OPTICAL REVIEW Vol. 10, No. I (2003) 24-30
~~j~"c 2003 The Optical Society of Japan
Calculation of the Ratio BetWeen the Second and First Harnlonic SignalS
in Wavelength-Modulation SpectroSCOpy for Absorption Measurement
Takaya ISEKI
Tec/~nical Research Institbtte, Tok)’o Gas Co., Ltd., Shibabtra 1-16-25, Minato-kbt. Tokyo l05-0023, Ja]’an
(Received ALlgLISt 20, 2002; Accepted November 8, 2002)
The dependence of the ratio between the second and first harmonic signals in wavelength-modulation spectroscopy is
calculated for the precise absorption measurement that is applied to gas sensing. In the calculation, the amplitude
modulation of the laser is considered as well as the frequency modulation. First, the dependence of signals on peak
optical depth is discussed for the Lorentzian and Gaussian lines. Next, the dependence of signals on gas concentration is
calculated in a weak absorption-limit. Finally, a numerical example is explained taking into consideration both peak
optical depth and gas concentration.
Key words: wavelength-modulation spectroscopy, tunable diode laser, absorption measurement, ratio detection, gas sensing
l. Introduction
Wavelength-modulation spectroscopy (WMS) using a
tunable diode laser (TDL) is one of the most common
techniques for precise absorption measurements. Recently,
this technique has been applied to gas sensing such as
environmental trace gas monitoringl) and gas leak detec-
tion.2) In the standard WMS using a TDL, the frequency
modulation (FM) is caused by a sinusoidal injection current
at frequency f, which is much smaller than the laser line
width, and the photodetector output is processed by the
phase-sensitive detection with reference to the fundamental
(I f), the second harmonic (2f), or higher harmonics of the
modulation. For high-speed measurement, the modulation
center is locked at an absorption center of the gas of interest,
and the quantitative information of the absorption is
calculated by the harmonic signals. Since the 2f signal at
the absorption center is proportional to the peak optical
depth, Do, in the weak-absorption limit (DO << 1), the 2f
signal is often used for gas sensing. This technique is
sometimes called "second harmomc detectron "3) On the
other hand, the I f signal due to the FM should vanish at the
absorption center. In fact, the I f signal arises because of the
amplitude modulation (AM) in the initial laser power.
Dividing the 2f signal by the If signal, we can cancel the
change in the initial laser power and the optics collection
efficiency.4’5) Thrs techmque rs sometunes called "ratro
detection. "
First, we discuss the dependence of the I f signal, the 2f
signal and the ratio between the two signals on peak optical
depth. Uehara obtained general expressions for the Of (DC)
signal, the 2f signal and the ratio between the 2f and Of
signals without considering the AM.6) However, the ratio
between the 2f and I f signals is more useful than the ratio
between the 2f and Of signals for a precise absorption
experiment, since the If signal has less noise than the Of
signal owing to phase sensitive detection. We calculate
analytical expressions for harmonic signals for the Lor-
entzian and Gaussian lines considering the AM of a TDL.
From the results of numerical calculation for the I f signal,
the 2f signal and the signal ratio, we evaluate the detection
limit of the ratio detection. In addition, we estimate the
accuracy of the approximation of weak absorption, in which
the signal ratio is assumed to be proportional to peak optical
depth.
Second, we calculate the dependence of the ratio between
the 2f and If signals on gas concentration for the
Lorentzian line. With increasing gas concentration, the
width of this line increases. Consequently, the harmonic
signals depend on the gas concentration. This dependence is
important in the case of gas leak detection, where the gas
concentration is high. We calculate the signal ratio between
the 2f and I f signals for the Lorentzian line as a function of
gas concentration, and evaluate the accuracy of the low-
concentration approximation in which the Lorentzian line
width is a constant.
Finally, we explain using a numerical example for a
portable remote methane sensor we have developed for gas
leak detection.2) The measurement principle of the sensor is
based on the ratio detection of the 2f and I f signals in the
WMS using the 2L,3-band R(3) Iine of l’-CH4 (~ =
1 .6537 um). The sensor outputs the path-integrated methane
concentration calculated by the linear approximation of
weak absorption and low concentration. For arbitrary values
of peak optical depth and methane concentration, we
calculate numerically the dependence of the ratio between
the 2f and I f signals on the path-integrated methane
concentration. In addition, we estimate the accuracy of the
sensor output for different methane concentrations.
2. Analytical Expression for the Ratio Detection of the
Second and First Harmonic Signals in Wavelength-
Modulation Spectroscopy with Amplitude Modula-
tion
The absorption of light through a medium is expressed by
Lambert-Beer law as
I Io exp(-D), ( I )
where I is the intensity after passing through the medium, Io
is the initial intensity, and D is termed the optical depth. For
the case where a laser light is absorbed by one molecular gas
in the atmosphere, Lambert-Beer law becomes
P = POK exp(-oc(co)C), (2)

OPTICAL REVIEW Vol. Io, No. I (2003)
where P is the received laser power, Po is the initial laser
power, K is the optics collection efficiency, ce(a)) is the
absorption coefficient, co is the angular frequency of the laser
light, and C is the path-integrated concentration of the
absorbing gas. The absorption coefficient is related to the
normalized line shape function, g(co - a)o), by
cl(co) = Sg((~) - coo)N, (3)
where S is the molecular line intensity, coo is the absorption
center, and N is the total number of molecules of the
absorbing gas per unit volume per atmosphere. The line
shape function is given by
gL((L) - a)o) = (-) y (4) 1
7T (a) - coo)2 + y2
for the Lorentzian (collision broadened) Iine and
gG((~) - (~)o) 7T l/2 1 In2(a) - a)o)~ (5) ) (
exp
for the Gaussian (Doppler broadened) Iine.
When the laser wavelength is modulated at frequency f,
and the modulation center is locked at an absorption center
of the gas of interest, the angular frequency of the laser light
is expressed as
~)(t) = a)o + coFM cos(ft) (f = 2Jrf), (6)
where a)FM rs the FM depth. The analytical expressions for
harmonic signals, which are related to Fourier series
expansion of P(co - (()o), were obtained in previous
studies,6~8) However, only the even harmonic signals was
calculated in these studies, since the odd harmonic signals
due to the FM should vanish at the absorption center. In fact,
odd harmonic signals arise because of the AM in the initial
laser power, which is expressed as
Po(t) = poK[1 + mAM cos(ft + ip)], (7)
where po is the DC component of the initial laser power,
mAM rs the AM ratio, and ip is the AM-FM phase difference.
Taking the AM into consideration, we can express the
received laser power as
oo
P(t) poK[1 + lnAM cos(ft + ip)] ~ c2n exp(12nft)
n=-oO
oo
= pOK ~ {c2n exp(12nft)
n=-oo
+ (112)kyA[c2n exp(iip) + c2n+2 exp(-iip)J
x exp[i(2n + 1)ft]},
(8)
where
c2n (117T) exp(- 2 20 ) cos(2nx)dx (9) = f A cos x+ 1
(n = O, 1, 2, . . .)
for the Lorentzian line and
f c2n = (117T) Jo exp(-Do exp(- In 2A2 cos2x)) cos(2nx)dx
T. ISEKI 25
(n = O, 1, 2, . . .) (10)
for the Gaussian line. In Eqs. (9) and (10), A is the relative
FM depth, Do is the peak optical depth, and k is the
coefficient between AM and FM, which is a constant
depending on the modulation characteristics of the laser.
These parameters are defined by
A = coFM/y, (1 1) SNC (Lorentzian line),
7T y Do = SNg(O)C (12) 1 1 12 SNC ( JT In 2 y (Gaussian line),
k = mAM/coFM. ( 13)
In the case where either the initial laser power or the optics
collection efficiency changes, the ratio between the 2f and
4,5)
If signals is utilized to cancel the change.
The harmonic signal amplitude of the received laser
power is expressed by the Fourier coefficient, c2n ’ as
po,f = P~CKco, (14)
p(2n+1) f pDCK(c2,1 + c2n+2 + 2c2nc2n+’- cos 2ip) 112lnAM,
, ), (15) (n = O, 1,2 . . .
p(2n+2)f _ 2pDCK (16) (n = O, 1, 2, . . .), - o c n+2,
From Eqs. (9)-(16), we can calculate the Of, If and 2f
signal amplitude for both the Lorentzian and Gaussian lines.
In the weak-absorption limit (DO << 1), by neglecting the
terms of Do higher than Do, the signal amplitude and the
ratio between the 2f and I f signals are expressed as
= poK[1 - I DoJ (DO << 1),
L (1 + A’-)1/2
pl,f ’ [1 + (1 + A2)112]2 _ 2A2 cos(2ip) Do}
L = poKmAM11 - (1 +A2)112[1 + (1 +A2)112]2
(DO << 1), (18)
p2.f - 2A2 pOK (1 + A2)11 [1 + (1 + A2)112]2 Do
L-
(DO << 1),
( L:= ) P 1 Plf ky (1+A2)1/2[1+(1+A’-)1/2]2Do
( 19)
(DO << 1), (20)
for Lorentzian line and
po.f = pOKLI Io(In 2A /2) exp( In 2A /2)Do]
G
(DO << 1), (21)
pl.f = poKkyA{1 - 2[Io(In 2A212)
G
- 211 (In 2A2/2) cos(2ip)] exp(- In 2A212)Do}
(DO << 1), (22)