## 2 Absorption Spectroscopy Fundamentals  
As mentioned above, TDLAT consists of both TDLAS and CT technology. TDLAS is used to obtain the spectral information in various positions around the measurement space, and the CT technology is then used to reconstruct 2D images of the temperature and gas concentration with above transmission signals. When a laser at frequency \(\
u\) [cm\({}^{-1}\)] enters a gas sample with a path length of \(L\) [cm], the absorbance \(\alpha_{\
u}\) is expressed as [19]  
\[\alpha_{\
u}=\int_{0}^{L}P(x)C(x)S[T(x)]\phi dl \tag{1}\]  
where \(x\) is the absorbing species, \(P(x)\) [atm] is the local total pressure, \(C(x)\) is the molar fraction of the absorbing species, \(T(x)\) [K] is the local temperature, and \(\phi\) [cm] is the normalized line shape function. For atmosphere pressure and a high temperature, the line shape is usually approximated by a Voigt profile [20; 21]. The line strength of molecular transition \(S\) [\(T(x)\)] [cm\({}^{-2}\)atm\({}^{-1}\)] is a function of temperature as follows [22]:  
\[S(T)=S(T_{0})\frac{Q(T_{0})}{Q(T)}\frac{T_{0}}{T}exp\left[-\frac{hcE_{i}^{ \prime\prime}}{k}\left(\frac{1}{T}-\frac{1}{T_{0}}\right)\right]\left[1-exp \left(\frac{-hcv_{0,i}}{kT}\right)\right]\left[1-exp\left(\frac{-hcv_{0,i}}{kT_ {0}}\right)\right]^{-1} \tag{2}\]where \(Q(T)\) is the partition function of the absorbing molecule [23], \(T_{0}\) [K] is the reference temperature, \(k\) [J/K] is the [PERSON]'s constant, \(h\) [J\(\cdot\)s] is [PERSON]'s constant, \(c\) [cm/s] is the speed of light, \(v_{0}\) [cm\({}^{-1}\)] is the line-center frequency, and \(E^{\prime}\) [cm\({}^{-1}\)] is the lower state energy of the transition \(v_{0}\). Because the line-shape function \(\phi\) is normalized \(\int\)\(\phi\)\(dv\)\(\equiv\) 1, the integrated absorbance \(A_{v}\) [cm\({}^{-1}\)] can be inferred from Equation (1). \[A_{v}=\int_{-\infty}^{+\infty}\alpha_{v}dv=\)\(\int_{0}^{L}P(x)C(x)S[T(x)]dl \tag{3}\]  
The integrated absorbance of two transition lines is measured simultaneously with the same species mole fraction, the same pressure, the same mole fraction, and the same path length. The above two absorbance ratios can be further simplified to the ratio of line strengths \(R\), which is a function of temperature only as expressed Equation (4) named double-line thermometry [24]. \[R(T)=\frac{A_{1}}{A_{2}}=\frac{\alpha_{v1}}{\alpha_{v2}}=\frac{S_{1}(T)}{S_{2 }(T)}=\frac{S_{1}(T_{0})}{S_{2}(T_{0})}exp\left[-\frac{hc}{k}(E_{1}^{\prime \prime}-E_{2}^{\prime\prime})\left(\frac{1}{T}-\frac{1}{T_{0}}\right)\right] \tag{4}\]  
Then, the temperature can be calculated at different temperature dependences:  
\[T=\frac{\frac{hc}{k}\left(E_{2}^{\prime\prime}-E_{1}^{\prime\prime}\right)}{ \ln R+ln\frac{S_{2}(T_{0})}{S_{1}(T_{0})}+\frac{hc}{k}\frac{E_{2}^{\prime \prime}-E_{1}^{\prime\prime}}{T_{0}}}. \tag{5}\]  
The integrated absorbance areas \(A_{v}\) for each line can be calculated by Voigt line-shape fitting methods. The gas mole fraction can in turn be obtained from the absorption of either transition with the known temperature, pressure, and path-length in Equation (3). However, this double-line thermometry actually obtains a path-averaged value because of the inherent assumption of a uniform temperature distribution along the line of sight. For the 2D distributions of temperature and gas concentration, the test section is divided into \(M\times N\) grids, as shown in Figure 1. Meanwhile, the target gas concentration and temperature are assumed to be uniform in each small grid. The optical path length \(L_{i,j}\) of the \(i\)-th laser beam within the \(j\)-th grid can be calculated according to the two intersecting position of the grid and beam. It should be noted that the numbers of discretized grids, views, and laser beams are important for the accuracy of the tomographic image. According to Equation (3), the integrated absorbance of the \(i\)-th laser beam \(A_{v,i}\) can be expressed as  
\[