# Laser Absorption Sensing Systems: Challenges, Modeling, and Design Optimization
## 2 Sensor Design Targets and Measurement Challenges  
The fundamental theory lying behind absorption spectroscopy is the Beer-Lambert law, presented in Equation (1), with multiple forms of expressions. The fractional transmission of light, given by the ratio between transmitted and incident light intensity, \(I_{t}\) [W/cm\({}^{2}\)] and \(I_{0}\) [W/cm\({}^{2}\)], can be expressed as an exponential function of absorbance \(\alpha_{\
u}\), with \(k_{\
u}\) [1/cm] being the spectral absorption coefficient, \(L\) [cm] the absorption pathlength, \(n\) [molecule/cm\({}^{3}\)] the number density of the absorbing species, \(\sigma_{\
u}\) [cm\({}^{2}\)/molecule] the absorption cross section, \(S\) [cm\({}^{-2}\)/atm] the absorption linestrength of an individual transition line, \(\phi_{\
u}\) [cm] the frequency-dependent lineshape function, \(P\) [atm] the total pressure, and \(X_{i}\) the mole fraction of the absorbing species \(i\). The subscript \(\
u\) is used to signify the spectral dependence of the parameter on the light frequency \(\
u\):  
\[ \begin{array}{ll}(I_{t}/I_{0})_{\
u}&=\exp(-\alpha_{\
u})\\ &=\exp(-k_{\
u}L)\\ &=\exp(-n\sigma_{\
u}L)\\ &=\exp(-S\phi_{\
u}PX_{i}L)\end{array}  \tag{1}\]  
This rather simple but nontrivial relationship roots from the energy balance derivation in the equation of radiative transfer. In a typical absorption measurement, the total absorbance \(\alpha_{\
u}\) is the ultimate \"signal\" to optimize, and the performance of LAS species sensor is commonly evaluated through detection limits and detection bandwidth. The limits of detection can be quantified as the noise-equivalent absorbance (NEA) or the minimal detectable concentration (e.g., in ppm), and the measurement bandwidth (Hz) can be defined as half of the measurement temporal resolution taking into account the Nyquist criterion. To directly compare the detectivity of LAS sensors under different measurement conditions, the pathlength-normalized detection limit (e.g., in ppm\(\cdot\)m) is often used, on the basis of the fact that the absorbance is linearly proportional to the absorption pathlength. Further normalization with bandwidth renders a detection limit in the unit of ppm\(\cdot\cdot\cdot\)Hz\({}^{-1/2}\) to account for the improvement of signal-to-noise ratio (SNR) with averaging. However, it needs to be noted that this inverse square-root dependence on averaging time can be applied if the signal carries only white noise. A more rigorous assessment of the validity of such normalization can be done with Allan variance analysis, which analyzes a sequence of data in the time domain. Although traditionally used as a measure of frequency stability of clocks and oscillators in metrology, the Allan variance analysis can also be adopted as a useful tool to identify and quantify different noise terms and indicate signal stability. As shown in the schematic of Figure 1, the root Allan variance, aka Allan deviation, of a time domain signal is computed as a function of different averaging times \(\tau\). By analyzing the characteristic regions and log-log scale slopes of the Allan deviation curve, different noise processes and modes may be identified. Even limiting our discussion to combustion-related LAS diagnostic studies, the demands and achieved performance of the laser sensors vary over a wide range. Figure 2 summarizes some representative results from a number of sensor development and implementation studies in terms of detection limits and bandwidths. It compares species concentration measurements in a variety of combustion systems including laboratory environments [1; 16; 17; 18; 19; 20; 21; 22], shock tube studies [23; 24; 25; 26; 27; 28], industry processes [29; 30; 31; 32; 33], and engines [34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46].