Supplemental Document
Optical-parametric-amplification-enhanced
background-free spectroscopy: supplement
M
INGCHEN
L
IU
,
R
OBERT
M. G
RAY
, A
RKADEV
R
OY
,
L
UIS
L
EDEZMA
,
AND
A
LIREZA
M
ARANDI
∗
Department of Electrical Engineering, California Institute of Technology, Pasadena, California, 91125,
USA
∗
marandi@caltech.edu
This supplement published with Optica Publishing Group on 17 May 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.25655223
Parent Article DOI: https://doi.org/10.1364/OL.520848
1
Supplementa
l
Materials for
Optical
-
parametric
-
amplification
-
enhanced
background
-
free spectroscopy
Mingchen Liu
1
, Robert M. Gray
1
, Arkadev Roy
1
, Luis Ledezma
1
, and Alireza Marandi
1, *
1
Department of Electrical Engineering
,
California Institute of Technology, Pasadena, C
alifornia,
91125, USA
*
m
arandi@caltech.edu
Full architecture of OPA
-
BFS
Supplementary
Fig. 1.
Optical
-
parametric
-
amplification
-
enhanced background
-
free spectroscopy (OPA
-
BFS). (a) short pulse generation. BPF: bandpass filter.
BPF and pulse shaper may be required to change and control the profile and pulse width of the original pump pulse because th
e short
-
pulse OPA in (c) may
need a pump pulse with a longer pulse width and
a
different profile. (b) Linear interferomet
ry
. While a Michelson
-
like interferometer is illustrated here, a
Mach
-
Zehnder
-
like interferometer can also work.
F
or clarity
,
we only p
resent the most important components of the interferometer
;
more details, especially
regarding
dispersion compensation and delay locking, can be found in Ref
[1
–
3]
. (c) Short
-
pulse OPA.
The
full
architecture of OPA
-
BFS is presented in
Supplementary Fig. 1
, which is composed of three parts: short pulse generation,
linear interferometry, and short
-
pulse OPA. While OPA
-
BFS does not require any specific technique for the pulse generation,
p
anel
(a) illustrates a sub
-
harmonic
optical paramet
ric oscillator
(OPO)
synchronously
pumped by a short
-
pulse mode
-
locked
laser (typically a fiber laser), which is a common way to generate short mid
-
IR pulses
[4
–
8]
. One important advantage of
synchronously
-
pumped OPO
s is that the timing and phase of signal pulses and pump pulses are intrinsically locked, which can
exempt additional efforts in
their control for the short
-
pulse OPA
[9]
. The second step is to use the signal pulses (generally
mid
-
IR) to interrogate the sample with a detection background suppressed by linear interferometry, as illustrated in
panel
(b).
The output of the interferometer consists of two parts
, the residual pulse center (background) which cannot be fully eliminated
by the interferometer and the
subsequent
FID signal which carries the spectral information of the sample.
Note that we make a
very short and clean separation between
the
excitation p
ulse (center) and FID radiation
for clarity of the illustration, which is
not
always
the case in practice. However, this will not influence our analysis and arguments
,
as there will always be part of
the
FID radiation that is far enough from the excitation
pulse center
and thus
can be separated well.
Compared to the residual
background (originally the excitation pulse) that is much more localized in the time domain (pulse width of ~10
-
100 fs), the
FID signal can typically last at least hundreds of ps and so
metimes have a local maxima at a relative delay of 10
-
100 ps
[
10
–
13]
. This can be understood equivalently in the frequency domain; while femtosecond pulses can have a bandwidth as broad
2
as tens of THz, a typical vibrational absorption has a linewidth on the order of magnitude of only 10 GHz
at
room
temperature
and
atmospheric pressure
, which can be even smaller at lower pressure or temperature. The output of the i
nterferometer
is then
sent to a short
-
pulse OPA (
Panel
(c)) as the signal to be amplified. The pump pulse is held at a chosen delay with respect to the
si
gnal
such that it overlaps with a strong portion of the FID but is
far away
from the excitation center.
Therefore, the FID
carrying useful sample signatures is amplified while the residual background is not, as it does not temporally overlap with t
he
pump
pulse. This can further improve the SNR of the absorption spectrum and make a trace sample detectable that cannot be
detected by DAS or linear BFS.
Here
,
we show the illustration of a
n
OPA based on nanophotonic periodically
-
poled lithium
niobate (PPLN) waveguides
[14,15]
,
which
was
recently demonstrated with unprecedented high
gain and broad bandwidth.
However, it can also be any other platform or material that can support short
-
pulse OPA with
high
parametric gain.
Theoretical model
of BFS based on linear interferometry
To further demonstrate OPA
-
BFS quantitatively, we conduct theoretical analysis and numerical simulation for different
schemes. The mathematical models for BFS of linear interferometry and noise analysis are based on ref
[1
–
3,16]
. Specifically,
some important parameters are adapted from a recent state
-
of
-
the
-
art experimental result reported in
re
f
[3]
, including the field
unbalanced factor
훿
=
10
−
2
and RIN ratio
휎
푟
=
10
−
2
. These numbers are very close to the experimental results in
[3]
but with
a simpler value for ease of presentation. More importantly, for linear BFS, the model using those two parameters gives a
th
eoretical LOD of absorbance equal to
훿
휎
푟
=
10
−
4
, which agrees with what is experimentally demonstrated in ref
[3]
.
The
amplitude
s
of
electric field
in the
sample arm (
“A
rm
1”
, subscript “spa”
)
and reference
arm (“Arm 2”
, subscript “rfa”
)
are
described as:
퐸
푠푝푎
=
퐸
0
푒
−
퐴
2
(
1
)
퐸
푟푓푎
=
−
퐸
0
(
1
+
훿
)
(
2
)
In the equation of
퐸
푠푝푎
,
A
denotes a small absorbance (
퐴
≪
1
)
.
In the equation of
퐸
푟푓푎
, t
he
negative
sign
denotes a
휋
phase
change
(destructive interference)
, and
훿
is
the
field unbalanced
factor, which is assumed real in this work.
Note that we assume
a
small
absorbance
퐴
≪
훿
.
Also, we
assume
a power
푃
0
=
푐
퐸
0
2
, where c is a proportional constant
and will be omitted in the
following derivation.
Therefore, the total optical power enter
ing
the interferometer is
approximately
2
푃
0
.
At the output of the
interferometer, the
amplitude and power of the combined field
are:
퐸
=
퐸
푠푝푎
+
퐸
푟푓푎
=
−
퐸
0
(
1
−
푒
−
퐴
2
+
훿
)
≅
−
퐸
0
(
퐴
2
+
훿
)
(
3
)
푃
=
퐸
2
=
퐸
0
2
(
퐴
2
+
훿
)
2
≅
푃
0
(
훿
2
+
퐴훿
)
(
4
)
Therefore, in BFS, the
optical power
incident on
the
detector
of
the
reference (
without sample, A=0
)
and absorbed
measurement
s
is:
푃
푟푒푓
퐵퐹푆
=
푃
0
(
훿
2
)
(
5
)
푃
푎푏푠
퐵퐹푆
=
푃
0
(
훿
2
+
퐴훿
)
(
6
)
The absorption signal, i.e., the difference between these two measurements, is
푃
푠
퐵퐹
푆
=
푃
0
(
퐴훿
)
(
7
)
Similarly, for
direct absorption spectroscopy (DAS),
we have:
푃
푟푒푓
퐷퐴푆
=
푃
0
(
8
)
푃
푎푏푠
퐷퐴푆
=
푃
0
푒
−
A
≅
푃
0
(
1
−
퐴
)
(
9
)
푃
푠
퐵퐹푆
=
푃
0
퐴
(
10
)
As for the noise
,
3
푃
푁
=
√
푃
퐷푁
2
+
푃
푆푁
2
+
푃
푅퐼푁
2
=
√
푃
퐷푁
2
+
푃
푡
ℎ
휈
(
Δ
푓
)
−
1
+
(
푃
푡
휎
푟
)
2
(
11
)
Here,
푃
퐷푁
,
푃
푆푁
, and
푃
푅퐼푁
denote
the
detector noise
(DN)
, shot noise
(SN)
, and
relative intensity noise (RIN), respectively.
푃
푡
denotes
the average total power incident on the
detector
,
and
휎
푟
denotes
the RIN
.
ℎ
is Planck
’s
constant,
휈
is the optical
frequency, and
Δ
푓
is the measurement
bandwidth
(
reciprocal
of the measurement
time for
each
spectral element
)
.
In this work
,
that
shot noise is negligible compared to
the
other two kinds of noise
under our assumptions
.
On one hand,
this is
generally the case for less advanced detectors, especially in the MIR region. On the other hand, this agrees with
the result
of
some previous expe
rimental works, for example,
r
ef.
[3]
.
Next, we define
a
“detectable
signal” as
having a
SNR=1
(
푃
푠
=
푃
푛
)
. Hence,
for BFS
at
high
power
s
,
when
RIN
dominates
,
we
have
:
푃
푠
=
푃
0
(
퐴
퐿푂퐷
퐵퐹푆
훿
)
=
푃
푁
=
푃
0
(
훿
2
)
휎
푟
(
12
)
퐴
퐿푂퐷
퐵퐹푆
=
훿
휎
푟
(
13
)
For DAS, we have
푃
푠
=
푃
0
퐴
퐿푂퐷
퐷퐴푆
=
푃
푁
=
푃
0
휎
푟
(
14
)
퐴
퐿푂퐷
퐷퐴푆
=
휎
푟
(
15
)
The a
bove
derivation shows that
, in the RIN
-
limited regime,
푃
푠
퐵퐹푆
=
훿
푃
푠
퐷퐴푆
and
푃
푁
퐵퐹푆
=
훿
2
푃
푁
퐷퐴푆
,
giving
the conclusion
퐴
퐿푂퐷
퐵퐹푆
=
훿
퐴
퐿푂퐷
퐷퐴푆
.
In other words, i
n linear BFS,
while the absorption signal
will be lowered by δ, the background will be
suppressed by δ
2
as will the RIN. Therefore, the SNR can be increased by 1/δ
.
This shows the
SNR
advantage of BFS in the
RIN
-
limited regime, as
훿
is generally
very small (
훿
≪
1
)
, and also highlights the
benefit
s of reducing the field unbalanced
factor or at least suppressing its effect
.
Simulation
Model and P
arameters
The simulation of OPA is based on solution of the coupled wave equations using the Fourier split step method
[
4,17]
, the
parameters of which are based on a recent experimental demonstration of a high
-
gain OPA by thin
-
film lithium niobate
nanophotonics
[14]
. The absorption of molecules is modeled based on data from the HITRAN database
[18]
, using a Lorentz
oscillator model for the line profile.
3.1
Simulation Model
Simulation of the optical parametric amplification (OPA) process
between the pump at frequency 2
ω
and
signal at frequency
ω
in the nonlinear crystal is based on solving the coupled wave equations
as in
ref
.
[4]
,
which are
given as:
휕
푧
퐸
휔
(
푧
,
푡
)
=
휅
퐸
2
휔
퐸
휔
∗
−
훼
휔
2
+
퐷
̂
휔
퐸
휔
(
16
)
휕
푧
퐸
2
휔
(
푧
,
푡
)
=
−
휅
퐸
휔
2
−
훼
2
휔
2
−
Δ
훽
′
휕
푡
퐸
2
휔
+
퐷
̂
2
휔
퐸
2
휔
(
17
)
where we have taken the time coordinate, t, to be co
-
moving with the signal. Additionally, the pump envelope phase has been
shifted by
π
/2 to ensure real
solutions if higher orders of dispersion are not considered.
In
these equations, the subscripts
ω and
2ω are used to denote the signal and pump, respectively.
퐸
푖
, i
ε
{ω,
2ω}, denotes the
electric field amplitude
, normalized such
that
|
퐸
푖
|
2
gives the instantaneous power.
The nonlinear coupling coefficient is given
by
휅
=
√
2
휂
0
휔
푑
푒푓푓
/
(
푤
푛
휔
√
휋
푛
2
휔
푐
)
,
where
휂
0
is the impedance of free space
,
푑
푒푓푓
is the
effective nonlinearity,
푤
is t
he beam radius
where a symmetric Gaussian
mode has been assumed
,
푛
푖
is the refractive index, and
푐
is the speed of light.
훼
푖
is the absorption coefficient
, which accounts
for losses incurred during propagation in the
waveguide
.
The group velocity mismatch between pump and signal is given by
Δ
훽
′
. Finally,
퐷
̂
푖
=
∑
[
(
−
푖
)
푚
+
1
훽
푚
(
푚
)
푚
!
]
∞
푚
=
2
is the dispersion operator, which descri
bes the material dispersion experienced by the
pump and signal during propagation in the waveguide.
In our simulation, we consider
up to third order dispersion.