# Dark solitons in Fabry-Pérot resonators with Kerr media and normal
dispersion

Graeme N. Campbell1<EMAIL_ADDRESS>Lewis Hill1,2 Pascal
Del’Haye2,3 Gian-Luca Oppo1 1SUPA and Department of Physics, University of
Strathclyde, Glasgow, G4 0NG, Scotland, UK
2Max Planck Institute for the Science of Light, 91058 Erlangen, Germany
3Department of Physics, Friedrich Alexander University Erlangen-Nuremberg,
91058 Erlangen, Germany

###### Abstract

Ranges of existence and stability of dark cavity-soliton stationary states in
a Fabry-Pérot resonator with a Kerr nonlinear medium and normal dispersion are
determined. The Fabry-Pérot configuration introduces nonlocal coupling that
shifts the cavity detuning by the round trip average power of the intracavity
field. When compared with ring resonators described by the Lugiato-Lefever
equation, nonlocal coupling leads to strongly detuned dark cavity solitons
that exist over a wide range of detunings. This shift is a consequence of the
counterpropagation of intracavity fields inherent to Fabry-Pérot resonators.
At difference with ring resonators, the existence and stability of dark
soliton solutions are dependent on the size and number of solitons in the
cavity. We investigate the effect of nonlocal coupling of Fabry-Pérot
resonators on multiple dark solitons and demonstrate long range interactions
and synchronization of temporal oscillations.

††preprint: APS/123-QED

##  I Introduction

Temporal cavity solitons (TCS) [1] have attracted significant interest for the
generation of broadband optical frequency combs [2] with many applications in
telecommunication [3, 4], spectroscopy [5, 6] and in the fundamental studies
of complex dissipative structures. TCS in high Q ring microresonators are now
routinely used to produce frequency combs. Light propagation in micro-ring
resonators is well described by the longitudinal version of the Lugiato-
Lefever equation (LLE) [7]. The LLE originally described the transverse,
dissipative spatial structures in passive optical systems with diffraction and
was later adapted to describe pattern formation along the cavity length [8,
9].

In recent years the generation of bright TCS within Fabry-Pérot (FP)
resonators in the anomalous dispersion regime has been first studied
analytically [10] and then experimentally demonstrated for continuous wave
[11] and pulsed [12] driving. This has resulted in increasing interest in
linear resonators for the generation of TCS as the FP geometry can offer
additional engineering possibilities when compared to a ring resonator for
tailoring dispersion and allowing for greater control over the bandwidth and
temporal duration of cavity solitons. Such possibilities include the
engineering of the core-cladding index [13], analogous to the engineering of
ring resonator geometry, or the design of the mirror dispersion [11].

Here we extend the work of [10] to the case of normal dispersion where stable
dark cavity-solitons (DCS) are found as opposed to bright ones. We model a FP
resonator filled with a Kerr nonlinear medium and investigate the inherent
counterpropagation of light under normal dispersion in Section II. We describe
first how the FP configuration results in many unstable stationary solutions,
which are stable in an equivalent ring resonator in Section III. In spite of
this fact, solutions containing moving fronts between bistable states exist
and are described in Section IV. It is these moving fronts that can lock with
each other and form DCS solutions. DCS stationary solutions are found to be
detuning shifted with respect to those in a ring resonator described by the
LLE by the average power of the field over a round trip of the cavity. This is
a result of an additional nonlocal coupling term exhibited by the FP resulting
from counterpropagation of the intracavity fields. To properly elucidate the
effects of the shift in detuning, stationary solutions of the FP model and
their stability are compared with those of ring resonators with normal
dispersion [14, 15], those of FP with anomalous dispersion [10] and those of
counterpropagating light in ring-resonators with normal dispersion [16].
Finally we investigate oscillatory DCS solutions in a FP and discuss the
effects of the nonlocal coupling on the oscillating solitons, homogeneous
background, and the interaction of two oscillating DCS. In particular, the
long range interaction between DCS is capable of synchronizing their
oscillations.

##  II The Fabry-Pérot model

We consider a high finesse FP resonator composed of highly reflective mirrors
and filled with a Kerr medium, see Fig. 1. The resonator is driven by linearly
polarized light, which is coupled through one of the cavity mirrors into the
resonator and the intracavity field is coupled out upon each reflection. This
system has been previously studied in the case of anomalous dispersion in
[10], where a so called Lugiato-Lefever equation for the Fabry-Pérot (FP-LLE)
was derived. For normal dispersion the dimensionless FP-LLE has the form

$\displaystyle\partial_{t}\psi$ $\displaystyle=$ $\displaystyle
S-(1+i\theta)\psi+i(|\psi|^{2}+2\langle|\psi|^{2}\rangle)\psi-i\partial^{2}_{\zeta}\psi$
(1)

where $\psi(\zeta,t)$ is the normalized field envelope composed of the
combined forward and backward counterpropagating fields in the FP resonator
defined over the extended domain $-L\leq\zeta\leq L$ where $L$ is the
dimensionless resonator length, $S$ is the amplitude of the normalized coupled
input field, here considered to be real and positive, and $\theta$ is the
normalized detuning to the near nearest cavity resonance. Here, $t$ is the
‘slow time’ temporal variable describing the evolution over many round trips
of the cavity whilst $\zeta$ is the ‘fast time’ spatial variable describing
the evolution over a single round trip of the linear cavity. The term notated
with angled brackets represents an integral

$\langle|\psi|^{2}\rangle=\frac{1}{2L}\int_{-L}^{L}|\psi|^{2}d\zeta$ (2)

corresponding to the average power of the intracavity field over a round trip
of the resonator. This term arises due to the counterpropagation of the fields
which is inherent in a FP configuration. Integral terms of this kind are also
present in bidirectionally pumped ring resonators where there are instead two
coupled equations, one for each counterpropagating field [17, 18, 16], and are
the result of rapid phase dynamics of cross-coupling terms due to the fields
seeing each other through an average intensity.

Figure 1: Setup: A Fabry Pérot resonator filled with a Kerr medium. An input
field pump enters the cavity on one side and part of the field is coupled out
upon each reflection at the output mirror. An example field power displaying a
DCS is shown.

A connection between Eq. (1) and a LLE for a ring resonator system has been
previously made in [10] and investigated in the context of anomalous
dispersion. By writing Eq. (1) as

$\displaystyle\partial_{t}\psi$ $\displaystyle=$ $\displaystyle
S-(1+i\theta_{\mathrm{eff}})\psi+i|\psi|^{2}\psi-i\partial^{2}_{\zeta}\psi$
(3)

we can see that the stationary solutions of Eq. 1 are also stationary
solutions of the LLE with an identical input field $S$ and an effective
detuning

$\theta_{\mathrm{eff}}=\theta-2\langle|\psi|^{2}\rangle.$ (4)

that is shifted by the average power of the intracavity field. The role of
this effective detuning in counterpropagation in a ring resonator was was
investigated in [16]. The shift in detuning introduced by the FP configuration
results in distinct features of the TCSs as discussed in the following
sections.

##  III Homogeneous states and plateaus

We describe stationary solutions composed of flat increments over the cavity
round trip of the FP model in the fast time, the normalized spatial coordinate
$\zeta$. We write Eq. (1) for $\partial_{t}\psi=0$ as

$\displaystyle\partial_{\zeta}U$ $\displaystyle=$
$\displaystyle\tilde{V},\quad\partial_{\zeta}V=\tilde{U},$
$\displaystyle\partial_{\zeta}\tilde{U}$ $\displaystyle=$
$\displaystyle-(\theta-\langle U^{2}+V^{2}\rangle)U-V+UV^{2}+U^{3}$ (5)
$\displaystyle\partial_{\zeta}\tilde{V}$ $\displaystyle=$
$\displaystyle-(\theta-\langle U^{2}+V^{2}\rangle)V+U+VU^{2}+V^{3}-S$

where $U,V$ are the real and imaginary components of $\psi$. We consider the
system of Eq. (5) to evolve in fast time $\zeta$ over a round trip of the
cavity and we adopt the terminology of [14, 19, 20, 21, 22]. Under this
construction, the fixed points $U_{0},V_{0}$ of Eq. (5) correspond to
$\partial_{\zeta}U=\partial_{\zeta}V=\partial_{\zeta}\tilde{U}=\partial_{\zeta}\tilde{V}=0$.
Due to the presence of the nonlocal terms, the fixed points of Eq. (5) depend
on the round trip average power of the field and require the full evolution
over the round trip of the resonator for their determination. Fixed points
corresponding to homogeneous stationary states (HSS) satisfy
$\langle|\psi_{\text{s}}|^{2}\rangle=|\psi_{\text{s}}|^{2}$ and can be found
by solving (see [10, 23])

$H^{3}-2(\theta-2H)H^{2}+\\{(\theta-2H)^{2}+1\\}H=S^{2}$ (6)

where $H^{2}=|\psi_{\text{s}}|^{2}=\langle|\psi_{\text{s}}|^{2}\rangle$.

The linear stability analysis of the FP HSSs can be found in [10] showing that
the middle power HSS in the simultaneous presence of three HSSs is always
unstable. The HSSs are fixed points of Eq. (5) when the average power over the
round trip is equal to the fixed point power. As such it is not possible for
solutions that start and return to a homogeneous state (including front and
DCS solutions, see Sections IV and V) to hang from the HSS as the presence of
a nonlocal fast time inhomogeneity would change the average power of the
field. Hence the HSS of the FP model are a subset of fixed points which do not
support exponentially localized solutions.

We consider now the fixed points of Eq. (5) corresponding to homogeneous
plateaus connected by step functions. These plateau states have uniform power
$Y_{\pm}$ of finite size $1-\Delta$ and $\Delta$, respectively, in the
normalized fast time variable $\zeta/(2L)$ and are then different from the
HSSs. The round trip average power is given by
$\langle|\psi|^{2}\rangle=\Delta Y_{-}+(1-\Delta)Y_{+}$ parameterized by the
two bistable plateau powers and the plateau durations with $0<\Delta<1$. The
plateau powers are then the solutions of the coupled cubic equations

$\displaystyle Y_{\pm}^{3}$ $\displaystyle-$ $\displaystyle 2(\theta-2\Delta
Y_{-}-2(1-\Delta)Y_{+})Y_{\pm}^{2}$ $\displaystyle+$
$\displaystyle\\{(\theta-2\Delta
Y_{-}-2(1-\Delta)Y_{+})^{2}+1\\}Y_{\pm}=S^{2}\,.$

For each value of the plateau duration $\Delta$, there are up to nine possible
solutions. When restraining to real solutions that satisfy $Y_{+}>Y_{-}$, at
most three pairs of $(Y_{+},Y_{-})$ solutions remain. For each of these pairs,
one can evaluate the effective detuning and find that only one pair
corresponds to the upper-lower HSS of the corresponding LLE, if these exist.
In this way one obtains two plateaus powers for each value of $\Delta$ unless
the effective detuning is outside the bistability region of the corresponding
LLE. This approximation has been used to great effectiveness to describe and
predict stable switching fronts between plateaus in bidirectionally pumped
ring resonators [16]. We will see in Section IV that solutions of plateau
connected by switching fronts are dynamical for the FP model while they have
large ranges of stability in bidirectional ring resonators.

The fast time stability of the plateau states can be understood by considering
the Jacobian matrix

$J=\begin{pmatrix}0&0&1&0\\\ 0&0&0&1\\\
V^{2}+3U^{2}-\theta_{\mathrm{eff}}&-1+2UV&0&0\\\
1+2UV&U^{2}+3V^{2}-\theta_{\mathrm{eff}}&0&0\end{pmatrix}_{(U_{s},V_{s})}$ (8)

where $U_{s}$ and $V_{s}$ are the steady states of Eqs. (5) and
$\theta_{\mathrm{eff}}$ is the effective detuning corresponding to the
stationary field. The Jacobian matrix assumes negligible change to the round
trip average power and provides eigenvalues of the form

$\lambda=\pm\sqrt{(2Y-\theta_{\mathrm{eff}})\pm\sqrt{(Y^{2}-1)}}$ (9)

where $Y=U_{0}^{2}+V_{0}^{2}$ is the plateau power of the stationary state
$U_{s},V_{s}$ from Eq. (5). The Jacobian (8) has a similar form to the fast
time analysis performed for a LLE [14], where $\theta$ replaces the effective
detuning and the plateaus correspond to the HSS. The eigenvalues (9) rule the
escape from and the approach to plateau states $Y_{\pm}$ along the stable and
unstable manifolds. Of key relevance is the transition that occurs at plateau
power $Y=1$, below which the eigenvalues (9) become complex. The lower power
plateau typically exist beneath this threshold, $Y_{-}<1$, displaying fast
time oscillation. This allows for the structurally stable intersection of
stable and unstable manifolds of the plateau corresponding to the formation of
dark cavity solitons [24]. These eigenvalues are essential in the
determination of the presence of local oscillations responsible for the
existence and stability of dark cavity solitons in Section V. We note that the
higher power plateau always has power $Y_{+}>1$ and yields four real
eigenvalues for the parameters we have considered.

###  III.1 Plateau stability in the slow time

The linear stability of the HSS of the FP has been been previously explored
[10] but due to the presence of the round trip average term in Eq. (1) the
existence and linear stability of a plateau solution depends now on the full
field variable along $\zeta$. Here we investigate the linear stability in the
slow time of solutions $\psi=\psi_{+}+\psi_{-}$ formed by two plateaus
$\psi_{\pm}$, coexisting on a round trip and joined by two step functions in
fast time. We determine the linear stability of these solutions to
perturbations of the form

$\psi_{\pm}=\psi_{\pm,\text{s}}+\left(\epsilon_{\pm}e^{ik_{\pm}\zeta+\Omega_{\pm}t}+c.c.\right)\,.$
(10)

By assuming that the wave numbers $k_{\pm}$ are periodic on their respective
plateau the effect of the perturbation on the round trip average becomes
$\langle|\psi_{\text{s}}+\epsilon|^{2}\rangle=\langle|\psi_{\text{s}}|^{2}\rangle$,
and we find that the growth rate of the perturbation on the higher power
$\Omega_{+}$ and lower power $\Omega_{-}$ plateaus are

$\displaystyle\Omega_{+}$ $\displaystyle=$
$\displaystyle-1\pm\sqrt{4Y_{+}\theta_{\mathrm{eff}}-3Y_{+}^{2}\theta_{\mathrm{eff}}^{2}-(4Y_{+}-\theta_{\mathrm{eff}})k_{+}^{2}-k_{+}^{4}}$
$\displaystyle\Omega_{-}$ $\displaystyle=$
$\displaystyle-1\pm\sqrt{4Y_{-}\theta_{\mathrm{eff}}-3Y_{-}^{2}\theta_{\mathrm{eff}}^{2}-(4Y_{-}-\theta_{\mathrm{eff}})k_{-}^{2}-k_{-}^{4}}$

where $\theta_{\mathrm{eff}}=\theta-2\langle|\psi_{\text{s}}|^{2}\rangle$ is
the effective detuning shifted by the average power of the stationary field.
The eigenvalues (LABEL:eq:plateaugrowth) have a similar form to that of the
HSS but now with a dependency on the fast time average. The dependence on the
round trip average power is explicit for the plateau states due to fast time
inhomogeneity over the round trip such that when $Y$ is equal to
$\langle|\psi|^{2}\rangle$ the eigenvalues of Eq. (LABEL:eq:plateaugrowth)
reduce to the eigenvalues of the HSS and are comparable to the stability
eigenvalues of the LLE [14]. The eigenvalues of Eq. (LABEL:eq:plateaugrowth)
predict a Turing instability of the lower power plateau solution starting at
the threshold $Y=1$ present for
$\theta-2\langle|\psi_{\text{s}}|^{2}\rangle\geq 2$. The critical wave number
associated with the maximum growth is given by
$k_{Y}^{2}=2(\theta-2\langle|\psi|^{2}\rangle-2Y_{-})$. The HSS of the FP also
displays a Turing instability starting at the threshold $H=1$ when
$\theta-2H\geq 2$ [10], and exhibits the critical wave number
$k_{H}^{2}=2(\theta-4H)$. Generally, the average intensity of the dark plateau
solutions is much larger than the lower power HSS, and as such the critical
wave number of the plateau is much smaller than that associated to the HSS,
i.e. $k_{Y}<k_{H}$.

If we now consider a time dependent homogeneous perturbation to the plateaus
of the form

$\psi_{\pm}=\psi_{\pm,\text{s}}+\epsilon_{\pm}^{\prime}(t)$ (12)

we can investigate the linear stability of the plateau solutions to
perturbations which change the average power of the field in the slow time
$\partial^{2}_{\zeta}\psi=0$. We derive a Jacobian matrix of the linearized
plateaus system as can be seen in Appendix A and evaluate the eigenspectrum of
this Jacobian matrix numerically.

Figure 2: Stability of coexisting plateaus in the parameter space of shifted
detuning, $\theta_{\mathrm{eff}}$, and input field, $S$, for a fixed length of
the lower power plateaus $\Delta=0.2$. The blue regions correspond to single
stable HSS of the FP model, the green regions to two stable plateaus
coexisting on the round trip of the resonator, and the yellow region to
unstable plateau solutions.

In Fig. 2 the stability of plateau states to perturbations of the average
power is reported in the parameter space $(\theta_{\mathrm{eff}},S)$ when the
lower power plateau has size $\Delta=0.2$. This allows for a direct comparison
with the solutions of bistable HSS as seen for a ring resonator [14]. We find
that near the onset of optical bistability, plateau solutions in the FP
configuration are unstable, as depicted by the yellow zone of Fig. 2. The
green zone shows stable plateaus, and the blue indicates solutions of a single
stable HSS. We find parameter regimes in which there is a bistability of the
HSS of the FP model but there are no stable solutions of coexisting plateaus.
In regions where plateau solutions are unstable, the system cannot support
exponentially localized structures which approach them such as switching
fronts (see next section) and indeed we do not observe the formation of stable
dark cavity-solitons in this regime.

##  IV Switching fronts and their dynamics

In the previous section we have studied the existence and stability of plateau
solutions connected with a step function. These are approximations to the
solutions of Eq. (5) that approach flat solutions along the fast time variable
$\zeta$. There are two kinds of such solutions: heteroclinic orbits, i.e.
trajectories that connect two separate fixed points of Eq. (5), and homoclinic
orbits corresponding to a trajectory that leaves and returns to the same fixed
point of Eq. (5). The former corresponds to switching fronts (SF) solutions
while the latter are here associated to cavity solitons: bright (dark) cavity
solitons in the case the fixed point corresponds to the high (low) power
plateau. As the boundary conditions of a FP resonator are periodic, SF
solutions exist as pairs with opposite orientation in the cavity. Taken
together, they form a heteroclinic cycle. DCS solutions described later in
this paper are themselves composed of oppositely oriented SFs which interact
and lock with each other through local oscillations close to the lower power
fixed point. An example of SF (heteroclinic) and DCS (homoclinic) in the
$(U,V)$ plane are presented in Fig. 3. These trajectories are anchored to the
plateau solutions discussed in Section III for a given value of the distance
$\Delta$. The blue solid lines in Fig. 3 correspond to the family of plateau
solutions when changing $\Delta$ while the circles mark the positions of the
HSS.

Figure 3: Examples of SF heteroclinic (black line top panel) and DCS homoclinc
(red dashed line bottom panel) trajectories in the $V$ vs $U$ plane of Eqs.
(5) where $U,V$ are the real and imaginary parts of the intracavity field
respectively. Solid blue lines correspond to upper and lower plateaus of
different size $\Delta$. Circles correspond to HSS.

Figure 4: (a) Solutions of the Fabry-Pérot model and equivalent ring-resonator
model for input field $S=2\sqrt{2}$ with FP length $2L=100$, and ring
circumference $L=100$. HSS of the FP (ring-resonator) model correspond to blue
(green) lines. Dashed curves correspond to unstable HSS. Stationary SF
solutions are plotted by using their round trip average power for the FP model
(black line) and for the ring-resonator model (black dot-dashed line). Stable
dark solitons solutions of different size correspond to the red lines and form
branches of distinct width. (b) Solutions of the ring-resonator model plotted
with respect to the ring-resonator detuning $\theta_{\mathrm{eff}}$ which are
related to the FP solutions through the effective detuning, Eq. (4), and
corresponding to a zoom of window (a). (c) Solutions of the FP model plotted
with respect to the detuning and corresponding to a zoom of window (a). (d)
Power profile of bistable stationary dark solitons for parameters
$S=2\sqrt{2}$, $\theta=18,\,2L=100$ and corresponding to the two diamonds in
(b). The solid blue line in (d) corresponds to the highest power HSS marked
with a circle in (b).

Figure 5: a) SFs moving away from the spatial point where their velocity
changes direction at separation of $\Delta\approx 0.5$ for parameters
$S=2\sqrt{2}$, $\theta=12.5$. The upper panel (lower panel) shows the
evolution of two SFs from a square wave initial condition with slightly
smaller (larger) SF distance than the turning point. In both cases the SFs
move away from the starting location, annihilate each other and collapse to
one of the bistable HSSs. b) SF velocities corresponding to the upper panel
(black line) and lower panel (red) of a).

We now discuss the stationary states and dynamics of SF (heteroclinic)
solutions found for the FP model and make comparison with similar solutions
found in ring resonators. The HSSs of the FP model are plotted in Fig. 4a as
blue curves. The HSSs of an equivalent ring-resonator model are plotted as
green curves. When compared to the case of a single field in a ring-resonator,
it becomes clear that the effect of the shift in the detuning is most
prominent for high power solutions, resulting in a large shift of the peak
resonance. SF solutions of Eq. (1) present as oppositely oriented pairs and
are generally dynamical, moving with identical speed in opposite directions.
The velocity of SF solutions depend on of the average power of the fields and
can display a turning point where the velocity of both SFs change sign. An
example evolution is shown in Fig. 5 for fixed parameters from an initial
condition near a velocity turning point at $\Delta\approx 0.5$. In the top
(bottom) panel of Fig. 5 the SF initial condition has separation slightly
narrower (wider) than at the velocity turning point. We can see that the SF
solutions either move towards each other until they annihilate (top panel), or
move away from each other until the effective detuning is shifted beyond the
bistability region of the equivalent LLE (bottom panel).

The velocity turning points of the SF of the FP resonator are plotted in Fig.
4a as solid black lines, and correspond to a specific value of $\Delta$ where
the effective detuning correspond to the Maxwell point of the LLE (see dash-
dot black line in Fig. 4a). The Maxwell point of the LLE corresponds to the
unique value of detuning $\Theta_{\mathrm{MP}}$ for the chosen parameters
where the velocity of a non interacting SF is zero. This then constitutes a
stationary state of the FP model with a value of $\Delta$ providing an
effective detuning that is equal to $\Theta_{\mathrm{MP}}$. Due to the
dependence of the effective detuning on the average power of the field there
exist a single separation of SF corresponding to a single turning point for
the FP. The turning point can be then located semi-analytically given that we
know $\Theta_{\mathrm{MP}}$ as shown for example for bidirectionally pumped
ring resonator in [16]. The direction of the SF motion is then determined by
the effective detuning being greater than $\Theta_{\mathrm{MP}}$ (Fig. 5a
upper panel) or smaller than $\Theta_{\mathrm{MP}}$ (Fig. 5a lower panel).

For counterpropagating light in ring resonators, there exists an abundance of
stable and robust light plateau stationary states composed of non-interacting
SFs that form in one field, while the counterpropaging field is flat in
profile [16]. The ring resonator system is described by two equations with
similar form to Eq. (1), one each for the forward and backward
counterpropagating fields. It is interesting to note that for
counterpropagation in a ring-resonator the SF with a given initial separation
move towards (instead of away from) the velocity turning point [16]. The
difference in direction of SF motion when compared with the FP model is due to
distinct backward and forward field profiles present for counterpropagation in
ring-resonators, such that the effective detuning of each field depends on the
power of the counterpropagating field and not explicitly on its own average
power. Unlike counterpropagating fields in ring-resonators, the stability of
SF solutions in a FP configuration depends critically on its own average power
resulting in SFs solutions that always move away from the turning point (see
Fig. 5). This explains why we do not observe the rich phenomenology of stable
SFs solutions of the counterpropagation in ring-resonators [16] and why only
DCS are achieved through the locking mechanism originating from the
interaction of SF through their oscillating tails as described in the next
section.

Finally, we note that although SF heteroclinic solutions are not stationary in
an FP configuration, the calculations of the plateau solutions in Section III
are still worth mentioning. During the motion displayed in Fig. 5, the
solution progresses through the plateau solutions of different size $\Delta$
instant by instant until the two SF interact with each other. During the SF
motion the cusp point of the heteroclinic trajectory of the top panel of Fig.
3 moves along the blue curves that are the plateau solutions of Eq. (III). If
the initial $\Delta$ is larger than that corresponding to the velocity turning
point, the cusp of the heteroclinic trajectory on the right hand side of the
the top panel of Fig. 3 moves leftward along the blue solid line until it
reaches the HSS marked by the circle. If the initial $\Delta$ is smaller than
the velocity turning point, the cusp of the heteroclinic trajectory on the
right hand side of the the top panel of Fig. 3 moves rightward along the blue
solid line until the SF annihilate and the system collapses to the lower HSS
(circle closer to the origin of the axes).

##  V Dark cavity solitons

For a FP resonator with normal dispersion, we observe the formation of DCS
steady states. Such states are composed of two SFs that lock with each other
because of the interaction of spatial oscillations present close to the lower
plateau, as shown for single field ring-resonators in [14] and in optical
parametric oscillators [25, 26]. Bistable DCS stationary solutions of Eq. (1)
are shown in Fig. 4d for parameter values $S=2\sqrt{2},\,\theta=18,\,2L=100$.
The two dark soliton solutions have different widths corresponding to distinct
cycles of the locked oscillatory tails. Such solitons can be obtained by a
perturbation of the HSS of suitable width.

It can be seen that the plateau power is different from that of the HSS (blue
line in Fig. 4d) and also for each DCS of different widths. As was discussed
in section III, the plateau power of an exponentially localized state depends
on the average power of the field over a round trip such that solitons of
different width display different plateau power due to their different average
power. The DCS of Fig. 4d correspond to the diamonds in 4b. The plateau
solutions from which a DCS hangs from correspond to the points on the solid
blue curve of the lower panel of Fig. 3 with the value of the shifted
detuning, $\theta_{\mathrm{eff}}$, of the corresponding distance $\Delta$.
Note that the bistable DCS and the HSS of the FP model are distributed along
the line $\langle|\psi|^{2}\rangle=-\theta_{\mathrm{eff}}/2+\theta/2$ as shown
in Figs. 4b and 4c as a pink dotted line, respectively.

The effect of the shift in detuning discussed throughout this paper can be
understood by making a comparison between the DCS of normal dispersion
described here and the FP bright solitons in the anomalous dispersion regime
[10]. Due to the large average power of DCS solutions, there is a much larger
shift in detuning when compared with the bright solitons in the anomalous
dispersion case. DCS solutions are also much further detuned than cavity
solitons in an equivalent ring resonator.

Figure 6: Approximate difference in detuning between dark soliton solutions of
the FP and equivalent ring-resonator models with respect to the common input
field. The black line corresponds to the SF turning point when $\Delta=0$ for
the FP model. The dashed line corresponds to the SF turning point when
$\Delta=0.1$.

To see this, we plot in Fig. 6 the difference in detuning values between the
FP resonator and an equivalent ring-resonator model $\theta_{\mathrm{eff}}$,
and corresponding to the location of DCS for given input powers $S$. The
location of a DCS is approximated by using the SF turning point line (an
example of which is shown in Fig. 4a) intersecting either the high power HSS
(solid black line in Fig. 6) or by selecting a point along this line
corresponding to a specific DCS size $\Delta$. The dashed black line in Fig. 6
shows solutions in which SFs are stationary with separation $\Delta=0.1$, the
approximate size of the DCS of Fig. 4d). These lines were calculated using the
numerical fit of the Maxwell point of the ring resonator model from [16]. We
see that as the input power is increased the shift becomes larger.
Furthermore, the range of detunings where DCS exist is much larger for the FP
model when compared to an equivalent ring-resonator system. For example, the
longest DCS solution branch for an FP resonator spans $16.75<\theta<19.37$,
see Fig. 4c, whilst DCS in the equivalent ring-resonator system are present
only in the range $5.38<\theta_{\mathrm{eff}}<6.43$, approximately $2.5$ times
smaller.

Figure 7: a) Soliton states containing a single soliton, red line, and two
solitons, black line for $S=2\sqrt{2}$,$\theta=18$. b)-c) FP solutions plotted
over detuning (or shifted detuning) with respect to their average power. HSS
are plotted as blue, b), green c), the black curves corresponds to the SF
turning point, and dark soliton solutions are plotted as red curves of one,
solid line, two, dashed line, and three, dot-dashed line solitons present in
the cavity. The pink dotted line correspond to the solutions with $\theta=18$.

The average power of the field is also affected by the number of solitons
present in the cavity such that the existence and stability of the solitons
depend also on the number of solitons in the cavity. Fig. 7a shows two
bistable DCS solutions for $S=2\sqrt{2}$, $\theta=18$ corresponding to one or
two solitons in the cavity. We see that the presence of an additional soliton
modifies the low power oscillations and the plateau power. In Figs. 7b and 7c
we plot the DCS solutions as red curves with one soliton, solid, two solitons,
dashed, and three solitons, dot-dashed, present in the cavity with an
identical number of fast-time oscillations. With each additional soliton in
the cavity, the average power of the field decreases. As such, solution
branches containing a large number of solitons appear at lower values of
detuning. For detuning $\theta=18$, input field $S=2\sqrt{2}$ and $2L=100$, we
find that states of one and two solitons are possible, but that further
perturbations of the system will not allow the formation of additional
solitons, leading instead to the destruction of pre-existing solitons.

##  VI Oscillatory dynamics

Figure 8: Oscillating solution a) one oscillating soliton, b) one oscillating
soliton and one ‘stationary’ soliton, c) two synchronized oscillating solitons
(displayed in the upper panels at $t=20$), all for identical parameters
$\theta=18$, $S=2\sqrt{2}$. The lower panels display the slow time evolution.
Figure 9: Oscillating solution with a) one oscillating soliton in a FP cavity
of length $2L=100$, b) one oscillating soliton and one locked soliton in a FP
cavity of length $2L=100$, c) two oscillating solitons in a FP cavity of
length $2L=100$, d) one oscillating soliton in a FP cavity of length $2L=50$,
all for identical parameters $\theta=18$, $S=2\sqrt{2}$.

We now investigate the dynamics of DCS states above a temporal instability of
Eq. (1). For the case of a ring-resonator, dynamical instabilities of DCS
solutions due to Hopf bifurcations result in local oscillations of the soliton
[14]. In what follows we demonstrate the effects of the nonlocal coupling of
Eq. (1) on the dynamics of oscillatory soliton solutions.

In Fig. 8a we show an example of an oscillatory solution of the FP model with
input field $S=2\sqrt{2}$ and detuning $\theta=18$. We can see that the
temporal dynamics of the system is not confined to the soliton but it extends
to an oscillation of the background plateau too. As the soliton moves through
its limit cycle, there is a change in the average power of the field. Hence,
due to the nonlocal self interaction term, regions of the cavity far from the
soliton exhibit oscillatory dynamics with identical period to that of the
soliton. These oscillations are small due to the small change in average power
originating from the oscillation of the soliton. A trace of the minimum power
of the oscillating soliton is plotted in Fig. 9a, showing a single period of
oscillation. In Fig. 8b we introduce an oscillating dark soliton to an initial
stationary solution containing a stable stationary dark soliton shown in Fig.
4d. The presence of the oscillating soliton induces small temporal
oscillations in the plateau power and tiny oscillations in the peaks of the
pre-existent DCS. The trace of the minimum power is plotted in 9b in black for
the oscillating soliton, and in red for the stable soliton. We note that the
oscillation period has decreased form Fig 8a. In Fig. 8c, we show the
evolution of two synchronized breathing solitons. Here we see a larger
oscillation amplitude of the plateau power than in the single soliton case,
due to the larger change in the average power of the field resulting from the
second oscillating soliton. In Fig. 9c we plot the trace of the minimum power
of the two oscillating solitons, seen in 8c, starting from an unsynchronized
initial condition. As the system evolves over slow time the soliton phases
begin to overlap, resulting in full synchronization. The resulting dynamics
has now experienced a period doubling with respect to the single oscillating
soliton. In Fig. 9d, we plot the evolution of a single soliton with half the
cavity length, $2L=50$, of the previous examples. We see that the dynamics of
this single soliton is identical to the synchronized dynamics of the two
soliton examples with $2L=100$ of Fig. 8c and 9c. In general we find that the
dynamics of $N$ well separated solitons in a cavity of length $L$ synchronizes
towards the dynamics of a single soliton in a cavity of length $L/N$.

We find that well separated solitons, located such that they do not interact
through the local dynamics at the tails, experience phase dependent
interaction through the nonlocal coupling. We note that oscillating solitons
of the normally dispersive LLE interact locally through their tails and as
such do not exhibit synchronization when well separated in a long cavity [14].
In the FP model, the change in average power of the field during an
oscillation of a single oscillating soliton is small, and as such the change
in the power of the background plateau is also small. By increasing the length
of the FP cavity, we can reduce the effects of soliton oscillation on the
average power such that we can approach the LLE dynamical and independent
solutions. The dynamics of the FP model is most distinct from the ring-
resonator LLE when the cavity length is small or the number of oscillating
solitons is large displaying long range interactions leading to
synchronization.

##  VII Conclusions

Figure 10: Bistable frequency combs for parameter $S=2\sqrt{2}$,
$\theta=18,2L=100$. These spectra correspond to the dark solitons shown in
Fig. 4d with lesser a) and greater b) width.

We have presented bistability, plateaus, switching fronts and dark soliton
states in an Fabry-Pérot model with normal dispersion. Through the use of the
effective detuning we demonstrated analogies and differences of these
solutions with the stationary states of the ring-resonator case, i.e. the LLE.
The homogeneous stationary states of the Fabry-Pérot model have a one to one
correspondence to the stationary states of a ring resonator described by the
LLE, with identical input field and a detuning that is shifted by the average
power of the intracavity field over a single round trip. We investigated the
linear stability of solutions formed by two plateaus that are different from
the homogeneous stationary states and connected by step functions. We
identified a region of instability near the onset of optical bistability of
the homogeneous stationary states. When compared with counterpropagating light
in a ring resonator [16], it turns out that plateau solutions in a Fabry-Pérot
cavity with normal dispersion are generally unstable. Their stability analysis
however is still useful since we can find Turing instabilities of the low
power plateaus and determine their critical wave number. Such wave number is
much smaller than those corresponding to Turing instabilities of the
homogeneous stationary state. It is this wave number that rules the local
oscillations close to the lower plateau when two switching fronts lock to form
a stable dark soliton. Due to the shift in detuning, a peculiar feature of
dark soliton solutions in the Fabry-Pérot model is that they do not connect to
the homogeneous stationary statuses (as they do in the ring-resonator model),
but instead to plateau solutions, which are dependent on the size (average
power) of the soliton.

For normal dispersion, the average power of the field, and hence the shift in
detuning, is comparatively large for dark solitons when compared with the
bright solitons of the anomalous dispersion case. The large shift results in
the dark soliton solution branches to be elongated, occupying a larger domain
of detuning values than an equivalent ring-resonator configuration, and found
in strongly detuned regimes.

Finally, we investigated the effects of nonlocal coupling on dynamics of
oscillatory solitons. We find that the nonlocal coupling induces temporal
oscillations to the homogeneous background power. In the presence of two
oscillating solitons, we observe phase dependent interaction, resulting in
synchronization and modifications of their oscillation periods. The resulting
synchronized dynamics approach that of a single soliton with half the cavity
length. It was found that the dynamics of the system approached that of an
equivalent ring-resonator in the limits of a long cavity and small soliton
number (small change in average power).

The predictions presented in this paper were obtained for parameter values
that are realistically feasible in existing experimental realizations [11].
When these systems operate at normal dispersion we expect to see the formation
of dark solitons steady states in experimental verifications. The Fabry-Pérot
system allows for additional design consideration of the cavity properties
allowing for control over frequency comb generation. Example frequency combs
of the Fabry-Pérot resonator are shown in Fig. 10 which correspond to the dark
solitons of Fig. 4d and show distinct modulations on the combs, corresponding
to the number of low intensity spatial oscillations of the dark soliton [27].
The engineering of frequency combs can be useful in applications in precision
spectroscopy, LIDAR, and channel generation for telecom systems.

## Acknowledgements

This research was supported by funding from the EPSRC DTA Grant No.
EP/T517938/1. P.D. acknowledges support by the H2020 European Research Council
(ERC) (756966,Counterlight), the Marie Sklodowska-Curie Innovative Training
Network (MSCA) (812818, Microcombs), and the Max Planck Society. L.H.
acknowledges funding provided by the CNQO group within the Department of
Physics at the University of Strathclyde, and the “Saltire Emerging
Researcher” scheme through the Scottish University’s Physics Alliance (SUPA)
and provided by the Scottish Government and Scottish Funding Council.

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##  Appendix A Appendix

Here we determine the linear stability of plateau solutions in slow time with
$\partial^{2}_{\zeta}\psi=0$. We do so by approximating the intracavity field
as a step function $\psi=\psi_{+}+\psi_{-}$ composed of higher power
$\psi_{+}$ and lower power $\psi_{-}$ plateau solutions such that the average
power of the field may be written as
$\langle|\psi|^{2}\rangle=\Delta|\psi_{-}|^{2}+(1-\Delta)|\psi_{+}|^{2}$ where
$\Delta$ is the size of the lower power plateau normalized to the round trip.
Hence the higher and lower power plateaus evolve according to the two coupled
equations

$\displaystyle\partial_{t}\psi_{\pm}$ $\displaystyle=$ $\displaystyle
S-(1+i\theta)\psi_{\pm}+i(|\psi_{\pm}|^{2}+2(\Delta|\psi_{-}|^{2}+(1-\Delta)|\psi_{+}|^{2}))\psi_{\pm}$
(13)

The linear stability of the plateau solutions can be understood by finding the
eigenspectrum of the Jacobian matrix

$J=\begin{pmatrix}-1-A_{+}-4(1-\Delta)U_{+}V_{+}&B_{+}-4(1-\Delta)V_{+}^{2}&-4\Delta
U_{-}V_{+}&-4\Delta V_{+}V_{-}\\\
-C_{+}+4(1-\Delta)U_{+}^{2}&-1+A_{+}+4(1-\Delta)U_{+}V_{+}&4\Delta
U_{+}U_{-}&4\Delta U_{+}V_{-}\\\
-4(1-\Delta)U_{+}V_{-}&-4(1-\Delta)V_{+}V_{-}&-1-A_{-}-4\Delta
U_{-}V_{-}&B_{-}-4\Delta V_{-}^{2}\\\
4(1-\Delta)U_{+}U_{-}&4(1-\Delta)U_{-}V_{+}&-C_{-}+4\Delta
U_{-}^{2}&-1+A_{-}+4\Delta U_{-}V_{-}\end{pmatrix}$ (14)

where $U_{\pm},V_{\pm}$ are the real and imaginary components of the plateaus
and $A_{\pm}=2U_{\pm}V_{\pm}$,
$B_{\pm}=\theta_{\mathrm{eff}}-U_{\pm}^{2}-3V_{\pm}^{2}$,
$C_{\pm}=\theta_{\mathrm{eff}}-3U_{\pm}^{2}-V_{\pm}^{2}$, where
$\theta_{\mathrm{eff}}=\theta-2\Delta(U_{-}^{2}+V_{-}^{2})-2(1-\Delta)(U_{+}^{2}+V_{+}^{2})$
is the effective detuning defined in Eq. (4). This formulation allows us to
investigate the stability of LLE stationary states for the equivalent FP
resonator. Numerical evaluation of the eigenspectrum of Eq. (14) yields the
results shown in Fig. 2 and are discussed in Section III.

In the limit $\Delta\rightarrow 0$ we can evaluate the eigenvalues of Eq. (14)
as

$\displaystyle\lambda$ $\displaystyle=$
$\displaystyle-1\pm\sqrt{A_{-}^{2}-B_{-}C_{-}}$ (15) $\displaystyle\lambda$
$\displaystyle=$
$\displaystyle-1\pm\sqrt{(A_{+}+4U_{+}V_{+})^{2}-(B_{+}-4V_{+}^{2})(C_{+}-4U_{+}^{2})}$
(16)

Eigenvalues (15) have an identical form to the eigenvalues of the homogeneous
stationary states and hence they have a negative real part for the low power
plateaus. Instability of the plateau solutions is therefore due to Eq. (16).


# Beyond Screens: Supporting Co-located Augmented Reality Experiences with
Smart Home Devices

Ava Robinson, Yu Jiang Tham, Rajan Vaish, Andrés Monroy-Hernández  Snap
Inc.USA<EMAIL_ADDRESS>
(2018)

###### Abstract.

We introduce Spooky Spirits, an AR game that makes novel use of everyday smart
home devices to support co-located play. Recent exploration of co-located AR
experiences consists mainly of digital visual augmentations on mobile or head-
mounted screens. In this work, we leverage widely adopted smart lightbulbs to
expand AR capabilities beyond the digital and into the physical world, further
leveraging the physicality of users’ shared environment.

Co-Located, Augmented Reality, IoT, Embodied, Social, Mobile AR, Smart Home
Devices

††copyright: acmcopyright††journalyear: 2018††doi:
XXXXXXX.XXXXXXX††conference: ; July 2017; Washington, DC, USA††price:
15.00††isbn: 978-1-4503-XXXX-X/18/06††ccs: Human-centered computing Ubiquitous
and mobile computing systems and tools; Ubiquitous and mobile computing

##  1\. Introduction

Recent research has shown how co-located AR experiences can enable people to
have fun together, but most consist of only digital visual augmentations using
phone-based AR (Dagan et al., 2022). Additionally, prior work has shown that
AR is well-suited for co-located experiences because it can be grounded in the
environment (Wetzel et al., 2008), for example by using physical objects as
enablers, or inputs, of the experience (Dagan et al., 2022). However, these
augmentations are constrained to pixels on a screen rather than augmenting the
physical space itself or using physical objects as outputs in the experience.
The recent increase of IoT devices in people’s homes (Koskela and Väänänen-
Vainio-Mattila, 2004) presents an opportunity to use smart home devices to
extend the augmentation of the physical world for co-located experiences.

Figure 1. Left: Users playing Spooky Spirits with the smart light in the
background and a user doing the T Pose body gesture. Middle: Game prompting
users to ask a question while the light illuminates the room white indicating
“spirits” listening. Right: Room getting illuminated red by the light,
representing a “no” answer.

Leveraging this opportunity, we explore using IoT devices, in particular a
smart light, as a core aspect of an experience and novel form of interaction.
We present Spooky Spirits, a mobile AR game111Implemented as a Snapchat Lens
using Lens Studio https://ar.snap.com/ built for two users to play together in
person along with a smart light 222Used Kasa Smart Lightbulbs
https://www.kasasmart.com/us/products/smart-lighting. Our system allows users
to easily connect their smart lights to the mobile experience via a web
interface.

We evaluated our system with 10 pairs of participants by observing how they
use the experience and interviewing them to learn more about the immersive
value and impact of augmenting a shared physical space using an IoT device. We
hope to motivate and inform future designs of mobile AR and IoT co-located
experiences.

##  2\. DESIGN AND EXPERIENCE OVERVIEW

The Spooky Spirits experience piggybacks off of existing spirit summoning and
fortune-telling games such as Ouija boards333Ouija board
game:https://en.wikipedia.org/wiki/Ouijaand Magic 8-Balls444Magic 8-Ball game:
https://en.wikipedia.org/wiki/Magic_8_Ball . This narrative embraces the magic
and spookiness of a smart light changing colors without manual input. To set
up the experience users must 1) have a smart light in their space 2) connect
their light to the mobile game using a website, and 3) find a partner to play
with in person.

Device arrangement is a key design consideration for co-located experiences
(Isbister et al., 2018)(Lundgren et al., 2015). Our experience involves two
users, a phone running the game, and a smart light. When playing Spooky
Spirits, one user, player 1, holds the phone with the game running and guides
their partner, player 2, through the playful “summoning ritual” by
communicating the instructions they receive via the mobile UI. When the
experience first begins, the game creates the spooky ambiance by turning the
smart light blue. Player 1 then directs their partner to stand in front of the
camera and position their body in a specific gesture, for example, a T Pose,
which is detected using Full Body Triggers 555https://docs.snap.com/lens-
studio/references/templates/object/full-body-triggers (see Fig. 1 - left).
After the body gesture is detected, the light turns white providing visual
feedback for both users indicating that the “spirits” are listening and users
are prompted to ask a yes-no question about the future aloud (see Fig. 1 -
Middle). For example, users might ask ”Will I win the lottery?”.

To make the light a salient part of the experience we chose to use it not only
as an enhancement to ambiance but also as a source of information in the
narrative of the experience. After users ask a yes-no question aloud, the
light will turn green or red representing the answer as yes or no respectively
(see Fig. 1 - Right), making the experience dependent on the information from
the state of the light. In this design, the smart light acts as the sole
source of immersion and augmentation for player 2. This allows us to explore
if the experience is immersive even without mobile visual augmentation for all
players.

This design encourages users to work together and leverages the physicality of
users’ space by augmenting their environment beyond pixels on a screen. This
provides an opportunity to observe the impact and immersive value that the
smart home device brings to the experience since only one user can see the UI
and AR on the mobile device, while the other user can only see the effects of
the light in their environment. This experience extends augmentation beyond
mobile for both users.

##  3\.  SMART DEVICE INTEGRATION SYSTEM

Figure 2. Website for users to launch the game and generate a 5-digit code to
connect to their smart light. Figure 3. Overview of the backend system for
integrating Kasa Devices into the game.

Our system involves a web app666https://spookyspirits.letsplayirl.com/ we
built for users to connect to their smart lights (see Fig. 2). We chose to use
Kasa Smart Light Bulbs as they have a public API, are easy for non-technical
users to set up, and are relatively inexpensive. As set up for our experience,
users use our web interface to log in to their existing Kasa account, select
the bulb they would like to use, and obtain a 5-digit randomly generated code
that is stored in DynamoDB. Next, users launch the game777They launched by
scanning a code in Snapchat and enter the 5-digit code to connect the game to
their smart light. By using an external interface for login we hoped to
simplify the connection process and reduce friction within the mobile
experience. The details of the system consist of AWS Amplify running a Next.js
app with 1) a server-side component that handles API calls to get and set the
settings of the smart light and 2) a client side component that displays the
website to the user (see Fig. 3). The Next.js app also polls from a DynamoDB
database periodically to check for changes for each device, represented by the
generated 5-digit code, and if there are changes, it sends the new data to the
Kasa Cloud using the Kasa public API, which modifies the state of the physical
devices. The Snapchat Lens will make get and set calls to the AWS Remote API
using the entered 5-digit code to change the state of the device.

##  4\. USER STUDY OBSERVATIONS

We performed a pilot study with 10 pairs of participants playing with the
Spooky Spirits experience together with a smart light and conducted semi-
structured interviews.

We aimed to explore the impact of a smart home device as a core aspect of a
co-located AR experience to help inform future designs of similar experiences.
Here are six main observations

  1. (1)

Augmentation of the physical world beyond mobile superseded the setup effort.
Users noted that the setup friction was high due to the smart light, however,
they felt that the light added to the overall immersive value.

  2. (2)

Augmentation via IoT devices enabled immersion beyond mobile and overcame
asymmetric device access. Although one user did not experience mobile visual
augmentation, generally they still felt immersed and involved in the
experience because of the augmentations from the smart light.

  3. (3)

Different setup decisions led to new interactions. We observed users who
placed the bulb in a ceiling socket try to reach up towards the light and pay
direct attention to the bulb itself, whereas users who placed the bulb in a
desk or standing lamp did not try to touch the physical bulb and instead paid
attention to the effect the bulb had on overall lighting.

  4. (4)

The state of the physical environment affected its augmentation. Users who
played in dark lit rooms felt the experience was more immersive than those who
played in rooms with more secondary light because the changes in the color and
brightness of the light were more salient.

  5. (5)

IoT devices could have played a more important role in giving feedback. Users
sometimes struggled to know what to expect from the smart light or when to
wait for the light to change.

  6. (6)

IoT devices were able to serve multiple roles, from creating ambiance to
providing information. During the game, blue light was used to create spooky
ambiance, and green or red light encoded information (yes or no answers to
users’ questions) that users easily decoded.

In the future, we hope to build and study more co-located AR experiences that
use a variety of IoT devices as novel forms of augmentation and interactions.
We hope this inspires future designers to continue to design more mobile AR
and IoT co-located experiences.

###### Acknowledgements.

Many thanks to Melissa Powers, Erica Principe Cruz, Samantha Reig, Tim Chong,
Jennifer He, and Eunice Kim for their help and support in designing, building,
and studying this experience. Also, many thanks to all our study participants
for their time.

## References

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  * Isbister et al. (2018) Katherine Isbister, Elena Márquez Segura, and Edward F Melcer. 2018\.  Social affordances at play: Game design toward socio-technical innovation. In _Proceedings of the 2018 CHI Conference on Human Factors in Computing Systems_. 1–10.
  * Koskela and Väänänen-Vainio-Mattila (2004) Tiiu Koskela and Kaisa Väänänen-Vainio-Mattila. 2004.  Evolution towards smart home environments: empirical evaluation of three user interfaces.  _Personal and Ubiquitous Computing_ 8 (2004), 234–240.
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