## Vortex solitons in lasers with feedback

Optics Express, Vol. 18, Issue 9, pp. 8859-8866 (2010)

http://dx.doi.org/10.1364/OE.18.008859

Acrobat PDF (1455 KB)

### Abstract

We report on the existence, stability and dynamical properties of two-dimensional self-localized vortices with azimuthal numbers up to 4 in a simple model for lasers with frequency-selective feedback. We build the full bifurcation diagram for vortex solutions and characterize the different dynamical regimes. The mathematical model used, which consists of a laser rate equation coupled to a linear equation for the feedback field, can describe the spatiotemporal dynamics of broad area vertical cavity surface emitting lasers with external frequency selective feedback in the limit of zero delay.

© 2010 Optical Society of America

## 1. Introduction

7. P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf

9. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. **95**, 053903 (2005). [PubMed]

## 2. Model

*E*(

*x*,

*y*,

*t*) and feedback

*F*(

*x*,

*y*,

*t*) fields:

*κ*is the decay rate of the field in the cavity,

*α*describes the nonlinear frequency shift,

*μ*is the pump current normalized to be 1 at the threshold of the solitary laser,

*ω*is the frequency detuning of the solitary laser (at threshold of the axial mode) with respect to the central frequency of Lorentzian filter [22], which is taken as reference frequency,

_{s}*σ*is the feedback strength, and

## 3. Vortex excitation

*E*(

*x*,

*y*) =

*A*(

*r*)

*e*. Here

^{imφ}*r*is the radial coordinate,

*φ*is the azimuthal angle and

*m*the azimuthal number in a polar coordinate system centered on the input vortex. Using initial conditions such that

*A*(

*r*) → 0 at

*r*= 0 is helpful for a faster approach to the vortex soliton, although direct integration can give vortex solutions for a very broad choice of initial

*A*(

*r*) function, see for example [10].

*m*= 1,2 vortices by choosing

*A*(

*r*) to correspond to a ring of radius and thickness approximately equal to the width of the fundamental (

*m*= 0) soliton. Figure 1 illustrates the coexistence of the three lowest

*m*self-localized states. Each of these structures was checked to be stable and, for the chosen distances between them, the interaction is so small that no significant changes are observed after integrating for a time up to four orders larger than the relaxation time of the system. For larger initial

*m*and ring radius (see Figure 2 and [12]) we were able to excite

*m*= 3,4 vortices. Further details about the observed vortices are presented in Fig. 2, where transverse sections of the amplitude (in semi-logarithmic scale) and the phase are shown for states with

*m*going from 0 to 4. The slope of the exponential decay of the amplitude is almost identical for all self-localized states while the vortex radius increases with the azimuthal number (Fig. 2a). The phase distribution of the

*m*= 0 soliton has one extremum in the center. For vortices there are three extrema, one of which is in the center (singular point), the other two corresponding to the maxima of the amplitude profile. As expected for an

*e*phase distribution, odd-

^{imφ}*m*vortices have a

*π*-discontinuity in the singular point, while for even

*m*the phase is continuous (Fig. 2b).

## 4. Bifurcation diagram of self-localized states

*E*=

*E*

_{0}(

*x*,

*y*)

*e*,

^{iωt}*F*=

*F*

_{0}(

*x*,

*y*)

*e*. Substituting into Eqs. (1), we can reduce the system to the following equation for

^{iωt}*E*

_{0}:

*A*and

*B*, which are easily expressed as a function of the parameters [15]. We solve Eq. (2) using a Newton method. As an initial guess, we use self-localized vortex solutions obtained from the direct integration of the full model (1). These form an array of

*N*by

*N*complex values (

*N*= 64 was used for most simulations, while some results were checked with

*N*= 128,256,512). We need one more condition, since solution of (2), on discretization, involves 2

*N*× 2

*N*+ 1 real coordinates, representing the complex field

*E*

_{0}(

*x*,

_{i}*y*) and the unknown frequency

_{j}*ω*. As such a condition, one can fix the free overall phase of the solution, by requiring at each iteration

*phase*(

*E*(

_{new}*x*

_{0},

*y*

_{0})) =

*phase*(

*E*(

_{prev}*x*

_{0},

*y*

_{0})), where (

*x*

_{0},

*y*

_{0}) can be chosen arbitrarily.

*μ*) in small steps we build the whole bifurcation diagram of each solution. The results corresponding to

*m*= 0,1,2 are presented in figure 3, which shows the total power as a function of the input pump. This quantity clearly distinguishes between the states with different vorticity. All three branches terminate at the same current values, marked by lines

*A*and

*B*in (Fig. 3). As in previous work [15], the

*E*= 0 “off” state is unstable between

*A*and

*B*, and so any soliton, or vortex soliton, is necessarily unstable in that range. In Fig. 4 we show some additional features of the branches presented in Fig. 3. For comparison, we include in Fig. 4 data for the homogenous lasing states which also bifurcate from (

*A*,

*B*) [15]. The maximum intensities ∣

*E*∣

_{max}^{2}of vortices with

*m*= 1,2… are almost indistinguishable (see also [10, 12]), therefore we present in Fig. 4a only the results for

*m*= 0 and

*m*=1. The frequency

*ω*of

*m*= 0 state is always similar to the frequency of

*m*= 1 state. We show their dependence on

*μ*in Fig. 4(b). The frequencies for higher

*m*are almost identical to

*m*= 1. The size of the self-localized states tends to infinity at the bifurcation points (see Fig. 4c for

*m*= 1 vortex or [15] for

*m*= 0 soliton), while the peak intensity simultaneously tends to zero (Fig. 4a). Interestingly, the self-localized states always have finite power. As shown in figure 3, the power does not vanish even as the bifurcation points are approached.

## 5. Stability of self-localized states

*E*= (

*E*

_{0}+

*δE*)

*e*,

^{iωt}*F*= (

*F*

_{0}+

*δF*)

*e*, and substituting into Eq. (1), we obtain, after linearizing, an eigenvalue problem

^{iωt}*Mê⃗*=

*λe⃗*to determine the stability of the solutions. The elements of matrix

*M̂*depend on the steady state

*E*

_{0},

*F*

_{0}and the parameters of the system, and

*e⃗*= [

*δE*(

*x*,

_{i}*y*),

_{j}*δF*(

*x*,

_{i}*y*)] is a vector containing the values of the perturbations

_{j}*δE*and

*δF*at each discretization point. In contrast to other works [8, 11], we solve here the full 2D problem. For the fundamental

*m*= 0 soliton there is only a drift instability, signaled as point M in Fig.3. This instability is displayed in movie (Fig.3, Media 1). The resulting drifting soli-tons were already studied in [15, 16] for the case of finite feedback delay. We note that this drift instability exists even at zero delay time, in contrast to the case studied in [23] where drift instability required the addition of a delay.

*m*= 1 vortex is stable between points

*C*

_{1}and

*M*

_{1}in Fig. 3, and bifurcates into a stable self-moving vortex at point

*M*

_{1}, as shown in movie (Fig.3,Media 2). This seems to be the only instability for the

*m*= 1 vortex soliton between the saddle-node

*C*

_{1}and the background instability at

*B*.

*m*= 2 vortex has a richer instability behavior. As the current

*μ*is decreased from its saddle-node

*C*

_{2}it is stable until the point

*G*

_{1}shown in Fig. 3, but it then undergoes several different instabilities as the pump current is further decreased. Fig. 5 shows the growth rates of the unstable modes, as a function of

*μ*, over the range (

*B*,

*C*

_{2}). There are three important modes, all with very different spatial structures, leading to qualitatively different dynamics. Interestingly, each of these three modes has the largest growth rate over a finite current range, within which it will dominate the dynamics when the

*m*= 2 vortex is subjected to a random perturbation. The first instability, at bifurcation (point

*G*

_{1}in Fig. 3) seems to be a “splitting” mode, which leads to a complex rotating self-localized structure (Fig. 5, Media 3), similar to the bound states of two

*m*= 1 vortices reported in [8, 9

9. N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. **95**, 053903 (2005). [PubMed]

*G*

_{2}, and is the most unstable mode over a range of currents somewhat below

*G*

_{2}. The drifting vortex seems to remain unstable to “splitting”, however, and undergoes a secondary instability leading to a structure similar to that in (Fig. 5, Media 3), but more complex, with the appearance of an additional rotation axis (see Fig. 5, Media 4). The third mode, which becomes undamped at

*G*

_{3}, leads to fission of the

*m*= 2 vortex, yielding two drifting

*m*= 0 solitons (see Fig. 5, Media 5).

## 6. Concluding remarks

*m*= 2 we have identified three potentially unstable modes with different structures, which lead to very different dynamical evolutions.

## Acknowledgments

## References and links

1. | T. Ackemann, W. J. Firth, and G.-L. Oppo, “Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices,” Advances in Atomic, Molecular, and Optical Physics , |

2. | S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tis-soni, T. Knodlk, M. Millerk, and R. Jagerk, “Cavity solitons as pixels in semiconductor microcavities,” Nature (London) |

3. | Y. Tanguy,, T. Ackemann, W. J. Firth, and R. Jager, “Realization of a Semiconductor-Based Cavity Soliton Laser,” Phys. Rev. Lett. |

4. | P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, “Cavity soliton laser based on mutually coupled semiconductor microresonators,” Phys. Rev. Let.. |

5. | A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, “Bifurcation analysis of laser autosolitons,” Quantum Electronics |

6. | T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, “Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber,” Appl. Phys. B |

7. | P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, “Bistable and addressable localized vortices in semiconductor lasers,” submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf |

8. | N. N. Rosanov, “Solitons in laser systems with saturable absorption,” in Dissipative Solitons, edited by
N. Akhmediev and A. Ankiewicz, Lect. Notes Phys. |

9. | N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, “Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling,” Phys. Rev. Lett. |

10. | L.-C Crasovan, B. A. Malomed, and D. Michalache, “Stable vortex solitons in the two-dimensional Ginzburg-Landau equation,” Phys. Rev. E |

11. | D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation,” Phys. Rev. A |

12. | W. J. Firth and D. V. Skryabin, “Optical solitons carrying orbital angular momentum,” Phys. Rev. Lett. |

13. | J. R. Salgueiro and Y. S. Kivshar, “Single- and double-vortex vector solitons in self-focusing nonlinear media,” Phys. Rev. E |

14. | A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, “Optical vortices and vortex solitons,” Progress in Optics , ed.
E. Wolf, |

15. | P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, “Self-localized structures in vertical-cavity surface-emitting lasers with external feedback,” Phys. Rev. E |

16. | P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth “Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback,” Phys. Rev. A |

17. | J. Atai and B. A. Malomed, “Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations,” Phys. Lett. A , |

18. | W. J. Firth and P. V. Paulau, “Soliton lasers stabilized by coupling to a resonant linear system,” submitted (2009). |

19. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

20. | S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser. and Photon Rev. |

21. | G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys. |

22. | A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, “Experimental and theoretical study of filtered optical feedback in a semiconductor laser,” IEEE Journal of Quantum Electronics |

23. | M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, “Spontaneous motion of cavity solitons induced by a delayed feedback,” Phys. Rev. Lett, |

**OCIS Codes**

(190.6135) Nonlinear optics : Spatial solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: January 26, 2010

Revised Manuscript: March 5, 2010

Manuscript Accepted: March 8, 2010

Published: April 13, 2010

**Citation**

P. V. Paulau, D. Gomila, P. Colet, N. A. Loiko, N. N. Rosanov, T. Ackemann, and W. J. Firth, "Vortex solitons in lasers with feedback," Opt. Express **18**, 8859-8866 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-9-8859

Sort: Year | Journal | Reset

### References

- T. Ackemann, W. J. Firth, and G.-L. Oppo, "Fundamentals and applications of Spatial Dissipative Solitons in Photonic Devices," Adv. Atom. Mol. Opt. Phys. 57, 323 (2009).
- S. Barland, J. R. Tredicce, M. Brambilla, L. A. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodlk, M. Millerk, and R. Jagerk, "Cavity solitons as pixels in semiconductor microcavities," Nature (London) 419, 699 (2002).
- Y. Tanguy, T. Ackemann, W. J. Firth, and R. Jäger, "Realization of a Semiconductor-Based Cavity Soliton Laser," Phys. Rev. Lett. 100, 013907 (2008).
- P. Genevet, S. Barland, M. Guidici, and J. R. Tredicce, "Cavity soliton laser based on mutually coupled semiconductor microresonators," Phys. Rev. Let. 101, 123905 (2008).
- A. G. Vladimirov, N. N. Rosanov, S. V. Fedorov, and G. V. Khodova, "Bifurcation analysis of laser autosolitons," Quantum Electron. 27, 949-952 (1997).
- T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, "Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber," Appl. Phys. B 98, 327 (2010).
- P. Genevet, S. Barland, M. Giudici, and J. R. Tredicce, "Bistable and addressable localized vortices in semiconductor lasers," submitted (2009) http://hal.archives-ouvertes.fr/docs/00/43/59/20/PDF/localizedvortices.pdf.
- N. N. Rosanov, "Solitons in laser systems with saturable absorption," in Dissipative Solitons, edited by N. Akhmediev and A. Ankiewicz, Lect. Notes Phys. 661, 101-130 (2004).
- N. N. Rosanov, S. V. Fedorov, and A. N. Shatsev, "Curvilinear motion of multivortex laser-soliton complexes with strong and weak coupling," Phys. Rev. Lett. 95, 053903 (2005). [PubMed]
- L.-C. Crasovan, B. A. Malomed, and D. Michalache, "Stable vortex solitons in the two-dimensional Ginzburg-Landau equation," Phys. Rev. E 63, 016605 (2000).
- D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, "Collisions between coaxial vortex solitons in the three-dimensional cubic-quintic complex Ginzburg-Landau equation," Phys. Rev. A 77, 033817 (2008).
- W. J. Firth and D. V. Skryabin, "Optical solitons carrying orbital angular momentum," Phys. Rev. Lett. 79, 2450 (1997).
- J. R. Salgueiro and Y. S. Kivshar, "Single- and double-vortex vector solitons in self-focusing nonlinear media," Phys. Rev. E 70, 056613 (2004).
- A. S. Desyatnikov, Y. S. Kivshar, and L. Torner, "Optical vortices and vortex solitons," Progress in Optics, ed. E. Wolf, 47, 291 (2005).
- P. V. Paulau, D. Gomila, T. Ackemann, N. A. Loiko, and W. J. Firth, "Self-localized structures in vertical-cavity surface-emitting lasers with external feedback," Phys. Rev. E 78, 016212 (2008).
- P. V. Paulau, D. Gomila, P. Colet, M. A. Matias, N. A. Loiko, and W. J. Firth "Drifting instabilities of cavity solitons in vertical-cavity surface-emitting lasers with frequency-selective feedback," Phys. Rev. A 80, 023808 (2009).
- J. Atai, and B. A. Malomed, "Exact stable pulses in asymmetric linearly coupled Ginzburg-Landau equations," Phys. Lett. A 246, 412 (1998).
- W. J. Firth and P. V. Paulau, "Soliton lasers stabilized by coupling to a resonant linear system," submitted (2009).
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826 (1995). [PubMed]
- S. Franke-Arnold, L. Allen, and M. Padgett, "Advances in optical angular momentum," Laser. Photon Rev. 2, 299 (2008).
- G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nat. Phys. 3, 305 (2007).
- A. P. A. Fisher, O. K. Andersen, M. Yousefi, S. Stolte, and D. Lenstra, "Experimental and theoretical study of filtered optical feedback in a semiconductor laser," IEEE J. Quantum Electron. 36, 375 (2000).
- M. Tlidi, A. G. Vladimirov, D. Pieroux, and D. Turaev, "Spontaneous motion of cavity solitons induced by a delayed feedback," Phys. Rev. Lett, 103,103904 (2009).

## 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.

### Supplementary Material

» Media 1: AVI (428 KB)

» Media 2: AVI (705 KB)

» Media 3: AVI (2455 KB)

» Media 4: AVI (1622 KB)

» Media 5: AVI (701 KB)

« Previous Article | Next Article »

OSA is a member of CrossRef.