**2. Materials and Methods**

#### *2.1. Transmission-Intensity-Normalized Second-Derivative Spectroscopy*

A direct tunable diode laser absorption spectroscopy (dTDLAS) is a reliable means for trace gas detections as it is relatively simple in construction, easy to handle, and reliable to use [25]. The technology is based on an attenuation of laser radiation due to absorption as descript by the Lambert–Beer's law, which can be written as:

$$I(\nu) = I\_0(\nu) \cdot \exp[-\varepsilon(\nu) \cdot L \cdot \mathbb{C}],\tag{1}$$

where *I*0(*ν*) is the incident intensity of the laser radiation of frequency *ν*. After passing through an absorbing medium, where optical path length is *L* and gas concentration is *C*, the transmitted intensity *I*(*ν*) is detected. The concentration-normalized absorption coefficient *ε*(*ν*) can be described by Equation (2):

$$
\varepsilon(\nu) = \mathcal{S}(T) \cdot P \cdot \phi(P, T, \nu),
\tag{2}
$$

where *P* is the total pressure, *S*(*T*) is the temperature-dependent line strength, *φ*(*P*, *T*, *ν*) is the line shape function which is pressure and temperature dependent [26,27].

However, the detection limit of dTDLAS is affected by noise contributions in the measurement signal. The data analysis during concentration inversion also involves numerical division, logarithmic calculations, and possible nonlinear least-squares fitting. This type of calculation-intensive analysis is a challenge for the simple microcontrollers typically used in such measurement instruments, and slows the data acquisition rate. To improve on this, a derivative spectroscopy technique [28,29] can be applied. By processing spectral signal with second-order differential, the derivative spectral signal is obtained, and correlated with gas concentration. The transmission-intensity-normalized first and second derivatives of measurement signals can be written by Equations (3) and (4), respectively.

$$\frac{dI}{d\nu}/I = \frac{dI\_0}{d\nu}/I\_0 - L \cdot \mathbb{C} \cdot \frac{d\varepsilon}{d\nu} \tag{3}$$

$$\frac{d^2I}{d\nu^2}/I = \frac{d^2I\_0}{d\nu^2}/I\_0 + \left(L \cdot \mathbb{C} \cdot \frac{d\varepsilon}{d\nu}\right)^2 - 2 \cdot L \cdot \mathbb{C} \cdot \frac{d\varepsilon}{d\nu} \times \frac{dI\_0}{d\nu}/I\_0 - L \cdot \mathbb{C} \cdot \frac{d^2\varepsilon}{d\nu^2},\tag{4}$$

When a linearly ramp is used as drive current to a diode laser, the first term of Equation (4) is zero in an ideal case when changes of laser intensity are proportional to changes in its drive current. The residual deviation from zero is not dependent on the gas absorption and can be treated as an offset background. The values of the second and third term are zero at the center frequency of an absorption line, where the curvature slope (i.e., first derivative) is zero. The fourth term (second derivative) reaches a maximum value at the line center. Therefore, the transmission-intensity-normalized second derivative spectra have a linear relationship with the concentration of the absorbing medium.
