of 7
Research Article
Vol. 7, No. 4 / April 2020 /
Optica
309
Interleaved difference-frequency generation for
microcomb spectral densification in the
mid-infrared
Chengying Bao,
1
,
Zhiquan Yuan,
1
,
Heming Wang,
1
Lue Wu,
1
Boqiang Shen,
1
Keeyoon Sung,
2
Stephanie Leifer,
2
Qiang Lin,
3
AND
Kerry Vahala
1
,
*
1
T. J. Watson Laboratory of Applied Physics, California Institute of Technology, Pasadena, California 91125, USA
2
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
3
Department of Electrical and Computer Engineering, University of Rochester, Rochester, New York 14627, USA
*Corresponding author: vahala@caltech.edu
Received 12 November 2019; revised 3 March 2020; accepted 5 March 2020 (Doc. ID 382992); published 8 April 2020
With their compact size and semiconductor-chip-based operation, frequency microcombs can be an invaluable light
source for gas spectrcoscopy. However, the generation of mid-infrared (mid-IR) frequency combs with gigahertz line
spacing as required to resolve many gas spectra represents a significant challenge for these devices. Here, a technique
referred to as interleaved difference-frequency generation (iDFG) is introduced that densifies the spectral line spacing
upon conversion of near-IR comb light into the mid-IR light. A soliton microcomb is used as both a comb light source
and microwave oscillator in a demonstration, and the spectrum of methane is measured to illustrate how the resulting
mid-IR comb avoids spectral undersampling. Beyond demonstration of the iDFG technique, this work represents an
important feasibility step towards more compact and potentially chip-based mid-IR gas spectroscopy modules.
© 2020
Optical Society of America under the terms of the OSA Open Access Publishing Agreement
https://doi.org/10.1364/OPTICA.382992
1. INTRODUCTION
Optical microresonator-based frequency combs (microcombs)
have received increasing interest over the past decade [1–4]. They
can operate with low power consumption and offer compact form
factors. Of special importance, soliton formation in microres-
onators provides an elegant method to generate fully coherent
frequency combs [3,5]. These soliton microcombs have been
tested in a wide range of applications including spectroscopy
[6–8], optical frequency synthesis [9], and optical clocks [10].
Since microcombs are fabricated at the wafer scale [see Fig. 1(a)],
a large number of portable functional modules can in principle be
produced efficiently.
For spectroscopy and gas sensing, the mid-infrared (mid-IR)
bands have been of keen interest for conventional frequency comb
development [11]. And mid-IR microcomb-based sensor modules,
on account of their compact form factor, could potentially be used
in the food industry [12], for human breath and health analysis
[13], for detection of chemical threats [14], and for monitoring of
greenhouse gases (such as CO
2
, CH
4
) [15,16]. Microcomb-based
spectroscopy has been demonstrated in both the near-IR [6,7]
and the mid-IR [8]. However, monolithic microcombs presently
feature very wide comb line spacings [3,17] in the mid-IR bands.
While such wide spacings can increase the acquisition rate in dual-
comb spectroscopy (DCS) [18], they also lead to undersampling
of gas spectra and require separate frequency tuning of the comb
to fill in the spectrum [19]. Reducing microcomb line spacing
has been a priority for opto-electrical interface in self-referenced
microcomb systems [9], and ultra-high-
Q
resonator platforms
can overcome the increased pumping volume of the narrower line
spacing combs. However, narrow gigahertz-rate line spacings have
been possible only in the near-IR [20,21]. Apart from this issue,
the development of complex self-referenced microcomb systems
in the mid-IR presents an even greater challenge. And while there
has been remarkable progress on mid-IR comb formation using
quantum cascade lasers, these devices so far also feature wide comb
line spacings [22,23].
For these reasons, the method of difference-frequency gener-
ation (DFG) presents an appealing alternative for generation of
microcomb-based coherent combs in the mid-IR. DFG conven-
iently leverages more mature visible and near-IR signal sources
to generate a wide range of mid-IR spectra and has an extensive
application history. An early use of DFG was to generate 3.39
μ
m
comb light to create a methane optical clock [24]. Generation of
mid-IR comb light for the earliest demonstration of DCS also
applied DFG by mixing Ti:sapphire lasers in GaSe crystals [25,26].
The use of fiber laser combs either alone (distinct spectral por-
tions of a single comb) [27] or in conjunction with a CW source
[28] for generation of mid-IR comb light was soon followed by
demonstration of spectrocopic measurements of methane [29].
When combined with DCS, it later enabled remarkably precise
spectral measurements of methane and other gases [30–33]. All
2334-2536/20/040309-07 Journal © 2020 Optical Society of America
Research Article
Vol. 7, No. 4 / April 2020 /
Optica
310
Fig. 1.
Interleaved difference-frequency generation (iDFG) experimental setup. (a) A 3 mm diameter soliton microcomb (fabricated on a 4 inch silicon
wafer) is pumped by a continuous wave (CW) 1.5
μ
m laser. The microresonator generates both the soliton optical pulses (green) with period
T
S
and, upon
photodetection (PD), the microwave signal at frequency
f
r
=
1
/
T
S
. This frequency is processed to create the EO-comb drive signal at frequency
f
EO
r
=
(
N
1
)
f
r
/
N
=
f
r
f
r
/
N
(i.e.,
1
f
r
=
f
r
f
EO
r
=
f
r
/
N
), which modulates a 1
μ
m CW laser to generate the EO-comb pulse stream (blue). The soli-
ton microcomb and the EO-comb are combined to pump a PPLN crystal to generate the mid-IR comb. Because the EO-comb is derived from the soliton
repetition rate the corresponding pulses temporally align with a period
T
MIR
=
(
N
1
)
T
EO
=
NT
S
, where
T
MIR
is the mid-IR pulse period. This creates a
mid-IR frequency comb having a line spacing of
f
r
/
N
=
1
f
r
. Larger
N
thereby enables finer spectral sampling of mid-IR absorption features (e.g., inset
illustrates how the black absorption spectra are sampled by combs having different line spacings). EOM, electro-optical modulator; BPF, bandpass filter;
WDM, wavelength division multiplexer. (b) Optical spectra of the near-IR EO-comb (left) and soliton microcomb (right). The inset shows the repetition
rate
f
r
of the soliton microcomb and the derived frequency of 15
f
r
/
16, which drives the EO-comb.
of these works used periodically poled lithium niobate (PPLN)
as the mixing crystal. More recently, super-octave mid-IR comb
spectra have been generated using fiber-comb intrapulse DFG in
GaP crystals [34]. These results dramatically extend the spectral
reach of the DFG technique, and intrapulse DFG has also been
applied using PPLN [35]. However, all comb DFG methods
demonstrated to date utilize narrow line spacing table-top combs.
It is therefore interesting to consider ways to generate mid-IR
combs with gigahertz line spacing using available near-IR wide line
spacing microcombs (10s of GHz). Such mid-IR combs would
avoid spectral undersampling while retaining a high acquisition
rate for DCS.
Here, we introduce interleaved DFG (iDFG) to generate
gigahertz line spacing mid-IR comb light at 3.3
μ
m. Similar to
conventional DFG, iDFG occurs between two combs, but now
having different repetition rates. The rates are specially chosen
so that a new mid-IR comb is formed with a line spacing equal to
an integer fraction of the repetition rate of either original comb
(i.e., the mid-IR comb spectrum is densified relative to the original
combs). In the work, a silica soliton microcomb [36] acts as both
a lightwave source and a microwave source [see Fig. 1(a)] and is
mixed with an electro-optic frequency comb (EO-comb) [37] to
produce mid-IR comb light. iDFG enables agile adjustment of the
mid-IR comb line spacing, and the generated 3.3
μ
m comb is used
to perform methane spectral measurement.
2. RESULTS
A. Interleaved Difference-Frequency Generation
In the measurement, a 1.5
μ
m soliton microcomb [36] and a 1
μ
m
EO-comb [37] having different repetition rates are mixed for
mid-IR comb generation. Due to their repetition rate difference
(
1
f
r
, subject to conditions described below; see the Appendix A),
mixing of the comb pulses repeats after 1
/1
f
r
[Fig. 1(a)], thereby
creating a temporal interleaving effect. Thus, iDFG not only
converts the near-IR combs into the mid-IR, but also reduces
the repetition rate (line spacing) of the mid-IR comb spectrum.
The line spacing is also tunable by adjustment of
1
f
r
, and to
demonstrate this feature, a range of mid-IR comb line spacings
(lowest 0.7 GHz) is generated. The soliton microcomb having a
comb line spacing of approximately 22 GHz uses a 3 mm diameter
ultra-high-
Q
wedge silica microresonator [36,38]. Its spectrum
[Fig. 1(b)] features a smooth and sech
2
-like envelope with a 3 dB
bandwidth of 1.5 THz (
70 lines within the 3 dB bandwidth).
As shown in the Appendix A, a uniform line spacing of the iDFG-
generated mid-IR comb requires that
1
f
r
=
m f
r
/
n
, where
m
,
n
are mutually prime. To ensure this condition, the soliton stream
is detected to generate a 22 GHz microwave signal
f
r
and is then
electrically processed to create the EO-comb drive signal frequency
equal to
f
EO
r
=
(
N
1
)
f
r
/
N
[Fig. 1(a) and inset of Fig. 1(b)].
This ensures
1
f
r
=
f
r
/
N
, where
N
is an integer (typically 16, 32)
and guarantees a strict frequency (and phase) relationship between
Research Article
Vol. 7, No. 4 / April 2020 /
Optica
311
the EO-comb and soliton comb repetition rates. Significantly, the
approach also leverages the excellent microwave stability of the
soliton microcomb [36,39–41] to replace a bulk microwave source
that is normally required to drive the EO-comb. As described else-
where [37], the EO-comb consists of cascaded phase and intensity
modulators. Figure 1(b) contains a representative optical spectrum
of the generated EO-comb. Current progress in on-chip EO mod-
ulators suggests that integrated solutions to this EO-comb will
be possible in the near future [42]. Alternatively, it is also possible
to replace the EO-comb with a 1
μ
m soliton microcomb using
fabrication methods demonstrated elsewhere [43].
B. Mid-IR Combs Generated by iDFG
The mid-IR comb at 3.3
μ
m is generated by pumping a 4 cm long
PPLN crystal (NTT Corporation) with the above soliton and
EO-combs. Integrated waveguide PPLN devices can offer much
smaller form factors [44,45]. When setting
N
=
16, a mid-IR
comb spanning about 80 nm can be generated [spectrum shown
in Fig. 2(a)]. The center frequency of the mid-IR comb can be
shifted by changing the temperature of the crystal to thereby adjust
the phase-matching condition. Due to the limited spectral reso-
lution of the mid-IR spectrometer (Horiba iHR 550), a discrete
comb structure is not resolvable in the spectrum. Therefore, to test
that the generated comb has a uniform line spacing that has been
reduced to
f
r
/
16 through the iDFG process, a fast mid-IR detector
is used to measure its line spacing by photodetection of the comb.
The electrical spectrum of the detected 3.3
μ
m comb shows a single
peak at
f
r
/
16
=
1.4 GHz [see Fig. 2(b)]. There are no additional
peaks in the spectrum, consistent with a uniform line spacing and
also showing reduction of the comb line spacing to 1.4 GHz from
that of the 22 GHz soliton microcomb. The resulting mid-IR
comb consists of more than 1500 lines. The measured phase noise
spectrum of the detected mid-IR comb signal is shown in the inset
of Fig. 2(b) and verifies excellent repetition-rate stability.
To further study the iDFG system, we replaced the soliton-
based microwave drive signal by an independent microwave signal
source (Agilent PSG). This allowed examination of the effect of
non-optimal drive frequencies on the detected mid-IR microwave
spectrum. As shown in Fig. 2(b), by using a non-optimal frequency
of 15
f
r
/
16
+
δ
f
(
δ
f
48 MHz) to drive the EO-comb, the
detected mid-IR electrical spectrum features peaks at the frequen-
cies of 16
δ
f
and
f
r
/
16
17
δ
f
in addition to the main peak at
f
r
/
16
δ
f
. This drive frequency arrangement thereby induces
a non-uniform line spacing in the mid-IR comb and is consistent
with our analysis in Appendix A. This non-uniformity is unde-
sirable, as it would increase the signal processing complexity in
applications such as DCS. As an aside, the ability to observe the
additional peaks also verifies that the mid-IR comb densification
occurs without the appearance of spectral gaps (i.e., there are
N
mid-IR lines generated for every 22 GHz spacing in the mid-IR).
Otherwise, these tones would not appear.
By adjusting the division ratio
N
, the iDFG system allows
generation of other mid-IR combs with different line spacings. For
example, when dividing
f
r
by
N
=
32, a mid-IR comb with a line
spacing of
f
r
/
32
=
0.7 GHz can be generated. The corresponding
optical spectrum is shown in Fig. 2(c) and is similar to the spectrum
in Fig. 2(a). The repetition rate of 0.7 GHz is verified in the inset of
Fig. 2(c). Other comb line spacings can also be readily generated,
but are not demonstrated here due to the limitations of the electri-
cal band pass filter employed. Hence, the line spacing of the 3.3
μ
m
comb can be selected in an agile way for different applications.
Residual gaps are believed to occur in the comb densification for
the
N
=
32 case. This is, however, not a fundamental limitation
but rather one caused by the limited phase-matching bandwidth of
the PPLN crystal as discussed in Section 2.C.
C. Methane Measurement
As a simple demonstration, the 3.3
μ
m iDFG comb was used for
absorption measurement of methane. This also tested its ability to
avoid spectroscopic undersampling in the mid-IR. Methane spec-
tra were measured by passing comb light from the 1.4 GHz mid-IR
comb through a 5 cm long single-pass gas cell containing a mixture
of 200 Torr methane and 560 Torr nitrogen, followed by spectral
analysis using the Horiba spectrometer. The absorption spectrum
is shown in Fig. 3(a). Six branches [P(2) to P(7)] in the
ν
3
band can
be observed in the spectrum and match the methane absorption
lines (included for comparison in the figure). Because only two
phase modulators and one intensity modulator were used for EO-
comb generation in this measurement (one phase modulator was
damaged), the mid-IR comb is spectrally narrower than the one in
Fig. 2, where three phase modulators and one intensity modula-
tor were used. Even with this narrower spectrum, three methane
absorption branches are observed in a single measurement. By tun-
ing the temperature of the PPLN crystal it was possible to extend
the spectrum over additional branches. Similar to Fig. 2(a), comb
lines and fine absorption features are not resolved due to the limited
resolution of the spectrometer.
To illustrate the benefit of the densified line spacing, a 22 GHz
mid-IR comb was also generated by conventional DFG of the
1.5
μ
m soliton with a CW 1
μ
m laser. Using these widely spaced
comb lines for measurement of the P(4) and P(5) branches,
the methane lines are spectrally undersampled [see Fig. 3(b)].
Moreover, the resulting 22 GHz mid-IR comb is narrower than
that shown in Fig. 2. This is because the PPLN crystal provides a
broader phase-matching bandwidth at 1
μ
m versus phase match-
ing at 1.5
μ
m [46]. Thus, a broader comb is obtained when having
a broadband 1
μ
m input. It is interesting to note that the CW
1
μ
m laser mixes with only
19 soliton microcomb lines within
the PPLN phase-matching bandwidth. To avoid the presence of
spectral gaps in the mid-IR comb, this number needs to be larger
than
N
. As a result, the
N
=
32 mid-IR comb is believed to contain
spectral gaps.
The mid-IR comb power generated here is about 200
μ
W
starting with
100 mW 1.5
μ
m soliton power and
100 mW
1
μ
m EO-comb power as inputs. Only a portion (less than 20%)
of the soliton spectrum contributes to iDFG due to the abovemen-
tioned phase-matching bandwidth. iDFG should decrease the
conversion efficiency compared to conventioanl DFG with iden-
tical repetition rates, since it reduces the overall temporal overlap
between the two near-IR combs. However, the iDFG efficiency
and output comb power can be enhanced relative to the current
results by using a thin-film PPLN waveguide, as the confinement
is higher [44,45], and dispersion engineering can be employed to
provide a more optimal design [42,47,48]. For example, significant
enhancement of second-harmonic generation (SHG) efficiency
in thin-film PPLN waveguides compared to conventional PPLN
waveguides has been reported [45]. These waveguides have also
enabled SHG over a bandwidth exceeding 10 THz via dispersion
engineering [48]. By replacing the EO-comb with a 1
μ
m soliton
microcomb [43], an increased efficiency can also result from the
Research Article
Vol. 7, No. 4 / April 2020 /
Optica
312
E
O
co
m
bfo
r iDF
G
3300
33
2
033
4
0
3360
3380
3
4
00
3
42
0
Wavelen
g
th
(
nm
)
3300
33
2
033
4
0
3360
3380
3
4
00
3
42
0
Wavelen
g
th
(
nm
)
Intensity (linear a.u.)
Intensity (linear a.u.)
f
r
soliton +
3
1
f
r
/32 EO-comb
00
.
5
1
1.
5
Frequenc
y
(G
Hz
)
Intensity (10 dB/div)
10
2
10
4
10
6
Frequenc
y
(
Hz
)
-1
00
-
80
-
60
-4
0
-2
0
0
Phase noise
(
dBc
/
Hz
)
1
.4
G
Hz mi
d
-IR
co
m
b
15f
r
/16
r
15f
r
/16+
r
f
16
f
f
r
soliton +
15f
r
/16 EO-comb
r
(
a
)
(
c
)
(
b
)
PPLN 31.7
°
C
PPLN 45.2
°
C
f
r
/16-
r
f
f
r
/16-17
r
f
f
r
/16
1
.4
G
Hz
0
0
.
5
1
Frequenc
y
(G
Hz
)
I
ntensit
y
(
10 dB
/
div
)
f
r
/32
r
0
.7
G
Hz
1
.
5
Fig. 2.
Mid-IR frequency combs generated by iDFG. (a) Optical spectra of the iDFG generated mid-IR comb. The center wavelength of the comb can
be shifted by changing the temperature of the PPLN crystal to vary the phase-matching condition. Due to the limited resolution of the spectrometer, the
individual comb lines (spaced by 1.4 GHz) are not resolved. (b) Electrical spectrum of the photodetected mid-IR comb in panel (a) showing a repetition rate
of
f
r
/
16
=
1.4 GHz (red line) resulting from driving the EO-comb at 15
f
r
/
16. When driving the EO-comb at a frequency slightly offset from 15
f
r
/
16
by an independent microwave oscillator, there will be additional peaks in the electrical spectrum (green spectral peaks). The inset shows the phase noise of
the generated mid-IR comb at 1.4 GHz. (c) The line spacing of the mid-IR comb generation can be varied. Here, a line spacing of
f
r
/
32
=
0.7 GHz, half
of that shown in panel (a), is generated by driving the EO-comb at a frequency of 31
f
r
/
32. The line spacing is verified by spectral analysis of the detected
comb (inset).
3340
3350
3360
3370
3380
3390
Wavelength (nm)
Transmission (a. u.)
JPL reference
3350
3354
3358
3362
3362
3366
3370
3374
Wavelength (nm)
w/o cell
w/ cell
w/o cell
w/ cell
(a)
P(2)
P(7)
P(3)
P(4)
P(5)
P(6)
0
6
9
20
5
9
2
0
8
9
20
7
9
2
0
0
0
30
9
9
2
Wavenumber (cm
-1
)
P(4)
P(5)
(b)
Intensity (a. u.)
Intensity (a. u.)
22 GHz
22 GHz
PPLN 32.8
°
C
PPLN 49.9
°
C
Fig. 3.
Methane absorption measurement using the 3.3
μ
m comb. (a) Measured absorption spectrum of methane over six branches [P(2) to P(7) in the
ν
3
band] using the 3.3
μ
m comb with a line spacing of 1.4 GHz. The absorption spectrum is obtained by normalizing the spectrum measured upon trans-
mission through the gas cell with the incident comb spectrum. Single comb lines are not resolved due to the limited resolution of the spectrometer (Horiba
iHR550). (b) When using the 22 GHz line spacing 3.3
μ
m comb to measure the P(4) and P(5) branches, the absorption features are spectrally undersam-
pled.
Research Article
Vol. 7, No. 4 / April 2020 /
Optica
313
higher peak pulse power relative to that afforded by the EO-comb.
However, a microcomb has a relatively fixed repetition rate, and
this will limit mid-IR line spacing tuning agility compared to the
EO-comb. To overcome this tuning limitation, multiple 1
μ
m
soliton microcombs, each having different repetition rates, could
be integrated on a single chip to allow discrete repetition rate
tuning. The repetition rate of a single 1
μ
m microcomb would be
locked to the
(
N
1
)
f
r
/
N
frequency by either active feedback or
injection locking [41]. This would require precise microfabrica-
tion control to obtain the desired free-spectral-range (FSR) so as to
facilitate locking. Control at the level of 1:20,000 of FSR has been
demonstrated [38], and the remaining fine tuning of FSR could be
accomplished using microheaters [49].
The absorption measurement demonstrated here is currently
limited by the resolution of our spectrometer. Moreover, the
spectrometer itself is large and bulky and employs a mechanical
scanning mechanism. These undesirable features of the mea-
surement can be eliminated by adding one mid-IR comb so as
to implement a DCS approach [6,18]. DCS can also leverage
the advantage of gigahertz mid-IR combs in terms of increased
acquisition rate.
3. DISCUSSION
The iDFG method was introduced and applied to generate mid-
IR frequency comb light from two near-IR combs: one a soliton
microcomb and the second an EO-comb whose drive frequency is
derived from the soliton microcomb. The iDFG method enabled
comb line densification of the mid-IR comb relative to the near-IR
combs so that spectral measurement of methane in the 3.3
μ
m
band was possible. This method should be easily extended to
produce denser combs at other spectral regions. Also, interleaved
sum-frequency generation (iSFG) is possible. Moreover, densifi-
cation of the sparse microcomb spectra is potentially useful in dual
microcomb lidar as a way to increase the ambiguity range [50,51].
The demonstrated iDFG module still consists of fiber-based
components. However, current progress towards complex
chip-integrated systems using soliton microcombs [9] as well
as significant progress in the area of integrated lithium niobate
components [42,45,47] bodes well for development of compact
iDFG modules. Moreover, soliton microcombs based entirely
upon lithium niobate and operating in the 1.5
μ
m band have
recently been demonstrated [52], so that all mid-IR comb gener-
ation components can in principle be monolithically integrated.
Other compact waveguide DFG (SFG) materials are also pos-
sible. For example, DFG to 3
μ
m has been demonstrated in
GaAs waveguides [53], and there has been impressive progress in
second-harmonic mixing using heterogeneously integrated GaAs
structures [54].
Mid-IR microcombs could also function as compact instru-
ment calibration sources in the field. Also, the iDFG microcomb
approach can provide a range of mid-IR comb wavelength bands by
leveraging the wide operational wavelength band of near-IR micro-
combs (currently 780 nm [43] through 2000 nm [55]). The soliton
microcomb range between 1000 nm and 1550 nm, in particular,
can be applied to generate mid-IR iDFG combs for chemical threat
detection. Finally, densified 3.3
μ
m microcomb and methane
measurement can be used to distinguish the biological and abiotic
contributions to methane formation on Mars [56].
APPENDIX A
Selection of
1
f
r
in iDFG. We write the near-IR EO and soliton
combs as
ν
EO
=
n
1
f
EO
r
+
f
EO
0
,
ν
S
=
n
2
f
r
+
f
S
0
, respectively.
The repetition rate of the soliton microcomb is denoted as
f
r
to
be consistent with the main text. The mid-IR comb generated in
iDFG can be written as
ν
MIR
=
n
1
,
n
2
(
n
1
f
EO
r
n
2
f
r
)
+
(
f
EO
0
f
S
0
)
.
(A1)
We further define
f
MIR
0
=
f
EO
0
f
S
0
(the mid-IR comb offset
frequency) and
1
f
r
=
f
r
f
EO
r
m f
r
/
n
+
δ
f
, where
m
,
n
are
mutually prime, and
δ
f
is zero or a small frequency compared to
f
r
(
δ
f
/
f
r
is an irrational number).
To analyze the condition for uniform line spacing generation in
iDFG, we focus on the virtual comb lines near the offset frequency
of the mid-IR comb (i.e., the comb does not extend to this range)
between
f
MIR
0
and
f
MIR
0
+
f
r
. Also, we consider separating the
overall mid-IR comb into sub-combs created by mixing one of
the comb lines of the EO-comb with the comb lines of the soliton
comb. The offset frequency for the
k
th
sub-comb is
f
MIR
0
(
k
)
=
f
MIR
0
+
mod
[
(
k
1
)1
f
r
,
f
r
]
=
f
MIR
0
+
mod
[
(
k
1
)
(
m
+
n
δ
f
f
r
)
,
n
]
f
r
n
.
(A2)
When
δ
f
=
0, all possible frequencies in the modulo term are
n
1
f
r
/
n
(
n
1
is an integer within [0,
n
1]). In other words, the
offset frequencies of different sub-combs will be spaced by
f
r
/
n
,
and a uniform line spacing of
f
r
/
n
in the mid-IR can be obtained.
Otherwise, non-zero
δ
f
will create additional line spacings.
Funding.
Defense
Threat
Reduction
Agency
(HD-
TRA11810047); Air Force Office of Scientific Research
(FA9550-18-1-035); Resnick Sustainability Institute for Science,
Energy and Sustainability, California Institute of Technology, and
the Kavli Nanoscience Institute.
Acknowledgment.
The authors thank Scott Diddams,
Christian Frankenberg, Neil Fromer, Qi-Fan Yang, and Xu Yi
for helpful discussions. The content of the information does not
necessarily reflect the position or the policy of the federal gov-
ernment, and no official endorsement should be inferred. Part
of the research was carried out at the Jet Propulsion Laboratory,
California Institute of Technology, under a contract with the
National Aeronautics and Space Administration. C. B. gratefully
acknowledges postdoctoral fellowship support from the Resnick
Sustainability Institute at Caltech.
Disclosures.
The authors declare no conflicts of interest.
These authors contributed equally to this work.
REFERENCES
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J. Kippenberg, “Optical frequency comb generation from a monolithic
microresonator,” Nature
450
, 1214–1217 (2007).
2. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-
based optical frequency combs,” Science
332
, 555–559 (2011).