The precision of laser-based sensors largely depends on the absorption strength of the selected target lines, which directly influences measurement accuracy. Therefore, selecting optimized absorption lines is critical. In this study, I will elaborate on the specific line selection criteria: 1) Target Molecule Selection: H₂O was chosen as the target molecule for TDLAS-based temperature measurement, with a designated temperature range of 500–2500 K. This range not only covers the operational temperatures of our laboratory tube furnace for experimental validation but also extends to meet the requirements of future combustion flow field studies in aerospace engines. 2) Band Selection: The chosen spectral band lacks interference from major combustion products such as CO and CO₂, and features uniformly distributed lower-state energy levels, enabling measurements across a wide temperature span. 3) Laser Selection and Initial Line Database: Based on the above analysis of Figure 1 and selection criteria (1) and (2), a DFB laser operating near 1,850 nm (1.8–2 μm) was selected, covering the 5,400–5,409 cm⁻¹ range. Absorption lines were initially chosen from HITEMP2010 with line strengths at 296 K meeting a threshold of S (T₀)≥1.0 × 10⁻³⁵ cm^-2 ∙ atm⁻¹. (4) Temperature-Based Line Screening: Within this initial database, line strengths were calculated at 100 K intervals, and line strengths meeting S(T)≥1.0 × 10⁻⁶ cm⁻² ∙ atm⁻¹ were selected to create temperature-specific databases from 500 to 2,500 K. (5) Signal-to-Noise Ratio (SNR) Optimization: For each temperature interval, lines with peak absorbance values yielding signal-to-noise ratios above 10 were further selected and consolidated to form an optimized absorption line database for temperature calculations over the 500–2,500 K range. Based on the criteria, H₂O absorption lines between 5,400.5 cm⁻¹ and 5,405.7 cm⁻¹ were selected. Because the line selection incorporated the temperature dependence of line strengths, each line in the final combined database exhibits varying intensities across temperatures, enabling accurate measurements from low to high temperatures.

Based on the line selection criteria, the simulated H₂O absorption spectra at 1,600 K (in blue) are shown in Figure 2A. Additionally, in combustion flow environments, such as aerospace engines and gas turbines, hydrocarbon-based fuels are commonly used, resulting in maximum potential interference concentrations of approximately 20% CO₂, 10% CO, and 10% CH₄. Hence, potential interference from carbon dioxide (CO₂), carbon monoxide (CO), and methane (CH₄) was considered, with their simulated absorption spectra displayed in black, red, and green, respectively. The H₂O absorption is significantly stronger than that of other gases, indicating that spectral interference from these gases can be ignored. Figure 2B further illustrates the normalized line strength distribution of the selected absorption lines for temperatures ranging from 500 to 2500 K. The variation in line strengths for each absorption line at different temperatures is distinguished by the color gradient in the heatmap, clearly indicating the applicable temperature calculation range for each line and demonstrating how the line strengths and detectable absorption lines vary with temperature, enabling temperature measurement over a wide range.

The most common methods for retrieving temperature in TDLAS combustion diagnostics are two-line thermometry [29, 30] and the Boltzmann Plot method [31–33]. However, these approaches involve computing the integrated absorbance over a wide spectral range, which can be computationally intensive. In this study, we developed a broadband spectral fitting model based on the univariate search method [34, 35] to determine temperature and H₂O concentration. This model fits multiple H₂O absorption lines across the absorption spectrum and continuous spectral bands, which has been reported in our previous work [36].

The flowchart for the broadband spectral fitting model is shown in Figure 3. First, a measured broadband spectral absorbance (denoted as Abs_mea) is recorded, and a model-based spectral absorbance (denoted as Abs_model), using Voigt profile, is simulated based on input fitting parameters such as H₂O line parameters from HITEMP2010 and experimental conditions (e.g., pressure, optical path length). A nonlinear univariate method (using the fminbnd function in MATLAB) is then employed to search for the optimal temperature (denoted as T_opt) under an initial concentration (denoted as X₀) and initial temperature (denoted as T_initial). A linear fitting between Abs_mea and Abs_model is performed to obtain the slope ( f ) and intercept ( b ) of the linear fit according to the relationship of Abs _ mea = f ∙ Abs _ model + b . The concentration is then updated as X opt = X 0 ∙ f . After jth iterations, the second norm of the fitting residuals (r _normr ) is minimized and the convergence condition (| T opt , j − 1 - T opt , j |<1、| X opt , j − 1 - X opt , j |<1 × 10⁻³) is satisfied, yielding the optimal temperature and concentration. Here, the r _normr value represents the second norm of the spectral fitting residuals, which is the square root of the sum of squared residuals. The r _normr is shown in Equation 1: r n o r m r =   ∑ j = 1 N Abs _ mea j − Abs _ mea j , f i t t e d 2 (1) Where Abs _ mea j is Abs_mea during the jth iteration, Abs _ mea j , f i t t e d is the fitting result according to Abs _ mea = f ∙ Abs _ model + b .

T opt and X opt are respectively the optimal temperature and concentration, then r _normr can be further expressed by Equation 2: arg   min T ∑ j = 1 N Abs _ mea j − f ∙ Abs _ model   j T o p t , X o p t − b 2 (2) Where Abs _ model   j is Abs _ model during the jth iteration, ∑ j = 1 N Abs _ mea j − f ∙ Abs _ model   j T o p t , X o p t − b 2 represents the objective function dependent on temperature T o p t . The term arg   min T indicates the value of T o p t that minimizes the objective function. Specifically, arg   min is an operator that finds the variable value at which the function reaches its minimum, and T o p t refers to this variable in the context.

The typical example illustrating this method is shown in Figures 4, 5. To fit the actual experimental conditions, an H₂O absorbance spectrum was simulated under the conditions of Pressure = 1 atm, X _H2O = 0.06, L = 23 cm and T = 1200 K. Gaussian noise with an amplitude of ±0.005 and a 20% error in H₂O absorption line parameters were added in the H₂O absorbance spectrum regarded as measured spectrum. Then it was input into the broadband spectral fitting model to obtain the trends for temperature, concentration, and r _normr during each iteration, as shown in Figure 4.

In the first iteration (blue curve in Figure 4A), the optimal temperature differs significantly from the true value of 1200 K, and the corresponding r _normr (green curve in Figure 4B) is also at its maximum. Figure 5A shows the Abs_mea (blue curve) and Abs_model (red curve) for the first iteration, where a poor fit between the measured and model-based spectrum is evident. Similarly, Figure 5B presents the linear fit of the spectral data for the first iteration, demonstrating a clear deviation between the measured data and the linear fit. These results indicate that the temperature and the concentration are not optimal in the first iteration. After several iterations, the temperature and concentration in the 11th iteration converge closely to the true values of 1200 K and 0.06, respectively. Furthermore, the r _normr is minimized. The spectra (Figure 5C) and the linear fit of the spectral data (Figure 5D) confirm a much-improved fit, indicating that the optimal temperature and concentration have been successfully determined.

A DFB laser (Nanoplus, S/N: 2899/03-02) operating near 1850.5 nm was selected as the laser source. A commercial laser controller (LDC501, Stanford Research Systems) was used to regulate the laser operating temperature, and a sawtooth wave signal generated by a commercial function generator (SDG2082X, SIGLENT) was applied to continuously scan the absorption lines of H₂O. The temperature measurement performance of the sensor was evaluated using a commercial tube furnace platform (GSL-1800X-III-₵50). The laser beam was split into two parts using a fiber splitter, as shown in Figure 6. Fifty percent of the beam was collimated by a fiber collimator, transmitted through the temperature test region in the furnace, and then detected by a photodetector (Thorlabs PDA10D-E-C). The other half of the beam passed through a quartz Fabry-Pérot etalon for frequency calibration and was detected by another photodetector. The generated photocurrents were converted into voltage signals using a custom-made transimpedance amplifier. The signals were then digitized using a 16-bit data acquisition card (NI PXIe-5172). To ensure a stable temperature region within the furnace, the sapphire tube was inserted into both ends of the furnace tube. Additionally, a pair of structures incorporating internal gas pipes was designed to mount the collimator. The structures allowed for purging the internal air gap with high-purity nitrogen (99.99%) to eliminate absorption interference from residual H₂O.

Seven equally spaced temperatures ranging from 600 K to 1800 K were set within the furnace’s maximum tolerance temperature of 2000 K. After reaching the preset temperatures, pure water vapor (100% purity) at three different pressures (P1–P3: 3.5–6.5 KPa) was introduced into the furnace, with the pressures monitored using a manometer (MKS628). For each temperature and pressure, 500 raw spectral signals were recorded once the pressure stabilized. Additional tests were conducted at higher pressures (P4–P6: 10–103 KPa) using a mixture of pure water vapor and high-purity nitrogen, with 500 spectral signals captured for each set condition. Non-absorbing regions of the raw spectral signals were fitted using a third-order polynomial to extract the H₂O absorbance. Figure 7A displays the measured raw data and the corresponding extracted H₂O absorbance. A spectral fitting result using the broadband spectral fitting model is shown in Figure 7B, demonstrating close agreement between the fitted and measured H₂O absorbance. But there is still a deviation between −0.02 and 0.02 as shown in Figure 7C. The reason of this deviation can be concluded as follows. DFB laser’s wide scanning range combined with spectral broadening and weak absorption lines, introduces errors in selecting non-absorbing regions, resulting in irregular baseline fluctuations that subsequently cause baseline shifts during spectral fitting, and this effect will be discussed in Section 3.2. Moreover, the spectral line parameters in HITEMP2010 have errors ranging from 5% to 20%, or even higher, and unavoidable spectral noise also contributes to the deviation. Calibration of spectral parameters and an increase in the SNR can reduce these errors, but due to the large number of lines involved, this process is still ongoing. However, as illustrated in Figure 7D, even with a deviation between −0.02 and 0.02, r _normr converges to a minimum after 13 iterations, allowing optimal temperature and concentration values to be obtained.