# Design of a Laser Driver and Its Application in Gas Sensing
## 3 Application of the Laser Driver
### Basic Theory  
Basic principles of absorption spectroscopy, such as the Beer-Lambert law and the modulation of the semiconductor laser have been well-described in earlier studies. The well-known contents will not be repeated here. Instead, we will focus on the improvements of TDLAS-WMS brought by logarithmic transformation and differential detection. After passing through the fiber beam splitter, the active beam experienced gas absorption and the auxiliary beam used as the reference were guided to detector D1 and D2, which generated the currents named \(I_{1}\) and \(I_{2}\):  
\[I_{1}(t)=K_{1}I_{0}(t)\exp[-PS(T)g_{L}(\
u)\chi L], \tag{5}\]  
\[I_{2}(t)=K_{2}I_{0}(t). \tag{6}\]  
In Equation (5), \(P\) is the total pressure, \(S(T)\) is the line strength at temperature \(T\), \(\chi\) is the species mole fraction and \(L\) is the length of the optical path. \(K_{1}\) and \(K_{2}\) respectively represent the total gain of the active beam and the auxiliary beam, taking the splitting ratio of the modulated light intensity \(I_{0}\), transmittance of the gas cell and responsivity of the detector into account.  
The circuit designed based on LOG114 was responsible for calculating the logarithmic ratio of \(I_{2}\) and \(I_{1}\), and generated the output as  
\[\begin{split} V_{LOGOUT}&=0.375\Big{(}1+\frac{R_{ 2}}{K_{1}}\Big{)}\ln\Big{(}\frac{I_{2}}{I_{1}}\Big{)}\\ &=0.375\Big{(}1+\frac{R_{2}}{K_{1}}\Big{)}\Big{[}\ln\Big{(}\frac {K_{2}}{K_{1}}\Big{)}+PS(T)g_{L}(\
u)\chi L\Big{]}\end{split} \tag{7}\]  
where \(R_{2}\) and \(R_{1}\) are resistors used to adjust the output amplitude to make it suitable for the subsequent signal processing. Theoretically, adjusting the VOA finely in the absence of a gas sample could entirely eliminate the DC bias that is related to \(K_{1}\) and \(K_{2}\), and thus obtains the optimal result. Actually, benefitting from the phase-sensitive detection of the lock-in amplifier, a tiny DC bias will not affect the final result. In the lock-in amplifier, \(V_{\text{LOGOUT}}\) is first multiplied by the in-phase and orthogonal components of the referenced signal, and then the AC terms are filtered out from the product. The retained in-phase and orthogonal DC terms are written as  
\[X_{\text{in-phase}}=\frac{A_{m}}{2}\cos(\gamma)PS(T)\chi LC_{2}, \tag{8}\]  
Figure 7: Schematic diagram of the experimental setup.  
\[Y_{\text{orthogonal}}=\frac{A_{m}}{2}\sin(\gamma)PS(T)\chi LC_{2}. \tag{9}\]  
In Equations (8) and (9), \(C_{2}\) is a Fourier coefficient of \(g_{L}\), which is shape function of the spectrum:  
\[g_{L}(\
u)=\sum_{k=0}^{\infty}C_{k}\cos(k\omega_{c}t). \tag{10}\]  
In Equation (10), \(\
u\) is the instantaneous wavenumber of laser and \(\omega_{c}\) is the angular frequency of wavelength modulation. For eliminating the influence of the detecting phase \(\gamma\), the sum of square roots of \(X_{\text{in-phase}}\) and \(Y_{\text{orthogonal}}\) is used to calculate \(R_{2f}\), which represents the amplitude of the second harmonic:  
\[R_{2f}=\sqrt{X_{\text{in-phase}}^{2}+Y_{\text{orthogonal}}^{2}}=\frac{A_{m}}{2} PS(T)\chi L|C_{2}|. \tag{11}\]  
Further analysis had been carried out for the relationship between \(\,\text{$1\,C_{2}$}\,\text{$\mid$}\) and \(\chi\). The result in Figure 8 shows that although \(\chi\) increased by 25 times (400 ppm to 1%), the change of \(C_{2}\) is almost negligible. Except for \(C_{2}\), the remaining parameters in Equation (11) are fixed and independent of \(\chi\). Therefore, an approximately linear relationship is established between \(R_{2f}\) and \(\chi\).