August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
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Chaotic behavior of cavity solitons induced by time delay feedback Krassimir Panajotov1,2,* and Mustapha Tlidi3 1
Brussels Photonics Team, Department of Applied Physics and Photonics (B-PHOT TONA), Vrije Universiteit Brussel, Pleinlaan 2, Brussels, Belgium 2
3
Institute of Solid State Physics, 72 Tzarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria
Optique Nonlinéaire Théorique, Université Libre de Bruxelles (U.L.B.), CP 231, Campus Plaine, B-1050 Bruxelles, Belgium *Corresponding author: kpanajot@b‑phot.org Received May 9, 2014; revised June 27, 2014; accepted July 2, 2014; posted July 3, 2014 (Doc. ID 211857); published August 7, 2014 We investigate spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subject to time-delayed optical feedback. We show that the inclusion of feedback leads to a period doubling route to chaos of spatially localized light structures. © 2014 Optical Society of America OCIS codes: (140.1540) Chaos; (140.5960) Semiconductor lasers; (190.1450) Bistability; (190.3100) Instabilities and chaos; (190.6135) Spatial solitons; (140.7260) Vertical cavity surface emitting lasers. http://dx.doi.org/10.1364/OL.39.004739
Cavity solitons (CSs) are localized light structures in the transverse plane of externally driven nonlinear resonators (for reviews see [1–6]). They appear in the coexisting region of parameters involving a periodic pattern and a stable homogeneous steady state [7–9]. In the transverse plane of the resonator, they consist of one or more regions in patterned state surrounded by a region in the homogeneous steady state [7]. Because of the large Fresnel number, the short cavity, and the mature technology, vertical-cavity surface-emitting lasers (VCSELs) are the most advanced systems for CS studies and applications. Indeed, with the injection of a holding beam with appropriate frequency, CSs have been demonstrated in broad area VCSELs both below [10,11] and above [12] the lasing threshold. A spatially localized structure has also been discovered in 40 μm diameter VCSELs by using their unique polarization properties [13], as well as in 80 μm diameter VCSELs lasing on a high transverse-order flower mode [14]. Of particular interest are VCSELs with saturable absorption as an external holding beam is not necessary, which significantly simplifies the system [15]. Therefore, VCSELs with a saturable absorber have been widely studied recently both theoretically [15,16] and experimentally [17,18]. Delayed optical feedback has been shown to strongly impact modal and dynamical behavior of semiconductor lasers [19,20] and has been widely studied for the case of VCSELs [21–25]. Recently, the impact of delayed feedback on CS dynamics has been theoretically investigated for the cases of a driven nonlinear optical resonator [26,27] and a broad-area VCSEL [28–30]. It has been shown that the time-delayed feedback leads to a spontaneous CS motion after a certain threshold of the feedback parameters is passed, which in the VCSEL case, critically depends on the feedback phase. In this Letter, we investigate the evolution of the single CS toward temporal chaos. We evidence the key bifurcations leading to these chaotic CSs. Oscillatory dynamics of localized structures has been predicted to occur in systems without optical feedback: in the Lugiato–Lefever model [31] of driven optical nonlinear cavity [32–34] and in a model of a VCSEL with a saturable absorber 0146-9592/14/164739-04$15.00/0
that goes beyond the mean-field approximation [35]. Localized structures, which become chaotic by period doubling have been predicted for a driven damped nonlinear Shrödinger equation [36,37], for a Josephson junction ladder and for a forced and damped van der Pol model [38]. Oscillatory dynamics of localized structures has been experimentally observed in an optically pumped VCSEL with a saturable absorber [39] and in a driven nonlinear fiber cavity [40]. Experimental studies of devices based on liquid crystals have shown that nonvariational effects can give rise to chaotic behavior of multiple-peaks localized solutions [41]. In all of these studies, time delay feedback was absent. To the best of our knowledge, the impact of optical delay feedback on the dynamics of the CSs in a broad area semiconductor laser with saturable absorber has not been reported. We consider the mean field model describing the space-time evolution of a broad area VCSEL with saturable absorption [16] and modify it by adding a delay optical feedback from a distant mirror in a self-imaging configuration, i.e., light diffraction in the external cavity is compensated (Fig. 1). As in [28], the optical feedback is considered for one round-trip in the external cavity according to [42] and leading to the following dimensionless partial differential equations:
Fig. 1. Schematic of broad-area VCSEL with both gain and saturable absorption sections and with an external cavity in a self-imaging configuration. © 2014 Optical Society of America
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dE 1 − jαN 1 − jβn − 1 j∇2⊥ E ζejϕ Et − τ; dt (1)
(3)
Fig. 2. Time traces of the CS intensity jExj2 for four different strengths of the optical feedback: (a) ζ 0.005, (b) ζ 0.015, (c) ζ 0.02, and (d) ζ 0.034.
is continuous and smooth, and the CS does not spread considerably as its peak amplitude changes in time. To better analyze transitions between different dynamical regimes of a single CS, we calculate the bifurcation diagram of the extrema of the CS peak amplitude as a function of the feedback strength. This analysis is summarized in Fig. 3(A). For small feedback strength ζ, the single CS is stationary in time as shown in zone (a) of Fig. 3(A). When increasing ζ, the CS becomes time dependent with a peak amplitude changing in a regular way corresponding to period one dynamics [zone (b) of Fig. 3(A)]. As we increase further ζ, a period doubling regime appears as shown in Fig. 3(A), zone (c). Two windows of spatially localized chaotic dynamics are clearly observed in Fig. 3(A). Figure 4 shows time traces (left column) and corresponding power spectra (right column) of the CS intensity for five different strengths of the optical feedback as denoted in Fig. 3(a). For low feedback strength of
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Here E is the slowly varying mean electric field envelope. N (n) is related to the carrier density, α (β) is the linewidth enhancement factor, and b1 (b2 ) is the ratio of photon lifetime to the carrier lifetime in the active layer (saturable absorber) (normalization is the same as in [16]). μ is the normalized injection current in the active material, γ measures absorption in the passive material, and s a2 b1 ∕a1 b2 is the saturation parameter with a12 the differential gain of the active (absorptive) material. The diffraction of intracavity light E is described by the Laplace operator ∇2⊥ acting on the transverse plane x; y, and carrier diffusion and bimolecular recombination are neglected. Time and space are scaled to the photon lifetime τp and diffraction length, respectively. The feedback is characterized by the time delay τ0 2Lext ∕c, the feedback strength ζ kf ∕κ, and phase ϕ ωτ0 . Here Lext is the external cavity length; c is the speed of light; and kf 1 − r 2 r ext ∕rτin , where r and r ext the VCSEL output mirror and external mirror amplitude reflectivities and κ 1∕τp is the photon decay rate. τin 2LC nC ∕c is the VCSEL cavity round-trip time with LC and nC the VCSEL effective cavity length and refractive index, respectively. Note that, as time is scaled to the photon lifetime in our model, τ in Eqs. (1–3) is in units of the photon lifetime τp , i.e., τ τ0 κ. We consider the same laser parameters as in [16]: α 2, β 0, b1 0.04, b2 0.02, γ 0.5, s 10, μ 1.42 and optical feedback with a time delay of τ 100 and phase ϕ 0. The current is chosen such that the laser without optical feedback resides in a bistable region between the zero homogeneous solution (E 0, p N μ, n −γ) and the lasing solution (E I eiwt , N μ∕1 I, n −γ∕1 sI). For this choice of parameters the upper branch exhibits a subcritical Turing type of bifurcation allowing for the formation of CSs [16]. We integrate numerically Eqs. (1)–(3) by using the standard split-step method with periodic boundary conditions. We fix all parameters mentioned above, and vary the feedback strength ζ. For a small ζ, the CS is stationary in time and the intensity profile jExj2 is localized as shown in Fig. 2(a). When increasing ζ, the CS exhibits regular time oscillations and displays period one dynamics as shown in Fig. 2(b). The frequency of the oscillations close to the onset of period one dynamics is measured to be ν ≈ 1∕72, which is quite close to the relaxation oscillations of the laser (at this injection current νRO ≈ 1∕74). Increasing further the feedback strength brings the CS to period two dynamics [Fig. 2(c)]. Finally, Fig. 2(d) presents an example of chaotic localized dynamics. The transition between different CS profiles
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Fig. 3. Bifurcation diagrams: (A) 1D CS solution of Eqs. (1)– (3) and (B) homogeneous solution of a small aperture laser.
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Fig. 4. Time traces (left column) and corresponding power spectra (right column) of the CS peak intensity jEj2max for different strengths of the optical feedback: (a) ζ 0.005, (b) ζ 0.015, (c) ζ 0.022, (d) ζ 0.0265, and (e) ζ 0.034.
ζ 0.005, the CS is stationary [Fig. 4(a), left column)] with strong zero frequency component and monotonically decreasing intensity in the power spectrum [Fig. 4(a), right column]. With increasing the feedback strength, ζ 0.015, the CS peak amplitude starts oscillating while it preserves the same spatial position [Fig. 4(b), left column]. The power spectrum reveals two strong peaks at the frequency of oscillations f ≈ f RO and its double f ≈ 2f RO [Fig. 4(b), right column]. For stronger feedback of ζ 0.022, the CS experiences a period doubling bifurcation [Fig. 4(c), left column] with power spectrum revealing components at half the fundamental frequency of oscillations and its multiples [Fig. 4(c), right column]. Period quadrupling is shown in [Figs. 4(d), left column, 4(d), right column] for ζ 0.0265. Finally, for feedback strength of ζ 0.034 the CS peak amplitude changes chaotically [Fig. 4(e), left column] with broad powers spectrum [Fig. 4(e), right column]. Despite the complicated chaotic dynamics, the single CS remains, firing a single spatially localized structure only. It does not broaden in spite of the available space along the transverse direction as evidenced in Fig. 2. To understand the link between the threshold and the frequency affecting the spatial profile of a single CS, we draw a bifurcation diagram of the homogeneous lasing solution of small aperture laser calculated by neglecting diffraction in Eq. (1); see Fig. 3(b). The obtained bifurcation diagram resembles the one of the single peak CS dynamics with Hopf and period doubling bifurcations occurring at similar but not exactly the same feedback strengths. Therefore, the CS complex dynamics seems to have its origin in the dynamics of the homogeneous lasing state, i.e., the CS are paths of such a state connected to the stable zero (nonlasing) state. We have checked that the main features of the reported dynamics are robust to noise. For example, in the case of Fig. 4(a)–4(d), adding to Eq. (1) a noise source with a mean value of zero and a strength of up to 5 × 10−4 preserves the main features of the dynamic, despite the peak’s heights in the time traces having somewhat varying amplitude. Thus, the spectra reported in
Fig. 5. Time traces of the peak intensities jEj21;2 max of two CSs separated by Δx 20. The first CS (blue dotted line) is settled at a period two dynamics. The second CS (red solid line) is written by injecting a Gaussian beam with beam waist of winj 2 and amplitude ainj 1.5 at t 2490 for Δt 200. The optical feedback rate is ζ 0.02.
Fig. 4, panel 2, show the same peaks, slightly broadened with respect to the noiseless case. We demonstrate in Fig. 5 that more than one CS can be simultaneously present in the system and experience the same kind of dynamics, however not synchronized with each other. As can be seen from the time traces of the peak intensities, a second CS, separated by a distance Δx 20 from the first one can be written by injecting an appropriate Gaussian beam and then settle to the same kind of period two dynamics. However, as the time of writing is taken arbitrary, there is a certain time shift of the two time traces, i.e., the two CSs do not synchronize their temporal dynamics. The reported dynamics always occurs, provided that Δx is larger than a certain critical distance ΔxC (for the parameters of Fig. 5 ΔxC 10.5). For Δx < ΔxC the two CSs merge into one (after a certain transient time of coexistence) and the final soliton then experiences the same kind of dynamics as if it were a single one from the beginning. a) 1 0.5
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Fig. 6. Two-dimensional simulations for μ 1.45: (left panel) time traces of the CS peak intensity for the cases of period one (ζ 0.01; ϕ 0), period two (ζ 0.024; ϕ 0), and chaotic dynamics (ζ 0.028; ϕ 1) and (right panel) snapshots of chaotic CS at three different times denoted in (c).
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Finally, Fig. 6 demonstrates that the time-delayed feedback induced complex temporal dynamics also takes place in two dimensional settings by showing three time traces of the CS peak intensity for the cases of period one, period two, and chaotic dynamics. Snapshots of a chaotic two-dimensional CS at three different times denoted in Fig. 6(c) are shown in the right panel of Fig. 6. In conclusion, we have numerically demonstrated a complex temporal dynamics of cavity solitons in a broad area VCSEL with saturable absorber when subject to time-delayed optical feedback. We show that the inclusion of feedback leads to period doubling route to chaos of a single or more spatially localized light structures. K. P. acknowledges the support of Fonds Wetenschappelijk Onderzoek—Vlaanderen, Geconcerteerds Onderzoeksactie and Onderzoeksraand of Vrije Universiteit Brussel. M. T. is a Senior Research Associate with the Fonds de la Recherche Scientifique F.R.S.-FNRS, Belgium. The financial support of the Interuniversity Attraction Poles program of Belgian Science Policy Office under grant IAP 7-35 is also acknowledged. References 1. L. Lugiato, Chaos, Solitons Fractals 4, 1251 (1994). 2. N. Rosanov, Prog. Opt. 35, 1 (1996). 3. K. Staliunas and V. J. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer, 2003). 4. Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003). 5. P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004). 6. T. Ackemann, W. Firth, and G. Oppo, Adv. At. Mol. Opt. Phys. 57, 323 (2009). 7. M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994). 8. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. Lugiato, Phys. Rev. A 58, 2542 (1998). 9. U. Bortolozzo, M. Clerc, and S. Residori, New J. Phys. 11, 093037 (2009). 10. V. Taranenko, I. Ganne, R. Kuszelewicz, and C. Weiss, Phys. Rev. A 61, 063818 (2000). 11. S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, Nature 419, 699 (2002). 12. X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, IEEE J. Sel. Top. Quantum Electron. 12, 339 (2006). 13. X. Hachair, G. Tissoni, H. Thienpont, and K. Panajotov, Phys. Rev. A 79, 011801(R) (2009). 14. E. Averlant, M. Tlidi, H. Thienpont, T. Ackemann, and K. Panajotov, Opt. Express 22, 762 (2014). 15. S. Fedorov, A. Vladimirov, G. Khodova, and N. Rosanov, Phys. Rev. E 61, 5814 (2000).
16. M. Bache, F. Prati, G. Tissoni, R. Kheradmand, L. Lugiato, I. Protsenko, and M. Brambilla, Appl. Phys. B 81, 913 (2005). 17. P. Genevet, S. Barland, M. Giudici, and J. Tredicce, Phys. Rev. Lett. 101, 123905 (2008). 18. T. Elsass, K. Gaulthron, G. Beaudoin, I. Sagnes, R. Kuzelewicz, and S. Barbay, Appl. Phys. B 98, 327 (2010). 19. D. Lenstra, B. Verbeek, and A. den Boef, IEEE J. Quantum Electron. 21, 674 (1985). 20. J. Mork, B. Tromborg, and P. Christiansen, IEEE J. Quantum Electron. 24, 123 (1988). 21. Y. Chung and Y. Lee, IEEE Photon. Technol. Lett. 3, 597 (1991). 22. P. Besnard, M. L. Chares, G. M. Stephan, and F. Robert, J. Opt. Soc. Am. B 16, 1059 (1999). 23. M. Arteaga, H. Unold, J. Ostermann, R. Michalzik, H. Thienpont, and K. Panajotov, IEEE J. Quantum Electron. 42, 89 (2006). 24. M. Arteaga, M. Valencia, M. Sciamanna, H. Thienpont, M. Lopez-Amo, and K. Panajotov, Phys. Rev. Lett. 99, 023903 (2007). 25. K. Panajotov, M. Sciamanna, M. Arizaleta, and H. Thienpont, IEEE J. Sel. Top. Quantum Electron. 19, 1700312 (2013). 26. M. Tlidi, A. Vladimirov, D. Pieroux, and D. Turaev, Phys. Rev. Lett. 103, 103904 (2009). 27. S. V. Gurevich and R. Friedrich, Phys. Rev. Lett. 110, 014101 (2013). 28. K. Panajotov and M. Tlidi, Eur. Phys. J. D 59, 67 (2010). 29. M. Tlidi, E. Averlant, A. Vladimirov, and K. Panajotov, Phys. Rev. A 86, 033822 (2012). 30. A. Pimenov, A. Vladimirov, S. Gurevich, K. Panajotov, G. Huyet, and M. Tlidi, Phys. Rev. A 88, 053830 (2013). 31. L. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987). 32. A. Stroggie, W. Firth, G. McDonald, M. Tlidi, R. Lefever, and L. Lugiato, Chaos, Solitons Fractals 4, 1323 (1994). 33. W. Firth, G. Harkness, A. Lord, J. McSloy, D. Gomila, and P. Colet, J. Opt. Soc. Am. B 19, 747 (2002). 34. D. Turaev, A. Vladimirov, and S. Zelik, Phys. Rev. Lett. 108, 263906 (2012). 35. L. Columbo, F. Prati, M. Brambilla, and T. Maggipinto, Eur. Phys. J. D 59, 115 (2010). 36. K. Nozaki and N. Bekki, Physica D 21, 381 (1986). 37. I. Barashenkov, E. Zemlyanaya, and T. van Heerden, Phys. Rev. E 83, 056609 (2011). 38. P. Martinez, L. Floria, F. Falo, and J. Mazo, Europhys. Lett. 45, 444 (1999). 39. T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, and S. Barbay, Eur. Phys. J. D 59, 91 (2010). 40. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt. Express 21, 9180 (2013). 41. N. Verschueren, U. Bortolozzo, M. G. Clerc, and S. Residori, Phys. Rev. Lett. 110, 104101 (2013). 42. R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 1, 347 (1982).
4739
Chaotic behavior of cavity solitons induced by time delay feedback Krassimir Panajotov1,2,* and Mustapha Tlidi3 1
Brussels Photonics Team, Department of Applied Physics and Photonics (B-PHOT TONA), Vrije Universiteit Brussel, Pleinlaan 2, Brussels, Belgium 2
3
Institute of Solid State Physics, 72 Tzarigradsko Chaussee Blvd., 1784 Sofia, Bulgaria
Optique Nonlinéaire Théorique, Université Libre de Bruxelles (U.L.B.), CP 231, Campus Plaine, B-1050 Bruxelles, Belgium *Corresponding author: kpanajot@b‑phot.org Received May 9, 2014; revised June 27, 2014; accepted July 2, 2014; posted July 3, 2014 (Doc. ID 211857); published August 7, 2014 We investigate spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subject to time-delayed optical feedback. We show that the inclusion of feedback leads to a period doubling route to chaos of spatially localized light structures. © 2014 Optical Society of America OCIS codes: (140.1540) Chaos; (140.5960) Semiconductor lasers; (190.1450) Bistability; (190.3100) Instabilities and chaos; (190.6135) Spatial solitons; (140.7260) Vertical cavity surface emitting lasers. http://dx.doi.org/10.1364/OL.39.004739
Cavity solitons (CSs) are localized light structures in the transverse plane of externally driven nonlinear resonators (for reviews see [1–6]). They appear in the coexisting region of parameters involving a periodic pattern and a stable homogeneous steady state [7–9]. In the transverse plane of the resonator, they consist of one or more regions in patterned state surrounded by a region in the homogeneous steady state [7]. Because of the large Fresnel number, the short cavity, and the mature technology, vertical-cavity surface-emitting lasers (VCSELs) are the most advanced systems for CS studies and applications. Indeed, with the injection of a holding beam with appropriate frequency, CSs have been demonstrated in broad area VCSELs both below [10,11] and above [12] the lasing threshold. A spatially localized structure has also been discovered in 40 μm diameter VCSELs by using their unique polarization properties [13], as well as in 80 μm diameter VCSELs lasing on a high transverse-order flower mode [14]. Of particular interest are VCSELs with saturable absorption as an external holding beam is not necessary, which significantly simplifies the system [15]. Therefore, VCSELs with a saturable absorber have been widely studied recently both theoretically [15,16] and experimentally [17,18]. Delayed optical feedback has been shown to strongly impact modal and dynamical behavior of semiconductor lasers [19,20] and has been widely studied for the case of VCSELs [21–25]. Recently, the impact of delayed feedback on CS dynamics has been theoretically investigated for the cases of a driven nonlinear optical resonator [26,27] and a broad-area VCSEL [28–30]. It has been shown that the time-delayed feedback leads to a spontaneous CS motion after a certain threshold of the feedback parameters is passed, which in the VCSEL case, critically depends on the feedback phase. In this Letter, we investigate the evolution of the single CS toward temporal chaos. We evidence the key bifurcations leading to these chaotic CSs. Oscillatory dynamics of localized structures has been predicted to occur in systems without optical feedback: in the Lugiato–Lefever model [31] of driven optical nonlinear cavity [32–34] and in a model of a VCSEL with a saturable absorber 0146-9592/14/164739-04$15.00/0
that goes beyond the mean-field approximation [35]. Localized structures, which become chaotic by period doubling have been predicted for a driven damped nonlinear Shrödinger equation [36,37], for a Josephson junction ladder and for a forced and damped van der Pol model [38]. Oscillatory dynamics of localized structures has been experimentally observed in an optically pumped VCSEL with a saturable absorber [39] and in a driven nonlinear fiber cavity [40]. Experimental studies of devices based on liquid crystals have shown that nonvariational effects can give rise to chaotic behavior of multiple-peaks localized solutions [41]. In all of these studies, time delay feedback was absent. To the best of our knowledge, the impact of optical delay feedback on the dynamics of the CSs in a broad area semiconductor laser with saturable absorber has not been reported. We consider the mean field model describing the space-time evolution of a broad area VCSEL with saturable absorption [16] and modify it by adding a delay optical feedback from a distant mirror in a self-imaging configuration, i.e., light diffraction in the external cavity is compensated (Fig. 1). As in [28], the optical feedback is considered for one round-trip in the external cavity according to [42] and leading to the following dimensionless partial differential equations:
Fig. 1. Schematic of broad-area VCSEL with both gain and saturable absorption sections and with an external cavity in a self-imaging configuration. © 2014 Optical Society of America
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dE 1 − jαN 1 − jβn − 1 j∇2⊥ E ζejϕ Et − τ; dt (1)
(3)
Fig. 2. Time traces of the CS intensity jExj2 for four different strengths of the optical feedback: (a) ζ 0.005, (b) ζ 0.015, (c) ζ 0.02, and (d) ζ 0.034.
is continuous and smooth, and the CS does not spread considerably as its peak amplitude changes in time. To better analyze transitions between different dynamical regimes of a single CS, we calculate the bifurcation diagram of the extrema of the CS peak amplitude as a function of the feedback strength. This analysis is summarized in Fig. 3(A). For small feedback strength ζ, the single CS is stationary in time as shown in zone (a) of Fig. 3(A). When increasing ζ, the CS becomes time dependent with a peak amplitude changing in a regular way corresponding to period one dynamics [zone (b) of Fig. 3(A)]. As we increase further ζ, a period doubling regime appears as shown in Fig. 3(A), zone (c). Two windows of spatially localized chaotic dynamics are clearly observed in Fig. 3(A). Figure 4 shows time traces (left column) and corresponding power spectra (right column) of the CS intensity for five different strengths of the optical feedback as denoted in Fig. 3(a). For low feedback strength of
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Here E is the slowly varying mean electric field envelope. N (n) is related to the carrier density, α (β) is the linewidth enhancement factor, and b1 (b2 ) is the ratio of photon lifetime to the carrier lifetime in the active layer (saturable absorber) (normalization is the same as in [16]). μ is the normalized injection current in the active material, γ measures absorption in the passive material, and s a2 b1 ∕a1 b2 is the saturation parameter with a12 the differential gain of the active (absorptive) material. The diffraction of intracavity light E is described by the Laplace operator ∇2⊥ acting on the transverse plane x; y, and carrier diffusion and bimolecular recombination are neglected. Time and space are scaled to the photon lifetime τp and diffraction length, respectively. The feedback is characterized by the time delay τ0 2Lext ∕c, the feedback strength ζ kf ∕κ, and phase ϕ ωτ0 . Here Lext is the external cavity length; c is the speed of light; and kf 1 − r 2 r ext ∕rτin , where r and r ext the VCSEL output mirror and external mirror amplitude reflectivities and κ 1∕τp is the photon decay rate. τin 2LC nC ∕c is the VCSEL cavity round-trip time with LC and nC the VCSEL effective cavity length and refractive index, respectively. Note that, as time is scaled to the photon lifetime in our model, τ in Eqs. (1–3) is in units of the photon lifetime τp , i.e., τ τ0 κ. We consider the same laser parameters as in [16]: α 2, β 0, b1 0.04, b2 0.02, γ 0.5, s 10, μ 1.42 and optical feedback with a time delay of τ 100 and phase ϕ 0. The current is chosen such that the laser without optical feedback resides in a bistable region between the zero homogeneous solution (E 0, p N μ, n −γ) and the lasing solution (E I eiwt , N μ∕1 I, n −γ∕1 sI). For this choice of parameters the upper branch exhibits a subcritical Turing type of bifurcation allowing for the formation of CSs [16]. We integrate numerically Eqs. (1)–(3) by using the standard split-step method with periodic boundary conditions. We fix all parameters mentioned above, and vary the feedback strength ζ. For a small ζ, the CS is stationary in time and the intensity profile jExj2 is localized as shown in Fig. 2(a). When increasing ζ, the CS exhibits regular time oscillations and displays period one dynamics as shown in Fig. 2(b). The frequency of the oscillations close to the onset of period one dynamics is measured to be ν ≈ 1∕72, which is quite close to the relaxation oscillations of the laser (at this injection current νRO ≈ 1∕74). Increasing further the feedback strength brings the CS to period two dynamics [Fig. 2(c)]. Finally, Fig. 2(d) presents an example of chaotic localized dynamics. The transition between different CS profiles
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Fig. 3. Bifurcation diagrams: (A) 1D CS solution of Eqs. (1)– (3) and (B) homogeneous solution of a small aperture laser.
August 15, 2014 / Vol. 39, No. 16 / OPTICS LETTERS
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Fig. 4. Time traces (left column) and corresponding power spectra (right column) of the CS peak intensity jEj2max for different strengths of the optical feedback: (a) ζ 0.005, (b) ζ 0.015, (c) ζ 0.022, (d) ζ 0.0265, and (e) ζ 0.034.
ζ 0.005, the CS is stationary [Fig. 4(a), left column)] with strong zero frequency component and monotonically decreasing intensity in the power spectrum [Fig. 4(a), right column]. With increasing the feedback strength, ζ 0.015, the CS peak amplitude starts oscillating while it preserves the same spatial position [Fig. 4(b), left column]. The power spectrum reveals two strong peaks at the frequency of oscillations f ≈ f RO and its double f ≈ 2f RO [Fig. 4(b), right column]. For stronger feedback of ζ 0.022, the CS experiences a period doubling bifurcation [Fig. 4(c), left column] with power spectrum revealing components at half the fundamental frequency of oscillations and its multiples [Fig. 4(c), right column]. Period quadrupling is shown in [Figs. 4(d), left column, 4(d), right column] for ζ 0.0265. Finally, for feedback strength of ζ 0.034 the CS peak amplitude changes chaotically [Fig. 4(e), left column] with broad powers spectrum [Fig. 4(e), right column]. Despite the complicated chaotic dynamics, the single CS remains, firing a single spatially localized structure only. It does not broaden in spite of the available space along the transverse direction as evidenced in Fig. 2. To understand the link between the threshold and the frequency affecting the spatial profile of a single CS, we draw a bifurcation diagram of the homogeneous lasing solution of small aperture laser calculated by neglecting diffraction in Eq. (1); see Fig. 3(b). The obtained bifurcation diagram resembles the one of the single peak CS dynamics with Hopf and period doubling bifurcations occurring at similar but not exactly the same feedback strengths. Therefore, the CS complex dynamics seems to have its origin in the dynamics of the homogeneous lasing state, i.e., the CS are paths of such a state connected to the stable zero (nonlasing) state. We have checked that the main features of the reported dynamics are robust to noise. For example, in the case of Fig. 4(a)–4(d), adding to Eq. (1) a noise source with a mean value of zero and a strength of up to 5 × 10−4 preserves the main features of the dynamic, despite the peak’s heights in the time traces having somewhat varying amplitude. Thus, the spectra reported in
Fig. 5. Time traces of the peak intensities jEj21;2 max of two CSs separated by Δx 20. The first CS (blue dotted line) is settled at a period two dynamics. The second CS (red solid line) is written by injecting a Gaussian beam with beam waist of winj 2 and amplitude ainj 1.5 at t 2490 for Δt 200. The optical feedback rate is ζ 0.02.
Fig. 4, panel 2, show the same peaks, slightly broadened with respect to the noiseless case. We demonstrate in Fig. 5 that more than one CS can be simultaneously present in the system and experience the same kind of dynamics, however not synchronized with each other. As can be seen from the time traces of the peak intensities, a second CS, separated by a distance Δx 20 from the first one can be written by injecting an appropriate Gaussian beam and then settle to the same kind of period two dynamics. However, as the time of writing is taken arbitrary, there is a certain time shift of the two time traces, i.e., the two CSs do not synchronize their temporal dynamics. The reported dynamics always occurs, provided that Δx is larger than a certain critical distance ΔxC (for the parameters of Fig. 5 ΔxC 10.5). For Δx < ΔxC the two CSs merge into one (after a certain transient time of coexistence) and the final soliton then experiences the same kind of dynamics as if it were a single one from the beginning. a) 1 0.5
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Fig. 6. Two-dimensional simulations for μ 1.45: (left panel) time traces of the CS peak intensity for the cases of period one (ζ 0.01; ϕ 0), period two (ζ 0.024; ϕ 0), and chaotic dynamics (ζ 0.028; ϕ 1) and (right panel) snapshots of chaotic CS at three different times denoted in (c).
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Finally, Fig. 6 demonstrates that the time-delayed feedback induced complex temporal dynamics also takes place in two dimensional settings by showing three time traces of the CS peak intensity for the cases of period one, period two, and chaotic dynamics. Snapshots of a chaotic two-dimensional CS at three different times denoted in Fig. 6(c) are shown in the right panel of Fig. 6. In conclusion, we have numerically demonstrated a complex temporal dynamics of cavity solitons in a broad area VCSEL with saturable absorber when subject to time-delayed optical feedback. We show that the inclusion of feedback leads to period doubling route to chaos of a single or more spatially localized light structures. K. P. acknowledges the support of Fonds Wetenschappelijk Onderzoek—Vlaanderen, Geconcerteerds Onderzoeksactie and Onderzoeksraand of Vrije Universiteit Brussel. M. T. is a Senior Research Associate with the Fonds de la Recherche Scientifique F.R.S.-FNRS, Belgium. The financial support of the Interuniversity Attraction Poles program of Belgian Science Policy Office under grant IAP 7-35 is also acknowledged. References 1. L. Lugiato, Chaos, Solitons Fractals 4, 1251 (1994). 2. N. Rosanov, Prog. Opt. 35, 1 (1996). 3. K. Staliunas and V. J. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer, 2003). 4. Y. Kivshar and G. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, 2003). 5. P. Mandel and M. Tlidi, J. Opt. B 6, R60 (2004). 6. T. Ackemann, W. Firth, and G. Oppo, Adv. At. Mol. Opt. Phys. 57, 323 (2009). 7. M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994). 8. L. Spinelli, G. Tissoni, M. Brambilla, F. Prati, and L. Lugiato, Phys. Rev. A 58, 2542 (1998). 9. U. Bortolozzo, M. Clerc, and S. Residori, New J. Phys. 11, 093037 (2009). 10. V. Taranenko, I. Ganne, R. Kuszelewicz, and C. Weiss, Phys. Rev. A 61, 063818 (2000). 11. S. Barland, J. Tredicce, M. Brambilla, L. Lugiato, S. Balle, M. Giudici, T. Maggipinto, L. Spinelli, G. Tissoni, T. Knodl, M. Miller, and R. Jager, Nature 419, 699 (2002). 12. X. Hachair, F. Pedaci, E. Caboche, S. Barland, M. Giudici, J. R. Tredicce, F. Prati, G. Tissoni, R. Kheradmand, L. A. Lugiato, I. Protsenko, and M. Brambilla, IEEE J. Sel. Top. Quantum Electron. 12, 339 (2006). 13. X. Hachair, G. Tissoni, H. Thienpont, and K. Panajotov, Phys. Rev. A 79, 011801(R) (2009). 14. E. Averlant, M. Tlidi, H. Thienpont, T. Ackemann, and K. Panajotov, Opt. Express 22, 762 (2014). 15. S. Fedorov, A. Vladimirov, G. Khodova, and N. Rosanov, Phys. Rev. E 61, 5814 (2000).
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