of 8
Soliton frequency comb at microwave rates
in a high-
Q
silica microresonator
X
U
Y
I
,
Q
I
-F
AN
Y
ANG
,
K
I
Y
OUL
Y
ANG
,
M
YOUNG
-G
YUN
S
UH
,
AND
K
ERRY
V
AHALA
*
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
*Corresponding author: vahala@caltech.edu
Received 23 September 2015; revised 28 October 2015; accepted 17 November 2015 (Doc. ID 250710); published 17 December 2015
Frequency combs are having a broad impact on science and technology because they provide a way to coherently link
radio/microwave-rate electrical signals with optical-rate signals derived from lasers and atomic transitions. Integrating
these systems on a photonic chip would revolutionize instrumentation, time keeping, spectroscopy, navigation, and
potentially create new mass-market applications. A key element of such a system-on-a-chip will be a mode-locked
comb that can be self-referenced. The recent demonstration of soliton mode locking in crystalline and silicon nitride
microresonators has provided a way to both mode lock and generate femtosecond time-scale pulses. Here, soliton
mode locking is demonstrated in high-
Q
silica resonators. The resonators produce low-phase-noise soliton pulse trains
at readily detectable pulse rates
two essential properties for the operation of frequency combs. A method for
the long-term stabilization of the solitons is also demonstrated, and is used to test the theoretical dependence of
the comb power, efficiency, and soliton existence power on the pulse width. The influence of the Raman process
on the soliton existence power and efficiency is also observed. The resonators are microfabricated on silicon chips
and feature reproducible modal properties required for soliton formation. A low-noise and detectable pulse rate sol-
iton frequency comb on a chip is a significant step towards a fully integrated frequency comb system.
© 2015 Optical
Society of America
OCIS codes:
(140.3460) Lasers; (190.4360) Nonlinear optics, devices; (140.3945) Microcavities; (140.7090) Ultrafast lasers.
http://dx.doi.org/10.1364/OPTICA.2.001078
1. INTRODUCTION
The optical frequency comb is revolutionizing a wide range of
subjects spanning spectroscopy to time standards [
1
9]. Since
their invention, a miniaturized approach to the formation of a
comb of optical frequencies has been proposed in high-
Q
micro-
resonators [10
,11]. These microcombs, or Kerr combs, have been
demonstrated in several material systems [
12
15], including cer-
tain planar systems suitable for monolithic integration [
16
20].
They have been applied in demonstrations of microwave gener-
ation [
12], waveform synthesis [
18], optical atomic clocks [
21]
and coherent communications [
22]. Microcombs were initially
realized through a process of cascaded four-wave mixing [
10]
driven by parametric oscillation [
23,24]. However, a recent
advance has been the demonstration of mode locking through
the formation of dissipative Kerr solitons in microresonators
[25,26]. While the initial work on microcombs demonstrated
phase-locked states [
19,27
30], including pulse generation [
31],
solitons are both phase locked and, being pulses, can be readily
broadened spectrally [
25]. Moreover, resonator dispersion can be
engineered so as to create coherent dispersive waves that broaden
the soliton comb spectrum within the resonator [
26,32]. Both of
these features can simplify the comb self-referencing process.
Finally, solitons have a very predictable spectral envelope, which
is useful in any application where individual comb teeth are
accessed for measurement. Dissipative Kerr solitons balance
dispersion with the Kerr nonlinearity while also balancing optical
loss with parametric gain from the Kerr nonlinearity [
25,33].
They have been observed in fiber resonator systems [
34]. In micro-
resonators, these solitons have been observed in crystalline
[25 ,35,36] and in silicon nitride-based systems [
26]. Crystalline
system solitons have also been externally broadened to 2/3 of an
octave, enabling the detection of the comb offset frequency [
37].
In a self-referenced frequency comb, the optical frequency
ω
m
of the
m
th comb tooth can be written as a simple function of two
frequencies, the pulse repetition frequency
ω
r
and the offset fre-
quency
ω
offset
such that
ω
m

m
ω
r

ω
offset
. To achieve self-ref-
erenced operation, these frequencies must be detectable and
ideally occur at rates for which low-noise electronics is available.
In this work, a comb of frequencies is generated by soliton mode
locking in a silica resonator. The soliton pulse trains occur at a
repetition frequency that is both readily detectable with commer-
cial photo detectors and that enables the use of low-noise electron-
ics. Moreover, the free-running phase noise of the detected pulse
train is comparable to a good K-band signal source. Aside from
the first observation of soliton frequency combs in a silica micro-
resonator, the control of soliton properties through a novel
locking technique is demonstrated. This technique is used to
controllably explore the soliton regime of operation, including
2334-2536/15/121078-08$15/0$15.00 © 2015 Optical Society of America
Research Article
Vol. 2, No. 12 / December 2015 /
Optica
1078
Corrected
18
December
2015
measuring the soliton efficiency and existence power, as well as
measuring the soliton power and power per line versus the soliton
pulse width for comparison with the theoretical values. Servo lock-
ing also enables stable mode locking indefinitely, as well as operation
at optimal detuning with respect to the comb pumping efficiency.
The influence of the Raman process on the soliton is also observed.
In addition to Raman-induced self-shifting of the soliton spectrum
[
38
], the soliton existence power and efficiency are observed to be
influenced by Raman interactions at short pulse durations. Finally,
the silica resonator used here is fabricated on a silicon chip, which
enables reproducible avoided-mode-crossing control through the
micro-lithographic control of the resonator geometry.
2. RESONATOR CHARACTERIZATION
The silica wedge resonators were fabricated using float-zone sil-
icon wafers [
39
]. The devices exhibit a nearly constant finesse over
a wide range of diameters and have previously been applied for
comb generation at free spectral range (FSR) values from 2.6 to
220 GHz, including the formation of stable phase-locked comb
states [
19
,
21
]. In the present work, 3 mm diameter devices with
an FSR of 22 GHz were prepared, and intrinsic
Q
-factors were
characterized by the linewidth measurement to lie near 400 mil-
lion. To both characterize the soliton tuning range (see discussion
below) and to provide a separate test of the
Q
-factor, the threshold
for parametric oscillation was measured and compared to the
theoretical value [
19
,
23
]:
P
th

π
n
ω
o
A
eff
4
η
n
2
1
D
1
Q
2
;
(1)
where
A
eff
60
μ
m
2
is the effective mode area,
n
is the refractive
index,
n
2
is the Kerr coefficient,
D
1
is the FSR in rad/s units,
η

Q
Q
ext
characterizes the waveguide to resonator loading,
where
Q
ext
is the external or coupling
Q
-factor and
Q
is the total
Q
-factor (intrinsic loss and loading included), and
ω
o
is the op-
tical frequency. A schematic of the experimental characterization
setup is provided in Fig.
1(a)
. The typical measured threshold
powers were around 2.5 mW and were consistent with the
measured
Q
-factors.
The mode family that is phase locked to form the soliton pulse
train must feature anomalous dispersion [
23
] and also minimal
distortion of the dispersion caused by other mode families within
the resonator [
40
]. The first of these requirements is straightfor-
ward in silica wedge resonators when operated in the 1.5
μ
m
band [
41
]. However, the second requirement is more compli-
cated, as wedge resonators feature many transverse mode families.
Minimizing avoided-crossing behavior is achieved by screening
wedge disks to find combinations of diameter, thickness, and
wedge angle that produce avoided-crossing free spectral regions.
In addition, it is observed that high-
Q
-factor mode families are
generally more immune to avoided-crossing distortion. To char-
acterize both mode family dispersion and avoided-mode-crossing
behavior, the mode frequencies were measured using a tunable
laser that was calibrated with a fiber Mach
Zehnder interferom-
eter. A measurement taken on a typical mode family used to
produce solitons is presented in Fig.
1(b)
. A parabolic fit to the
data featuring an anomalous dispersion of
17 kHz
FSR
is pro-
vided for comparison to the data. Two avoided mode crossings
are apparent in the spectrum. Numerical simulation of the reso-
nator dispersion allowed the identification of the mode families
believed to be responsible for the avoided mode crossing.
Generally, the dispersion modeling was found to be in excellent
agreement with measured dispersion curves for the soliton mode
family and these other mode families.
3. MEASUREMENT OF COMB PROPERTIES
Modes belonging to families featuring both anomalous dispersion
and well-behaved spectra (i.e., few avoided crossings) were
pumped using a scanning fiber laser. Solitons form when the
pump frequency is red detuned relative to an optical mode, and
give rise to characteristic steps in the pump power transmission
versus tuning [
25
]. As has been described elsewhere, the excita-
tion of stable soliton trains is complicated by the thermal non-
linearity of the resonator [
26
], which is well known to destabilize
a red-detuned pump wave [
42
]. Fortuitously, solitons feature
power dependence with the tuning of the pump wave that reverses
this behavior and that will stabilize the pump on the red side of
the resonance once the nonsoliton transients have died away. To
induce stability, the two-step protocol was employed [
26
]. Both
single and multiple soliton states were stably excited in different
resonators. Figure
1(c)
shows the spectrum measured for a single-
soliton state. The square of a hyperbolic secant function is also
overlaid onto the spectrum to verify the characteristic single-
soliton spectral shape. From this fitting, the soliton pulse width
τ
is inferred to be 130 fs where the pulse shape is
sech
2

t
τ

.
(Note: this definition of pulse width is
0.57
× the FWHM width
of the soliton pulse). The presence of small spurs in the spectrum
of Fig.
1(c)
correlate with the appearance of avoided crossings in
the mode dispersion spectrum in Fig.
1(b)
.
Direct confirmation of single-soliton generation is provided by
frequency-resolved optical gating (FROG) and autocorrelation
traces [see Fig.
1(d)
]. In these measurements, the pump laser was
suppressed by fiber Bragg-grating filters, and a dispersion compen-
sation of -
1.5 ps
nm
was applied using a programmable optical
filter before the comb was amplified by an erbium-doped fiber am-
plifier (EDFA). A pulse width of 250 fs with a pulse period of 46 ps
is inferred from this data. The measured pulse width is larger than
that fitted from the optical spectrum (130 fs) due to the limited
wavelength bandwidth of the optical pre-amplifier used in this
measurement. The FROG data was also used to reconstruct the
phase of the spectrum and showed a constant phase across the comb
lines.
An important feature of the soliton states generated in this
work is their detectable and stable repetition rate. Figure
2(a)
con-
tains the phase noise spectra of the detected soliton fundamental
repetition frequency measured using single solitons generated
with three different resonators. The upper right inset of Fig.
2(a)
is a typical radio frequency spectrum of the fundamental repeti-
tion frequency. The repetition frequency can be seen to be
21.92 GHz (resolution bandwidth is 10 kHz), and has an excel-
lent stability that is comparable to a good K-band microwave
oscillator. For example, one of the devices measured has a phase
noise level of
100 dBc
Hz
at a 10 kHz offset (referenced to a
10 GHz carrier frequency). We believe that some of the variations
observable in these spectra are not fundamental, but instead
are related to the pump laser noise. For example, the step feature
in the spectrum near the 20 kHz offset frequency also appears in
the frequency noise of the pump laser [see green curve in
Fig.
2(a)
].
The ability to reproduce mode family dispersion characteristics
in different resonators was investigated. Figures
2(b)
and
2(c)
give
Research Article
Vol. 2, No. 12 / December 2015 /
Optica
1079
the results of mode family dispersion measurements and soliton
optical spectra on two resonators that are close in size and shape to
the device in Fig.
1
. There is high level of consistency in both the
magnitude of the dispersion and also the minimal presence of
mode crossings in the measured spectra. This level of consistency
was sufficient across many devices to provide the ready observa-
tion of soliton steps. Nonetheless, there are observable variations
in the nature of the soliton steps formed in scanning the resona-
tors. The corresponding soliton steps under the pump frequency
scan are shown in Fig.
2(d)
. Also, the microwave phase noise spec-
tra in Fig.
2(a)
were measured on soliton trains using these same
three devices, and exhibit slight differences.
4. PUMP-RESONATOR DETUNING LOCK
For steady-state soliton generation, it was possible to pump the
resonator without any frequency locking control of the pump
laser (i.e., open loop). In this mode of operation, solitons were
generated for several hours provided that the pump power was
high (see comment in next section). However, several soliton
properties depend upon the pump-resonator detuning,
δω
ω
o
ω
p
, where
ω
o
is the resonator frequency and
ω
p
is the pump
frequency. It is therefore desirable to stabilize the detuning of
the pump relative to the cavity. For example, the soliton power
and pulse width are given by the following expressions
[
25
,
43
,
44
]:
Fig. 1.
Experimental setup, soliton mode family dispersion, optical spectrum, and autocorrelation. (a) Experimental setup. A continuous-wave fiber
laser is amplified by an EDFA. The laser frequency is separately monitored using a fiber Mach
Zhender interferometer (not shown). To implement the
two-step protocol, the laser is modulated by an acousto-optic modulator (AOM). The laser is coupled to a high-
Q
wedge resonator using a fiber taper. The
power transmission and microwave beatnote are detected by photo detectors (PD) and sent to an oscilloscope and electrical spectrum analyzer (ESA). T
he
soliton optical spectra are measured using an optical spectrum analyzer (OSA). Dispersion compensation (not shown) is employed before the solitons
are
amplified for autocorrelation and the FROG measurement. Polarization controllers, optical isolators, and optical Bragg-grating filters are not s
hown.
(b) Measured frequency dispersion (blue points) belonging to the soliton-forming mode family is plotted versus the relative mode number,
μ
. To construct
this plot, the mode frequency relative to a
μ

0
mode (mode to be pumped) is measured using a calibrated Mach
Zehnder interferometer (fiber optic-
based). To second order in the mode number, the mode frequency is given by the Taylor expansion,
ω
μ

ω
0

μ
D
1

1
2
μ
2
D
2
, and the dashed red curve
is a fit using
D
1
2
π

21.92 GHz
and
D
2
2
π

17 kHz
. In the plot, the mode frequencies are offset by the linear term in the Taylor expansion to make
clear the second-order group dispersion. The measured modes span wavelengths from 1520 to 1580 nm, and
μ

0
corresponds to a wavelength close to
1550 nm. The presence of nonsoliton-forming mode families can be seen through the appearance of avoided mode crossings (spur-like features) that
perturb the parabolic shape. Simulations of the nonsoliton mode families believed to be responsible for these spurs are provided (see mode 1 and mode 2
dashed curves). In addition, the normalized transverse intensity profiles for the soliton and nonsoliton spatial modes are provided at the top of the
panel
(red indicates higher mode intensity). The simulation used the Sellmeier equation for the refractive index of silica. The oxide thickness, wedge ang
le, and
radius were finely adjusted to produce the indicated fits. (c) The optical spectrum of a single soliton state is shown with a
sech
2
envelope (red dashed line)
superimposed for comparison. The pump laser is suppressed by 20 dB with an optical Bragg-grating filter. (d) FROG (upper) and autocorrelation trace
(lower) of the soliton state in (c). The optical pulse period is 46 ps and the fitted pulse width is 250 fs (red solid line).
Research Article
Vol. 2, No. 12 / December 2015 /
Optica
1080
P
sol

2
η
A
eff
n
2
Q
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
nc
β
2
δω
p
;
(2)
τ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
c
β
2
2
n
δω
r
;
(3)
where
β
2

nD
2
cD
2
1
is the group velocity dispersion and is
negative for anomalous dispersion. Because the amount of detun-
ing,
δω
, will typically exceed tens of cavity linewidths, it is not
immediately clear how standard locking methods can be applied.
Moreover, such techniques may interfere with the mode-locking
process itself. The technique demonstrated here uses the theoreti-
cal dependence of the average soliton comb power on the pump-
resonator detuning [Eq. (
2
)]. This dependence means that
δω
can
be servo locked by requiring the comb power to hold a fixed set
point. Indeed, the implementation of this locking method is a test
of the theoretical independence of the comb power and the pump
power. The servo control also has the effect of fixing the soliton
pulse width [see Eq. (
3
)].
To demonstrate the servo control method and to also test our
theoretical predictions, several soliton properties were measured.
The comb power and the maximum power per comb line were
measured at a series of detuning values,
δω
2
π
, estimated to
range from 12.8 MHz (
τ

187 fs
) to 29.6 MHz (
τ

123 fs
),
or approximately 13 to 30 cavity linewidths. At each point, the
soliton spectrum was also recorded, which enables the calculation
of the soliton pulse width
τ
. The results of these measurements are
compiled in Fig.
3(a)
. Also, Fig.
3(b)
shows the soliton spectrum
measured at the detuning limits presented in Fig.
3(a)
. The
dashed lines in Fig.
3(a)
give the predicted comb power and maxi-
mum comb line power based on the following expressions for the
comb power and the comb power spectral envelope [
25
,
43
,
44
]:
P
sol

2
c
η
A
eff
β
2
n
2
Q
1
τ
;
(4)
P

Δ
ω

π
c
2
η
A
eff
β
2
n
2
D
1
Q
sech
2

πτ
2
Δ
ω

;
(5)
where Eq. (
4
) is obtained by eliminating
δω
in Eqs. (
2
) and (
3
),
and
Δ
ω
denotes the comb tooth frequency relative to the comb's
center frequency. Note that the peak power of the spectral
envelope (i.e., the maximum comb tooth power) is determined
entirely by the cavity properties. This feature is apparent in both
the data in Fig.
3(a)
and the spectra in Fig.
3(b)
. As an aside,
Eqs. (
2
) and (
3
) are derived under an assumption of large fre-
quency detuning in units of cavity linewidth, which is satisfactory
for the current measurements.
An important benefit of the feedback control presented here is
that it provides a convenient way to stabilize the mode-locked
system indefinitely. For example, the system was operated contin-
uously for over 24 h. For over 19 h of this period, comb properties
were recorded and are presented in Fig.
4
.
(a)
(b)
(c)
(d)
Fig. 2.
Detected phase noise and electrical spectra for three devices with corresponding mode dispersion and soliton data. (a) Phase noise spectral
density function plotted versus offset frequency from the detected soliton repetition frequency of three different devices. A Rohde & Schwarz phase
noise
analyzer was used in the measurement. Inset shows the electrical spectrum of the soliton repetition frequency (21.92 GHz) for one device. The other
devices had similar spectra, with repetition frequencies of 22.01 and 21.92 GHz. The phase noise of the fiber pump laser is shown in green and was
generated by mixing two nominally identical pump lasers to create a 2.7 GHz electrical beatnote. Several features in the pump laser phase noise are
reproduced in the soliton phase noise (see features near and above 20 kHz). The black line connecting the square dots is the measurement floor of the
phase noise analyzer. (b),(c) The mode dispersion spectra and soliton optical spectra are presented for two of the devices measured in (a). The third d
evice
in (a) is from Fig.
1
. (d) Pump power transmission is plotted versus tuning across a resonance used to generate the soliton spectra in (c). The data show the
formation of steps as the pump tunes red relative to the resonance. Both the blue-detuned and the red-detuned operation of the pump relative to the
resonance are inferred from generation of an error signal using a Pound
Drever
Hall system-operated open loop.
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5. EXISTENCE POWER AND RAMAN
In units of cavity linewidth, the predicted maximum detuning for
soliton existence is given by
δω
max
π
2
P
in
16
P
th
[
25
,
43
,
44
].
This power dependence explains why increased pump power is
effective in providing a stable, nonlocked soliton operation. At
higher pumping powers, the solitons can survive over a wider
range of tuning values, as the pump laser frequency drifts relative
to the cavity resonant frequency. However, as noted above, it
is preferrable to prevent this relative drift. In addition to the rea-
sons mentioned in the previous section, locking the detuning fre-
quency,
δω
, enables operation at a lower (and hence more
efficient) pump power setting that is close to the existence power
limit for solitons of the desired pulse width.
To measure the minimum existence power,
δω
was held con-
stant while the pump power was reduced until the soliton pulse
train disappeared. Both the power level and the soliton spectrum
were recorded near the point of disappearance.
δω
was then reset
to a new value, and the measurement was repeated. Figure
3(c)
gives the minimum power measured this way plotted versus the
pulse width (as calculated earlier from the soliton spectrum).
Substituting for
δω
in Eq. (
3
) using
δω
max
gives the following
expression for the minimum pump power for soliton existence
as a function of the pulse width [
25
,
43
,
44
]:
P
min
in

2
c
π
A
eff
β
2
η
n
2
τ
2
1
QD
1
:
(6)
This expression is plotted in Fig.
3(c)
(dashed red line). A
deviation from the predicted dependence is observed. Improved
agreement with the data is provided by a simulation (small dashed
red line) using the Lugiato
Lefever equation (LLE) [
25
,
44
47
]
augmented by Raman terms [
48
,
49
]. In this analysis, a Raman
time constant of 2.4 fs was extracted by fitting the data. This time
Power (dBm)
Wavelength (nm)
1500
1520
1540
1560
1580
1600
-30
-50
-70
Detuning A (measured)
Detuning A (simulated)
Detuning B (measured)
Detuning B (simulated)
-10
Detuning A (simulated
w/o Raman terms)
1540
1550
1560
-30
-15
-25
-20
0.8 nm
4.3 nm
(a)
(b)
(c)
Soliton power (mW)
Minimum pump power (mW)
Power per comb line (
μ
W)
Efficiency (%)
120
20
40
60
80
200
100
10
0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.5
0.25
1.0
1.5
2.0
5
10
15
20
25
130
140
150
160
170
180
190
Pulse width (fs)
120
130
140
150
160 170 180 190
Pulse width (fs)
Simulation w/
Raman terms
Measurement
Equation 4
Measurement
Equation 6
Simulation w/
Raman terms
Measurement
Equation 7
Measurement
Equation 5
A
B
Fig. 3.
Control of soliton properties. (a) Measured soliton comb output power is plotted versus measured soliton pulse width (red points) with com-
parison to Eq. (
4
) (dashed red line). The measured power per central comb line is plotted versus pulse width (blue points) with comparison to Eq. (
5
)
(dashed blue line). (b) The observed soliton spectra at the limits of the measurement in Fig.
3(a)
are shown [see arrows A and B in Fig.
3(a)
]. Solid orange
and green curves are simulations using the Lugiato
Lefever equation, which includes Raman terms. The indicated wavelength shifts between the pump
and the centroid of the soliton spectrum result from Raman interactions with the soliton. The location of the pump line for both spectra is indicated by
the
dashed black line and has been suppressed by filtering. The inset shows a magnified view near the central region of the blue spectrum. The green (purple
)
envelope provides the Lugiato
Lefever simulation with (without) Raman terms. The green spike is the location of the pump. (c) The measured minimum
pump power for soliton existence is plotted versus the measured soliton pulse width (red points) with comparison to Eq. (
6
) (dashed red line). The
measured efficiency is plotted versus the measured soliton pulse width (blue points) with comparison to Eq. (
7
) (dashed blue line). The simulation using
the Lugiato-Lefever equation including Raman terms improves the agreement with data (small dashed red and blue lines).
Research Article
Vol. 2, No. 12 / December 2015 /
Optica
1082
constant closely agrees with a value of 3 fs measured for a silica
optical fiber [
50
]. The contribution of the Raman terms is both
predicted and measured to be stronger for shorter pulses (or
equivalently larger
δω
). These observations are also consistent
with the modeling, which shows modifications to soliton effi-
ciency as a result of the Raman process [
49
].
It is interesting that the predicted soliton power and power per
line do not require a Raman correction [see Fig.
3(a)
]. The Raman
correction primarily influences the temporal phase of the soliton
pulse and not the amplitude [
38
]. It would therefore be expected
to alter the existence condition, which is associated with the
phase of the soliton field, and not the power per line or overall
comb power.
All of the soliton spectra observed in this work were red shifted
relative to the pump wavelength. Raman interaction with dissi-
pative Kerr solitons has recently been shown to cause such self-
shifting of the spectrum [
38
,
51
,
52
]. The effect has been observed
in silicon nitride resonators [
38
]. Like the correction to the mini-
mum power described above, this shift is stronger for shorter
pulses and larger detuning frequencies. Shifted soliton spectra
can be produced by several mechanisms [
53
] besides Raman, in-
cluding soliton recoil caused by dispersive wave generation [
26
].
There was no evidence of dispersive wave generation at shorter
wavelengths in this work. Consistent with other reports [
38
],
the Raman-augmented LLE simulation explains the soliton spec-
tral shift observed here [see orange and green curves in Fig.
3(b)
].
Control of the pump-resonator detuning frequency enables
stable operation near an optimal pumping efficiency limit.
Defining the efficiency [
49
] as the soliton power divided by
the minimum pump power for soliton existence gives the follow-
ing simple expression:
Γ
πη
2
D
1
τ
:
(7)
A comparison of this prediction with the measurement is pro-
vided in Fig.
3(c)
(
η

0.29
in this measurement). A deviation
between the simple formula [Eq. (
7
)] and the data at small pulse
widths occurs on account of the Raman effects and has been cor-
rected in the figure (small dashed blue line) using the Raman-
augmented LLE result.
6. DISCUSSION
Reduced soliton repetition rates (
D
1
2
π
) are accompanied by an
increased mode volume, and therefore require greater pumping
0
5
10
15
20
0.0
0.1
0.2
0.3
0.4
0.5
Hours
Soliton power (mW)
0
5
10
15
Power per comb line (
μ
W)
0
50
100
150
200
Pulse width (fs)
0
2
4
6
8
Raman wavelength shift (nm)
0
5
10
15
20
Hours
(a)
(b)
Fig. 4.
Continuous soliton measurement over 19 h. (a) The soliton
power and central comb line power are plotted versus time in hours.
The soliton power experiences a slow drift to lower values, which is attrib-
uted to a slow variation in either the power set point of the electronic con-
trol or in the detected power (perhaps due to temperature drift). The central
comb line power is approximately constant over the measured period, with
the exception of a small step increase around 6 h. This could be associated
with anything that increases the resonator coupling efficiency, decreases the
resonator
Q
, or both. Since the resonator is taper coupled, any kind of
mechanical shock could introduce such a step change in these quantities.
(b) Pulse width and Raman self-shift (defined as the wavelength center of
the soliton spectrum less the pump wavelength) are plotted versus time in
hours. A step in these quantities is recorded at the identical moment to that
recorded for the central comb line power. It is interesting to note that be-
cause the soliton power is feedback controlled, from Eq. (
2
), a step increase
(decrease) in coupling
η
(
Q
) would be accompanied by a compensating
decrease in pump detuning. From Eq. (
3
), this feedback-controlled step
in detuning would cause an increase in the soliton pulse width (and de-
crease in the Raman shift), as observed in the recorded data of panel b.
Finally, the operation of the soliton comb proceeded for an additional 5 h
beyond the record presented in this figure. During the additional 5 h, other
measurements were performed that required disabling the data recording.
Repetition frequency (GHz)
0
0
1
110
10
5
10
8
10
7
10
6
10
9
10
10
Total quality factor
-50
Power per comb line (dBm)
1mW
10W
1W
100mW
10mW
20
Efficiency (%)
0
2
2
.
02
Fig. 5.
Required pumping power and maximum power per comb line.
Linear contours give the minimum pump power required for existence of
200 fs solitons assuming the silica resonator of this work with the indi-
cated
Q
-factor and repetition frequency (note that scales change for other
material systems). These are derived using Eq. (
6
). Coloration indicates
the maximum power per comb line given by Eq. (
5
). The upper hori-
zontal axis is the efficiency found using Eq. (
7
). This axis does not depend
on the material.
η

0.7
is assumed in making the plot.
Research Article
Vol. 2, No. 12 / December 2015 /
Optica
1083