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Periodic and chaotic solitons in a semiconductor laser with saturable absorber H. Vahed, F. Prati, M. Turconi, S. Barland and G. Tissoni Phil. Trans. R. Soc. A 2014 372, 20140016, published 22 September 2014
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Periodic and chaotic solitons in a semiconductor laser with saturable absorber H. Vahed1 , F. Prati2,4 , M. Turconi3,† , S. Barland3 rsta.royalsocietypublishing.org
and G. Tissoni3 1 School of Engineering Emerging Technologies, University of Tabriz,
Research Cite this article: Vahed H, Prati F, Turconi M, Barland S, Tissoni G. 2014 Periodic and chaotic solitons in a semiconductor laser with saturable absorber. Phil. Trans. R. Soc. A 372: 20140016. http://dx.doi.org/10.1098/rsta.2014.0016
One contribution of 19 to a Theme Issue ‘Localized structures in dissipative media: from optics to plant ecology’.
Tabriz, Iran 2 Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy 3 Université de Nice Sophia Antipolis, Institut Non Linéaire de Nice, CNRS UMR 7335, 1361 Route des Lucioles, 06560 Valbonne, France 4 CNISM, Research Unit of Como, Via Valleggio 11, 22100 Como, Italy In a semiconductor laser with saturable absorber, solitons may spontaneously drift and/or oscillate. We study three different regimes characterized by strong intensity oscillations, both periodic and chaotic. We show that (i) soliton dynamics may be similar to that of passively Q-switched lasers, (ii) solitons may drift and oscillate simultaneously, and (iii) chaotic solitons may coexist with stationary ones and with the laser off solution.
Subject Areas: optics Keywords: solitons, passive Q-switching, chaos Author for correspondence: F. Prati e-mail: [email protected]
†
Present address: Observatoire de la Côte d’Azur-Laboratoire ARTEMIS, Boulevard de l’Observatoire CS 34229, 06304 Nice, France.
1. Introduction Dissipative solitons are a universal feature of nonlinear science [1,2]. Although they appear in many different systems such as vertically driven layers of sand [3], highly dissipative fluids [4] and biology [5], they have been studied most extensively in optics. A particular type of optical solitons are the so-called cavity solitons (CSs), two-dimensional spatial localized structures that arise in the plane perpendicular to the light propagation direction in nonlinear optical systems driven by an external coherent field [6,7]. CSs can also be produced in nonlinear cavities which are bistable even in the absence of a driving field. Such systems are called CS lasers [8]. The interest in CSs stems from their promising applications in optical information processing and storage. Under this respect, the functionalities of CSs
2014 The Author(s) Published by the Royal Society. All rights reserved.
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We consider a vertical-cavity surface-emitting laser with an integrated absorber whose dynamics in the transverse plane is described by the equations [29]
and
2 ]F, F˙ = [(1 − iα)D + (1 − iβ)d − 1 + i∇⊥
(2.1)
˙ = −b[D(1 + |F|2 ) + B1 D2 − μ] D
(2.2)
d˙ = −rb[d(1 + s|F|2 ) + B2 d2 + γ ].
(2.3)
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2. The model
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are enhanced if they can move in the transverse plane, oscillate at fixed position or even oscillate while they move. For instance, the mobility of CSs under an external control beam was used to implement an all-optical delay line [9] and a low-energy optical switch [10]. An even larger amount of information could be probably stored in the CSs if they were oscillating localized structures and their dynamics could be controlled by varying an external parameter (e.g. the pump). Spontaneous oscillations can appear in solitons as a result of an Andronov–Hopf bifurcation of the stationary solitons. In optics, this was shown in systems with a saturable defocusing nonlinearity [11], a saturable absorber [12,13] and a Kerr nonlinearity [14,15]. More generally, oscillating solitons were demonstrated in reaction–diffusion systems [16], in a magnetic wire forced by a transversal uniform and oscillatory magnetic field [17] and in parametric systems where one parameter is modulated periodically [18,19]. Here, we consider a CS laser formed by a laser with saturable absorber. CSs have been observed in such systems both in extended [20–22] and monolithic resonators [23,24]. As long as the simple single-mode temporal dynamics is considered, it is known that the Andronov–Hopf bifurcation in a laser with saturable absorber leads to passive Q-switching, where short pulses are produced with a repetition time of about 1 ns [25]. The main idea of the present paper is to combine the subcritical bifurcation of a laser with saturable absorber, which allows for the existence of CSs, with its intrinsic dynamical behaviour owing to the Andronov–Hopf bifurcation to create passively Q-switched CSs, i.e. pulses that are localized both in space and in time. Being pulses separated by long intervals during which the field intensity is close to zero, these dynamical CSs differ from the oscillating solitons predicted in a similar system in [13]. They may be instead related to the observations of [24,26], although in those experiments it was not possible to demonstrate that the self-pulsing structures coexist with the laser off solution. In a system with translational symmetry, it is possible to switch on at the desired positions several passively Q-switched CSs thus forming an array of localized Q-switched lasers that coexist in the same device. The natural question is how do they interact one with the other. Although the interaction may occur only during the short duration of the pulses, it can be effective, because the pulses are very intense. We expect to observe pulse synchronization, because the interaction is nonlinear [27], and the appearance of forces related to phase locking, because the localized lasers, unlike in conventional arrays, are free to move [28]. In this paper, we also show that other dynamical CSs may exist in a laser with saturable absorber under different parametric conditions. Pulsating CSs drifting in a non-uniform way appear when carrier dynamics is much slower in the absorber than in the amplifier. When radiative recombination is included in the model equations, a wide branch of stable chaotic CSs similar to those found in [24] is predicted. In §2, we present the dynamical model and its stationary solutions, homogeneous and localized. In §3, we calculate the existence domain of self-pulsing CSs and show how their intensity and period depend on the pump parameter. The interaction of four passively Q-switched CSs in a 2 × 2 array is analysed in §4. The travelling pulsating CSs are shown in §5, whereas the chaotic CSs are characterized in §6. The conclusions are drawn in §7.
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Figure 1. Peak intensity of stable stationary CSs (filled circles) and unstable homogeneous steady state solution (dashed line) as a function of μ.
The dimensionless variables F, D and d are respectively the slowly varying envelope of the electric field, the carrier density of the amplifier and the carrier density of the absorber. The parameters α, β are the linewidth enhancement factors in the amplifier and in the absorber, respectively; μ is the pump parameter, γ is the unsaturated absorption, s is the saturation parameter, i.e. the ratio of saturation intensity in the amplifier and in the absorber. Time is scaled to the cavity lifetime (1 time unit (t.u.) ∼ a few picoseconds), b is the ratio of photon lifetime to carrier lifetime in the amplifier and r is the ratio of carrier lifetimes in the amplifier and in the absorber. Space is scaled to the diffraction length (1 space unit (s.u.) ∼ a few micrometres). The coefficients B1 and B2 are the rates of radiative recombination. For a precise definition of all variables and parameters, see the appendix of Vahed et al. [10]. The analysis of equations (2.1)–(2.3) is simpler if radiative recombination is neglected setting B1 = B2 = 0 as in [30]. The homogeneous stationary solution is bistable if the condition s > 1 + 1/γ is satisfied. In the plane wave approximation (i.e. when the variables are assumed uniform in the transverse field, and the transverse Laplacian is neglected), part of the upper branch of the homogeneous solution is destabilized through an Andronov–Hopf bifurcation provided the ratio r of carrier lifetimes in the two media is larger than a critical value rc which is approximately given, in the limit b 1, by √ √ s−1+ γ . (2.4) rc = √ γs Beyond the plane wave approximation, localized structures (CSs) are also solutions of equations (2.1)–(2.3) provided r < rc , i.e. the upper branch is stable with respect to the Andronov– Hopf bifurcation. In this case, the transverse Laplacian induces instead a modulational instability in the upper branch, associated with a real eigenvalue, which extends to the whole branch if α > β. This makes possible the coexistence of the CSs with the laser off solution for a range of pump values below the laser threshold μthr = 1 + γ . By choosing s = 10 and γ = 0.5, we have rc 0.58. Figure 1 shows the branch of stable CSs for r = 0.45, and b = 0.01, α = 2, β = 0.2. The CSs can either arise spontaneously or be excited in a predetermined position using different techniques [31]. Optical pumping allows for an optical control of carrier distribution in the transverse profile and incoherent switching [23,24]. Here, we consider instead coherent switching by a Gaussian pulse of width w centred in (x0 , y0 ) Finj = F0 e−[(x−x0 )
2
+(y−y0 )2 ]/2w2
,
(2.5)
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stationary CS 1.43
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Figure 2. Stability diagram of CSs in the (μ, r) plane. The horizontal dashed line corresponds to the value r = 0.5 used in figure 3.
which is injected into the laser for a time interval tinj . No explicit time dependence appears in Finj, because the pulse is assumed to be resonant with the cold cavity whose frequency is assumed as reference frequency; the amplitude F0 is real, its phase is irrelevant because the CSs are (blue) detuned with respect to the cold cavity. For a given value of the pump parameter μ below laser threshold, the CSs are stable only for a finite range of r. The stability domain is limited from below by a drift instability originated by the Goldstone modes of the system, which induce a spontaneous motion of the CSs. The properties of these moving CSs have already been investigated in the presence of circular [32] and squared [33] reflecting boundaries. The upper boundary of the stability domain is instead associated with an Andronov–Hopf bifurcation whose effect is to makes the CSs oscillate without a translational motion. Figure 2 shows the stability domain of the oscillating CSs in the plane (μ, r). Below the lower boundary, the stationary CSs are stable. As the upper boundary is crossed, the laser either jumps to the off state for small μ or develops complex spatio-temporal dynamics for larger μ.
3. Localized passive Q-switching The CS oscillations present the typical features of passive Q-switching in a laser with saturable absorber [34]. Yet, a notable difference with respect to [34] is that the Q-switched CSs are bistable with the laser off solution. This occurs, even in the plane wave model, when r < 1. Let us fix r = 0.5 (dashed line in figure 2) and vary μ from right to left crossing the two boundaries of the stability region of oscillating CSs. As shown in figure 3, after crossing the right boundary, which is placed at μ 1.4635, the CS starts oscillating, and already very close to threshold the oscillations are highly nonlinear, because the action of the saturable absorber favours the formation of narrow pulses between which the emitted intensity falls very close to zero. A portion of the pulse train for μ = 1.45 is shown in the inset. By further decreasing the pump parameter μ, the amplitude of the pulses increases slowly, whereas the period grows more rapidly as μ approaches the right boundary of the stability domain, μ 1.4472.
4. 2 × 2 arrays of passively Q-switched solitons Once localized Q-switching is demonstrated, it is interesting to study how several passively Q-switched CSs interact. To this aim, we analysed 2 × 2 arrays of such structures for μ = 1.47 and
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Figure 3. (a) Maximum and minimum value of the peak intensity of an oscillating CS as function of μ for r = 0.5. The inset shows a time trace for μ = 1.45. (b) Corresponding period of the intensity oscillation.
r = 0.51. The dynamics of the CSs can be monitored by calculating their positions and the field intensity and phase at the peak. In this way, we can determine whether the CSs are correlated in intensity and/or phase, and if their mutual distances evolve in time under the action of some force. The results can be summarized by plotting the four intensities, the relative phase of two CSs on the same side and on a diagonal and the average distance between first neighbours. We considered an array where the initial distance between the first neighbours, i.e. two CSs on the same side of the square, is 22 s.u., and analysed the time evolution of the CSs both when they are switched on simultaneously and when they are switched on in sequence. The time evolution of the array when the CSs are switched on simultaneously is shown in the left column of figure 4. The pulses, which are initially perfectly synchronized and locked in phase, lose synchronization and phase-locking at about t = 3000 t.u. Yet, phase-locking and synchronization are re-established at about t = 30 000 t.u. The average distance between first neighbours at the beginning falls slightly below the initial value of 22 s.u., but very soon it starts to grow at an almost constant rate independent of phase-locking and synchronization. In the phase-locked state, two CSs placed on the same side of the square are locked at π , whereas two CSs on a diagonal are locked at 0. This configuration agrees with the findings of [28] where it was shown that the relative phase of two stationary CSs behaves differently depending on their distance. For moderate distances, the CSs initially lock at π , although for longer times they partially unlock and reach a state of phase entrainment, where the relative phase remains bounded but it oscillates around π . For larger distances, the stationary CSs lock in phase. In the right column of figure 4, the time evolution of the array is shown when the four CSs are switched on in the same positions as before, but with a delay of 500 t.u. one from the other. Initially, the pulses are well separated, but at t = 10 000 t.u., the delays among pulses are comparable to those of the previous case, and again phase-locking and pulse synchronization are achieved at about t = 30 000 t.u. The average distance of first neighbours is never smaller than 22 s.u., and it grows at the same rate found when the CSs are simultaneously switched on. These results indicate that the CSs interact at very large distance, much larger than one soliton diameter and even larger than its phase profile [28]. Phase-locking is robust against noise, that
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Figure 4. Dynamics of a 2 × 2 array of passively Q-switched CSs for μ = 1.47 and r = 0.51. In (a–f ), the time evolution of the peak intensity of the fours CSs is represented by different colours (black, red, green, blue). (a,c,e,g,i) The CSs are switched on simultaneously. (b,d,f ,h,j) The CSs are switched on in sequence at intervals of 500 t.u. (a–f ) Intensity of the four CSs at different stages of the evolution. (g,h) Relative phase of two CSs belonging to the same row of the array (φs , red) and placed on the diagonal (φd , black). (i,j) Average distance of the first neighbours. (Online version in colour.)
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(a)
(b)
(c)
5. Stop and go solitons By decreasing by an order of magnitude the ratio r of carrier lifetimes, i.e. assuming that carriers are much slower in the absorber, a new phenomenon is observed. As in [13,32], the localized structures are subjected to a drift instability which makes them move along straight lines whose direction, if not controlled by means of an address beam, is random. Yet, unlike in [32], the peak intensity of the CS pulsates while it moves, and the motion is not uniform. An example is shown in figure 5 which captures the motion of the CS at three different times. Figure 5a,c shows two consecutive pulses, taken at their maximum intensity, which is about |F|2 = 0.46. In figure 5b, the intensity is minimum, and it is about |F|2 = 0.017 at the peak. Here, the intensity profile is much broader than at the maxima. Note that the positions of the intensity peak at the minimum (figure 5b) and at the second maximum (figure 5c) coincide. Initially, the CS moves rapidly but it is damped because carrier dynamics, in particular in the absorber, is too slow to follow that fast motion. Then, the motion stops while the CS is regenerated, and only when the CS recovers its maximum intensity does it start again to move. This behaviour is a further manifestation of the dual wave–material character of the CS, already stressed in [36]. The self-propelled solitons lose radial symmetry and they are elongated in the direction of motion [32,37], as can be better appreciated in the central density plot of figure 5. The time interval between the first and the second snapshots is 4718 t.u., and the distance between the maxima of the two intensity distributions is 4.5 s.u. Assuming (1 s.u.)/(1 t.u.) = 1000 km s−1 = 1000 µm ns−1 , the speed of the CS is about 1 µm ns−1 , comparable to that of [32].
6. Chaotic solitons Here, we take into account the effects of carrier recombination and consider the parameter set B1 = B2 = 0.1, s = 1, γ = 2, α = 2, β = 1, b = 0.01 and r = 1 [10,28,29,31–33]. For this set of parameters, the stationary CSs are stable for a wide range of pump current values μ (see fig. 2 of [32]). We focus on the complex spatio-temporal dynamical state which is usually observed beyond the laser threshold μthr and find that it can coexist with the stationary CS. Therefore, there exists a large region of coexistence of three stable states in the parameter space: — laser off state, stable between μ = 0 and μthr , — stationary CS, stable between μ = 4.56 and μ = 5.1, and
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was always included in our simulations. This phenomenon could be related to the enhancement of solitons’ interaction owing to the Andronov–Hopf bifurcation that makes them oscillate, which was demonstrated for Kerr CSs in [35].
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Figure 5. Three consecutive snapshots of the intensity in the transverse section of the laser illustrating self-propelled Q-switching. (a) t = 0, (b) t = 4718 and (c) t = 8192. The grey scale ranges 0–0.46 in (a) and (c), and 0–0.017 in (b).
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Figure 6. (a) Branch of the chaotic state (maximum intensity) as a function of the pump current μ. Points indicate the mean values and bars the standard deviations. (b) Snapshot of a numerical simulation showing the intensity profile, obtained for μ = 6. (Online version in colour.)
— extended spatio-temporal chaos: the field intensity is rapidly varying in space and time, spanning between almost 0 and a very high maximum value, the typical spatial size of the chaotic pattern being of the order of the CS dimension.
In figure 6a, we display for a wide interval of pump current μ the mean values and the standard deviations of the maximum intensity of the chaotic state, whose typical instantaneous intensity profile is shown in figure 6b, for μ = 6. To the left of the branch, the chaotic solution loses its stability in favour of the laser off solution. To the right of the branch, it seems to remain stable indefinitely. In this regime of coexistence between the homogeneous off solution and the extended chaotic state, we looked for chaotic localized states (the so-called chaoticons), as recently reported experimentally and numerically in a different system [38]. To this aim, we started from the laser off solution and injected a localized Gaussian writing beam of very strong amplitude F0 = 15 (much higher than in the case of CS switching) and width w = 2 s.u. during 50 t.u. When the writing beam is switched off, the system spontaneously evolves towards a CS that displays chaotic internal dynamics. Figure 7 shows the instantaneous transverse distribution of the field intensity and carrier density in the amplifier for one of these structures obtained for μ = 4.92. We verified the stability and localization of this chaotic CS for very long integration times (50 000 t.u.), and different values of the pump current μ. The CS remains stable for 4.67 ≤ μ ≤ 4.94. If μ is further reduced to (or below) the value of μ = 4.65 its internal dynamics starts freezing and finally it collapses onto a stationary CS (figure 8a). On the other hand, when μ is increased up to (or above) μ = 4.96, the chaotic CS starts expanding very slowly and finally the extended chaotic state is reached. The effective independence, multistability and localization of the chaotic CS were checked by creating two chaotic CSs in the transverse plane (figure 8b) and by switching on simultaneously a stationary and a chaotic CS. All these simulations were performed keeping equal the carrier lifetimes in the amplifier and in the absorber (r = 1). Actually, the stability of the chaotic CS depends on r, in a very similar way as the stationary CS (see for example fig. 2 of ref. [32]). By keeping μ fixed at the value of 4.85 and by reducing r, at r = 0.7 the chaotic CS collapses to a CS which then starts moving but dies very soon. We did not observe stable drifting chaotic CSs. For values of r higher than or equal to 1.8, the chaotic CS starts to expand very slowly and finally the extended chaotic state is reached.
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The same happens if we try to switch on a larger chaotic CS by increasing the width w of the writing beam. The CS apparently remains localized, but actually it expands very slowly and finally it invades the whole integration grid becoming the extended chaotic state. The entire process of completely filling the integration grid can take up to 10 000 t.u. This feature makes our chaotic CS deeply different from the chaoticon described in [38]. In that case, the chaotic structures are shown to remain localized independently of their size, that is, they do not vary their initial size. The origin of this sharply different behaviour is still unknown but even with very large integration windows (thus making it impossible for the initial localized structure to interact with its image through the boundaries) in our simulations localization is lost if the CS’s size passes a critical value of about two to three stationary CS diameters.
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In this paper, we have predicted the existence of different kinds of dissipative spatial solitons in a semiconductor laser with saturable absorber where the field intensity displays strong oscillations in time. When the carrier lifetimes in the amplifier and in the absorber are comparable, an Andronov–Hopf instability of the stationary solitons leads to the appearance of what we called passively Q-switched solitons, which present features very similar to those of passive Q-switching extensively studied in the time domain. The fact that passive Q-switching is localized in space allows studying the interaction of different oscillating solitons which coexist in the same laser. In a 2 × 2 array, we have shown that in the long-term evolution the four solitons synchronize their intensity pulses and lock their phases in such a way that two solitons in the same row or column have a phase difference of π , whereas two solitons on the diagonal have the same phase. The four solitons also experience a repulsive force which makes them move apart one from the other. If the carriers in the absorber are much slower than in the amplifier a different instability causes the soliton to drift along straight lines in a stop and go manner, so that while it moves its peak intensity decreases until it reaches a minimum, then it stops until it recovers the maximum intensity, and then it moves again along the same line. Chaotic solitons appear when radiative recombination is included in the model equations and the laser off state is perturbed with a writing beam whose intensity is much larger than that needed to excite a stationary soliton. These states are a portion of an extended chaotic state that coexists with the laser off solution. Because with the same parameters the stationary solitons are also stable, chaotic and stationary solitons may coexist in the same laser. Funding statement. F.P. acknowledges support from the Italian Ministry of Research (MIUR) through the ‘Futuro in Ricerca’ FIRB-grant PHOCOS-RBFR08E7VA. S.B. and G.T. acknowledge support from the ANR-Jeunes Chercheurs Project MOLOSSE (ANR-12-JS04-0002-01, www.molosse.org) and from the CNRS-Region PACA Project E-Buffo.
References 1. Akhmediev N, Ankiewicz A (eds). 2005 Dissipative solitons. Berlin, Germany: Springer. 2. Akhmediev N, Ankiewicz A (eds). 2008 Dissipative solitons: from optics to biology and medicine. Berlin, Germany: Springer. 3. Umbanhowar PB, Melo F, Swinney HL. 1986 Localized excitations in a vertically vibrated granular layer. Nature 382, 793–796. (doi:10.1038/382793a0) 4. Lioubashevski O, Arbell H, Fineberg J. 1996 Dissipative solitary states in driven surface waves. Phys. Rev. Lett. 76, 3959–3962. (doi:10.1103/PhysRevLett.76.3959) 5. Lejeune O, Tlidi M, Couteron P. 2002 Localized vegetation patches: a self-organized response to resource scarcity. Phys. Rev. E 66, 010901. (doi:10.1103/PhysRevE.66.010901) 6. Barland S et al. 2002 Cavity solitons as pixels in semiconductor microcavities. Nature 419, 699–702. (doi:10.1038/nature01049) 7. Lugiato LA. 2003 Introduction to the feature section on cavity solitons: an overview. IEEE J. Quantum Electron. 39, 193–196. (doi:10.1109/JQE.2002.807195) 8. Ackemann T, Firth W, Oppo G-L. 2009 Fundamentals and applications of spatial dissipative solitons in photonic devices. Adv. Atom. Mol. Opt. Phys. 57, 323–421. (doi:10.1016/ S1049-250X(09)57006-1) 9. Pedaci F et al. 2008 All-optical delay line using semiconductor cavity solitons. Appl. Phys. Lett. 92, 011101. (doi:10.1063/1.2828458) 10. Vahed H, Prati F, Tajalli H, Tissoni G, Lugiato LA. 2012 Low-energy switch based on a cavity soliton laser. Eur. Phys. J. D 66, 148. (doi:10.1140/epjd/e2012-30109-2)
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7. Conclusion
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We also stress the very large spatial coherence that characterizes all our structures, owing to the huge phase profile which extends to regions where the field intensity is very close to 0 (see, for example, the phase profile of the stationary CS shown in fig. 2 of [28]).
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11. Michaelis D, Peschel U, Lederer F. 1998 Oscillating dark cavity solitons. Opt. Lett. 23, 1814–1816. (doi:10.1364/OL.23.001814) 12. Vladimirov AG, Fedorov SV, Kaliteevskii NA, Khodova GV, Rosanov NN. 1999 Numerical investigation of laser localized structures. J. Opt. B 1, 101–106. (doi:10. 1088/1464-4266/1/1/019) 13. Rozanov NN, Fedorov SV, Shatsev AN. 2001 Pulsating solitons in a laser with relaxation of gain and saturable absorption. Opt. Spectrosc. 91, 232–234. (doi:10.1134/1.1397844) 14. Firth WJ, Harkness GK, Lord A, McSloy JM, Gomila D, Colet P. 2002 Dynamical properties of two-dimensional Kerr cavity solitons. J. Opt. Soc. Am. B 19, 747–752. (doi:10. 1364/JOSAB.19.000747) 15. Leo F, Gelens L, Emplit P, Haelterman M, Coen S. 2013 Dynamics of one-dimensional Kerr cavity solitons. Opt. Express 21, 9180–9191. (doi:10.1364/OE.21.009180) 16. Gurevich SV, Amiranashvili Sh, Purwins H-G. 2006 Breathing dissipative solitons in three-component reaction–diffusion system. Phys. Rev. E 74, 066201. (doi:10.1103/ PhysRevE.74.066201) 17. Urzagasti D, Laroze D, Clerc MG, Pleiner H. 2013 Breather soliton solutions in a parametrically driven magnetic wire. Europhys. Lett. 104, 40001. (doi:10.1209/0295-5075/ 104/40001) 18. Barashenkov IV, Zemlyanaya EV, van Heerden TC. 2011 Time-periodic solitons in a damped-driven nonlinear Schrödinger equation. Phys. Rev. E 83, 056609. (doi:10.1103/ PhysRevE.83.056609) 19. Barashenkov IV, Zemlyanaya EV. 2011 Soliton complexity in the damped-driven nonlinear Schrödinger equation: stationary to periodic to quasiperiodic complexes. Phys. Rev. E 83, 056610. (doi:10.1103/PhysRevE.83.056610) 20. Genevet P, Barland S, Giudici M, Tredicce JR. 2008 Cavity soliton laser based on mutually coupled semiconductor microresonators. Phys. Rev. Lett. 101, 123905. (doi:10.1103/ PhysRevLett.101.123905) 21. Genevet P, Columbo L, Barland S, Giudici M, Gil L, Tredicce JR. 2010 Multistable monochromatic laser solitons. Phys. Rev. A 81, 053839. (doi:10.1103/PhysRevA.81.053839) 22. Genevet P, Turconi M, Barland S, Giudici M, Tredicce JR. 2010 Mutual coherence of laser solitons in coupled semiconductor resonators. Eur. Phys. J. D 59, 109–114. (doi:10.1140/epjd/e2010-00138-0) 23. Elsass T, Gauthron K, Beaudoin G, Sagnes I, Kuszelewicz R, Barbay S. 2010 Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber. Appl. Phys. B 98, 327–331. (doi:10.1007/s00340-009-3748-9) 24. Elsass T, Gauthron K, Beaudoin G, Sagnes I, Kuszelewicz R, Barbay S. 2010 Control of cavity solitons and dynamical states in a monolithic vertical cavity laser with saturable absorber. Eur. Phys. J. D 59, 91–96. (doi:10.1140/epjd/e2010-00079-6) 25. Antoranz JC, Gea J, Velarde MG. 1981 Oscillatory phenomena and Q switching in a model for a laser with a saturable absorber. Phys. Rev. Lett. 47, 1895–1898. (doi:10.1103/ PhysRevLett.47.1895) 26. Chen YF, Lan YP. 2004 Formation of repetitively nanosecond spatial solitons in a saturable absorber Q switched laser. Phys. Rev. Lett. 93, 013901. (doi:10.1103/PhysRevLett.93.013901) 27. Barsella A, Lepers C, Dangoisse D, Glorieux P, Erneux T. 1999 Synchronization of two strongly pulsating CO2 lasers. Opt. Commun. 165, 251–259. (doi:10.1016/S0030-4018(99)00237-0) 28. Vahed H, Kheradmand R, Tajalli H, Tissoni G, Lugiato LA, Prati F. 2011 Phase-mediated longrange interactions of cavity solitons in a semiconductor laser with a saturable absorber. Phys. Rev. A 84, 063814. (doi:10.1103/PhysRevA.84.063814) 29. Prati F, Caccia P, Tissoni G, Lugiato LA, Aghdami KM, Tajalli H. 2007 Effects of carrier radiative recombination on a VCSEL-based cavity soliton laser. Appl. Phys. B 88, 405–410. (doi:10.1007/s00340-007-2711-x) 30. Bache M, Prati F, Tissoni G, Kheradmand R, Lugiato LA, Protsenko I, Brambilla M. 2005 Cavity soliton laser based on VCSEL with saturable absorber. Appl. Phys. B 81, 913–920. (doi:10.1007/s00340-005-1997-9) 31. Aghdami KM, Prati F, Caccia P, Tissoni G, Lugiato LA, Kheradmand R, Tajalli H. 2008 Comparison of different switching techniques in a cavity soliton laser. Eur. Phys. J. D 47, 447–455. (doi:10.1140/epjd/e2008-00069-3)
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32. Prati F, Tissoni G, Lugiaton LA, Mahmoud Aghdami K, Brambilla M. 2010 Spontaneously moving solitons in a cavity soliton laser with circular section. Eur. Phys. J. D 59, 73–79. (doi:10.1140/epjd/e2010-00123-7) 33. Prati F, Lugiato LA, Tissoni G, Brambilla M. 2011 Cavity soliton billiards. Phys. Rev. A 84, 053852. (doi:10.1103/PhysRevA.84.053852) 34. Dubbeldam JLA, Krauskopf B. 1999 Self-pulsations of lasers with saturable absorber: dynamics and bifurcations. Opt. Commun. 159, 325–338. (doi:10.1016/S0030-4018(98)00568-9) 35. Turaev D, Vladimirov AG, Zelik S. 2012 Long-range interaction and synchronization of oscillating dissipative solitons. Phys. Rev. Lett. 108, 263906. (doi:10.1103/PhysRevLett. 108.263906) 36. Barland S et al. 2012 Solitons in semiconductor microcavities. Nat. Photon. 6, 204. (doi:10.1038/nphoton.2012.50) 37. Rosanov NN, Fedorov SV, Shatsev AN. 2009 Collisions of one-dimensional asymmetric spatial laser solitons. Opt. Spectrosc. 106, 869–874. (doi:10.1134/S0030400X09060137) 38. Verschueren N, Bortolozzo U, Clerc MG, Residori S. 2013 Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback. Phys. Rev. Lett. 110, 104101. (doi:10.1103/PhysRevLett.110.104101)
Periodic and chaotic solitons in a semiconductor laser with saturable absorber H. Vahed, F. Prati, M. Turconi, S. Barland and G. Tissoni Phil. Trans. R. Soc. A 2014 372, 20140016, published 22 September 2014
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Periodic and chaotic solitons in a semiconductor laser with saturable absorber H. Vahed1 , F. Prati2,4 , M. Turconi3,† , S. Barland3 rsta.royalsocietypublishing.org
and G. Tissoni3 1 School of Engineering Emerging Technologies, University of Tabriz,
Research Cite this article: Vahed H, Prati F, Turconi M, Barland S, Tissoni G. 2014 Periodic and chaotic solitons in a semiconductor laser with saturable absorber. Phil. Trans. R. Soc. A 372: 20140016. http://dx.doi.org/10.1098/rsta.2014.0016
One contribution of 19 to a Theme Issue ‘Localized structures in dissipative media: from optics to plant ecology’.
Tabriz, Iran 2 Dipartimento di Scienza e Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy 3 Université de Nice Sophia Antipolis, Institut Non Linéaire de Nice, CNRS UMR 7335, 1361 Route des Lucioles, 06560 Valbonne, France 4 CNISM, Research Unit of Como, Via Valleggio 11, 22100 Como, Italy In a semiconductor laser with saturable absorber, solitons may spontaneously drift and/or oscillate. We study three different regimes characterized by strong intensity oscillations, both periodic and chaotic. We show that (i) soliton dynamics may be similar to that of passively Q-switched lasers, (ii) solitons may drift and oscillate simultaneously, and (iii) chaotic solitons may coexist with stationary ones and with the laser off solution.
Subject Areas: optics Keywords: solitons, passive Q-switching, chaos Author for correspondence: F. Prati e-mail: [email protected]
†
Present address: Observatoire de la Côte d’Azur-Laboratoire ARTEMIS, Boulevard de l’Observatoire CS 34229, 06304 Nice, France.
1. Introduction Dissipative solitons are a universal feature of nonlinear science [1,2]. Although they appear in many different systems such as vertically driven layers of sand [3], highly dissipative fluids [4] and biology [5], they have been studied most extensively in optics. A particular type of optical solitons are the so-called cavity solitons (CSs), two-dimensional spatial localized structures that arise in the plane perpendicular to the light propagation direction in nonlinear optical systems driven by an external coherent field [6,7]. CSs can also be produced in nonlinear cavities which are bistable even in the absence of a driving field. Such systems are called CS lasers [8]. The interest in CSs stems from their promising applications in optical information processing and storage. Under this respect, the functionalities of CSs
2014 The Author(s) Published by the Royal Society. All rights reserved.
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We consider a vertical-cavity surface-emitting laser with an integrated absorber whose dynamics in the transverse plane is described by the equations [29]
and
2 ]F, F˙ = [(1 − iα)D + (1 − iβ)d − 1 + i∇⊥
(2.1)
˙ = −b[D(1 + |F|2 ) + B1 D2 − μ] D
(2.2)
d˙ = −rb[d(1 + s|F|2 ) + B2 d2 + γ ].
(2.3)
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2. The model
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are enhanced if they can move in the transverse plane, oscillate at fixed position or even oscillate while they move. For instance, the mobility of CSs under an external control beam was used to implement an all-optical delay line [9] and a low-energy optical switch [10]. An even larger amount of information could be probably stored in the CSs if they were oscillating localized structures and their dynamics could be controlled by varying an external parameter (e.g. the pump). Spontaneous oscillations can appear in solitons as a result of an Andronov–Hopf bifurcation of the stationary solitons. In optics, this was shown in systems with a saturable defocusing nonlinearity [11], a saturable absorber [12,13] and a Kerr nonlinearity [14,15]. More generally, oscillating solitons were demonstrated in reaction–diffusion systems [16], in a magnetic wire forced by a transversal uniform and oscillatory magnetic field [17] and in parametric systems where one parameter is modulated periodically [18,19]. Here, we consider a CS laser formed by a laser with saturable absorber. CSs have been observed in such systems both in extended [20–22] and monolithic resonators [23,24]. As long as the simple single-mode temporal dynamics is considered, it is known that the Andronov–Hopf bifurcation in a laser with saturable absorber leads to passive Q-switching, where short pulses are produced with a repetition time of about 1 ns [25]. The main idea of the present paper is to combine the subcritical bifurcation of a laser with saturable absorber, which allows for the existence of CSs, with its intrinsic dynamical behaviour owing to the Andronov–Hopf bifurcation to create passively Q-switched CSs, i.e. pulses that are localized both in space and in time. Being pulses separated by long intervals during which the field intensity is close to zero, these dynamical CSs differ from the oscillating solitons predicted in a similar system in [13]. They may be instead related to the observations of [24,26], although in those experiments it was not possible to demonstrate that the self-pulsing structures coexist with the laser off solution. In a system with translational symmetry, it is possible to switch on at the desired positions several passively Q-switched CSs thus forming an array of localized Q-switched lasers that coexist in the same device. The natural question is how do they interact one with the other. Although the interaction may occur only during the short duration of the pulses, it can be effective, because the pulses are very intense. We expect to observe pulse synchronization, because the interaction is nonlinear [27], and the appearance of forces related to phase locking, because the localized lasers, unlike in conventional arrays, are free to move [28]. In this paper, we also show that other dynamical CSs may exist in a laser with saturable absorber under different parametric conditions. Pulsating CSs drifting in a non-uniform way appear when carrier dynamics is much slower in the absorber than in the amplifier. When radiative recombination is included in the model equations, a wide branch of stable chaotic CSs similar to those found in [24] is predicted. In §2, we present the dynamical model and its stationary solutions, homogeneous and localized. In §3, we calculate the existence domain of self-pulsing CSs and show how their intensity and period depend on the pump parameter. The interaction of four passively Q-switched CSs in a 2 × 2 array is analysed in §4. The travelling pulsating CSs are shown in §5, whereas the chaotic CSs are characterized in §6. The conclusions are drawn in §7.
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3
0.4
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0
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Figure 1. Peak intensity of stable stationary CSs (filled circles) and unstable homogeneous steady state solution (dashed line) as a function of μ.
The dimensionless variables F, D and d are respectively the slowly varying envelope of the electric field, the carrier density of the amplifier and the carrier density of the absorber. The parameters α, β are the linewidth enhancement factors in the amplifier and in the absorber, respectively; μ is the pump parameter, γ is the unsaturated absorption, s is the saturation parameter, i.e. the ratio of saturation intensity in the amplifier and in the absorber. Time is scaled to the cavity lifetime (1 time unit (t.u.) ∼ a few picoseconds), b is the ratio of photon lifetime to carrier lifetime in the amplifier and r is the ratio of carrier lifetimes in the amplifier and in the absorber. Space is scaled to the diffraction length (1 space unit (s.u.) ∼ a few micrometres). The coefficients B1 and B2 are the rates of radiative recombination. For a precise definition of all variables and parameters, see the appendix of Vahed et al. [10]. The analysis of equations (2.1)–(2.3) is simpler if radiative recombination is neglected setting B1 = B2 = 0 as in [30]. The homogeneous stationary solution is bistable if the condition s > 1 + 1/γ is satisfied. In the plane wave approximation (i.e. when the variables are assumed uniform in the transverse field, and the transverse Laplacian is neglected), part of the upper branch of the homogeneous solution is destabilized through an Andronov–Hopf bifurcation provided the ratio r of carrier lifetimes in the two media is larger than a critical value rc which is approximately given, in the limit b 1, by √ √ s−1+ γ . (2.4) rc = √ γs Beyond the plane wave approximation, localized structures (CSs) are also solutions of equations (2.1)–(2.3) provided r < rc , i.e. the upper branch is stable with respect to the Andronov– Hopf bifurcation. In this case, the transverse Laplacian induces instead a modulational instability in the upper branch, associated with a real eigenvalue, which extends to the whole branch if α > β. This makes possible the coexistence of the CSs with the laser off solution for a range of pump values below the laser threshold μthr = 1 + γ . By choosing s = 10 and γ = 0.5, we have rc 0.58. Figure 1 shows the branch of stable CSs for r = 0.45, and b = 0.01, α = 2, β = 0.2. The CSs can either arise spontaneously or be excited in a predetermined position using different techniques [31]. Optical pumping allows for an optical control of carrier distribution in the transverse profile and incoherent switching [23,24]. Here, we consider instead coherent switching by a Gaussian pulse of width w centred in (x0 , y0 ) Finj = F0 e−[(x−x0 )
2
+(y−y0 )2 ]/2w2
,
(2.5)
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0.53
4
r
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stationary CS 1.43
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1.45
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1.48
m
Figure 2. Stability diagram of CSs in the (μ, r) plane. The horizontal dashed line corresponds to the value r = 0.5 used in figure 3.
which is injected into the laser for a time interval tinj . No explicit time dependence appears in Finj, because the pulse is assumed to be resonant with the cold cavity whose frequency is assumed as reference frequency; the amplitude F0 is real, its phase is irrelevant because the CSs are (blue) detuned with respect to the cold cavity. For a given value of the pump parameter μ below laser threshold, the CSs are stable only for a finite range of r. The stability domain is limited from below by a drift instability originated by the Goldstone modes of the system, which induce a spontaneous motion of the CSs. The properties of these moving CSs have already been investigated in the presence of circular [32] and squared [33] reflecting boundaries. The upper boundary of the stability domain is instead associated with an Andronov–Hopf bifurcation whose effect is to makes the CSs oscillate without a translational motion. Figure 2 shows the stability domain of the oscillating CSs in the plane (μ, r). Below the lower boundary, the stationary CSs are stable. As the upper boundary is crossed, the laser either jumps to the off state for small μ or develops complex spatio-temporal dynamics for larger μ.
3. Localized passive Q-switching The CS oscillations present the typical features of passive Q-switching in a laser with saturable absorber [34]. Yet, a notable difference with respect to [34] is that the Q-switched CSs are bistable with the laser off solution. This occurs, even in the plane wave model, when r < 1. Let us fix r = 0.5 (dashed line in figure 2) and vary μ from right to left crossing the two boundaries of the stability region of oscillating CSs. As shown in figure 3, after crossing the right boundary, which is placed at μ 1.4635, the CS starts oscillating, and already very close to threshold the oscillations are highly nonlinear, because the action of the saturable absorber favours the formation of narrow pulses between which the emitted intensity falls very close to zero. A portion of the pulse train for μ = 1.45 is shown in the inset. By further decreasing the pump parameter μ, the amplitude of the pulses increases slowly, whereas the period grows more rapidly as μ approaches the right boundary of the stability domain, μ 1.4472.
4. 2 × 2 arrays of passively Q-switched solitons Once localized Q-switching is demonstrated, it is interesting to study how several passively Q-switched CSs interact. To this aim, we analysed 2 × 2 arrays of such structures for μ = 1.47 and
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(a)
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0
200
400
time
600
800
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Figure 3. (a) Maximum and minimum value of the peak intensity of an oscillating CS as function of μ for r = 0.5. The inset shows a time trace for μ = 1.45. (b) Corresponding period of the intensity oscillation.
r = 0.51. The dynamics of the CSs can be monitored by calculating their positions and the field intensity and phase at the peak. In this way, we can determine whether the CSs are correlated in intensity and/or phase, and if their mutual distances evolve in time under the action of some force. The results can be summarized by plotting the four intensities, the relative phase of two CSs on the same side and on a diagonal and the average distance between first neighbours. We considered an array where the initial distance between the first neighbours, i.e. two CSs on the same side of the square, is 22 s.u., and analysed the time evolution of the CSs both when they are switched on simultaneously and when they are switched on in sequence. The time evolution of the array when the CSs are switched on simultaneously is shown in the left column of figure 4. The pulses, which are initially perfectly synchronized and locked in phase, lose synchronization and phase-locking at about t = 3000 t.u. Yet, phase-locking and synchronization are re-established at about t = 30 000 t.u. The average distance between first neighbours at the beginning falls slightly below the initial value of 22 s.u., but very soon it starts to grow at an almost constant rate independent of phase-locking and synchronization. In the phase-locked state, two CSs placed on the same side of the square are locked at π , whereas two CSs on a diagonal are locked at 0. This configuration agrees with the findings of [28] where it was shown that the relative phase of two stationary CSs behaves differently depending on their distance. For moderate distances, the CSs initially lock at π , although for longer times they partially unlock and reach a state of phase entrainment, where the relative phase remains bounded but it oscillates around π . For larger distances, the stationary CSs lock in phase. In the right column of figure 4, the time evolution of the array is shown when the four CSs are switched on in the same positions as before, but with a delay of 500 t.u. one from the other. Initially, the pulses are well separated, but at t = 10 000 t.u., the delays among pulses are comparable to those of the previous case, and again phase-locking and pulse synchronization are achieved at about t = 30 000 t.u. The average distance of first neighbours is never smaller than 22 s.u., and it grows at the same rate found when the CSs are simultaneously switched on. These results indicate that the CSs interact at very large distance, much larger than one soliton diameter and even larger than its phase profile [28]. Phase-locking is robust against noise, that
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Figure 4. Dynamics of a 2 × 2 array of passively Q-switched CSs for μ = 1.47 and r = 0.51. In (a–f ), the time evolution of the peak intensity of the fours CSs is represented by different colours (black, red, green, blue). (a,c,e,g,i) The CSs are switched on simultaneously. (b,d,f ,h,j) The CSs are switched on in sequence at intervals of 500 t.u. (a–f ) Intensity of the four CSs at different stages of the evolution. (g,h) Relative phase of two CSs belonging to the same row of the array (φs , red) and placed on the diagonal (φd , black). (i,j) Average distance of the first neighbours. (Online version in colour.)
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(a)
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(c)
5. Stop and go solitons By decreasing by an order of magnitude the ratio r of carrier lifetimes, i.e. assuming that carriers are much slower in the absorber, a new phenomenon is observed. As in [13,32], the localized structures are subjected to a drift instability which makes them move along straight lines whose direction, if not controlled by means of an address beam, is random. Yet, unlike in [32], the peak intensity of the CS pulsates while it moves, and the motion is not uniform. An example is shown in figure 5 which captures the motion of the CS at three different times. Figure 5a,c shows two consecutive pulses, taken at their maximum intensity, which is about |F|2 = 0.46. In figure 5b, the intensity is minimum, and it is about |F|2 = 0.017 at the peak. Here, the intensity profile is much broader than at the maxima. Note that the positions of the intensity peak at the minimum (figure 5b) and at the second maximum (figure 5c) coincide. Initially, the CS moves rapidly but it is damped because carrier dynamics, in particular in the absorber, is too slow to follow that fast motion. Then, the motion stops while the CS is regenerated, and only when the CS recovers its maximum intensity does it start again to move. This behaviour is a further manifestation of the dual wave–material character of the CS, already stressed in [36]. The self-propelled solitons lose radial symmetry and they are elongated in the direction of motion [32,37], as can be better appreciated in the central density plot of figure 5. The time interval between the first and the second snapshots is 4718 t.u., and the distance between the maxima of the two intensity distributions is 4.5 s.u. Assuming (1 s.u.)/(1 t.u.) = 1000 km s−1 = 1000 µm ns−1 , the speed of the CS is about 1 µm ns−1 , comparable to that of [32].
6. Chaotic solitons Here, we take into account the effects of carrier recombination and consider the parameter set B1 = B2 = 0.1, s = 1, γ = 2, α = 2, β = 1, b = 0.01 and r = 1 [10,28,29,31–33]. For this set of parameters, the stationary CSs are stable for a wide range of pump current values μ (see fig. 2 of [32]). We focus on the complex spatio-temporal dynamical state which is usually observed beyond the laser threshold μthr and find that it can coexist with the stationary CS. Therefore, there exists a large region of coexistence of three stable states in the parameter space: — laser off state, stable between μ = 0 and μthr , — stationary CS, stable between μ = 4.56 and μ = 5.1, and
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was always included in our simulations. This phenomenon could be related to the enhancement of solitons’ interaction owing to the Andronov–Hopf bifurcation that makes them oscillate, which was demonstrated for Kerr CSs in [35].
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Figure 5. Three consecutive snapshots of the intensity in the transverse section of the laser illustrating self-propelled Q-switching. (a) t = 0, (b) t = 4718 and (c) t = 8192. The grey scale ranges 0–0.46 in (a) and (c), and 0–0.017 in (b).
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Figure 6. (a) Branch of the chaotic state (maximum intensity) as a function of the pump current μ. Points indicate the mean values and bars the standard deviations. (b) Snapshot of a numerical simulation showing the intensity profile, obtained for μ = 6. (Online version in colour.)
— extended spatio-temporal chaos: the field intensity is rapidly varying in space and time, spanning between almost 0 and a very high maximum value, the typical spatial size of the chaotic pattern being of the order of the CS dimension.
In figure 6a, we display for a wide interval of pump current μ the mean values and the standard deviations of the maximum intensity of the chaotic state, whose typical instantaneous intensity profile is shown in figure 6b, for μ = 6. To the left of the branch, the chaotic solution loses its stability in favour of the laser off solution. To the right of the branch, it seems to remain stable indefinitely. In this regime of coexistence between the homogeneous off solution and the extended chaotic state, we looked for chaotic localized states (the so-called chaoticons), as recently reported experimentally and numerically in a different system [38]. To this aim, we started from the laser off solution and injected a localized Gaussian writing beam of very strong amplitude F0 = 15 (much higher than in the case of CS switching) and width w = 2 s.u. during 50 t.u. When the writing beam is switched off, the system spontaneously evolves towards a CS that displays chaotic internal dynamics. Figure 7 shows the instantaneous transverse distribution of the field intensity and carrier density in the amplifier for one of these structures obtained for μ = 4.92. We verified the stability and localization of this chaotic CS for very long integration times (50 000 t.u.), and different values of the pump current μ. The CS remains stable for 4.67 ≤ μ ≤ 4.94. If μ is further reduced to (or below) the value of μ = 4.65 its internal dynamics starts freezing and finally it collapses onto a stationary CS (figure 8a). On the other hand, when μ is increased up to (or above) μ = 4.96, the chaotic CS starts expanding very slowly and finally the extended chaotic state is reached. The effective independence, multistability and localization of the chaotic CS were checked by creating two chaotic CSs in the transverse plane (figure 8b) and by switching on simultaneously a stationary and a chaotic CS. All these simulations were performed keeping equal the carrier lifetimes in the amplifier and in the absorber (r = 1). Actually, the stability of the chaotic CS depends on r, in a very similar way as the stationary CS (see for example fig. 2 of ref. [32]). By keeping μ fixed at the value of 4.85 and by reducing r, at r = 0.7 the chaotic CS collapses to a CS which then starts moving but dies very soon. We did not observe stable drifting chaotic CSs. For values of r higher than or equal to 1.8, the chaotic CS starts to expand very slowly and finally the extended chaotic state is reached.
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Figure 7. (a) Field intensity distribution and (b) amplifier carrier density distribution of a chaotic CS, obtained for μ = 4.92. (Online version in colour.)
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Figure 8. Field intensity evolution of CSs. Space (in one transverse dimension) varies along the horizontal axis, time grows upwards along the vertical axis. (a) Chaotic CS collapsing onto a stationary CS for μ = 4.65. (b) Two independent chaotic CSs are simultaneously present in the integration grid, obtained for μ = 4.85. The whole time duration is 1500 t.u. (Online version in colour.)
The same happens if we try to switch on a larger chaotic CS by increasing the width w of the writing beam. The CS apparently remains localized, but actually it expands very slowly and finally it invades the whole integration grid becoming the extended chaotic state. The entire process of completely filling the integration grid can take up to 10 000 t.u. This feature makes our chaotic CS deeply different from the chaoticon described in [38]. In that case, the chaotic structures are shown to remain localized independently of their size, that is, they do not vary their initial size. The origin of this sharply different behaviour is still unknown but even with very large integration windows (thus making it impossible for the initial localized structure to interact with its image through the boundaries) in our simulations localization is lost if the CS’s size passes a critical value of about two to three stationary CS diameters.
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In this paper, we have predicted the existence of different kinds of dissipative spatial solitons in a semiconductor laser with saturable absorber where the field intensity displays strong oscillations in time. When the carrier lifetimes in the amplifier and in the absorber are comparable, an Andronov–Hopf instability of the stationary solitons leads to the appearance of what we called passively Q-switched solitons, which present features very similar to those of passive Q-switching extensively studied in the time domain. The fact that passive Q-switching is localized in space allows studying the interaction of different oscillating solitons which coexist in the same laser. In a 2 × 2 array, we have shown that in the long-term evolution the four solitons synchronize their intensity pulses and lock their phases in such a way that two solitons in the same row or column have a phase difference of π , whereas two solitons on the diagonal have the same phase. The four solitons also experience a repulsive force which makes them move apart one from the other. If the carriers in the absorber are much slower than in the amplifier a different instability causes the soliton to drift along straight lines in a stop and go manner, so that while it moves its peak intensity decreases until it reaches a minimum, then it stops until it recovers the maximum intensity, and then it moves again along the same line. Chaotic solitons appear when radiative recombination is included in the model equations and the laser off state is perturbed with a writing beam whose intensity is much larger than that needed to excite a stationary soliton. These states are a portion of an extended chaotic state that coexists with the laser off solution. Because with the same parameters the stationary solitons are also stable, chaotic and stationary solitons may coexist in the same laser. Funding statement. F.P. acknowledges support from the Italian Ministry of Research (MIUR) through the ‘Futuro in Ricerca’ FIRB-grant PHOCOS-RBFR08E7VA. S.B. and G.T. acknowledge support from the ANR-Jeunes Chercheurs Project MOLOSSE (ANR-12-JS04-0002-01, www.molosse.org) and from the CNRS-Region PACA Project E-Buffo.
References 1. Akhmediev N, Ankiewicz A (eds). 2005 Dissipative solitons. Berlin, Germany: Springer. 2. Akhmediev N, Ankiewicz A (eds). 2008 Dissipative solitons: from optics to biology and medicine. Berlin, Germany: Springer. 3. Umbanhowar PB, Melo F, Swinney HL. 1986 Localized excitations in a vertically vibrated granular layer. Nature 382, 793–796. (doi:10.1038/382793a0) 4. Lioubashevski O, Arbell H, Fineberg J. 1996 Dissipative solitary states in driven surface waves. Phys. Rev. Lett. 76, 3959–3962. (doi:10.1103/PhysRevLett.76.3959) 5. Lejeune O, Tlidi M, Couteron P. 2002 Localized vegetation patches: a self-organized response to resource scarcity. Phys. Rev. E 66, 010901. (doi:10.1103/PhysRevE.66.010901) 6. Barland S et al. 2002 Cavity solitons as pixels in semiconductor microcavities. Nature 419, 699–702. (doi:10.1038/nature01049) 7. Lugiato LA. 2003 Introduction to the feature section on cavity solitons: an overview. IEEE J. Quantum Electron. 39, 193–196. (doi:10.1109/JQE.2002.807195) 8. Ackemann T, Firth W, Oppo G-L. 2009 Fundamentals and applications of spatial dissipative solitons in photonic devices. Adv. Atom. Mol. Opt. Phys. 57, 323–421. (doi:10.1016/ S1049-250X(09)57006-1) 9. Pedaci F et al. 2008 All-optical delay line using semiconductor cavity solitons. Appl. Phys. Lett. 92, 011101. (doi:10.1063/1.2828458) 10. Vahed H, Prati F, Tajalli H, Tissoni G, Lugiato LA. 2012 Low-energy switch based on a cavity soliton laser. Eur. Phys. J. D 66, 148. (doi:10.1140/epjd/e2012-30109-2)
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We also stress the very large spatial coherence that characterizes all our structures, owing to the huge phase profile which extends to regions where the field intensity is very close to 0 (see, for example, the phase profile of the stationary CS shown in fig. 2 of [28]).
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11. Michaelis D, Peschel U, Lederer F. 1998 Oscillating dark cavity solitons. Opt. Lett. 23, 1814–1816. (doi:10.1364/OL.23.001814) 12. Vladimirov AG, Fedorov SV, Kaliteevskii NA, Khodova GV, Rosanov NN. 1999 Numerical investigation of laser localized structures. J. Opt. B 1, 101–106. (doi:10. 1088/1464-4266/1/1/019) 13. Rozanov NN, Fedorov SV, Shatsev AN. 2001 Pulsating solitons in a laser with relaxation of gain and saturable absorption. Opt. Spectrosc. 91, 232–234. (doi:10.1134/1.1397844) 14. Firth WJ, Harkness GK, Lord A, McSloy JM, Gomila D, Colet P. 2002 Dynamical properties of two-dimensional Kerr cavity solitons. J. Opt. Soc. Am. B 19, 747–752. (doi:10. 1364/JOSAB.19.000747) 15. Leo F, Gelens L, Emplit P, Haelterman M, Coen S. 2013 Dynamics of one-dimensional Kerr cavity solitons. Opt. Express 21, 9180–9191. (doi:10.1364/OE.21.009180) 16. Gurevich SV, Amiranashvili Sh, Purwins H-G. 2006 Breathing dissipative solitons in three-component reaction–diffusion system. Phys. Rev. E 74, 066201. (doi:10.1103/ PhysRevE.74.066201) 17. Urzagasti D, Laroze D, Clerc MG, Pleiner H. 2013 Breather soliton solutions in a parametrically driven magnetic wire. Europhys. Lett. 104, 40001. (doi:10.1209/0295-5075/ 104/40001) 18. Barashenkov IV, Zemlyanaya EV, van Heerden TC. 2011 Time-periodic solitons in a damped-driven nonlinear Schrödinger equation. Phys. Rev. E 83, 056609. (doi:10.1103/ PhysRevE.83.056609) 19. Barashenkov IV, Zemlyanaya EV. 2011 Soliton complexity in the damped-driven nonlinear Schrödinger equation: stationary to periodic to quasiperiodic complexes. Phys. Rev. E 83, 056610. (doi:10.1103/PhysRevE.83.056610) 20. Genevet P, Barland S, Giudici M, Tredicce JR. 2008 Cavity soliton laser based on mutually coupled semiconductor microresonators. Phys. Rev. Lett. 101, 123905. (doi:10.1103/ PhysRevLett.101.123905) 21. Genevet P, Columbo L, Barland S, Giudici M, Gil L, Tredicce JR. 2010 Multistable monochromatic laser solitons. Phys. Rev. A 81, 053839. (doi:10.1103/PhysRevA.81.053839) 22. Genevet P, Turconi M, Barland S, Giudici M, Tredicce JR. 2010 Mutual coherence of laser solitons in coupled semiconductor resonators. Eur. Phys. J. D 59, 109–114. (doi:10.1140/epjd/e2010-00138-0) 23. Elsass T, Gauthron K, Beaudoin G, Sagnes I, Kuszelewicz R, Barbay S. 2010 Fast manipulation of laser localized structures in a monolithic vertical cavity with saturable absorber. Appl. Phys. B 98, 327–331. (doi:10.1007/s00340-009-3748-9) 24. Elsass T, Gauthron K, Beaudoin G, Sagnes I, Kuszelewicz R, Barbay S. 2010 Control of cavity solitons and dynamical states in a monolithic vertical cavity laser with saturable absorber. Eur. Phys. J. D 59, 91–96. (doi:10.1140/epjd/e2010-00079-6) 25. Antoranz JC, Gea J, Velarde MG. 1981 Oscillatory phenomena and Q switching in a model for a laser with a saturable absorber. Phys. Rev. Lett. 47, 1895–1898. (doi:10.1103/ PhysRevLett.47.1895) 26. Chen YF, Lan YP. 2004 Formation of repetitively nanosecond spatial solitons in a saturable absorber Q switched laser. Phys. Rev. Lett. 93, 013901. (doi:10.1103/PhysRevLett.93.013901) 27. Barsella A, Lepers C, Dangoisse D, Glorieux P, Erneux T. 1999 Synchronization of two strongly pulsating CO2 lasers. Opt. Commun. 165, 251–259. (doi:10.1016/S0030-4018(99)00237-0) 28. Vahed H, Kheradmand R, Tajalli H, Tissoni G, Lugiato LA, Prati F. 2011 Phase-mediated longrange interactions of cavity solitons in a semiconductor laser with a saturable absorber. Phys. Rev. A 84, 063814. (doi:10.1103/PhysRevA.84.063814) 29. Prati F, Caccia P, Tissoni G, Lugiato LA, Aghdami KM, Tajalli H. 2007 Effects of carrier radiative recombination on a VCSEL-based cavity soliton laser. Appl. Phys. B 88, 405–410. (doi:10.1007/s00340-007-2711-x) 30. Bache M, Prati F, Tissoni G, Kheradmand R, Lugiato LA, Protsenko I, Brambilla M. 2005 Cavity soliton laser based on VCSEL with saturable absorber. Appl. Phys. B 81, 913–920. (doi:10.1007/s00340-005-1997-9) 31. Aghdami KM, Prati F, Caccia P, Tissoni G, Lugiato LA, Kheradmand R, Tajalli H. 2008 Comparison of different switching techniques in a cavity soliton laser. Eur. Phys. J. D 47, 447–455. (doi:10.1140/epjd/e2008-00069-3)
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32. Prati F, Tissoni G, Lugiaton LA, Mahmoud Aghdami K, Brambilla M. 2010 Spontaneously moving solitons in a cavity soliton laser with circular section. Eur. Phys. J. D 59, 73–79. (doi:10.1140/epjd/e2010-00123-7) 33. Prati F, Lugiato LA, Tissoni G, Brambilla M. 2011 Cavity soliton billiards. Phys. Rev. A 84, 053852. (doi:10.1103/PhysRevA.84.053852) 34. Dubbeldam JLA, Krauskopf B. 1999 Self-pulsations of lasers with saturable absorber: dynamics and bifurcations. Opt. Commun. 159, 325–338. (doi:10.1016/S0030-4018(98)00568-9) 35. Turaev D, Vladimirov AG, Zelik S. 2012 Long-range interaction and synchronization of oscillating dissipative solitons. Phys. Rev. Lett. 108, 263906. (doi:10.1103/PhysRevLett. 108.263906) 36. Barland S et al. 2012 Solitons in semiconductor microcavities. Nat. Photon. 6, 204. (doi:10.1038/nphoton.2012.50) 37. Rosanov NN, Fedorov SV, Shatsev AN. 2009 Collisions of one-dimensional asymmetric spatial laser solitons. Opt. Spectrosc. 106, 869–874. (doi:10.1134/S0030400X09060137) 38. Verschueren N, Bortolozzo U, Clerc MG, Residori S. 2013 Spatiotemporal chaotic localized state in liquid crystal light valve experiments with optical feedback. Phys. Rev. Lett. 110, 104101. (doi:10.1103/PhysRevLett.110.104101)
