Wang et al.
Light: Science & Applications
(2020) 9:205
Of
fi
cial journal of the CIOMP 2047-7538
https://doi.org/10.1038/s41377-020-00438-w
www.nature.com/lsa
ARTICLE
Open Access
Dirac solitons in optical microresonators
Heming Wang
1
,Yu-KunLu
1,2
,LueWu
1
, Dong Yoon Oh
1,3
, Boqiang Shen
1
, Seung Hoon Lee
1,4
and
Kerry Vahala
1
Abstract
Mode-coupling-induced dispersion has been used to engineer microresonators for soliton generation at the edge of
the visible band. Here, we show that the optical soliton formed in this way is analogous to optical Bragg solitons and,
more generally, to the Dirac soliton in quantum
fi
eld theory. This optical Dirac soliton is studied theoretically, and a
closed-form solution is derived in the corresponding conservative system. Both analytical and numerical solutions
show unusual properties, such as polarization twisting and asymmetrical optical spectra. The closed-form solution is
also used to study the repetition rate shift in the soliton. An observation of the asymmetrical spectrum is analysed
using theory. The properties of Dirac optical solitons in microresonators are important at a fundamental level and
provide a road map for soliton microcomb generation in the visible band.
Introduction
Soliton mode locking in microresonators
1
provides a
pathway for the miniaturization of frequency comb sys-
tems
2
. The dissipative solitons
3
formed in the resulting
microcombs
4
are coherently pumped
5
and were
fi
rst
observed in optical
fi
bre cavities
6
. In microresonators,
such Kerr solitons (KSs) have been realized in a wide
range of geometries and material systems
7
–
14
. Soliton
microcomb devices have been tested in diverse system
demonstrations, including spectroscopy
15
–
17
, coherent
communications
18
, range detection
19
–
21
, optical fre-
quency synthesis
22
, exoplanet studies
23
,
24
, and optical
clocks
25
. Progress towards integration of the microcomb
with pump and other control functions is also being
made
26
–
28
. Modal coupling, wherein distinct mode
families experience frequency degeneracy analogous to an
energy level crossing
29
, is an important feature of soliton
formation in microresonators. Such crossings impart
structure to the soliton spectral envelope
30
and are
responsible for an intriguing range of microcomb phe-
nomena of both scienti
fi
c and technical importance,
including dispersive wave emissions
9
,
31
, dark soliton for-
mation
32
, pump noise isolation
33
, improved pumping
ef
fi
ciency
34
,
35
, and dispersion engineering for near-visible
emissions
36
,
37
.
Here, a new type of soliton in microresonators, termed
Dirac solitons (DSs), is shown to result from broadband
modal coupling. The name originates from the nonlinear
Dirac equations, which govern the dynamics of these
solitons and are discussed below. A similar soliton has
been theoretically studied in
fi
bre Bragg gratings
38
,
39
and
later experimentally observed
40
. In these Bragg solitons,
forward and backward propagating waves are coupled by
a periodic structure, and a Dirac-like model has been
applied to understand these systems
41
,
42
. As shown
recently for broadband coupling in a dimer resonator
system
43
, the co-existence of coupling and nonlinearity
changes the solution behaviour qualitatively, and a full
understanding requires a non-perturbative approach. We
show that broadband nonlinear coupling results in a range
of new phenomena in the Dirac soliton system, including
polarization twisting along the soliton and asymmetrical
soliton comb spectra. A closed-form expression for DSs is
derived by solving the Lugiato-Lefever equation (LLE)
5
,
44
augmented with mode coupling. Curiously, the requisite
coupling conditions for DS generation have been obtained
experimentally for near-visible
36
and 1-
μ
m-band
37
soliton
© The Author(s) 2020
Open Access
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ional License, which permits use, sharing, adaptation, distribution and
reproduction
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nktotheCreativeC
ommons license,
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’
s Creative Commons license, unless indicated
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’
s Creative Commons license and your intended use is not permitted by sta
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.
Correspondence: Kerry Vahala (
vahala@caltech.edu
)
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology,
Pasadena, CA 91125, USA.
2
Present address: Research Laboratory of Electronics,
MIT-Harvard Center for Ultracold Atoms, Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA, USA
1234567890():,;
1234567890():,;
1234567890():,;
1234567890():,;
generation. As shown here, these experiments were per-
formed in a regime where Dirac solitons collapse into
conventional Kerr solitons. New data from the 778-nm-
band measurement featuring asymmetrical spectra will be
presented, showing initial deviation away from conven-
tional Kerr soliton behaviour.
Results
Polarization mode coupling and coupled LLEs
To illustrate the features of DSs, we consider the speci
fi
c
case of a circularly symmetric (whispering-gallery) reso-
nator that has an initial re
fl
ection plane of symmetry in the
plane of the resonator. The resulting geometry supports
transverse-electric (TE) and transverse-magnetic (TM)
mode families (Fig.
1
a) that are symmetrical and anti-
symmetrical, respectively, with respect to the re
fl
ection
plane. A pair of TE and TM modes can have accidental
degeneracy for a particular wavenumber. By breaking
the re
fl
ection geometry, it becomes possible to lift the
degeneracy and create an avoided crossing
36
. In effect, the
original modes are strongly coupled, and the eigenmodes
of the non-symmetric system are hybrid modes, as shown
in Fig.
1
b. Loosely speaking, the resulting hyperbolic shape
of the eigenfrequency dispersion creates an anomalous
dispersion window that is suitable for soliton generation.
However, we note that this dispersion is not associated
with a single mode family across the entire avoided
crossing. Indeed, the mode composition changes when the
wavenumber increases, evolving from TM to TE mode for
the upper branch or vice versa for the lower branch.
The standard form of the LLE for one transverse mode
family (denoted by mode 1) describes the temporal soliton
dynamics in a microresonator
44
:
∂
E
1
∂
t
¼ð
i
δω
κ
1
2
Þ
E
1
i
^
L
1
E
1
þ
ig
11
j
E
1
j
2
E
1
þ
f
1
ð
1
Þ
Here,
E
1
is the slowly varying amplitude in a co-moving
frame normalized to optical energy, de
fi
ned via
E
1
=
E
1
A
1
,where
E
1
is the electric
fi
eld and
A
1
is the
normalized vector
fi
eld distribution. The frequency
detuning
δω
=
ω
c
−
ω
p
is the frequency difference
between the cavity resonant frequency
ω
c
and pump
frequency
ω
p
. The Kerr nonlinear coef
fi
cient is
g
11
¼
n
ð
2
Þ
ω
c
c
=
ð
n
2
V
11
Þ
, with speed of light in vacuum
c
,
refractive index
n
, Kerr nonlinear index
n
(2)
and mode
volume
V
11
¼ð
R
j
A
1
j
2
dV
Þ
2
=
R
j
A
1
j
4
dV
.
κ
1
is the energy
loss rate, and
f
1
is the pumping term for mode 1. The
linear dispersion operator
~
L
1
describes mode dispersion
and can be Taylor expanded as
^
L
1
iD
1
;
1
∂
θ
D
2
;
1
∂
2
θ
=
2,
where
θ
is the angular coordinate and
D
1,1
/(2
π
)and
D
2,1
/
(2
π
) are the free spectral range (FSR) and second-order
dispersion (proportional to the group velocity dispersion
(GVD)), respectively, for mode 1. In the case of a
conservative system (
κ
1
=
0and
f
1
=
0) and
D
1,1
=
0 (i.e.,
choosing a co-moving reference frame), the exact soliton
solution to Eq. (
1
) is given by:
E
1
¼
ffiffiffiffiffiffiffiffiffi
2
δω
g
11
s
sech
ffiffiffiffiffiffiffiffiffi
2
δω
D
2
;
1
s
θ
!
ð
2
Þ
which is also commonly used as an ansatz to describe a
KS
7
.
To generalize the above LLE to the coupled pair of TE
and TM modes (modes 1 and 2, respectively), we intro-
duce mode coupling as well as cross-phase modulation
into the equations. The following two-mode coupled LLE
results:
∂
E
1
∂
t
¼ð
i
δω
κ
1
2
Þ
E
1
þ
ig
c
E
2
δ
D
1
∂
E
1
∂
θ
þ
i
ð
g
11
j
E
1
j
2
E
1
þ
g
12
j
E
2
j
2
E
1
Þþ
f
1
∂
E
2
∂
t
¼ð
i
δω
κ
2
2
Þ
E
2
þ
ig
c
E
1
þ
δ
D
1
∂
E
2
∂
θ
þ
i
ð
g
22
j
E
2
j
2
E
2
þ
g
12
j
E
1
j
2
E
2
Þþ
f
2
ð
3
Þ
Here,
g
c
> 0 gives the coupling rate between the two
(originally uncoupled) modes,
g
ij
¼
n
ð
2
Þ
ω
c
c
=
ð
n
2
V
ij
Þ
(
i
,
j
=
1, 2), and the mode volumes
V
11
and
V
22
and cross mode
volume
V
12
are de
fi
ned as
V
ij
¼ð
R
j
A
i
j
2
dV
R
j
A
j
j
2
dV
Þ
=
R
j
A
i
j
2
j
A
j
j
2
dV
. A reference frequency
(relative to which all
Group velocity
Coupling
Degeneracy
Hybridization
a
Wavenumber
Wavenumber
Frequency
b
Frequency
c
d
Nonlinear
Nonlinear
+
Dispersion
Kerr
Soliton
+
+
Dirac
Soliton
Fig. 1 Principle of mode hybridization and Dirac solitons. a
For a
symmetric resonator cross section (top-left insets), TE and TM modes
within the resonator can become accidentally degenerate at the same
wavenumber. The bottom (right) inset depicts the TE (TM) mode
electric
fi
eld directions.
b
For an asymmetric resonator cross section
(top-left inset), degeneracy is lifted, and an avoided crossing is
created. The left and right insets depict the hybridized mode electric
fi
eld directions.
c
Schematic of balancing nonlinear and dispersion
effects for a KS.
d
Schematic of balancing nonlinear, coupling and
group velocity difference effects for one of the components in a DS
Wang et al.
Light: Science & Applications
(2020) 9:205
Page 2 of 15
frequencies are measured) is chosen as the degeneracy
frequency. In the symmetric co-moving frame (moving at a
speed corresponding to
D
1
¼ð
D
1
;
1
þ
D
1
;
2
Þ
=
2), the group
velocities of the two resulting hybrid modes become anti-
symmetric with
δ
D
1
¼j
D
1
;
1
D
1
;
2
j
=
2. Here, second- and
higher-order dispersions are ignored, as the coupling-
induced dispersion of the eigenfrequency is typically one
order of magnitude larger than the intrinsic mode dispersion.
The nonlinear terms include se
lf-phase and cross-phase
modulation. Other four-wave mixing terms that induce
nonlinear coupling, such as
j
E
1
j
2
E
1
E
2
and
E
2
1
ð
E
2
Þ
2
,have
been dropped because these are either forbidden by
re
fl
ection symmetry or strongly suppressed by the phase
mismatching of the underlying modes.
The LLE Eq. (
1
) (without loss and pump terms) is known
as the nonlinear Schrödinger equation in theoretical physics.
Similarly, the coupled LLE Eq. (
3
) presented here are a
generalization of the nonlinear Dirac equations. When only
cross-phase modulation is considered (
g
11
=
g
22
=
0), Eq. (
3
)
reduces to the massive Thirring model in quantum
fi
eld
theory
45
, which is known to support Dirac solitons
46
,
47
.
With equal but nonzero self-phase modulations (
g
11
=
g
22
),
the Bragg soliton solution
38
,
39
is recovered, which has been
realized in
fi
bre Bragg grating systems
40
.Ontheotherhand,
when second-order dispersion is present and is much
stronger than the effect induced by linear inter-mode cou-
pling within the band be
ing considered, Eq. (
3
)becomesthe
vector soliton model in birefringent systems, where soliton
solutions are also known
48
–
50
.Wenotethatthevector
soliton relies on both modes having anomalous dispersion,
while anomalous dispersion is not required in the DS model.
Before proceeding to solve Eq. (
3
), it is helpful to
understand why a DS solution exists in the absence of
second-order dispersion. The conventional KS is a delicate
balance of the Kerr nonlinear e
ffect, which creates chirping
within a pulse, and the anomalous mode dispersion, which
cancels the chirping effect (Fig.
1
c). In the DS system, the
hyperbolic-shaped upper-bran
ch eigenfrequency spectrum
(as in Fig.
1
b) resembles a spectrum with anomalous dis-
persion. While this
“
dispersion
”
plays the same role as
conventional dispersion and constitutes the foundation for
the generation of the DS, this viewpoint only holds when
the pumping frequency (and soliton spectrum) is close to
this branch. Indeed, in this case, it will be shown that it
reduces to the conventional KS. In general, and as noted in
the introduction, dispersion is only locally well-de
fi
ned in
this spectrum, because the mode composition of the hybrid
mode can change rapidly with respect to the wavenumber.
Correspondingly, the dispersion interpretation fails for the
general DS, and the coupling effect must be treated non-
perturbatively. These rapid composition changes in the
hybridized modes redistribute pulse energy in the frequency
domain and produce a new contribution to chirping within
the pulse, which manifests as p
hase differences between the
two-mode components of the pulse. Coupling then makes
the two components interfere d
ifferently at different posi-
tions and leads to both chirping and pulse shifting. These
effects are delicately cancelled by nonlinear effects and
group velocity differences, respectively (Fig.
1
d), and
maintain the DS pulse shape without anomalous dispersion.
Inthelanguageof
fi
eld theory, anomalous mode dispersion
provides a positive
“
mass
”
for the KS
fi
eld, which then
becomes a well-de
fi
ned non-relativistic
fi
eld theory. The
“
mass
”
of the DS
fi
eld is the inter-mode coupling, and the
mode spectrum corresponds to a relativistic
fi
eld theory.
Closed-form soliton solutions
To obtain an analytical solution for the coupled LLE, we
consider a conservative system by setting
κ
1
=
κ
2
=
0 and
f
1
=
f
2
=
0. Equation (
3
) can then be solved by
fi
nding the
invariants associated with the system (see
‘
Materials and
methods
’
). The closed-form single bright soliton solution
for Eq. (
3
), without periodic boundary conditions, can be
obtained as:
E
1
;
2
¼
±
ffiffiffiffiffi
2
g
c
G
q
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1
~
ξ
2
p
ð
δ
D
1
±
v
δ
D
1
v
Þ
1
=
4
cosh 2
ffiffiffiffiffiffiffiffi
1
~
ξ
2
p
~
θ
~
ξ
±
γ
=
2
ffiffiffiffiffiffi
1
~
ξ
p
cosh
ffiffiffiffiffiffiffiffi
1
~
ξ
2
p
~
θ
i
ffiffiffiffiffiffi
1
þ
~
ξ
p
sinh
ffiffiffiffiffiffiffiffi
1
~
ξ
2
p
~
θ
1±
γ
exp
i
v
δ
D
1
~
ξ
~
θ
ð
4
Þ
G
¼
δ
D
1
þ
v
δ
D
1
v
g
11
2
þ
δ
D
1
v
δ
D
1
þ
v
g
22
2
þ
g
12
;
γ
¼
1
G
δ
D
1
þ
v
δ
D
1
v
g
11
δ
D
1
v
δ
D
1
þ
v
g
22
~
ξ
¼
δ
D
1
ffiffiffiffiffiffiffiffiffiffiffiffi
δ
D
2
1
v
2
p
δω
g
c
;
~
θ
¼
g
c
ffiffiffiffiffiffiffiffiffiffiffiffi
δ
D
2
1
v
2
p
θ
where
E
1
(
E
2
) takes the upper (lower) sign in all instances of
±or
∓
,
v
is the repetition rate shift in the symmetric co-
moving frame,
G
is the combined nonlinear coef
fi
cient,
γ
is
a phase exponent related to
v
,
~
ξ
is the reduced detuning and
~
θ
is the reduced coordinate. W
hile dark soliton solutions
and bright soliton on a background solutions can also be
found in the same conservative system (see Supplementary
Information), we will focus on this bright soliton solution
and refer to it as the DS. In the following discussion of DS
properties, we take the special case
g
11
=
g
22
(i.e., additional
exchange symmetry between the modes) and
v
=
0(i.e.,the
pulse is stationary in the symmetric co-moving frame), and
the general solution simpli
fi
es to:
E
1
;
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
g
2
c
δω
2
Þ
ð
g
11
þ
g
12
Þ
g
c
s
1
±
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
c
δω
p
cosh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
2
c
δω
2
p
θ
=
δ
D
1
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
c
þ
δω
p
sinh
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
2
c
δω
2
p
θ
=
δ
D
1
ð
5
Þ
Figure
2
a shows the offset frequency for the hybrid
modes, de
fi
ned as
ω
off
¼
ω
k
ω
c
k
D
1
where
k
is the
relative wavenumber (the difference between the absolute
wavenumber and the wavenumber at the degeneracy
Wang et al.
Light: Science & Applications
(2020) 9:205
Page 3 of 15
point) and
ω
k
is the mode eigenfrequency at
k
. Due to the
square roots in the special solution (Eq. (
5
)), the soliton
detuning range can lie only in the band gap created by the
avoided crossing. This phenomenon can be intuitively
understood, as none of the comb lines can have the same
frequency as the resonator modes, which would otherwise
create in
fi
nite amplitudes on the modes due to perfect
resonance with no loss. Geometrically, the resonance of
the soliton can be described by the linear equation
ω
DS
=
−
δω
+
vk
, and the line cannot intersect the two hyper-
bolas of mode frequencies on the mode spectrum plot.
The same also holds for the general solution (Eq. (
4
)) and
is depicted in Fig.
2
b (see
‘
Materials and methods
’
). As a
result, the soliton cannot have a group velocity faster than
the
fi
rst mode or slower than the second mode. This
argument does not apply to dark solitons and bright
solitons on a background, where the resonance line always
intersects the mode spectrum (see Supplementary
Information).
To understand the properties of the DS, we consider
some special cases of detuning (marked in Fig.
2
a). The
fi
rst case is when
δω
approaches
−
g
c
, where the resonance
line is close to the upper branch. By taking appropriate
limits, Eq. (
5
) reduces to:
E
1
;
2
±
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
g
c
þ
δω
Þ
g
11
þ
g
12
s
sech
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
g
c
ð
g
c
þ
δω
Þ
δ
D
2
1
s
θ
!
ð
6
Þ
which is exactly in the form of a conventional KS. A
comparison of the exact DS near
δω
=
−
g
c
and the
limiting KS is shown in Fig.
2
c. The appearance of the
sech-shaped KS here is not a coincidence. The effective
nonlinear coef
fi
cient for a single component is
g
11
+
g
12
(using
j
E
1
j
2
¼j
E
2
j
2
), the effective detuning is
g
c
+
δω
, and
the curvature of the eigenfrequency (the hybrid-mode
equivalent of the second-order dispersion) is
δ
D
2
1
=
g
c
,as
derived from coupled mode theory. The reduction of a DS
to a KS is straightforward when these quantities are
substituted into the KS solution. Thus, if the DS is close to
the resonance (which is usually the case when the
hybridization coupling
g
c
is large), the eigenfrequency
spectrum is locally equivalent to a single mode in terms of
dispersion, and mode composition differences for differ-
ent wavenumbers can be ignored. This phenomenon also
applies to the general solution (Eq. (
4
)) by explicitly
reducing the coupled LLE to a single-mode LLE, and the
E
1
E
2
θ
Amplitude
–2
0
2
Amplitude
–2
0
2
I
II
III
Phase
Phase
0
–
π
/4
π
/4
0
–
π
/4
π
/4
δω =
g
c
δω =
0
δω =
–
g
c
δω
Resonance
v
= 0
v
> 0
a
b
Relative wavenumber
Offset frequency
Relative wavenumber
Offset frequency
c
Amplitude (normlaized unit)
0.4
0.2
0
θ
(Normalized unit)
–4
–2
0
2
4
d
Amplitude (normlaized unit)
1.5
1
–0.5
0.5
0
θ
(Normalized unit)
–4
–2
0
2
4
e
Amplitude (normlaized unit)
E
2
–1
0
1
–1
0
1
–1
0
1
E
2
E
2
–1
–1
0
E
1
I
II
III
f
Amplitude (normlaized unit)
1
–1
0
2
θ
(Normalized unit)
–4
–2
0
2
4
g
Amplitude
(normlaized unit)
2
0
4
Power
k
(Normalized unit)
–2
–1
0
1
2
10 dB
Fig. 2 Closed-form solution of Dirac solitons in microresonators. a
Resonance diagram showing two branches of hybrid modes, the allowed
range for the soliton resonance line when
v
=
0 (shaded area), and the three special cases of detuning discussed in the text.
b
Same as (
a
) but shows
only the range for the soliton resonance line with an arbitrary
fi
xed positive repetition rate shift,
v
.
c
Real part (orange line) and imaginary part (green
line) of the
E
1
component of the DS at
δω
=
−
0.9
g
c
. The normalization scheme used for the plot is
g
c
=
1,
δ
D
1
=
1 and
g
11
+
g
12
=
1. The black
dashed curve shows the corresponding KS pro
fi
le for comparison.
d
Real part (orange line), imaginary part (green line) and norm (black line) of the
E
1
component of the DS at
δω
=
0. The normalization scheme used for the plot is the same as that in (
c
).
e
Polarization twist of DS at
δω
=
0. The two
projections show the slowly varying amplitude envelope (grey dashed lines, left scale) and phase relative to the pulse centre (red solid lines, right
scale) of the
fi
eld components. The right insets show the polarization states at the three different spatial slices marked on the plot. The arrow
indicates the direction of state change over time.
f
Real part (orange line), imaginary part (green line) and norm (black solid line) of the
E
1
component
of the DS at
δω
=
g
c
. The normalization scheme used for the plot is the same as that in (
c
). The black dashed line shows the 1/
θ
asymptote for the
envelope.
g
Upper panel: frequency domain amplitudes for the
E
1
component of the DS at
δω
=
−
0.8
g
c
(blue line),
δω
=
0 (yellow line), and
δω
=
0.8
g
c
(red line), plotted on a linear scale. Lower panel: frequency domain power distribution for the
E
1
component (yellow line),
E
2
component (cyan
line) and combined (black line) DS at
δω
=
0, plotted on a log scale. The normalization scheme used for the plot is the same as that in (
c
)
Wang et al.
Light: Science & Applications
(2020) 9:205
Page 4 of 15
truncation errors can also be estimated (see
‘
Materials and
methods
’
).
The second case is when
δω
=
0, where the resonance
line passes through the degeneracy point. In this case, Eq.
(
5
) simpli
fi
es to:
E
1
;
2
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
g
c
g
11
þ
g
12
s
1
± cosh
g
c
θ
=
δ
D
1
ðÞ
i
sinh
g
c
θ
=
δ
D
1
ðÞ
ð
7
Þ
and begins to deviate from a hyperbolic secant shape
(Fig.
2
d). With this analytical solution, it becomes
apparent that each component of the wave packet has
an overall phase shift when
θ
goes from
−
∞
to
∞
(
π
/2 in
this special case). This phase twist within the pulse
contributes to the chirping and shifting of the soliton
pulse when they are coupled together, as discussed in the
previous section. If pulse polarization is considered, it
also twists from the start of the pulse to the end of the
pulse (Fig.
2
e).
The last case we consider is when
δω
approaches
g
c
,
where the resonance line moves towards the lower branch
and maximum red detuning is approached. As this phe-
nomenon occurs, the exponential tails of the soliton decay
at an increasingly slower rate until
fi
nally in the limit
δω
→
g
c
Eq. (
5
) becomes:
E
1
;
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
c
g
11
þ
g
12
r
2
±1
2
ig
c
θ
=
δ
D
1
ð
8
Þ
showing that the solution decays polynomially rather than
exponentially when
θ
→
∞
(Fig.
2
f). The resulting poly-
nomial tails can potentially enable long-range interactions
of the DS.
We now turn to the frequency domain pro
fi
le for Eq. (
5
)
by the Fourier transform, which can also be expressed in
closed form using contour integration:
Z
1
1
E
1
;
2
e
ik
θ
d
θ
¼
±
π
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
δ
D
2
1
ð
g
11
þ
g
12
Þ
g
c
s
sech
~
k
exp ±
arccos
ð
δω
=
g
c
Þ
π
~
k
ð
9
Þ
where
k
is the relative wavenumber and
~
k
¼
πδ
D
1
k
=
ð
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
2
c
δω
2
p
Þ
. Apart from the usual sech-
shaped envelope, the extra exponential factor causes the
spectrum of each component to become asymmetrical
around
k
=
0 (Fig.
2
g). This phenomenon can be
explained by different mode compositions on the different
sides of the spectrum. We note that the spectrum for the
total power is still symmetric, which is consistent with the
soliton carrying no total momentum in the symmetric co-
moving frame. In the general
v
≠
0 case, the spectrum for
the total power is expected to become asymmetric around
the centre frequency of the soliton.
DS with dissipation and repetition rate shifts
In addition to being an exact solution to the conservative
system, the solution (Eq. (
4
)) also serves as a DS ansatz for
dissipative cases, similar to how a KS can be approximated
by a sech-shaped soliton on a background
7
. As an exam-
ple, we study the repetition rate shifts associated with the
DS when externally pumped by continuous waves.
Accordingly, we do not require
g
11
=
g
22
or
v
=
0 and
return to work with the general solution (Eq. (
4
)).
The repetition rates of ideal KSs remain constant when
pumped in the same mode while the detuning changes, as
formulated by the standard LLE. In contrast, real-world
KSs may experience additional nonlinear effects, includ-
ing dispersive wave backactions
33
or Raman effects
51
,
which lead to centre frequency changes and repetition
rate shifts. The mode hybridization process is similar to a
mode crossing in dispersive wave generation where two
modes are strongly coupled, and therefore, the DS is also
expected to experience repetition rate shifts. As the
repetition rate shift parameter
v
is free in the conservative
case, we need to
fi
nd the conditions that determine
v
when dissipation is present.
By calculating the momentum integral of the solution,
the following criterion is obtained (see
‘
Materials and
methods
’
):
Z
κ
1
j
E
1
j
2
∂
arg
E
1
∂
θ
þ
κ
2
j
E
2
j
2
∂
arg
E
2
∂
θ
d
θ
¼
0
ð
10
Þ
where arg is the argument function and
E
1,2
should be
substituted by the DS solution. According to the above
criterion, the phase twist of each component is essential in
determining the repetition rate shift. Intuitively, this
concept can be understood as follows: the pulse cannot
carry any net momentum in the reference frame of the
pumping, and any additional momentum will be damped
out by the dissipation. All the above integrations can be
carried out in closed form, leading to an equation in
v
,
which can then be solved as the repetition rate shift.
For a
fi
xed detuning
δω
, the repetition rate shift
v
depends on the ratios of the nonlinear coef
fi
cients
g
11
,
g
22
and
g
12
. Figure
3
a plots the special case of
δω
=
0. When
the nonlinearity on the second mode increases, the optical
fi
eld will shift to the
fi
rst mode to compensate, leading to
an increased overall speed of the pulse, and vice versa. As
the repetition rate shift results from the imbalance of self-
phase modulations, increasing the proportion of
g
12
leads
to more stability in the repetition rate, while decreasing
the proportion of
g
12
allows more tunability. We note
that, depending on the nonlinear nature of the resonator
material and the mode overlap, the cross-phase modula-
tion may be larger or smaller than the self-phase mod-
ulation
52
,
53
. Moreover, theoretical DS solutions exist for
almost all combinations of nonlinear coef
fi
cients.
Wang et al.
Light: Science & Applications
(2020) 9:205
Page 5 of 15
For the tunability of the DS repetition rate, Fig.
3
bshows
the repetition rate shift as the detuning changes. Near
δω
=
−
g
c
, the repetition rate shift approaches zero, which is con-
sistent with the local KS equivalence argument in the pre-
vious section. With a more red-detuned
δω
, the effect of
imbalance in the nonlinear coef
fi
cients is more apparent,
leading to repetition rate changes in the corresponding
direction. Simulations of t
he coupled LLE have also been
performed and show that both the simulated pulse shape and
the repetition rate shifts agree with the analytical solutions
(Fig.
3
b). A graphical representation of the repetition rate
shift is also shown in Fig.
3
c. As an aside, breather-like
states
54
have also been observed in the simulations, but the
origin of breathing and whether it behaves in the same way as
for KS breathers
55
–
58
is not yet fully understood.
Although the discussion so far has focused on pumping
at the central mode, it can be readily generalized to off-
centre pumping by introducing additional detunings into
each of the mode families and shifting the spectral centres
of the solutions accordingly. The DS offers a novel and
controllable way to tune the repetition rate of the fre-
quency combs. Together with existing nonlinear pro-
cesses for the resonator, the hybridization-induced shift
can be tailored to enhance or suppress the overall repe-
tition rate shift with respect to the pump detuning and
may
fi
nd application in optical frequency division or for
pump noise isolation, such as what is performed using
quiet point operation
33
.
Implementation of Dirac solitons
The wedge resonator
59
is used to induce mode hybridiza-
tion and Dirac soliton formation. This resonator offers very
high-quality factors
60
and independent control over key
parameters during the fabrication process (Fig.
4
a). A wedge
is entirely characterized by three geometric parameters: the
diameter
D
, which depends on the lithographic pattern; the
thickness
t
, which depends on the oxidation growth time of
the silicon wafer; and the wedge angle
α
, which depends on
the adhesion between silica and the photoresist used for
patterning. In the following, we will
fi
xtheresonatordia-
meter as
D
=
3.2 mm (corresponding to a resonator FSR of
~20 GHz at ~1550 nm), but we note that this can be readily
generalized to resonators of other sizes.
For a symmetrical wedge resonator (
α
=
90°), the typical
simulated effective refractive index
n
eff
versus wavelength is
showninFig.
4
b. At shorter wavelengths, TE1 and TM1 have
the highest indices, followed by TE2, TE3 and other high-
order modes. Since the electrical
fi
elds of the TM modes are
along the thickness direction, t
heir indices are more sensitive
to changes in the wavelength scale, and the index of TM1
decreases faster than TE2 as the wavelength increases.
Eventually, TM1 and TE2 cross, and their relative positions
are interchanged at longer wavelengths. However, for
α
=
90°, no hybridization occurs, as the re
fl
ection symmetry
prohibits interactions between
modes of different parities.
On the other hand, if we explicitly break the re
fl
ection
symmetry of the resonator by decreasing the wedge angle
(
α
< 90°), the original modes will see an asymmetric change in
the refractive index pro
fi
le, which causes them to couple.
Such couplings lift the degeneracy, leading to avoided
crossing. The two cases are compared in Fig.
4
c, where
n
eff
is
fi
rst converted to the mode number
m
via
m
¼
n
eff
D
ω
m
=
ð
2
c
Þ
,where
ω
m
is the resonance (angular)
frequency, and then plotted as offset (angular) frequencies
ω
off
¼
ω
m
ω
X
ð
m
m
X
Þ
D
1
versus the relative mode
number
m
−
m
X
, where the subscript X indicates the
quantity at the degeneracy point. We note that the relative
mode number has the same role as the relative wavenumber
k
in the theoretical analyses, except that it is restricted to
integer values for periodic boundary conditions.
In view of perturbation theory
61
, the wedge part of the
resonator perturbs the underlying symmetrical structure
and induces polarization coupling similar to the coupling
g
11
g
11
+ 2
g
12
+
g
22
g
22
g
11
+ 2
g
12
+
g
22
0
0.2
0.4
0.6
0.8
1
0
0.2
0.4
0.6
0.8
1
–1
1
a
b
c
–1
–0.5
0
0.5
1
0
0.05
0.1
0.15
v
/
δ
D
1
δω
/
g
c
Offset frequency
Relative wavenumber
δω
/
g
c
–0.8
–0.4
0
0.4
0.8
θ
(Arbitrary unit)
|
E
1
|
(Arbitrary unit)
Fig. 3 Repetition rate shifts in the DS. a
Ternary plot of the
normalized repetition rate shift
v
/
δ
D
1
versus the proportions of
nonlinear coef
fi
cients for
δω
=
0.
κ
1
=
κ
2
is assumed. The black dashed
curve (
g
11
g
22
¼
g
2
12
) separates the parameter space into two regions;
cross-phase modulation is dominant in the upper region, while self-
phase modulation is dominant in the lower region.
b
Plot of the
repetition rate shift versus the detuning.
κ
1
=
κ
2
is assumed. The
parameters are
g
11
=
g
12
and
g
22
=
2
g
12
. The black curve is the
analytical result, and the dots are simulated data that use a modi
fi
ed
split-step Fourier algorithm adapted to the hybrid system. The inset
shows a comparison of the simulated (orange solid line) and analytical
(black dashed line) pulse shapes of
j
E
1
j
at
δω
=
0.
c
Plot of repetition
rate shifts on the mode spectrum. Each line indicates a soliton
resonance line with different detunings (negative y-intercept). The
parameters are the same as those in (
b
)
Wang et al.
Light: Science & Applications
(2020) 9:205
Page 6 of 15
obtained in directional couplers
62
. Therefore, we expect
that the centre wavelength of hybridization
λ
X
is deter-
mined by the thickness
t
, while the wedge angle controls
the coupling strength
g
c
. A plot of
λ
X
versus
t
is shown as
the black curve in Fig.
4
d. As
t
is the only geometry scale
close to optical wavelengths in the system, we expect that
λ
X
will scale linearly with
t
, which can be visually veri
fi
ed
in the plot. This scaling allows for hybridization to occur
at short wavelengths where the dispersion of the original
modes (for example, the TE1 mode shown in the
fi
gure) is
typically normal. A plot of
g
c
versus
α
is shown in Fig.
4
e.
While only a particular wavelength (778 nm) is shown,
g
c
depends on the wavelength very weakly, varying less than
5% from wavelengths of 400
–
1600 nm. The coupling
strength scales linearly with
α
near
α
=
90°, which can also
be independently veri
fi
ed by
fi
rst-order perturbation
theory (see
‘
Materials and methods
’
), but the coupling
effect eventually saturates at shallow wedge angles
because mode pro
fi
les cannot
“
squeeze
”
into the wedge
tip as
α
decreases. The calculated GVD
β
2
is shown in Fig.
4
f, which is related to
D
2
via
β
2
¼
nD
2
=
ð
cD
2
1
Þ
. Using
suitably designed thicknesses and wedge angles greater
than 30°, an anomalous dispersion window can be created
all the way down to the blue side of the visible spectrum,
where simple geometrical dispersion fails to compensate
for normal material dispersion.
Demonstration of Dirac solitons
Guided by these design principles, devices that target
1550 nm and 778 nm as their hybridization wavelengths
were fabricated. The mode spectra are measured for each
device using a tuneable laser and a calibrated Mach-
Zehnder interferometer
8
. As expected, each of the devices
shows a pair of modes with hyperbolic dispersion and
large curvatures (Fig.
5
a, c). The local
D
2
of the anom-
alous branch can be
fi
t to give
D
2
=
2
π
× 401 kHz
(1550 nm) (Fig.
5
b) and
D
2
=
2
π
× 132 kHz (778 nm),
corresponding to
β
2
=
−
790 ps
2
km
−
1
and
β
2
=
−
255 ps
2
km
−
1
, respectively, which are orders of magnitude larger
than the mode intrinsic GVD without hybridization.
Finally, to demonstrate the existence of DSs in wedge
resonators with hybridized modes, we generated solitons at
778 nm. The detailed experimental setup and measurement
procedures can be found elsewhere
36
. The optical spec-
trum of the soliton is shown in Fig.
5
. A direct sech
2
fi
tto
the spectrum reveals that the frequency components are
Silica
Silicon
a
t
D
/2
α
1.425
1.43
1.435
740
820
TE1
TM1
TE2
TE3
TE1
TM1
TE2
TE3
b
c
–20
20
–500
500
400
760
780
800
Wavelength (nm)
Frequency (THz)
n
eff
390
380
370
Offset frequency (GHz)
Relative mode number
0
–10
0
10
–20
20
–500
500
Offset frequency (GHz)
Relative mode number
0
–10
0
10
0
1
1.0
1.5
500
1000
1500
Wavelength (nm)
Anomalous
Normal
2.0
2.5
t
(
μ
m)
Bulk
silica
d
TE1
e
15
10
5
0
g
c
/2
π
(GHz)
α
(Degree)
90
80
70
60
50
40
30
f
β
2
(ps
2
km
–1
)
100
0
–100
–200
–300
Normal
Anomalous
Bulk
TE2
TM1
Hybrid, 30°
Hybrid, 60°
200
300
400
500
600
700
Wavelength (nm)
1500
1000
800
700
600
500
400
Frequency (THz)
1200
Fig. 4 Implementation of mode hybridization. a
Cross-sectional view of a silica wedge resonator on a silicon pillar (not to scale). The parameters
that de
fi
ne the wedge geometry are also shown.
b
Plot of
n
eff
for the
fi
rst four modes (TE1, TM1, TE2, and TE3) versus wavelength (740
–
820 nm) for a
wedge resonator. The parameters are
t
=
1.47
μ
m and
α
=
90°.
c
Left panel: mode spectrum plot for the boxed region in (
b
). The insets are simulated
mode pro
fi
les (electric
fi
eld norm). Right panel: same as left panel but with
α
=
30°.
d
Relationship between
t
and
λ
X
(black curve). Additionally, the
zero-dispersion wavelength of bulk silica (green dashed line) and the zero-dispersion boundary for the TE1 mode (purple curve) are shown.
e
Coupling
g
c
versus wedge angle
α
at
λ
X
=
778 nm. The dashed line is the result from perturbation theory (see
‘
Materials and methods
’
) and is
tangent to the
g
c
curve at
α
=
90°.
f
Effective GVD
β
2
that can be achieved using mode hybridization across the infrared and visible spectra. The
parameters are
α
=
30° and
α
=
60°. The dispersion of bulk silica, TE2, and TM1 modes is also shown for comparison. The colour bar shows the
approximate colour of light in the visible band
Wang et al.
Light: Science & Applications
(2020) 9:205
Page 7 of 15