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Home Search Collections Journals About Contact us My IOPscience Dark solitons of the power-energy saturation model: application to mode-locked lasers This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2013 J. Phys. A: Math. Theor. 46 095201 (http://iopscience.iop.org/1751-8121/46/9/095201) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 76.118.79.222 The article was downloaded on 13/02/2013 at 03:49 Please note that terms and conditions apply. IOP PUBLISHING JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL J. Phys. A: Math. Theor. 46 (2013) 095201 (18pp) doi:10.1088/1751-8113/46/9/095201 Dark solitons of the power-energy saturation model: application to mode-locked lasers M J Ablowitz 1 , S D Nixon 2 , T P Horikis 3 and D J Frantzeskakis 4 1 Department of Applied Mathematics, University of Colorado, 526 UCB, Boulder, CO 80309-0526, USA 2 Department of Mathematics and Statistics, University of Vermont, Burlington, VT 05401, USA 3 Department of Mathematics, University of Ioannina, Ioannina 45110, Greece 4 Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece E-mail: [email protected], [email protected], [email protected] and [email protected] Received 4 November 2012, in final form 16 January 2013 Published 12 February 2013 Online at stacks.iop.org/JPhysA/46/095201 Abstract The generation and dynamics of dark solitons in mode-locked lasers is studied within the framework of a nonlinear Schrödinger equation which incorporates power-saturated loss, as well as energy-saturated gain and filtering. Modelocking into single dark solitons and multiple dark pulses are found by employing different descriptions for the energy and power of the system defined over unbounded and periodic (ring laser) systems. Treating the loss, gain and filtering terms as perturbations, it is shown that these terms induce an expanding shelf around the soliton. The dark soliton dynamics are studied analytically by means of a perturbation method that takes into regard the emergence of the shelves and reveals their importance. PACS numbers: 42.65.Tg, 42.65.Sf, 05.45.Yv, 02.30.Ik (Some figures may appear in colour only in the online journal) 1. Introduction Dark solitons, namely envelope solitons having the form of density dips with a phase jump across their density minimum, are fundamental nonlinear excitations of the defocusing nonlinear Schrödinger (NLS) equation. They are termed black if the density minimum is zero and gray otherwise. These structures are extremely robust and interact elastically with each other [1]. Their discovery, which dates back to early 70’s [2, 3], was followed by intensive study both in theory and in experiment: in fact, the emergence of dark solitons on a modulationally stable background is a fundamental phenomenon arising in diverse physical settings. Indeed, dark solitons have been observed and studied in numerous contexts including: 1751-8113/13/095201+18$33.00 © 2013 IOP Publishing Ltd Printed in the UK & the USA 1 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al discrete mechanical systems [4], electrical lattices [5, 6], magnetic films [7, 8], plasmas [9, 10], fluids [11, 12], atomic Bose–Einstein condensates [13] and nonlinear optics [14]. In nonlinear optics, dark solitons are predicted to have some advantages as compared to their bright counterparts (which are supported by the focusing NLS model). Amongst others, dark solitons can be generated by a threshold-less process [15], are less affected by loss [16, 17] and background noise [16], and are more robust against Gordon–Haus jitter [18], higher order dispersion [19] etc. Dark optical solitons have potential application in, for example, inducing steerable waveguides in optical media [20–22], or for ultradense wavelength-division-multiplexing [23] (see also [14] and references therein). Recently, there has been an interest in ‘dark pulse lasers’, namely laser systems emitting trains of dark solitons on top of the continuous wave (cw) emitted by the laser; various experimental results have been reported utilizing fiber ring lasers [24–28] (see also [29, 30]), quantum dot diode lasers [31] and dual Brillouin fiber lasers [32]. These works, apart from introducing a method for a systematic and controllable generation of dark solitons, can potentially lead to other important applications related, e.g., with optical frequency combs, optical atomic clocks, and others. An important aspect in these studies is the ability of the laser system to induce a fixed phase relationship between the modes of the laser’s resonant cavity, i.e. to mode-lock; in such a case, interference between the laser modes in the normal dispersion regime causes the formation of a sequence of dark pulses on top of the stable cw background emitted by the laser [24, 31]. A mode-locked (ML) laser, much like any optical oscillator, requires two basic constituents, namely amplification and feedback. From a theoretical point of view, this poses a particularly challenging problem in modeling pulses in the cavity of a ML laser: pulse propagation in this setting should be studied in the presence of dispersion, nonlinearity, as well as loss, gain and filtering. Furthermore, for dark pulses, the model should be subject to nontrivial boundary conditions at large distances from the source—as in the case of dark soliton solutions of the defocusing NLS equation. Additionally, the dynamics and modelocking ability of the system should be demonstrated in settings of practical relevance as, e.g., in the case of a finite periodic domain, so as to take into regard the periodic round trips of light in the laser cavity. Finally, the dark soliton dynamics should be studied in the physically relevant situation where all the above physical mechanisms and mathematical requirements are present. In this work, we employ the so-called power-energy saturation (PES) equation [33–36], with nonvanishing boundary conditions at infinity. This model was recently used in [37], where it was shown that—under experimentally relevant requirements—general initial conditions mode-lock into dark solitons, resembling corresponding solutions of the pertinent unperturbed NLS model. Here, we study and compare two variants of the PES equation, corresponding to two alternative definitions of the energy (and power) of the system. For each model, we investigate the generation of single and multiple dark solitons, considering also the case of periodic domains with appropriate boundary conditions. We also study analytically the evolution of dark solitons by means of a perturbation method, recently introduced in [38]. This method allows us to reveal the role of the shelf (that expands from each dark pulse) on the dynamics and interactions of solitons. Our analytical findings are corroborated by direct numerical simulations. This paper is organized as follows. In section 2 the PES model and its two variants are introduced and mode-locking into both single and multiple dark solitons is demonstrated. In section 3, first we briefly describe our perturbative approach, and then apply it to the two variants of the PES model. Direct numerical simulations, are in good agreement with the 2 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al analytical findings. Finally, in section 4, we summarize our findings and discuss further future studies. 2. The model and mode-locking 2.1. The PES equation and its variants Dark solitons in ML lasers, within the framework of an NLS type equation, namely the PES equation [33–36], are analyzed using the methodology developed in [37]. The PES equation can be expressed in the following dimensionless form: 1 ig iτ il u+ utt − u, iuz − utt + |u|2 u = 2 1 + E/Esat 1 + E/Esat 1 + P/Psat (1) where the complex electric field envelope u(z, t ), z being the direction of propagation and t retarded time, is subject to the boundary conditions |u(z, t )| → u∞ (z) as |t| → ∞. In equation (1), P and E represent the power and energy of the system, while Psat and Esat denote corresponding saturation values, respectively. Furthermore, g, τ , and l are positive real constants, with the corresponding terms representing gain, spectral filtering (both saturating with energy), and loss (saturating with power). Typical dimensional numbers can be found in [35]. The PES equation extends the so-called Haus master laser equation [39] to allow for power saturation. This is discussed in some detail in [33, 34]. We also note that different types of NLS equations with different applications from those studied in this paper (ML lasers) have been considered in the context of bright [40–43] and dark [44–46] pulses. Assuming an unbounded domain, it is clear that the right-hand side of equation (1) does not vanish at t → ±∞ and, thus, the background wave (supporting the dark soliton) should have a nontrivial dependence on the evolution variable z. To determine the evolution of the background wave, we seek solutions of equation (1) in the form u(z) = u∞ (z) exp[iθ (z)], where u∞ (z) and θ (z) are real functions denoting the amplitude and phase of the background respectively. Substituting this ansatz into equation (1), and separating real and imaginary parts, we obtain an equation connecting the phase with the amplitude, namely dθ /dz = u2∞ , as well as the following equation for the evolution of the amplitude u∞ (z): du∞ l g u∞ . − (2) = dz 1 + E/Esat 1 + P/Psat To proceed further, E and P must be expressed in terms of u∞ . Since non-zero boundary conditions +∞ 2 at infinity are imposed, it is clear that the common energy definition Es = −∞ |u| dt is not finite and, thus, a different definition for E and P (the energy is the integral of power) is required. Below, we investigate two different alternatives: first, the energy is taken as the drop in energy associated with the dip in the background (i.e. dark energy); second, the standard definition of energy is used, but a finite domain is considered. The first model is based on the dark or renormalized energy (RE), which is natural +∞in order to discuss conserved quantities of the pure NLS equation [47], given by E(z) = −∞ (u2∞ − |u|2 ) dt; accordingly, the instantaneous power, P(z, t ) = u2∞ − |u|2 , follows consistently from +∞ E = −∞ Pdt. This definition of the pulse’s energy, although unconventional, can be construed in terms of physical properties of the system. Indeed, dark pulses can be thought of as focusing the vacuum. As the peak-to-background intensity ratio can achieve dramatic numbers in bright pulses, so can the focused-to-unfocused vacuum ratio in dark pulses [48]. We refer to this model as the RE model. 3 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al In the framework of this model, we assume an approximate solution of equation (1) in the form of a stationary (black) dark soliton, i.e. u(z, t ) = u∞ tanh(u∞t ) exp(iu2∞ z); in this case, E = 2u∞ , while P → 0 as t → ±∞. For these values of E and P, equation (2) reads: g du∞ − l u∞ , (3) = dz 1 + 2u∞ /Esat and, thus, the equilibrium of equation (3) corresponding to du∞ /dz = 0 is: Esat g −1 . (4) u∞ = 2 l The above expression sets the background amplitude of the dark soliton, and implies that locking onto dark solitons is only achieved if the gain is large enough to counter balance the loss (g/l > 1). The simple physical picture stemming from this analysis is that the combined perturbation of gain, filtering and loss in the RE model, mode-locks to a black soliton of the pure NLS equation, with the appropriate background—cf equation (4). This approximate result was recently confirmed by direct numerical simulations [37]. Nevertheless, at a later section (section 3.2), we will further investigate the above approximations by means of the perturbation theory for dark solitons devised in [38] and we will consider stability over long distances. To study the mode-locking capabilities of our model, we integrate equation (1) starting with a step profile of the form u(0, t ) = (1/2)[H(t − c) − H(−(t + c))], c > 0 small, where H(t ) is the Heavyside function. The parameters are kept fixed with typical values, namely Esat = Psat = 1 and τ = l = 0.1 and g = 0.5. In the top panel of figure 1 we show the evolution of the background amplitude, which sets the dark soliton amplitude for these values. In order to lock onto dark soliton solutions the gain term needs to be sufficiently strong, i.e. the parameter g be large enough, to counter balance the losses. As it is also clear from equation (4), mode-locking occurs when g > l and when that happens the resulting pulse is a soliton of the unperturbed system, as shown in figure 1. The second consideredmodel is based on assuming a finite domain and using averaged T/2 power, namely E = (1/T ) −T/2 P(z, t ) dt, where P = |u(z, t )|2 and T is the averaged time through the cavity. We term this the average power (AP) model. In this case, considering again the above mentioned approximate dark soliton solution of equation T/2(1), we may approximate the energy and power, for sufficiently large T , as follows: E = T1 −T/2 |u|2 dt ≈ T1 u2∞ T = u2∞ and P = u2∞ . Thus, in this case, equation (2) becomes g du∞ l u∞ . = − (5) dz 1 + u2∞ /Esat 1 + u2∞ /Psat Let us assume a finite domain defined by the round trip time T . In this setting, it is T/2 physically relevant to consider the AP Pav = T1 −T/2 |u|2 dt which, for sufficiently large T , can be approximated as: Pav ≈ T1 u2∞ T = u2∞ . Then, the energy results consistently from the usual definition, E = Pav T = u2∞ T [49]. This way, we can now observe that where P̃sat E u2 T Pav = ∞ = , Esat Esat P̃sat = Esat /T . Thus, equation (2) becomes g du∞ l = − u∞ , dz 1 + u2∞ /Psat 1 + u2∞ /P̃sat leading to the equilibrium: g−l . u2∞ = l/P̃sat − g/Psat 4 (6) (7) (8) J. Phys. A: Math. Theor. 46 (2013) 095201 PES soliton NLS soliton 2 1 1 0 0 u u 2 M J Ablowitz et al −1 −1 −2 −2 −20 −10 0 t 10 −20 20 initial input locked pulse −10 0 t 10 20 3 |u| 2 1 0 20 10 300 200 100 0 t −10 −20 0 z Figure 1. Top left panel: the initial step profile (dashed (red) line) and the resulting ML dark soliton profile (solid (blue) line) are depicted. Top right panel: comparison between the profiles of the resulting soliton (solid (blue) line) and the corresponding soliton of the unperturbed NLS equation (dashed (red) line), i.e. u(z, t ) = u∞ tanh(u∞ t ) exp(iu2∞ z). Bottom panel: evolution of a step initial profile under the RE version of the PES equation. From equation (8) we see that for mode-locking to occur either g > l and g/l < Psat /P̃sat , or g < l and g/l > Psat /P̃sat . Equation (8) is a slightly different condition for mode-locking then equation (4)], as it also depends on Esat , Psat . 2.2. Shelves and periodic domains Though it is less prominent in the RE model, both the RE model and the AP model exhibit shelves [37] that are developed on the soliton background and expand away from the dark pulse. Such shelves are explained in the perturbation theory for dark solitons [38] and, as we will discuss in more detail below, they play an important role on dark soliton dynamics. An example of the development of a (small) shelf in the RE model is shown in figure 2. The AP description also results in more complicated dynamics and the development of a shelf on the soliton background, which is more pronounced as compared to the RE case. In fact shelves occur naturally in the perturbation theory for dark solitons. We illustrate the resulting shelf in figure 3, where now g = 0.2, l = 0.1, Esat = 1 and Psat = 5. Due to a noisy background which occurs in experiments [24] this feature maybe difficult to observe in an experiment. However, the shelf also affects the phase of the resulting pulse (see inset in figure 3). In a physical system, where the domain is both finite and periodic (i.e. a ring laser), the shelves will begin interacting. In this section we consider multiple dark solitons on a 5 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al 1.05 Predicted Shelf Height 1.04 1.5 |u| |u| 1.045 1.035 1.03 1 0.5 0 −20 −20 −10 0 t 10 t 20 20 Figure 2. The temporal profile of the amplitude of a black soliton in the RE model. Starting with the initial condition of a black soliton with predicted equilibrium background, the development of a small shelf around the soliton occurs. The inset shows the magnitude of the soliton; the scale is too large for the shelf to be seen. Here, g = 0.3, τ = 0.05, l = 0.1 and Esat = Psat = 1. PES NLS 1.5 |u (t)| moving shelf moving shelf 1 0.5 phase 4 π −4 0 −30 π −15 0 t 15 30 Figure 3. The development of a shelf in the solution to the PES equation using the AP model (solid line) as compared to the solution of the NLS equation (dashed line). Here, g = 0.2, τ = 0.05, l = 0.1, Esat = 1, and Psat = 5 and the resulting shelf size is approximately 0.05. finite domain, [−T/2, T/2], with periodic boundary conditions. The period with respect to the retarded time corresponds to the time it takes light to traverse to the laser ring once and thus the boundaries represent the same point in the real space-time. In such a case, a single soliton is insufficient to satisfy the periodic boundary conditions, since the phase change across the pulse is between zero (no soliton) and π (a black soliton); in the physical system the total phase change must be a multiple of 2π . Initial conditions which lead to a chain of N dark solitons are given by u(0, t ) ≈ u∞ N uk (t ), (9) k=1 uk (0, t ) = cos αk + i sin αk tanh[u∞ sin αk (t − tk )], (10) where u∞ is the background height, −2π αk 2π is the phase change across the k-th soliton and tk is the center of the kth soliton. To satisfy the periodic boundary conditions we require N k=1 6 αk = mπ , for some m ∈ Z. (11) J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al |u| 1 0 3 2 1 0 −1 −30 0.5 0 6 4 2 0 −30 φ 0.5 φ |u| 1 −20 −10 0 t 10 20 30 −20 −10 0 t 10 20 30 Figure 4. Left panel: the temporal profiles of the initial amplitude and phase of two solitons with opposite phase change, respectively. Right panel: same as in left, but for a chain of three solitons whose phase changes add up to 2π . 8 z = 300 6 Phase 1.5 |u | 1 2 0.5 0 −50 4 −25 300 200 100 0 t 25 50 0 z 0 −50 −25 0 t 25 50 Figure 5. Left panel: evolution of the amplitude of a two-black-soliton state in the RE model with periodic boundary conditions. Right panel: the temporal profile of the phase of this state at z = 300. Some ways to satisfy this condition are two solitons with opposite phase change (including the degenerate case of two black solitons) and a chain of three solitons whose phase changes have the same sign and add up to 2π . This is illustrated in figure 4. We begin by considering initial conditions consisting of two black solitons. In the RE model the evolution settles on an equilibrium solution which is close to two black solitons. A slight difference can be seen in the phase profile which exhibits curvature between the solitons (for unperturbed solitons the phase would be constant as seen in figure 4). An example is shown in figure 5 for parameter values g = 0.5, l = τ = 0.1 and Esat = Psat = 1. This equilibrium is found under slight variations in the soliton centers tk and slight variations in the phase change over the solitons (giving gray solitons for initial conditions instead of black). When this equilibrium solution is slightly perturbed, we find numerically that the solution is unstable: the pulses eventually vanish and the background magnitude begins to grow exponentially. In figure 6 the equilibrium was perturbed by random noise on the order of 10−4 ; as is observed, by z = 150 the solution has moved noticeably away from the black soliton equilibrium and by z = 300 the background is growing exponentially. In the AP model no equilibrium emerges; the moving shelves emanating from the two solitons continue to interact with each other, resulting in continuously increasing fluctuations in the background height. The shelf fronts move at a constant speed and eventually form a 7 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al |u| 2 1 300 0 −50 150 −25 0 t 25 50 0 z Figure 6. Evolution of the amplitude of the equilibrium state shown in figure 5 when perturbed by a small random noise. It is readily observed that the state is unstable. 1.5 z = 300 |u| 1 0.5 0 −50 −25 0 t 25 50 Figure 7. Left panel: contour plot showing the evolution of the norm of a two-black-soliton state in the AP model with periodic boundary conditions. Right panel: temporal profile of the norm at z = 300; the interacting shelves are clearly visible. 7 z = 300 6 Phase 5 4 3 2 1 0 −1 −50 −25 0 25 50 t Figure 8. Left panel: same as in figure 7, but for three gray solitons. Right panel: the temporal profile of the phase at z = 300. diamond-like grid structure—see, e.g., the contour plot in the top panel of figure 7. Though fluctuations in the background increase as z increases, the average height remains close to 1 and the essence of black solitons persist. In figures 8 and 9 we consider an example consisting of three initial solitons with αi = π /3 for i = 1, 2, 3, the same grid structure appears in the contour plot. While the background height oscillates around 1, the soliton troughs decrease and 8 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al 1.5 |u| at boundary min(|u|) |u| 1 0.5 0 0 100 z 200 300 Figure 9. The norm of the background measured at the boundary (top curve), and soliton trough norm (bottom curve) in the case of the evolution of three gray solitons. The dips in the background occur as the solitons pass across the boundary. the phase changes across the solitons increase as z increases. This illustrates the general trend for gray solitons to become black solitons in the AP model. The grid structure still appears in the contour plot shown in the top panel of figure 8. Furthermore, in the bottom panel of the same figure, it is shown that the phase change over the solitons alone no longer need to sum to a multiple of 2π since there are now variations in the phase between the solitons. Note that in the above numerical simulations for the AP model we have used the parameter values g = 0.18, l = τ = 0.1, Esat = 1, and Psat = 10. 3. Dynamics of dark solitons 3.1. Dark soliton perturbation theory Perturbation theory as applied to bright NLS solitons which decay at infinity has been developed over many years [50–52]. However, the non-vanishing boundary conditions characterizing the dark solitons introduce serious complications when applying the perturbation methods developed for bright solitons. Earlier works [53, 54] (see also the reviews [14, 13] and references therein) were able to determine adiabatic changes in the intensity dip, but did not fully determine the behavior of the perturbed dark soliton, especially when it is subject to dissipative effects. Recently, a perturbation method was developed in [38], which permits the detailed study of the effect of small perturbations (including dissipative ones) on dark soliton solutions of the defocusing NLS equation. We present a brief summary of the method here and, in the next subsection, we apply this method to the two variants of the PES model (cf section 2.1). We start by considering a perturbed defocusing NLS equation of the form: iuz − 12 utt + |u|2 u = F[u], (12) with boundary conditions |u| → u∞ as t → ±∞; in equation (12), F[u] is a general functional perturbation and || 1 is a small parameter. We assume that F[ueiφ ] = F[u] eiφ , which is the true in the PES equation to simplify the equations, however this is not necessary to the analysis. We note that in the periodic ring laser configuration we are assuming that the solution satisfies this boundary condition for T 1. After the interaction begins this will no longer hold and the perturbation theory cannot be expected to maintain accuracy. First, we 9 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al find the solution of equation (12) corresponding to the (cw) background; assuming that the latter has the form u = u∞ (z) exp[iφb (z)] (where φb (z) denotes the varying phase of the cw wave), we obtain from equation (12) the following equations describing the evolution of the cw background: d d u∞ = Im {F[u∞ ]} , φ∞ = 0, (13) dz dz where φ∞ ≡ limt→+∞ φb (z) − limt→−∞ φb (z) is the change in phase from −∞ to +∞. Equations (13) imply that the amplitude of the background wave evolves adiabatically, while its phase difference at t → ±∞ remains unaffected by the perturbation. Following the methodology of [38], we now break the problem into two regions: an outer region consisting of the boundary conditions at infinity, characterized by equations (13), and an inner region consisting of the dark soliton and the shelf which develops around it; a representative illustration of these regions, is depicted in figure 3 above. The solution in the inner region is found by employing a multi-scale expansion. It is found that boundaries between the inner and outer region propagate with a velocity V (z) = ±u∞ (z). The unknown function u(z, t ) is broken into its magnitude q and phase φ, namely u = q exp(iφ), and then q and φ are expanded in series of as: q = q0 +q1 +O( 2 ) and φ = u2∞ dz+φ0 +φ1 +O( 2 ). Note that this expansion is valid only in the inner region. At the leading order, the equations for q0 and φ0 take the form of the hydrodynamic equations pertinent to the unperturbed NLS equation (12) and, thus, possess a dark soliton solution of the form: u0 = q0 eiφ0 = {A + iB tanh [B (t − Az − t0 )]} eiσ0 , (14) where A and B are connected to the soliton velocity and depth, respectively, and satisfy the constraint A2 + B2 = u2∞ , while t0 and σ0 denote the soliton center and a phase, respectively. Note that the case A = 0 corresponds to a black soliton, which has a π phase jump, while for A = 0 the solution (14) describes a gray soliton. The soliton parameters A, B, t0 , σ0 are taken to vary adiabatically in z. This way, introducing a slow evolution scale, Z = z, we assume that the soliton parameters are functions of Z. There − + are also four parameters which describe the shape of the shelf, namely q− 1 (Z), q1 (Z), φ1t (Z), + φ1t (Z), arising from the asymptotic limits of the order correction terms, i.e. q1 (Z, t ) → q± 1 (Z) and φ1t (Z, t ) → φ1t± (Z) as t → ±∞. Of these eight total slowly evolving parameters, seven may be solved for in a closed form. Next, we consider the integrals of motion of the unperturbed NLS equation (12) (for = 0): ∞ 2 1 2 1 ∂u 2 2 (15a) H= + 2 (u∞ − |u| ) dt, −∞ 2 ∂t ∞ 2 u∞ − |u|2 dt, (15b) ED = −∞ I= ∞ −∞ R= Im uut∗ dt, (15c) t u2∞ − |u|2 dt, (15d) ∞ −∞ where the Hamiltonian H, the RE of the dark soliton ED (cf section 2.1) and the momentum I are conserved quantities, while the last can be written in term of the momentum, i.e. I = − dR . dz 10 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al In the presence of the perturbation ( = 0), evolution equations for these integrals may be obtained from equations (12) and (13): ∞ dH d 2 = ED u∞ + 2Re F[u](uz − u2∞ u)∗ dt , (16a) dz dZ −∞ ∞ dED ∗ = 2Im F[u∞ ]u∞ − F[u]u dt , dz −∞ ∞ dI ∗ = 2Re F[u]ut dt , dz −∞ ∞ dR ∗ = −I + 2Im t F[u∞ ]u∞ − F[u]u dt . dz −∞ (16b) (16c) (16d) We now use the modified conservation equations (3.1) to solve for the shelf parameters and φ1t± , as well as the slow evolution variables A, σ0Z . More work is required in order to q± 1 find t0 which is not needed for the present discussion; see [38] for details. Note that if we find A, then B = (u2∞ − A2 )1/2 . As mentioned above, the edge of the shelf propagates with velocity V (Z) = u∞ (Z). The basic type of argument used on these modified conservation laws is next illustrated by example. The evolution equation for energy (16b) remains the same after transforming to z the moving frame of reference T = t − 0 A(s) ds − t0 and ζ = z: ∞ ∞ 2 d u∞ − |u|2 dT = 2Im F[u∞ ]u∞ − F[u]u∗ dT . (17) dζ −∞ −∞ At O(1) the equations are satisfied, while at O() we have: ∞ − ∗ BZ − u∞ (u∞ − A)q+ = Im F[u dT . (18) + (u + A)q ]u − F[u ]u ∞ ∞ ∞ 0 0 1 1 −∞ To this end, the following set of evolution equations are derived for the soliton and shelf parameters: d u∞ = Im {F[u∞ ]} , (19a) dZ ∞ d F[u0 ]u∗0T dT , (19b) 2B A = Re dZ −∞ ∞ d u∞ σ0 = BZ + Re {F[u∞ ]} − Im F[u∞ ]u∞ − F[u0 ]u∗0 dT , (19c) dZ −∞ q+ 1 = 1 2 (σ0Z + φ0Z ) / (u∞ − A) , (19d) q− 1 = 1 2 (σ0Z − φ0Z ) / (u∞ + A) , (19e) + = −2q+ φ1T 1, (19f) − = 2q− φ1T 1, (19g) BZ = (u∞ u∞Z − AAZ ) /B, (19h) φ0Z = (2ABZ − 2BAZ ) /u2∞ . (19i) 11 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al 3.2. Application of perturbation theory to the PES model Now we apply the above perturbation theory to the PES equation (1). Particularly, to put the problem in the notation of section 3.1, we set: g τ utt lu , (20) F[u] = i + − 1 + E/Esat 1 + E/Esat 1 + P/Psat where it is assumed that the small parameter is implicitly contained in the right hand side of equation (20); thus, the term F[u] is small and we can apply the results of section 3.1. Notice that in both RE and AP models (cf section 2.1), the energy appears explicitly in the perturbation term, so it is convenient to frame the discussion in terms of the energy as its own parameter. From equations (16b) and (19c) we have: ∞ d (0) E = 2Im F[u∞ ]u∞ − F[u0 ]u∗0 dt , (21) dZ −∞ d 1 (0) /u∞ , σ0 = BZ − EZ dZ 2 (22) where E (0) is the first order approximation for the energy, i.e. E = E (0) + E (1) + O( 2 ), and we have also used the fact that Re {F[u∞ ]} = 0 for this perturbation. Notice that the energy E (0) at O(1), has contributions from both the core of the soliton and the shelf. We first analyze the RE model. Utilizing equations (3.1), we find that the evolution of the above subset of soliton parameters is described by the following closed system of equations (recall: A2 + B2 = u2∞ ), d g u∞ = u∞ − lu∞ , (23a) dz 1 + E (0) /Esat d lPsat g A− tanh−1 A= dz 1 + E (0) /Esat B B2 + Psat d (0) 4g 2τ /3 E = B+ B3 (0) dz 1 + E /Esat 1 + E (0) /Esat u2∞ + Psat B −1 − 4l tanh . B2 + Psat B2 + Psat B B2 + Psat A, (23b) (23c) Note that the above equations are expressed in terms of z, since the small parameter is implicitly contained in the right-hand side of equation (20). From u∞ , A and E (0) , one can then determine σ0 from equation (22) and, finally, the other soliton and shelf parameters may be calculated from equations (3.1). Let us now take a closer look at equation (23a) (which should also be compared with the approximate result of equation (3) for the RE model): assuming that the soliton energy is approximately the same with the one of the unperturbed system, i.e. E = 2u∞ , it follows that the equilibrium background for black solitons (B = u∞ , A = 0) given by equation (4) does not satisfy equation (23c) in general. This leads to a discrepancy between the soliton energy and the total energy and indicates that a shelf will develop around the soliton. The shelf height can be calculated as part of the perturbation analysis and, for the considered form of the perturbation, it turns out that it has a small size. Indeed, in figure 2, using typical parameter values, it can be seen that the shelf height is O(10−3 ), which is too small to be observed in a plot of the soliton (see the inset in this figure); that is why we zoom in to make our comparison 12 J. Phys. A: Math. Theor. 46 (2013) 095201 2 M J Ablowitz et al Energy E 1.5 1 Background u∞ 0.5 0 0 Numerics Asymptotics Shelves begin interacting 10 20 z 30 40 50 Figure 10. Evolution of the RE and background height for the RE model: solid (blue) lines and dashed (red) lines correspond to the numerical results and the asymptotic analytical predictions, respectively. The vertical line indicates the spatial distance at which the shelves begin interacting. Here, the parameter values are g = 0.5, τ = 0.1, l = 0.1 and Esat = Psat = 1. 4 |u| 3 2 1 0 −20 50 0 20 40 0 z t Figure 11. Evolution of a gray soliton, decaying into a cw with renormalized or ‘dark’ energy E = 0, in the RE model. Here, parameter values are g = 0.3, τ = 0.05, l = 0.1 and Esat = Psat = 1. between numerics and asymptotic prediction. The small size of the shelf amplitude, in this particular setting, explains the agreement found in [37] between the prediction of equation (4) and the numerical simulations without consideration of the shelf. Though the shelf may seem small, when considered on a periodic computational domain, the interaction of shelves can eventually have a significant effect on the RE. In figure 10, we provide a comparison between the numerics and the asymptotic results, indicating also the emergence of shelf interactions. Analysis shows that the equilibrium solutions of equations (3.2) with A = 0 (recall that this value corresponds to a black soliton) can be shown to be unstable. In fact, any deviation from a purely stationary, black soliton state (i.e. any gray soliton) is found to eventually degenerate into a cw with RE E = 0. At this point, the equation for the background becomes d u∞ = (g − l) u∞ , (24) dz which implies exponential growth. This behavior is observed in figure 11. In the case of periodic boundary conditions, the deviation from the purely black soliton state may occur due to shelf interaction. 13 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al 2 Numerics Asymptotics |u| 1.5 1 0.5 0 0 5 10 z 15 20 Figure 12. Evolution of the background height (top curves) and soliton trough (bottom curves) as found numerically (solid (blue) lines) and analytically (dashed (red) lines) compared to asymptotic results for the AP model. Here g = 0.3, τ = 0.05, l = 0.1, Esat = 1 and Psat = 10. 2.2 2 u∞ 1.8 1.6 1.4 1.2 0 0.05 0.1 A 0.15 0.2 Figure 13. The phase plane for A and u∞ . The dotted (red) line indicates the equilibrium for u∞ . Here, g = 0.3, τ = 0.05, l = 0.1, Esat = 1 and Psat = 10. Next, we consider the AP model. In this case, the dark soliton energy can be defined as ED (z) = T (u2∞ − E ) and, for T 1, we may use the approximation E = u2∞ − T1 ED ≈ u2∞ (with small error for large T ). Employing again equations (3.1), we find that the evolution of the relevant soliton parameters is described by the following equations: d g l u∞ = u∞ − u∞ , (25a) 2 2 dz 1 + u∞ /Esat 1 + u∞ /Psat d g lPsat A= A− tan−1 2 dz 1 + u∞ /Esat A2 + Psat B A2 + Psat A, 4g 8τ /3 d (0) E = B+ B3 dz D 1 + u2∞ /Esat 1 + u2∞ /Esat B Psat 1 −1 − 4l +1 tan . u2∞ A2 + Psat A2 + Psat 14 (25b) (25c) J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al Figure 14. Contour plot showing the interaction between two gray solitons with opposite phase change in the AP model. Here, g = 0.18, τ = 0.1, l = 0.1, Esat = 1 and Psat = 10. 1.5 |u| at Boundary min(|u|) 1 0.5 0 0 100 z 200 300 Figure 15. The amplitude of the background measured at the boundary (top curve) and its minimum value (bottom curve) associated with figure 14. Here, g = 0.18, τ = 0.1, l = 0.1, Esat = 1 and Psat = 10. The above analytical predictions are compared to direct numerical simulations in figure 12. It is observed that there is a good agreement between the two, at least for propagation distances up to z = 20 as shown in the figure. Having approximated the dark soliton energy as mentioned above, equation (25c) decouples from equations (25a) and (25b). The latter equations possess an equilibrium point at A = 0 (corresponding to a black soliton) and (26) u∞ = (g − l)/[(l/Esat ) − (g/Psat )] (when sign(g − l) = sign [(l/Esat ) − (g/Psat )]), which is the same with the result of equation (8). As opposed to the RE model, this equilibrium is found to be stable for a wide range of parameters, as is also indicated by the typical phase portrait shown in figure 13. Since the perturbation is over a finite domain, it is natural to consider the problem on a periodic domain, as in the case of a fiber ring laser. In this case, the shelves continue to interact with each other, affecting also the soliton interaction dynamics. In figure 14, where the interaction between two gray solitons with opposite phases is shown, the shelf interaction is also visible and creates a diamond-like grid structure in the contour plot. Despite the 15 J. Phys. A: Math. Theor. 46 (2013) 095201 M J Ablowitz et al interaction of the shelves, we can still observe evolution toward black solitons—note the state that is formed after the quasi-elastic collision between the gray solitons. Here we should note that once the shelves begin interacting, the perturbation theory is no longer valid; however, we see that the pulses still maintain the characteristics of a black soliton with a trough near zero and a phase change near π . This is further illustrated in figure 15, where the soliton amplitude at the boundary and soliton trough amplitude are plotted for the evolution with parameters given and dynamics depicted in figure 14. 4. Conclusions and discussion While bright solitons have been extensively studied theoretically and their dynamical properties have been verified in numerous experiments over the past decades, dark solitons are less well understood both theoretically and experimentally. This has been particularly true in ML lasers. The purpose of this work is to address the theoretical problem of the dark solitons dynamics and introduce a model that captures its properties. In so doing, we have used the PES equation, a variant of the NLS equation, including power saturating loss and energy saturating gain and filtering. We considered two different versions of the PES model, depending on the definition of the energy (and power): the first version is based on the RE (RE model), while the second one was based on the AP (AP model). The RE model explicitly builds in a periodic domain. It is pertinent to round trips of light in the laser cavity; i.e. a ring laser. For both versions of the PES equation, we have found that single and multiple dark solitons can mode-lock to states containing dark solitons or dips of light on a continuous background. Dark solitons are exact solutions of the pure NLS equation; they are approximate solutions of the two PES equations. In each case, the amplitudes of the background waves at equilibrium were determined analytically and conditions for mode-locking were found; the latter, are in qualitative agreement with experimental observations [24]. We also used a perturbation method, recently devised in [38], to study the dynamics of these dark solitons. Assuming that the loss, gain and filtering terms are sufficiently small, we described analytically the evolution of both the dark solitons and the shelves that are expanded from them. In the case of single dark solitons, and for the specific type of the perturbation in the RE model, the shelves were found to be small (sizes of order O(10−3 )); this explains the a priori accurate predictions for the values of the equilibrium amplitudes of the background wave in this model. Although being of small size, the shelves were also found to be import in the case of multiple dark solitons, as they can significantly distort a ML multiple black soliton state. On the other hand, in the AP model the shelves are larger in size and there is an additional large effect due to numerous shelf interactions which occur because of the finite size of the lasing region. Another result stemming from our analytical approach concerns the stability of the dark solitons: We have shown that the RE model exhibits mode-locking to equilibrium solutions consisting of black solitons, but these equilibria are weakly unstable: when a black soliton ceases to be quiescent (i.e. when it is perturbed so as to become a gray soliton with a small velocity), it decays into its supporting background, which in turn grows exponentially. On the other hand, the AP model exhibits stable equilibria, i.e. stable black solitons; in this case, we have shown that gray solitons (and even interacting ones) evolve into quiescent black soliton states. Based on these considerations, the AP interpretation of the energy is more suitable for modeling the mode-locking dynamics of dark pulses. There are many future directions that can be considered that follow along the lines of this work. 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