Supplemental Document
Cavity-enhanced dual-comb spectroscopy in the
molecular fingerprint region using free-running
quantum cascade lasers: supplement
C
HARLES
R. M
ARKUS
,
1
J
AKOB
H
AYDEN
,
2
D
ANIEL
I. H
ERMAN
,
2
P
HILIP
A. K
OCHERIL
,
1
D
OUGLAS
C. O
BER
,
1
T
ERMEH
B
ASHIRI
,
1
M
ARKUS
M
ANGOLD
,
2
AND
M
ITCHIO
O
KUMURA
1,
∗
1
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena,
California 91106, USA
2
IRsweep AG, Stäfa, 8712, Switzerland
∗
mo@caltech.edu
This supplement published with Optica Publishing Group on 15 November 2024 by The Authors
under the terms of the Creative Commons Attribution 4.0 License in the format provided by the
authors and unedited. Further distribution of this work must maintain attribution to the author(s)
and the published article’s title, journal citation, and DOI.
Supplement DOI: https://doi.org/10.6084/m9.figshare.27034477
Parent Article DOI: https://doi.org/10.1364/JOSAB.534286
Cavity-enhanced dual-comb spectroscopy in
the molecular fingerprint region using free-
running quantum cascade lasers:
supplemental document
1. Characterization of the PZT mounted cavity mirror
Knowledge of the piezoelectric transducer (PZT) position throughout a sweep was needed in
order to accurately determine the cavity finesse from the transmission profiles. A simple
interferometer was used to characterize the PZT, as shown in Fig. S1. Measurements were
performed using the same optical mount that was used for the cavity-enhanced measurements.
A 650 nm diode laser was sent to a 50/50 beam splitter, where half was reflected towards the
PZT-mounted mirror 6 cm away, and the transmitted light reflected off of a static mirror. The
two paths recombined on the splitter and were sent to a Si detector.
Fig. S1. Layout of the interferometer used for the PZT characterization.
The same conditions used during measurement, a 9 V
pp
triangle wave with a 5 V offset, were
sent to the PZT controller. This is multiplied by 15 with high voltage PZT driver and was used
to drive the PZT. The applied voltage and detector output were recorded on an oscilloscope.
The results are shown in Fig. S2. As the voltage increases the interference fringes squeeze
together indicating that it is moving faster. Since the optical path length (OPL) changes by
twice the distance, the peaks of each fringe were used to mark 325 nm of mirror displacement.
Fig. S2. Results from the interferometer measurement. Above: Voltage applied to the PZT.
Below: The signal from the photodiode. The transmission peaks used to determine the velocity
are shown as red dots.
Fitting the peak positions to a 5th degree polynomial provides the mirror position at each point
in time. Then, the first derivative provides the velocity, both are shown in Fig. S3. The mirror
traverses 4.225
휇
m over the sweep, and the velocity is initially -100
휇
m/s and increases to -300
휇
m/s.
Fig. S3. Above: Position as a function of time over the sweep of the PZT. Red dots are the
measured positions from counting fringes, the blue line represents a fit to a 5th order
polynomial. Below: The first derivative with respect to time of the polynomial, representing
mirror velocity.
In the bow-tie cavity, the change in OPL is also twice the distance the mirror moves. In the data
analysis shown in the main text, the OPL is set to be twice the distance shown in the polynomial
fit in Fig. S3. This is what ultimately determined the mirror reflectivity which allowed for
quantitative measurements.
2. Finesse of a bow-tie cavity
For a bow-tie cavity, the cavity transmission function
푇
was shown in the main text to be:
푇
=
푡
2
푟
푚
―
2
푒
―
푖휔
푐
푑
1푚
1
―
푟
4
푒
―
푖휔
푐
푑
Finesse is defined as the cavity the free spectral range (FSR) divided by the mode width
Δ
휈
. To
get the transmitted power,
푇
is multiplied by its complex conjugate
푇
∗
. With
푟
2
=
푅
and
푡
2
=
푇
,
this gives:
푇
푇
∗
=
푇
2
푅
푚
―
2
1
+
푅
4
―
푅
2
(
푒
―
푖휔
푐
푑
+
푒
푖휔
푐
푑
)
=
푇
2
푅
푚
―
2
1
+
푅
4
―
푅
2
(
cos
휔
푐
푑
)
Then, when the cavity is near resonance (
휔
푐
푑
=
2휋푚
), the approximation
cos
푥
≈
1
―
푥
2
2
is used
and the substitution
휔
=
2휋휈
gives:
푇
푇
∗
≈
푇
2
푅
푚
―
2
(
1
―
푅
2
)
2
+
푅
2
휔
푐
푑
2
=
푇
2
푅
푚
―
2
/(
푅
2
2휋푑
푐
2
)
(
1
―
푅
2
)
2
푅
2
2휋푑
푐
2
―
휈
2
Which is a Lorentzian line shape with linewidth:
Δ
휈
=
1
―
푅
2
휋푅푑/푐
=
퐹푆푅(1
―
푅
2
)
휋푅
by considering the FSR =
푐
푑
when the index of refraction
푛
=
1
. Then the finesse which is equal
to FSR/
Δ
휈
is given by:
퐹
=
휋푅
1
―
푅
2
≈
2휋
1
―
푅
4
which is what is given in the main text.