A carbon dioxide (CO2) Differential Absorption Lidar (DIAL) for accurate CO2 concentration measurement requires a frequency locking system to achieve high frequency locking precision and stability. We describe the frequency locking system utilizing Frequency Modulation (FM), Phase Sensitive Detection (PSD), and Proportional Integration Derivative (PID) feedback servo loop, and report the optimization of the sensitivity of the system for the feed back loop based on the characteristics of a variable path-length CO2 gas cell. The CO2 gas cell is characterized with HITRAN database (2004). The method can be applied for any other frequency locking systems referring to gas absorption line

Bai, Yingsin

Beyon, Jeffrey

Chen, Songsheng

Kavaya, Michael J.

Koch, Grady

Petros, Mulugeta

Petzar, Paul

Singh, Upendra N.

Trieu, Bo

Yu, Jirong

NASA Technical Reports Server

 OPTIMIZATION OF A 2-MICRON LASER FREQUENCY STABILIZATION SYSTEM FOR A 
DOUBLE-PULSE CO2 DIFFERENTIAL ABSORPTION LIDAR 
Songsheng Chen1, Jirong Yu2, Yingxin Bai1, Grady Koch2, Mulugeta Petros3,  
Bo Trieu2, Paul Petzar4, Upendra N. Singh2, Michael J. Kavaya2, Jeffrey Beyon2 
1SSAI, One Enterprise Parkway, Suite 200, Hampton, VA 23666 USA 
757-864-1105, songsheng.chen-1@nasa.gov 
2NASA Langley Research Center, MS 474, Hampton, VA  23681 USA 
3STC, 10 Basil Sawyer Drive, Hampton, VA  23666 USA 
4National Institute of Aerospace, Hampton, VA 23666 USA 
 
ABSTRACT 
A carbon dioxide (CO2) Differential Absorption Lidar 
(DIAL) for accurate CO2 concentration measurement 
requires a frequency locking system to achieve high 
frequency locking precision and stability. We describe 
the frequency locking system utilizing Frequency 
Modulation (FM), Phase Sensitive Detection (PSD), 
and Proportional Integration Derivative (PID) feedback 
servo loop, and report the optimization of the sensitivity 
of the system for the feed back loop based on the 
characteristics of a variable path-length CO2 gas cell. 
The CO2 gas cell is characterized with HITRAN 
database (2004). The method can be applied for any 
other frequency locking systems referring to gas 
absorption line.  
1. INTRODUCTION  
Active sensing of carbon dioxide (CO2) in the 
atmosphere, such as a pulsed Differential Absorption 
Lidar (DIAL) or a continuous wave Laser Absorption 
Spectrometer (LAS), has become a very promising 
technique for accurate measurement of the CO2 
concentration, which has a significant contribution to 
the global warming and climate change. The active 
sensing of the CO2 concentration promises a full day 
and night coverage during all seasons and a quantitative 
study of CO2 surface sources and sinks [1]. Similar to 
all the pulsed differential absorption lidars, a double-
pulsed 2-micron differential absorption lidar CO2 under 
development requires frequency-stabilized and narrow 
linewidth continuous wave (CW) seeder lasers as 
injection seeding sources. It has been theoretically 
estimated that the frequency variation for the CO2 
differential absorption lidar at 2 micron has to be within 
~2 MHz, or frequency stability has to be less than ~1E-
8, to achieve CO2 concentration with the accuracy of 
one ppm [2].  
2. BASIC PRINCIPLE AND THEORY 
2.1 Frequency modulation 
The single frequency 2-micron laser beam is directed 
into a single frequency modulator made of 
MgO:LiNbO3 and the output laser beam is modulated. 
In case of a simple sine-wave modulation, the 
modulation signal is given, 
{ }mmtmtm ϕπν += 2sin}{ 0                (1) 
where m0, νm, and ϕm are the amplitude, frequency, and 
initial phase of the harmonic modulation signal 
respectively. 
The electric field of the laser beam from the frequency 
modulator, can be expressed, 
{ }( ){ } ..2sinexp}{ 00 cctMtiEtE mmc ++++= ϕπνϕπν       (2) 
where E0, νc, and ϕ0 are the amplitude, frequency, and 
initial phase of the laser carrying beam respectively (νm 
<<νc). M is the modulation index related to the 
amplitude, the frequency, and the efficiency of the 
modulation. Under a normal weak modulation, or 
M<<1, the E{t} in (2) can be expanded as a Fourier 
series, 
..)}})(2(exp{
)})(2({exp{
22
)}2(exp{
2
..)}})(2(exp{
)})(2(}{exp{{
2
)}2(exp{}{
2
..)})(2(exp{}{
2
}{
0
0
0
0
0
0
01
0
00
0
0
0
ccti
tiM
E
ti
E
ccti
tiMJ
E
tiMJ
E
ccntniMJ
E
tE
mmc
mmc
c
mmc
mmc
c
mmc
n
n
+−+−−
++++
+=
+−+−−
++++
+≈
++++= ∑
+∞
−∞=
ϕϕννπ
ϕϕννπ
ϕπν
ϕϕννπ
ϕϕννπ
ϕπν
ϕϕννπ
 
                    (3) 
where Jn{M} is Bessel Function (first kind and integer 
order n)  and the terms of order M2 and higher are 
dropped.  
To lock the laser frequency to the center of a specified 
CO2 gas absorption line, the modulated laser beam is 
directed into a variable path-length CO2 gas cell and the 
laser frequency is tunable within the width of the CO2 
gas absorption line with a driving of the feedback servo 
loop. The wavelength of the CO2 absorption line is 
2.050967 µm.  
https://ntrs.nasa.gov/search.jsp?R=20100026002 2019-08-30T10:24:29+00:00Z
 2.2 CO2 gas cell 
For optimization of the frequency locking purposes, a 
variable pass-length CO2 gas cell is used and the gas in 
the cell is refillable to different pressures. For the 
frequency-locking purpose, the CO2 gas cell is 
characterized in amplitude attenuation,δ, and phase 
shift, ϕ, similar to a frequency modulation spectroscopy 
[3], due to the resonant absorption and resonant 
dispersion at the specified absorption line. The δ and ϕ 
are defined for three frequencies, (νc-νm), νc, and 
(νc+νm) as 
);(
2
;
2
;
2
;
2
mc
mc
mc
mc
mc
cc
c
c
c
c
LnL
c
LnL
νν
π
ϕ
α
δ
πν
ϕ
α
δ
±==
==
±
±
±
±
      (4) 
where c is the speed of light, αc-m, αc, and αc+m, are 
intensity absorption coefficients and nc-m, nc, and nc+m, 
are indexes of refraction for the laser beams at three 
different frequencies. The intensity absorption 
coefficients and the indexes of refraction are directly 
related to the specified absorption line shape of the CO2 
gas in the cell. The electric field of the beam after 
passing through the cell can be expressed by, 
..)}})(2(exp{)}(exp{
)})(2(exp{)}({exp{
22
)}2(exp{)}(exp{
2
}{
0
0
0
0
0
cctii
tiiM
E
tii
E
tE
mmcmcmc
mmcmcmc
ccc
+−+−+−−
++++−+
++−=
−−
++
ϕϕννπϕδ
ϕϕννπϕδ
ϕπνϕδ
                (5) 
The intensity of laser beam to be detected is  
)}2)2(2cos(}exp{
2
)2)2(2cos(}exp{
2
)2cos(}exp{
)2cos(}exp{
)224cos(1}{2exp{
2
1)(
8
}{}{
8
|}{|
8
}{
0
0
0
2
0
*2
mccmmcmcc
mccmmcmcc
mcmcmmcc
mcmcmmcc
ccc
tM
tM
tM
tM
tEc
tEtEctEctI
−−
++
−−
++
−−++−−+
−−+++−+
−++−+
−++−+
−++−≈
==
ϕϕϕϕννπδδ
ϕϕϕϕννπδδ
ϕϕϕπνδδ
ϕϕϕπνδδ
ϕϕπνδ
π
ππ
  
                   (6) 
where the terms of order M2 are ignored due to M<<1. 
Generally, a photo-detector can only response to signals 
from zero to radio frequency (rf), including modulation 
frequency, so the terms, νc, 2νc-νm, 2νc+νm, are 
averaged to a direct current (dc) signal. The detected 
signal is proportional to the intensity of the laser beam 
and can be separated into two signals; one is a dc signal 
and the other a signal at modulation frequency. The 
output current signal from the photo-detector at the 
modulation frequency is more interesting for a phase 
sensitive detection and can be expressed by 
 
)}2sin()2cos({
2
)(
8
}{ sincos
2
0mod mmmm tt
MEctS ϕπνϕπν
π
+−+∝ FF  
)}sin(}exp{
)sin(}}{exp{2exp{
)}cos(}exp{
)cos(}}{exp{2exp{
sin
cos
mccmcc
mccmccc
mccmcc
mccmccc
−−
++
−−
++
−−+
−−−=
−−+
−−−=
ϕϕδδ
ϕϕδδδ
ϕϕδδ
ϕϕδδδ
F
F
                  (7) 
or 
 
)}2sin()2cos({}{ sincosdetmod mmmm ttMCtS ϕπνϕπν +−+= FF  
                           (8) 
where Cdet is a constant of the photo-detector and 
related to the optical power incident on the photo-
detector, the quantum efficiency, the area, and gain of 
the photo-detector, as well as the energy of the 
modulation wave. The terms, cosF  and sinF , are 
determined by the specified absorption line and the line 
shape of the CO2 gas in the cell. 
2.3 Phase sensitive detection 
Phase sensitive detection technique is an effective way 
to recover the small signal buries in lager ambient 
noises by narrow bandwidth amplification. It requires a 
phase sensitive detector, or lock-in amplifier, to realize 
extremely narrow bandwidth detection, normally 1e-3 
Hz, through a phase-locked loop. By multiplying a local 
reference signal, )2cos(}{ 0 LLL tStS ϕπν += , where S0, 
νL, and ϕL are the amplitude, frequency, and phase of 
the local reference signal respectively, on the photo-
detector signal, equation (8), and applying a low-pass 
filter to the multiplied signal, a phase-sensitive dc 
signal is achieved especially when νL =νm,   
)}sin()cos({
2
1
sincos0det mLmLpsd MSCS ϕϕϕϕ −+−= FF
     (9) 
Typically, 
 
2
;
2
1
0;
2
1
sin0det2
cos0det1
π
ϕϕ
ϕϕ
=−==
=−==
mLpsd
mLpsd
MSCSS
MSCSS
F
F              (10) 
These dc signals are directly proportional to the 
amplitude of the detected modulation signal and can be 
used to lock the frequency of the laser beam based 
on
cosF  and sinF , the characteristics of CO2 absorption 
line. Phase dependency of the phase-sensitive dc signal, 
in case of 
2
,0 πϕϕ ≠− mL , can be eliminated by a so-
called dual-phase lock-in technique, in which another 
local reference signal, )2sin(}{ 0
'
LLL tStS ϕπν += , 
 which has only a phase shift of 
2
π
 from the first one, is 
applied to photo-detector signal, equation (8), 
separately. The second PSD signal is, 
)}sin()sin({
2
1
sincos0det
'
mLmLpsd MSCS ϕϕϕϕ −−−= FF     (11) 
so the dual-phase lock-in signal, which eliminates the 
phase dependency, will be, 
2'2 )()( psdpsd SSS +=               (12)
  
2.4 Calculation of S1, S2, S 
For the purposes of comparison and optimization of our 
current frequency-locking system [4], the calculation is 
limited to a modulation frequency of 175 MHz.  A 
Gaussian absorption line shape is a good approximation 
of the characteristics of the CO2 gas cell with a pressure 
of several Torres and a variable length of 6 to 12 
meters. The normalized amplitude attenuation at the 
specified CO2 absorption line and the related phase shift 
can be expressed by [5], 
ξξ
ν
ν
π
νϕ
ν
ν
νδ
ν
ν
dD
D
D
∫ ∆ −∆−−=
∆
−=
)2ln(
0
2
2
2
2
2
)exp(}
)(
)2ln(exp{2}{
}
)(
)2ln(exp{}{
 
                (13) 
whereν is relative frequency shifted from the center 
frequency of the specified CO2 absorption line and ∆νD 
is the full width half maximum (FWHM) of the 
specified CO2 absorption line. A typical normalized 
amplitude attenuation line with a full width half 
maximum of 350MHz, ∆νD=350MHz, and the related 
phase shift are shown in Fig.1.  
 
Fig. 1. Typical Gaussian absorption and dispersion with 
a full width half maximum of 350MHz  
Fig.2 shows the calculated PSD signals, S1, S2, S for 
different line width of the absorption line, which can be 
obtained by changing the pressure and length of the 
CO2 absorption cell. 
 
                 (a)                  (b)  (c)  
Fig. 2. Calculated PSD signals for different absorption 
line widths referring to the modulation frequency, (a) 
νm/∆νD=0.4, (b) νm/∆νD=0.5, (c) νm/∆νD=0.6 
Both S1 and S2 signals around the center of absorption 
line can be utilized as error signals of the PID feedback 
servo loop for the frequency locking. The sensitivity for 
a certain detection and feed back servo loop system 
depends on the slope of the signals, S1 or S2, around the 
center of absorption line. 
3. SENSITIVITY ANALYSIS OF THE FEED 
BACK LOOP 
Practically both the amplitude attenuation and the line 
width of the gas absorption line vary when the pressure 
of the gas cell changes. The variable pass-length CO2 
gas cell is characterized with HITRAN database (2004) 
and shown in Fig. 3. 
 
(a) 
 
(b) 
Fig. 3. Amplitude attenuation (a) and line width (b) of 
the CO2 absorption line for different path-length of the 
gas cell as a function of the pressure in the gas cell 
 The slopes of the PSD signals, S1 and S2 around the 
center of the absorption line present the sensitivity of 
the feed back loop to drive the frequency of the laser 
frequency and are calculated within a range of ±5 
MHz at the center. These calculated results are shown 
in Fig. 4.  
 
(a) 
 
(b) 
Fig. 4. Calculated PSD signals, S1 (a) and S2 (b) around 
center of the CO2 absorption line for different path-
length of the gas cell as a function of the pressure in the 
gas cell 
Both PSD signals, S1 and S2, show the optimized 
sensitivities at different pressures of the gas cell can be 
obtained for different lengths of the cell. 
4. EXPERIMENT 
The verification of the experiment has been set up and 
the diagram of the experiment and displays of the 
results are shown in Fig. 5.  
 
(a) 
 
  
(b)  
Fig. 5. (a) Diagram of the experiment for verification of 
the optimization of the frequency-locking system; (b) 
PSD signal (S2) and absorption of the specified CO2 line 
for a 8-m fixed length of the gas cell at different 
pressures, 1.5 Torr (left), 3.0 Torr (middle), 7.0 
Torr (right).  
The PSD signal, S2, at the pressure of 3.2 Torr was 
chosen for the frequency locking and the frequency 
variation of less than 1 MHz was achieved and will 
be reported separately in details.  
5. CONCLUSIONS 
The optimization of a 2-micron laser frequency locking 
system with a FM, PSD, and PID feedback servo loop 
has been simulated and verified in terms of 
characteristics of the CO2 gas absorption cell. The 
results of the simulations can be applied to any other 
frequency locking system referring to a gas absorption 
line. 
REFERENCES 
[1] Jerome Caron and Yannig Durand, 2009: 
Operating wavelengths optimization for a spaceborne 
lidar measuring atmospheric CO2,  Appl. Opt., 48, No. 
28, pp. 5413-5422 
[2] E. Ehret, C. Kiemle, M. Wirth, A. Amediek, A. 
Fix, and S. Houweling, 2008: Space-borne remote 
sensing of CO2, CH4, and N2O by integrated path 
differential absorption lidar: a sensitivity analysis, Appl. 
Phys. B, 90, pp. 593-608. 
[3] G. C. Bjorklund and M. D. Levenson, 1983: 
Frequency Modulation (FM) Spectroscopy, Appl. Phys. 
B, 32, pp. 145-152. 
[4] G. J. Koch, J. Y. Beyon, F. Gibert, B. W. Barnes, 
S. Ismail, M. Petros, P. J. Petzar, J. Yu, E. A. Modlin, 
K. J. Davis, and U. N. Singh, 2008: Side-line tunable 
laser transmitter for differential absorption lidar 
measurements of CO2: design and application to 
atmospheric measurements, Appl. Opt., 47, No. 7,  pp. 
944-956. 
[5] S. W. North, X. S. Zheng, R. Fei, and G. E. Hall, 
1996: Line shape analysis of Doppler broadened 
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Optimization of A 2-Micron Laser Frequency Stabilization System for a Double-Pulse CO2 Differential Absorption Lidar

Optimization of A 2-Micron Laser Frequency Stabilization System for a Double-Pulse CO2 Differential Absorption Lidar

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