of 9
Article
https://doi.org/10.1038/s41467-024-53447-3
Quadratic-soliton-enhanced mid-IR
molecular sensing
Robert M. Gray
, Mingchen Liu, Selina Zhou, Arkadev Roy, Luis Ledezma
&
Alireza Marandi
Optical solitons have long been of in
terest both from a fundamental per-
spective and because of their applica
tion potential. Both cubic (Kerr) and
quadratic nonlinearities
can lead to soliton formation, but quadratic solitons
can practically bene
fi
t from stronger nonlinearity and achieve substantial
wavelength conversion. However, despi
te their rich physics, quadratic cavity
solitons have been used only for broadband frequency comb generation,
especially in the mid-infrared. Here, we show that the formation dynamics of
mid-infrared quadratic cavity solitons, speci
fi
cally temporal simultons in
optical parametric oscillators, can be
effectively leveraged to enhance mole-
cular sensing. We demonstrate signi
fi
cant sensitivity enhancement while cir-
cumventing constraints of tradition
al cavity enhancement mechanisms. We
perform experiments sensing CO
2
using cavity simultons around 4
μ
mand
achieve an enhancement of 6000. Additi
onally, we demonstrate large sensi-
tivity at high concentrations of CO
2
, beyond what can be achieved using an
equivalent high-
fi
nesse linear cavity by orders of magnitude. Our results
highlight a path for utilizing quadratic
cavity nonlinear dynamics and solitons
for molecular sensing beyond what can be achieved using linear methods.
Since their discovery, optical solitons
1
,
2
have been the subject of
intense study due to the rich physics underlying their dynamics
3
6
,
relying on a delicate interplay of linear and nonlinear effects, as well as
their broad application in areas including low-noise frequency
synthesis
7
,astronomy
8
, and spectroscopy
9
,
10
, among others. Quadratic
solitons
11
14
can bene
fi
t from the inherent strength of the quadratic
nonlinearity, which relaxes the requirement on resonator
fi
nesse or
pump power for achieving soliton formation, as well as the ability to
achieve ef
fi
cient conversion between disparate spectral bands.
Temporal simultons are one such quadratic soliton, which consist
of a co-propagating bright-dark soliton pair at the fundamental and
second harmonic frequencies, respectively
15
,
16
. More recently, cavity
simultons have been demonstrated in synchronously-pumped degen-
erate optical parametric oscillators (OPOs) operating in the high-gain,
low-
fi
nesse regime
17
. Such temporal cavity simultons are shown to have
several favorable properties including broader bandwidths, which
increase with increasing pump power, and higher ef
fi
ciencies. When
extended to the mid-infrared (mid-IR) regime
18
, where many important
molecules have their strongest absorption features
19
, these properties
make the simulton OPO a powerful frequency comb source for mole-
cular sensing and spectroscopy.
In this work, we utilize the formation dynamics of quadratic
cavity simultons for molecular sensing, in particular, the uniquely
high sensitivity of simulton formation to the intracavity loss (Fig.
1
a,
b). In a proof-of-principle experiment sensing CO
2
in a 1.2-m-long
OPO operating in the simulton regime at around 4.18
μ
m
18
,we
measure an equivalent path length enhancement of up to 6000 and
additionally show a maximum sensitivity at concentrations of CO
2
as
high as atmospheric levels that is orders of magnitude larger than
what can theoretically be achieved through linear methods using a
source of equivalent power and bandwidth to the output of our
broadband OPO. We additionally extend our experimental results to
estimate a detector-limited normalized noise equivalent absorption
(NEA) of 1.05 × 10
10
cm
-1
/
ffiffiffiffiffiffi
Hz
p
for realistic system parameters.
Finally, we use numerical simulation to investigate the unique
dynamics responsible for this sensing behavior and show the
Received: 16 March 2024
Accepted: 14 October 2024
Check for updates
Department of Electrical Engineering, California Institute of Technology, Pasadena, CA, USA.
e-mail:
rmgray@caltech.edu
;
marandi@caltech.edu
Nature Communications
| (2024) 15:9086
1
1234567890():,;
1234567890():,;
potential of the method to achieve high linearity across a dynamic
range of 10
7
.
Sensing based on simulton formation dynamics enables a funda-
mentally different scaling behavior compared to typical linear absorp-
tion sensing following the Beer-Lambert Law
20
, as illustrated in Fig.
1
c, d.
In particular, although passive cavity enhancement
21
23
can offer extre-
mely high sensitivities at low analyte concentrations, the dynamic range
is limited. For example, recent works
24
have demonstrated normalized
NEA values on the order of 10
-13
cm
-1
/
ffiffiffiffiffiffi
Hz
p
, while their dynamic range is
constrained to about 4 orders of magnitude
25
if not extended through a
frequency
26
,
27
or path-length multiplexed
28
approach. By contrast, cavity
soliton dynamics can achieve high sensitivity and signi
fi
cant signal
enhancement even at large sample concentrations, thereby promising
precision and extended dynamic range for mid-IR gas sensing while
avoiding the typical requirements of high-
fi
nesse and high-power
operation. Furthermore, in contrast to intracavity laser absorption
sensing techniques
29
34
, cavity-simulton enhancement mitigates the
limitations in sensitivity imposed by spontaneous emission and dif
fi
-
culty in measuring the low signal powers required for near-threshold
operation
29
. Moreover, simultons can be achieved at arbitrary wave-
lengths, paving the way towards a un
iversal molecular sensing scheme,
especially in wavelength ranges where lasers are not readily available.
Results
Theory of cavity simulton formation
The simulton solution may be readily derived from the coupled wave
equations describing a degenerate traveling-wave optical parametric
ampli
fi
er for a pump at frequency 2
ω
and a signal at frequency
ω
,
where only the nonlinear interaction and walk-off are considered (see
Supplementary Note 7)
15
17
. The evolution of the bright-dark soliton
pair in the signal and pump
fi
elds,
E
ω
and
E
2
ω
, in the crystal is given by:
E
ω
ð
z
,
t
Þ
=
a
ffiffiffiffiffi
2
τ
p
sech
t

T
τ

,
ð
1a
Þ
E
2
ω
ð
z
,
t
Þ
=

E
2
ω
,0
tanh
t

T
τ

:
ð
1b
Þ
Here,
E
j
,
j
ε
{
ω
,2
ω
}, is the
fi
eld amplitude of the
j
th
wave,
E
2
ω
,0
is the pump
amplitude,
a
is the simulton signal amplitude,
T
is the simulton cen-
troid position, and
τ
is the simulton pulse width. We may use the
nonlinear manifold projection method to
fi
nd the evolution of
T
,
τ
,and
a
35
. We assume an OPO with a nonlinear crystal of length
l
,atotalcavity
length of
L
, a round-trip group delay of
Δ
T
RT
, and a round-trip loss of
coef
fi
cient of
α
ω
=
α
R
+
α
samp
+
α
oth
,where
α
R
is the loss due to the
output coupler re
fl
ectivity
R
=
e

α
R
L
,
α
samp
is loss due to the sample,
and
α
oth
encapsulates all other round-trip losses. Then, the evolution of
the system from the
n
throundtriptothe
n
+ 1th round trip is given by:
T
ð
n
+1
Þ
=
Δ
T
ð
n
Þ
+
T
ð
n
Þ
+
Δ
T
RT
,
ð
2a
Þ
τ
ð
n
+1
Þ
=
τ
0
,
ð
2b
Þ
a
ð
n
+1
Þ
=
a
ð
n
Þ
e
κ
E
2
ω
,0
l

α
ω
L
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1+
a
ð
n
Þ
2
a
2
sim
ð
e
2
κ
E
2
ω
,0
l

1
Þ
r
,
ð
2c
Þ
where
a
sim
is the steady-state simulton amplitude,
τ
0
is the simulton
pulse width,
κ
is the nonlinear coupling coef
fi
cient, and
Δ
T
ð
n
Þ
=

τ
0
2
ln
ð
1+
a
ð
n
Þ
2
a
2
sim
ð
e
2
κ
E
2
ω
,0
l

1
ÞÞ
is the simulton centroid shift due to
nonlinear acceleration caused by pump depletion and subsequent
back conversion.
From these equations, we see the requirements for stable cavity
simulton formation, as depicted in Fig.
1
b. Firstly, we see that achieving
a non-zero steady-state amplitude requires that the gain balance the
loss; here, this demands. Secondarily, the simulton centroid shift,
Δ
T
,
must balance the round-trip group delay due to the cavity length
detuning,
Δ
T
RT
. This allows the simulton to re-synchronize with the
pump, such that signal pulse circulation time
T
circ
=
T
RT
+
Δ
T
RT
+
Δ
T
equals the pump repetition period,
T
rep
. Further, we see the inter-
dependence of these two conditions, as achieving a suf
fi
cient timing
advance to compensate the round-trip delay requires a suf
fi
cient
amount of gain and pump depletion, which results in the dynamics
responsible for the simulton sensing mechanism.
Principle of simulton enhancement
The simulton-based sensing mechanism exploits the interplay
between energy and timing in the simulton regime to attain high
sensitivity to the sample of interest. The theoretical principle of
threshold sensing as leveraged by the simulton sensing mechanism is
depicted in Fig.
2
a. For a given pump power, the addition of a small
amount of loss due to the sample causes a threshold increase, resulting
in a corresponding decrease in the output power,
Δ
P
.Theabsolute
change in power is proportional to the local slope ef
fi
ciency at the
sensing point, meaning a higher slope ef
fi
ciency results in a higher
sensitivity.
Fig. 1 | Enhanced sensing using quadratic cavity simultons. a
Schematic depic-
tion of sensing in the simulton regime of a synchronously-pumped optical para-
metric oscillator at degeneracy. The bright soliton in the signal interacts with the
sample every round trip, and the resulting competing nonlinear dynamics generate
the measured signal response.
b
Speci
fi
cally, stable simulton operation requires the
simulton group advance,
Δ
T
, to balance the round-trip group delay,
Δ
T
RT
,andthe
parametric gain to balance the sample loss,
α
samp
, and output coupling.
c
Schematic
representation of linear absorption sensing governed by the Beer-Lambert Law for
light interacting with a sample over a path length
L
.
d
Linear methods (light blue
region) face limitations in the achievable sensitivity at high sample concentrations.
In contrast, active cavity sensing with quadratic cavity (orange) can achieve high
sensitivities at high sample concentrations.
T
rep
, pump repetition period;
T
circ
,pulse
circulation time in the cavity;
Δ
T
, simulton group advance;
T
RT
, cold cavity round-
trip time
Δ
T
RT
, round-trip group delay;
χ
(2)
, second-order susceptibility;
ω
, angular
frequency;
α
samp
, sample absorption coef
fi
cient; OC, output coupling;
P
in
,input
power;
P
out
, output power;
L
, path length;
, reduced Planck
sconstant.
Article
https://doi.org/10.1038/s41467-024-53447-3
Nature Communications
| (2024) 15:9086
2