## Longitudinal mode multistability in Ring and Fabry-Pérot lasers: the effect of spatial hole burning |

Optics Express, Vol. 19, Issue 4, pp. 3284-3289 (2011)

http://dx.doi.org/10.1364/OE.19.003284

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### Abstract

We theoretically discuss the impact of the cavity configuration on the possible longitudinal mode multistability in homogeneously broadened lasers. Our analysis is based on the most general form of a Travelling-Wave Model for which we present a method that allows us to evaluate the monochromatic solutions as well as their eigenvalue spectrum. We find, in agreement with recent experimental reports, that multistability is more easily reached in Ring than in Fabry-Pérot cavities which we attribute to the different amount of Spatial-Hole Burning in each configuration.

© 2011 Optical Society of America

5. M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguides lasers,” Appl. Phys. Lett. **80**(17), 3051–3053 (2002). [CrossRef]

7. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A **33**, 1842 (1986). [CrossRef] [PubMed]

8. L. A. Kotomtseva, “Steady states for longitudinal modes and dynamics of a laser with a saturable absorbent,” Quantum Semiclass. Opt. **10**, 331 (1998). [CrossRef]

9. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection-laser properties,” *IEEE J. Quantum Electron.*16, 347 (1980). [CrossRef]

10. A. Loose, B. K. Goswami, H.-J. Wünsche, and F. Henneberger, “Tristability of a semiconductor laser due to time-delayed optical feedback,” Phys. Rev. E **79**, 036211 (2009). [CrossRef]

11. K. P. Komarov, “Multistable single-mode emission from solid-state state lasers,” Quantum Electron. **24**(11), 975–976 (1994). [CrossRef]

_{2}laser [12

12. J. R. Tredicce, L. M. Narducci, N. B. Abraham, D. K. Bandy, and L. A. Lugiato, “Experimental-evidence of mode competition leading to optical bistability in homogenously broadened lasers,” Opt. Commun. **56**, 435 (1986). [CrossRef]

13. M. Yamada, “Theory of mode competition noise in semiconductor injection-lasers,” IEEE J. Quantum Electron. **22**, 1052 (1986). [CrossRef]

14. F. Pedaci, S. Lepri, S. Balle, G. Giacomelli, M. Giudici, and J. R. Tredicce, “Multiplicative noise in the longitudinal mode dynamics of a bulk semiconductor laser,” Phys. Rev. E **73**, 041101 (2006). [CrossRef]

7. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A **33**, 1842 (1986). [CrossRef] [PubMed]

17. H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A **43**, 2446–2454 (1991) [CrossRef] [PubMed]

18. L. A. Lugiato, L. M. Narducci, and M. F. Squicciarini, “Exact linear-stability analysis of the plane-wave Maxwell-Bloch equations for a ring laser,” Phys. Rev. A **34**, 3101 (1986). [CrossRef] [PubMed]

19. G. J. de Valcárcel, E. Roldán, and F. Prati, “Risken-Nummedal-Graham-Haken instability in class B lasers,” Opt. Commun. **163**, 5–8 (1999). [CrossRef]

7. L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A **33**, 1842 (1986). [CrossRef] [PubMed]

18. L. A. Lugiato, L. M. Narducci, and M. F. Squicciarini, “Exact linear-stability analysis of the plane-wave Maxwell-Bloch equations for a ring laser,” Phys. Rev. A **34**, 3101 (1986). [CrossRef] [PubMed]

21. R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. **213**, 420 (1968). [CrossRef]

22. A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional Ring Lasers,” Phys. Rev. A **81**, 043817 (2010). [CrossRef]

*A*

_{±}are the scaled slowly varying amplitudes of the counter-propagating electric fields,

*B*

_{±}are their respective polarizations,

*D*

_{0}is the quasi-homogeneous inversion density and

*D*

_{±2}are the spatially-dependent contributions to the grating in the population inversion density that arise from standing wave effects and lead to saturation of the gain. Space and time (

*s*,

*τ*) are scaled by the length

*L*and the time of flight

_{c}*τ*of the cavity, respectively.

_{c}*α*are the internal losses per unit length,

*γ*determines the spectral width of the gain spectrum,

*δ*is the detuning, and

*ɛ*and

*η*are the decay times for

*D*

_{0}and

*D*

_{±2}respectively, which differ due to the impact of diffusion on the decay of the grating terms. Although this model does not correctly describe the asymmetric gain curve typical of semiconductor materials as it lacks the strong amplitude-phase coupling denoted by Henry’s linewidth enhancement factor

*α*, it still can shed some light on the different behavior of ring and FP lasers regarding multistability. One can expect

_{H}*α*to induce an asymmetry of the multistability band around the dominant mode, but a precise analysis of semiconductor devices requires modelling the material response as in e.g. [23

_{H}23. J. Javaloyes and S. Balle, “Quasiequilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A **81**, 062505 (2010). [CrossRef]

*δ*= 0), hence the boundary conditions are written in Fig. 1, where

*r*

_{±}and

*t*

_{±}denote the reflectivity and transmissivity of the forward and backward waves.

*D*

_{0}=

*J, A*

_{±}=

*B*

_{±}=

*D*

_{±2}= 0 [22

22. A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional Ring Lasers,” Phys. Rev. A **81**, 043817 (2010). [CrossRef]

24. S. Fürst, A. Pérez-Serrano, A. Scirè, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiments and theory,” Appl. Phys. Lett. **93**, 251109 (2008). [CrossRef]

*r*

_{±}= 0), the frequency (

*m*in each of the counter-propagating directions read where

*t*

_{±}= 0), we have —with

*A*

_{±}(0), we solve for the spatial dependence of (1)–(4) using standard integration techniques with a spatial step

*h*= 1/

*N*towards the other end of the cavity, where the propagated values

*A*

_{±}(1) must verify the boundary conditions. By using a Newton-Raphson algorithm a new guess for the field amplitudes

*A*

_{±}(0) and the modal frequency is proposed and the process is repeated until one reaches convergence. The final trajectory generated by this shooting method provides a discretized representation of the modal profile as a spatial mesh of

*N*points.

**V**

_{n}_{+1}=

**U**(

*h,*

**V**

*) described in [22*

_{n}22. A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional Ring Lasers,” Phys. Rev. A **81**, 043817 (2010). [CrossRef]

**V**a time step

*h*while verifying the Courant condition [25

25. R. J. LeVeque, *Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (2007). [CrossRef] [PubMed]

**V**hereby finding the matrix

**M**=

*∂*

**U**

*/∂*

**V**representing the linear operator governing the time evolution for the perturbations around one given monochromatic solution. We finally compute the 11×

*N*Floquet multipliers

*z*of

_{i}**M**, which determine the eigenvalues as

*λ*=

_{i}*h*

^{−1}ln

*z*. We used

_{i}*N*= 256 mesh points; in this case, determining the spatial profile of the monochromatic solution, generating the matrix

**M**and diagonalizing it using the QR decomposition method takes 1, 10 and 60 seconds, respectively, on an standard PC using C++ routine based on

*Octave*[26]. Stability results have been controled by direct integration of the TWM [22

**81**, 043817 (2010). [CrossRef]

*L*= 2.4 mm and

_{c}*τ*= 25 ps, a modal gain of 33 cm

_{c}^{−1}, a gain width of 13 nm, a carrier lifetime 0.5 ns and a diffusion coefficient of 5 cm

^{2}/s.

*m*= 2 of a symmetric, bidirectional ring laser. In panel a) we show that just above the threshold current

*J*≃ 0.51, this solution corresponds to an unstable bidirectional state. At

*J*

_{∈}≃ 1.5, a pitchfork bifurcation into unidirectional emission occurs, but the degenerate (almost) unidirectional states are also unstable, as evidenced by the eigenvalues shown in panel b) for

*J*= 3. However, for currents above

*J*> 3.5, they become stable and all the eigenvalues have

*Re*(

*λ*) < 0 (see panel c) for

*J*= 4).

*m*.

*g*,

*γ*,

*ɛ*, and

*η*of a ring laser have to be twice their equivalent FP values. For the parameters of Fig. 2, the ring laser just above threshold has only one stable solution

*m*= 0. Upon increasing

*J*, this bidirectional solution becomes unstable first via a Hopf bifurcation at

*J*∼ 0.7 and then via a pitchfork bifurcation at

*J*

_{∈}∼ 1.5 that leads to two symmetrical, almost unidirectional, solutions. Although the solutions corresponding to

*m*= 3 remain unstable over the interval of

*J*shown, solutions

*m*= 1 and

*m*= 2 become stable for high enough

*J*, hence the system easily displays multistability once in the almost unidirectional regime. The equivalent FP laser behaves remarkably different from the ring laser regarding multistability (see Fig. 3b). Above threshold, the mode

*m*= 0 starts lasing stably, but when the pump is increased it quickly becomes unstable through a multimode instability [17

17. H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A **43**, 2446–2454 (1991) [CrossRef] [PubMed]

*D*

_{2}averaged along the cavity at different pumping levels for mode

*m*= 0 of the lasers in Figs. 3 and 4, respectively.

*D*

_{2}is almost the same for all modes of a given laser due to the minute modal gain differences. In the ring laser, |

*D*

_{2}| saturates at a comparatively low value as soon as the pitchfork bifurcation leading to unidirectional operation occurs; for FP configurations, instead, the necessarily higher reflectivity of the facets makes |

*D*

_{2}| larger than in the equivalent ring, and it increases continuously with the pump level.

*D*

_{2}(see Eq. (4)). In fact, as shown in Fig. 5, now both FP configurations display multistability among longitudinal modes because now the spatial average of

*D*

_{2}(Fig. 5 panel c) is half that in Fig. 4 (panel d).

## Acknowledgments

## References and links

1. | C. J. Born, S. Yu, M. Sorel, and P. J. R. Laybourn, “Controllable and stable mode selection in a semiconductor ring laser by injection locking,” in CLEO Proceedings, paper CWK4, (2003). |

2. | C. J. Born, M. Hill, S. Yu, and M. Sorel, “Study of longitudinal mode coupling in a semiconductor ring laser,” in Proceedings of the 17th Annual Meeting of the IEEE-LEOS, pp. 27–28 (2004). |

3. | C. J. Born, M. Sorel, and S. Yu, “Linear and nonlinear mode interactions in a semiconductor ring laser,” IEEE J. Quantum Electron.41, 261 (2005). [CrossRef] |

4. | Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, “A novel semiconductor ring laser device aimed for all-optical signal processing,” in ECOC Proceedings, paper Th.1.C.4 (2008). |

5. | M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, “Unidirectional bistability in semiconductor waveguides lasers,” Appl. Phys. Lett. |

6. | K. Huybrechts, B. Maes, G. Morthier, and R. Baets, “Tristable all-optical flip-flop using coupled non linear cavities,” in |

7. | L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, “Mode-mode competition and unstable behavior in a homogeneously broadened ring laser,” Phys. Rev. A |

8. | L. A. Kotomtseva, “Steady states for longitudinal modes and dynamics of a laser with a saturable absorbent,” Quantum Semiclass. Opt. |

9. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection-laser properties,” |

10. | A. Loose, B. K. Goswami, H.-J. Wünsche, and F. Henneberger, “Tristability of a semiconductor laser due to time-delayed optical feedback,” Phys. Rev. E |

11. | K. P. Komarov, “Multistable single-mode emission from solid-state state lasers,” Quantum Electron. |

12. | J. R. Tredicce, L. M. Narducci, N. B. Abraham, D. K. Bandy, and L. A. Lugiato, “Experimental-evidence of mode competition leading to optical bistability in homogenously broadened lasers,” Opt. Commun. |

13. | M. Yamada, “Theory of mode competition noise in semiconductor injection-lasers,” IEEE J. Quantum Electron. |

14. | F. Pedaci, S. Lepri, S. Balle, G. Giacomelli, M. Giudici, and J. R. Tredicce, “Multiplicative noise in the longitudinal mode dynamics of a bulk semiconductor laser,” Phys. Rev. E |

15. | T. Acsente, “Laser diode intensity noise induced by mode hopping,” Romanian Rep. Phys. |

16. | Ya. I. Khanin, |

17. | H. Fu and H. Haken, “Multifrequency operations in a short-cavity standing-wave laser,” Phys. Rev. A |

18. | L. A. Lugiato, L. M. Narducci, and M. F. Squicciarini, “Exact linear-stability analysis of the plane-wave Maxwell-Bloch equations for a ring laser,” Phys. Rev. A |

19. | G. J. de Valcárcel, E. Roldán, and F. Prati, “Risken-Nummedal-Graham-Haken instability in class B lasers,” Opt. Commun. |

20. | H. Risken and K. Nummedal, “Self-pulsing in lasers,” J. Appl. Phys. |

21. | R. Graham and H. Haken, “Quantum theory of light propagation in a fluctuating laser-active medium,” Z. Phys. |

22. | A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional Ring Lasers,” Phys. Rev. A |

23. | J. Javaloyes and S. Balle, “Quasiequilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A |

24. | S. Fürst, A. Pérez-Serrano, A. Scirè, M. Sorel, and S. Balle, “Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiments and theory,” Appl. Phys. Lett. |

25. | R. J. LeVeque, |

26. | J. W. Eaton, |

**OCIS Codes**

(140.3410) Lasers and laser optics : Laser resonators

(140.3430) Lasers and laser optics : Laser theory

(140.3560) Lasers and laser optics : Lasers, ring

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: October 18, 2010

Revised Manuscript: December 20, 2010

Manuscript Accepted: December 23, 2010

Published: February 4, 2011

**Citation**

A. Pérez-Serrano, J. Javaloyes, and S. Balle, "Longitudinal mode multistability in Ring and Fabry-Pérot lasers: the effect of spatial hole burning," Opt. Express **19**, 3284-3289 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-4-3284

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### References

- C. J. Born, S. Yu, M. Sorel, and P. J. R. Laybourn, "Controllable and stable mode selection in a semiconductor ring laser by injection locking," in CLEO Proceedings, paper CWK4, (2003).
- C. J. Born, M. Hill, S. Yu, and M. Sorel, "Study of longitudinal mode coupling in a semiconductor ring laser," in Proceedings of the 17th Annual Meeting of the IEEE-LEOS, pp. 27-28 (2004).
- C. J. Born, M. Sorel, and S. Yu, "Linear and nonlinear mode interactions in a semiconductor ring laser," IEEE J. Quantum Electron. 41, 261 (2005). [CrossRef]
- Z. Wang, G. Yuan, G. Verschaffelt, J. Danckaert, and S. Yu, "A novel semiconductor ring laser device aimed for all-optical signal processing," in ECOC Proceedings, paper Th.1.C.4 (2008).
- M. Sorel, P. J. R. Laybourn, G. Giuliani, and S. Donati, "Unidirectional bistability in semiconductor waveguides lasers," Appl. Phys. Lett. 80(17), 3051-3053 (2002). [CrossRef]
- K. Huybrechts, B. Maes, G. Morthier, and R. Baets, "Tristable all-optical flip-flop using coupled non linear cavities," in Winter Topical Meeting Series (IEEE, New York, 2008), p. 1617.
- L. M. Narducci, J. R. Tredicce, L. A. Lugiato, N. B. Abraham, and D. K. Bandy, "Mode-mode competition and unstable behavior in a homogeneously broadened ring laser," Phys. Rev. A 33, 1842 (1986). [CrossRef] [PubMed]
- L. A. Kotomtseva, "Steady states for longitudinal modes and dynamics of a laser with a saturable absorbent," Quantum Semiclass. Opt. 10, 331 (1998). [CrossRef]
- R. Lang, and K. Kobayashi, "External optical feedback effects on semiconductor injection-laser properties," IEEE J. Quantum Electron. 16, 347 (1980). [CrossRef]
- A. Loose, B. K. Goswami, H.-J. Wünsche, and F. Henneberger, "Tristability of a semiconductor laser due to time-delayed optical feedback," Phys. Rev. E 79, 036211 (2009). [CrossRef]
- K. P. Komarov, "Multistable single-mode emission from solid-state state lasers," Quantum Electron. 24(11), 975-976 (1994). [CrossRef]
- J. R. Tredicce, L. M. Narducci, N. B. Abraham, D. K. Bandy, and L. A. Lugiato, "Experimental-evidence of mode competition leading to optical bistability in homogenously broadened lasers," Opt. Commun. 56, 435 (1986). [CrossRef]
- M. Yamada, "Theory of mode competition noise in semiconductor injection-lasers," IEEE J. Quantum Electron. 22, 1052 (1986). [CrossRef]
- F. Pedaci, S. Lepri, S. Balle, G. Giacomelli, M. Giudici, and J. R. Tredicce, "Multiplicative noise in the longitudinal mode dynamics of a bulk semiconductor laser," Phys. Rev. E 73, 041101 (2006). [CrossRef]
- T. Acsente, "Laser diode intensity noise induced by mode hopping," Romanian Rep. Phys. 59, 87 (2007).
- Ya. I. Khanin, Fundamentals of Laser Dynamics, Cambridge Int. Sci. Pub. Ltd., Cambridge, UK (2006).
- H. Fu, and H. Haken, "Multifrequency operations in a short-cavity standing-wave laser," Phys. Rev. A 43, 2446-2454 (1991). [CrossRef] [PubMed]
- L. A. Lugiato, L. M. Narducci, and M. F. Squicciarini, "Exact linear-stability analysis of the plane-wave Maxwell-Bloch equations for a ring laser," Phys. Rev. A 34, 3101 (1986). [CrossRef] [PubMed]
- G. J. de Valcárcel, E. Roldán, and F. Prati, "Risken-Nummedal-Graham-Haken instability in class B lasers," Opt. Commun. 163, 5-8 (1999). [CrossRef]
- H. Risken, and K. Nummedal, "Self-pulsing in lasers," J. Appl. Phys. 39, 4662 (1968). [CrossRef]
- R. Graham, and H. Haken, "Quantum theory of light propagation in a fluctuating laser-active medium," Z. Phys. 213, 420 (1968). [CrossRef]
- A. Pérez-Serrano, J. Javaloyes, and S. Balle, "Bichromatic emission and multimode dynamics in bidirectional Ring Lasers," Phys. Rev. A 81, 043817 (2010). [CrossRef]
- J. Javaloyes, and S. Balle, "Quasiequilibrium time-domain susceptibility of semiconductor quantum wells," Phys. Rev. A 81, 062505 (2010). [CrossRef]
- S. Fürst, A. Pérez-Serrano, A. Scirè, M. Sorel, and S. Balle, "Modal structure, directional and wavelength jumps of integrated semiconductor ring lasers: Experiments and theory," Appl. Phys. Lett. 93, 251109 (2008). [CrossRef]
- R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations, Steady State and Time Dependent Problems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, (2007). [CrossRef] [PubMed]
- J. W. Eaton, GNU Octave Manual, Network Theory Limited, (2002).

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