Extension of wavelength-modulation spectroscopy to
large modulation depth for diode laser absorption
measurements in high-pressure gases
Hejie Li, Gregory B. Rieker, Xiang Liu, Jay B. Jeffries, and Ronald K. Hanson
Tunable diode laser absorption measurements at high pressures by use of wavelength-modulation
spectroscopy (WMS) require large modulation depths for optimum detection of molecular absorption
spectra blended by collisional broadening or dense spacing of the rovibrational transitions. Diode lasers
have a large and nonlinear intensity modulation when the wavelength is modulated over a large range
by injection-current tuning. In addition to this intensity modulation, other laser performance parameters
are measured, including the phase shift between the frequency modulation and the intensity modulation.
Following published theory, these parameters are incorporated into an improved model of the WMS
signal. The influence of these nonideal laser effects is investigated by means of wavelength-scannedWMS
measurements as a function of bath gas pressure on rovibrational transitions of water vapor near
1388 nm. Lock-in detection of the magnitude of the 2f signal is performed to remove the dependence on
detection phase. We find good agreement between measurements and the improved model developed for
the 2f component of the WMS signal. The effects of the nonideal performance parameters of commercial
diode lasers are especially important away from the line center of discrete spectra, and these contribu-
tions become more pronounced for 2f signals with the large modulation depths needed for WMS at
elevated pressures. © 2006 Optical Society of America
OCIS codes: 120.0120, 120.1740, 300.0300, 300.1030, 300.6260, 300.6380.
1. Introduction
Tunable diode laser (TDL) absorption sensors for
temperature, concentration, and velocity measure-
ments in gases have been studied extensively by var-
ious researchers.
1–11
Most of these sensors are based
on direct absorption methods because of the rela-
tively simple interpretation of the results.
2–6
When
absorption is weak, wavelength-modulation spectros-
copy (WMS) is a well-known technique for improving
the signal-to-noise ratio through phase-sensitive
detection.
7–11
With regard to WMS theory, 40 years
ago Wilson
12
employed numerical integration to ob-
tain the first three harmonics for Gaussian and
Lorentzian absorption line shapes. Arndt
13
developed
an analytical solution based on Fourier analysis for
all harmonics of a Lorentzian line shape, with explicit
expressions for the 1f and 2f components. Reid and
Labrie
14
performed the first experimental TDL WMS
experiments and measured the second-harmonic sig-
nals for Lorentzian, Voigt, and Gaussian line shapes.
However, all these early approaches assumed the la-
ser intensity to be independent of laser frequency,
and thus they are suitable for only small modulation
depths for injection-current-tuned diode lasers.
There is extensive literature on TDLWMS, and we
do not attempt a further general review but focus our
attention on extending TDL WMS to the large mod-
ulation depths needed for measurements with
blended and overlapped transitions. Such spectra are
found for large polyatomic molecules with densely
spaced spectra or from smaller target molecular spe-
cies at high pressures at which spectra are blended by
collisional broadening. Figure 1 illustrates absorp-
tion of 1% water vapor in air as a function of pressure
near 7204 cm

1
. Even at atmospheric pressure the
three transitions near 7204 cm

1
are blended, and
pressures of 20 atm blend features more than 5 cm

1
away. Optimal WMS detection of such features re-
quires large modulation depths.
Typical TDL WMS is performed by modulation of
The authors are with the High Temperature Gasdynamics Lab-
oratory, Department of Mechanical Engineering, Stanford Univer-
sity, Stanford, California 94305. H. Li’s e-mail address is
hejieli@stanford.edu.
Received 13 June 2005; accepted 23 August 2005.
0003-6935/06/051052-10$15.00/0
© 2006 Optical Society of America
1052 APPLIED OPTICS
Vol. 45, No. 5
10 February 2006

the laser wavelength (frequency) with a sinusoidal
injection current,
15–23
which produces a simultaneous
modulation of the laser intensity. We show here that
quantitative TDL WMS with large modulation
depths requires consideration of diode laser perfor-
mance characteristics including simultaneous fre-
quency modulation (FM) and intensity modulation
(IM), the phase shift between FM and IM, and non-
linear IM.
A number of refinements to early WMS models
have been made in the literature to include these
effects. Philippe and Hanson
15
extended WMS theory
to account for linear laser IM and numerically calcu-
lated the 1f and 2f signals by using a Fourier decom-
position of a Voigt profile. Kluczynski et al.
16,17
derived an expression to include the effect of the
FM
IM phase shift for WMS with frequency-doubled
light. Independently of their research and using a
slightly different formalism, Schilt et al.
18
developed
a theoretical model of WMS for a Lorentzian line
shape in the general case of combined IM and FM
with an arbitrary FM
IM phase shift. Recently,
Gharavi and Buckley
20
considered the intensity non-
linearity for a low-frequency current ramp used to
tune the laser wavelength. This nonlinearity can also
be empirically accounted for by performing polyno-
mial fits to the nonabsorbing portions of the laser
scans.
21
Using Fourier analysis, Kluczynski and
Axner
24
developed a general theoretical description
of WMS that includes the effect of the FM
IM phase
shift, the nonlinear IM associated with the sinusoi-
dal current modulation, and wavelength-dependent
transmission. However, their theoretical approach
needs to be simplified and extended for practical
gas-sensing applications. To our knowledge, no
work has been done on implementations of large-
modulation-depth WMS, including real diode laser
performance parameters.
In this paper, large-modulation-depth WMS with
2f detection is extended, following the theoretical
work of Kluczynski and Axner,
24
to account for the
real diode laser characteristics of FM
IM phase shift
and a nonlinear IM. Their equations for the ampli-
tude components of the 2f signal are rewritten to
provide the magnitude of the 2f signal, thereby
eliminating the dependence of the signal on the de-
tection phase. To test the extendedWMS theory, TDL
WMS validation experiments are performed using
pressure-broadened water-vapor rovibrational tran-
sitions near 1388 nm. We find improved agreement
between these measurements and the extended
model developed for the 2f component of the WMS
signal. The effects of the nonideal performance pa-
rameters of real diode lasers are found to be espe-
cially important away from the line center of discrete
spectra. These contributions become more pro-
nounced for 2f signals with the large modulation
depths needed for WMS at elevated pressures at
which pressure broadening blends the spectrum. For
optically thin measurements with characterized la-
sers, we find that normalization of the 2f signal with
the 1f signal can remove the need for calibration.
2. Theory: Extension of Wavelength-Modulation
Spectroscopy to Account for Real Diode Laser
Performance
Following the general theory of Kluczynski and Ax-
ner,
24
we rewrite the equations for the WMS signal
into a form providing the magnitude of the 2f signal.
This allows direct comparison with laboratory mea-
surements by use of a lock-in amplifier. The diode
laser injection current is sinusoidally modulated with
angular frequency 2f to produce laser FM
(t)
a cos(t), (1)
where 
is the center laser frequency and a is the
modulation depth. The diode laser intensity is simul-
taneously modulated with a FM
IM phase shift,
15–20
and the instantaneous laser intensity I
0
t varies non-
linearly with the injection current
24
:
I
0
(t) I


0
[1 i
0
cos(t
1
)
Ç
1f term
 i
2
cos(2t
2
)].
Ç
2f term
(2)
The average laser intensity at 
is given by I
0


, i
0
is the
linear 1f and i
2
the nonlinear 2f IM amplitude
(normalized by I
0


), 
1
is the FM
IM phase shift, and

2
is the phase shift of the nonlinear IM. Equation (2)
could be generalized to include higher-order harmon-
ics, but our experimental characterization of commer-
cial diode lasers shows that the IM is well described
by a combination of 1f and 2f terms (Section 3).
The transmission coefficient  of monochromatic
radiation through a uniform medium of length L (in
centimeters) is given by the Beer–Lambert relation
()

I
t
I
0


 exp


()

. (3)
Here I
t
and I
0
are the transmitted and incident laser
intensities, respectively, and  represents the
Fig. 1. Spectral simulation of 1% H
2
O in air at 1000 K, 1 cm path
length.
10 February 2006
Vol. 45, No. 5
APPLIED OPTICS 1053