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Dark solitons of the power-energy saturation model: application to mode-locked lasers
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2013 J. Phys. A: Math. Theor. 46 095201
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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
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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
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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.
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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
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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
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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
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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)
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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
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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.
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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
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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. First, an interesting theme for investigation would be the study of dispersion managed
dark solitons in ML lasers (for a relevant work for bright solitons see [34]). Additionally,
16
J. Phys. A: Math. Theor. 46 (2013) 095201
M J Ablowitz et al
since the same (PES) model can also be used to model bright solitons in the normal dispersion
regime [35], it would be interesting to study trains of antisymmetric solitons of [35] in the
context of dark pulse lasers.
Acknowledgments
This research was partially supported by the US Air Force Office of Scientific Research, under
grant FA9550-12-1-0207. The work of DJF was partially supported by the Special Account
for Research Grants of the University of Athens.
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