# Research on the Application of THz-TDS in Coal-Rock Interface Recognition
## 2 Experimental Detection
### Methods for Calculation of Optical Parameters  
As shown in Figure 1, when THz wave \(E(\omega)\) is irradiated onto the sample from air, it can be seen from [PERSON]'s law [22; 23; 24] that reflected wave \(E_{r}(\omega)\) and transmitted wave \(E_{t}(\omega)\) will be generated on the surface of the sample, and transmitted wave \(E_{t}(\omega)\) can have the same phenomenon on another surface of the sample, resulting in transmitted wave \(E_{t0}(\omega)\) and reflected wave \(E_{r0}(\omega)\). The reflected wave \(E_{r0}(\omega)\) is reflected several times inside the sample, resulting in a series of transmitted waves such as \(E_{t1}(\omega)\) and \(E_{t2}(\omega)\), which are known as echoes. All the transmitted waves superimposed together are received as THz time-domain signals containing sample information. In this experiment, THz waves are vertically incident on the surface of the sample. According to [PERSON], [PERSON] et al., who reported the physical and mathematical model of optical parameters in the THz band [25; 26; 27], when the sample thickness d is thick enough, echo signals such as \(E_{t1}(\omega)\) and \(E_{t2}(\omega)\) in the time-domain spectrum are far away from transmitted wave \(E_{t0}(\omega)\), so separation can be realized. In this experiment, the main peak signal and the echo signal \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Proportion of Coal**} & \multirow{2}{*}{**Mass (g)**} & \multicolumn{4}{c}{**Thickness (mm)**} \\ \cline{3-6}  & & **Quartz Sandstone-** & **Limestone-** & **Quartz Sandstone-** & **Limestone-** \\  & & **Anthracite (Q-A)** & **Anthracite (L-A)** & **Lignite (Q-L)** & **Lignite (L-L)** \\ \hline
0.2 & 0.04 & 1.00 & 0.98 & 0.95 & 0.95 \\ \hline
0.4 & 0.08 & 0.99 & 0.95 & 0.95 & 0.95 \\ \hline
0.6 & 0.12 & 0.99 & 1.00 & 1.01 & 0.96 \\ \hline
0.8 & 0.16 & 1.09 & 1.02 & 0.96 & 1.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coal and rock mixture information. (coal and rock mixed proportionally). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Anthracite Flakes**} & \multirow{2}{*}{**Mass (g)**} & \multicolumn{4}{c}{**Thickness (mm)**} \\ \cline{3-6}  & & **First Measure** & **Second Measure** & **Third Measure** & **Average Value** \\ \hline Af1 & 4.10 & 3.72 & 3.77 & 3.83 & 3.77 \\ \hline Af2 & 2.90 & 3.48 & 3.43 & 3.49 & 3.47 \\ \hline Af3 & 3.57 & 3.80 & 4.02 & 3.78 & 3.87 \\ \hline Af4 & 4.53 & 4.73 & 4.68 & 4.54 & 4.65 \\ \hline Af5 & 3.35 & 3.91 & 3.87 & 3.97 & 3.9 can be separated. Therefore, the \(E_{t0}(\omega)\) can be used to approximately represent the sum of the THz signals that penetrate the sample \(E_{s}(\omega)\). Two preconditions should be noted for using this model: (1) The sample compression structure in the experiment is uniform, the front and back surfaces of sample are smooth, and the samples remain parallel. (2) The response function of THz-TDS cannot change with time. In this study, the powder was thoroughly mixed to produce a smooth and parallel surface tablet. The refractive index of air is approximately set to 1. According to [PERSON]'s law:  
\[T_{1}(\omega)=\frac{2}{\widetilde{n}(\omega)+1} \tag{1}\]  
\[T_{2}(\omega)=\frac{2\widetilde{n}(\omega)}{\widetilde{n}(\omega)+1} \tag{2}\]  
\[R_{2}(\omega)=\frac{\widetilde{n}(\omega)-1}{\widetilde{n}(\omega)+1} \tag{3}\]  
\[P(\omega,d)=exp\left[-i\frac{\widetilde{n}(\omega)\omega d}{c}\right] \tag{4}\]  
where \(\widetilde{n}(\omega)\): sample complex refractive index; \(\omega\): angular frequency; d: sample thickness; \(T_{1}(\omega)\): THz wave transmission coefficient into the sample; \(T_{2}(\omega)\), \(R_{2}(\omega)\): THz wave transmission coefficient from the sample to the air, reflection coefficient; \(P(\omega,\,d)\): THz wave propagation coefficient in the sample. \(\widetilde{n}(\omega)\) can be obtained by \(n(\omega)\), \(\kappa(\omega)\):  
\[\widetilde{n}(\omega)=n(\omega)-i\kappa(\omega) \tag{5}\]  
The time-domain signal is obtained by THz detection.