Theory of Laser Absorption Spectroscopy and Quartz Tuning Fork Detector

When detecting the concentration of chemical gases using optical spectroscopy sensors, the mutual absorption process between incident laser light and gas absorption species satisfies the well-known Lambert–Beer law, which is usually used for theoretical analysis and data processing [7]. When an excited laser with an emitting wavelength λ� passes through a uniform gas medium, the relationship between the incident light intensity I0I0 and transmitted light intensity II can be described with the Lambert–Beer law, and the corresponding mathematical expression is:

where α(λ)�� is the absorption coefficient of a specific substance at a wavelength λ�, C C is the concentration of the chemical gas to be measured, and LL is the effective light path for the interaction between the laser light and the chemical gas. For evenly distributed gas molecules, the absorption coefficient, the absorption line shape, the line strength and the number of molecules satisfy:

where S(T)ST is the gas molecular absorption line intensity at temperature T(K), N(T,P) NT,P is the number density of the gas molecule, and ϕ(λ−λ0)��−�0 is the gas molecular absorption line shape. Generally, the gas molecule number density N(T,P)NT,P can be described as:

where N₀ = 2.678 × 10¹⁹ (mol/cm³ ·atm) is the Avogadro constant, T_ref is the reference temperature, which is usually taken as 276 K, and T is the actual laboratory temperature. P₀ is the reference pressure, which is usually taken as 1 atm, and P is the actual sample gas pressure. The molecular absorption line shape usually depends on experimental conditions, especially for the broadening effects. Typically, both Doppler and collisional broadening effects are significant, and neither can be neglected, while natural broadening is much less significant than collisional broadening and can be completely neglected. Thus, a representative line shape that is a convolution of Doppler and collisional broadening called the Voigt function is widely used. Moreover, the gas molecular absorption line shape ϕ(λ−λ0)��−�0 satisfies the normalization condition:

According to the Lambert–Beer law described above, the integral absorbance area AA of a single specific molecule transition can be described as:

When the relevant experimental conditions (such as temperature T, pressure P, optical path L, and spectral line parameters) are known and the molecule absorption spectrum is measured, the number or concentration of absorption gas molecules can be calculated by combing with a line shape-fitting algorithm and Equation (5).

Moreover, the quartz tuning fork is used as a light detector by employing its resonant effect and piezoelectric effect [8]. The high resonant frequency (typically ~32.768 kHz) allows for a good noise suppression effect. To realize the laser spectral signal detection, the excited laser source should be pulsed or modulated for a continuous-wave laser, and a pulsed repetitive rate or modulated frequency will be set to match with the resonant frequency of the quartz tuning fork detector. Because of the same frequency condition, the transmitted light beam can excite the resonant effect of the quartz tuning fork, and the mechanical resonance process of the quartz tuning fork will produce a piezoelectric current due to its piezoelectric effect. The piezoelectric current can be measured and converted into a voltage signal using a low-noise preamplifier. Theoretically, the mechanical model of the quartz tuning fork can be simplified as a second-order damping-mass-spring system; its effective mass m can be expressed by its own density ρ�, length l�, width w�, and thickness hℎ. The mathematical relationship between mass and geometric parameters is expressed as follows:

Moreover, the relationship between the resonance frequency of the quartz tuning fork and the mechanical parameters is:

where Y represents Young’s modulus of the quartz crystal and k represents the elastic coefficient of the quartz crystal, which is described as:

When the relevant parameters of the quartz tuning fork are constant, the corresponding resonance frequency of the quartz tuning fork can be calculated and analyzed using finite element simulation software. Moreover, the quality factor Q (i.e., Q-value) is another key parameter to judge the performance of the quartz tuning fork-based detector, which expresses the loss of vibration energy or the amount of the damping process. The Q-value can be determined from the experimentally measured resonant profile according to the following equation:

where Δf� is the frequency bandwidth (full width at half maximum) at 1/2–√1/2 of the maximum signal amplitude, typically a few Hz.