Carolin P. Bauer

1] Department of Physics, Institute for Quantum Electronics, ETH Zurich,
Switzerland

# High-sensitivity dual-comb and cross-comb spectroscopy across the infrared
using a widely-tunable and free-running optical parametric oscillator

<EMAIL_ADDRESS>Zofia A. Bejm  Michelle K. Bollier  Justinas Pupeikis
Benjamin Willenberg  Ursula Keller  Christopher R. Phillips  [

###### Abstract

Coherent dual-comb spectroscopy (DCS) enables high-resolution measurements at
high speeds without the trade-off between resolution and update rate inherent
to mechanical delay scanning approaches. However, high system complexity and
limited measurement sensitivity remain major challenges for DCS. Here, we
address these challenges via a wavelength-tunable dual-comb optical parametric
oscillator (OPO) combined with an up-conversion detection method. The OPO is
tunable in the short-wave infrared (1300-1670 range) and mid-infrared
(2700-5000 range) where many molecules have strong absorption bands. Both OPO
pump beams are generated in a single spatially-multiplexed laser cavity, while
both signal and idler beams are generated in a single spatially-multiplexed
OPO cavity. The near-common path of the combs in this new configuration
enables comb-line-resolved and aliasing-free measurements in free-running
operation. By limiting the instantaneous idler bandwidth to below 1 , we
obtain a high power per comb line in the mid-infrared of up to 160 . With a
novel intra-cavity nonlinear up-conversion scheme based on cross-comb
spectroscopy, we leverage these power levels while overcoming the sensitivity
limitations of direct mid-infrared detection, leading to a high signal-to-
noise ratio (50.2 1/2) and record-level dual-comb figure of merit ($3.5\times
10^{8}$ 1/2). As a proof of concept, we demonstrate the detection of methane
with 2-ppm concentration over 3- path length. Our results demonstrate a new
paradigm for DCS compatible with high-sensitivity and high-resolution
measurements over a wide spectral range.

## Introduction

Dual-comb spectroscopy (DCS) has advanced molecular absorption spectroscopy
measurements across the infrared spectrum, including gas detection over open
paths for environmental monitoring [1, 2, 3, 4, 5], combustion analysis [6, 7,
8], plasma spectroscopy [9], and the detection of individual isotopes in gas
mixtures [10, 11, 12]. Many dual-comb technologies target the mid-infrared
(mid-IR) wavelength region since various molecules exhibit strong and
distinctive rovibrational absorption lines at these long wavelengths. However,
the practicality of DCS measurements can be constrained by limited signal-to-
noise ratio (SNR) and high system complexity. Here, we address both of these
challenges with a new source based on a tunable optical parametric oscillator
(OPO) and a novel up-conversion detection that uses the OPO as a spectroscopic
transceiver to obtain sensitivity exceeding the fundamental shot-noise limited
performance accessible via direct DCS detection.

Existing sources in the mid-infrared include direct laser sources such as
quantum-cascade lasers [13] and inter-band cascade lasers [14], as well as
parametric sources such as microresonators [15], difference-frequency
generation [16, 17], optical parametric oscillators [18, 19] and intra-pulse
difference-frequency generation using broadband lasers such as Cr2+-doped
ZnS/ZnSe [20, 21, 22].

A critical consideration when comparing these different technologies is the
spectral coverage. A broad spectral bandwidth allows for highly flexible
spectroscopic measurements on a wide range of targets. Direct laser-based
sources have a limited bandwidth or tuning capacity, implying many emitters
are needed to cover a large fraction of the mid-IR wavelength range. In
contrast, some parametric sources can cover a broad instantaneous spectral
bandwidth with one device. A number of approaches in recent years have
utilized dual-comb sources to access octave-spanning infrared wavelength
ranges [10, 15, 23, 24, 25, 21, 22]. However, a broad instantaneous optical
bandwidth implies that the total laser power gets distributed over a larger
number of comb lines, which impairs the achievable sensitivity [26]; we follow
the approach of [26] where the SNR is proportional to the magnitude of the
Fourier transform of the time-domain interferogram. So far, for high-
resolution octave-spanning DCS experiments in the mid-IR, this led to either
integration times in the minute-to-hour time scale or an SNR limited to $<$20
1/2 [10, 21, 23, 24, 25]. One method to decrease the number of comb lines is
to operate at high repetition rates, but this comes with two challenges: It
decreases the available pulse energy for nonlinear wavelength conversion
processes, and for high-resolution measurements it becomes necessary to
carefully interleave the DCS measurement by sweeping the comb line frequencies
[13, 27].

A promising alternative approach is a dual-comb source with a narrow
instantaneous bandwidth where the broad spectral coverage is obtained via
wavelength tunability. In this configuration, there are only a few thousand
comb lines in the optical spectrum at a time, in contrast to octave-spanning
100- lasers with of order a million comb lines. Therefore, it supports high
power per comb line, and -level repetition rate differences can be used
without aliasing of the DCS spectrum. The large repetition rate difference
enables phase tracking required for comb-resolved (and hence fully coherent)
measurements [28] with simple free-running dual-comb oscillators [29, 30, 31,
32]. To realize such a spectrometer configuration, here we demonstrate for the
first time a spatially-multiplexed single-cavity dual-comb OPO synchronously
pumped by a spatially-multiplexed single-cavity dual-comb laser. We thereby
obtain a simple yet powerful dual-comb source emitting in the SWIR and mid-IR
using two free-running cavities (one laser cavity, and one OPO cavity)
enabling high-sensitivity measurements. The concept of the light source is
illustrated in Fig. 1.

Figure 1:  Block diagram of the system used for dual-comb spectroscopy.
Single-cavity narrow-band dual-comb laser oscillator synchronously pumps a
wavelength-tunable single-cavity dual-comb optical parametric oscillator.

Singly-resonant OPOs are ideal for obtaining tunable coherent light with
narrow instantaneous bandwidth. While leveraging the high conversion
efficiency of parametric processes, they transfer the low-noise and high
powers achievable with near-infrared bulk-crystal solid-state pump sources
[33, 34] to the SWIR and mid-IR over a broad spectral range without having to
rely on spectral broadening [35, 36, 37]. To take advantage of these
capabilities, we have developed a new Yb:YAG dual-comb pump laser oscillator
which operates at a repetition rate of 250 and delivers $\approx 2~{}\unit{}$
average power per comb, which enables several hundred of signal and idler
power. The moderate comb-line spacing is able to resolve the absorption
features of various gases at low pressure (e.g. $<$100 ¯), and the combination
of picosecond pulse durations and hundreds of average powers yields more than
100 per comb line, compatible with high-sensitivity measurements.

While tunable OPOs offer high performance, a major concern is system
complexity since potentially four cavities are needed with corresponding
feedback loops and matched tuning servos to keep them mutually stabilized and
spectrally overlapped. Here we address this complexity issue by using single-
cavity dual comb concepts, a topic which has been discussed in recent reviews
[38, 39]. In the spatial multiplexing approach of [40], one of the flat intra-
cavity mirrors is replaced with a reflective biprism in order to create two
closed cavity paths (one for each comb). Tuning the lateral biprism position
adjusts the cavity round-trip time difference between the two combs. Here we
use this technique for both the laser and OPO cavities simultaneously.

A variety of shared-cavity approaches have been explored for dual-comb OPOs.
There are demonstrations based on two pump lasers pumping a single OPO cavity
with either two nonlinear crystals [41] or asynchronously with one nonlinear
crystal [42, 43]. These systems are partly stabilized and had limitations to
their sensitivity related either to decoherence over short time scales or to
low power per comb line [42, 43]. A notable recent demonstration showed a cw-
OPO cavity pumped by an electro-optic dual-comb source [44]. In previous work
[45], we demonstrated a simple free-running single-cavity approach for both
the pump laser and the OPO. In the OPO ring cavity with counter-propagating
beams, the intra-cavity dispersion led to a compensation of the optical path
length difference for the two combs via a shift of the center wavelength.
However, this reduced spectral overlap between the two combs limited the
repetition rate difference $\Delta f_{\mathrm{rep}}$ to the 100- range.

In this work, by using spatial multiplexing of a linear OPO cavity, we can
freely adjust the optical path length difference between the two resonant
signal beams in order to match their round-trip times with the inverse of the
repetition rate of their respective pump pulse trains. Accordingly, the
synchronous pumping condition can be satisfied for both combs simultaneously
at repetition rate differences $\Delta f_{\mathrm{rep}}$ of several kilohertz,
allowing high-resolution and high-sensitivity coherently-averaged
measurements. The OPO is tunable over a wavelength range 1300 to 1670 (signal)
and 2700 to 5000 (idler), with up to 580 (signal) and 320 (idler). To validate
that the configuration is compatible with coherent DCS, we conduct
spectroscopy measurements using the OPO signal combs around 1540 . Our results
show good agreement with the HITRAN database [46] and a high SNR of 42.3
within a measurement time of 2.15 (figure of merit of $3.2\times 10^{7}$ 1/2
[47]).

For practical spectroscopy measurements in the mid-IR, photodetection poses an
additional challenge. Mid-IR detectors have limited saturation power, and
typically exhibit lower quantum efficiency, lower speed, higher cost, and
higher noise compared to InGaAs detectors. Our new dual-comb OPO architecture
enables an elegant solution to this issue by using nonlinear up-conversion to
operate the OPO cavity as a spectroscopic dual-comb transceiver. To accomplish
this, we mix the idler of one comb with the intra-cavity signal of the other
comb, thereby efficiently up-convert the mid-IR spectroscopic information
(encoded in the free induction decay (FID) of the idler pulses) to the near-IR
in a time-gated fashion. Now the up-converted wave can be detected with a
standard InGaAs detector in a heterodyne scheme. Our approach builds on the
recently demonstrated cross-comb spectroscopy (CCS) technique [48] with two
new and critical milestones: use of a tunable dual-comb OPO (which also
removes the need for any spectral broadening stage), and the development of an
efficient and experimentally simple intra-cavity up-conversion method.

Rather than having to attenuate the mid-IR beam to avoid detector saturation,
we use the full $>$300- average power of the idler combs in this process. Due
to this fundamental boost in sensitivity compared to direct dual-comb
detection, we can obtain a high normalized SNR of 50.2 1/2 in a proof-of-
principle measurement of ambient air at 3320 . Also for the corresponding
figure of merit (FOM) [49, 26, 47] a high value of $3.5\times 10^{8}$ 1/2 is
achieved, which is well above the fundamental quantum-noise-limited value that
could be obtained by DCS using the same local oscillator power [26]. The high
sensitivity enables the trace-gas detection of methane at its natural
abundance of about 2 ppm over a path length of 3 . We conduct two further CCS
measurements across the OPO wavelength tuning range on a low-pressure
acetylene sample cell to demonstrate the broad spectral coverage and high
resolution that can be obtained with this system. These results represent, to
the best of our knowledge, the highest SNR and the highest FOM obtained to
date in high-resolution mid-IR DCS or CCS in a traveling-wave configuration
[10, 23, 50, 27, 21, 48, 22].

## Results

### Experimental set-up and operating conditions

The laser source consists of a dual-comb diode-pumped solid state laser (SSL)
from a single cavity pumping a dual-comb OPO from a single cavity (Fig. 1).
Both cavities are spatially multiplexed with an intra-cavity biprism that
introduces a spatial separation on the optically active elements while still
sharing all optical elements. Since we operate the cavities in free-running
mode, it is necessary to measure the phase fluctuations fast enough in order
to track them. There is a natural phase-sampling rate of $\Delta
f_{\mathrm{rep}}$ because a single carrier-envelope phase value can be
obtained from each dual-comb interferogram. Therefore, to conduct coherent
measurements without the aid of cw lasers [51], $\Delta f_{\mathrm{rep}}$
should be sufficiently high to sample the free-running fluctuations of the
dual-comb laser. The shared cavity approach leads to correlated fluctuations
of the combs, which enables phase tracking at smaller $\Delta
f_{\mathrm{rep}}$. This is beneficial since $\Delta f_{\mathrm{rep}}$ also
needs to be small enough to avoid spectral aliasing. The RF bandwidth occupied
by the interferograms is

$\displaystyle\Delta f_{RF}=\Delta\nu_{opt}\frac{\Delta
f_{\mathrm{rep}}}{f_{\mathrm{rep}}}$ (1)

where $\Delta\nu_{\mathrm{opt}}$ is the full optical bandwidth of the comb. To
avoid aliasing, this RF bandwidth must not exceed $f_{\mathrm{rep}}/2$. The
condition is fulfilled with an optical bandwidth constrained according to:

$\displaystyle\Delta\nu_{\mathrm{opt,max}}=\frac{f_{\mathrm{rep}}^{2}}{2}\frac{1}{\Delta
f_{\mathrm{rep}}}$ (2)

Therefore, high $\Delta f_{\mathrm{rep}}$ requires either increasing
$f_{\mathrm{rep}}$ or narrowing the optical bandwidth. Here, we choose a
configuration with moderate $f_{\mathrm{rep}}$=250 and picosecond pulse
durations for several reasons: (i) this provides high spectral resolution
without comb line interleaving, (ii) optical pulse parameters well-suited for
efficient frequency conversion, and (iii) high power per comb line enabling
high-sensitivity. Consequently, Yb:YAG is selected as a gain medium for the
OPO pump since its modest gain bandwidth naturally leads to transform-limited
pulses of several hundred femtoseconds duration. When at the same time the
tuning potential of an OPO is exploited, the spectrum can be positioned at a
specific set of absorption lines to efficiently capture the spectroscopic
signatures of various molecules over a broad wavelength range.

### Spatially-multiplexed dual-comb Yb:YAG laser

Figure 2:  Schematic of the dual-comb Yb:YAG laser and its pumping
arrangement. The laser is end-pumped by a multi-mode diode laser delivering
18.7 at 969 (DILAS). The pump mirror is dichroic for the diode pump light and
the solid-state laser light and serves as the output coupler (OC) with a
transmission rate of 3.45%. The combination of an undoped YAG Brewster plate
introducing self-phase modulation and dispersive mirrors introducing an
estimated round-trip group delay dispersion (GDD) of -13000 2 provides the
conditions for stable soliton modelocking. The biprism is mounted on a
translation stage to allow for tuning of the repetition rate difference
$\Delta f_{\mathrm{rep}}$.

The Yb:YAG pump laser (see Fig. 2) generates transform-limited picosecond
pulses via soliton modelocking with a semiconductor saturable absorber mirror
(SESAM) [52, 53]. A dual-comb oscillator configuration is realized by
inserting a biprism-like optic with a high-reflectivity coating as a turning
mirror into the cavity [40]. The biprism apex angle of 179 leads to a
horizontal mode center separation of 0.7 on the gain medium and 2.0 on the
biprism and the SESAM. Lateral translation of the biprism changes the cavity
length difference between the two combs, resulting in a tuning range of
$\Delta f_{\mathrm{rep}}$ from -100 to 3.9 without impairing the laser
performance. The laser uses a single multimode-fiber-coupled pump diode that
is split into two beams of equal power to pump each comb. The laser is
implemented in a robust aluminum housing to reduce its susceptibility to
environmental fluctuations. For comparable lasers, a stability of $\Delta
f_{\mathrm{rep}}$ in the range was obtained with the help of a slow piezo-
based feedback loop to correct for small long-term drifts in $\Delta
f_{\mathrm{rep}}$ that were measured with a frequency counter approach [40].
Figure 3 shows the performance of the laser at a nominal operation point at
which it pumps the OPO. The average output powers are 2.0 (comb 1) and 1.9
(comb 2), which corresponds to optical-to-optical efficiencies of 23% (comb 1)
and 22% (comb 2).

Figure 3:  Characterization of the single-cavity dual-comb Yb:YAG pump laser.
(a) Optical spectrum at a center wavelength of 1031 for comb 1 (blue) and comb
2 (red); resolution bandwidth (RBW): 0.08 , measured with a diffraction
grating-based spectrum analyzer (Keysight-Agilent 70951A).
(b) Second-harmonic generation autocorrelation traces with corresponding
deconvoluted full-width half-maximum (FWHM) pulse durations of 920 (comb 1)
and 860 (comb 2).
(c) RF spectrum of the pump laser. The nominal repetition rate difference is
$\Delta f_{\mathrm{rep}}$=3.4 at a repetition rate of 246.9 , where comb 1 is
defined to be at the lower repetition rate; RBW: 3 .

### Spatially-multiplexed dual-comb OPO

The OPO is synchronously and collinearly pumped by the two output beams of the
Yb:YAG laser. The OPO cavity layout is depicted in Fig. 4a. Analogous to the
SSL, the OPO is spatially multiplexed with a biprism (see Fig. 4c). This
allows to implement a linear cavity that is singly resonant for both signal
beams. The beams are spatially separated in the nonlinear crystal (see Fig.
4b), which avoids nonlinear coupling between the two combs. The round-trip
time difference between the two signal combs in the OPO cavity is modified by
tuning the biprism position transverse to the beam propagation direction.
Thus, the round-trip time of each signal comb can be matched to the inverse
repetition rate of each pump pulse train. By this means, both combs can be
synchronously pumped simultaneously without requiring a shift of the center
wavelength to compensate for the relative round trip delay from $\Delta
f_{\mathrm{rep}}$.

Figure 4:  (a) Schematic of the dual-comb OPO.
(b) Illustration of the spatial multiplexing in the PPLN crystal, where the
two combs are spatially separated in the direction vertical relative to the
optical table and interact with the same QPM grating period.
(c) Illustration of the spatial multiplexing on the biprism in the direction
vertical relative to the optical table.

The OPO is pumped with 1.7 (comb 1) and 1.6 (comb 2) in the periodically-poled
LiNbO3 (PPLN) crystal. Both combs have the same polarization to utilize the
$\mathrm{d}_{33}$ nonlinear coefficient. The measured pump 1/e2 radius at the
position of the crystal is 60 and the calculated mode radius for the signal
beam is 80 . The idler beam is transmitted by a 45 dichroic designed for
s-polarization: it has transmission $T>70$% from 2500-5000 and reflectivity
$R>$99.85% from 1290-1650 . The s-polarization requires the orientation of the
PPLN crystal as illustrated in Fig. 4b. To ensure both combs experience the
same quasi-phase-matching (QPM) grating period, the multiplexing of the OPO
cavity is implemented in the vertical, see Fig. 4b-c. The 4--long crystal, 5 .
% MgO doped, has a QPM fan-out grating with periods ranging from 24.8 to 31.2
(HC Photonics) and is operated at room temperature. The crystal is anti-
reflection (AR) coated on both facets with specifications $R<1\%$ for
1020-1040 and 1290-1620 , and $R<10\%$ for 3000-5000 . In order to optimize
the oscillation threshold to obtain high pump depletion, we use a wedged OC
with an out-coupling rate $T_{\mathrm{OC}}=12$% (Layertec, 1250-1700 ). At the
nominal operation point, the OPO runs 2.7x (comb 1) and 2.8x (comb 2) above
the oscillation threshold. The flat OPO end mirror is mounted on a motorized
picomotor translation stage to adjust the average cavity round-trip time.
Synchronous pumping for both combs simultaneously is obtained by adjusting the
end mirror and biprism positions.

A dispersive mirror adding -800 2 per round-trip (Ultrafast Innovations, R
$>99.8$%, 1325-1725 ) is included in the cavity. The increased magnitude of
the round-trip dispersion decreases the sensitivity of the center wavelength
to cavity length fluctuations. In a measurement where we tune the center
wavelength over the full tuning range and determine the required cavity length
change to maintain synchronicity, we estimate a total round-trip intra-cavity
GDD of -600 2.

The OPO cavity was optimized to support a small spatial separation between the
two signal combs as this leads to more correlated path length fluctuations and
hence low noise, thereby permitting operation at small $\Delta
f_{\mathrm{rep}}$. The cavity design, which maintains a small vertical
separation between the two combs below 1.6 throughout the cavity, required an
inverted biprism-like optic that reflects the incoming beams towards a common
optical axis (seeFig. 4c); the required optic was assembled by gluing two
separate angled pieces together.

### Characterization of the dual-comb system

In this section, we present the performance characteristics of the dual-comb
system. For the relative intensity noise (RIN) and SWIR spectroscopy
measurements, the OPO is tuned to about 1545 . The results of the OPO
characterization at this tuning point are summarized in Table 1.

Table 1: Summary of the characterization of the OPO output beams. Parameter | Unit | Signal 1 | Signal 2 | Idler 1 | Idler 2
---|---|---|---|---|---
Center wavelength, $\lambda_{0}$ |  | 1546.8 | 1546.7 | 3091 | 3093
Bandwidth, $\Delta\lambda$ (FWHM) |  | 5.4 | 6.2 | 20.6 | 21.9
Bandwidth, $\Delta\nu$ (FWHM) |  | 0.68 | 0.78 | 0.65 | 0.69
$P_{\mathrm{out}}$ |  | 580 | 585 | 320 | 322
Power per comb line |  | 212 | 186 | 122 | 116
RIN at $f_{\mathrm{offset}}>1$ |  c/  | -162.8 | -160.1 | - | -
|  |  |  |  |

To determine the quantum efficiency of the parametric amplification process,
we account for transmission losses experienced by the two beams on the way to
the power meter to estimate the average pump and idler powers inside the PPLN
crystal. This estimation indicates internal idler powers of 366 (for comb 1)
and 369 (for comb 2), yielding quantum efficiencies of $>$60%.

Another important performance aspect for high-sensitivity DCS measurements is
RIN. SSLs achieve ultra-low noise performance at the high offset frequencies
relevant for DCS measurements due to their powerful low-loss cavities and long
upper state lifetime gain media [54]. OPOs can inherit this property from
their pump laser [35, 36, 37], making them a compelling choice for low-noise
tunable sources. The noise characterization was performed on the signal beams
(see supplementary information). For both the pump laser and the OPO the
measured RIN is lower than –160 c/ for offset frequencies beyond 1 , which
corresponds to shot-noise limited performance given the illumination levels on
the photodiode.

Figure 5 displays the tuning range of the OPO accessed by scanning the lateral
position of the fan-out PPLN crystal. The tuning range for the idler reaches
from 2700 to 5000 (60-110 ) and for the signal from 1300 to 1670 (180-230 ).
Due to the dispersion in the cavity, its length is adjusted at each tuning
step to maintain synchronicity with the pump. The output power exhibits some
variation with respect to wavelength, which is likely due to a combination of
effects including wavelength dependence of the coatings (for
$\lambda_{s}<1350~{}\unit{\nano}$ and $\lambda_{s}>1600~{}\unit{\nano}$),
water absorption (for $\lambda_{s}$ from 1350-1415 ), and the parametric gain.

Figure 5:  Wavelength tunability of the OPO.
(a) Optical spectra of signal 1 (colored lines) and signal 2 (black dashed
lines). RBW: 0.08 , measured with a diffraction grating-based spectrum
analyzer (Keysight-Agilent 70951A). The corresponding power levels for signal
1 (circles) and for signal 2 (crosses).
(b) Optical spectra of idler 1 (colored lines) and idler 2 (black dashed
lines). The corresponding power levels for idler 1 (circles) and for idler 2
(crosses). The idler power is calculated to the position of the idler
transmitting dichroic by considering the losses from a coated Germanium
window, an uncoated CaF2 lens, and three silver mirrors. RBW: 0.09 , measured
with a Fourier transform-based spectrum analyzer (Thorlabs OSA205). The
spectra of the same color correspond to a signal-idler pair generated at a
distinct phase-matching configuration. The shaded areas indicate the water
absorption windows in the signal and idler tuning range.

### SWIR dual-comb spectroscopy of acetylene

In this section, we present a SWIR DCS measurement with the OPO signal combs.
The measurements are taken at the nominal OPO configuration as described
above. As illustrated in Fig. 6a, the two signal combs are combined on an
uncoated ultra-violet fused silica (UVFS) wedged window. The combined beams
are sent through a 14--long sample cell filled with acetylene (C2H2) at a
nominal pressure of 65 ¯.

The data structure in each channel corresponds to interferograms (IGMs): in
time intervals defined by $1/\Delta f_{\mathrm{rep}}\approx
300~{}\unit{\micro}$, the pulses from the two combs become temporally
overlapped and an interferometric signal is generated. To help with the idler
measurements that will be discussed in the next subsection, we wanted to
operate with $\Delta f_{\mathrm{rep}}$ as small as possible while still
supporting coherent averaging. The pump laser supports coherent averaging at
$\Delta f_{\mathrm{rep}}\approx 3~{}\unit{\kilo}$ due to its robust prototype
housing, and the OPO inherits its low timing jitter [37, 45]. However, since
the OPO is constructed on a breadboard setup it is more sensitive to noise and
would need $\Delta f_{\mathrm{rep}}\approx 10~{}\unit{\kilo}$. Therefore, to
operate at a $\Delta f_{\mathrm{rep}}$=3.4 while still sampling the OPO phase
fluctuations fast enough, we implemented the simple pulse stacking setup shown
in Fig. 6a. A small portion of the signal beams are split 50/50 on a beam
splitter, and one port is delayed by about a third of the round-trip cavity
length before being recombined with the other port, resulting in three IGM
patterns within each $\Delta f_{\mathrm{rep}}$ period. Details of the data
acquisition and the processing steps to make use of these additional IGMs are
explained in the Methods (see Fig. 11). The data from both channels (one with
and one without the pulse stacking) are recorded on a data acquisition card.

In the coherent averaging process, each 1/$\Delta f_{\mathrm{rep}}$ period $k$
of the data is frequency-shifted and interpolated onto an optical delay grid
of duration 1/$f_{\mathrm{rep}}$. This yields a complex array
$y_{\mathrm{k}}(t)$ for a data period $k$. The coherently averaged signal is
given by

$\displaystyle
y_{\mathrm{ETS}}(t)=\frac{1}{N}\sum_{k=1}^{N}y_{\mathrm{k}}(t).$ (3)

where ETS stands for equivalent time sampling. To obtain the noise floor we
use the same set of arrays, but summed with alternating sign so that coherent
signals cancel and only the noise remains:

$\displaystyle
y_{\mathrm{noise}}(t)=\frac{1}{N}\sum_{k=1}^{N}(-1)^{k}y_{\mathrm{k}}(t),$ (4)

where $N$ is implicitly even in Eq. 4. The DCS spectrum is given by
$S_{\mathrm{ETS}}(\nu)=|\mathscr{F}\left[y_{\mathrm{ETS}}\right](\nu)|$ and
the noise spectrum is given by
$S_{\mathrm{noise}}(\nu)=|\mathscr{F}\left[y_{\mathrm{noise}}\right](\nu)|$.
This approach to calculating the noise floor is a generalization of Fourier
transforming a long time trace consisting of multiple periods and looking mid-
way between the RF comb lines[17, 55].

Figure 6b shows the coherently averaged results from a 2.15- duration
measurement. The nominal noise floor is determined as the mean of
$S_{\mathrm{noise}}$ for frequencies within the FWHM bandwidth of
$S_{\mathrm{ETS}}$. We thereby obtain a peak sensitivity of
$\sigma_{H}=5.9\times 10^{-5}$, which corresponds to a normalized SNR of 40.7
1/2. The transmission dips due to absorption in the acetylene cell are also
clearly imprinted on the DCS spectrum. The full range of frequencies shown in
the figure, where the spectrum is above its noise floor, is $\approx
4.5~{}\unit{\tera}$. Given the repetition rate difference $\Delta
f_{\mathrm{rep}}$=3.4 , this translates to about 60 RF bandwidth (see Eq. 1),
which is significantly below $f_{\mathrm{rep}}$/2. Therefore, it was
straightforward to position the RF spectrum in a way to avoid spectral
aliasing.

In Fig. 6c the transmission features are converted to absorbance
$\alpha_{\mathrm{measured}}$ using the approach to infer $\alpha_{\mathrm{}}$
from $y_{\mathrm{ETS}}(t)$ described in the Methods. A comparison between
measured and predicted absorption $\alpha_{\mathrm{HITRAN}}$ is also shown.
The prediction is based on the HITRAN database [46, 56] and accounts for the
cell’s pressure $p$, temperature $T$, and an off-center beam path $L$ through
the sample cell that falls within the provided tolerances. As our measurements
are conducted without an optical reference or exact knowledge of $\Delta
f_{\mathrm{CEO}}$, we cannot independently link RF frequencies to optical
frequencies. Therefore, we rely on the unique pattern of the absorption
features to align the measured and the predicted absorbance by a center
frequency shift.

We obtain generally good agreement in the shape and strength of the absorption
features, as indicated by the residuals. There is a periodic disturbance in
the residuals that is not due to absorption. This effect is likely due to
intra-cavity beam height changes in the OPO cavity which lead to small
polarization rotations on the combs, since beam rotations can turn into
etalon-like delayed electric field components upon propagation through the
birefringent PPLN crystal. We used a relatively high OPO output coupling rate
(12%) to mitigate intracavity enhancement of these and any other etalon
effects. The residuals have a standard deviation of $7.7\times 10^{-3}$ for
the spectral range 193.2 -1 to 195.8 -1. This standard deviation becomes
$4.6\times 10^{-3}$ when the depolarization peaks in the above range are
excluded.

Figure 6:  Coherently-averaged 2.15-second-long DCS measurement of a low-
pressure acetylene (C2H2) sample cell (Thorlabs, GCA20) with nominal values
$L_{\mathrm{nom}}=14\pm 3$ and $p_{\mathrm{nom}}=66\pm 13$ ¯.
(a) Experimental set-up of the DCS measurement. The components are listed in
the Methods. (b) Measured transmission spectrum $S_{\mathrm{ETS}}$ (blue), and
the corresponding noise floor $S_{\mathrm{noise}}$ (green).
(c) Comparison of the predicted absorbance based on HITRAN data
$\alpha_{\mathrm{HITRAN}}$ (black) and the measured absorbance
$\alpha_{\mathrm{measured}}$ (blue). $T=22.8$ , $L=13.4$ , $p=67.9$ ¯.
Residuals (red), plotted with a vertical offset for better visibility, are
calculated by subtracting $\alpha_{\mathrm{HITRAN}}$ from
$\alpha_{\mathrm{measured}}$.
(d) Zoom-in to a single absorption line of acetylene to illustrate the
measurement resolution of 250 . Each marker corresponds to an optical
frequency comb line.

We calculate the FOM of this measurement according to Eq. (5) in [47] and
obtain a competitive FOM = $3.5\times 10^{7}$ 1/2. For the given average power
level and amplification the shot-noise is estimated to be 1440 2/. With 3000
comb lines ($\Delta\nu_{\mathrm{FWHM}}/f_{\mathrm{rep}}$), a detector noise-
equivalent power (NEP) of $2.0\times 10^{-14}$ /Hz1/2, RIN below -160 c/ at
the relevant offset frequencies, and noise contributions from the digitization
card of 1380 2/ (inferred from the datasheet), a theoretical sensitivity
$\sigma_{H}$ of $1\times 10^{-4}$ 1/2 (normalized SNR=$1/\sigma_{H}=40.0$ 1/2)
can be expected, where the noise contribution from the digitization unit and
shot-noise are of similar magnitude. The calculated normalized SNR is in good
agreement with the measured value.

### Intracavity cross-comb spectroscopy setup

Next, we discuss the mid-IR spectroscopy measurements and show how the high
average powers provided by the OPO can be directly translated into enhanced
sensitivity. To allow for a compact description of the new approach described
in this subsection, we denote the two pump combs as P-1 and P-2, the signal
combs as S-1 and S-2, and the idler combs as I-1 and I-2.

As mid-IR DCS measurements are typically limited by the low saturation power
and high noise of Peltier-cooled mid-IR detectors, direct mid-IR DCS does not
take full advantage of the low-noise and high-power idler beams the OPO
provides. To overcome this limitation we use CCS, which is a heterodyne up-
conversion technique first demonstrated in [48]. To help explain our new
approach to this technique, we first illustrate in Fig. 7a a minimal
implementation of CCS that is enabled by our tunable dual comb OPO. The
temporal profiles of the interacting optical waves are illustrated in Fig. 7b.
The detection process involves the following main steps:

  1. 1.

I-1 is sent through an absorbing gas, leading to temporally-delayed features
on the electric field (the so-called free induction decay, FID. See top of
Fig. 7b).

  2. 2.

I-1 is mixed with S-2 in a sum frequency generation (SFG) crystal, thereby
sampling the FID of I-1.

  3. 3.

The generated SFG wave $E_{\mathrm{\mathrm{SFG}}}$ has the same optical
carrier frequency as the pump. Therefore, for subsequent heterodyne detection,
it is combined on a beam splitter with a small portion of P-2 that acts as the
local oscillator (LO) wave.

  4. 4.

The combined beam (SFG and LO) is detected on an InGaAs photodiode. This
results in three terms: the intensity of the LO wave, the intensity of the SFG
wave, and the heterodyne beat signal corresponding to the product between the
SFG and LO electric fields. The heterodyne signal is the one of interest as it
contains the spectroscopic information.

A theoretical description of the CCS signal is provided in the Methods. The RF
spectrum of the photodiode signal is illustrated in Fig. 7c, and the
corresponding time-domain features are shown in Fig. 7d. The black lines
correspond to the intensity of the SFG wave, which has a strong time-domain
peak near $T=0$ and decays quickly away from $T=0$. The grey line indicates
the LO including its white shot noise contribution. The blue lines show the
heterodyne signal of interest, which is analogous to a dual-comb
interferometry signal. With our repetition rate difference of 3.4 , the
heterodyne signal and SFG signal are narrow-band enough to occupy separate
non-overlapping regions of the RF space without spectral aliasing issues, and
can therefore be easily extracted using either analog or digital band-pass
filters.

Figure 7:  Concept of the OPO-based CCS approach.
(a) Minimal implementation of CCS on the basis of two OPOs.
(b) Temporal profiles of the interacting waves. From top to bottom: the FID of
the idler trace; the signal pulse at a delay $T$ probing the corresponding
portion of the FID; the SFG wave from the product of the idler and signal wave
generated at delay $T$; the temporally overlapped P-2 LO.
(c) Components of the RF spectrum: the SFG intensity (black), the LO shot
noise (grey), digital band-pass filter (light grey), and the product between
the LO and SFG fields (dark blue; heterodyne signal).
(d) Conceptual illustration of the time-domain oscilloscope traces (compare
equivalences in c). The absorption features have negative delays with respect
to the center burst because we assume the idler FID is sampled by the “slower”
signal comb. Red: original trace from the SFG+LO mixing; Black: low-pass
filtered trace containing only the SFG part; Dark blue: band-pass filtered
trace containing only the SFG+LO heterodyne signal, which contains the
spectroscopic information.

Our dual-comb OPO enables a nearly ideal architecture for CCS, which is
depicted in Fig. 8a-b. The key idea is that I-1 is sent through an absorbing
target (step 1) and then simply routed back along the path of I-2. In this
configuration, I-1 passes back through the intra-cavity PPLN crystal at the
same position as I-2 but counter-propagating. As the collinearly-pumped OPO
cavity is linear, S-2 is also counter-propagating intra-cavity at the same
position. Hence, SFG occurs between I-1 and S-2 (step 2). This SFG process is
inherently efficient, phase-matched, aligned, and spectrally overlapped with
P-2 since it simply represents the reverse of the optical parametric
amplification process. Furthermore, since SFG corresponds to the time-domain
product between the electric fields of I-1 and S-2, the generated SFG wave is
always synchronized with the intra-cavity S-2 pulses. Given that the timing of
P-2 and S-2 is locked together by the OPO process, the SFG wave is also
temporally overlapped with the incoming P-2 pulses. Thus, a small residual
reflection of P-2 by the HTpump / HRsignal-coated surface of the input mirror
($R_{pump}<1$%) inherently functions as the P-2 LO beam (step 3). Thus, no
additional components or alignment is needed to obtain interference between
the SFG wave and the P-2 LO. In short, all six important constraints to obtain
the heterodyne beating are inherently satisfied (SFG phase-matching and
efficiency; LO generation; spatial, spectral and temporal overlap with the
LO). Since the (SFG+LO) beam of interest is counter-propagating to the
incoming pump light, an isolator is added to the setup to extract the SFG+LO
beam via the isolator’s rejection port. Note that a second beam is present
with the roles of the combs interchanged (i.e. P-1 acting as the LO instead of
P-2). This beam is blocked in our setup, but it could potentially be used to
improve the SNR by a factor of 2.

Figure 8:  (a) Side view of the experimental implementation of the CCS scheme.
Note that we only show a simplified illustration of the OPO cavity (compare
Fig. 4). After the dual-comb OPO cavity, a Germanium window blocks any signal
or pump beam components co-propagating with the output idler beams. Each idler
beam passes through the gas target either on its path from or towards the OPO.
Lenses are used to create a focus of each idler beam at a common position
serving as a point of symmetry where the idler propagation is reversed. Each
idler propagates on the path of the other idler beam to the intra-cavity PPLN
crystal where the idler and signal combs are mixed in an SFG process. At the
pump mirror, the SFG wave (dashed, dark blue) is interfered with a small
residual pump reflection serving as LO. The rejection port of a 5--aperture
isolator (InPut Optica, HPTG) is used to separate the backwards propagating
CCS signal from the incoming pump. One of the two comb paths is blocked, and
the other SFG+LO field is used for detection.
(b) Top-view of the detection set-up where the CCS wave is split by a
partially transmitting optic into a low-power (LP) and high-power (HP)
channel.

### CCS measurements

To show the ultra-high sensitivity measurements supported by combining our
dual-comb OPO with intracavity CCS detection, in this section we present a
measurement on ambient air with the idler tuned to 3320 to probe strong
methane absorption features. The light reflected by the isolator is split into
two free space paths: a high-power (HP) path containing $>$95% of the power,
and a low-power (LP) path that is attenuated to about 0.5%. The paths are
illustrated conceptually in Fig. 8b. Both paths are coupled into separate
single-mode fibers to implement a customized high-sensitivity detection
scheme. In the HP path, we measure a fiber-coupled LO average power of 4.65
and, when the I-1 and S-2 pulses are synchronous at a delay $T=0$, we estimate
a fiber-coupled SFG power of 55 (0.22 ).

The strong yet low noise signals in our setup present a unique measurement
challenge: a high photocurrent of order 100 must be detected while supporting
a high bandwidth $>$100 and a measurement noise floor of around 100 (to
resolve shot noise of the 4- LO beam). Most photodiodes cannot handle such a
large current without saturation, and high-speed data acquisition systems
generally do not have a dynamic range of $10^{6}$. To overcome these issues we
combine data from the LP and HP paths as follows. The LP path is attenuated in
order to accurately measure the center burst without saturation. The HP path
is configured so that the $T=0$ center burst is strongly saturated, but with
enough sensitivity to reach shot-noise limited performance away from $T=0$. To
avoid degrading the FID signal due to time-delayed saturation effects on the
HP channel, we configure the system so that the raw time trace is “backwards”
(see Fig. 7d), as this means that saturation-induced distortions occur at
negative optical delays and can be discarded. The two channels are coherently
averaged individually and then combined digitally to obtain the full signal
(composite trace). Details of the stitching process and the detection setup
are in the Methods. The coherently averaged LP and HP signals are shown in
Fig. 9a. The same terminology from Eqs. 3 and 4 are used for the signal and
noise floor, respectively. The amplitude and phase of the LP channel have been
carefully scaled to match the HP channel within the FID region, which permits
digital stitching of the two signals together. The stitched signal and its
noise floor are shown in Fig. 9b, which illustrates the decay of the FID
towards the signal’s noise floor.

Figure 9:  Coherently averaged CCS signals from the LP and HP channels.
(a) Magnitude of the LP, HP, and composite traces zoomed in to the center
burst. The peak of the LP channel is scaled such that its FID matches that of
the HP channel. The composite trace is derived by summing the product of the
$y_{\mathrm{ETS}}(t)=y_{\mathrm{LP}}(t)F_{\mathrm{stitch}}(t)+y_{\mathrm{HP}}(t)(1-F_{\mathrm{stitch}}(t))$.
(b) Magnitude of $y_{\mathrm{ETS}}$ (blue) and its noise floor
$y_{\mathrm{noise}}$ away from $T=0$ (green).

Figure 10 shows the spectroscopic features of the measurement. The spectrum
has an SNR of 40.2 ($\sigma_{H}=9.7\times 10^{-5}$) as shown in Fig. 10a.
Given the short measurement time of 10-, this corresponds to a high normalized
SNR of 50.2 1/2. Accounting for the spectral shape of the signal [47], we
obtain an ultra-high FOM of $3.5\times 10^{8}$ 1/2. The shot-noise limited
noise floor is shown by the grey dashed line, indicating that the measurement
is close to its fundamental sensitivity limit.

The absorption spectrum is determined as described in the Methods and is shown
in Fig. 10b. The measured spectrum is in good agreement with the HITRAN
database, as indicated by the small residuals. The standard deviation of the
residuals is $5.6\times 10^{-4}$ in a 1.2 wide segment centered about the
peak. To obtain the best overlap, we account for the temperature $T$, pressure
$p$, as well as the length $L$ of the idler beam path and the relative
abundances $F$ of the detected molecules: $T=25$ , $p=0.95$ ¯, $L=271$ ,
$F_{\mathrm{H_{2}O}}=0.01$, $F_{\mathrm{CH_{4}}}=2.5$ ppm. Figure 10c shows an
inset of the absorption spectrum in the region of the methane absorption band,
highlighting the trace-gas detection of methane at the 1 ppm level. The figure
shows a coherently averaged measurement over 10 , but the achieved sensitivity
is high enough such that the absorption features can already be clearly
resolved in a single IGM period of 300 . Figure 10d shows an inset of the
residuals in a spectral region where there is negligible absorption from water
vapor or methane. The directly calculated residuals are close to the noise
floor-limited residuals. This shows that even the absorbance, despite its
sensitivity to small distortions in the coherent signal from unsuppressed
etalons, has close to shot-noise limited sensitivity.

Figure 10:  Coherently-averaged 10--long CCS measurement of the ambient
laboratory air.
(a) Measured spectrum (blue) and its noise floor (green). The blue curve is
obtained from $y_{\mathrm{ETS}}(t)$ over the time range from -20 to 2800 . The
green curve is obtained from $y_{\mathrm{noise}}(t)$ over the full time range
except for the small region around $T=0$ where the stitching occurs at 5 .
Identical temporal filtering has been applied to $y_{\mathrm{ETS}}(t)$ and
$y_{\mathrm{noise}}(t)$ to ensure that the latter curve represents the noise
floor of the former curve. Averaged noise floor (black). Theoretical noise
floor (grey) calculated from shot noise of the LO, with a correction to
account for the reduced detector responsivity at the center RF frequency of
the CCS interferogram.
(b) Measured absorption spectrum (blue) compared with HITRAN calculations
(black). The residuals are shown by the red, which is displaced from 0 for
visibility.
(c) Zoom-in on the absorption near the methane features to highlight the
trace-gas detection aspects of the measurement. $\alpha_{\mathrm{HITRAN}}$ is
calculated with a sampling rate corresponding to the resolution of the
measured data.
(d) Zoom-in on the residuals where there are no water or methane features, for
comparison to the noise floor.

In addition to the above measurement of ambient air, which was chosen to
highlight the achievable sensitivity and trace-gas detection of methane, two
further measurements are presented in the Supplementary information. There, we
present measurements on a low-pressure acetylene cell at two different
wavelength tuning points (2960 and 3790 ) to show the high-resolution and
wavelength-tunability capabilities of the system.

## Discussion

We demonstrate a novel platform for high-resolution spectroscopy in the SWIR
and mid-IR spectral regions that combines a simple system architecture with
high sensitivity. Our results are based on a new light source consisting of a
tunable dual-comb OPO pumped by a picosecond dual-comb solid-state laser. The
laser and the OPO cavities use spatial multiplexing to generate both combs in
a single cavity. This means all the optical elements are shared both in the
laser cavity and in the OPO cavity, leading to high mutual coherence between
the two combs from a simple setup. The implementation of the spatial
multiplexing in the OPO cavity allows to synchronously pump both OPO combs
simultaneously at a repetition rate difference of several while maintaining
high spectral overlap between the two combs. Hence, phase tracking of the
heterodyne dual-comb signals and subsequent coherent averaging is possible
even in fully-free running operation of both cavities.

While maintaining a narrow instantaneous bandwidth, the OPO covers a broad
spectral range due to its tuning capability. The OPO signal tuning range is
from 1300 to 1670 , and the OPO idler tunes from 2700 to 5000 . This wide
spectral range in the SWIR and mid-IR allows for the investigation of
absorption features from numerous molecules. The high output power in
combination with the narrow spectral bandwidth results in high power per comb
line of up to 210 in the SWIR and 160 in the mid-IR. Furthermore, the dual-
comb system has low relative-intensity noise measured to be below -160 c/ at
megahertz offset frequencies. In the context of direct DCS measurements, the
high power of the OPO makes it particularly compelling for measurements
involving high losses such as long open paths or in broadband multi-pass
cells.

We carried out two spectroscopic measurement campaigns. In the SWIR
measurements, we employ the signal to show the compatibility of the OPO with
high-resolution and high-sensitivity direct DCS measurements. In the other
campaign, we focus on the idler to show the fundamental boost in sensitivity
achievable by operating the OPO as a spectroscopic transceiver while using the
full average power available.

The shot-noise limited SWIR measurements on a low-pressure acetylene cell had
a normalized SNR of 40.7 1/2 and a FOM of $3.5\times 10^{7}$ 1/2. For
comparison, a normalized SNR of 41 1/2 was obtained with electro-optic dual-
combs [57], but that system had fewer comb lines (115) and a limited tuning
range of 100 . In another experiment, a FOM of $7.2\times 10^{7}$ 1/2 was
obtained by sending 23.5 to a photodetector that is fast enough to allow for
correction of the introduced non-linearity [47]. The measurement in [47] was
limited by the RIN of the used laser source, so there is room for further
improvement by combining high-power detectors with bulk solid-state laser-
based light sources that are not limited by RIN.

In order to utilize the high idler powers provided by the OPO and overcome the
limitations posed by mid-IR detectors, we demonstrated a new cross-comb
spectroscopy scheme based on up-conversion between the idler and the intra-
cavity signal. The approach, which requires only a few extra parts compared to
the source itself, yielded a record-high FOM of $3.5\times 10^{8}$ 1/2 and
trace-gas detection of methane in ambient air. We showed coherently averaged
results, but the sensitivity is high enough to already resolve methane
absorption in a single interferogram period. Our results represent the first
implementation of OPO-based CCS and the first intracavity CCS architecture.
Here, the OPO is operated as a transceiver, such that the intracavity signal
automatically up-converts the idler free induction decay (FID) to
intrinsically be spectrally and temporally overlapped with the pump beam which
then can serve as the local oscillator. To the best of our knowledge, we
demonstrated the highest FOM for high-resolution mid-IR DCS without relying on
cavity enhancement [10, 23, 50, 27, 21, 48, 22]. The sensitivity far exceeds
the fundamental shot-noise limited performance that would be possible with
direct DCS using the same 4- local oscillator power. The improvement comes
from the time-gating effect of the up-conversion, where away from $T=0$, the
main idler pulses do not get up-converted and therefore they do not contribute
to shot noise [58]; the dominant contribution to shot noise is instead the
lower power P-2 LO beam. In fact, reaching similar performance in a direct DCS
configuration would require using the full mid-IR power (300 ) on a photodiode
with near-ideal quantum efficiency. This exceeds the capabilities of mid-IR
detectors by two to three orders of magnitude. One other important point is
that, while measurements with the OPO signal exhibited small spectral
modulations attributable to intracavity depolarization effects, these effects
are not visible on our measurements with the OPO idler since any time-delayed
components on the signal do not get efficiently down- or up-converted.

To take full advantage of the scheme we needed to split the measurements into
separate low- and high-power channels. This two-channel detection may be
necessary in general when targeting shot-noise-limited performance due to the
very stringent demands placed on the photodiode and electronics in terms of
power, dynamic range, and bandwidth. In our measurements, the two channels can
be stitched together with small residuals because we carefully calibrated the
electronic transfer functions of both channels. Also, the demands on the
matching of the two channels are reduced when the goal is to detect very weak
features resulting in a weak FID, which is our primary motivation. As soon as
high sensitivity is not critical, e.g. when sampling strong absorption
features, it is not necessary to push the measurement to its fundamental shot
noise limits. In that case, a single detector operated at moderate power would
be sufficient. For measurements using the full power of the OPO, further
sensitivity improvements could be obtained by OPO power scaling. Eventually,
the damage threshold of the photodiode or the electronic amplifier would set a
limit to the useable power, but there is no indication that we are close to
such a limit with the current measurements.

In conclusion, our results validate the use of tunable OPOs as an attractive
way forward for dual-comb spectroscopy applications. Future work will involve
extending to longer wavelengths and deploying the set-up for the detection of
trace-gas mixtures in multi-pass cells or longer air paths. The obtained SNR
would support combline-resolved measurements of absorbance at the $<1~{}$ppm
level after coherent averaging timescales of one minute. For measurements in
ambient conditions, higher repetition rates of around 1 could be used to
obtain even higher SNR. For high-resolution applications, the repetition rate
could be reduced, or spectral interleaving techniques could be deployed. The
potential of low-noise dual-comb solid-state lasers and OPOs to reach
repetition rates from below 100 to $>1$ while maintaining ultra-low noise
performance make them ideally suited to access this parameter space.

## Methods

### Theoretical description of CCS signal

In the following, we use the same notation as introduced in the experimental
results section. To extract an absorption spectrum from a CCS measurement, it
is useful to express the electronic signal in terms of the properties of the
combs and the sample. For this, we determine the charge generated per pulse on
the photodiode as a function of delay between the two mixing combs. The SFG
electric field $E_{\mathrm{SFG}}$ can be approximated as:

$\displaystyle
E_{\mathrm{SFG}}^{(n)}(t)=e^{in\delta_{is}}E_{\mathrm{I-1}}(t+T_{n})E_{\mathrm{S-2}}(t),$
(5)

where the idler field $E_{\mathrm{I-1}}$ implicitly contains the spectroscopic
information, $n$ represents a pulse index,
$\delta_{C}=\phi_{\mathrm{I-1}}+\phi_{\mathrm{S-2}}-\phi_{\mathrm{P-2}}$ is a
carrier envelope phase (CEP) shift term with $\phi_{\mathrm{j-k}}$ being the
CEP shift of wave $j$ of comb $k$ from one pulse to the next, and $T_{n}$ is
the delay between the combs at pulse pair $n$. The envelopes $E_{j}$ are the
positive-frequency electric fields of each wave with the CEP set to zero. To
obtain a photodiode signal, we interfere this SFG beam with a (small) replica
of pump P-2 (LO), which results in a time-varying photocurrent source varying
on ultrafast timescales:

$\displaystyle
i_{\mathrm{\mathrm{source}}}^{(n)}=\eta\left[\left|E_{\mathrm{SFG}}^{(n)}(t)\right|^{2}+\left|E_{\mathrm{P-2}}(t)\right|^{2}+2\mathrm{Re}\left(e^{in\delta_{C}}E_{\mathrm{SFG}}^{(n)}(t)E_{\mathrm{P-2}}^{*}(t)\right)\right]$
(6)

where $\eta$ is the photodiode responsivity. Since the photodiode has a “slow”
response, we end up integrating this photocurrent source to obtain a charge
per pulse-pair:

$\displaystyle
Q_{n}=Q_{\mathrm{SFG}}(T_{n})+Q_{\mathrm{pump}}+2\mathrm{Re}\left(Q_{\mathrm{CCS,n}}\right).$
(7)

Here, $Q_{\mathrm{SFG}}$ corresponds to a convolution between the envelopes of
the idler and signal fields; it is strong when the main pulses of each wave
are temporally overlapped ($T_{n}=0$). $Q_{\mathrm{pump}}$ is a constant term
corresponding to the pump pulse energy at the photodiode. The CCS term
$Q_{\mathrm{CCS}}$ is phase sensitive and given by:

$\displaystyle Q_{\mathrm{CCS,n}}=e^{in\delta_{C}}\int
E_{\mathrm{I-1}}(t+T_{n})E_{\mathrm{S-2}}(t)E_{\mathrm{P-2}}^{*}(t)dt,$ (8)

where the integration limits of infinity implicitly assume that neighboring
pulses in the train are neglected. To elucidate further, note that the signal-
pump product within the integrand constitutes a gate with which the idler
field is convolved. By approximating the phase and delay variables as
continuous (a valid assumption provided the repetition rate difference is
small enough to avoid aliasing) and multiplying by the repetition rate to
obtain a current, we obtain:

$\displaystyle i_{\mathrm{CCS}}(T)\approx f_{\mathrm{rep}}\exp(2\pi
i\nu_{CCS}T)\int
E_{\mathrm{I-1}}(t+T)E_{\mathrm{S-2}}(t)E_{\mathrm{P-2}}^{*}(t)dt,$ (9)

where the effective carrier frequency of this function on optical timescales
is $\nu_{CCS}=(f_{\mathrm{rep}}/\Delta
f_{\mathrm{rep}})(f_{\mathrm{rep}}\delta_{C}/(2\pi))$.

Performing a Fourier transform, we obtain:

$\displaystyle\tilde{i}_{CCS}(\nu)$
$\displaystyle=\tilde{E}_{I-1}(\nu-\nu_{CCS})\tilde{G}(\nu-\nu_{CCS})^{*}$
$\displaystyle=\tilde{E}_{I-1,init}(\nu-\nu_{CCS})\tilde{H}(\nu-\nu_{CCS})\tilde{G}(\nu-\nu_{CCS})^{*}$
(10)

where tilde denotes a frequency domain quantity, $\tilde{E}_{I-1,init}$ is the
idler electric field before passing the absorptive sample, $\tilde{H}$ is the
absorption transfer function, $\tilde{G}(\nu)=\mathscr{F}\left[G(t)\right]$,
and the gate $G(t)$ is given by:

$\displaystyle G(t)=E_{\mathrm{S-2}}(-t)^{*}E_{\mathrm{P-2}}(-t).$ (11)

Equation Theoretical description of CCS signal shows that the CCS spectrum
contains the spectroscopic information of the sample, weighted by the spectrum
of the idler and the effective gate pulse defined by the product of the signal
and pump of the other comb. Therefore, spectroscopic information that lies
within the bandwidth of the idler can be retrieved, provided the signal and
pump pulses are of comparable duration (which is usually the case for an OPO).

### Phase tracking with low $\Delta f_{\mathrm{rep}}$

Several data processing steps are needed to extract absorption values from the
raw data. The overall procedure consists of (i) IGM phase tracking, (ii)
coherent averaging, (iii) digital stitching (for CCS measurements only), and
(iv) HITRAN fitting, envelope extraction, and etalon removal. Here we discuss
step (i); steps (ii)-(iv) are discussed later in the Methods.

We operate at $\Delta f_{\mathrm{rep}}$=3.4 , which is large enough to
coherently average the pump but not the OPO signal [28]. The OPO-signal pulse
stacking setup of Fig. 6 mimics the role of a larger $\Delta f_{\mathrm{rep}}$
by sampling the phase fluctuations three times more rapidly.

Figure 11:  Example signal interferograms. (a) DCS interferogram used for
spectroscopy analysis. (b) Reference signal containing three interferograms
per $\Delta f_{\mathrm{rep}}$ period, used to effectively triple the phase
sampling rate.

More specifically, we obtain three arrays of phase samples
$\underline{\phi}_{j}$, where $\underline{\phi}_{j}[n]$ is the carrier
envelope phase (CEP) of pulse $j\in\\{1,2,3\\}$ within overall IGM period $n$.
The timing difference between these IGMs is stable but not known precisely.
Therefore, we use differences between their CEP values to determine the phase
fluctuations of the “main” IGM’s $\underline{\phi}_{1}$. To unwrap
$\underline{\phi}_{1}$ we calculate
$\underline{\delta}_{21}=\mathrm{unwrap}(\underline{\phi}_{2}-\underline{\phi}_{1})$,
$\underline{\delta}_{32}=\mathrm{unwrap}(\underline{\phi}_{3}-\underline{\phi}_{2})$,
and
$\underline{\delta}_{13}[n]=\mathrm{unwrap}(\underline{\phi}_{1}[n+1]-\underline{\phi}_{3}[n])$.
$\underline{\phi}_{1}$ is then obtained by taking the sum
$(\underline{\delta}_{21}+\underline{\delta}_{32}+\underline{\delta}_{31})$
and removing any overall $2\pi$ phase slope offset. This procedure works for
all the traces, so none have to be discarded. In general, the pulse stacking
approach could be avoided, or deployed successfully for much lower
$f_{\mathrm{rep}}$ systems, by using mechanically robust cavity assemblies
instead of breadboard setups.

For the idler data (i.e. the CCS measurements) an additional step is required
because the idler phase has fluctuations of similar magnitude to those of the
signal while also being influenced by the pump phase. To find the idler phase,
we calculate $\varphi=\mathrm{unwrap}(\phi_{i}+\phi_{s}-\phi_{p})$ on each IGM
period (which has sufficiently small fluctuations to unwrap without
ambiguities), and replace the directly calculated idler CEP values with
$\varphi-\phi_{s}+\phi_{p}$. This procedure preserves the original value
modulo $2\pi$ while removing phase-wrapping ambiguities. Now having an IGM
phase array for all channels, coherent averaging is enabled.

### Data processing of the interferometric measurements

Coherent averaging (step (ii)) proceeds as usual by interpolating the phases
as obtained in the previous Methods section on the whole time grid, frequency
shifting, and interpolating on the optical delay grid. The last step benefits
from the fact that the timing grid is inherited from the pump laser, which is
in a robust mechanical prototype with low timing jitter. Coherent averaging is
applied to all traces. For the CCS signals, the phase array is obtained from
the low-power (LP) channel, and that phase array is used to average both LP
and high-power (HP) channels.

The digital stitching step (step (iii)) involves identifying features on the
coherently averaged signals that are shared between the LP and HP channels.
For the HP channel, the limiting factor is saturation, while for the LP
channel, it is the noise floor. Therefore, the first few hundred picoseconds
of the free induction decay (FID) are suitable for this purpose as those
features are neither saturated in the HP channel nor within the noise floor of
the LP channel. Since the LP and HP electronics are not identical, we carried
out a measurement of their electronic transfer functions $H_{\mathrm{LP}}$ and
$H_{\mathrm{HP}}$ with a separate dual comb laser (a prototype version of the
one reported in [32]); this captures spectral oscillations arising from
reflections of the electronic signals. The LP channel is multiplied by
$H_{\mathrm{HP}}/H_{\mathrm{LP}}$ prior to stitching. The final output of the
above procedures is denoted $y_{\mathrm{ETS}}(t)$: it is a signal of duration
$1/f_{rep}$ whose Fourier series coefficients correspond to comb line pairs.

### Extracting the absorbance from DCS and CCS measurements

In this section, we describe the data processing performed on
$y_{\mathrm{ETS}}(t)$ to obtain absorbance data (step (iv)). We perform
measurements without a reference and therefore need to use
$y_{\mathrm{ETS}}(t)$ itself to infer the underlying absorption-free envelope
that gives rise to the FID. This process is optimized to reduce the residuals
between the measured data and the database prediction. The analysis has three
interconnected steps involving digital removal of the absorption features,
calculating an envelope, and removing “etalons”. Provided all strong
absorption features originate from gases whose lines are well tabulated in a
database, the FID can be removed accurately and without information loss:

$\displaystyle\tilde{y}_{\mathrm{corrected}}(\nu)=\tilde{y}_{\mathrm{ETS}}(\nu)H(\nu+\nu_{0};T,P,L)^{-1}$
(12)

where $\nu$ is the optical frequency variable of the coherently averaged
signal and $\nu_{0}$ a frequency shift parameters used to match the observed
absorption features to the predicted ones (i.e. to estimate absolute optical
frequencies). Other fit parameters included in the absorption transfer
function $H(\nu)=\exp(-\alpha_{\mathrm{}}(\nu)L)$ are temperature $T$,
pressure $p$ and sample length $L$. Once the spectroscopic features are well
fit, $y_{\mathrm{corrected}}(t)$ has weak or negligible FID, and a few-
picosecond-long time gate $g(t)$ can be applied to it to obtain an envelope;
we use a hyperbolic tangent shaped filter for this purpose:

$\displaystyle y_{\mathrm{env}}(t)=y_{\mathrm{corrected}}(t)g(t)$ (13)

The time gate ensures that no sharp spectroscopic features are left, and the
FID removal before applying the gate suppresses background fluctuations
arising from convolution of the gate with the absorption features. The
(complex) absorption can now be obtained by division:

$\displaystyle\alpha_{\mathrm{}}(\nu+\nu_{0})=-\ln\left(\frac{\tilde{y}_{\mathrm{ETS}}(\nu)}{\tilde{y}_{\mathrm{env}}(\nu)}\right)$
(14)

Initially, $\nu_{0}$ and the environmental parameters of the transfer function
$H$ are not known accurately. The calculated absorption
$\alpha_{\mathrm{}}(\nu+\nu_{0})$ can be used to estimate these parameters as
well. Etalons are removed coherently from the input to Eq. 12 by applying an
inverse optical transfer function approach with amplitude, phase, and delay
parameters adjusted to suppress the time-domain peaks. Some etalons that are
far from the FID region are removed with a time gate.

### Detection system used for the DCS and CCS measurements

In the DCS measurements, the combined combs are detected by a fast photodiode
(modified Thorlabs, DET10N2). The photodiode is connected to an 80 bias-tee
(DC output terminated with 50 Ohm) followed by a 98- lowpass filter, a 30-dB
amplifier (ENA-220T, RFBay), and analyzed with a 14-bit data-acquisition card
(Spectrum instrumentation, M4i.4451-x8).

To maximize the SNR of the CCS measurement, we split the CCS signal in a LP
and a HP part. The HP trace is configured to measure the FID with high
sensitivity, but the center burst is strongly saturated. Conversely, the LP
trace is configured to capture the center burst information that is lost in
the HP trace (including its carrier-envelope phase for use in coherent
averaging).

For the LP trace, we attenuate the light rejected by the isolator by a factor
of about 200 to avoid saturation and detect the fiber-coupled beam with a
gigahertz photodiode (DET01CFC). The attenuation is implemented with a wedged
laser output coupler ($T<5\%$) followed by an ND filter. The electronic signal
generated by the photodiode is first split with an SMA-T piece. One port is
terminated with a 50-Ohm resistor, while the other is low-pass filtered
(passband frequency 140 ), amplified by 30 (ENA-220T, RFBay), and then
connected to the oscilloscope (WavePro 404HD, Teledyne Lecroy) by a short BNC
cable. The termination resistor directly after the SMA-T is used instead of a
bias tee in order to reduce oscillations on the electronic transfer function
that arise due to reflections from the low-pass filter.

The HP path uses the full power available (apart from $<5\%$ split off for the
LP path). The light is fiber coupled and sent to a higher-power photodiode
(modified Thorlabs DET10N2). The DC part of the signal is split off with a
bias-T (cut-off frequency 80 ) and terminated with a 50-Ohm resistor. The AC
port of the bias-T is connected to a high-pass filter (passband frequency 27.5
) followed by a low-pass filter (passband frequency 98 ), and then through a
23-dB amplifier (LNA-580T, RFBay). The resulting center burst far exceeds the
saturation level of the amplifier (which is $\approx\pm 3~{}\unit{}$), but is
within the absolute maximum ratings. On the oscilloscope, the vertical zoom
level is optimized to reach as high sensitivity as possible while staying
below the absolute maximum input voltage limits. Little center burst
information is left on the HP channel, but the FID is resolved to shot-noise
limited sensitivity. Subsequent stitching of the LP and HP traces yields the
underlying signal that could be obtained with an ideal high dynamic range
photodiode and electronics chain.

Funding This work was supported by a BRIDGE Discovery Project Nr.
40B2-0_180933 a joint research programme of the Swiss National Science
Foundation (SNSF) and Innosuisse – the Swiss Innovation Agency. This project
has received funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (966718).

Acknowledgements The authors thank Walter Bachmann from the ETH Physics
Department Engineering office for the mechanical design of the dual-comb
solid-state laser aluminum housing.

Contributions C.P. developed the concept of the project. C.B. planned and
built the dual-comb OPO cavity. J.P. planned and built the dual-comb solid-
state laser. B.W. and J.P. assisted in experimental optimization steps of the
laser. B.W. developed the inverted-biprism. M.B. explored suitable
experimental configurations of the OPO. C.B. and Z.B. carried out the final
OPO alignment and performed the non-spectroscopic experiments. C.P. developed
the computer program for coherent averaging and adaptive sampling of the
spectroscopy measurements, and developed the electronics set-up for the CCS
measurements. C.B., C.P., and Z.B. performed the spectroscopy measurements
with the OPO. C.P., Z.B., and C.B. carried out the data analysis of the DCS
and CCS measurements. C.B. and C.P. wrote the manuscript. C.P. and U.K.
supervised the project. All authors discussed the results and read and
approved the final manuscript.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper is
available at ETH Zurich Research Collection library [59].

## Supplementary Information

### Low-pressure CCS measurement

To demonstrate the suitability of the source for high-resolution measurements,
we conducted two more measurements on a 65-¯ acetylene cell where the
absorption features have a width $\sim 1~{}\unit{\giga}$. Firstly, the idler
center wavelength is set to 3792 , at the wings of a strong acetylene
absorption feature. In this spectral band also water has some weak absorption
lines. The results of this measurement are shown in Fig. 12a-c. This
measurement has a SNR of 36.5 ,$\sigma_{H}=2.2\times 10^{-4}$ for a 10--long
measurement, corresponding to 46.6 1/2, and a FOM of $1.3\times
10^{8}~{}\unit{}^{1/2}$. The SNR is limited by the available idler power,
which is reduced because the sample cell is not ideally suited for mid-IR
measurements: it has fused silica windows which are absorbing in the
wavelength range $>$2400 . The measurement shows good agreement with the
HITRAN database. The standard deviation of the residuals is $7.1\times
10^{-4}$ in a 1.2 wide segment centered about the peak.

In a second measurement, the idler center wavelength is set to 2959 , where
acetylene has a weak absorption feature. The results of this measurement are
shown in Fig. 12d-f. The measurement has a SNR of 40.1 , $\sigma_{H}=9.7\times
10^{-5}$, corresponding to 50.2 1/2, and a FOM of $3.4\times
10^{8}~{}\unit{}^{1/2}$. The measurement shows good agreement with the HITRAN
database, as well. The standard deviation of the residuals is $1.6\times
10^{-3}$ in a 1.2 wide segment centered about the peak.

Figure 12:  CCS measurements on a low-pressure acetylene cell. Left column:
Measurement around 3792 . Right column: Measurement around 2959 .
(a) Measured spectrum (blue) and its noise floor (green). The blue curve is
obtained from $y_{\mathrm{ETS}}(t)$ over the time range from -20 to 2000 . The
green curve is obtained from $y_{\mathrm{noise}}(t)$ over the full time range
except for the small region around $t=0$ where the stitching occurs at 6 .
(b) Measured absorption spectrum (blue) compared with HITRAN calculations
(black). The residuals are shown by the red curve, which is displaced from 0
for visibility.
(c) Zoom-in on the absorption features to highlight the good agreement of the
measurement with HITRAN. $\alpha_{HITRAN}$ is calculated with a sampling rate
corresponding to the resolution of the measured data.
Corresponding information for (d-f). The blue curve is obtained from
$y_{\mathrm{ETS}}(t)$ over the time range from -50 to 2800 . The stitching
occurs at 20 .

To obtain the best overlap with the HITRAN data base, we account for the cell
temperature, deviations from the manufacturer-specified pressure, as well as
an off-center beam path through the sample cell that falls within the provided
tolerances. In the CCS measurements, we also account for atmospheric
absorption of methane and water. A summary of the relevant parameters can be
found in Tab. 2 for all spectroscopic measurements presented. The nominal
values for the cell are: $L_{\mathrm{nom}}=14\pm 3$ and
$p_{\mathrm{nom}}=66\pm 13$ ¯

Table 2: Summary of the parameters for the HITRAN-based absorbance calculation for all spectroscopic measurements performed. Parameter | Symbol | Unit | DCS 1540  | CCS 2959  | CCS 3320  | CCS 3792
---|---|---|---|---|---|---
Cell pressure | $p_{\mathrm{cell}}$ |  ¯ | 67.9 | 78.5 | - | 79
Cell temperature | $T_{\mathrm{cell}}$ |  | 23 | 42.9 | - | 43.9
Cell distance | $L_{\mathrm{cell}}$ |  | 13.4 | 13.2 | - | 13.6
Air pressure | $p_{\mathrm{air}}$ |  ¯ | - |  $\approx$950 |  $\approx$950 |  $\approx$1000
Air temperature | $T_{\mathrm{air}}$ |  | - | 23 | 25 | 23
Air distance | $L_{\mathrm{air}}$ |  | - | 2.68 | 2.71 | 2.68
Fraction $\mathrm{H}_{2}$O | $F_{\mathrm{H_{2}O}}$ | % | - | 1.01 | 1.05 | 1.52
Fraction $\mathrm{CH}_{4}$ | $F_{\mathrm{CH_{4}}}$ | ppm | - | - | 2.5 | -
|  |  |  |  |  |

We attribute the elevated temperature and pressure of the acetylene cell in
the CCS measurements to the strong absorption of UVFS at the idler wavelength.
The fractions for water and methane are found by keeping the interaction
length of the idler with the atmosphere fixed to a value found by measuring
the path length with a ruler. All measurements have been conducted on
different days such that variations in the air humidity can be expected. For
the measurements using the acetylene cell, the optical alignment and cell
placement have changed in between the measurements, such that the exact path
through the cell was not preserved.

### Coherence of the SWIR DCS measurement

In Fig. 13, we show how the SNR of our measurement scales by
$\sigma_{H}\propto T^{-1/2}$ [26], where $T$ is the measurement time.

Figure 13:  SNR as a function of the measurement time $T$ (blue crosses),
compared to the SNR of a single IGM scaled with $\sqrt{T}$ (red).

### Relative intensity noise measurements

The RIN measurement scheme, which is adapted from the one we described in
[40], uses two measurements (one for low noise frequencies and one for high
noise frequencies) and stitches them together to get high sensitivity across
the whole frequency range. The baseband noise on the photocurrent generated by
an InGaAs photodiode is measured on a signal source analyzer (SSA) (Keysight
Technologies E5052B). To illustrate the relation of the pump and OPO RIN, the
measurements of both the pump beams and the signal beams are given. For the
RIN measurements, the following average powers were sent to the photodiode:
6.9 (pump 1), 5.4 (pump 2), 4.6 (signal 1), and 3.0 (signal 2). The RIN power
spectral density (PSD) for both the SSL and the OPO spanning from 10 to 10 is
displayed in Fig. 14a, and the corresponding integrated RIN spectrum is shown
in Fig. 14b. We see that the RIN reaches the measurement noise floor at about
1 . For both the pump laser and the OPO, the RIN is lower than –160 1/2 beyond
1 offset frequencies. For the OPO, the integrated RIN is 0.031% (signal 1) and
0.030% (signal 2) when integrated from 10 to 10 . Due to the absence of
available data in literature, a comparison of the RIN performance to other
dual-comb OPO approaches is not possible.

Figure 14: Intensity noise characterization of the single-cavity dual-comb
system based on baseband measurements for the Yb:YAG laser (dash-dotted: pump
1 in blue, pump 2 in red) and for the signal beams (solid: signal 1 in blue,
signal 2 in red). To determine the measurement noise floor, the optical input
to the photodiode was blocked and the resulting power spectral density was
scaled according to the typical incident power in the unblocked case (4.5 ).
The resulting effective RIN noise floor is at around -175 1/2 (grey); it is
well below the measured RIN values. (a) RIN-PSD. The straight solid lines
indicate the shot noise limit for the signal measurements: -163 1/2 (signal 1,
blue), -160.5 1/2(signal 2, red). (b) Integrated RIN-PSD with the integration
interval from 10 to 10 . Stitching frequency: 220 .

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