of 4
Universal isocontours for dissipative Kerr solitons
X
INBAI
L
I
,
1,2,
B
OQIANG
S
HEN
,
1,
H
EMING
W
ANG
,
1,
K
I
Y
OUL
Y
ANG
,
1,
X
U
Y
I
,
1
Q
I
-F
AN
Y
ANG
,
1
Z
HIPING
Z
HOU
,
2
AND
K
ERRY
V
AHALA
1,
*
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
2
State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Electronics Engineering and Computer Science,
Peking University, Beijing 100871, China
*Corresponding author: vahala@caltech.edu
Received 19 February 2018; revised 24 April 2018; accepted 26 April 2018; posted 26 April 2018 (Doc. ID 323414); published 23 May 2018
Dissipative Kerr solitons can be generated within an exist-
ence region defined on a space of normalized pumping
power versus cavity-pump detuning frequency. The con-
tours of constant soliton power and constant pulse width
in this region are studied through measurement and simu-
lation. Such isocontours impart structure to the existence
region and improve understanding of soliton locking and
stabilization methods. As part of the study, dimensionless,
closed-form expressions for soliton power and pulse width
are developed (including Raman contributions). They pro-
vide isocontours in close agreement with those from the full
simulation, and, as universal expressions, can simplify the
estimation of soliton properties across a wide range of
systems.
© 2018 Optical Society of America
OCIS codes:
(190.5530) Pulse propagation and temporal solitons;
(140.3945) Microcavities; (190.0190) Nonlinear optics.
https://doi.org/10.1364/OL.43.002567
Temporal optical solitons resulting from the balance of
dispersion with the Kerr nonlinearity have long been studied
in optical fiber systems [
1,2]. In addition to their many remark-
able properties, these nonlinear waves are important in mode-
locking [
3], continuum generation [
4], and were once considered
as a means to send information over great distances [
5,6].
Recently, a new type of dissipative temporal soliton [
7] was ob-
served in optical fiber resonators [
8]. These coherently driven
cavity solitons (CSs) were previously considered a theoretical
possibility [
9], and related soliton phenomena including breather
solitons and Raman interactions have also been reported in this
system [
10
12]. While leveraging the Kerr effect to balance
dispersion, this soliton also regenerates using Kerr-induced para-
metric amplification [
13]. Their recent demonstration in micro-
cavity systems [
14
20] has made possible highly stable frequency
microcombs [
21,22]. Referred to as dissipative Kerr solitons
(DKs) in the microcavity system, soliton phenomena including
the Raman self-shift [
23
25], optical Cherenkov radiation
[16,25
28], multisoliton systems [
29
31], and the cogeneration
of new types of solitons [
32] have been reported. Moreover,
the compact soliton microcomb devices are being studied for
systems-on-a-chip applications such as dual-comb spectroscopy
[33,34], precision distance measurement [
35,36], optical com-
munications [
37], and optical frequency synthesis [
38].
Regions of stability and existence are well known in driven
soliton systems [
39]. These properties of DKs and CSs have
been studied using the Lugiato
Lefever (LL) equation [
9,40]
in a space of normalized pumping power and cavity-pump fre-
quency detuning [
14,41
43]. In analogy with thermodynamic
phase diagrams, this soliton existence diagram also contains
other regions of existence including those for breather solitons
as well as more complex dynamical phenomena [
44
46].
Figure
1 is
a typical diagram showing only the stable soliton
region. In thermodynamic phase diagrams another useful con-
struct is the isocontour for processes performed with a state
variable held constant (e.g., isochors and isotherms) [
47].
These contours not only provide a way to understand processes
within the framework of the phase diagram but impart struc-
ture to the phase diagram that improves intuition of thermo-
dynamical processes. In this Letter, contours of constant soliton
power and constant pulse width are measured and compared to
theory. Closed-form expressions for normalized power and pul-
sewidth that include the Raman process are also developed.
The normalized LL equation is shown below as Eq. (
1)[14].
The slowly-varying field envelope
ψ
is defined such that
j
ψ
j
2


2
g
κ

N
where
N
is photon number,
κ
is the cavity mode
power damping rate, and
g

ω
2
c
cn
2

n
2
V
0

is the Kerr coef-
ficient with material refractive index
n
,Kerrnonlinearindex
n
2
,
optical mode volume
V
0
, cavity resonant frequency
ω
c
,Planck
s
constant
,andspeedoflight
c
.
τ
κ
t
2
and
θ

φ
ffiffiffiffiffiffiffiffiffiffiffiffiffi
κ
2
D
2
p
are the normalized time and cavity polar coordinate (
φ
)where
D
2
is the second-order dispersion parameter [
14, 15].
f
2
P
P
th
is the ratio of the input pump power and parametric threshold
power [
13,15], and
ζ

ω
c
ω
p

2
κ

is the normalized fre-
quency detuning between cavity resonant frequency
ω
c
and pump
frequency
ω
p
.
γ
D
1
τ
R
ffiffiffiffiffiffiffiffiffiffiffiffiffi
κ
2
D
2
p
is the normalized Raman co-
efficient where
τ
R
is the material Raman constant [
24]and
D
1
2
π
is the cavity free-spectral range [
14,15]. Eq. (1
) is as follows:
ψ
τ

j
1
2
2
ψ
θ
2

j
j
ψ
j
2
ψ

1

j
ζ

ψ

j
γ
j
ψ
j
2
θ
ψ

f:
(1)
Isopower contours found by solving Eq. (
1) are shown
in Fig.
1 as red contours. The analysis is performed for a
Letter
Vol. 43, No. 11 / 1 June 2018 /
Optics Letters
2567
0146-9592/18/112567-04 Journal © 2018 Optical Society of America
Provided under the terms of the OSA Open Access Publishing Agreement
high-
Q
silica resonator, and parameters used in the calculation are
provided below and in the Fig.
1
caption. Numerical simulation is
based on propagating the LL equat
ion from an initial soliton seed
until steady state is achieved [
24
].
The following simplified analytical solution for the soliton
is also used to study soliton behavior [
14
]:
ψ

A

B
sech

θ
τ
θ

e
j
φ
0
, where
A
is the soliton background field,
B
is the amplitude,
φ
0
is the soliton phase, and
τ
θ
τ
s
τ
0
is
the normalized soliton pulse width (
τ
s
is the physical pulse
width and
τ
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
D
2

κ
D
2
1

p
). By Fourier transform the soli-
ton spectrum in optical frequency
ν
varies as
sech

ν
ν
s

where
ν
s
τ
s

π
2
. Approximate expressions giving the Raman-free
dependence of amplitude and pulse width on detuning and
pump power have been developed [
14
,
48
]. By including high-
order corrections and Raman corrections, improved expressions
result as well as an expression for soliton average power,
B

ζ
,
f
,
γ

ffiffiffiffiffi
2
ζ
p
0
@
1

5
8
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π
2
f
2
8
ζ
2
ζ
3
s
1
A

1
64
225
γ
2
ζ
3

,
(2)
τ
θ

ζ
,
f
,
γ

1
ffiffiffiffiffi
2
ζ
p
0
@
1
1
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π
2
f
2
8
ζ
2
ζ
3
s
1
A

1

64
225
γ
2
ζ
3

,
(3)
p

ζ
,
f
,
γ

ffiffiffiffiffi
2
ζ
p
0
@
1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
π
2
f
2
8
ζ
2
ζ
3
s
1
A

1
64
225
γ
2
ζ
3

,
(4)
where
p
P
sol
P
0
is the normalized time-averaged soliton
power
P
sol
and
P
0

κ
E
ω
c
π
g

ffiffiffiffiffiffiffiffiffiffiffiffiffi
κ
D
2
2
p
with
κ
E
the optical
loss rate from waveguide-resonator coupling [
15
]. As an aside,
the requirement of the square root to be real in these expressions
(
ζ
<
π
2
f
2
8
) gives the approximate upper bound of detuning
for soliton existence in the phase diagram [
14
,
39
,
49
]. Dotted
lines in Fig.
1
are the isopower contours using Eq. (
4
)(equiv-
alently
p

ζ
,
f
,
γ

Constant
for
γ

2
.
1
×
10
3
), and they are
in excellent agreement with the simulation contours. Raman
contributions become especially important at larger detuning
values where the soliton spectrum increases in width [
23
,
24
].
To illustrate this point, the dashed curves in Fig.
1
result by using
Eq. (
4
)exceptwith
γ

0
.
The experimental setup is shown in Fig.
2(a)
. The resonator
was an ultrahigh-
Q
silica wedge resonator having a diameter of
approximately 3 mm (free spectral range
D
1
2
π

21
.
9GHz
).
Further details on its fabrication are described in Ref. [
50
].
4.0
4.5
5.0
5.5
6.0
6.5
7.0
7.5
7.9
7.4
7.0
6.8
6.4
5.5
5.1
5 1015202530
Normalized detuning
10
20
30
40
50
Normalized Pump Power
f
²
10
15
20
30
5.4
6.0
5.7
5.5
6.0
Simulation
Experiment
Theory
24681012
Cavity-pump detuning (MHz)
20
40
60
80
Pump power (mW)
Fig. 1.
Dissipative Kerr soliton phase diagram and isopower con-
tours. The phase diagram features normalized pump power
f
2
along
the vertical axis and normalized detuning
ζ
along the horizontal axis.
The green region contains stable soliton states. Black-dotted lines
(gray-dashed lines) are isopower contours using Eq. (
4
) with Raman
term (w/o Raman).
p
is incremented from 4.0 to 8.0 in steps of 0.5.
Red lines are simulated isopower contours using Eq. (
1
). Blue dots
give the measured soliton isopower contours at the soliton powers
93, 99.5, 117.5, 125, 129, 136, and 145
μ
W (left to right), which
correspond to
p
values of 5.1, 5.5, 6.4, 6.8, 7.0, 7.4, and 7.9.
For these measurements
Q

197
million (
κ
2
π

0
.
98 MHz
),
κ
E
κ

0
.
26
, and
γ

2
.
1
×
10
3
. Large green and blue data points
correspond to spectra in Fig.
2(b)
. The inset shows the measured iso-
power contours using another similar device, with soliton powers
of 299, 320, and 335
μ
W (left to right), which correspond to
p
values
of 5.4, 5.7, and 6.0. For these measurements
Q

115
million
(
κ
2
π

1
.
69 MHz
),
κ
E
κ

0
.
39
, and
γ

2
.
8
×
10
3
.
EOM
EDFA
AOM
PC
Resonator
FBG
PM
PM
Oscilloscope
OSA
ESA
Comb
PD
Servo
FG
FG
FG
Lock point
Triggering and
Attenuation
sech² fitting
(a)
(b)
Fig. 2.
Measurement system and low power operation.
(a) Measurement setup. EOM, electro-optical phase modulator;
PC, polarization controller; PM, inline power meter; PD, photodetec-
tor; FG, function generator. (b) Soliton spectra at normalized detuning
and pumping power (
ζ

21
,
f
2

53
) (blue) and (
ζ

8
.
0
,
f
2

9
.
0
) (green). The corresponding phase diagram locations are
marked in Fig.
1
. Red curve: squared hyperbolic-secant fitting.
2568 Vol. 43, No. 11 / 1 June 2018 /
Optics Letters
Letter
The measurement used the
TE
1
mode family pumped at
1550 nm and the second-order dispersion was measured
to be
D
2
2
π

12
.
1kHz
at 1550 nm by a method re-
ported elsewhere [
15
]. The mode area was calculated to be
A
eff

40
μ
m
2
, and the silica Raman constant
τ
R

2
.
4fs
was also used [
24
]; this is valid when the soliton spectral width
is below 13 THz [
51
]. Finally, the resonator used in this mea-
surement featured minimal avoided mode crossings and disper-
sive waves. Their presence would interfere with the ideal power
dependence predicted by the LL equation. To measure pump
detuning, weak phase-modulation of the pump light and de-
tection of converted amplitude modulation sidebands were per-
formed [
43
,
52
]. In this method, pump light reflected by the
fiber Bragg grating (FBG) contains the modulation information
and is analyzed by an electrical spectrum analyzer (ESA) to re-
trieve the detuning frequency. The soliton spectrum transmit-
ted past the FBG is sent to a detector and optical spectrum
analyzer (OSA) for analysis. To determine soliton power the
FBG filtered line was manually reinserted.
Triggering and locking of single soliton states used the sol-
iton average power to servo control the pump laser frequency
[
53
]. Because this soliton locking method maintains a constant
soliton power it provides a convenient way to map out the iso-
power contours. Specifically, as opposed to varying

ζ
,
f
2

in
the phase diagram and monitoring soliton power, the isopower
measurement proceeded by varying only the pumping power
with the soliton locked at constant output power. The servo
control then compensates for these variations by adjusting
the pumping frequency. The corresponding detuning was then
recorded as described above. Pump power was varied using
a combination of an acousto-optical modulator (AOM) and
erbium-doped fiber amplifier (EDFA). Upon completion of
an isopower contour, the soliton power setpoint was adjusted
and the measurement repeated. The measured isopower data
points are shown in the main panel of Fig.
1
. Each measure-
ment proceeded until it was no longer possible to reliably lock
the soliton state. There is overall good agreement between
measurement and theory. Errors are largest at lowest detuning
values; however, even here they are relatively small (
10%
).
The ability to measure the contours over such large ranges
and their good agreement with theory and simulations show-
case the systems robustness and quality. As an additional test,
a second loading condition was also measured. The inset to
Fig.
1
shows this data, which are in reasonable agreement with
the simulation and Eq. (
4
). Measured soliton powers have ex-
perienced a
1
.
2dB
insertion loss between the resonator and
the detector. It is also noted that breather solitons could be
stably locked near the upper boundary in Fig.
1
. However,
the region was small. As a result, breathers were not studied
in this work.
Stable generation of solitons at small detuning is of practical
importance for low pumping power operation of the soliton
system. To this end, the green data point (
ζ

8
.
0
,
f
2

9
.
0
)
in Fig.
1
shows both the lowest detuning and the lowest power
soliton state observed in this study. The corresponding unnor-
malized quantities are 4.2 MHz and 10.8 mW. This is, to the
authors
knowledge, the lowest operating power reported for
any soliton microcomb platform. Making this result equally
important is that the repetition rate is detectable (21.9 GHz)
requiring large mode volume and hence higher pumping power
levels compared to, for example, terahertz-rate microcombs.
Corresponding soliton spectra are presented in Fig.
2(b)
.
The result was achieved by both the use of a high-quality-factor
resonator sample as well as the improved understanding gained
through these measurements of the stability regional bounda-
ries [
53
]. For comparison, a soliton spectrum produced at
(
ζ

21
,
f
2

53
) is also shown in Fig.
2(b)
. These values
are plotted as the light blue data point in Fig.
1
and correspond
to unnormalized quantities 11.3 MHz and 63.5 mW. The cav-
ity loading condition for these two spectra is: loaded
Q

182
million and
κ
E
κ

0
.
44
.
In parallel with the isopower data point collection, the sol-
iton pulse width was also measured by fitting of the optical
spectral envelope [
15
]. Then the data set

ζ
,
f
2
,
τ
θ

was linearly
interpolated to determine isocontours of pulse width (blue con-
tours in Fig.
3
). It was not possible to interpolate isopulse-
width contours at lower detuning values where there are fewer
isopower data points. For comparison, simulated pulse width
(red) and the analytical expression, Eq. (
3
) (dotted black),
are plotted. The interpolated pulse width isocontours are less
accurate than the directly measured power contours but none-
theless show reasonable agreement between the data and theory.
Overall, the pulse width contours are more weakly dependent
upon normalized pumping power (i.e., more vertical) com-
pared to the soliton power contours.
In summary, contours of constant power and constant pulse
width have been measured for dissipative Kerr solitons.
Measurements were found to be in good agreement with
the LL equation numerical model augmented by Raman inter-
actions. There was also good agreement with the predictions of
closed-form expressions that include the Raman interaction.
Compared to the large-detuning approximation that predicts
that soliton power depends only upon resonator-pump detun-
ing (i.e., vertical isopower contours), it is found that soliton
power depends both upon pumping power and detuning.
The resulting tilt of isopower contours at low detuning suggests
24681012
Cavity-pump detuning (MHz)
20
40
60
80
Pump power (mW)
5 1015202530
10
20
30
40
50
Normalized Power
f ²
Exp. interp.
Simulation
Theory
0.210
Normalized detuning
0.195
0.180
0.165
0.150
0.135
0.168
0.137
0.150
0.128
0.124
Fig. 3.
Isocontours of soliton pulse width. The device is unchanged
from the Fig.
1
main panel. Red solid lines (black-dotted lines) are
simulated [Eq. (
3
) theory] isocontours of normalized pulse width
τ
θ
ranging from 0.21 to 0.135 (equidistant steps of 0.015). Blue solid
lines are the linear interpolation from measurement of isocontours at
190, 170, 155, 145, and 140 fs, which correspond to
τ
θ
: 0.168, 0.150,
0.137, 0.128, and 0.124.
Letter
Vol. 43, No. 11 / 1 June 2018 /
Optics Letters
2569
that soliton locking by servo control of pumping power could
potentially be an option for low-detuning ranges just as servo
control of pump frequency is used at larger detuning ranges.
Stable soliton operation for pump powers as low as 10.8 mW
was also demonstrated in this work. These measurements pro-
vide structure to the phase diagram picture of soliton existence.
The universal nature of the closed-form expressions should
make them suitable for use in other CS and DK soliton plat-
forms. Future work could consider incorporating higher-order
dispersion into the analysis to include, for example, the impact
of phenomena such as dispersive waves.
Funding.
Defense Advanced Research Projects Agency
(DARPA) (HR0011-16-C-0118); Kavli Nanoscience Institute.
Acknowledgment.
The authors thank Chengying Bao,
Yu-Hung Lai, and Myoung-Gyun Suh for helpful discussions.
X. Li appreciates the financial support from China Scholarship
Council (CSC).
These authors contributed equally to this work.
REFERENCES
1. Y. S. Kivshar and G. Agrawal,
Optical Solitons: From Fibers to
Photonic Crystals
(Academic, 2003).
2. G. P. Agrawal,
Nonlinear Fiber Optics
(Academic, 2007).
3. S. Cundiff,
Dissipative Solitons
(Springer, 2005), pp. 183
206.
4. J. M. Dudley, G. Genty, and S. Coen, Rev. Mod. Phys.
78
, 1135
(2006).
5. A. Hasegawa and Y. Kodama, Proc. IEEE
69
, 1145 (1981).
6. H. A. Haus and W. S. Wong, Rev. Mod. Phys.
68
, 423 (1996).
7. N. Akhmediev and A. Ankiewicz,
Dissipative Solitons: From Optics to
Biology and Medicine
(Springer, 2008).
8. F. Leo, S. Coen, P. Kockaert, S.-P. Gorza, P. Emplit, and M.
Haelterman, Nat. Photonics
4
, 471 (2010).
9. S. Wabnitz, Opt. Lett.
18
, 601 (1993).
10. J. K. Jang, M. Erkintalo, S. G. Murdoch, and S. Coen, Opt. Lett.
39
,
5503 (2014).
11. M. Anderson, F. Leo, S. Coen, M. Erkintalo, and S. G. Murdoch,
Optica
3
, 1071 (2016).
12. Y. Wang, M. Anderson, S. Coen, S. G. Murdoch, and M. Erkintalo,
Phys. Rev. Lett.
120
, 053902 (2018).
13. T. Kippenberg, S. Spillane, and K. Vahala, Phys. Rev. Lett.
93
,
083904 (2004).
14. T. Herr, V. Brasch, J. Jost, C. Wang, N. Kondratiev, M. Gorodetsky,
and T. Kippenberg, Nat. Photonics
8
, 145 (2014).
15. X. Yi, Q.-F. Yang, K. Y. Yang, M.-G. Suh, and K. Vahala, Optica
2
,
1078 (2015).
16. V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. Pfeiffer, M.
Gorodetsky, and T. Kippenberg, Science
351
, 357 (2016).
17. C. Joshi, J. K. Jang, K. Luke, X. Ji, S. A. Miller, A. Klenner, Y.
Okawachi, M. Lipson, and A. L. Gaeta, Opt. Lett.
41
, 2565 (2016).
18. P.-H. Wang, J. A. Jaramillo-Villegas, Y. Xuan, X. Xue, C. Bao, D. E.
Leaird, M. Qi, and A. M. Weiner, Opt. Express
24
, 10890 (2016).
19. M. Yu, Y. Okawachi, A. G. Griffith, M. Lipson, and A. L. Gaeta, Optica
3
, 854 (2016).
20. Q. Li, T. C. Briles, D. A. Westly, T. E. Drake, J. R. Stone, B. R. Ilic, S. A.
Diddams, S. B. Papp, and K. Srinivasan, Optica
4
, 193 (2017).
21. P. Del
Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, and
T. Kippenberg, Nature
450
, 1214 (2007).
22. T. J. Kippenberg, R. Holzwarth, and S. Diddams, Science
332
, 555
(2011).
23. M. Karpov, H. Guo, A. Kordts, V. Brasch, M. H. Pfeiffer, M. Zervas, M.
Geiselmann, and T. J. Kippenberg, Phys. Rev. Lett.
116
, 103902
(2016).
24. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, Opt. Lett.
41
, 3419 (2016).
25. X. Yi, Q.-F. Yang, X. Zhang, K. Y. Yang, X. Li, and K. Vahala, Nat.
Commun.
8
, 14869 (2016).
26. N. Akhmediev and M. Karlsson, Phys. Rev. A
51
, 2602 (1995).
27. C. Bao, Y. Xuan, D. E. Leaird, S. Wabnitz, M. Qi, and A. M. Weiner,
Optica
4
, 1011 (2017).
28. A. Cherenkov, V. Lobanov, and M. Gorodetsky, Phys. Rev. A
95
,
033810 (2017).
29. Q.-F. Yang, X. Yi, K. Yang, and K. Vahala, Nat. Photonics
11
, 560
(2017).
30. C. Joshi, A. Klenner, Y. Okawachi, M. Yu, K. Luke, X. Ji, M. Lipson,
and A. L. Gaeta,
Counter-rotating cavity solitons in a silicon nitride
microresonator,
arXiv: 1711.04849 (2017).
31. D. C. Cole, E. S. Lamb, P. Del
Haye, S. A. Diddams, and S. B. Papp,
Nat. Photonics
11
, 671 (2017).
32. Q.-F. Yang, X. Yi, K. Y. Yang, and K. Vahala, Nat. Phys.
13
, 53 (2017).
33. M.-G. Suh, Q.-F. Yang, K. Y. Yang, X. Yi, and K. Vahala, Science
354
,
600 (2016).
34. A. Dutt, C. S. Joshi, X. Ji, J. Cardenas, Y. Okawachi, A. L. Gaeta, and
M. Lipson,
Conference on Lasers and Electro-Optics (CLEO): Science
and Innovations
(Optical Society of America, 2017), paper STh3L.2.
35. M.-G. Suh and K. J. Vahala, Science
359
, 884 (2018).
36. P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf,
J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W.
Freude, T. J. Kippenberg, and C. Koos, Science
359
, 887 (2018).
37. P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H.
P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R.
Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C.
Koos, Nature
546
, 274 (2017).
38. D. T. Spencer, T. Drake, T. C. Briles, J. Stone, L. C. Sinclair, C.
Fredrick, Q. Li, D. Westly, B. R. Ilic, A. Bluestone, N. Volet, T.
Komljenovic, L. Chang, S. H. Lee, D. Y. Oh, M. G. Suh, K. Y. Yang,
M. H. P. Pfeiffer, T. J. Kippenberg, E. Norberg, L. Theogarajan,
K. Vahala, N. R. Newbury, K. Srinivasan, J. E. Bowers, S. A.
Diddams, and S. B. Papp, Nature
557
, 81 (2018).
39. I. Barashenkov and Y. S. Smirnov, Phys. Rev. E
54
, 5707 (1996).
40. L. A. Lugiato and R. Lefever, Phys. Rev. Lett.
58
, 2209 (1987).
41. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, Opt.
Express
21
, 9180 (2013).
42. C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, Phys. Rev. A
89
, 063814 (2014).
43. H. Guo, M. Karpov, E. Lucas, A. Kordts, M. Pfeiffer, V. Brasch, G.
Lihachev, V. Lobanov, M. Gorodetsky, and T. Kippenberg, Nat.
Phys.
13
, 94 (2017).
44. C. Bao, J. A. Jaramillo-Villegas, Y. Xuan, D. E. Leaird, M. Qi, and A. M.
Weiner, Phys. Rev. Lett.
117
, 163901 (2016).
45. E. Lucas, M. Karpov, H. Guo, M. Gorodetsky, and T. Kippenberg,
Nat. Commun.
8
, 736 (2017).
46. M. Yu, J. K. Jang, Y. Okawachi, A. G. Griffith, K. Luke, S. A. Miller, X.
Ji, M. Lipson, and A. L. Gaeta, Nat. Commun.
8
, 736 (2017).
47. R. E. Sonntag, C. Borgnakke, G. J. Van Wylen, and S. Van Wyk,
Fundamentals of Thermodynamics
(Wiley, 2003).
48. H. Guo, M. Karpov, E. Lucas, A. Kordts, M. Pfeiffer, V. Brasch, G.
Lihachev, V. Lobanov, M. Gorodetsky, and T. Kippenberg, Nat.
Phys.
13
, 94 (2016).
49. S. Coen and M. Erkintalo, Opt. Lett.
38
, 1790 (2013).
50. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J.
Vahala, Nat. Photonics
6
, 369 (2012).
51. M. Erkintalo, G. Genty, B. Wetzel, and J. M. Dudley, Opt. Express
18
,
25449 (2010).
52. E. Lucas, H. Guo, J. D. Jost, M. Karpov, and T. J. Kippenberg, Phys.
Rev. A
95
, 043822 (2017).
53. X. Yi, Q.-F. Yang, K. Y. Yang, and K. Vahala, Opt. Lett.
41
, 2037
(2016).
2570 Vol. 43, No. 11 / 1 June 2018 /
Optics Letters
Letter