Dissipative Light Bullets in Kerr Cavities: Multistability, Clustering, and Rogue Waves
S. S. Gopalakrishnan ,
1,2,*
K. Panajotov ,
3,4
M. Taki,
2
and M. Tlidi
1
1
Facult´e des Sciences, Universit´e libre de Bruxelles (U.L.B), CP. 231, 1050 Brussels, Belgium
2
Universit´e Lille, CNRS, UMR 8523—PhLAM— Physique des Lasers, Atomes et Mol´ecules, F-59000 Lille, France
3
Department of Applied Physics and Photonics (IR-TONA), Vrije Universiteit Brussels, Pleinlaan 2, 1050 Brussels, Belgium
4
Institute of Solid State Physics, 72 Tzarigradsko Chaussee Boulevard, 1784 Sofia, Bulgaria
(Received 17 November 2020; revised 3 February 2021; accepted 19 March 2021; published 16 April 2021)
We report the existence of stable dissipative light bullets in Kerr cavities. These three-dimensional (3D)
localized structures consist of either an isolated light bullet (LB), bound together, or could occur in clusters
forming well-defined 3D patterns. They can be seen as stationary states in the reference frame moving with
the group velocity of light within the cavity. The number of LBs and their distribution in 3D settings are
determined by the initial conditions, while their maximum peak power remains constant for a fixed value of
the system parameters. Their bifurcation diagram allows us to explain this phenomenon as a manifestation
of homoclinic snaking for dissipative light bullets. However, when the strength of the injected beam is
increased, LBs lose their stability and the cavity field exhibits giant, short-living 3D pulses. The statistical
characterization of pulse amplitude reveals a long tail probability distribution, indicating the occurrence of
extreme events, often called rogue waves.
DOI: 10.1103/PhysRevLett.126.153902
Localized structures (LSs) and localized patterns are
dissipative structures found far from equilibrium that have
been observed in many areas of natural science, such as
chemistry, physics, biology, and plant ecology [1].
Nonlinear optics and laser physics provide classic examples
of LSs that can be experimentally investigated in a reliable
way [2]. In one-dimensional (1D) dispersive driven
cavities, they have been theoretically predicted [3] and
experimentally demonstrated [4,5]. Their link to frequency
combs was established in [6] and has reinforced the interest
in this field of research. Their spectral contents correspond
to optical frequency combs, which have applications in
metrology and spectroscopy [7].
In broad area cavities where two-dimensional (2D)
diffraction cannot be ignored, LSs have been experi-
mentally observed as well [8]. They are addressable
structures with a possibility for applications in all-optical
control of light, optical storage, and information processing
[8]. Apart from these applications in 1D and 2D settings,
LSs are of general interest in other fields of active research
such as hydrodynamics, Bose–Einstein condensates, biol-
ogy, and plant ecology [8].
From a theoretical point of view, the challenging
aspect lies in the collapse of LSs for the case of the
nonlinear Schrödinger equation when the dimensionality of
the system is at least 2 [9]. To prevent wave collapse in
three-dimensional settings, it is necessary to introduce
additional physical effects, such as optical cavity [10],
semiconductor active media [11], saturable absorbers [12],
parametric oscillators [10], or twisted waveguide arrays
[13] (see recent reviews [2,8,14]).
The purpose of this Letter is to predict the occurrence of
stable stationary light bullets (LBs) in the dispersive and
diffractive driven Kerr resonators. These three-dimensional
(3D) dissipative structures can be either isolated or bound
together or form self-organized clusters. Some of these
structures have been reported close to the nascent bist-
ability regime [15]. We characterize these 3D solutions by
constructing their bifurcation diagram and determine their
range of stability. This diagram exhibits multistability,
which clearly indicates that clustering phenomenon
belongs to a homoclinic snaking type of bifurcation.
When increasing the injected field power, we observe
transition via period doubling to a complex dynamical
regime. In this regime, statistical analyses show that the
amplitude probability distribution possesses a long tail with
pulse intensity height well beyond 2 times the significant
wave height (SWH) [16]. The SWH is defined as the mean
height of the highest third of waves. From these two
characteristics, we can infer that this complex behavior
belongs to the class of rogue waves. In optics, rogue waves
have been first observed in one-dimensional settings by
Solli et al. [17]. Finally, we provide experimentally relevant
parameters based on chalcogenide glass, which possesses a
very strong Kerr effect and responds instantaneously to
electrical excitation.
We consider a Fabry–Perot cavity filled with Kerr
medium and coherently driven by an external injected field
E
i
. The transmitted part of this field will interact with the
nonlinear media and suffer from nonlinearity, diffraction,
chromatic dispersion, and losses. The schematic setup of an
optical cavity filled with a Kerr medium is shown in Fig. 1.
PHYSICAL REVIEW LETTERS 126, 153902 (2021)
0031-9007=21=126(15)=153902(6) 153902-1 © 2021 American Physical Society
When dispersion and diffraction have the same influence,
the formulation of this problem leads to the generalized
Lugiato–Lefever equation (LLE) [18], describing the
dynamics of nonlinear optical cavities as the one shown
in Fig. 1. In its 3D form, the LLE reads [10,18]
t
r
∂E
∂t
¼
ffiffiffi
θ
p
E
i
− ðκ þ
_
{ϕ −
_
{γljEj
2
ÞE
þ
_
{
l
2q
∇
2
⊥
þ
β
2
l
2
∂
2
∂τ
2
E; ð1Þ
where E ¼ Eðx; y; τ;tÞ is the normalized slowly varying
envelope of the intracavity field, and E
i
is the input field
amplitude. The time t corresponds to the slow-time
evolution of E over successive round-trips, whereas τ
accounts for the fast time in a reference frame traveling
at the group velocity of light in the Kerr medium. The
cavity round-trip time is denoted by t
r
and the total losses
by κ. The 2D diffraction is described by the Laplace
operator ∇
2
⊥
¼ ∂
xx
þ ∂
yy
acting on the transverse
plane ðx; yÞ. The diffraction coefficient is inversely propor-
tional to the wave number in the cavity material
q ¼ ω
0
n=c ¼ 2πn=λ
0
, where ω
0
is the injected field
frequency, λ
0
is the wavelength in vacuum, n is the linear
refractive index, and c is the speed of light. The second
derivative with respect to τ describes the group velocity
dispersion. The chromatic dispersion coefficient β
2
is
considered to be positive assuming that the Kerr cavity
operates in the anomalous dispersion regime. The non-
linearity coefficient is γ ¼ n
2
ω
0
=ðcA
eff
Þ, with n
2
being the
nonlinear refractive index of the Kerr material considered.
The effective area A
eff
is the surface illuminated in the
transverse plane. The transmission coefficient θ is the same
for the input and the output mirrors M
1;2
, and θ is supposed
to be much smaller than unity. The length of the cavity is
denoted by l and ϕ ¼ 2πl=λ
0
is the linear phase shift
accumulated by the intracavity field over the cavity length
l. To simplify further and to reduce the number of
parameters describing the time evolution of the intracavity
field, we introduce the following changes:
ðx; yÞ →
ffiffiffiffiffiffiffiffi
l
2qκ
s
ðx; yÞ; ðt; τÞ →
t
r
κ
t;
ffiffiffiffiffiffi
β
2
l
2κ
r
τ
;
E
i
→ κ
ffiffiffiffiffiffiffi
κ
γθl
r
E
i
; and E →
ffiffiffiffi
κ
γl
r
E: ð2Þ
Under these changes, the generalized LLE (1) takes its
dimensionless form
∂E
∂t
¼ E
i
− ð1 þ
_
{δÞE þ
_
{
∇
2
⊥
þ
∂
2
∂τ
2
E þ
_
{jEj
2
E; ð3Þ
where δ ¼ ϕ=κ is the cavity detuning parameter. The
homogeneous stationary solutions of LLE E
s
are given
by E
2
i
¼jE
s
j
2
½1 þðδ − jE
s
j
2
Þ
2
.Forδ <
ffiffiffi
3
p
, the trans-
mitted intensity as a function of the input intensity E
2
i
is
single valued, whereas bistability occurs for δ >
ffiffiffi
3
p
. The
steady-state homogeneous solution undergoes a modula-
tional instability at the threshold value, E
2
ic
¼ 1 þðδ − 1Þ
2
for the injected field intensity. The corresponding intra-
cavity intensity is jE
c
j
2
¼ 1 [18]. At this bifurcation point,
the wavelength of 3D patterns is Λ ¼ 2π=
ffiffiffiffiffiffiffiffiffiffi
2 − δ
p
.
A weakly nonlinear theory has been performed in 3D
settings [10]. This analysis has revealed the predominance
of the body-centered-cubic (bcc) lattice structure over a
variety of 3D structures in the cavity field intensity [10].
The validity of this analysis is restricted to the values of the
detuning parameter within the range δ < 41=30. In what
follows, we focus on the strongly nonlinear regime where
modulational instability is subcritical, i.e., δ > 41=30.
Remarkably, besides the emergence of bcc structures, the
same mechanism predicts the possible existence of stable
dissipative LBs. An example of a single light bullet is
shown in Fig. 2(a). This structure is obtained by a
numerical simulation of Eq. (3) by using periodic boundary
conditions in all directions. The initial condition used
consists of a Gaussian function added to the value of the
homogeneous steady state with an amplitude comparable to
one of the periodic bcc structures. The cross section of the
single LB along the transverse plane is shown in Fig. 2(i).
The spatial profile of this 3D solution indicates that
LBs possess a damped oscillatory tail. This single LB is
obtained by a direct numerical simulation of Eq. (3),
which can also be solved by assuming a spherical
approximation. In this case, the stationary LB has the
form EðrÞ¼E
s
½1 þAðrÞ with r ¼ðx
2
þ y
2
þ τ
2
Þ
1=2
and
with boundary conditions Að0Þ¼A
0
ð≠ 0Þ, ∂
r
Að0Þ¼0,
and Aðr → ∞Þ¼0. The two profiles, one obtained from a
direct numerical simulation and the other using the
FIG. 1. Schematic setup of an optical cavity filled with a Kerr
medium. An injected field E
i
is launched inside the cavity by
means of the mirror M
1
. The input M
1
and the output M
2
mirrors
are separated by a distance l and are identical. The resulting field
circulating within the cavity after several round-trips is denoted
by E. A possible output consisting of a cluster involving 14 light
bullets is shown. This 3D isosurface is obtained by a direct
numerical simulation of Eq. (3) with parameters E
i
¼ 1.21;
δ ¼ 1.7.
PHYSICAL REVIEW LETTERS 126, 153902 (2021)
153902-2
spherical symmetry approximation, are in good agreement,
as shown in Fig. 2(h). The width of the LB is close to
≈π=
ffiffiffiffiffiffiffiffiffiffi
2 − δ
p
, which is half the critical wavelength at the
modulational instability.
Numerically, the challenge posed by the 3D LLE arises
from the strongly nonlinear term, which when discretized
leads to a large system of coupled strongly nonlinear stiff
ordinary differential equations [19]. Because of this reason,
the 3D simulations of LLE are still missing in the literature.
It is to be noted that finite-difference methods can some-
times lead to spurious solutions that are nonphysical [19],
which is where higher-order spectral methods come to the
fore. In the present Letter, the spatial discretization of the
LLE is done using a Fourier spectral method with periodic
boundary conditions [19,20], and the time stepping is
carried out with a fourth-order exponential time differenc-
ing Runge–Kutta scheme [20]. The main advantage of
using a Fourier spectral method lies in the fact that the
linear term of the discretized set of equations is diagonal,
and more importantly, the nonlinear term is evaluated in
physical space and then transformed to Fourier space.
Further details on these methods can be found in these
excellent books [19,20]. All the numerical simulations in
the present Letter are carried out on a periodic domain of
size 80 units in each direction, resolved using 128 grid
points, with a time step of 0.01.
When initial conditions are set to two Gaussian shells
centered at different positions in Eðx; y; τÞ space, these two
perturbations evolve toward the formation of two bounded
LBs, as can be seen on Fig. 2(b). By placing different LBs
close to each other in the cavity, clusters of stable bounded
states of LBs can be formed. When the individual LBs are
well separated from one another, they are independent.
However, when they are brought closer to each other, they
start to interact via their oscillatory, exponentially decaying
tails. The number of LBs and their distribution in the
ðx; y; τÞ space is large. Some examples of closely packed
LBs are shown in Figs. 2(c)–2(f). More precisely, Eq. (3)
admits an infinite set of 3D solutions if the size of the
system is infinite, with the limiting case corresponding to
the infinitely extended periodic pattern distribution. These
solutions are obtained for the same values of parameters,
and they differ only by the seeded initial conditions. All
these LBs coexist as stable solutions with the bcc lattice in
the range P shown in Fig. 2, often called the pinning range
[21]. These LBs are confined in ðx; y; τÞ, and can be seen as
a cluster of the elemental structure (a single LB) with a
well-defined size, which are robust and stable.
The bifurcation diagram of these closely packed clus-
ters of LBs is shown in Fig. 2(g) where the difference in
intensity of the LB with respect to the background state,
N ¼
R
jE − E
s
j
2
dxdydτ is shown as a function of the
injected field E
i
for a fixed value of the detuning
parameter δ. The associated label is denoted next to the
bifurcation curves. In the pinning r ange P,thesystem
exhibits a high degree of multistability. The multiplicity of
these 3D solutions of the LLE is strongly reminiscent of
homoclinic snaking, indicating that the formation of
bounded LB states and clusters of them is indeed a very
robust phenomenon. Homoclinic snaking type of bifur-
cation has been investigated in one-dimensional settings
[22]. It was first studied in relation with the Swift-
Hohenberg equation [23] and is also a well-documented
issue in spatially extended systems (s ee overviews [24]).
This phenomen on, however, has never been documented
in 3D settings. Note, however, that the 3D LBs in the LLE
with a simple driving term does not admit vortex sol-
utions, while other 3D vortices with embedded vorticity
have been reported in [25]. We now provide a possible set
of experimental parameter values. The link between the
physical and the nondimensional parameter s and variables
are given by Eq. (2). The experimental setup consists of a
broad ar ea Fabry–Perot cavity filled with a material
FIG. 2. Clusters of closely packed light bullets involving 1–6
LBs are shown respectively by the 3D isosurface in (a)–(f). (g) 3D
bifurcation diagram associated with the LBs. The continuous
black line denotes the stationary steady state. The pinning range
is indicated by P. (h) Comparison between the LB obtained by a
spherical symmetry (red dotted line) and the cross section along
1D direction of the single LB denoted by (a) represented by a
continuous black line. (i) Cross section along the transverse
plane of the single LB denoted by (a) with the profiles
exhibiting decaying spatial oscillations. Parameter settings are
E
i
¼ 1.21; δ ¼ 1.7.
PHYSICAL REVIEW LETTERS 126, 153902 (2021)
153902-3
having high Kerr coefficient and driven by a coherently
injected beam. We suggest using chalcogenide glass,
which is characterized by a very strong Ker r effect. To
be more concrete, let us consider some typical physical
parameters values. The nonlinear refractive index n
2
coefficient is as high as n
2
≈ 2.3 × 10
−17
m
2
=W. This
results in a nonlinearity coefficient of γ ≈
0.144 W
−1
km
−1
for an effective (illuminated) area of
A
eff
¼ 25 × 10
4
μm
2
. The length of the cavity is l ¼
1000 μm and t he reflectivity of the mirrors
1 − θ ¼ 0.95, so that the optical losses are determined
by the mirror transmission as the i ntrinsic m aterial
absorption loss can be as small as 40 dB=km. T he
wavelength λ
0
¼ 4 μm is chosen c lose to the zero
dispersion wavelength. With these realistic physical
parameters, the intensity of the injected field should be
on the order of 10 MW=cm
2
.Thisisreasonablesinceitis
well below the damage threshold of chalcogenides, which
can be as high as 1 GW=cm
2
. The choice of these physical
parameters are from [26]. In these settings, we expect that
the spatial width in ðx; yÞ plane will be ≈270 μmandthe
temporal width of the LB will be ≈0.08 ps for a value
of β
2
≈ 20 ps
2
=km.
When increasing the injected field intensity E
i
beyond
the pinning range P, LBs are observed to undergo transition
via period doubling to a complex spatiotemporal regime.
In the remainder of this Letter, we will characterize this
complex regime. At high values of E
i
, the occurrence of
rogue waves in one- [27] and two-dimensional optical Kerr
cavities has been recently shown [28]. The research on
extreme events or rogue waves has gathered significant
interest, especially in the fields of hydrodynamics and
nonlinear optics, as witnessed by recent review papers
[29,30]. Rogue waves are rare pulses with amplitudes
significantly larger than the average one that arise due to
self-focusing of energy of a wave group into one majestic
event. We investigate this phenomenon in 3D settings, and
we assume that the cavity operates in the monostable
regime.
Figure 3(a) shows a 3D isosurface of the intensity in the
cavity. Figure 3(b) shows a two-dimensional cut along the
transverse plane, demonstrating an event with optical
intensity significantly larger than the average amplitude
of the background state. The three-dimensional perspective
[Fig. 3(a)] shows the complex spatiotemporal regime at this
high value of E
i
. To quantify such extreme events, starting
from a random initial condition, a statistical analysis based
on the pulse amplitudes observed within the optical cavity
is registered. Figure 3(c) shows the probability distribution
of the number of events as a function of the intensity of the
pulses in semilogarithmic scale for two different values of
E
i
. When E
i
is increased beyond E
ic
, though the system
enters into a complex dynamical regime, the tail of the
pulse height distribution stays below the threshold of twice
the significant wave height. This can be seen in Fig. 3(c),
where the statistical analysis at E
i
¼ 1.5 is shown in red.
As E
i
is further increased, there is a considerable number of
events with the maxima of intracavity intensity more than
2x the SWH, with even events of pulse amplitude as high as
4x the SWH. This can be seen from the statistical analysis
shown in Fig. 3(c) for E
i
¼ 5. In this complex spatiotem-
poral regime, the non-Gaussian statistical distribution of the
wave intensity can be clearly seen with a long tail
probability distribution. As remarked earlier, this is the
main signature typical of rogue wave formation. These
large intensity pulses belong to the class of rogue waves or
extreme events [28,29]. We would like to emphasize that
rogue waves are only formed in the LLE model when the
spatiotemporal complexity is well developed, i.e., when the
neighboring pulses in the oscillating pattern are interacting
strongly. Even when the peak amplitude of the pulses
clearly display complex dynamics, as in the case of
E
i
¼ 1.5 [Fig. 3(c)], no rogue waves are formed in the
system. This is reflected in the tail of the pulse height
distribution, which stays below the threshold of 2x SWH.
In conclusion, we have explicitly demonstrated the
existence of robust three-dimensional dissipative solitons
in the form of light bullets, leading to a striking light power
confinement. The LBs result from the combined action of
dispersion and diffraction mediated by Kerr nonlinearity,
dissipation, and pumping. They could exist as isolated light
bullets or in clusters of bounded states. They are dissipative
three-dimensional structures that travel with the group
velocity of light within the Kerr cavity. We have established
FIG. 3. Three-dimensional rogue waves. (a) A 3D isosurface of
rogue waves in the cavity obtained for E
i
¼ 5. (b) The cross
section along the transverse ðx; yÞplane reveals an extreme event.
(c) The probability distribution of the number of events as a
function of the intensity of the pulses in semilogarithmic scale for
two different values of E
i
. The dashed line indicates events of
amplitudes twice the SWH for E
i
¼ 5. Parameter settings are
δ ¼ 1.7.
PHYSICAL REVIEW LETTERS 126, 153902 (2021)
153902-4
the link of this phenomenon with the homoclinic snaking
type of bifurcation. When increasing further the intensity of
the injected field, we have observed the transition from a
stationary distribution of LBs to a complex regime. The
statistical analysis has revealed evidences of 3D rogue
waves with abnormal amplitudes larger than twice the
significant wave height. Finally, we have provided realistic
parameters for an experimental observation in an optical
resonator filled with chalcogenide glass. These individually
addressable LBs can bring a significant benefit for 3D
optical storage, paving the way for future experimental
research.
This work was supported in part by the
Fonds Wetenschappelijk Onderzoek-Vlaanderen FWO
(G0E5819N). We also acknowledge the support from the
French National Research Agency (LABEX CEMPI, Grant
No. ANR-11- LABX-0007), as well as the French Ministry
of Higher Education and Research, Hauts de France
Council, and European Regional Development Fund
(ERDF) through the Contrat de Projets Etat-Region
(CPER Photonics for Society P4S). M. T. acknowledges
financial support from the Fonds de la Recherche Scientific
FNRS under Grant CDR No. 35333527 “Semiconductor
optical comb generator.” A part of this work was supported
by the “Laboratoire Assoc´e International” University of
Lille—ULB on “Self-organisation of light and extreme
events” (LAI-ALLURE).”
*
Corresponding author.
shyam7sunder@gmail.com
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PHYSICAL REVIEW LETTERS 126, 153902 (2021)
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