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Dirac Solitons in Optical Microresonators: Supplementary information
Heming Wang
1
, Yu-Kun Lu
1
, Lue Wu
1
, Dong Yoon Oh
1
, Boqiang Shen
1
, Seung Hoon Lee
1
, and Kerry Vahala
1
,
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA.
Corresponding author: vahala@caltech.edu
The conservative coupled Lugiato-Lefever equations may admit solutions with nonzero backgrounds, where the
fields do not vanish when
θ
→ ±∞
. In the following we will show the existence of these solutions with the help of
a phase space and then derive some special cases of such solutions. We note that, while these solutions are valid
for the conservative hybrid-mode system, the addition of loss or other broadband effects may change the solutions
in a qualitative way. The background fields also make the solutions difficult to satisfy the periodic conditions for a
resonator. It is not known if soliton solutions with backgrounds can exist in a lossy resonator in the form given below.
The equations for the Dirac soliton reads
(
δD
1
v
)
θ
E
1
=
iδωE
1
+
ig
c
E
2
+
i
(
g
11
|
E
1
|
2
E
1
+
g
12
|
E
2
|
2
E
1
)
(S1)
(
δD
1
+
v
)
θ
E
2
=
iδωE
2
+
ig
c
E
1
+
i
(
g
22
|
E
2
|
2
E
2
+
g
12
|
E
1
|
2
E
2
)
(S2)
As in the main text, we introduce the following quantities:
̄
H
=
δω
(
|
E
1
|
2
+
|
E
2
|
2
) +
g
c
(
E
1
E
2
+
E
2
E
1
) +
1
2
(
g
11
|
E
1
|
4
+
g
22
|
E
2
|
4
+ 2
g
12
|
E
1
|
2
|
E
2
|
2
)
(S3)
̄
N
= (
δD
1
v
)
|
E
1
|
2
(
δD
1
+
v
)
|
E
2
|
2
(S4)
G
=
δD
1
+
v
δD
1
v
g
11
2
+
δD
1
v
δD
1
+
v
g
22
2
+
g
12
(S5)
We begin by obtaining the background (continuous-wave) solutions in the system. To eliminate the global phase
dependence, we rewrite the equations of motion using two amplitude variables,
|
E
1
|
and
|
E
2
|
, and a phase difference
variable,
χ
arg(
E
1
E
2
):
(
δD
1
v
)
θ
|
E
1
|
=
g
c
|
E
2
|
sin
χ
(S6)
(
δD
1
+
v
)
θ
|
E
2
|
=
g
c
|
E
1
|
sin
χ
(S7)
θ
χ
=
2
δD
1
δω
δD
2
1
v
2
+
(
g
c
δD
1
v
|
E
2
|
|
E
1
|
+
g
c
δD
1
+
v
|
E
1
|
|
E
2
|
)
cos
χ
+
(
g
11
|
E
1
|
2
+
g
12
|
E
2
|
2
δD
1
v
+
g
22
|
E
2
|
2
+
g
12
|
E
1
|
2
δD
1
+
v
)
(S8)
We denote the background solutions as
|
E
1
|
0
,
|
E
2
|
0
and
χ
0
, and at these points all three derivatives should vanish.
This happens when
|
E
1
|
0
and
|
E
2
|
0
are both zero, or are both nonzero. As we have solved the first case in the previous
section, we will focus on the case where
|
E
1
|
0
>
0 and
|
E
2
|
0
>
0. In this case sin
χ
0
= 0, and
χ
0
= 0 or
π
, i.e. the
two components in the background are completely in-phase or out-of-phase relative to the mode coupling.
A two-dimensional phase space can be constructed from the real and imaginary parts of
E
1
E
2
(Fig. S1a). The
fields at each
θ
correspond to a point in the diagram, and follow a contour defined by constant
̄
H
and
̄
N
as
θ
varies. Background solutions appear in the diagram as fixed points on the real axis. Soliton solutions converge to the
background for
θ
→ ±∞
, and therefore are homoclinic orbits connecting the background state to itself (Fig. S1b).
The shape of the orbit is a lima ̧con and is described by the following equation:
[
zz
+
a
2
(
z
+
z
)
]
2
=
b
2
zz
, z
=
E
1
E
2
−|
E
1
|
0
|
E
2
|
0
cos
χ
0
(S9)
a
=
2
g
c
G
(1 +
G
|
E
1
|
0
|
E
2
|
0
cos
χ
0
/g
c
)
, b
=
g
c
|
G
|
(
δD
1
v
)
|
E
1
|
2
0
+ (
δD
1
+
v
)
|
E
2
|
2
0
|
E
1
|
0
|
E
2
|
0
1 +
G
|
E
1
|
0
|
E
2
|
0
cos
χ
0
/g
c
δD
2
1
v
2
(S10)
According to the properties of a lima ̧con, when
b <
|
a
|
the curve has a inner loop, and the background solution
becomes a saddle point (Fig. S1b). The inner loop and the outer loop each correspond to a soliton solution, where the
2
Re(
E
1
E
2
*)
Im(
E
1
E
2
*)
Re(
E
1
E
2
*)
Im(
E
1
E
2
*)
Re(
E
1
E
2
*)
Im(
E
1
E
2
*)
a
b
c
FIG. S1: Phase space portraits of solitons in the hybrid-mode system. For simplicity we choose
g
11
=
g
22
= 0 (
G
=
g
12
)
in these plots. The length of one grid unit in the plot represents 2
g
c
/G
. Arrows indicate the direction of state change when
θ
increases. (a) The phase space portrait for bright solitons with
v
= 0,
δω
=
g
c
/
2 (dashed line) and
v
= 0,
δω
=
g
c
/
2
(solid line). (b) The phase space portrait for dark soliton and soliton-on-background solutions, with a component-in-phase
background. Parameters are
v
= 0,
δω
= 2
g
c
and
|
E
1
|
2
0
=
|
E
2
|
2
0
=
g
c
/G
. (c) The phase space portrait for dark soliton and
soliton-on-background solutions, with a component-out-of-phase background. Parameters are
v
=
5
/
3
δD
1
,
δω
= 3
g
c
and
|
E
1
|
2
0
=
|
E
2
|
2
0
= 4
g
c
/G
. In both (b) and (c) the saddle point topology is present near the background state.
inner loop resembles the conventional dark soliton and the outer loop is a soliton-on-background solution. If
b >
|
a
|
the lima ̧con is a simple closed curve that does not pass through the background state, and the solution becomes
a Turing roll. For the critical case
b
=
|
a
|
, the lima ̧con reduces to a cardioid, and only the soliton-on-background
solution remains.
The sign of cos
χ
0
determines if the background components are in-phase or out-of-phase, and how the lima ̧con
is oriented. For
|
v
|
< δD
1
, the
b
≤ |
a
|
condition results in
χ
0
= 0. In this case the reduced detuning is restricted
to
̃
ξ
1, and the resonance line of the soliton intersects the bottom branch twice. For
|
v
|
> δD
1
, cos
χ
0
has the
opposite sign to
G
, which may become negative. No particular restrictions have been found for the detuning
δω
, and
the resonance line of the soliton intersects both branches once. Typical phase spaces of these two cases are illustrated
in Figs. S1b and S1c. The case
|
v
|
=
δD
1
does not correspond to solitons, as one of the
|
E
1
,
2
|
loses its dynamics, and
all solutions are continuous waves.
In the following, we derive the analytical solutions for these solitons. We restrict ourselves to the case
|
v
|
< δD
1
to
avoid the discussions on parameters that may change sign, but the technique can be readily generalized. We introduce
additional reduced variables to simplify the expressions:
̃
E
1
δD
1
vE
1
,
̃
E
2
δD
1
+
vE
2
,
̃
G
G
g
c
|
E
1
|
0
|
E
2
|
0
(S11)
Similarly,
|
̃
E
1
|
0
and
|
̃
E
2
|
0
are the values of the corresponding variable at the background.
We extend the definition of
ψ
2
as
ψ
2
1
2
(
|
̃
E
1
|
2
+
|
̃
E
2
|
2
) =
1
2
[
(
δD
1
v
)
|
E
1
|
2
+ (
δD
1
+
v
)
|
E
2
|
2
]
(S12)
which has the same meaning as the
ψ
2
in the main text when
̄
N
= 0. The value of
ψ
2
at the background reads
ψ
2
0
[
(
δD
1
v
)
|
E
1
|
2
0
+ (
δD
1
+
v
)
|
E
2
|
2
0
]
/
2. The differential equation for
ψ
2
reads
θ
ψ
2
= 2
|
E
1
||
E
2
|
sin
χ
(S13)
=
δD
1
δD
2
1
v
2
(
ψ
2
ψ
2
0
)
4(1 +
̃
G
)
[
̃
G
(
ψ
2
ψ
2
0
)
2
ψ
2
0
]
2
|
̃
E
1
|
2
0
|
̃
E
2
|
2
0
(S14)
where we have used the conservation of
̄
H
and
̄
N
and substituted their values at the background. Integration gives
ψ
2
=
ψ
2
0
+
2
[
ψ
4
0
(1 +
̃
G
)
|
̃
E
1
|
2
0
|
̃
E
2
|
2
0
]
̃
G
[
ψ
2
0
+
σ
1 +
̃
G
|
̃
E
1
|
0
|
̃
E
2
|
0
cosh(
β
̃
θ
)
]
, β
4
̃
G
̄
N
2
|
̃
E
1
|
2
0
|
̃
E
2
|
2
0
,
̃
θ
=
g
c
δD
2
1
v
2
θ
(S15)
3
The saddle point criterion from the lima ̧con ensures that
β
is a real number. The
σ
before the cosh function is
determined by how the square root is taken. For dark-soliton-like solutions (inner loop of the lima ̧con) we take
σ
= 1,
and for soliton-on-background solutions (outer loop of the lima ̧con) we take
σ
=
1.
The rest of the solution process is identical to the bright soliton case, which proceeds by finding the equation for
arg
E
1
,
2
followed by integration. Combining all results above, the field solution can be written as
E
1
=
|
E
1
|
2
0
|
̃
E
1
|
0
|
̃
E
2
|
0
β
2
cosh(
β
̃
θ
) +
i
(
̄
N
+ 2
̃
G
|
̃
E
1
|
2
0
)
β
sinh(
β
̃
θ
)
(
δD
1
v
)
̃
G
[
2
σ
1 +
̃
G
+ 2
ψ
2
0
/
(
|
̃
E
1
|
0
|
̃
E
2
|
0
) cosh(
β
̃
θ
) +
sinh(
β
̃
θ
)
]
1
/
2
×
[
2
σ
1 +
̃
G
+ 2
ψ
2
0
/
(
|
̃
E
1
|
0
|
̃
E
2
|
0
) cosh(
β
̃
θ
)
sinh(
β
̃
θ
)
2
σ
1 +
̃
G
cosh(
β
̃
θ
) + 2
ψ
2
0
/
(
|
̃
E
1
|
0
|
̃
E
2
|
0
)
]
γ/
2
exp(
ik
0
θ
)
(S16)
E
2
=
±
|
E
2
|
2
0
|
̃
E
1
|
0
|
̃
E
2
|
0
β
2
cosh(
β
̃
θ
) +
i
(
̄
N
+ 2
̃
G
|
̃
E
2
|
2
0
)
β
sinh(
β
̃
θ
)
(
δD
1
+
v
)
̃
G
[
2
σ
1 +
̃
G
+ 2
ψ
2
0
/
(
|
̃
E
1
|
0
|
̃
E
2
|
0
) cosh(
β
̃
θ
) +
sinh(
β
̃
θ
)
]
1
/
2
×
[
2
σ
1 +
̃
G
+ 2
ψ
2
0
/
(
|
̃
E
1
|
0
|
̃
E
2
|
0
) cosh(
β
̃
θ
)
sinh(
β
̃
θ
)
2
σ
1 +
̃
G
cosh(
β
̃
θ
) + 2
ψ
2
0
/
(
|
̃
E
1
|
0
|
̃
E
2
|
0
)
]
γ/
2
exp(
ik
0
θ
)
(S17)
k
0
1
2
δD
1
(
g
c
|
E
2
|
2
0
−|
E
1
|
2
0
|
E
1
|
0
|
E
2
|
0
+ (
g
11
g
12
)
|
E
1
|
2
0
(
g
22
g
12
)
|
E
2
|
2
0
)
(S18)
where the sign of
E
2
is negative if the lima ̧con loop encloses the origin, or positive if the origin is not enclosed.
|
E
1
|
0
and
|
E
2
|
0
are the background field amplitudes, i.e. the positive solutions to the following equation:
2
δD
1
δω
=
g
c
(
δD
1
+
v
)
|
E
2
|
0
|
E
1
|
0
+
g
c
(
δD
1
v
)
|
E
1
|
0
|
E
2
|
0
+(
g
11
|
E
1
|
2
0
+
g
12
|
E
2
|
2
0
)(
δD
1
+
v
)+(
g
22
|
E
2
|
2
0
+
g
12
|
E
1
|
2
0
)(
δD
1
v
) (S19)
A special case can be obtained by setting
g
11
=
g
22
,
v
= 0, and
|
E
1
|
0
=
|
E
2
|
0
=
(
δω
g
c
)
/
(
g
11
+
g
12
). In this
case
E
1
=
E
2
=
δω
g
c
g
11
+
g
12
δω
g
c
δω
sinh(2
(
δω
g
c
)
g
c
θ/δD
1
)
δω
cosh(2
(
δω
g
c
)
g
c
θ/δD
1
) +
σ
(S20)