# Laser Absorption Sensing Systems: Challenges, Modeling, and Design Optimization
## 3 Fundamentals of Different LAS Measurement Schemes
### 2f-WMS Method  
By adding a high-frequency modulation to the laser injection current, the WMS scheme raises the signal detection band to a designated frequency range and later extracts the signal with band-pass filtering processes. This allows the method to have better noise rejection capability, and a 2-100 improvement in SNR has been demonstrated over DAS [55].  
When the absorption features introduce distortions to the sinusoidally modulated laser intensity, harmonic components at integer multiples of the modulation frequency would appear. Due to the symmetry of an absorption feature about its line center, even orders of harmonic signals would peak near the line center position. For small modulation depths, the nonlinear laser intensity modulation can be neglected, and the peak of the second harmonic signal, or the 2\(f\) peak height, is a function of species concentration, pressure, and temperature and therefore can be taken as the target signal to measure [56; 57]. The dependence on multiple non-precalibrated parameters is the major limitation of 2_f_-WMS method, however, for two-line thermometry, the 2\(f\) peak height ratio of the two selected lines can be directly reduced to linestrength ratio with the use of suitable modulation depth and line pair. Figure 4 shows the flow chart of the fitting procedure of the 2_f_-WMS method, including 2\(f\) peak ratio thermometry and species concentration extraction. To account for the unknown proportionality to environmentally dependent factors such as transmission losses and instantaneous laser intensity, the 2\(f\) peak magnitudes need to be pre-calibrated on the basis of HITRAN/HITEMP database line parameters with known pressure and temperature conditions and a nominal value of gas concentration.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Technique** & **Parameter Initialization** & **Parameter Predetermined** & **Feature/Limitation** \\ \hline DAS LM algorithm & \(v_{0},T,\ X\) & no & Easy to fit multiple absorption lines \\ \hline DAS AI algorithm & no & \(\Delta v\) & Complicated to fit multiple absorption lines, a factor of 3-4 faster than the LM algorithm \\ \hline
2_f_-WMS & \(T,\ X\) & \(P\) & Calibrated on the basis of database, hard to calculate absolute \(T\) \\ \hline Calibration-free & & & Wavelength-dependent laser characteristics may lead to measurement errors \\ WMS-2_f_/1\(f\) with fixed-wavelength laser characterization & \(v_{0},T,\ X\) & \(v(t),\ M_{1}(t)\) & Non-linear laser intensity variation along measurement beam path \\ \hline Calibration-free & & & \\ WMS-2_f_/1\(f\) with scanned-wavelength laser characterization & & & \\ \hline Recovery of & & & \\ DAS lineshapes & \(v_{0},T,\ X\) & \(\varphi_{1},X_{nf},Y_{nf}\) & Higher-order harmonics come with low SNR \\ \hline DAS-calibrated WMS & no & & Direct, on-the-fly calibration, increased precision and SNR for trace \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of fitting algorithms in direct absorption spectroscopy (DAS) and wavelength modulation spectroscopy (WMS) techniques. LM: Levenberg–Marquardt, AI: advanced integrative, SNR: signal-to-noise-ratio.