The two first generations of PA cells rely on the differential Helmholtz resonator (DHR) architecture [16], composed of two identical chambers linked by two capillaries (Fig. 1). The gas inlet and outlet are connected to the middle of the capillaries. Photoacoustic excitation is ensured by lighting one of the chambers with a laser source. Although only one chamber is illuminated, the acoustic waves, opposite in phase at the Helmholtz resonance frequency, are generated in both chambers and the output signal consists in the difference of the measurements provided by two microphones, each one sensing the pressure in one chamber. This differential pressure measurement strategy provides a partial immunity to external acoustic noise.
3D-printed Cell
The acoustic model used for the design of the miniaturized cells has already been presented elsewhere [17] and only the main features are reminded in this paper. The calculations are based on the resolution of the harmonic version of the Full Linearized Navier–Stokes (FLNS) coupled equations system, composed of the linearized versions of the momentum, mass, and energy conservation equations. Unlike the simplified acoustic wave equation, this framework enables handling losses of viscous or thermal origin. Indeed, in the useful 1 to 20 kHz acoustic frequency range of interest to us, dissipative phenomena cannot be neglected because the viscous and thermal boundary layers, spanning on several tens of micrometers in air, occupy a significant proportion of the volume of the narrowest components of the acoustic network composing the photoacoustic cell.
The resolution is performed, by the finite element method (FEM), with the commercial software COMSOL Multiphysics (Comsol AB, Stockholm, Sweden). The finite element mesh of the study domain is refined in the vicinity of the walls so that several elements, usually three, are placed across the thickness of the viscous boundary layer. The heat source is obtained by the Beer–Lambert exponential attenuation law, assuming a perfectly collimated illumination of the entrance optical window. The cell is filled with air at standard temperature and pressure, whose physical properties are extracted from the material database of the software. Regarding boundary conditions, all walls are considered rigid and isothermal. The gas inlet and outlet are subject to atmospheric pressure and are not the site of any heat exchange. The pressure signal is defined as the average, on the microphone membrane, of the amplitude of the pressure field.
In this design, all the cavities are cylindrical. The diameter and length of the chamber are 750 µm and 20 mm, and those of the capillaries are 500 µm and 20 mm. The expected resonance frequency is 2.3 kHz, which is close to the 2.2 kHz measured one.
The cell, made of stainless steel, is 3D-printed by direct metal laser sintering. The cell hosts a pair of commercial top port capacitive MEMS microphones (Fig. 2), directly glued on the metal block. The output analogical signal is processed by an interface discrete electronic circuit placed on a PCB.
Various measurements, based on set-ups involving slightly different versions of the 3D-printed cell, capacitive microphones and readout configurations, and addressing several trace gases, consistently lead to Normalized Noise Equivalent Absorption (NNEA) values in the order of 10−8·cm−1·W/Hz1/2 (Table 1).
Several examples of trace gas measurements are presented. In the first set-up, an interband cascade laser (Nanoplus, Gerbrunn, Germany) was adjusted on the 2979 cm−1 absorption line of methane [13]. The methane concentration in nitrogen was varied between 1 ppm and 2000 ppm. The 2.2 mW laser power was modulated in amplitude and the integration time was fixed to 2.7 s. The metal cell was equipped with transparent barium fluoride windows. A linear regression was then performed on the measurements, leading to an estimated limit of detection LOD1σ ≈ 126 ppb. An additional measurement, made with a 0.5 ppm dilution, confirmed the ability of the component to detect sub-ppm concentrations (Fig. 3).
A second set of experiments was performed with a mixture of formaldehyde in nitrogen. The set-up involved a 80 mW quantum cascade laser (QCL) (mirSense, Palaiseau, France) tuned on the 1765 cm−1 peak of formaldehyde. The cell was equipped with sapphire windows. The gas mixture was injected, from a gas diluter developed in our laboratory, in a range from 0 ppm to 10 ppm, by steps of 1 ppm (Fig. 4). The QCL current, and thus the optical wavelength, was modulated to carry out 1-f and 2-f detection schemes. The integration time was set to 900 ms and, in both cases, based on a linear fit of the concentration range measurements, the LOD1σ was determined at ≈ 25 ppb.
Finally, a variant of the PA cell has been built in PMMA, with a geometry slightly adapted to match the lower resolution of the 3D-printing hardware (same overall dimensions but 1.5 mm diameter for all chambers and capillaries). The resonance frequency is 4.5 kHz (vs. 4.2 kHz obtained from the FEM model) and the quality factor is ≈ 3.2. The cell was equipped with silicon windows covered with a silicon nitride antireflective coating. The results of experiments performed on a range of nitric oxide mixtures in nitrogen are presented on Fig. 5. Since then, we used a GasMix commercial diluter (AlyTech, Juvisy-sur-Orge, France) to prepare the gas samples. The light source was a QCL with 100 mW optical power at 1906 cm−1 (Thorlabs, Newton, NJ, USA). A 2-f detection scheme and a short integration time of 0.2 s led to a LOD1σ ≈ 650 ppb, estimated from a linear fit of the measured data (Fig. 5 left). A basic correction for the much larger volume (152 mm3 vs. 40 mm3) and much shorter integration time (0.2 s vs. 7 s) leads to a LOD with slightly reduced performances than that of the initial metal cell.
Allan deviation plot provides information about the temporal stability of the experimental set-up as a whole [18, 19]. It presents the signal standard deviation as a function of the integration time τ and allows assessing the optimal integration time and thus the ultimate detection limit of a system. Indeed, two regimes can be noticed on the graph of Fig. 5 right: the first one, where the instrument is stable and noise decreases as \(1/\sqrt \tau\) (typical of predominant white noise source), until an optimum, marking the occurrence of a second drift regime, where longer averaging becomes detrimental. Many sources of drift can be identified as, for instance, QCL instability, thermal drift, etc. Here, we observe a signal stability up to ≈ 25 s integration time and an associated ultimate LOD1σ ≈ 15 ppb.
Microfabricated Silicon Cell
The same DHR architecture, still using commercial microphones, has then been employed to create the second-generation device [14]. Ten times smaller in volume than the previous one (Fig. 1), the PA cell has been produced from two silicon wafers patterned, etched, and bonded by the means of techniques commonly used in MEMS foundries. The layout has been adapted in order to match the specific constraints of microfabrication, such as for instance vertical etching in planar silicon substrates and limited depth-to-width aspect ratio (Fig. 6).
To design the miniaturized cells, we relied on the acoustic model mentioned in the previous section. However, the lengthy resolution of the FLNS equation system by the FEM can hardly be used directly within a numerical optimization algorithm, which requires multiple evaluations of the “black-box” simulator. Thus, in order to limit the use of computational resources by the optimization process, we decided to resort to the use of metamodels [20].
The process of metamodeling successively implies evaluating the high-fidelity FEM model at points carefully chosen in the input parameter space (design of experiment) and then training the metamodel on the obtained dataset. Once the metamodel is built, the response at a new evaluation point can be predicted instantaneously. The metamodel can thus be used as a surrogate of the expensive simulator in costly procedures, such as inversion or optimization. Kriging metamodels, based on the theory of Gaussian processes, and known for their efficiency and adaptability, have been used in this work. In addition, the efficient global optimization adaptive algorithm [21], that improves the accuracy of the metamodel during the optimization by adding new points to the initial design of experiment, has been implemented.
The design problem has been posed as a constrained optimization problem: find the set of geometrical parameters maximizing the differential signal while keeping the cell resonance frequency under a given upper bound. The length of the chamber, which governs the overall size of the photoacoustic cell, as well as its cross-section, which is imposed by the angular divergence and location of the laser source, are fixed a priori by the designer. Only four dimensions are thus left in the parameter space, namely the distance between the chambers and between the capillaries, and the height and width of the capillaries. The free software toolbox SBDO, available at https://github.com/freeSBDO/SBDOT, allowed an easy implementation of the constrained optimization problem in MATLAB (The MathWorks Inc., Natick, U.S.A.).
Four configurations, corresponding to different chamber lengths (5 mm and 8 mm) and resonance frequencies (10 kHz and 15 kHz), have been obtained. As can be seen on Fig. 7 (left), the expected signal produced by the novel silicon cells is slightly larger than that of the 3D-printed metal cell, while still keeping a resonance frequency sufficiently low to allow for the relaxation of the molecules of interest.
Two 550 µm thick silicon wafers were used to create the PA cell. The key points of the fabrication, carried out in our 200 mm MEMS production facility, consisted in (i) etching, in both wafers, the deep trenches constituting the chambers and capillaries and (ii) airtight molecular bonding of the two wafers. Metal contacts were patterned on the upper wafer in order to allow soldering the microphones and outputting the electrical signal. After soldering the two analogical AKU350 MEMS microphones (Akustika, Bosch Germany) and dicing, the PA chip was mounted on a flexible PCB and the electrical contacts were wire-bonded. Finally, fused silica capillaries were connected to the gas inlet and outlet to provide calibrated gas samples.
A specialized electronic board, including a digital lock-in amplifier and signal processing functions, was developed at this occasion to insure a high signal-to-noise ratio and to guarantee the consistency of the miniaturization of the sensor and of its associated electronics.
In a first step, the resonance frequency of the silicon cells was measured using a 5.5 mW distributed feedback QCL from mirSense (Palaiseau, France), targeted at the 2302 cm−1 carbon dioxide absorption line. Measurements were performed directly on ambient air. As can be seen on Fig. 7, all four variants resonate close to the expected resonance frequencies (maximum discrepancy is less than 9 %) and the values of the measured quality factors are comparable to the computed ones (2.2 to 3.3 vs. 2.5 to 3.9).
The microfabricated silicon cells have then been thoroughly characterized and the main operating parameters, such as LOD, stability, and NNEA, have been determined in different conditions of operation.
The first series of experiments presented here used a mixture of carbon monoxide in nitrogen. The continuous wave QCL (Alpes laser, Neuchâtel, Switzerland) emitted 75 mW at 2127.7 cm−1. Both 1-f and 2-f wavelength modulation schemes were used (Fig. 8) and the LOD1σ reached with 2.1 s integration time was, respectively, estimated at ≈ 420 ppb and ≈ 450 ppb.
A second series of measurements was carried out on nitric oxide with the experimental set-up described previously for the 3D-printed PMMA cell. This time, we were interested in the spectroscopic capabilities of the cell. Several peaks corresponding to the absorption of water (≈ 7000 ppm) and nitric oxide (20 ppm) can clearly be seen on the left part of Fig. 9. A concentration range of nitric oxide in nitrogen was also measured with a QCL optical power of 123 mW at 1903.1 cm−1 (Fig. 9, right) and using a 2-f modulation scheme. Let us note that the signal returns to zero after several increasing or decreasing concentration steps. This suggests that surface adsorption is negligible. Here, we obtained a LOD1σ ≈ 89 ppb.
In parallel, measurements, presented on Fig. 10, were conducted to assess the signal stability of the two set-ups used for the detection of carbon monoxide (blue dots) and nitric oxide (green dots). Both Allan deviation plots show a similar behavior, with a maximum integration time larger than 10 s. Regarding nitric oxide, the ultimate LOD1σ is around 9 ppb for a 12 s averaging time. Let us note that, when compared with the experiments performed with the 3D-printed PMMA cell in similar conditions (same QCL), the level of noise is increased by almost an order of magnitude.
A last set of experiments was performed on a mixture of fluorocarbon in nitrogen with a silicon cell thermally regulated at 25 °C. The QCL (Alpes laser, Neuchâtel, Switzerland) emitted 67 mW at 1283 cm−1. A 2-f wavelength modulation scheme was used and the LOD1σ was estimated at ≈ 18 ppb with an integration time of 1 s. In order to identify some sources of drift, Allan deviation plots are presented on Fig. 11 for three different experimental configurations, respectively, with the laser off, with the laser on in air, and with the laser on and gas mixture present. As can be seen, switching on the laser (blue dots) adds a significant level of white noise and constitutes a first source of drift, with a stability up to ≈ 1000 s. With the gas of interest present (green dots), the system is still quite stable, with an optimal integration time ≈ 50 s and an ultimate LOD1σ ≈ 2 ppb.