Universal power law for front propagation in all fiber resonators ∗
S. Coulibaly,1 M. Taki,1 and M. Tlidi 2, 1 Laboratoire
de Physique des Lasers, Atomes et Molécules, CNRS UMR8523, Université des Sciences et Technologies de Lille, Batiment P5, 59655 Villeneuve d’Ascq Cedex, France 2 Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), Campus Plaine CP. 231, B-1050 Bruxelles, Belgium ∗ [email protected]
Abstract: We consider a bistable system consisting of all fiber cavity driven by an external injected continuous wave. We report on front propagation in a high finesse cavity. We study the asymptotic behavior of the front velocity. We show that the front velocity is affected by the distance from the critical point associated with bistability. We provide a scaling low governing its evolution near the up-switching point of the bistable curve. We show also that the velocity of front propagation obeys a generic power law when the front velocity approaches asymptotically its linear growing value. © 2014 Optical Society of America OCIS codes: (190.4420) Nonlinear optics; (190.4370) Nonlinear optics, fibers; (060.5530) Pulse propagation and temporal solitons; (070.5753) Resonators.
References 1. L.A. Lugiato and R. Lefever, "Spatial Dissipative Structures in Passive Optical Systems," Phys. Rev. Lett. 58, 2209–2211 (1987). 2. M. Tlidi, R. Lefever, and P. Mandel, "Pattern selection in optical bistability," Quantum Semiclass. Opt. 8 931–938 (1996). 3. F. Mitschke, G. Steinmeyer, and A. Schwache, "Generation of one-dimensional optical turbulence," Physica D 96, 251–258 (1996). 4. S. Coen and M. Haelterman, "Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber," Phys. Rev. Lett. 79, 4139-4142 (1997). 5. M. Stratmann, T. Pagel, and F. Mitschke, "Experimental observation of temporal soliton molecules," Phys. Rev. Lett. 95, 143902 (2005). 6. M. Tlidi, M. Haelterman, and P. Mandel,"3D patterns and pattern selection in optical bistability," Europhys. Lett. 42, 505–509 (1998). 7. K. Staliunas, "Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators," Phys. Rev. Lett. 81, 81–85 (1998). 8. M. Tlidi, "Three-dimensional crystals and localized structures in diffractive and dispersive nonlinear ring cavities," J. Opt. B: Quantum Semiclass. Opt. 2, 438–442 (2000). 9. M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity light bullets: Three-dimensional localized structures in a nonlinear optical resonator," Phys. Rev. Lett. 93, 203901 (2004). 10. P. Tassin, G. Van der Sande, N. Veretenov, P. Kockaert, I. Veretennicoff, and M. Tlidi, "Three-dimensional structures in nonlinear cavities containing left-handed materials", Optics Express, 14, 9338–9343 (2006). 11. M. Tlidi, M. Taki, and T. Kolokolnikov, "Introduction: Dissipative localized structures in extended systems", Chaos 17, 037101 (2007). 12. N. Akhmediev and A. Ankiewicz, Eds. Dissipative Solitons: From Optics to Biology and Medicine (SpringerVerlag, Berlin, Heidelberg, 2008). 13. O. Descalzi, M.G. Clerc, S. Residori, and G. Assanto, Localized States in Physics: Solitons and Patterns (Springer, 2010).
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 483
14. K. Staliunas, and V.J. Sanchez-Morcillo "Transverse Patterns in Nonlinear Optical Resonators" Springer Verlag, Springer Tracts in Modern Physics, Vol.183, 2003. 15. A.J. Scroggie, W.J. Firth, G.S McDonald, M. Tlidi, R. Lefever, and L.A Lugiato, "Pattern formation in a passive Kerr cavity," Chaos, Solitons and Fractals 4, 1323–1354 (1994). 16. M. Tlidi, P. Mandel, and R. Lefever, "Localized structures and localized patterns in optical bistability," Phys. Rev. Lett. 73, 640–643 (1994). 17. G. Slekys, K. Staliunas, C.O. Weiss, "Spatial localized structures in resonators with saturable absorber," Opt. Commun. 149, 113–116 (1998). 18. V.B. Taranenko, K. Staliunas, and C. O. Weiss, "Spatial soliton laser: Localized structures in a laser with a saturable absorber in a self-imaging resonator," Phys. Rev. A56, 1582–1591 (1997). 19. V. Odent, M. Taki, and E. Louvergneaux, "Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity," New J. Phys. 13, 113026 (2011). 20. F. Leo, S. Coen, P. Kockaert, S-P. Gorza, P. Emplit, and M. Haelterman, "Temporal cavity solitons in onedimensional Kerr media as bits in an all-optical buffer," Nature Photon. 4, 471–476 (2010). 21. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, "Dynamics of one-dimensional Kerr cavity solitons," Opt. Express 21, 9180–9191 (2013). 22. D. Gomila, M. A. Matias, and P. Colet, "Excitability mediated by localized structures in a dissipative nonlinear optical cavity," Phys. Rev. Lett. 94, 063905 (2005). 23. D. Turaev, A. G. Vladimirov, and S. Zelik, "Long-range interaction and synchronization of oscillating dissipative solitons", Phys. Rev. Lett. 108, 263906-263909 (2012). 24. O.A Egorov and F. Lederer, "Spontaneously walking discrete cavity solitons," Opt. Lett. 38, 1010–1012 (2013). 25. S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, "Convection versus dispersion in optical bistability," Phys. Rev. Lett. 83, 2328–2331 (1999). 26. K.Staliunas and V.J. Sanchez-Morcillo, "Dynamics of phase domains in the Swift-Hohenberg equation," Phys. Lett. A241 28–34 (1998). 27. M. Tlidi, P. Mandel, and R. Lefever, "Kinetics of localized pattern formation in optical systems," Phys. Rev. Lett. 81, 979–982 (1998). 28. M. Tlidi, Paul Mandel, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, "Phase-separation dynamics of circular domain walls in the degenerate optical parametric oscillator," Optics Lett. 25, 487–489 (2000). 29. V. B. Taranenko, K. Staliunas, and C.O. Weiss, "Pattern formation and localized structures in degenerate optical parametric mixing", Phys. Rev. Lett. 81 2236–2239 (1998). 30. G.T. Dee and J.S. Langer, "Propagating pattern selection," Phys. Rev. Lett. 50, 383–386 (1983). 31. G.T. Dee and W. van Saarloos, "Bistable Systems with Propagating Fronts Leading to Pattern Formation," Phys. Rev. Lett. 60, 2641–2644 (1988). 32. M. Taki, M. N. Ouarzazi, H. Ward, and P. Glorieux, "Nonlinear front propagation in optical parametric oscillators," J. Opt. Soc. Am. B 17, 997–1003 (2000). 33. W. van Saarloos, "Front propagation into unstable states," Phys. Rep. 286, 29–222 (2003). 34. A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, "Optical fiber systems are convectively unstable", Phys. Rev. Lett. 101, 113904-113907 (2008). 35. Kestutis Staliunas, Oleg Egorov, Yuri S. Kivshar, and Falk Lederer, "Bloch Cavity Solitons in Nonlinear Resonators with Intracavity Photonic Crystals," Phys. Rev. Lett. 101, 153903-153906 (2008). 36. P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M.Tlidi, "Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials," Phys. Rev. A 74, 033822 (2006). 37. L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, "Impact of nonlocal interactions in dissipative systems: Towards minimal-sized localized structures," Phys. Rev. A 75, 063812 (2007). 38. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, "Noise-sustained convective structures in nonlinear optics," Phys. Rev. Lett. 79, 3633–3636 (1997). 39. S. Coulibaly, C. Durniak, and M. Taki, "Spatial Dissipative Solitons Under Convective and Absolute Instabilities in Optical Parametric Oscillators," Lecture Notes in Physics, 751, 261–285 (2008). 40. H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, "Influence of walkoff on pattern formation in nondegenerate optical parametric oscillators," Phys. Rev. E 63, 016604 (2000).
1.
Introduction
In recent years a considerable progress has been realized in the understanding of localized structures (LS) in nonlinear optical systems. These structures are often called cavity solitons. They have been found in optical cavity subject to continuous injection and energy dissipation. They can be classified into three types (i) spacial cavities: LS appear thanks to the interplay between diffraction and nonlinearity [1, 2], (ii) temporal cavities: LS result from the interplay between
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 484
dispersion and nonlinearity [3–5], (iii) spatio-temporal cavities: three-dimensional light bullets appear as a result of combined influence of diffraction and dispersion, and nonlinearity [6–10]. These simple and robust devices have attracted a considerable attention both from fundamental as well as from applied point of views (see latest overviews on this issue [11–14]). The prerequisite condition for light confinement leading to stabilization of LSs is the occurrence of subcritical modulational instability where a coexistence between a homogeneous background and a self-organized periodic structure occurs [15, 16]. These dissipative solitons have been evidenced in photorefractive oscillator with intracavity saturable absorber [17] and in a laser with saturable absorber in a self-imaging resonator [18]. However, it is only recently that experimental evidence of cavity solitons in a passive cavity has been demonstrated in the spatial domain by using the Kerr-like medium [19]. In the temporal domain, temporal cavity solitons have also been observed in passive fiber cavity and an estimation of the capacity to operate as all-optical memories is given: a very high value of 45 Kbits at 25 Gbits/s [20]. For large intensities, temporal cavity solitons could exhibit a self-pulsation or chaotic behavior [21–24]. The formation of cavity solitons is relatively a well-understood issue. However, front propagation in driven optical fiber cavities has received only a scant attention [25]. Note that spatial fronts can also exist in 2D configuration giving rise to domain walls. The latter have been theoretically studied in Swift-Hohenberg model in [26–28], and observed in degenerate four-wave-mixing oscillators in [29]. In this paper, we study front propagation into an unstable intermediate homogeneous steady state in a driven all fiber cavity. In Sect. 2, we present the Lugiato-Lefever model and we summarize a linear stability analysis of the homogeneous steady states. In Sect. 3, we show that during time evolution t, the transient velocity of single propagating flat front varies according to the universal power law v(t) − v∗ ∼ 1/t, where v∗ is the asymptotic linear growing velocity. Such a generic transition law has been predicted in real order parameter equations for fluid mechanics [30,31,33]. We study analytically and numerically the asymptotic behavior of the front velocity in the neighborhood of the nascent optical bistability and close to the up-switching point of the bistability curve. We show that the value of the asymptotic linear growing velocity v∗ is corrected. This correction depends on the light intensity as v∗ ∝ |I − Ic |1/2 . We conclude in Sect. 4. 2.
Lugiato-Lefever model
We consider an optical bistable system consisting of a simple all-fiber cavity pumped by a continuous wave [1, 34]. A theoretical study of these devices has shown that the slowly varying envelope of the electric field E within the fiber cavity is governed by the dimensionless partial differential equation (the Lugiato-Lefever model [1]) ∂ E(t, ξ ) ∂2 = S + −1 − i(∆ − |E(t, ξ )|2) − iα 2 E(t, ξ ) (1) ∂t ∂ξ where S is the input field amplitude assumed to be real and positive to fix the reference phase, and ∆ is the detuning parameter. The slow time t is proportional to round-trip time in the cavity. In the fiber resonator the ξ coordinate plays the role of the fast time τ in a reference frame traveling at the group velocity of light in the Kerr material. The second order derivative with respect to ξ appearing in Eq. 1, describes the chromatic dispersion effect. The α parameter is the dispersion coefficient: α < 0 for anomalous dispersion and α > 0 for normal dispersion. Equation 1 has been derived first to describe the dynamics of diffractive Kerr cavity where the spatial coupling is provided by diffraction in the ξ = (x, y) transverse plane. The α parameter is then a diffraction coefficient that can be either positive or negative. The later situation has been theoretically predicted in resonators filled with photonic crystals [35] and in devices with a #195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 485
left-handed material, i.e., materials with simultaneously negative permittivity and permeability [36, 37]. The homogeneous steady states (HSS); (∂ E/∂√t = ∂ 2 E/√∂ ξ 2 = 0) of Eq. (1) are given by S2 = Is [1 + (∆ − Is )2 ] where Is = |Es |2 . For ∆ < 3 (∆ > 3), the transmitted intensity as a function of the input intensity S2 is monostable (bistable). The linear stability analysis shows that the HSS solution Is exhibits a modulational instability at IM = 1 and the critical frequency at threshold is given by Ω2M = (∆ − 2)/α . We shall focus our analysis in the bistable case. The linear analysis may be broken up into two cases, (i) when operating in the anomalous dispersion regime, i.e., α < 0, the lower HSS√is always unstable and the upper HSS exhibits a modulational instability when Is > 1 and for 3 < ∆ < 2. For ∆ = 2, the threshold associated with the modulational instability coincides with the up-switching point Iu,+ . For ∆ > 2, the modulational instability domain is bounded from below by the up-switching point Iu,+ = (2∆ + √ 2 ∆ − 3)/3, and extends to infinity for large intensities. (ii) when operating in the normal dispersion regime, i.e., α > 0, the upper HSS is always stable, and the modulationnal instability affects a portion of the lower HSS of the bistability curve. This happens only for ∆ > 2 and front propagation is drastically affected by modulational instability. Therefore, stable flat fronts, that we are interested with, exist only for ∆ < 2. Here, we only analyze the situation where α is positive (normal dispersion). 3.
Front velocity
√ We focus on the dynamics of fronts propagation in bistable regime ∆ > 3. In this regime, the modulational instability appears subcritically. So there exist a large domain of the injected field from which we can generate temporal cavity solitons [15, 19, 20]. These solutions are homoclinic solutions occurring in the coexistence regime involving a lower HSS and train of periodic solutions which are both linearly stable. In the following we consider heteroclinic solutions, namely a front connecting an unstable HSS (the intermediate HSS of the bistable) and the lower HSS. In the sequel we focus on the front dynamics in normal dispersion regime where no modulational instability occurs. To study how front propagate into an unstable homogeneous state, we perturb the system around the intermediate HHS of the bistable curve. The initial condition consists of a small localized perturbation in the form of Gaussian beam around the unstable intermediate (us , vs ) as (us − 0.01 exp(ξ /5)2 , vs ). us = Re(E), vs = Im(E). The time evolution of two flat fronts connecting the intermediate state and the lower HSS is shown in Figure 1. The temporal profile of flat fronts is shown in this figure where we have plotted, for clarity, the bistability cycle showing the unstable and the stable branches connected by the front. At the first stage, the localized perturbation grows gradually in time until it reaches its minimum value corresponding to the lower stable HSS. This spreading process is accompanied by a propagation of flat fronts into the region occupied by the unstable HSSs. As can be seen from the t − ξ map (Fig. 1), the spreading velocities to the left and to the right of the front are exactly the same. This is due to the fact that the model Eq. 1 preserves the reflection symmetry ξ → −ξ . Front propagation has been reported in systems subjected to convection or walk-off where the reflection symmetry is broken [25, 32, 38, 39]. In the long time evolution, numerical simulations of the model Eq. 1, show indeed that the transient velocity approaches asymptotically from below the linear spreading velocity v⋆ as shown in Figure 2a . From this figure we see that the kinetic laws governing flat front propagating varies according to the universal power law v(t) − v∗ ∼ 1/t, where v∗ is the asymptotic linear growing speed. The asymptotic form of the linear spreading velocity v⋆ is estimated analytically in terms of the dynamical parameters. To characterize the motion of the front connecting unstable HSS with stable one we decompose the electric field into its real and imaginary parts and consider the following perturbation around the homogeneous steady states
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 486
I Id
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Figure 1. Left panel: lower figure represents t − ξ map that shows the dynamics of a flat propagating front spreading into an unstable HSS. The black color indicates the intermediate unstable state and the white color represents the stable lower HSS solution. The up figure shows a cross section at a fixed time t as indicated in the figure. Right panel: top figure shows the bistability cycle with the branches connected by the front. The lower figure display the transient evolution of the front before reaching its asymptotic velocity. The parameters are α = 1, S = 1.27, and ∆ = 1.78.
(E, E ∗ ) = (Es , Es∗ ) + (δ E, δ E ∗ ) exp [−i (λ t − Ωξ )], where δ E and δ E ∗ are small perturbations. The frequency Ω satisfies the relation (∂ 2 /∂ ξ 2 + Ω2 ) exp (iΩξ ) = 0. Only one of the roots of the characteristic equation gives rise to the instability and it reads q 2 2 2 λ = i −1 + I − ((∆ − 2I) − α Ω ) . (2) We emphasize here that the classical linear stability theory is insufficient, as it stands, to explain the front motion because it applies to extended perturbations characterized by a single wave number. By contrast, in order to determine the linear response of the system to a localized perturbation, it is necessary to include a finite band of modes in the dynamical description. This can be achieved by reformulating the linear stability analysis as an initial-value problem. A detailed description of the concept and techniques of instabilities in terms of an initial-value problem analysis can be found in [40] (See also [30] and [33]). Here we just recall the conditions for a stable front to exist. ∂λ ℑ = 0, (3a) ∂Ω ℑ (λ ) ℑ (λ ) ∂λ and = v. (3b) ℜ = ℑ (Ω) ℑ (Ω) ∂Ω The first Eq. (3a) determines the most unstable career frequency that dominates the dynamics while the second Eqs. (3b) express the requirement that the front propagates into the unstable region without changing amplitude. This happens when the group velocity is identical to the envelope velocity of the front [first equality of Eq. (3b)] where the latter is defined by the second #195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 487
0.25 ∗ vnum
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Figure 2. (a) Transient velocity as a function of time that approaches for a long time evolution its asymptotic value v∗ obtained from numerical simulation of Eq. (1). (b) Asymptotic front velocity v∗ vs the intensity distance to critical point I − Ic . Dots are measured velocities and solid line reproduces the predicted velocities by Eq. (8). The parameters are the same as in Fig. 1.
equality in (3b). Since λ and Ω are in general complex, the above front equations determine the complex frequency and the front velocity. The use of a complex frequency is not surprising in this context: its real part accounts for the envelope oscillations and the imaginary part for the envelope front. Close to threshold, the dispersion relation reads 2 i ih2 I − 1 − δ − α Ω2 , λ≈ (4) 2 where δ = ∆ − 2I. By taking into account Eq. 4, we found ∂λ ℜ = −2α Ωi δ − α 3Ω2r + Ω2i , ∂Ω
(5)
where Ωr and Ωi are respectively the real and the imaginary part of Ω. The ℜ(.) denotes the real part. The value of Ωr characterizes the temporal modulation of the front. If Ωr = 0 then the front is flat and the asymptotic linear spreading velocity v⋆ which describes the rate at which an initial perturbation spreads into an unstable state is explicitly given from Eqs. 3 by v⋆ = −2α Ωi δ + α Ω2i± with (6) i h p 1 (7) Ω2i± = −δ ± 13I 2 − 16∆I + 4∆2 + 3 . 3α In these expressions of the frequency Ω, only Ω2i,− is relevant since Ω2i,+ < 0 in the region where the front exists. For flat fronts (Ωr = 0 ), the above expressions determine completely the characteristics of the front. However, these expressions are cumbersome for a physical interpretation. In order to get more insight about the front dynamics, we restrict our analysis to nascent bistability where the critical slowing down occurs. To this end, we introduce a small parameter ε ≪ 1 that measures the distance from√the critical point. This small parameter is defined as 1 2 I = Iu (1 + ε ), where I = Iu = 3 2∆ − ∆ − 3 represents the limiting curve for the front to i h √ exist. Indeed, Ω2i,− = − 31α (∆ − 2I) + 13I 2 − 16∆I + 4∆2 + 3 > 0 when I > Iu and vanishes #195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 488
together with the front velocity for I = Iu . Close to the lower switching point, the middle HSS is always unstable. To capture the asymptotic behavior of front velocity, we explore the vicinity of the up-switching point Is = Iu by expanding the expression of v∗ in Eq. (6) in the small parameter ε . First, we obtain at the leading order in ε the approximate expression of the frequency Ωi,− from Eq. (7) as √ Iu 2∆ − 3Iu 2 Ωi,− = ε + O(ε ε ) α 2Iu − ∆
Hence the velocity v∗ = v (Ωi,− ), can be obtained from Eq. (6) after substituting ε by its expression as function of I (see the definition of ε above), as v∗ = ±2α ∆2 − 3
1/4
√ (2Iu − ∆)1/2 I − Iu .
(8)
The last expression is the main result and allows to estimating the asymptotic propagation velocity of the front. The ± correspond to two opposite velocities as shown in Fig. 1 where two symmetric fronts are generated from a localized perturbation and √ propagate in the opposite directions. Note also that we recover the bistability condition (∆ > 3) for the existence of the front. However, the most interesting result is that the velocity varies with the distance from the intensity critical point leading to an intensity dependence. We have checked this analytical prediction by numerical simulations. We have plotted in Fig. 2b the asymptotic front velocity v∗ as function of I − Ic where solid line corresponds to the analytical expression (8) and dots denote the corresponding numerical result. As can be seen from the figure the agreement is excellent. We have also checked the accuracy of the analytical predictions by measuring the dimensionless numerical and theoretical velocities for the particular value of I − Ic = 1.1 (see ∗ = 0.2294 leading to an accuracy less than 1%. Of Fig. 2b). We found v∗num = 0.2286 and vth course this asymptotic velocity is reached after a transient as shown in Fig. 1 (the zoom left panel) where we have plotted the first stage of the front evolution. 4.
Conclusion
In conclusion, by using the Lugiato-Lefever model, we have investigated the front propagation in nonlinear all fiber cavities subject to injection. We investigate front propagation into an unstable state. We show that during time evolution, the velocity of flat propagating front evolves according to the universal power law , i.e., v(t) − v∗ ∼ 1/t, where v∗ is the asymptotic linear growing speed. We established analytically the formula for v∗ as a function of the intensity showing that the front velocity is propositional to |I − Ic |1/2 where Ic is the existence threshold of the propagating front. Acknowledgments M.T received support from the Fonds National de la Recherche Scientifique (Belgium). This research was supported by the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P7-35 and the French Project "ANR Blanc N12-BS04-0011-02".
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 489
S. Coulibaly,1 M. Taki,1 and M. Tlidi 2, 1 Laboratoire
de Physique des Lasers, Atomes et Molécules, CNRS UMR8523, Université des Sciences et Technologies de Lille, Batiment P5, 59655 Villeneuve d’Ascq Cedex, France 2 Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), Campus Plaine CP. 231, B-1050 Bruxelles, Belgium ∗ [email protected]
Abstract: We consider a bistable system consisting of all fiber cavity driven by an external injected continuous wave. We report on front propagation in a high finesse cavity. We study the asymptotic behavior of the front velocity. We show that the front velocity is affected by the distance from the critical point associated with bistability. We provide a scaling low governing its evolution near the up-switching point of the bistable curve. We show also that the velocity of front propagation obeys a generic power law when the front velocity approaches asymptotically its linear growing value. © 2014 Optical Society of America OCIS codes: (190.4420) Nonlinear optics; (190.4370) Nonlinear optics, fibers; (060.5530) Pulse propagation and temporal solitons; (070.5753) Resonators.
References 1. L.A. Lugiato and R. Lefever, "Spatial Dissipative Structures in Passive Optical Systems," Phys. Rev. Lett. 58, 2209–2211 (1987). 2. M. Tlidi, R. Lefever, and P. Mandel, "Pattern selection in optical bistability," Quantum Semiclass. Opt. 8 931–938 (1996). 3. F. Mitschke, G. Steinmeyer, and A. Schwache, "Generation of one-dimensional optical turbulence," Physica D 96, 251–258 (1996). 4. S. Coen and M. Haelterman, "Modulational instability induced by cavity boundary conditions in a normally dispersive optical fiber," Phys. Rev. Lett. 79, 4139-4142 (1997). 5. M. Stratmann, T. Pagel, and F. Mitschke, "Experimental observation of temporal soliton molecules," Phys. Rev. Lett. 95, 143902 (2005). 6. M. Tlidi, M. Haelterman, and P. Mandel,"3D patterns and pattern selection in optical bistability," Europhys. Lett. 42, 505–509 (1998). 7. K. Staliunas, "Three-Dimensional Turing Structures and Spatial Solitons in Optical Parametric Oscillators," Phys. Rev. Lett. 81, 81–85 (1998). 8. M. Tlidi, "Three-dimensional crystals and localized structures in diffractive and dispersive nonlinear ring cavities," J. Opt. B: Quantum Semiclass. Opt. 2, 438–442 (2000). 9. M. Brambilla, T. Maggipinto, G. Patera, and L. Columbo, "Cavity light bullets: Three-dimensional localized structures in a nonlinear optical resonator," Phys. Rev. Lett. 93, 203901 (2004). 10. P. Tassin, G. Van der Sande, N. Veretenov, P. Kockaert, I. Veretennicoff, and M. Tlidi, "Three-dimensional structures in nonlinear cavities containing left-handed materials", Optics Express, 14, 9338–9343 (2006). 11. M. Tlidi, M. Taki, and T. Kolokolnikov, "Introduction: Dissipative localized structures in extended systems", Chaos 17, 037101 (2007). 12. N. Akhmediev and A. Ankiewicz, Eds. Dissipative Solitons: From Optics to Biology and Medicine (SpringerVerlag, Berlin, Heidelberg, 2008). 13. O. Descalzi, M.G. Clerc, S. Residori, and G. Assanto, Localized States in Physics: Solitons and Patterns (Springer, 2010).
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 483
14. K. Staliunas, and V.J. Sanchez-Morcillo "Transverse Patterns in Nonlinear Optical Resonators" Springer Verlag, Springer Tracts in Modern Physics, Vol.183, 2003. 15. A.J. Scroggie, W.J. Firth, G.S McDonald, M. Tlidi, R. Lefever, and L.A Lugiato, "Pattern formation in a passive Kerr cavity," Chaos, Solitons and Fractals 4, 1323–1354 (1994). 16. M. Tlidi, P. Mandel, and R. Lefever, "Localized structures and localized patterns in optical bistability," Phys. Rev. Lett. 73, 640–643 (1994). 17. G. Slekys, K. Staliunas, C.O. Weiss, "Spatial localized structures in resonators with saturable absorber," Opt. Commun. 149, 113–116 (1998). 18. V.B. Taranenko, K. Staliunas, and C. O. Weiss, "Spatial soliton laser: Localized structures in a laser with a saturable absorber in a self-imaging resonator," Phys. Rev. A56, 1582–1591 (1997). 19. V. Odent, M. Taki, and E. Louvergneaux, "Experimental evidence of dissipative spatial solitons in an optical passive Kerr cavity," New J. Phys. 13, 113026 (2011). 20. F. Leo, S. Coen, P. Kockaert, S-P. Gorza, P. Emplit, and M. Haelterman, "Temporal cavity solitons in onedimensional Kerr media as bits in an all-optical buffer," Nature Photon. 4, 471–476 (2010). 21. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, "Dynamics of one-dimensional Kerr cavity solitons," Opt. Express 21, 9180–9191 (2013). 22. D. Gomila, M. A. Matias, and P. Colet, "Excitability mediated by localized structures in a dissipative nonlinear optical cavity," Phys. Rev. Lett. 94, 063905 (2005). 23. D. Turaev, A. G. Vladimirov, and S. Zelik, "Long-range interaction and synchronization of oscillating dissipative solitons", Phys. Rev. Lett. 108, 263906-263909 (2012). 24. O.A Egorov and F. Lederer, "Spontaneously walking discrete cavity solitons," Opt. Lett. 38, 1010–1012 (2013). 25. S. Coen, M. Tlidi, Ph. Emplit, and M. Haelterman, "Convection versus dispersion in optical bistability," Phys. Rev. Lett. 83, 2328–2331 (1999). 26. K.Staliunas and V.J. Sanchez-Morcillo, "Dynamics of phase domains in the Swift-Hohenberg equation," Phys. Lett. A241 28–34 (1998). 27. M. Tlidi, P. Mandel, and R. Lefever, "Kinetics of localized pattern formation in optical systems," Phys. Rev. Lett. 81, 979–982 (1998). 28. M. Tlidi, Paul Mandel, M. Le Berre, E. Ressayre, A. Tallet, and L. Di Menza, "Phase-separation dynamics of circular domain walls in the degenerate optical parametric oscillator," Optics Lett. 25, 487–489 (2000). 29. V. B. Taranenko, K. Staliunas, and C.O. Weiss, "Pattern formation and localized structures in degenerate optical parametric mixing", Phys. Rev. Lett. 81 2236–2239 (1998). 30. G.T. Dee and J.S. Langer, "Propagating pattern selection," Phys. Rev. Lett. 50, 383–386 (1983). 31. G.T. Dee and W. van Saarloos, "Bistable Systems with Propagating Fronts Leading to Pattern Formation," Phys. Rev. Lett. 60, 2641–2644 (1988). 32. M. Taki, M. N. Ouarzazi, H. Ward, and P. Glorieux, "Nonlinear front propagation in optical parametric oscillators," J. Opt. Soc. Am. B 17, 997–1003 (2000). 33. W. van Saarloos, "Front propagation into unstable states," Phys. Rep. 286, 29–222 (2003). 34. A. Mussot, E. Louvergneaux, N. Akhmediev, F. Reynaud, L. Delage, and M. Taki, "Optical fiber systems are convectively unstable", Phys. Rev. Lett. 101, 113904-113907 (2008). 35. Kestutis Staliunas, Oleg Egorov, Yuri S. Kivshar, and Falk Lederer, "Bloch Cavity Solitons in Nonlinear Resonators with Intracavity Photonic Crystals," Phys. Rev. Lett. 101, 153903-153906 (2008). 36. P. Kockaert, P. Tassin, G. Van der Sande, I. Veretennicoff, and M.Tlidi, "Negative diffraction pattern dynamics in nonlinear cavities with left-handed materials," Phys. Rev. A 74, 033822 (2006). 37. L. Gelens, G. Van der Sande, P. Tassin, M. Tlidi, P. Kockaert, D. Gomila, I. Veretennicoff, and J. Danckaert, "Impact of nonlocal interactions in dissipative systems: Towards minimal-sized localized structures," Phys. Rev. A 75, 063812 (2007). 38. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, "Noise-sustained convective structures in nonlinear optics," Phys. Rev. Lett. 79, 3633–3636 (1997). 39. S. Coulibaly, C. Durniak, and M. Taki, "Spatial Dissipative Solitons Under Convective and Absolute Instabilities in Optical Parametric Oscillators," Lecture Notes in Physics, 751, 261–285 (2008). 40. H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, "Influence of walkoff on pattern formation in nondegenerate optical parametric oscillators," Phys. Rev. E 63, 016604 (2000).
1.
Introduction
In recent years a considerable progress has been realized in the understanding of localized structures (LS) in nonlinear optical systems. These structures are often called cavity solitons. They have been found in optical cavity subject to continuous injection and energy dissipation. They can be classified into three types (i) spacial cavities: LS appear thanks to the interplay between diffraction and nonlinearity [1, 2], (ii) temporal cavities: LS result from the interplay between
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 484
dispersion and nonlinearity [3–5], (iii) spatio-temporal cavities: three-dimensional light bullets appear as a result of combined influence of diffraction and dispersion, and nonlinearity [6–10]. These simple and robust devices have attracted a considerable attention both from fundamental as well as from applied point of views (see latest overviews on this issue [11–14]). The prerequisite condition for light confinement leading to stabilization of LSs is the occurrence of subcritical modulational instability where a coexistence between a homogeneous background and a self-organized periodic structure occurs [15, 16]. These dissipative solitons have been evidenced in photorefractive oscillator with intracavity saturable absorber [17] and in a laser with saturable absorber in a self-imaging resonator [18]. However, it is only recently that experimental evidence of cavity solitons in a passive cavity has been demonstrated in the spatial domain by using the Kerr-like medium [19]. In the temporal domain, temporal cavity solitons have also been observed in passive fiber cavity and an estimation of the capacity to operate as all-optical memories is given: a very high value of 45 Kbits at 25 Gbits/s [20]. For large intensities, temporal cavity solitons could exhibit a self-pulsation or chaotic behavior [21–24]. The formation of cavity solitons is relatively a well-understood issue. However, front propagation in driven optical fiber cavities has received only a scant attention [25]. Note that spatial fronts can also exist in 2D configuration giving rise to domain walls. The latter have been theoretically studied in Swift-Hohenberg model in [26–28], and observed in degenerate four-wave-mixing oscillators in [29]. In this paper, we study front propagation into an unstable intermediate homogeneous steady state in a driven all fiber cavity. In Sect. 2, we present the Lugiato-Lefever model and we summarize a linear stability analysis of the homogeneous steady states. In Sect. 3, we show that during time evolution t, the transient velocity of single propagating flat front varies according to the universal power law v(t) − v∗ ∼ 1/t, where v∗ is the asymptotic linear growing velocity. Such a generic transition law has been predicted in real order parameter equations for fluid mechanics [30,31,33]. We study analytically and numerically the asymptotic behavior of the front velocity in the neighborhood of the nascent optical bistability and close to the up-switching point of the bistability curve. We show that the value of the asymptotic linear growing velocity v∗ is corrected. This correction depends on the light intensity as v∗ ∝ |I − Ic |1/2 . We conclude in Sect. 4. 2.
Lugiato-Lefever model
We consider an optical bistable system consisting of a simple all-fiber cavity pumped by a continuous wave [1, 34]. A theoretical study of these devices has shown that the slowly varying envelope of the electric field E within the fiber cavity is governed by the dimensionless partial differential equation (the Lugiato-Lefever model [1]) ∂ E(t, ξ ) ∂2 = S + −1 − i(∆ − |E(t, ξ )|2) − iα 2 E(t, ξ ) (1) ∂t ∂ξ where S is the input field amplitude assumed to be real and positive to fix the reference phase, and ∆ is the detuning parameter. The slow time t is proportional to round-trip time in the cavity. In the fiber resonator the ξ coordinate plays the role of the fast time τ in a reference frame traveling at the group velocity of light in the Kerr material. The second order derivative with respect to ξ appearing in Eq. 1, describes the chromatic dispersion effect. The α parameter is the dispersion coefficient: α < 0 for anomalous dispersion and α > 0 for normal dispersion. Equation 1 has been derived first to describe the dynamics of diffractive Kerr cavity where the spatial coupling is provided by diffraction in the ξ = (x, y) transverse plane. The α parameter is then a diffraction coefficient that can be either positive or negative. The later situation has been theoretically predicted in resonators filled with photonic crystals [35] and in devices with a #195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 485
left-handed material, i.e., materials with simultaneously negative permittivity and permeability [36, 37]. The homogeneous steady states (HSS); (∂ E/∂√t = ∂ 2 E/√∂ ξ 2 = 0) of Eq. (1) are given by S2 = Is [1 + (∆ − Is )2 ] where Is = |Es |2 . For ∆ < 3 (∆ > 3), the transmitted intensity as a function of the input intensity S2 is monostable (bistable). The linear stability analysis shows that the HSS solution Is exhibits a modulational instability at IM = 1 and the critical frequency at threshold is given by Ω2M = (∆ − 2)/α . We shall focus our analysis in the bistable case. The linear analysis may be broken up into two cases, (i) when operating in the anomalous dispersion regime, i.e., α < 0, the lower HSS√is always unstable and the upper HSS exhibits a modulational instability when Is > 1 and for 3 < ∆ < 2. For ∆ = 2, the threshold associated with the modulational instability coincides with the up-switching point Iu,+ . For ∆ > 2, the modulational instability domain is bounded from below by the up-switching point Iu,+ = (2∆ + √ 2 ∆ − 3)/3, and extends to infinity for large intensities. (ii) when operating in the normal dispersion regime, i.e., α > 0, the upper HSS is always stable, and the modulationnal instability affects a portion of the lower HSS of the bistability curve. This happens only for ∆ > 2 and front propagation is drastically affected by modulational instability. Therefore, stable flat fronts, that we are interested with, exist only for ∆ < 2. Here, we only analyze the situation where α is positive (normal dispersion). 3.
Front velocity
√ We focus on the dynamics of fronts propagation in bistable regime ∆ > 3. In this regime, the modulational instability appears subcritically. So there exist a large domain of the injected field from which we can generate temporal cavity solitons [15, 19, 20]. These solutions are homoclinic solutions occurring in the coexistence regime involving a lower HSS and train of periodic solutions which are both linearly stable. In the following we consider heteroclinic solutions, namely a front connecting an unstable HSS (the intermediate HSS of the bistable) and the lower HSS. In the sequel we focus on the front dynamics in normal dispersion regime where no modulational instability occurs. To study how front propagate into an unstable homogeneous state, we perturb the system around the intermediate HHS of the bistable curve. The initial condition consists of a small localized perturbation in the form of Gaussian beam around the unstable intermediate (us , vs ) as (us − 0.01 exp(ξ /5)2 , vs ). us = Re(E), vs = Im(E). The time evolution of two flat fronts connecting the intermediate state and the lower HSS is shown in Figure 1. The temporal profile of flat fronts is shown in this figure where we have plotted, for clarity, the bistability cycle showing the unstable and the stable branches connected by the front. At the first stage, the localized perturbation grows gradually in time until it reaches its minimum value corresponding to the lower stable HSS. This spreading process is accompanied by a propagation of flat fronts into the region occupied by the unstable HSSs. As can be seen from the t − ξ map (Fig. 1), the spreading velocities to the left and to the right of the front are exactly the same. This is due to the fact that the model Eq. 1 preserves the reflection symmetry ξ → −ξ . Front propagation has been reported in systems subjected to convection or walk-off where the reflection symmetry is broken [25, 32, 38, 39]. In the long time evolution, numerical simulations of the model Eq. 1, show indeed that the transient velocity approaches asymptotically from below the linear spreading velocity v⋆ as shown in Figure 2a . From this figure we see that the kinetic laws governing flat front propagating varies according to the universal power law v(t) − v∗ ∼ 1/t, where v∗ is the asymptotic linear growing speed. The asymptotic form of the linear spreading velocity v⋆ is estimated analytically in terms of the dynamical parameters. To characterize the motion of the front connecting unstable HSS with stable one we decompose the electric field into its real and imaginary parts and consider the following perturbation around the homogeneous steady states
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 486
I Id
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Figure 1. Left panel: lower figure represents t − ξ map that shows the dynamics of a flat propagating front spreading into an unstable HSS. The black color indicates the intermediate unstable state and the white color represents the stable lower HSS solution. The up figure shows a cross section at a fixed time t as indicated in the figure. Right panel: top figure shows the bistability cycle with the branches connected by the front. The lower figure display the transient evolution of the front before reaching its asymptotic velocity. The parameters are α = 1, S = 1.27, and ∆ = 1.78.
(E, E ∗ ) = (Es , Es∗ ) + (δ E, δ E ∗ ) exp [−i (λ t − Ωξ )], where δ E and δ E ∗ are small perturbations. The frequency Ω satisfies the relation (∂ 2 /∂ ξ 2 + Ω2 ) exp (iΩξ ) = 0. Only one of the roots of the characteristic equation gives rise to the instability and it reads q 2 2 2 λ = i −1 + I − ((∆ − 2I) − α Ω ) . (2) We emphasize here that the classical linear stability theory is insufficient, as it stands, to explain the front motion because it applies to extended perturbations characterized by a single wave number. By contrast, in order to determine the linear response of the system to a localized perturbation, it is necessary to include a finite band of modes in the dynamical description. This can be achieved by reformulating the linear stability analysis as an initial-value problem. A detailed description of the concept and techniques of instabilities in terms of an initial-value problem analysis can be found in [40] (See also [30] and [33]). Here we just recall the conditions for a stable front to exist. ∂λ ℑ = 0, (3a) ∂Ω ℑ (λ ) ℑ (λ ) ∂λ and = v. (3b) ℜ = ℑ (Ω) ℑ (Ω) ∂Ω The first Eq. (3a) determines the most unstable career frequency that dominates the dynamics while the second Eqs. (3b) express the requirement that the front propagates into the unstable region without changing amplitude. This happens when the group velocity is identical to the envelope velocity of the front [first equality of Eq. (3b)] where the latter is defined by the second #195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 487
0.25 ∗ vnum
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Figure 2. (a) Transient velocity as a function of time that approaches for a long time evolution its asymptotic value v∗ obtained from numerical simulation of Eq. (1). (b) Asymptotic front velocity v∗ vs the intensity distance to critical point I − Ic . Dots are measured velocities and solid line reproduces the predicted velocities by Eq. (8). The parameters are the same as in Fig. 1.
equality in (3b). Since λ and Ω are in general complex, the above front equations determine the complex frequency and the front velocity. The use of a complex frequency is not surprising in this context: its real part accounts for the envelope oscillations and the imaginary part for the envelope front. Close to threshold, the dispersion relation reads 2 i ih2 I − 1 − δ − α Ω2 , λ≈ (4) 2 where δ = ∆ − 2I. By taking into account Eq. 4, we found ∂λ ℜ = −2α Ωi δ − α 3Ω2r + Ω2i , ∂Ω
(5)
where Ωr and Ωi are respectively the real and the imaginary part of Ω. The ℜ(.) denotes the real part. The value of Ωr characterizes the temporal modulation of the front. If Ωr = 0 then the front is flat and the asymptotic linear spreading velocity v⋆ which describes the rate at which an initial perturbation spreads into an unstable state is explicitly given from Eqs. 3 by v⋆ = −2α Ωi δ + α Ω2i± with (6) i h p 1 (7) Ω2i± = −δ ± 13I 2 − 16∆I + 4∆2 + 3 . 3α In these expressions of the frequency Ω, only Ω2i,− is relevant since Ω2i,+ < 0 in the region where the front exists. For flat fronts (Ωr = 0 ), the above expressions determine completely the characteristics of the front. However, these expressions are cumbersome for a physical interpretation. In order to get more insight about the front dynamics, we restrict our analysis to nascent bistability where the critical slowing down occurs. To this end, we introduce a small parameter ε ≪ 1 that measures the distance from√the critical point. This small parameter is defined as 1 2 I = Iu (1 + ε ), where I = Iu = 3 2∆ − ∆ − 3 represents the limiting curve for the front to i h √ exist. Indeed, Ω2i,− = − 31α (∆ − 2I) + 13I 2 − 16∆I + 4∆2 + 3 > 0 when I > Iu and vanishes #195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 488
together with the front velocity for I = Iu . Close to the lower switching point, the middle HSS is always unstable. To capture the asymptotic behavior of front velocity, we explore the vicinity of the up-switching point Is = Iu by expanding the expression of v∗ in Eq. (6) in the small parameter ε . First, we obtain at the leading order in ε the approximate expression of the frequency Ωi,− from Eq. (7) as √ Iu 2∆ − 3Iu 2 Ωi,− = ε + O(ε ε ) α 2Iu − ∆
Hence the velocity v∗ = v (Ωi,− ), can be obtained from Eq. (6) after substituting ε by its expression as function of I (see the definition of ε above), as v∗ = ±2α ∆2 − 3
1/4
√ (2Iu − ∆)1/2 I − Iu .
(8)
The last expression is the main result and allows to estimating the asymptotic propagation velocity of the front. The ± correspond to two opposite velocities as shown in Fig. 1 where two symmetric fronts are generated from a localized perturbation and √ propagate in the opposite directions. Note also that we recover the bistability condition (∆ > 3) for the existence of the front. However, the most interesting result is that the velocity varies with the distance from the intensity critical point leading to an intensity dependence. We have checked this analytical prediction by numerical simulations. We have plotted in Fig. 2b the asymptotic front velocity v∗ as function of I − Ic where solid line corresponds to the analytical expression (8) and dots denote the corresponding numerical result. As can be seen from the figure the agreement is excellent. We have also checked the accuracy of the analytical predictions by measuring the dimensionless numerical and theoretical velocities for the particular value of I − Ic = 1.1 (see ∗ = 0.2294 leading to an accuracy less than 1%. Of Fig. 2b). We found v∗num = 0.2286 and vth course this asymptotic velocity is reached after a transient as shown in Fig. 1 (the zoom left panel) where we have plotted the first stage of the front evolution. 4.
Conclusion
In conclusion, by using the Lugiato-Lefever model, we have investigated the front propagation in nonlinear all fiber cavities subject to injection. We investigate front propagation into an unstable state. We show that during time evolution, the velocity of flat propagating front evolves according to the universal power law , i.e., v(t) − v∗ ∼ 1/t, where v∗ is the asymptotic linear growing speed. We established analytically the formula for v∗ as a function of the intensity showing that the front velocity is propositional to |I − Ic |1/2 where Ic is the existence threshold of the propagating front. Acknowledgments M.T received support from the Fonds National de la Recherche Scientifique (Belgium). This research was supported by the Interuniversity Attraction Poles program of the Belgian Science Policy Office, under grant IAP P7-35 and the French Project "ANR Blanc N12-BS04-0011-02".
#195233 - $15.00 USD (C) 2014 OSA
Received 5 Aug 2013; revised 4 Oct 2013; accepted 5 Oct 2013; published 3 Jan 2014 13 January 2014 | Vol. 22, No. 1 | DOI:10.1364/OE.22.000483 | OPTICS EXPRESS 489
