Wavelength Modulation Spectroscopy
333
absorbing spectral wings. The time-indexed relative transmission τ (ν), is obtained
by normalizing the absorption-dependent signal by the baseline as shown in Fig. 6(c).
The intensity normalization compensates for the variation of the laser intensity across
the scan range and yields the absolute absorption profile.
The time-indexed data captured by the data acquisition system is now wavelengthreferenced to convert the time axis to a wavelength axis. To achieve this, the second
part of the laser light is passed through a resonator whose output is a periodically
varying signal as shown in Fig. 6d. The resonator is essentially any system with a
feed-forward or a feedback path between the input and the output. The output of such
a system exhibits a series of sharp peaks that are equally spaced in the frequency,
and can be used as a frequency ruler with the frequency spacing termed as the free
spectral range (FSR). Note that a linear time-domain current scan of the laser does not
lead to a linear time-domain wavelength scan because the wavelength does not tune
linearly with the injection current. Equal current increments therefore do not produce
equal wavelength increments. The key point is that the resonator peaks are equally
spaced in frequency, although they are not equally spaced in time. The time axis of
the data acquisition system, which is the common factor, may be disregarded and
the gas absorption line may instead be superimposed on the resonator output. Next,
the resonator peaks are identified and a polynomial fit is obtained. This polynomial
is then evaluated for the number of points that are present in the gas absorption
signal (e.g. 10,000 points for a typical digital oscilloscope). A 10,000-point gas
absorption signal is therefore now superimposed on a 10000-point frequency scale
with each interval of the scale being equal to the resonator’s FSR. The minimum of
the normalised gas line corresponds to the line centre wavelength (λ 0 ) of the line.
This point can therefore be taken as an absolute wavelength marker. The points to the
left and right of the line centre frequency ν 0 (not wavelength λ 0 ) are replaced by the
corresponding frequency values by adding and subtracting integer multiples of the
FSR. The set of frequency values are inverted to yield the corresponding wavelength
values.
Finally, Fig. 6e shows the wavelength-referenced relative transmission τ (ν) that
bears the signature of the absolute gas absorption line shape. The relative transmission
that follows the Beer-Lambert law is written as,
τ (ν) =
I out
I in
= exp[−α(ν)] = exp
−
i
PxS i (T )φ i (ν)
L
(11)
where, I in and I out are the incident and transmitted laser intensities, respectively,
α(ν) is the absorbance at optical frequency ν, T is the gas temperature, P is the gas
pressure, x is the mole fraction of the gas, L is the path length through the absorbing
gas, φ(ν) is the line shape function of the transition i and S i (T) is the line strength at
temperature T of the i
th absorption transition. The relative transmission is a function
of the mole fraction, pressure and temperature of the gas. Now, a simulated line shape
is generated using the spectral parameters provided in the HITRAN database and an
appropriate line shape function. The simulated relative transmission is fitted to the
333
absorbing spectral wings. The time-indexed relative transmission τ (ν), is obtained
by normalizing the absorption-dependent signal by the baseline as shown in Fig. 6(c).
The intensity normalization compensates for the variation of the laser intensity across
the scan range and yields the absolute absorption profile.
The time-indexed data captured by the data acquisition system is now wavelengthreferenced to convert the time axis to a wavelength axis. To achieve this, the second
part of the laser light is passed through a resonator whose output is a periodically
varying signal as shown in Fig. 6d. The resonator is essentially any system with a
feed-forward or a feedback path between the input and the output. The output of such
a system exhibits a series of sharp peaks that are equally spaced in the frequency,
and can be used as a frequency ruler with the frequency spacing termed as the free
spectral range (FSR). Note that a linear time-domain current scan of the laser does not
lead to a linear time-domain wavelength scan because the wavelength does not tune
linearly with the injection current. Equal current increments therefore do not produce
equal wavelength increments. The key point is that the resonator peaks are equally
spaced in frequency, although they are not equally spaced in time. The time axis of
the data acquisition system, which is the common factor, may be disregarded and
the gas absorption line may instead be superimposed on the resonator output. Next,
the resonator peaks are identified and a polynomial fit is obtained. This polynomial
is then evaluated for the number of points that are present in the gas absorption
signal (e.g. 10,000 points for a typical digital oscilloscope). A 10,000-point gas
absorption signal is therefore now superimposed on a 10000-point frequency scale
with each interval of the scale being equal to the resonator’s FSR. The minimum of
the normalised gas line corresponds to the line centre wavelength (λ 0 ) of the line.
This point can therefore be taken as an absolute wavelength marker. The points to the
left and right of the line centre frequency ν 0 (not wavelength λ 0 ) are replaced by the
corresponding frequency values by adding and subtracting integer multiples of the
FSR. The set of frequency values are inverted to yield the corresponding wavelength
values.
Finally, Fig. 6e shows the wavelength-referenced relative transmission τ (ν) that
bears the signature of the absolute gas absorption line shape. The relative transmission
that follows the Beer-Lambert law is written as,
τ (ν) =
I out
I in
= exp[−α(ν)] = exp
−
i
PxS i (T )φ i (ν)
L
(11)
where, I in and I out are the incident and transmitted laser intensities, respectively,
α(ν) is the absorbance at optical frequency ν, T is the gas temperature, P is the gas
pressure, x is the mole fraction of the gas, L is the path length through the absorbing
gas, φ(ν) is the line shape function of the transition i and S i (T) is the line strength at
temperature T of the i
th absorption transition. The relative transmission is a function
of the mole fraction, pressure and temperature of the gas. Now, a simulated line shape
is generated using the spectral parameters provided in the HITRAN database and an
appropriate line shape function. The simulated relative transmission is fitted to the