## Appendix A Detailed Derivation  
The total light radiation, \(\Phi_{\mathrm{out}}\), received by the detector is proportional to the sum of the output radiation:  
\[\begin{array}{ll}\Phi_{\mathrm{out}}&=\sum\limits_{n=1}^{n}\Phi_{\mathrm{ not}}=\ f\rho\Phi_{0}\exp(-\alpha(L_{0}+L_{\mathrm{ave}}))[1+(1-f)\rho\exp(- \alpha L_{\mathrm{ave}})+(1-f)^{2}\rho^{2}\exp(-2\alpha L_{\mathrm{ave}})\\ &+(1-f)^{2}\rho^{2}\exp(-2\alpha L_{\mathrm{ave}})+(1-f)^{3}\rho^{3}\exp(-3 \alpha L_{\mathrm{ave}})+\cdot\cdot\cdot+(1-f)^{n}\rho^{n}\exp(-n\alpha L_{ \mathrm{ave}})]\\ &=\ f\rho\Phi_{0}\exp(-\alpha(L_{0}+L_{\mathrm{ave}}))\frac{1-\rho^{n}(1-f)^{ n}\exp(-n\alpha L_{\mathrm{ave}})}{1-\rho(1-f)\exp(-\alpha L_{\mathrm{ave}})}\\ &=f\Phi_{0}\exp(-\alpha(L_{0}+L_{\mathrm{ave}}))\frac{1}{1/\rho-(1-f)\exp(- \alpha L_{\mathrm{ave}})}.\end{array} \tag{1}\]  
Here, \(\rho(1-f)\) exp(\(-\alpha L_{\mathrm{ave}}\)) is less than 1 and \(\rho^{n}(1-f)^{n}\) exp(\(-n\alpha L_{\mathrm{ave}}\))\(\sim\)0. Thus, we ignored it.  
Then, \(\alpha L_{\mathrm{ave}}\) and \(\alpha L_{0}\) are usually less than 1 in gas detection (Equation (6) in the main text). The following approximation was obtained mathematically as Equation (A2) (same as Equation (8) in the main text):  
\[\exp(-\alpha L_{\mathrm{ave}})\approx 1-\alpha L_{\mathrm{ave}},\ \exp(-\alpha L_{0})\approx 1- \alpha L_{0}. \tag{2}\]  
Taking Equation (A2) into (1), the expression can be written as follow:  
\[\begin{array}{ll}\Phi_{\mathrm{out}}&=f\Phi_{0}[1-\alpha(L_{0}+L_{\mathrm{ave }})]\frac{1}{1/\rho-(1-f)(1-\alpha L_{\mathrm{ave}})}\\ &=f\Phi_{0}[1-\alpha(L_{0}+L_{\mathrm{ave}})]\frac{1/\rho-(1-f)-\alpha L_{ \mathrm{ave}}(1-f)}{[1/\rho-(1-f)]^{2}-\alpha^{2}L_{\mathrm{ave}}^{2}(1-f)^{2} }.\end{array} \tag{3}\]As \(\alpha\)Lave is less than 1, \(\alpha^{2}\)Lave\({}^{2}(1-f)^{2}\) was ignored. Then Equation (A3) can be converted into Equation (A4).  
\[\begin{array}{ll}\Phi_{\text{out}}&\approx f\[PERSON]_{0}[1-\alpha(L_{0}+L_{\text{ave }})]\frac{1/\rho-(1-f)-\alpha L_{\text{ave}}(1-f)}{[1/\rho-(1-f)]^{2}}\\ &=\frac{f\[PERSON]_{0}}{1/\rho-(1-f)}[1-\alpha(L_{0}+L_{\text{ave}})]\Big{[}1-\frac {\alpha L_{\text{ave}}(1-f)}{1/\rho-(1-f)}\Big{]}\\ &\approx\frac{f\[PERSON]_{0}}{1/\rho-(1-f)}\Big{[}1-\alpha(L_{0}+L_{\text{ave}})- \frac{\alpha L_{\text{ave}}(1-f)}{1/\rho-(1-f)}\Big{]}\\ &=\frac{f\[PERSON]_{0}}{1/\rho-(1-f)}\Big{[}1-\alpha L_{0}-\alpha L_{\text{ave}} \frac{1/\rho}{1/\rho-(1-f)}\Big{]}\\ &=\Phi^{\prime}{}_{0}\Big{[}1-\alpha\Big{(}L_{0}+\frac{1}{1-\rho(1-f)}L_{\text {ave}}\Big{)}\Big{]}.\end{array}\] (A4)  
Then, we defined a condition. If Equation (A5) (same as Equation (7) in the main text) is satisfied,  
\[\alpha\bigg{(}L_{0}+\frac{1}{1-\rho(1-f)}L_{\text{ave}}\bigg{)}<<1.\] (A5)  
Thus,  
\[1-\alpha\bigg{(}L_{0}+\frac{1}{1-\rho(1-f)}L_{\text{ave}}\bigg{)}\approx\exp \bigg{(}-\alpha\bigg{(}L_{0}+\frac{1}{1-\rho(1-f)}L_{\text{ave}}\bigg{)}\bigg{)}.\] (A6)  
Hence, \(\Phi_{\text{out}}\) can be expressed as follows, which is same as the Equation (10) in the main text.  
\[\Phi_{\text{out}}=\Phi^{\prime}{}_{0}\exp\bigg{[}-\alpha(L_{0}+\frac{1}{1- \rho(1-f)}L_{\text{ave}})\bigg{]}.\] (A7)  
Thus, the output radiation can be expressed in the same form as the Beer-Lambert law.