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Laser Absorption Spectroscopy

Authors:
  • NASA Jet Propulsion Laboratory

Abstract and Figures

Laser diagnostic spectroscopy is a non-intrusive method that has gained importance to in situ study ultrafast gas-phase reactions with fast time response and high time resolution and it can be understood from principles of quantum mechanics. The theory quantizes the energy of the atoms and molecules by describing the existences of atoms and molecules in specific quantum states with discrete values of energy and angular momentum.
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Chapter 2
Laser Absorption Spectroscopy: Theory
Laser based spectroscopy being a non-intrusive method has gained importance for in-situ
study of gas-phase reactions due to fast time response and high time resolution, and it can
be understood from principles of quantum mechanics. The theory of quantum mechanics
describes the existence of atoms and molecules in specific quantum states having discrete
energy and angular momentum values and hence quantizes the energy of atoms and molecules
[
35
]. The discrete energy levels for a particle with mass
m
moving in a potential field by
V(x)can be determined using the time-independent Schrödinger equation.
d2ψ(x)
dx2+8mπ2
h2[EV(x)]ψ(x) = 0 (2.1)
where
h
is Planck’s constant,
ψ
is wave-function.
E
is the total internal energy of the system
excluding kinetic energy, as the sum of electronic, vibrational, and rotational energy, i.e.,
E=Eelec +Evib +Erot (2.2)
According to Born-Oppenheimer approximation (discussed later), change in discrete quantum
energy states of atoms or molecules leads to discrete transitions which is a direct consequence
of quantization of internal energy states [
35
]. These differences in energy can be directly
related to the energy of emitted or absorbed photons in discrete transition in emission or
absorption spectra. The energy differences,
E
, of the photon-induced radiative transitions
between two molecular quantum states can be described by the Planck’s law.
E=Eupper Elower =hν=hc
λ=h˜
ν(2.3)
8Laser Absorption Spectroscopy: Theory
where
c
is the speed of the light [m/s] in vacuum,
ν
,
λ
, and
˜
ν
are the frequency [s
1
], the
wavelength [nm], and the wavenumber [s
1
] of the corresponding electromagnetic wave
associated with the photons, respectively.
The total energy change in Equation 2.3 can be seen as sum of the individual changes in
electronic, vibrational, and rotational energy in Equation 2.4. Hence, it defines the molecular
transitions (in emission or absorption spectra) into three different domains of electromagnetic
spectrum [35].
E=Eelec +Evb +Erot (2.4)
where:
Erot: Microwave Region (1 mm – 1 m)
Erot +Evib: Infrared Region (700 nm – 1 mm)
Erot +Evib +Eelec: UV / Visible Region (10 – 700 nm)
2.1 Beer-Lambert’s Law
Consider a collimated monochromatic light beam with intensity
I0
is shone through an
absorbing gas sample having number density
n
[molecules/cm
3
]. When the beam frequency
ν
[cm
1
] resonates with the frequency of a transition for a species present in the gas sample, the
beam (laser, in our case) energy will get absorbed. In the Figure 2.1 the extent of attenuation
of beam intensity along a differential length of
dx
is governed by the Einstein theory of
radiation as follows [10]:
dIν
Iν
=Pχabs(x)Si(T(x))φ(ν)dx (2.5)
Fig. 2.1 Change of beam intensity across a gas slab of infinitesimally small length
where
Iν
[V] is the beam intensity,
T
[K] the local temperature,
P
[atm] the total pressure,
χabs(x)
the mole fraction of the absorbing species of interest,
φ(ν)
[cm] the lineshape
function, and
Si(T(x))
[cm
2
atm
1
] the linestrength of
ith
transition.
Si(T(x))
is a function
2.1 Beer-Lambert’s Law 9
of the temperature as [35]:
S(T) = S(T0)Q(T0)
Q(T)
T0
TexphcE′′
k1
T1
T01exp(hcν0
kT )
1exp(hcν0
kT0)(2.6)
where
h
[m
2
kg/s] is Planck’s constant,
c
[cm/s] the speed of light,
k
[m
2
kg/s
2
K] is Boltz-
mann’s constant,
Q(T)
the partition function of the absorbing species,
T0
[K] the reference
temperature,
ν0
[cm
1
] the line-center frequency, and
E′′
[cm
1
] the lower state energy of
the transition.
The parameters at the reference temperature
T0
can be obtained from the HITRAN
database [
41
], or can be calibrated in a static cell under controlled conditions. It should be
remembered that the reference temperature
T0
in the HITRAN database is always set to 296
K.
The conventional units for linestrength is either a number-density dependent version used in
the HITRAN database which is generally represented as
S
[cm
1
/mol
×
cm
2
] or a pressure-
dependent version [cm
2
atm
1
] used further in this thesis. As reference value of linestrength
S(T0)
at 296 K is directly obtained from HITRAN database for all further works discussed in
this thesis and hence a conversion between Sand Sis crucial [35]:
Scm2atm1=Scm1/ mol ×cm2×nmol/cm3
P[atm](2.7)
where
n
is the number density of the absorbing species. Now by using the ideal gas law
and converting the unit of pressure into [Pa], a relationship without dependence on number
density can be obtained as follows:
Scm2atm1=Scm1/ mol ×cm2×101325[Pa/atm)
kT (2.8)
where
k
is the Boltzmann constant having the value of 1.38054
×
10
23
[J/K]. For the
linestrength at reference temperature of 296 K (as tabulated in HITRAN) the above conversion
can be simplified in the form of following equation
S=S× 2.488 ×1019cm2atm1(2.9)
The partition function
Q(T)
mentioned in Equation 2.6 can be defined as the product of
rotational, vibrational, and electronic partition functions and can be approximately estimated
10 Laser Absorption Spectroscopy: Theory
from a third order polynomial:
Q(T) = a+bT +cT 2+dT 3(2.10)
The coefficients of above polynomial expression
a
,
b
,
c
, and
d
depend solely on temperature
and species. The above coefficients of the polynomial approximation for the partition function
of carbon monoxide (CO) molecule for example have been listed in Table 2.1 [
26
]. However,
partition functions for different species have been tabulated in HITRAN for a wide range of
temperature, which can be directly used for analysis.
Table 2.1 Coefficients of the approximated polynomial (2.10) to calculate the partition
function for CO
Coefficients 70 K <T<500 K 500 K <T<1500 K 1500 K <T<3005 K
a 0.27758 ×1000.90723 ×1010.63418 ×102
b 0.36290 ×1000.33263 ×1000.20760 ×100
c –0.74669 ×1050.11806 ×1040.10895 ×103
d 0.14896 ×1070.27035 ×1070.19844 ×108
The fractional transmission
τν
for a total sample length of
L
[cm] can be deduced from
Equation 2.5 as
τν=It
I0ν
=exp PZL
0
χabs(x)Si(T(x))φ(ν)dx)(2.11)
The quantity ’absorption’ can be defined as
aν=1τν=1It
I0ν
=1exp PZL
0
χabs(x)Si(T(x))φ(ν)dx)
In most cases, to study the transition parameters of absorption spectrum such as linestrength
and lineshape, it is opportune to transform the transmission or absorption into absorbance
to obtain direct proportionate relation with linestrength, total pressure, mole fraction of
absorbing species, lineshape function, and path length.
The absorbance ανis defined as
αν=lnIt
I0ν
=PZL
0
χabs(x)Si(T(x))φ(ν)dx (2.12)
2.1 Beer-Lambert’s Law 11
The estimation of lineshape function in the above expression is not straight-forward in
most of the cases as it strongly depends on experimental conditions and the species of
interest. However, the lineshape
φ(ν)
being a probability density function by definition can
be normalized as
R
φ(ν)dν=1
, hence spectrally-integrated absorbance
A
[cm
1
] defined
as area underneath the absorption spectrum can be obtained from Equation 2.12 as
A=Z
ανdν=PZL
0
χabs(x)Si(T(x))dx (2.13)
In the absence of any non-uniformity along the line of sight in the gas medium of interest
(i.e. sampling length having uniform temperature
T
and species mole fraction
χabs
), Equation
2.11 reduces to the most commonly used form of Beer-Lambert’s law, which is the very
fundamental equation used in Direct Absorption Spectroscopy (DAS),
τν=It
I0ν
=exp PχabsSi(T)φ(ν)L(2.14)
However, there are several forms for the linestrength and Beer-Lambert’s law having their
own units and notation. One of the most important forms in terms of absorption cross-
section (
σν
) which is useful in the cases of broader spectral lines (applicable for large sized
molecules or broadened line, discussed later) is
τν=It
I0ν
=exp(nσνL) = exp(kνL)(2.15)
where
σν
[cm
2
/molecules] is the frequency-dependent absorption cross-section, and
kν
[cm
1
] is the spectral absorption coefficient. Intuitively, absorption cross-section can be
imagined as the effective size of an absorbing molecule if it simply blocked light from
passing through a volume. Imagine molecules being replaced by opaque spheres of cross
sectional area
Am
, where
Am
is not the molecules actual size, but rather a measure of how
much light it absorbs. The absorption cross-section then expresses the sum over all molecular
cross-sections,
Am
, per mole of molecules. For a particular isolated transition
i
with frequency
ν,
kν=PχabsSi(T)φ(ν)
The spectral absorbance (2.12) is thus simplified as
αν=lnIt
I0ν
=PχabsSi(T)φ(ν)L(2.16)
12 Laser Absorption Spectroscopy: Theory
and the spectrally-integrated absorbance Aof an isolated transition ican now be defined as
A=PχabsSi(T)L(2.17)
which is nothing but the product of partial pressure of the absorbing species, temperature-
dependent linestrength, and total path length. Hence for direct absorption measurements
without concentration information, the concentration of the target species inside the gas
medium can be obtained from the measured absorbance of the transition as
χabs =αν
PSi(T)φ(ν)L=A
PSi(T)L(2.18)
Line position, linestrength and lineshape at a given pressure and temperature are charac-
teristics of an absorbing molecule. Thus the absorption spectra is unique to an absorbing
molecule and can be characterized by position, strength and shape of the line. Hence it
becomes absolutely necessary to understand molecular spectra of different types of molecules,
which is explained in next section.
2.2 Molecular Spectra
As mentioned earlier, three parameters of absorption line can uniquely determine the
macro and micro states of the species of interest. According to the principles of quantum
mechanics the energy levels of most molecules (and atoms) are discrete or quantized and
optically allowed absorptive and emissive transitions can occur only in some very particular
cases. Consequently, the absorption and emission spectra are typically discrete in most cases.
The molecular internal energies of interest are: rotational, vibrational, and electronic, with a
gradual trend of increasing energy spacings:
Erot <Evib <Eelec
Because of this trend there are rotational, vibrational, electronic, rovibrational (coupling
of rotational and vibrational excitation) and rovibronic (coupling of all three excitations)
spectra. Near infrared region is not capable of capturing electronic excitations, hence our
interest lies on rotational and vibrational transitions only.
To understand and simulate spectra associated with an actual absorption or emission
transition, some physical models must be discussed to interpret the underlying processes.
The simplest physical models to explain rotational and vibrational transition for the diatomic
molecule is Rigid Rotor (RR) and Simple Harmonic Oscillator (SHO) models, respectively.
2.2 Molecular Spectra 13
Some of the assumptions used in these two approximated models can be relaxed to form more
improved models to consider the coupling between rotation and vibration: the Non-rigid
Rotor and the Anharmonic Oscillator (AHO). Essentially, these more complex models are
revised version of the original, simpler models with some corrections. Though these models
have improved over the years, still the Born-Oppenheimer approximation is commonly used
for predicting rovibrational excitation.
2.2.1 Diatomic Molecule
The optical phenomena of emission, absorption, and scattering are of paramount conse-
quences for primary interactions of light and matter. Among various possibilities for the
interaction, the most common modes are:
Electric dipole moment (Absorption/Emission transition)
Induced polarization (Raman scattering)
Elastic scattering (Rayleigh scattering)
Heteronuclear diatomic molecules, with permanent opposite net charges on either ends
(e.g., NO, CO), have a permanent electric dipole moment. The rotational or vibrational
motion of this electric dipole moment, brings the prospect of electromagnetic absorption or
emission. The strength and probability of absorption or emission strongly depends on the
electric dipole moment and how it varies with internuclear spacing. However, EM radiation
is able to interact with diatomic molecules by rearranging its electron distribution.
Figure 2.2 depicts potential well curves of two different electronic levels for a diatomic
molecule. Basically, potential well of an electronic energy state is a plot of potential
energy with internuclear distance, which describes the variation of electronic force with the
separation between two nuclei. It is clear from Figure 2.2 that, each electronic energy level
can be divided into a number of vibrational levels, which further contains several discrete
rotational states. Line spacings between the rotational transitions are less, which falls in the
IR region. Hence, light-matter interaction with ro-vibrational motion can be detected using
IR waves. In the IR region, absorption and emission happen when the incoming photon
energy resonates with the energy spacing between two discrete ro-vibrational states. This
results in discrete absorption or emission within a specific vibrational band and each of these
transitions denotes unique change in rotational energy. But for electronic spectra, energy
transitions are related to the electronic distribution in the molecule’s shell.
14 Laser Absorption Spectroscopy: Theory
Fig. 2.2 Potential well curves (potential energy vs internuclear distance) of two electronic
levels for a diatomic molecule [31]
Rotational Spectra
Rigid Rotor (RR) Model:
The RR model approach combines principles of both classical
and quantum mechanics. Here, the atoms are assumed to be point masses with a constant
equilibrium separation distance, which makes the bonds "rigid" as shown in Figure 2.3.
Using this model, the rotational energy (in units of wavenumber) is given by [56]:
F(J) = BJ(J+1)(2.19)
where,
F(J)
[cm
1
] is rotational energy,
J
(= 0,1,2,...) the rotational quantum number, and
B
[cm
1
] the rotational constant. It is to be noted that, rotational constant is species-specific,
and is defined in terms of Planck’s constant
h
[m
2
kg/s], speed of light
c
[m/s] and the moment
of inertia of the molecule I[Kg.m2]
B=h
8π2Ic (2.20)
The solution to Schrödinger equation [2.1] is based on theory of quantum mechanics and
it also yields “selection rules” for allowed rotational transitions, which restricts the change
in rotational quantum number
(Jfinal Jinitial )
for a diatomic rigid rotor, to only
±1
. For
the cases of pure rotational transitions (without any changes in vibrational or electronic
configuration), the change in Jcan be restricted to +1 if the change in Jis defined as [56]:
J=JJ′′ = +1
2.2 Molecular Spectra 15
Fig. 2.3 Rigid rotor approximation for rotational transition of a diatomic molecule having a
permanent dipole moment
where, Jand J′′ are rotational quantum numbers of upper and lower state respectively.
Non-rigid Rotor Model:
The model to explain molecular rotation can be improved by
relaxing the assumption of rigidity made in the previous model. There are two governing
effects leading to non-rigidity of rotation and which further affect Band F(J):
With increase in vibrational energy
Evib
, the average nuclear separation also increases
which results in increase in the moment of inertia
I
, and hence the rotational constant
Bdecreases (Equation 2.20).
As rotational energy (
J
) increases, due to centrifugal distortion the average nuclear
separation increases, and hence the rotational constant Bdecreases.
However, the effects of vibrational stretching dominates over those of centrifugal distor-
tion. The effects of these non-rigidities are considered in the modified expression for the
rotational energy, Fv(J)[56]:
Fv(J) = BvJ(J+1)DvJ2(J+1)2(2.21)
where,
Dv
is the centrifugal distortion constant, and
Bv
is the vibrationally dependent ro-
tational constant. It is important to note that the subscript
v
denotes the dependence on
vibrational quantum number. Now the rotational energy becomes function of both rotational
and vibrational quantum numbers. The constants Bvand Dvare defined by [56]:
Bv=Beαe(v+1/2)
Dv=De+βe(v+1/2)
16 Laser Absorption Spectroscopy: Theory
where, both αeand βeare positive constants.
Vibrational Spectra
Simple Harmonic Oscillator (SHO):
The fundamental model to understand diatomic
vibration is the SHO, which considers two point masses separated by an equilibrium distance
as shown in Figure 2.4. Two point masses oscillate about the equilibrium distance as if the
bond between them were a spring. Following this model, vibrational energy (in units of
wavenumber) is given by:
G(v) = ωe(v+1/2)(2.22)
where,
G(v)
[cm
1
] is vibrational energy,
v
(= 0,1,2,...) the vibrational quantum number,
ωe
[cm1] the vibrational constant.
Fig. 2.4 Simple harmonic oscillator approximation for vibrational transition of a diatomic
molecule; reand rmin represent equilibrium and greatest compression distance
It is interesting to note that, under this approach of SHO, adjacent vibrational quantum
states have equal energy spacing between them, which is direct consequence of Equation
2.22,
G(v+1)G(v) = ωe
independent of
v
. Considering the assumptions of SHO model,
quantum mechanical solution to Schrödinger equation [2.1] leads to a very simple selection
rule for absorption and emission of a heteronuclear diatomic molecule, which restricts the
change in vibrational quantum number to only 1 [56]:
v=vv′′ = +1
where, vand v′′ are vibrational quantum numbers of upper and lower state respectively.
Anharmonic Oscillator (AHO):
In actual cases diatomic molecules do not stick exactly
to the idealized SHO model, but rather have anharmonicities and due to these, the energy
spacing between adjacent quantum states and the shape of the potential well (refer Figure
2.2 Molecular Spectra 17
2.2) get affected. Hence, the model for vibrational energy can be improved by considering
the effects of anharmonicities in oscillations. The total vibrational energy for an oscillating
diatomic, corrected for higher order anharmonicities can be written as [35]:
G(v) = ωe(v+1/2)ωexe(v+1/2)2(2.23)
where, xeis the anharmonicity constant.
After this correction for anharmonicity, energy spacing in Morse potential well decreases.
Moreover, this correction modifies the selection rule for vibrational transitions which guar-
antees finite probabilities for
v=2,3,...
and higher transitions, though with increase in
magnitude of vthese probabilities drop rapidly.
Rovibrational Spectra
Born-Oppenheimer Approximation:
For rudimentary understandings on rovibration, the
vibrating rigid rotor model coming from the Born–Oppenheimer approximation can be dis-
cussed, which considers vibration and rotation as two independent phenomena. Rovibrational
transitions are associated with simultaneous changes in vibrational and rotational quantum
numbers,
v
and
J
respectively. Both the phenomena being independent the total energy for
such an Rovibrational transition,
T(v,J)
, is just the sum of the energy for a rigid rotor,
F(J)
(2.19), and that for a SHO, G(v)(2.22) [35].
T(v,J) = ERR +ESHO
=F(J) + G(v)
=BJ(J+1) + ωe(v+1/2)
(2.24)
The same set of selection rules applied for the rigid rotor and SHO models, are also valid for
the combined rovibrational transition under Born–Oppenheimer approximation:
v= +1
J=±1
Spectral Branches:
According to the selection rule discussed before, the rotational quan-
tum number
J
can either decrease or increase by 1 for an allowed transition and hence, two
different branches of line positions turn up as shown in Figure 2.5. A decrease in rotational
quantum number (
J<J′′
,
J=1
) leads to the
P
branch, whereas an increase (
J>J′′
,
J= +1
) results in the
R
branch. And the gap between the lowest lines in the P and R
branches is commonly known as the “null gap”.
18 Laser Absorption Spectroscopy: Theory
Fig. 2.5 Energy level diagram for P and R absorption transitions from the ground vibrational
state for a typical heteronuclear diatomic molecule [35]
2.2.2 Polyatomic Molecule
The interaction of electromagnetic radiation with polyatomic molecules is almost similar
to that with diatomic molecules as discussed previously. At certain characteristic frequencies,
also known as resonant frequencies, molecular vibrations and rotations lead to changes
in electric dipole moments of the molecules. At these resonant frequencies, molecules
can essentially have interaction with the incident radiation through absorption, emission,
or scattering. The major difference from diatomics is that polyatomic molecules have
more number of rotational and vibrational modes due to their structural complexities, and
consequently each of these modes result in further plausible resonances.
Rotational Spectra
A molecule is distinguished by three mutually orthonormal principal axes of molecular
rotation and three principal moments of inertia,
IA
,
IB
, and
IC
, are defined about these axes
as shown in Figure 2.6. Hence three rotational constants
A,B,C
exist corresponding to each
moment of inertia as [35]:
A=h
8π2IAc,B=h
8π2IBc,C=h
8π2ICccm1
There are several arrangements with which the atoms can construct a polyatomic molecule.
Depending on different atomic configuration, relative magnitudes of moments of inertia,
net rotational energy and hence the line position get altered. That means, this arrangement
is an important deciding factor of the nature of rotational transition we can expect from a
polyatomic molecule.
2.2 Molecular Spectra 19
Fig. 2.6 Three orthonormal principal axes and moments of inertia for a polyatomic molecule,
ammonia (The molecule’s defining symmetry lies along the
A
-axis, which is unique for a
molecule) [35]
Table 2.2 summarizes all the important quantities associated with rotational spectra of
polyatomic molecules having different atomic arrangements. It should be noted that the
quantity
K
used in Table 2.2 is angular momentum quantum number which can take any
integral values from
J
to
+J
including zero, and is a consequence of
2J+1
degeneracy
of each
J
rotational level. Readers are encouraged to refer to [
35
] for more details in this
context.
Table 2.2 Various parameters of rotational spectra of polyatomic molecule with different
atomic configurations [35], [56]
Type Description Energy F(J,K)Line Position ¯vJ,K, cm1Examples
Linear IB=IC,IA=0BJ(J+1)DJ 2(J+1)22B(J′′ +1)4D(J′′ +1)3OCS, HCN, HC2Cl
Symmetric Top IA6=IB=IC,IA6=0BJ(J+1) + (AB)K22B(J+1)CH3F (Prolate), BCl3(Oblate)
Spherical Top IA=IB=IC6=0 N/A N/A CH4
Asymmetric Rotor IA6=IB6=IC6=0 Tabulated Tabulated H2O, NO2
Vibrational Spectra
Depending on the arrangement and number of atoms, polyatomic molecules have several
different vibrational modes. The existence and nature of these distinguished modes have
strong influences on the vibrational bands in the spectra of these molecules, as shown in
Figure 2.7. For a polyatomic molecule with
N
atoms,
3N
modes can be specified, out of
which
3
are translational,
3
are rotational (
2
if linear) and thus
3N6
are vibrational (
3N5
if linear). The different types of bands appearing in a particular spectrum are all based on the
vibrational modes discussed above. These different types of bands are classified as follows:
Fundamental Bands vi, the ith vibrational mode; v=vv′′ =1 for the ith mode
20 Laser Absorption Spectroscopy: Theory
First Overtone 2vi;v=vv′′ =2 for the ith mode
Second Overtone 3vi;v=vv′′ =3 for the ith mode
Combination Bands Changes in multiple quantum numbers, eg.
v1+v2:v1=v2=1
2v1+v2:v1= +2 and v2= +1
Difference Bands Quantum number generally changes with mixed sign, eg.
v1v2:v1=±1 and v2=1, and vice versa
Fig. 2.7 Different vibrational modes of water molecule (Note that, in Parallel (
k
) and perpen-
dicular (
) bands, the vibrations take place parallel and perpendicular to the main axis of
symmetry respectively) [35]
It is quite obvious that the fundamental vibrational bands are generally much dominant
compared to the combination, difference, and overtone bands. For proper harmonic molecules
like CO, the fundamental bands are stronger approximately two orders of magnitude than
overtone bands. Though for typical anharmonic molecules like NH
3
, the difference in relative
strength between the fundamental and overtone or combination bands is roughly just one
order of magnitude or less.
Rovibrational Spectra
As discussed for diatomic molecules (2.2.1), when both rotational and vibrational quantum
number change then rovibrational spectra are obtained with different branches (
P,Q,R
) for
polyatomic molecules also. For polyatomic rotational transitions atomic arrangement plays
an important role. Again, for each atomic configuration, there are parallel and perpendicular
modes of vibration, and for each parallel band there can be symmetric and asymmetric stretch.
2.3 Spectral Lineshape 21
These make this particular analysis highly intricate and an elaborated description on this can
be obtained in [35].
Fig. 2.8 Generated absorption spectra (in terms of linestrengths) of pure water molecule in
the range of 1.3-1.5 µm (6666.7-7692.3 cm1) at 300 K and 1 atm
Figure 2.8 shows the absorption transitions of pure water molecule in wavelength range
of 1.3-1.5
µ
m, where the
v1+v3
combination and
2v1
and
2v3
overtone bands of H
2
O
vibrational absorption spectra coincide with the widely used telecommunication bands. This
line survey shows only the transitions having linestrength more than 104cm2atm1.
2.3 Spectral Lineshape
Though the absorption should ideally occur only at a particular frequency, but due to vari-
ous non-ideal effects (discussed in the next subsection), a distribution of spectral absorption
coefficient is obtained about the center frequency. The lineshape function
φ(ν)
can be taken
as primary measure of how the spectral absorption coefficient varies with frequency (i.e. the
shape of an isolated absorption line) and appears directly in Beer-Lambert’s Law (2.14) as:
φ(ν) = kν
Si(T)kνdν(2.25)
22 Laser Absorption Spectroscopy: Theory
This variation with frequency is essentially caused by various physical mechanisms in the
medium due to molecular interactions which caused the broadening of lines. To predict the
accurate form of lineshape functions these broadening mechanisms should be understood.
By definition, lineshape is nothing but the probability density function and hence its integral
over entire frequency domain is unity, given by:
Z
φ(ν)dν=1 (2.26)
Fig. 2.9 Sample line shape of an absorption line centered at ν0[35]
Figure 2.9 is a classical example of a lineshape of an absorption transition centered at
frequency
ν0
. It is to be noted that, the lineshape attains the maximum value of
φ(ν0)
at
its linecenter frequency
ν0
. The width of a lineshape profile is generally characterized by
the
ν
which is its width (in units of frequency) at its half maximum, known as full width
at half maximum (FWHM). Nevertheless the half width at half maximum, HWHM, is also
sometimes considered to describe the width of the profile. To decide on the unit of lineshape,
the integral in Equation 2.26 is considered having no dimensions. As the units of
dν
are
generally either cm1or s1,φ(ν)must have units of cm or s, respectively.
2.3.1 Line Broadening Mechanisms
Broadening of absorption lines happens when the transition energy levels are affected by
the perturbations caused by physical mechanisms occurring in the medium or the individual
2.3 Spectral Lineshape 23
species start interacting with light. Line broadening mechanisms can be broadly seen as
either homogeneous (effect is same for all the species present) or inhomogeneous (interaction
varies for different classes or subgroups).
Among all, there are four major mechanisms responsible for line-broadening in a medium,
which are Natural, Collisional, Doppler, and Stark broadening. For all the measurement
conditions involved in the current studies, only Collisional and Doppler broadening mecha-
nisms are relevant and will be discussed in the next few subsections. Detailed discussions on
other mechanisms such as Dicke narrowing [
19
], Collisional line mixing [
23
], and Speed-
dependent collisional broadening [
66
], which can’t be neglected for some other special cases
like ultra high or ultra low pressure conditions, can be obtained in the literature cited.
Collisional Broadening (Pressure Broadening)
The energy difference between two quantum states involved in an optical transition is the
measure of the energy of that transition. For homogeneous broadening [
35
], the uncertainty
associated with these discrete energy levels can be correlated with their individual life-times
by the Heisenberg Uncertainty Principle, which puts a restriction on the uncertainty in energy
Eiof ith level by [40]:
Eih
2πτi
(2.27)
where,
τi
is the uncertainty in life-time of level
i
. The overall uncertainty associated with
a particular transition in units of frequency
ν
(the frequency uncertainty is nothing but
FWHM, discussed earlier) can be expressed in terms of the uncertainties in life-times of
upper and lower energy levels, τand τ′′ respectively, as
ν=1
2π1
τ+1
τ′′(2.28)
The life-time broadening is homogeneous because the Uncertainty Principle is applicable
to all the atoms present in a similar way. The atomic system can be modeled as a damped
oscillator assuming a Lorentzian Form to derive the resulting lineshape function φ(ν):
φL(ν) = 1
2π
ν
(νν0)2+ ( ν
2)2(2.29)
Life-time of an energy level can be shrinked due to perturbations that occur in the course of
an inelastic collision. From Equation 2.28, it is clear that as the lifetime
τi
of a molecule in
ith
level gets shortened, uncertainty becomes greater, and hence a broader lineshape is obtained
for the transition. That means, an increase in collision frequency shortens the lifetimes to
24 Laser Absorption Spectroscopy: Theory
Fig. 2.10 Figure showing a typical pressure broadening phenomenon on absorption spectra
of CH4obtained from Spectraplot simulation [90]
a greater extent and consequently transitions get broadened. Collisional broadening can
also occur when the molecular rotation and/or vibration are perturbed by elastic de-phasing
collisions, or the angular momentum vector of the dipole gets reoriented by elastic angular-
momentum altering collisions [
96
]. There are two types of broadening which are defined as
self-broadening, where collisions that take place between identical species, and collisional
broadening, where different species collide with each other. In the Figure 2.10, it is observed
that, due to an increase in pressure at a constant temperature (293 K), species concentration
(1% CH4in air) and sampling path length (20 cm), all the absorption lines get broadened.
In the current studies, interest lies on the measurement conditions well within the impact
collision limit, which considers binary and instantaneous collisions. The impact approxima-
tion (the assumption that the collisions are instantaneous) is valid when number densities are
approximately less than 5 Amagat, depending upon the species [
38
]. Under the assumptions
of
Impact Theory
, the pressure-broadened lineshape undoubtedly follows a Lorentzian
profile. The collisional FWHM given by Equation 2.28 can be expressed as:
νC=ZB
π(2.30)
2.3 Spectral Lineshape 25
where,
ZB
is the net collision frequency for different collision partners present in the entire
system. In a more practical way, the net uncertainty
νC
for the species of interest
i
in a
gaseous medium is generally modeled as the product of the total pressure and the sum of the
mole fractions χjfor each collisional partners j, and their collisional coefficients 2γijas:
νC=P
j
χj2γij(2.31)
From Equation 2.31, it is seen that collisional broadening FWHM gets scaled linearly with
the total pressure of the gas system. Therefore, this broadening mechanism is dominant at
elevated pressure conditions. Hence, collisional broadening is sometimes termed as pressure
broadening. The dependence of pressure broadening coefficient
2γ
on temperature is often
roughly modeled using power law approximation:
2γ(T) = 2γ(T0)T0
Tn
(2.32)
where
T0
is the reference temperature,
2γ(T0)
the pressure broadening coefficient at the refer-
ence temperature
T0
, and
n
is the temperature exponent, which is less than unity in general.
In this thesis, the values of
2γ(T0)
and
n
are directly obtained from HITRAN database, with
296 K as the value of reference temperature
T0
. It is to be noted that, these broadening
parameters can be experimentally obtained as well and then 300 K can also be taken as the
value of
T0
. But there are some species whose collisional broadening parameters cannot
be experimentally measured directly, then an approximation of
2γ(300K)0.1cm1bar1
and
n0.5
can be used for preliminary analysis [
35
]. So with this value of
n
in Equation
2.32, it is clear that the broadening coefficient is higher at lower temperature as it is inversely
proportional to
T
and thus, Equation 2.31 indicates that collisional broadening is higher at
lower temperature. It must be remembered that, the cases with high number densities, the
previously discussed impact approximation can break-down.
Doppler Broadening
Random thermal motion of absorbing molecules in a gas medium results in Doppler
broadening of an absorption line. If a molecule in the absorbing state is observed to have a
velocity component along the laser propagating path (i.e., in the same, or opposite direction of
the laser propagation), then this broadening phenomenon takes place. In such cases, instead
of real frequency, the absorbing species see a Doppler-shifted frequency and absorption takes
place when this Doppler-shifted frequency resonates with the absorption transition.
26 Laser Absorption Spectroscopy: Theory
Random velocities of the molecules of any gas in constant motion can be described by
the Maxwell velocity distribution function. Each set of gas molecules having same velocity
components is termed as a
velocity class
. The Maxwell velocity distribution determines the
fraction of molecules present in each velocity class. Hence, each velocity class has different
values of Doppler shift, and this makes Doppler broadening an inhomogeneous mechanism.
Thus the distribution function results in a lineshape function assuming a
Gaussian form
[35]:
φD(ν) = 2
νDrln2
πexp"4 ln 2 νν0
νD2#(2.33)
where
ν0
is the linecenter frequency for the transition and
νD
is the Doppler FWHM defined
as,
νD=ν0r8kT ln 2
mc2(2.34)
which can be simplified further by substituting the constants in the above expression,
νD7.1623 ×107ν0rT
M(2.35)
where
M
[g/mol] is the molecular mass of the absorbing species. Equation 2.35 is a conve-
nient form of Doppler FWHM which is used in all measurement calculations.
Equation 2.35 clearly suggests that the Doppler FWHM is directly proportional to the transi-
tion line-center frequency (
ν0
), and
pT/M
indicating that Doppler broadening is maximum
for high-frequency transitions of light-weight absorbing species at elevated temperatures.
Combined Collisional and Doppler Broadening (Voigt Profile)
Lorentzian and Gaussian lineshapes describe the homogeneous and inhomogeneous mech-
anisms of line-broadening respectively. Following the discussions above, it is understood that
collisional broadening usually dominates under high pressure conditions whereas, Doppler
broadening becomes dominant under high temperature conditions. But in combustion en-
vironment, where both pressure and temperature can be significantly high, neither of these
mechanisms can be neglected as both are prominent here. If a statistical independence can
be established between the collisional broadening and the thermal motion (i.e. each point on
a collision-broadened profile is further broadened by Doppler effects), the overall lineshape
function will assume a
Voigt profile
, which is the convolution of Lorentzian and Gaussian
lineshape functions [35]:
φV(ν) = Z
φD(u)φC(νu)du (2.36)
2.3 Spectral Lineshape 27
To simplify the above Equation 2.36, let us define certain parameters:
a=ln2 (νC+νN)/νDln2νC/νD
w=2ln2 (νν0)/νD
φD(ν0) = 2
νDqln2
π
y=2uln2
νD
(2.37)
Here,
a
is the ratio of Collisional width to Doppler width, called Voigt parameter and it is
an indication of the relative significance of Doppler and collisional effects in broadening,
with
a
increasing as the collisional broadening become more dominant;
w
is the distance
from the linecenter frequency normalized by the Doppler FWHM;
φD(ν0)
is the maximum
amplitude at the linecenter frequency of the Doppler lineshape profile and
y
is the integral
variable used for simplification. Using these parameters Equation 2.36 can be simplified to a
more convenient form as:
φV(ν) = 2ln 2
πνD
a
πZ
exp y2dy
a2+ (wy)2(2.38)
φV(ν) = φD(ν0)V(a,w)(2.39)
Equation 2.38 is the Voigt lineshape and
V(a,w)
in Equation 2.39 is the well-known Voigt
function. This Voigt function is tabulated in different domains or it can be calculated using
standard mathematical subroutines. For example, Humlicek algorithm [
44
] (later improved
by Martin Kuntz [
55
] and amended by Wim Ruyten [
83
]) and algorithm developed by
McClean et al.[
64
] are some of the most widely used approximations to estimate Voigt
function. Usually the voigt function doesn’t have any closed form analytical expression, but
when the applied laser is set to the linecenter (
w=0
) of a particular transition, voigt function
can be expressed in the following analytical expression:
V(a,0) = exp a2erfc(a)
=exp a2[1erf(a)] (2.40)
2.3.2 Line Shifting Mechanisms
There are physical mechanisms by which the absorption lines get shifted in frequency
domain just as the physical line-broadening mechanisms discussed in previous subsection.
Similar to broadening, pressure shift and Doppler shift are two dominant factors responsible
for line-shifting.
28 Laser Absorption Spectroscopy: Theory
Pressure Shift
Intermolecular potential of the species of interest can be perturbed by the interactions
between two collision partners. These perturbations in the intermolecular potential affect
the spacings between discrete quantum energy levels, and consequently the frequencies of
various transitions of the molecule in question also gets altered. And these differences from
the equilibrium linecenter frequencies are the result of the
pressure shift
phenomenon. It
is important to note that, the collisions with small impact parameters contributes to the line
broadening, whereas the line-shift is more significant for large impact parameters. This
implies that the elastic collisions with large impact parameters do not cause appreciable
broadening effects but can still very effectually shift the linecenter [
56
]. Similar to collisional
halfwidth (
νc
), which is proportional to pressure (
P
), mole fraction (
χ
) and a broadening
coefficient (
2γ
) as mentioned in Equation 2.31, the pressure shift (
νs
) is directly proportional
to partial pressure (pA=P
AχA), and pressure shift coefficient (δ) as:
νs=P
A
χAδA(2.41)
Note that, both
2γ
and
δ
have units of cm
1
atm
1
. From the Equation 2.41, it is clear that,
the shift
νs
is dominant at elevated pressure. The shift coefficient (
δ
) can be estimated from
a power law approximation in terms of a reference temperature
T0
similar to the broadening
coefficient (2γ) (as expressed in Equation 2.32):
δA(T) = δA(To)To
Tm
(2.42)
where,
m
is the temperature exponent. Note that while
2γ>0
, the pressure shift (
δ
) can be
either positive or negative. For example, for IR H
2
O spectra, the average value of
δ
is -0.017
cm1atm1[35].
Doppler Shift
For gas molecules having a mean velocity component relative to the propagating laser
beam, we observe a shift in the entire profile (lineshape function) governed by:
δ ν =νo
u
c(2.43)
2.4 Direct Absorption Spectroscopy (DAS) 29
where
u
is the component of the mean velocity in the direction of laser propagation and
c
the velocity of light in vacuum. By measuring this shift in frequency, gas velocity can be
measured non-intrusively.
2.4 Direct Absorption Spectroscopy (DAS)
Among the various TDLAS techniques developed over the past decades, DAS (Direct
Absorption Spectroscopy) is the most elementary laser diagnostic technique which has
achieved a huge popularity in the field of species-specific measurements of temperature,
species concentration, and flow velocity. These parameters are important to understand
underlying physics in a typical combustion process. DAS technique is generally used in
the measurement conditions having a comparatively large signal-to noise ratio (SNR). It
refers to the cases when the variations of the baseline (zero-absorption line) don’t cause any
significant problems to evaluate reference line intensity (
I0
). In order to enhance the temporal
resolution and accuracy of the temperature measurement, two different strategies (fixed- and
scanned-wavelength techniques) have been established for direct absorption method.
2.4.1 Fixed-Wavelength Direct Absorption Spectroscopy
In this technique, the frequency of the laser beam is set to the linecenter frequency of the
transition of interest, or to an appropriate position for broad absorption regions of a spectrum.
By acquiring the intensities of laser beams before and after entering the sampling region (gas
medium), the information in the course of the acquisition period can be calculated. Though
this particular technique is very simple to design and execute various aspects, discussed next,
should be taken care of.
Accuracy of this measurement technique has strong dependence on the wavelength in
question, so accurate line position must be checked ahead of each measurement. Though
in the stable laser sources provided by the modern laser techniques the wavelength can be
uniquely fixed by proper choice of operating case temperature and current (laser diode current
for diode lasers), line position can shift when the laser is operated for too long. Therefore,
instruments like wavemeters or calibration cells are required simultaneously to monitor the
real time wavelength value.
Selection of robust absorption lines with known spectral parameters such as linestrength
and lineshape enhances signal-to-noise ratios and simplifies the process of line character-
30 Laser Absorption Spectroscopy: Theory
ization. Still, keeping various measurement conditions in mind, weak transitions are also
favored in many cases. One such condition is to perform DAS measurements for species
having very high concentration, where to ensure linearity of the absorption, weak lines are to
be chosen.
Though fixed-wavelength DAS technique is substantially and successfully used in shock-
tube experiments to study chemical kinetics, the method has less robustness due to its narrow
spectral information. Consequently,this technique is usually used only when ultimate time
resolution (order of MHz) is necessary and losses due to non-absorbing transmissions can
either be neglected or corrected for.
2.4.2 Scanned-Wavelength Direct Absorption Spectroscopy
In recent days most of the laser-absorption sensor for typical combustion flow diagnostics
are developed using scanned-wavelength techniques due to their enhanced robustness and
adaptability, increased prospective for tracking the actual combustion systems.
Unlike the Fixed-Wavelength technique, in Scanned-wavelength DAS strategy the wavelength
or frequency of the laser beam is tuned over a definite range, which is executed either by
altering the operating case temperature or the current of a diode laser. An etalon can be used
to monitor this scanning range of wavelength, which depends on the type of laser used in
experiments. For example, scanning range of frequency (wavenumber) for Quantum Cascade
Lasers is normally less than 1 cm
1
and it has strong dependence on the scanning frequency,
usually scanning range of laser gets reduced with increase in scanning frequency.
Fig. 2.11 Schematic showing a typical scanned wavelength experiment
2.4 Direct Absorption Spectroscopy (DAS) 31
Fig. 2.12 Three different types of signals obtained in a typical scanned wavelength experiment
Scanned wavelength strategy can be explained by the typical schematic shown in Figure
2.11. A diode laser is driven by the laser controller, which comprises of temperature and
current controller modules. A function generator modulates the laser injection current and
thus laser wavelengths get tuned over the absorption features of interest. In order to obtain
absorption information from the spectrum, three signals are required in total, the reference
(
I0
), the transmitted (
It
) and the etalon (
Ietalon
) signals as shown in Figure 2.12. Time scale
is correlated to absolute frequency domain using the etalon signal. By selecting accurate
frequency and scanning range, the entire lineshape of an absorption transition line of a species
in gaseous phase at pressure below
20 bar can be obtained throughout the scan without
any compromise in temporal resolution.
Transformation of the intensity data from time-domain into laser frequency/wavelength
domain requires a solid etalon with a known free spectral range (FSR) as discussed before. An
etalon is the simplest version of a Fabry-Perot interferometer, where a light-beam experiences
several internal reflections between two reflecting faces, and its resultant optical transmission
(and hence reflection) is periodic in wavelength. The FSR of an etalon can be evaluated as:
FSR cm11
2nd (2.44)
where nis the refractive index of the medium between the reflecting surfaces, and dis the
normal distance by which two parallel mirrors are separated. Since the FSR is constant (the
32 Laser Absorption Spectroscopy: Theory
peak-to-peak distance in the etalon signal), it allows to establish a simple transformation
relation between time and frequency domains.
A general conclusion drawn from the discussions in last section is that the scanned-
wavelength technique compensates for some issues affecting the fixed-wavelength strategy.
But at the same time, due to the limitations in scanning frequency, this technique is not
advisable for the measurements requiring high time resolution. Moreover, in some extreme
experimental conditions at elevated pressure (40 bar) and elevated temperature (2000 K)
the scanned-wavelength method loses the detailed absorption feature of the transition in
question due to dominant line-broadening and barely exhibits any advantages over the fixed-
wavelength method. Actually in such extreme measurement conditions, the robustness of
DAS method becomes doubtful and hence another strategy, known as
Wavelength Modula-
tion Frequency (WMS)
, has been developed for such extremities [
70
], discussion of which
is beyond the scope of current research.
2.5 Absorption-based Thermometry
Temperature is the most important and fundamental parameter in any chemical kinetic
problem including combustion processes. A large number of diagnostic techniques have been
developed and applied to different combustion systems to evaluate gas-phase temperature
[
28
], [
85
], [
102
]. In past few decades, various optical and laser-based diagnostics such as laser
induced fluorescence (LIF) [
18
] and spectrum-line reversal have emerged and widely used
for temperature measurement because of their non-intrusive nature and fast time response
capability. In recent days, absorbance-based thermometry using the principles of tunable
diode-laser absorption spectroscopy (TDLAS) has firmly established its capability and
robustness for sensitive and precise temperature measurement with fine temporal resolution
(up to several micro seconds) [
59
], [
104
]. Hence, using this technique the temperature
information can be determined for various rapid combustion processes such as ignition in IC
engines, shock tube experiments, and unsteadiness in scramjet combustors.
Laser absorption spectroscopy-based line of sight (LOS) temperature measurement
techniques (thermometry) can be broadly classified into one-line, two-line and multi-line
thermometries, in accordance with the number of absorption transition(s) used in the tech-
nique. Among these, one-line and two-line thermometry strategies need the assumption of
uniform temperature distribution along the line of sight, and thus path-averaged temperature
values are obtained in these techniques. However, to infer information on the temperature
distribution with non-uniformity along the measurement path, multi-line thermometry can
2.5 Absorption-based Thermometry 33
be used. It is to be noted that for each of the strategies either the fixed-wavelength or the
scanned-wavelength scheme can be used.
2.5.1 One-line Thermometry
As the name suggests, this method needs only one transition data. There are two ways to
interpret temperature from DAS spectra obtained from a transition.
In the first way, it can be said that the one-line thermometry is based on the temperature
dependence of Doppler broadening of a particular transition. Temperature can be calculated
by rearranging Equation 2.35 once Doppler FWHM
νD
is estimated from the obtained
spectra:
T=MvD
7.1623 ×107v02
(2.45)
Essentially, this one-line thermometry gives promising results only at low pressure conditions
where the overall linewidth of the spectrum is dominated by the Doppler width. But at
atmospheric or elevated pressures, the pressure broadening dominates, hence it is extremely
difficult to get precise measurement of Doppler FWHM and measure temperature accurately.
The second and a more robust way is to get temperature from integrated area under the
DAS spectra obtained from a transition using Equation 2.17. But to use this approach the
information of pressure
P
, length of optical path
L
and mole fraction of absorbing species
χabs
is needed. Now,
P
is readily available in most of the cases,
L
can also be obtained,
though it might be a source of error in some cases, but
χabs
is rarely available. Assuming
all three parameters to be known, Equation 2.17 can be solved for linestrength
S(T)
, which
then solely depends on temperature. Then temperature can be obtained numerically using
Equation 2.6. This approach is more robust as it is less sensitive to quality of voigt fit when
compared to the former. Also, in cases where multiple linewidths exist, leading to almost
same voigt fit, the temperature deduced using first method might be entirely wrong.
2.5.2 Two-line Thermometry
The major drawback of one-line thermometry was the requirement of prior knowledge
of
χabs
which is unavailable in most of the critical conditions (eg. scramjet exhaust, rocket
combustor etc) at which these methods are supposed to be implemented. To overcome these
difficulties, two-line thermometry is developed which is an absorption ratio-based method.
Basically, by selecting two absorption lines whose linestrengths have different temperature
34 Laser Absorption Spectroscopy: Theory
dependence, the temperature can be evaluated by taking ratio of the linestrengths without
any prior information of background loss.
Using Fixed-Wavelength Scheme:
To obtain temperature using the fixed-wavelength two-
line thermometry method, two laser wavelengths are set to two different transition line-centers.
The maximum amplitudes of absorbances
α1
and
α2
at line-centers of both the transitions
can be experimentally obtained (Figure 2.13). Then Beer-Lambert law (Equarion 2.16) can
be used to define the ratio Ras:
R=α1
α2
=kv1L
kv2L=PχabsS1(T)φ(v01)L
PχabsS2(T)φ(v02)L=S1(T)φ(v01)
S2(T)φ(v02)=f(T,P)(2.46)
It is important to note that the peak absorbance ratio
R
in Equation 2.46 is not only a function
Fig. 2.13 Ratio of peak absorbances for two-line thermometry using fixed wavelength strategy
of temperature, but there is dependence on pressure and mole fraction of target species
as well through the involvement of lineshape functions in Equation 2.46. The differences
between both lineshape functions can be considered as small enough to neglect their effects
if the selected transitions are very close to each other. In that particular case Equation 2.46
can be simply expressed as the ratio of linestrengths of the transitions. By the substitution of
Equation 2.6 into 2.46, the expression of Ris obtained as follows:
R(T) = α1
α2S1(T)
S2(T)=S1(T0)
S2(T0)exp hc
k E′′
2E′′
11
T1
T0 (2.47)
2.5 Absorption-based Thermometry 35
Now the absorbance ratio
R
in Equation 2.47 depends only on temperature and the following
expression can be used to calculate:
T= hc
k(E′′
2E′′
1)
ln(R(T)) + lnS2(T0)
S1(T0)+ hc
kEn
2En
1
T0(2.48)
The line pair must be properly chosen (line selection criteria are discussed later) to
ensure that they have similar lineshape functions which further makes the peak absorbance
ratio (
R
) insensitive to
χabs
. In such cases, a calibration database in terms of the ratio (
R
)
versus temperature (
T
) and pressure (
P
) can be produced prior to the measurements using
some reference value of
χabs
. Once pressure is independently measured, the experimentally
obtained absorbance ratio can be compared with previously generated calibration database to
get the temperature value. However, at elevated pressure conditions, while calculating peak
absorbance at the selected laser set-point the contributions from nearby lines should be taken
into consideration due to dominant line broadening and mixing. Under such cases,
R=α1(ν1)
α2(ν2)=
n
i=1
Si(T)·φi(ν1)
m
j=1
Sj(T)·φj(ν2)
(2.49)
Using Scanned-Wavelength Scheme:
In this strategy, the integrated absorbances (areas)
of the selected transitions are measured under the same conditions of pressure, mole fraction
and sampling length (Figure 2.14). Hence, the ratio of the integrated absorbances becomes
independent of lineshape functions and gets scaled linearly with the linestrength ratio, which
depends on temperature only:
R=A1
A2
=S1(T)
S2(T)=f(T)(2.50)
So, the gas temperature is now evaluated using the ratio of experimentally measured
integrated absorbances (area ratio) for two transitions satisfying a set of selection criteria,
discussed later. The terms in the integrated absorbance ratio coming from the last term
of Equation 2.6 can be ignored only if the line-center frequencies of both the transitions
are close enough and in that case an analytic expression for temperature can be obtained.
The area ratio
R
in Equation 2.50 can thus be reduced to Equation 2.47, and as discussed
before, temperature is obtained from this ratio using Equation 2.48. Once gas temperature is
calculated from the linestrength ratio, mole fraction of the target species can be evaluated
36 Laser Absorption Spectroscopy: Theory
Fig. 2.14 Ratio of integrated absorbances for two-line thermometry using scanned-wavelength
strategy
from the estimated integrated absorbance (area) for any of the transition as:
χabs =Ai
PSi(T)L(2.51)
Scanned-wavelength scheme is able to capture the whole absorption feature. The baseline
(line without absorption) intensity
I0
can be estimated from the non-absorbing wings of
the spectrum (in the either side of an isolated peak) to consider various non-linearities like
laser intensity variation, background absorbance, detection gain and different non-resonant
transmission loss. On that account, the uncertainty of the evaluated temperature gets reduced
in this scheme compared to the fixed-wavelength technique.
Line Selection Criteria
To design a tunable diode laser-based (TDL) sensor for thermometry applications transition,
line selection is considered to be the most important step, and the choice of optimum
transitions can greatly improve the senor performance. Any species can have millions of
absorption transitions in the entire spectral range. Even if the wavelength range of interest is
restricted to a definite vibrational band, still thousands of potential candidate lines will exist.
It is a cumbersome and non-viable exercise to choose a transition from such a huge set of
lines manually. Hence to select the suitable transitions, an efficient and methodical computer-
based scheme, which depends on certain spectroscopic criteria, is developed in this portion
of thesis. To interpret effective design criteria and concepts, this portion focuses on line
selection rules for absorption spectroscopy-based two-line thermometry technique. However,
it is very crucial to note that, many interlinked factors bring additional complications to
2.5 Absorption-based Thermometry 37
the design rules for optimum line selection for two-line thermometry and these factors play
important role in determining the overall sensor performance for a selected line pair [
104
],
[
105
]. The important factors which must be taken into account in selecting the transitions
are: (a) strength of absorption, (b) proper spectral separation, (c) absence of interference
from nearby transitions, (d) temperature sensitivity, and (e) non-uniform effects like thermal
boundary layers. Overall selection process is significantly influenced by the interactions
of all the above mentioned factors and hence, the optimum line pair should be selected
case-by-case. Though the following criteria are valid for two-line thermometry using any
absorbing species, occasionally examples to explain some criteria are provided based on
H2O transitions.
1. The candidate lines must lie inside the spectral region rendered by the laser emis-
sion. (C1)
The selection of candidate H
2
O transition lines is restricted to the wavelength region
of 1.3-1.5
µ
m, where the
v1+v3
combination and
2v1
and
2v3
overtone bands of H
2
O
vibrational absorption spectra coincide with the widely used telecommunication bands,
and hence required diode lasers and other optical components are readily accessible
[5].
2. Both lines should have adequate absorption over the entire temperature range of
interest. (C2)
The peak absorbance for a transition occurs at ν0and from Equation 2.16,
αv,peak =Si(T)·P·χabs ·L·φ(ν0)
The empirical relation to the Voigt profile discussed before (Equation 2.40) can be
used here to estimate the lineshape function at linecenter ν0of the transition.
This criterion suggests that the peak absorption must be greater than the quantity
(NL)
·
(SNR) and for that an expected noise level (NL), and a desired signal to noise
ratio (SNR) are to be assumed. Moreover, the maximum absorption should be less than
about
0.8
to evade experimental hassles caused by optically-thick measurements [
105
].
For a sampling path length of
L
[cm], with mole fraction of target species lying in the
range of χabs,min to χabs,max, and at a total pressure of P[atm],
αv,peak =Si(T)·P·χabs,min ·L·φ(ν0)(NL)·(SNR)
and, αv,peak =Si(T)·P·χabs,max ·L·φ(ν0)0.8
38 Laser Absorption Spectroscopy: Theory
These constraint the product of linestrength and lineshape function in the temperature
range Tmin Tmax(K)as:
(NL)·(SNR)
P·χabs,min ·LSi(T)·φ(ν0)0.8
P·χabs,max ·L(2.52)
3. The potential lines must reduce the effects of non-uniformities caused by thermal
boundary layers. (C3)
The sensitivity of linestrength to temperature is influenced by thermal boundary effects,
and this can be deduced from Equation 2.6 as
dS/S
dT /T=hc
k
[E′′ E(T)]
T(2.53)
where,
E(T)
is some characteristic energy of the absorbing species and which is a
function of temperature as follows [68]:
E(T) = k
hc
T
Q(T)
d[T Q(T)]
dT (2.54)
The non-uniformity arising from the cold boundary layer has a significant impact on
the accuracy of the hot core (relatively uniform) temperature measurement. This can
be estimated from the difference in integrated absorbance along the measurement path
in the cold boundary layer [68]:
A=AcAb=Pχabs S(Tc)δZδ
0
Sdy=Pχabs ZSc
Sb
ydS
=Pχabs
hc
kZTc
Tb
yS(T)[E′′ E(T)]
T2dT
(2.55)
where
Ab
is the integrated absorbance coming from the portion of cold boundary layer,
δ
is the thickness of boundary layer and
y
is the integration variable along the boundary
layer. The subscript of
c
is used to denote quantities at the uniform core region and
b
represents variables on the boundary.
When
A<< Ab
the boundary layer does not influence the measurement. In Equation
2.55 to minimize
A
, lines whose
E′′
values lie in close vicinity of
E(T)
in the
temperature range of interest must be selected.
2.5 Absorption-based Thermometry 39
4. The selected lines must not be significantly affected by interferences coming from
nearby transitions. (C4)
It is very important to screen the candidate lines by spectral isolation from nearby
absorption transitions in order to reduce the uncertainty associated with the direct
absorption analysis. Only transitions free from significant interferences within certain
spectral range (upper limit of the frequency interval) of their line-center frequencies
are taken. In addition, if the two lines are close enough together (within certain lower
limit of frequency interval) that they get merged into a single absorption feature at
decent pressure conditions then those lines are retained.
5. Absorption ratio yielded by the selected line pair must be sensitive to the temper-
ature range of interest. (C5)
According to Equation 2.50, the linestrength ratio can be acquired from the integrated
absorbance ratio for two transitions. The ratio
R(T)
depends on the values of integrated
absorbances
A1
and
A2
, which are measured as areas under the best Voigt fit to the
experimentally obtained spectra. The standard deviation
σR
of
R
, which is a measure
of its uncertainty, is now estimated from the error propagation equation,
σ2
R
=σ2
A1R
A12
+σ2
A2R
A22
+2σ2
A1A2R
A1R
A2(2.56)
where,
R
A1
=R
A1
and R
A2
=R
A2
Neglecting the correlation between integrated absorbances
A1
and
A2
, following ap-
proximation can estimate dR/Ras:
dR
RσR
R
=sσA1
A12
+σA2
A22
(2.57)
High temperature sensitivity makes a sensor more accurate. By differentiating Equation
2.47, the relative sensitivity of linestrength ratio Rwith respect to temperature can be
obtained as:
dR/R
dT /T
=hc
k|E′′
1E′′
2|
T(2.58)
If the integrated absorbance
A
can be obtained within X
%
, the criteria to estimate
temperature with an accuracy of Y
%
in the entire range of
Tmin Tmax(K)
, constrains
40 Laser Absorption Spectroscopy: Theory
the minimum lower state energy difference:
E′′
i=E′′
1E′′
2
dR/R
dT /T
Tk
hc =X%2
Y%Tmax 1
1.4388 cm1(2.59)
It can also be seen from the Equation 2.59 that a line pair having large value of
lower state energy difference leads to high temperature sensitivity. However, this is
restricted by two practical concerns. Firstly, as temperature is estimated from the
ratio of measured absorbances for two absorption features, equal SNR is preferred for
both measurements. Lines having very high
E′′
value exhibit very small absorbance
and hence the potential for measuring the absorbance becomes the upper limit on
E′′
. Secondly, transitions with small
E′′
value have large absorbance in cold thermal
boundary layers which becomes a pragmatic lower limit on
E′′
as mentioned in
Criterion 3.
6. The absorption ratio must be single-valued in the entire temperature range and
the linestrength values of the line pair should be comparable. (C6)
If the measurement uncertainty is comparable for each of the selected absorption tran-
sitions, the absorbance ratio is best estimated. Hence, linestrength ratio is constrained
to be in the range of
R=0.25
to
R=4
throughout the entire temperature range [
65
],
though this accessible range of
R
is rather arbitrary. This criterion guarantees similar
signal-to-noise ratio (SNR) for the measured absorbances using the selected transitions.
According to Criterion 5, it is clear that larger the difference in the lower-state
energies of the candidate lines, higher the temperature sensitivity. However, lines
having high
E′′
value exhibit weak absorbances and hence small SNR. So, there is
a trade-off between the SNR and the temperature sensitivity for a given condition to
obtain the optimum measurements.
7. Candidate lines must be apparently free from interference of ambient H2O. (C7)
For a transition having strong absorbance at room temperature, purging with nitrogen
or dry air must be performed with utmost care outside the target measurement zone
so that interference from ambient water molecules get eliminated. This argument can
be generalized for any potential absorbing species present outside the measurement
zone. Occasionally even a short unpurged path in the line of sight may lead to greater
measurement uncertainty and inaccuracy. Choice of appropriate transitions having
desirable lower state energy E′′ values can mitigate this issue easily.
2.5 Absorption-based Thermometry 41
The ratio of linestrength at a particular temperature to that at reference temperature is
obtained from Equation 2.6:
S(T)
S(T0)=Q(T0)
Q(T)
T0
TexphcE′′
k1
T1
T01exp(hcν0
kT )
1exp(hcν0
kT0)(2.60)
It is noticed that linestrength ratio at different temperature in Equation 2.60 depends on
Fig. 2.15 Linestrength scaled by values at reference temperature (296 K) as a function of
temperature for H
2
O transition centered at 7185.5973 cm
1
for different lower state energy
values
lower state energy (
E′′
) values of transitions. With increase in
E′′
value, the linestrength
ratio is also increased at elevated temperature as shown in Figure 2.15. It is to be
remembered that, this criterion is applicable only for high temperature measurements.
If the linestrength value in the temperature range of 1000 - 2500 K has to be at least
3 times stronger than that at room temperature, minimum lower state energy is then
constrained as (for H2O molecule):
E′′ 1700cm1
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