

*t*′). The carrier density

*N*obeys the rate equation for semiconductor lasers,

*μ*is the dimensionless pump (

*μ*~ 1 at laser threshold). In the set (1)-(2) the dimension-less time is rescaled by the photon lifetime

*τ*. The parameter

_{p}*γ*is the ratio of

*τ*over the carrier lifetime

_{p}*τ*.

_{s}## 3. Steady state solution and bistability

*μ*m-wide single-transverse-mode ridge waveguides in a double-quantum-well GaAs/AlGaAs structure with 1-mm ring radius. The estimated parameter values for this device are [7

7. M. Sorel, J. P. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. **27**, 1992 (2002). [CrossRef]

*α*= 3.5,

*s*= 0.005,

*c*= 0.01,

*k*= 0.0044,

_{c}*k*= 0.000327,

_{d}*γ*= 0.002,

*τ*= 10 ps and

_{p}*τ*= 0.6 ps.

_{in}*E*

_{1,2}=

*Q*

_{1,2}exp(

*iωt*+

*iϕ*

_{1,2}), and the corresponding value for

*N*. Fig. 2 shows the bifurcation diagram for the parameters considered here. At

*μ*~ 1 laser oscillation takes place. The presence of dissipative backscattering (

*k*) favors the presence of two steady state symmetric solutions (

_{d}*Q*

_{1}=

*Q*

_{2}=

*Q*) just above threshold, despite the presence of strong cross-saturation between the two modes [7

**27**, 1992 (2002). [CrossRef]

*ϕ*

_{2}-

*ϕ*

_{1}). The In-Phase Symmetric Solution (IPSS) corresponds to Φ = 0 while the Out-of-Phase Symmetric Solution (OPSS) corresponds to Φ =

*π*. Since we are considering a positive value of

*k*, the OPSS is the stable symmetric solution. The OPSS solution exists for any value of the pump above the threshold. However, it is not always stable. In Fig. 2 we have plotted the OPSS with a solid black line from the laser threshold to

_{d}*μ*~ 1.5. At this current the OPSS is destabilized to a Hopf bifurcation. We have indicated the unstable OPSS with a straight grey dashed line. The Hopf bifurcation leads to the emergence of a oscillatory behavior in which the CW and the CCW modes coexist. The intensity of both modes oscillate with the same amplitude but are in antiphase. The oscillations are driven by the conservative part of the back-scattering coefficient

*k*, and represent a dynamic competition between the two counterpropagating modes. We have determined the limit cycle from numerical integration of Eq. (2)-(3). We represent the maxima (minima) of the oscillating modal intensities by open circles (squares). This oscillatory behavior is stable up to

_{c}*μ*~ 2.6.

*μ*~ 2.0, two unstable asymmetric solutions emerge at a pitchfork bifurcation from the unstable OPSS and they coexist with the stable limit cycle. We have obtained the asymmetric steady states by solving the right hand sides of Eq. (2)-(3) equated to zero with a Newton-Raphson method. These two stationary states consist in laser emission mainly concentrated in one propagation direction, i.e. quasi Clockwise (qCW) or quasi Counterclockwise (qCCW). In the qCW steady state the CW intensity takes a value on the upper branch while the CCW intensity takes a value in the lower branch. The qCCW steady state corresponds to the opposite situation. The contrast factor between the two modes

*μ*from the pitchfork bifurcation. Because the asymmetric stationary states are initially unstable, we have plotted them with grey dashed lines from

*μ*~ 2.0 up to

*μ*~ 2.6. At this current value the stable limit cycle loses its stability and the asymmetric stationary solution becomes stable (denoted as solid lines). For high pump values, the strong cross-saturation between the two counterpropagating waves tends to favor the quasi-unidirectional behavior. In this regime the device shows bistability between the two asymmetric solutions, and we refer to it as the

*bistable regime*. This bistability between counterpropagating modes can be used for data storage when the SRL is addressed with an optical pulse. In the next section, we quantify the speed of and the necessary pulse energy to achieve successful data storage. We should note that as the cross-saturation parameter

*c*tends to

*s*the pitchfork bifurcation moves towards infinite pumping values. Therefore, a large cross-saturation value is required to have bistability.

## 4. Switching

*F*

_{1,2}=

*A*exp(-

*t*/

*τ*), characterized by the pulse amplitude

*A*, and the pulse decay time

*τ*. The trigger amplitude

*A*is in general complex, due to the (constant) dephasing accumulated by the external field in the optical wave guide outside the laser cavity. However, it is known that such associated phase does not affect the injection properties [15

15. V. Annovazzi-Lodi, A. Scirè, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. **34**, 2350 (1998). [CrossRef]

*A*to be real.

*τ*/

*T*< 0.01) the oscillations may persist after the pulse ends (~ 1.5 ns). When a pulse is applied the response of the system is very fast. This allows for high speed optical data storage. However, please note that the time the system takes to reach the steady state after the switching event is longer than the response or switching time. This will limit the time between consecutive successful switching events. In this work, we concentrate only on characterizing the switching event.

*P*

_{1}= |

*E*

_{1}|

^{2}and

*P*

_{2}= |

*E*

_{2}|

^{2}, evaluated at a time delay equal to the trigger period T, that is

*T*= 1000

_{max}*T*is the total integration time,

*P*̄

_{1}~

*μ*- 1 is the average values of the intensity |

*E*

_{1}|

^{2}when the (solitary) qCW mode is active (the same holds for the CCW mode intensity

*P*̄

_{2}). Note that the normalization procedure that we use allows

*X*(

*T*) to be slightly larger than one, due to the power injected by the trigger. If all the trigger pulses induce a switching

*X*(

*T*) ~ 1, whereas if some switching events fail

*X*(

*T*) decreases and approaches to zero if no switchings occur. We have computed the value of

*X*(

*T*) for different values of the trigger amplitude and decay time, and the result is shown in Fig. 4. The numerical simulations show that the shape of the pulse is not critical for the switching to occur, being the relevant magnitude its energy, given by

*ε*= ∫

^{∞}

_{0}|

*F*

_{1,2}|

^{2}

*dt*=

*A*

^{2}

*τ*/2. Figure 4 shows that a transition to successful switching events occurs in correspondence of an iso-energy curve for the trigger pulse, corresponding to a minimum (critical) switching energy

*ε*~ 10

_{c}*fJ*, which has been calculated assuming 100 mW of optical power inside the cavity, in agreement with the output optical power and coupler efficiency measured in real devices. With the same data, and after each pulse is applied, we calculate the time it takes the system to reach one half of the value of the intensity of the final state (

*P*̄

_{1,2}). This time (

*t*from now on) is an estimation of the response time of the system to the trigger pulse, and characterizes the switching speed (see Fig. 5). Due to the presence of noise,

_{R}*t*undergoes a statistic distribution, that we characterize through its mean value <

_{R}*t*> and variance

_{R}_{R}, are comprised inside the symbol sizes which indicates that, for the parameter we have considered, the fluctuations in the switching time are very small. The obtained response time is throughout much shorter than the inverse of the relaxation oscillation frequency (

*τ*=

_{rel.ox}*f*

^{-1}

_{rel.ox}, where

## 5. Conclusion

*ps*or sub-

*ps*) time scales would require more sophisticated (e.g. traveling-wave) modeling approach, and will be the subject of future investigations.

## Acknowledgments

## References and links

1. | C. O. Weiss and R. Vilaseca, “Dynamics of lasers,” Weinheim, New York (1991), and refs therein. |

2. | Q. L. Williams and R. Roy, “Fast polarization dynamics of an erbium-doped fiber ring laser,” Opt. Lett , |

3. | H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of bifurcation to chaos in an all-optical bistable system,” Phys. Rev. Lett , |

4. | E. J. D’Angelo, E. Izaguirre, G. B. Mindlin, L. Gil, and J. R. Tredicce, “Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry,” Phys. Rev. Lett , |

5. | W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys. |

6. | T. Krauss, P. J. R. Laybourn, and J. S. Roberts, “CW operation of semiconductor ring lasers,” Electron. Lett. |

7. | M. Sorel, J. P. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina, and S. Donati, “Alternate oscillations in semiconductor ring lasers,” Opt. Lett. |

8. | M. Sorel, G. Giuliani, A. Scirè, R. Miglierina, J. P. R. Laybourn, and S. Donati, “Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model,” IEEE J. Quantum Electron. |

9. | J. J. Liang, S. T. Lau, M. H. Leary, and J. M. Ballantyne, “Unidirectional operation of waveguide diode ring lasers,” Applied Phys. Lett. |

10. | S. Zhang, Y. Liu, D. Lenstra, M.T. Hill, H. Ju, G.D. Khoe, and H.J.S. Dorre, “Ring-laser optical flip-flop memory with single active element,” J. Sel. Top. Q. Electron. |

11. | V. R. Almeida, C. A. Barrios, R. P. Panepucci, M. Lipson, M. A. Foster, D. G. Ouzounov, and A. L. Gaeta, “All-optical switching on a silicon chip,” Opt. Lett. |

12. | V. R. Almeida and M. Lipson, “Optical bistability on a silicon chip,” Opt. Lett. |

13. | M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge, Y. S Oei, H. Binsma, G. D. Khoe, and M. K. Smit, “A fast low-power optical memory based on coupled micro-ring lasers,” Nature |

14. | G. H. M. van Tartwijk and D. Lenstra, “Semiconductor lasers with optical injection and feedback,” Quantum Semi-class. Opt. |

15. | V. Annovazzi-Lodi, A. Scirè, M. Sorel, and S. Donati, “Dynamic behavior and locking of a semiconductor laser subjected to external injection,” IEEE J. Quantum Electron. |

16. | M. San Miguel and R. Toral, “Stochastic Effects in Physical Systems,” Instabilities and Nonequilibrium Structures V, edited by E. Tirapegui, J. Martinez, and R. Tiermann, Netherlands: Kluwer Academic Publishers, (1999). |

**OCIS Codes**

(140.3560) Lasers and laser optics : Lasers, ring

(140.5960) Lasers and laser optics : Semiconductor lasers

(200.4660) Optics in computing : Optical logic

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: May 11, 2007

Revised Manuscript: July 31, 2007

Manuscript Accepted: August 1, 2007

Published: September 24, 2007

**Citation**

Toni Pérez, Alessandro Scirè, Guy Van der Sande, Pere Colet, and Claudio R. Mirasso, "Bistability and all-optical switching in semiconductor ring lasers," Opt. Express **15**, 12941-12948 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-20-12941

Sort: Year | Journal | Reset

### References

- C. O. Weiss and R. Vilaseca, "Dynamics of lasers," Weinheim, New York (1991), and refs therein.
- Q. L. Williams, and R. Roy, "Fast polarization dynamics of an erbium-doped fiber ring laser," Opt. Lett, 21, 1478 (1996). [CrossRef] [PubMed]
- H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, "Observation of bifurcation to chaos in an all-optical bistable system, " Phys. Rev. Lett, 50, 109 (1983) [CrossRef]
- E. J. D’Angelo, E. Izaguirre, G. B. Mindlin, L. Gil, and J. R. Tredicce, "Spatiotemporal dynamics of lasers in the presence of an imperfect O(2) symmetry," Phys. Rev. Lett, 68, 3702 (1992). [CrossRef] [PubMed]
- W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich and M. O. Scully, "The ring laser gyro," Rev. Mod. Phys. 5761 (1985). [CrossRef]
- T. Krauss, P. J. R. Laybourn and J. S. Roberts, "CW operation of semiconductor ring lasers," Electron. Lett. 26, 2095 (1990). [CrossRef]
- M. Sorel, J. P. R. Laybourn, A. Scirè, S. Balle, G. Giuliani, R. Miglierina,S. Donati, "Alternate oscillations in semiconductor ring lasers," Opt. Lett. 27, 1992 (2002). [CrossRef]
- M. Sorel, G. Giuliani,A. Scirè, R. Miglierina, J. P. R. Laybourn, S. Donati, "Operating regimes of GaAs-AlGaAs semiconductor ring lasers: experiment and model," IEEE J. Quantum Electron. 39, 1187 (2003). [CrossRef]
- J. J. Liang, S. T. Lau, M. H. Leary and J. M. Ballantyne, "Unidirectional operation of waveguide diode ring lasers," Applied Phys. Lett. 70, 1192 (1997). [CrossRef]
- S. Zhang, Y. Liu, D. Lenstra, M.T. Hill, H. Ju, G.D. Khoe, H.J.S. Dorre, "Ring-laser optical flip-flop memory with single active element," J. Sel. Top. Q. Electron. 10, 1093 (2004). [CrossRef]
- V. R. Almeida, C. A. Barrios, R. P. Panepucci, M. Lipson, M. A. Foster, D. G. Ouzounov, A. L. Gaeta, "All-optical switching on a silicon chip," Opt. Lett. 29, 2867 (2004). [CrossRef]
- V. R. Almeida and M. Lipson, "Optical bistability on a silicon chip," Opt. Lett. 29, 2387 (2004). [CrossRef] [PubMed]
- M. T. Hill, H. J. S. Dorren, T. de Vries, X. J. M. Leijtens, J. H. den Besten, B. Smalbrugge,Y. S Oei, H. Binsma, G. D. Khoe, M. K. Smit, "A fast low-power optical memory based on coupled micro-ring lasers," Nature 432, 206 (2004). [CrossRef] [PubMed]
- G. H. M. van Tartwijk, D. Lenstra, "Semiconductor lasers with optical injection and feedback," Quantum Semiclass. Opt. 7, 87 (1995). [CrossRef]
- V. Annovazzi-Lodi, A. Scirè, M. Sorel, S. Donati, "Dynamic behavior and locking of a semiconductor laser subjected to external injection," IEEE J. Quantum Electron. 34, 2350 (1998). [CrossRef]
- M. San Miguel, R. Toral, "Stochastic Effects in Physical Systems," Instabilities and Nonequilibrium Structures V, edited by Tirapegui E., Martinez J., and Tiermann R., Netherlands: Kluwer Academic Publishers, (1999).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.