*2.2. Basics of Laser Absorption*

The theory of direct absorption spectroscopy is well-understood and is only briefly reviewed to clarify the notation and units [26,27]. For a more detailed introduction, we refer to textbooks on the subject [28,29]. Measurements are based on the Beer–Lambert law, which describes the relation between incident laser light intensity *I*<sup>0</sup> and transmitted laser light intensity *It* at wavenumber *<sup>ν</sup>* as it passes through a gas medium on a path length *<sup>L</sup>* as:

$$\left(\frac{I\_l}{I\_0}\right) = \mathbf{e}^{-a\_{\overline{v}}} = \mathbf{e}^{-k\_{\overline{v}}L},\tag{1}$$

where the spectral absorbance *<sup>α</sup>v* is the product of *<sup>L</sup>* and the spectral absorption coefficient *kv* (cm<sup>−</sup>1), which is defined according to:

$$k\_{\overline{\upsilon}} = p \propto\_i S(T, \widetilde{\upsilon}\_0) \,\,\phi\_{\overline{\upsilon}} \tag{2}$$

where *<sup>p</sup>* (bar) is the pressure, *xi* the mole fraction of the absorbing species *<sup>i</sup>*, *<sup>S</sup>*(*T*, *<sup>v</sup>*0) (cm−<sup>2</sup> bar−1) the line strength dependent on temperature *<sup>T</sup>* and line-center wavenumber *<sup>υ</sup>*<sup>0</sup> (cm<sup>−</sup>1), and *φυ* (cm) the line-shape function, which is normalized with +∞ <sup>−</sup><sup>∞</sup> *φυdν* <sup>≡</sup> 1.

For a single transition, the absorbance can be integrated as:

$$A\_{\bar{i}} = \int\_{-\infty}^{+\infty} a\_{\bar{i}\bar{l}} \, d\tilde{\nu} = p \text{ } \mathbf{x}\_{\bar{i}} \text{ } \mathbf{S}\_{\bar{i}}(T) \text{ } L. \tag{3}$$

The line strength is given by:

$$S\_i(T) = S\_i(T\_0) \frac{Q(T\_0)}{Q(T)} \left(\frac{T\_0}{T}\right) \exp\left(-\frac{hcE\_i''}{k}\left(\frac{1}{T} - \frac{1}{T\_0}\right)\right) \frac{1 - \exp\left(\frac{-hc\overline{v}\_{0i}}{kT}\right)}{1 - \exp\left(\frac{-hc\overline{v}\_{0i}}{kT\_0}\right)},\tag{4}$$

where *T* is an arbitrary temperature, *T*<sup>0</sup> is a reference temperature, *h* is the Planck constant (J s), *c* is the speed of light (cm/s), *k* is the Boltzmann constant (J/K), *E <sup>i</sup>* is the lower-state energy (cm−1), and *Q*(*T*) is the partition function, which is also temperature-dependent and can be approximated with the following polynomial:

$$Q(T) = a + bT + cT^2 + dT^3.\tag{5}$$

Coefficients (*a*, *b*, *c*, *d*) for CO, CO2, and H2O from Ref. [30] are used.

The less congested absorption spectrum of water vapor (as compared to CO2) is used to infer temperature from the ratio of two absorption transitions with different ground state internal energy. Along a common optical path, the water mole fraction and pressure are the same for both transitions; thus, the ratio of the two integrated absorbance values can be simplified to the ratio of the respective line strengths:

$$R = \frac{A\_1}{A\_2} = \frac{S\_1(T)}{S\_2(T)} = \frac{S\_1(T\_0)}{S\_2(T\_0)} \exp\left(-\frac{h\varepsilon}{k}\left(E\_1'' - E\_2''\right)\left(\frac{1}{T} - \frac{1}{T\_0}\right)\right) \frac{1 - \exp\left(\frac{-h\bar{\alpha}\_{01}}{kT}\right)}{1 - \exp\left(\frac{-h\bar{\alpha}\_{01}}{kT\_0}\right)} \frac{1 - \exp\left(\frac{-h\bar{\alpha}\_{02}}{kT}\right)}{1 - \exp\left(\frac{-h\bar{\alpha}\_{02}}{kT}\right)}.\tag{6}$$

If the two selected transitions have similar wavelengths, the ratio of two integrated absorbances can be approximated while maintaining high accuracy as follows:

$$R = \frac{S\_1(T\_0)}{S\_2(T\_0)} \exp\left(-\frac{hc}{k}\left(E\_1'' - E\_2''\right)\left(\frac{1}{T} - \frac{1}{T\_0}\right)\right). \tag{7}$$

To infer the highest temperature measurement accuracy, the value should be as large as possible over the expected temperature range.

$$
\begin{vmatrix}
\frac{dR}{R} \\
\frac{dT}{T}
\end{vmatrix} = \left(\frac{hc}{k}\right) \frac{|E\_1'' - E\_2''|}{T} \tag{8}
$$

From Equation (7), the temperature can be obtained by:

$$T = \frac{\frac{hc}{k} \left(E\_2'' - E\_1''\right)}{\frac{hc}{kT\_0} \left(E\_2'' - E\_1''\right) + \ln \frac{S\_2(T\_0)}{S\_1(T\_0)} + \ln R}. \tag{9}$$

In Equation (2), the line-shape function *φν* (cm) is a convolution of Doppler and collisional broadening [31]:

$$\phi(\hat{\upsilon}) = \phi\_{\rm D}(\hat{\upsilon}\_0) \frac{a}{\pi} \int\_{-\infty}^{+\infty} \frac{e^{-y^2}}{a^2 + (w - y)^2} dy = \phi\_{\rm D}(\hat{\upsilon}\_0) V(a, w), \tag{10}$$

where *V*(*a*, *w*) is the Voigt function that can be numerically approximated [32]. The Voigt parameter as a measure for the relative significance of Doppler and collisional broadening is defined as *<sup>w</sup>* indicates the non-dimensional line position *<sup>φ</sup>*D(*ν*0) is the Doppler line center magnitude at *<sup>ν</sup>*<sup>0</sup> and *<sup>y</sup>* is an integration variable:

$$n = \frac{\sqrt{\ln 2} \Delta \tilde{v}\_{\mathbb{C}}}{\Delta \tilde{v}\_{\mathbb{D}}}.\tag{11}$$

$$w = \frac{2\sqrt{\ln 2}(\tilde{\nu} - \tilde{\nu}\_0)}{\Delta \tilde{\nu}\_{\rm D}},\tag{12}$$

$$\phi\_{\rm D}(\tilde{v}\_0) = \frac{2}{\Delta \tilde{v}\_{\rm D}} \sqrt{\frac{\ln 2}{\pi}},\tag{13}$$

$$y = \frac{2\mu\sqrt{\ln 2}}{\Delta \hat{v}\_{\text{D}}} \tag{14}$$

The collision-broadened line width <sup>Δ</sup>*ν*<sup>c</sup> depends on the pressure and the product of the sum of the mole fraction for each collision partner species B and its collisional broadening coefficient 2*γ*B:

$$
\Delta \hat{\nu}\_{\text{c}} = P \sum\_{\text{B}} x\_{\text{B}} 2\gamma\_{\text{B}} \tag{15}
$$

which varies with temperature

$$2\gamma\_\mathcal{B}(T) = 2\gamma\_\mathcal{B}(T\_0) \left(\frac{T\_0}{T}\right)^N,\tag{16}$$

where *T*<sup>0</sup> is the reference temperature and *N* is the temperature coefficient. The temperaturedependent Doppler broadening is defined by:

$$
\Delta \tilde{\nu}\_{\rm D} = \tilde{\nu}\_0 \left( 7.1623 \times 10^{-7} \right) \left( \frac{T}{M} \right)^{0.5} \text{ }^{\prime} \tag{17}
$$

where *M* is the molecular mass in g/mol.

When temperature has been determined by the two-color ratio method and pressure and optical path length are known, the species mole fractions can be determined from the absorption of a single transition of known spectroscopic parameters. In cases, where no isolated transition can be measured, the concentration of the target species is inferred from model calculations matching the measured absorption spectrum.

#### **3. Wavelength Selection and Data Analysis**

An important part of wavelength-multiplexed sensor design is the selection of laser wavelengths to target sections of the absorption spectrum of the target gas. In the ideal case, each laser is chosen to scan in wavelength across an individual transition of the target species that is free of interference from other chemical components of the gas along the laser line-of-sight. As noted before, in high-pressure, high-temperature hydrocarbon combustion gases, such isolated absorption transitions are not found for CO, CO2, and H2O detection. For water vapor, the pressure-broadened target transitions are blended with other water vapor transitions. For CO and CO2, the situation is more complicated, as the fundamental absorption bands of these two species strongly overlap, and the CO concentration is typically much lower than CO2 in combustion product gases. For all three species, determination of the zero-absorption transmitted laser intensity is not straightforward. Center wavenumbers and wavelengths, line strengths, and lower-state energies of the four target transitions utilized are listed in Table 1. The two NIR water vapor transitions have been previously used by several authors, e.g., [1,7,8,33], and the two MIR transitions have been used for CO and CO2 by the Hanson group [27].


**Table 1.** Spectroscopic parameters of chosen main transitions.

Using the HITEMP database [25], Figure 1 shows spectra simulations in the region of the selected wavelengths at 1 bar with a temperature of 2000 K and an optical pathlength of 60 mm. The assumed mole fractions of CO, CO2, and H2O are 0.1, 8, and 18%, respectively, typical for the burned gas effluent from hydrocarbon combustion. Figure 1a,c show the region around the two target water lines used for thermometry. The water vapor lines near 1391.67 nm (7185.59 cm<sup>−</sup>1) and 1469.29 nm (6806.03 cm−1) are free of significant interfering absorption from other components of the gas flow such as CO and CO2. However, several water transitions overlap within the depicted scan range. Most importantly, note the absorbance never goes to zero even at pressures as low as 1 bar, making the determination of the zero-absorption baseline an important task. The absorbance shown in the scan range of Figure 1b is primarily CO absorption with only modest contributions from CO2, while the scan range in Figure 1d is the reverse. Thus, simultaneous fitting of the two MIR scan ranges can return the CO and CO2 concentrations, especially since the gas temperature is known from the ratio of the two water vapor measurements.

The zero-absorption baseline is determined for the sensor by first measuring the laser transmission over the scanned-wavelength region, with nitrogen purging of the measurement line-of-sight before and after the measurement. However, detection efficiency and laser intensity transmitted for the benign no-combustion measurement can vary from the combustion gas measurement by differences in optical alignment and potential beam steering in the hot combustion gases (especially in the target gas turbine combustor test rig). Therefore, we assume a linear loss term *η* (baseline scaling factor) for each laser intensity, which is determined by iterative spectral fitting over the scan range using the algorithm described in Figure 2.

**Figure 1.** Simulated absorbance spectra for each of the four wavelength ranges used at 1.01 bar, 2000 K, 0.1% CO, 8% CO2, 18% H2O mole fraction with an optical path length of 60 mm for the spectral regions near 7185.59, 6806.03, 2059.91, and 2190.02 cm−1, respectively. The NIR spectra (**a**,**c**) consist of several water absorption lines, and each spectrum is dominated by a single strong transition. The MIR spectrum (**b**) is dominated by an isolated CO transition that builds on a CO2 and water background. MIR spectrum (**d**) consists mainly of several CO2 absorption lines, with some overlap of isolated CO and water transitions. The large number of CO2 absorption lines in this region and their superposition results in a significant baseline for the CO measurement.

The purpose of the algorithm in Figure 2 is the calculation of the transmitted laser intensity. The algorithm determines the integrated absorbance *A* for each line, the line-shape function *<sup>φ</sup>*(*ν*), and the baseline scaling factor *<sup>η</sup>*. Initial guesses are needed. For the integrated absorbance *A* (Equation 3) for each individual transition, line-center wavenumber *<sup>ν</sup>*0, and the collisional broadening coefficient <sup>Δ</sup>*ν<sup>c</sup>* values from the HITEMP database [25] are used. The relationship between laser scanning time and frequency is characterized by using an etalon placed in a third beam path (cf. Figure 4), so that the line-shape function, absorbance, and the measured incident and transmitted laser intensity can be converted from the time domain to the frequency domain. The simulated transmitted intensity versus frequency <sup>S</sup>*I*t(*ν*) is obtained with the Beer-Lambert law and baseline measured without combustion <sup>M</sup>*I*0(*ν*); to account for non-absorption losses with combustion, this baseline intensity is scaled by a fit parameter, *η*. After comparison of the simulated and the measured transmitted intensities versus frequency <sup>S</sup>*I*t(*ν*) and <sup>M</sup>*I*t(*ν*), the gas properties are determined from best-fit parameters.

**Figure 2.** Algorithm for iterative spectra fitting by comparing simulation and measurement of the transmitted intensity versus time (i.e., wavelength) of a wavelength-scanned laser.
