## Multimode dynamics in bidirectional laser cavities by folding space into time delay |

Optics Express, Vol. 20, Issue 8, pp. 8496-8502 (2012)

http://dx.doi.org/10.1364/OE.20.008496

Acrobat PDF (819 KB)

### Abstract

Multimode dynamics in bidirectional laser cavities can be accurately described by folding space into time delay. This results in a set of delayed algebraic equations that preserve the dynamics of all cavity modes while drastically reducing number of degrees of freedom. This reduction allows for both linear stability analysis and bifurcation diagram reconstruction, as well as integration times reduced by orders of magnitude.

© 2012 OSA

1. R. Linke, B. Kasper, C. Burrus, I. Kaminow, J.-S. Ko, and T. Lee, “Mode power partition events in nearly single-frequency lasers,” J. Lightwave Technol. **3**, 706–712 (1985). [CrossRef]

2. M. Ohtsu, Y. Teramachi, Y. Otsuka, and A. Osaki, “Analyses of mode-hopping phenomena in an AlGaAs laser,” IEEE J. Quantum. Elect. **22**, 535–543 (1986). [CrossRef]

3. L. Furfaro, F. Pedaci, M. Giudici, X. Hachair, J. Tredicce, and S. Balle, “Mode-switching in semiconductor lasers,” IEEE J. Quantum. Elect. **40**, 1365–1376 (2004). [CrossRef]

4. L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of he-ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. **5**, 4–5 (1964). [CrossRef]

5. H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. **6**, 1173–1185 (2000). [CrossRef]

8. J. A. Fleck, “Emission of pulse trains by Q-switched lasers,” Phys. Rev. Lett. **21**, 131–133 (1968). [CrossRef]

9. J. Javaloyes and S. Balle, “Mode-locking in Fabry-Pérot lasers,” IEEE J. Quantum Electron. **46**, 1023–1030 (2010). [CrossRef]

14. A.G. Vladimirov, A.S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron **45**, 462 –468 (2009). [CrossRef]

*δt*, while the Courant-Friedrichs-Lewy (CFL) condition for stability imposes an accordingly fine spatial discretization

*δz*.

*L*≪

_{a}*λ*, the dynamics can be described by a set of ODEs complemented by Delayed-Differential Equations (DDEs) or Delayed-Algebraic Equations (DAEs) [16

16. J. Mulet and S. Balle, “Mode locking dynamics in electrically-driven vertical-external-cavity surface-emitting lasers,” IEEE J. Quantum Electron. **41**, 1148–1156 (2005). [CrossRef]

17. L. A. Lugiato and F. Prati, “Difference differential equations for a resonator with a very thin nonlinear medium,” Phys. Rev. Lett. **104**, 233902 (2010). [CrossRef] [PubMed]

18. R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. **16**, 347–355 (1980). [CrossRef]

*L*≫

_{a}*λ*, a tractable model has been developed only for unidirectional cavities under strong approximations; linear gain and recombination, no internal losses and the neglect of gain dispersion due to the presence of a external frequency filter [19

19. A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A **72**, 033808 (2005). [CrossRef]

20. M. Rossetti, P. Bardella, and I. Montrosset, “Modeling passive mode-locking in quantum dot lasers: A comparison between a finite-difference traveling-wave model and a delayed differential equation approach,” IEEE J. Quantum. Electron **47**, 569 –576 (2011). [CrossRef]

*E*

_{±}read where space and time (

*z,t*) are normalized to the cavity length

*L*and to the cavity time of flight

*τ*=

_{c}*L/υ*, respectively. The source terms contain both the internal losses

_{g}*α*and the active medium polarization

*P*

_{±}(

*z,t*) whose dynamical evolution is given by a local equation determined by the type of active material. The solution of Eq. (1) reads where

_{±}is the advected integral of the source. Notice that the results in [17

17. L. A. Lugiato and F. Prati, “Difference differential equations for a resonator with a very thin nonlinear medium,” Phys. Rev. Lett. **104**, 233902 (2010). [CrossRef] [PubMed]

*S*

_{±}do not strongly vary along the characteristic lines

*ξ*

_{±}=

*z*∓

*t*in the integration interval, one can approximate and then fields at different locations are coupled through a DAE of the form

- In the particular case of a unidirectional ring laser, Eq. (4) corresponds to the models of [19, 20
19. A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A

**72**, 033808 (2005). [CrossRef]] but without eliminating adiabatically the polarization of the active medium.20. M. Rossetti, P. Bardella, and I. Montrosset, “Modeling passive mode-locking in quantum dot lasers: A comparison between a finite-difference traveling-wave model and a delayed differential equation approach,” IEEE J. Quantum. Electron

**47**, 569 –576 (2011). [CrossRef] - In the case of a bidirectional Fabry-Pérot (FP) cavity, Eq. (4) corresponds to the model developed in [14] but without eliminating adiabatically the polarization of the active medium as well as neglecting the half wavelength spatial hole burning.
14. A.G. Vladimirov, A.S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron

**45**, 462 –468 (2009). [CrossRef] - In the case of a quasi-monochromatic field and under the UFL, it is natural to expand
*E*(*z,t*−*τ*) =*E*(*z,t*) −*τĖ*(*z,t*) + ··· which leads to the RE approximation, whose particular form depends on both the boundary conditions and the material description.

*z*= 0 and

*z*= 1 are considered, i.e.

*τ*= 1. The reason is that the intermediate “missing” spatial points have been folded into the time delays. As an example, we consider a FP cavity containing an active medium composed of homogeneously broadened two-level atoms, whose evolution equations are those given in [10

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

*E*

_{+}(0,

*t*) =

*R*

_{1}

*E*

_{−}(0

*,t*) and

*E*

_{−}(1

*,t*) =

*R*

_{2}

*E*

_{+}(0

*,t*) where

*R*

_{1,2}denote the complex reflection coefficients of the facets. The LSA of the “off” solution

*E*

_{±}= 0 yields the threshold current

*J*and lasing frequency

_{m}*ω*of the cavity modes through the implicit equation where

_{m}*g*(

*ω*) =

*γJ/*(

*γ*

*− iω*) −

*α*. The solutions for different values of

*R*

_{1}=

*R*

_{2}=

*R*are plotted in Fig. 1 (dots) together with the exact result (lines) in the TWM [10

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

*e*≈ (1 +

^{g}*g*/2)/(1−

*g*/2), which is accurate provided that

*g*< 0.5 [21]. Indeed Eq. (5) gives an excellent approximation to

*all*cavity modes for high reflectivities

*R*≳ 0.75, getting worse as

*R*reduces, the net threhsold gain increases and the field amplitudes become non uniform in space. We emphasize that in obtaining Fig. 1 we have considered only the endpoints of the cavity,

*z*= 0 and

*z*= 1, but we recovered the correct information for all cavity modes in the UFL.

*N*sections short enough for the UFL to apply within each of them. The fields in section

*j*∈ [1..

*N*] are then coupled to their nearest neighbors by a temporal DAE of the form The number of spatial points in Eq. (7) is dictated by the amount of non uniformity of the field only while the time step is chosen only considering the fast evolution of the material.

12. P. Stolarz, J. Javaloyes, G. Mezosi, L. Hou, C. Ironside, M. Sorel, A. Bryce, and S. Balle, “Spectral dynamical behavior in passively mode-locked semiconductor lasers,” IEEE Photon. J. **3**, 1067–1082 (2011). [CrossRef]

22. J. Javaloyes and S. Balle, “Freetwm: a simulation tool for semiconductor lasers.” (2012). Available at http://nova.uib.es/ONL/Softwares/Softwares.html.

*L*= 1mm, with a Saturable Absorber length of 3.2%, and the roundtrip is

*τ*=

_{c}*L/υ*= 12.5ps. In order to correctly describe the polarization of the active medium, the time step is

_{g}*δt*= 23.5fs. The CFL condition thus imposes

*δz*=

*υ*= 1.89

_{g}δt*μ*m for the full TWM, hence the gain and SA sections are discretized over

*N*

_{1}= 513 and

*N*

_{2}= 17 points, respectively. Figure 3 depicts the bifurcation diagrams for the output intensity (pulse height, duration and skewness) as the current in the gain section is scanned in the full TWM (left column) and for two different DAE descriptions (center and right columns). For the full TWM we observe that immediately after threshold, the laser emits in pulsed mode. The pulse train is unstable due to a Q-switching instability which remains up to

*J*≈ 3

*J*; above this value, the pulse train becomes stable and regular. As the current increases, the pulses shorten down to ≃ 1 ps, and at the same time they become more asymmetrical. The results in the second and third columns —obtained with our DAE approach with

_{th}*N*

_{1}= 33 and

*N*

_{2}= 5, and

*N*

_{1}= 9 and

*N*

_{2}= 2, respectively— completely agree with those for the full TWM for all the variables. However, they have been obtained much more easily: in the third column, the number of DOF and the calculation time have been divided by ∼ 50 and while the temporal increment is still 23.5fs, the distance between points in the gain section is close to Δ

*z*∼ 100

*μ*m. In the three cases, the number of modes is identical (∼ 1000) as determined by 2/

*δt*.

23. 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). [CrossRef] [PubMed]

*N*= 9, 17 and 65. We use a method similar to the one in [23

23. 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). [CrossRef] [PubMed]

*N*= 9 are obtained in a few seconds, those for

*N*= 65 required several hours as well as several Gigabytes of memory; the LSA of the full TWM (with

*N*= 513) could not be performed. The results are shown in Fig. 4, where we plot the real part of the eigenvalues versus its corresponding imaginary part. Each eigenvalue corresponds to a laser mode, with the imaginary part giving the modal frequency and the sign of the real part determining whether the mode is stable (ℜ(

*λ*) < 0) or unstable. One recognizes the typical asymmetric gain spectrum of an active semiconductor as given by the susceptibility of [24

24. S. Balle, “Simple analytical approximations for the gain and refractive index spectra in quantum well lasers,” Phys. Rev. A **57**, 1304–1312 (1998). [CrossRef]

*N*= 9 and those obtained with

*N*= 65.

## Acknowledgments

## References and links

1. | R. Linke, B. Kasper, C. Burrus, I. Kaminow, J.-S. Ko, and T. Lee, “Mode power partition events in nearly single-frequency lasers,” J. Lightwave Technol. |

2. | M. Ohtsu, Y. Teramachi, Y. Otsuka, and A. Osaki, “Analyses of mode-hopping phenomena in an AlGaAs laser,” IEEE J. Quantum. Elect. |

3. | L. Furfaro, F. Pedaci, M. Giudici, X. Hachair, J. Tredicce, and S. Balle, “Mode-switching in semiconductor lasers,” IEEE J. Quantum. Elect. |

4. | L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of he-ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett. |

5. | H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron. |

6. | E. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang, “Auto97: Continuation and bifurcation software for ordinary differential equations,” (2011). |

7. | K. Engelborghs, T. Luzyanina, and G. Samaey, “Dde-biftool v. 2.00: a matlab package for bifurcation analysis of delay differential equations,” Tech. Rep., Department of Computer Science, K.U.Leuven, Belgium. (2001). |

8. | J. A. Fleck, “Emission of pulse trains by Q-switched lasers,” Phys. Rev. Lett. |

9. | J. Javaloyes and S. Balle, “Mode-locking in Fabry-Pérot lasers,” IEEE J. Quantum Electron. |

10. | A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional ring lasers,” Phys. Rev. A |

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

12. | P. Stolarz, J. Javaloyes, G. Mezosi, L. Hou, C. Ironside, M. Sorel, A. Bryce, and S. Balle, “Spectral dynamical behavior in passively mode-locked semiconductor lasers,” IEEE Photon. J. |

13. | J. Javaloyes and S. Balle, “All-optical directional switching of bistable semiconductor ring lasers,” IEEE J. Quantum Electron |

14. | A.G. Vladimirov, A.S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron |

15. | L. Narducci and N. B. Abraham, |

16. | J. Mulet and S. Balle, “Mode locking dynamics in electrically-driven vertical-external-cavity surface-emitting lasers,” IEEE J. Quantum Electron. |

17. | L. A. Lugiato and F. Prati, “Difference differential equations for a resonator with a very thin nonlinear medium,” Phys. Rev. Lett. |

18. | R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron. |

19. | A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A |

20. | M. Rossetti, P. Bardella, and I. Montrosset, “Modeling passive mode-locking in quantum dot lasers: A comparison between a finite-difference traveling-wave model and a delayed differential equation approach,” IEEE J. Quantum. Electron |

21. | W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, |

22. | J. Javaloyes and S. Balle, “Freetwm: a simulation tool for semiconductor lasers.” (2012). Available at http://nova.uib.es/ONL/Softwares/Softwares.html. |

23. | 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 |

24. | S. Balle, “Simple analytical approximations for the gain and refractive index spectra in quantum well lasers,” Phys. Rev. A |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(250.5980) Optoelectronics : Semiconductor optical amplifiers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 16, 2012

Revised Manuscript: March 20, 2012

Manuscript Accepted: March 20, 2012

Published: March 27, 2012

**Citation**

J. Javaloyes and S. Balle, "Multimode dynamics in bidirectional laser cavities by folding space into time delay," Opt. Express **20**, 8496-8502 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-8-8496

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### References

- R. Linke, B. Kasper, C. Burrus, I. Kaminow, J.-S. Ko, and T. Lee, “Mode power partition events in nearly single-frequency lasers,” J. Lightwave Technol.3, 706–712 (1985). [CrossRef]
- M. Ohtsu, Y. Teramachi, Y. Otsuka, and A. Osaki, “Analyses of mode-hopping phenomena in an AlGaAs laser,” IEEE J. Quantum. Elect.22, 535–543 (1986). [CrossRef]
- L. Furfaro, F. Pedaci, M. Giudici, X. Hachair, J. Tredicce, and S. Balle, “Mode-switching in semiconductor lasers,” IEEE J. Quantum. Elect.40, 1365–1376 (2004). [CrossRef]
- L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of he-ne laser modes induced by synchronous intracavity modulation,” Appl. Phys. Lett.5, 4–5 (1964). [CrossRef]
- H. A. Haus, “Mode-locking of lasers,” IEEE J. Sel. Top. Quantum Electron.6, 1173–1185 (2000). [CrossRef]
- E. Doedel, A. R. Champneys, T. F. Fairgrieve, Y. A. Kuznetsov, B. Sandstede, and X. Wang, “Auto97: Continuation and bifurcation software for ordinary differential equations,” (2011).
- K. Engelborghs, T. Luzyanina, and G. Samaey, “Dde-biftool v. 2.00: a matlab package for bifurcation analysis of delay differential equations,” Tech. Rep., Department of Computer Science, K.U.Leuven, Belgium. (2001).
- J. A. Fleck, “Emission of pulse trains by Q-switched lasers,” Phys. Rev. Lett.21, 131–133 (1968). [CrossRef]
- J. Javaloyes and S. Balle, “Mode-locking in Fabry-Pérot lasers,” IEEE J. Quantum Electron.46, 1023–1030 (2010). [CrossRef]
- A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Bichromatic emission and multimode dynamics in bidirectional ring lasers,” Phys. Rev. A81, 043817 (2010). [CrossRef]
- J. Javaloyes and S. Balle, “Quasiequilibrium time-domain susceptibility of semiconductor quantum wells,” Phys. Rev. A81, 062505 (2010). [CrossRef]
- P. Stolarz, J. Javaloyes, G. Mezosi, L. Hou, C. Ironside, M. Sorel, A. Bryce, and S. Balle, “Spectral dynamical behavior in passively mode-locked semiconductor lasers,” IEEE Photon. J.3, 1067–1082 (2011). [CrossRef]
- J. Javaloyes and S. Balle, “All-optical directional switching of bistable semiconductor ring lasers,” IEEE J. Quantum Electron47, 1078 –1085 (2011). [CrossRef]
- A.G. Vladimirov, A.S. Pimenov, and D. Rachinskii, “Numerical Study of Dynamical Regimes in a Monolithic Passively Mode-Locked Semiconductor Laser,” IEEE J. Quantum Electron45, 462 –468 (2009). [CrossRef]
- L. Narducci and N. B. Abraham, Laser Physics and Laser Instabilities (World Scientific, Singapore, 1988).
- J. Mulet and S. Balle, “Mode locking dynamics in electrically-driven vertical-external-cavity surface-emitting lasers,” IEEE J. Quantum Electron.41, 1148–1156 (2005). [CrossRef]
- L. A. Lugiato and F. Prati, “Difference differential equations for a resonator with a very thin nonlinear medium,” Phys. Rev. Lett.104, 233902 (2010). [CrossRef] [PubMed]
- R. Lang and K. Kobayashi, “External optical feedback effects on semiconductor injection laser properties,” IEEE J. Quantum Electron.16, 347–355 (1980). [CrossRef]
- A. G. Vladimirov and D. Turaev, “Model for passive mode locking in semiconductor lasers,” Phys. Rev. A72, 033808 (2005). [CrossRef]
- M. Rossetti, P. Bardella, and I. Montrosset, “Modeling passive mode-locking in quantum dot lasers: A comparison between a finite-difference traveling-wave model and a delayed differential equation approach,” IEEE J. Quantum. Electron47, 569 –576 (2011). [CrossRef]
- W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University Press, 2007.
- J. Javaloyes and S. Balle, “Freetwm: a simulation tool for semiconductor lasers.” (2012). Available at http://nova.uib.es/ONL/Softwares/Softwares.html .
- A. Pérez-Serrano, J. Javaloyes, and S. Balle, “Longitudinal mode multistability in ring and Fabry-Pérot lasers: the effect of spatial hole burning,” Opt. Express19, 3284–3289 (2011). [CrossRef] [PubMed]
- S. Balle, “Simple analytical approximations for the gain and refractive index spectra in quantum well lasers,” Phys. Rev. A57, 1304–1312 (1998). [CrossRef]

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