QEPAS sensor using a radial resonator

Abstract

We report on the use of a radial acoustic resonator to increase the acoustic signal of a quartz enhanced photoacoustic spectroscopy (QEPAS) sensor. This approach is an attractive alternative to usual configurations based on microtube longitudinal resonators since it enables to substantially relax laser beam alignment and positioning and thus paves the way towards more cost-effective industrial production. This new QEPAS configuration is first investigated and designed by finite element simulation. It is then experimentally implemented and characterised by detecting acetylene around 1.5 µm. The combination of the radial acoustic resonator with a custom quartz tuning fork (QTF) leads to a QEPAS sensor with a normalised noise equivalent absorption (NNEA) of $3.9 \times 10^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$ which is close to the performance of state-of-the-art QEPAS sensors based on microtube resonators.

1 Introduction

Laser based spectroscopy techniques are widely used in the gas detection industry. These techniques enable a good selectivity to discriminate molecular species and excellent sensitivities down to parts-per-billion in volume (ppbv) [1]. Among these laser-based techniques, photoacoustic spectroscopy offers many advantages for local gas sensing. First, since it does not rely on direct photodetection but rather on the detection of acoustic waves generated by gas absorption/relaxation cycles [2], it can be implemented at any optical wavelength with similar acoustic detectors. In particular, photoacoustic spectroscopy enables to carry out gas sensing in the midwave and longwave infrared where many molecules exhibit strong absorption lines while direct optical detection would usually require cryogenic cooling of the photodetector. Other advantages include a balance between compactness and sensitivity and a high dynamic range.

To detect the acoustic waves, different techniques have been investigated. Microphone based photoacoustic spectroscopy (MPAS) was introduced first. To reach high sensitivity with MPAS, the pressure has generally been amplified using acoustic cells in which the laser beam passes [3]. In a very similar way, cantilevers have also been designed to detect acoustic waves in cantilever enhanced photoacoustic spectroscopy (CEPAS) [4]. Eventually, in 2002 it was proposed to replace microphones by quartz tuning forks (QTF) [5] in quartz enhanced photoacoustic spectroscopy (QEPAS). In QEPAS, the main configuration, called “on-beam”, consists in focusing the laser beam between the fork prongs.

To compare these techniques sensitivity, a figure of merit referred as the normalised noise equivalent absorption (NNEA) was introduced. The NNEA corresponds to the signal-to-noise ratio normalised by the generated signal and measurement bandwidth. While the photoacoustic signal is usually used for normalisation, we rather use the generated heat H to avoid possible discrepancies while taking the different modulation schemes into account as detailed in Appendix A. It is defined here as follows:

$$\begin{aligned} \mathrm {NNEA} = \frac{H}{\mathrm {SNR} \,\times \sqrt{ {\Delta } f}}. \end{aligned}$$

(1)

H is the heat amplitude generated in the medium in W cm$^{-1}$ due to gas excitation and relaxation. It takes into account the laser mean power, the overlap of the laser emission spectrum with the gas absorption but also the gas relaxation rate [6].

Using this figure of merit, MPAS and CEPAS literature setups reached a state-of-the-art NNEA around 10$^{-10}$ W cm$^{-1}$ Hz$^{-1/2}$ [4, 7, 8]. In QEPAS, watch QTFs were used at first [5, 9], due to their off-the-shelf availability and low cost. For such setups, sensitivities as good as $3 \times 10^{-8}$ W cm$^{-1}$ Hz$^{-1/2}$ have been reported [10, 11] which is still two orders of magnitude worse than for MPAS and CEPAS. Later, custom QTFs have been specially designed for QEPAS. Indeed, there is a priori no reason to assume that watch QTFs are particularly well suited to photoacoustics, as they are mostly used in vacuum for time reference applications. Moreover, they present a small gap between the prongs, near 300 µm which requires tiny laser beam sizes, which can be difficult to achieve, particularly for long wavelengths. Eventually, they present a high resonance frequency near 32,768 Hz that reduces the generated heat when measuring gases with slow relaxation rates [12]. To circumvent these issues, custom QTFs were designed with a larger gap between the prongs, up to 2 mm, and smaller resonance frequencies, down to a few kHz [13, 14]. In addition, parametric studies maximised their quality factor at atmospheric pressure or their sensitivity [11, 15]. Nonetheless, the sensitivity was not significantly improved and the best reported ones also reached around $1 \times 10^{-8}$ W cm$^{-1}$ Hz$^{-1/2}$ [11, 16].

While this sensitivity is sufficient for many applications, it is still not for others such as air quality analysis. To enhance the sensor detection limit, acoustic resonators have been used in QEPAS in a similar way to MPAS. To increase the pressure amplitude seen by the QTF, longitudinal acoustic resonators, under the shape of tubes in which the laser beam passes, are placed around the QTF (see Fig. 1 Left). This main configuration called “on-beam”[5] has enabled typical sensitivity gains of 15–60 [10, 17], leading to a state-of-the-art NNEA around $1 \times 10^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$ [6, 10, 18, 19]. However, this configuration presents significant drawbacks, particularly for industrialisation. Indeed, since the optimal parameters correspond to tubes with a diameter of typically 500 µm and a length of about 1 cm, focusing the laser beam inside such tubes is challenging for industrial system integration. It is even more challenging in the mid-infrared, which is a wavelength range of interest where many gas species display strong and specific absorption features. The laser beam can illuminate the tubes and generate an undesirable acoustic background but also a loss of power transmitted through the tubes. Furthermore, these tubes are difficult to machine, to align, and render the system less robust. Nonetheless, to our best knowledge, these resonators are the only ones used in QEPAS. Only variations are found in terms of tubes positioning, such as “off-beam” [20, 21] or “T-shape” [22] setups. These last configurations enable to avoid QTF illumination by the laser beam by placing the acoustic resonators next to the QTF and demonstrated similar NNEA as for “on-beam”. However, even if microtubes with slightly larger diameters (less than 1 mm) can be used, comparable difficulties still occur concerning laser beam alignment.

Fig. 1

(Left) Schematic of QEPAS on-beam configuration. Tubes (grey) are placed from each side of the QTF (brown) and the laser in focused inside them (red). (Right) Schematic of the investigated QEPAS configuration combining a radial acoustic resonator and a custom QTF

In the following, we investigate a new kind of resonator for QEPAS which combines a radial acoustic resonator that has the advantage of being less sensitive to laser beam alignment and a custom QTF (see Fig. 1 Right).

2 Previous developments: custom QTF and acoustic recovery cavity

This first acoustic resonator is adapted to one of our custom QTF named C2 [15] for which several studies have been reported [11, 15, 23]. To underline this QTF and the acoustic recovery cavity specifics, we will summarise here the main results we obtained as well as comparisons with classic configurations.

2.1 Custom QTF description

C2 QTF was designed to present a good coupling with acoustic waves. In order to maximise the pressure amplitude differential between external and internal QTF faces, the prong width was chosen equal to half the acoustic wavelength, i.e. $\lambda _{\mathrm {ac}} / 2$ [15]. Moreover, to facilitate laser beam alignment at long wavelengths or for strongly spatially diverging sources, it presents the largest reported gap of 2 mm. Furthermore, in compliance with these considerations, the QTF was designed to resonate around 21 kHz to mitigate the limitations due to gas relaxation rate compared to watch QTFs whose resonant frequency is typically 32 kHz, leading to a prong width of 8 mm, a thickness of 2 mm and a length of 13.6 mm.

While this geometry enables a good pressure differential compared to a watch QTF [6], we demonstrated that it also induces important acoustic losses [23] as illustrated in Fig. 2. If any QTF emits acoustic waves, the particular geometry of C2 makes this loss source dominant, reducing its quality factor down to 8,000 in air at atmospheric pressure (compared to 10,000 for watch QTFs [5, 10]) and thus its sensitivity which we measured equal to $1.8 \times 10^{-8}$ W cm$^{-1}$ Hz$^{-1/2}$.

Fig. 2

Simulation of acoustic wave emission of QTF C2. The QTF displacement amplitude is maximal in red. The positive relative pressure is in red, the negative in blue

2.2 Acoustic recovery cavity

To reduce the acoustic losses and improve the sensitivity, an acoustic recovery cavity was designed to be coupled to this QTF [15, 24]. Its geometry was adapted to the quadrupolar emission mode of the QTF [25] and chosen to be a cylinder which distance to the external face of the prongs is equal to $\lambda _{\mathrm {ac}} / 2$ as shown in Fig. 3. As a consequence the emitted acoustic wave is sent back to the external QTF faces. However, it is important to underline that this cavity was not an acoustic resonator for the acoustic wave emitted by the photoacoustic effect at the laser beam location.

Fig. 3

Schematic of acoustic recovery cavity with QTF C2. The distance between the cavity inner surface and the external face of the QTF prongs is equal to $\lambda _{\text {ac}} / 2$

With this acoustic recovery cavity, an enhancement of the QTF quality factor was demonstrated, passing from 8,000 up to 75,000 in air at atmospheric pressure [15]. This improvement resulted in a twice better sensitivity compared to the bare QTF C2, reaching $9 \times 10^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$ [6].

While such a sensitivity is better than for any other QTF without acoustic resonator, it is still one order of magnitude below the QEPAS state-of-the-art sensitivity. As a consequence, we also added longitudinal acoustic resonators similarly to classical QEPAS setups [23]. The tubes were embedded in the acoustic recovery cavity. An optimal coupling was obtained for tubes of length 7.3 mm each and diameter 500 µm yielding a sensitivity of $2 \times 10^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$ [6] close to the state-of-the-art.

Nonetheless, using such tubes presents important drawbacks as already mentioned above. They are hard to machine, to place around the QTF and to align one relatively to the other. In addition, laser beam focusing in such long and thin tubes is also tricky. As a consequence, we consider the use of such tubes not desirable for a potential industrial system and we had to come up with another solution.

3 New radial resonator theoretical study

3.1 A radial resonator as a new paradigm

While the recovery cavity enabled to increase the quality factor, we demonstrated that it does not enhance the pressure amplitude, resulting in a limited sensitivity improvement [6]. Furthermore, the tubes enable a very high pressure amplification but this pressure is very badly distributed on the QTF prongs, also limiting the sensitivity. Based on these findings, we concluded that it would be interesting to find a compromise between pressure generation and coupling between the acoustic and mechanical resonators.

After analysing different acoustic modes of resonance, we determined that a radial mode could provide for a good pressure overlap on C2 prongs and good pressure amplification. We thus adapted our recovery cavity to use a radial mode and then to resonate this time with the acoustic wave generated by the photoacoustic effect. This theoretically occurs for a radius equal to $\lambda _{\mathrm {ac}} / 2$ as shown in Fig. 4.

Fig. 4

Schematic of our new radial resonator with QTF C2. The distance between the cavity inner surface and the laser beam is equal to $\lambda _{\text {ac}} / 2$

However, this condition should be adapted because the QTF and the radial resonator are coupled. Generally, the coupling of a high quality factor resonator to a low quality factor one leads to the drop of the coupled system quality factor. In QEPAS, the QTF exhibits a typical quality factor of 10,000 while the acoustic resonator one is around 10 [6], leading to the same phenomenon. In QEPAS this coupling between both resonators is widely known [10, 26]. When the acoustic resonator frequency is adapted to the QTF one, an optimal coupling is reached. At this optimum, the quality factor of the coupled system is minimal. Despite this drop of the quality factor, the SNR is maximal thanks to the pressure amplification due to the acoustic resonance. In addition, one could notice that the system frequency presents an anti-resonance around this optimum.

The frequency matching between resonators is usually obtained by tuning the acoustic resonator length when using longitudinal modes or the radius in our case. The tolerance around the length or radius value yielding to the optimal matching condition is usually of a few hundreds of micrometers, hence requiring a careful adjustment.

To determine the optimal coupling as well as to estimate the sensor sensitivity, we modelled it through finite element modelling (FEM) as detailed hereafter.

3.2 Modelling based on finite element method

Numerical modelling was used to investigate the dependence of the QEPAS sensor key parameters, i.e. resonance frequency, quality factor, and SNR, on the radial resonator radius. FEM modelling of the QTF and acoustic resonator was carried out using the OOFELIE:: Multiphysics solver (version 4.4) which includes most of the involved phenomena: QTF acoustic and structural losses, vibroacoustic coupling and acoustic resonator acoustic losses [27]. However, thermoviscous losses, which are not yet modelled by our FEM solver, are usually limiting. As a consequence, we computed these analytically to correct the FEM results as we will detail hereafter. This approach enables us to evaluate the sensor performance and determine the optimal resonator parameters.

Preliminary FEM modeling was performed to compute QTF structural losses. However, for more accurate computations we forced the QTF structural quality factor to the one measured under vacuum, i.e. 100,000. We considered the air to be non-viscous to facilitate simulations and took a sound speed of 340 m s$^{-1}$ and density of 1.225 kg m$^{-3}$. The radial acoustic resonator has a 16 mm length and radius of 10 mm and was centered at 75% of the QTF prong length. This height is the optimal laser beam height for this bare QTF [11]. Using FEM, we derived this resonator quality factor to 246, quantifying only the losses through openings. The acoustic resonator thermoviscous quality factor can be computed analytically or through finite element method [28, 29]. We evaluated it to 500, leading to a total quality factor of 165 for the radial resonator of 10 mm radius. We later corrected the resulting coupled quality factor Q^* and SNR to take into account thermoviscous losses. Eventually, we studied the resonators parameters with FEM modelling by varying the acoustic resonator radius and obtained the results presented in Fig. 5.

Fig. 5

FEM results on the frequency, coupled quality factor and SNR when varying the radial resonator radius

We obtain an optimal coupling for a radius of 10 mm. Around this radius, we observe a frequency antiresonance, a minimum of the coupled quality factor and a maximum of the signal-to-noise ratio. This behavior is similar to what is generally obtained in QEPAS [10]. The coupled quality factor drops from 62,000 for a radius of 12 mm to 335 for a radius of 10 mm. This important decrease of the system quality factor shows that the coupling between the two resonators is very efficient. Meanwhile, the SNR (we chose to compute in a 1/8 Hz bandwidth to match with experiments) increases from 66 for the bare QTF to 1,027 at the best coupling. This corresponds to a 15 times improvement. This value is analog to typical SNR enhancements reported in the literature using acoustic microresonators [10], confirming the proper expected performance of this setup.

Eventually, we derived the sensor NNEA. We considered a heat density similar to the experimental one of $1.04 \times 10^{-6} {\text {W cm}}^{-1}$ whose computation is detailed in Appendix A and a SNR of 1027 in a 1/8 Hz bandwidth. We derived a NNEA of $3 \times 10^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$ which is close to the state-of-the-art for QEPAS sensors. This result is very promising as the system exhibits a good sensitivity while offering a more convenient implementation.

This good sensitivity originates both from the adaptation of the QTF prong width to the acoustic wavelength and from the good matching with the acoustic radial mode. Conversely to previous QEPAS implementations where the use of a custom QTF did not yield discriminating benefits, the use of a custom QTF is of prime importance here. Indeed, we also evaluated the use of a radial resonator with a watch QTF. We simulated a resonator of 3 mm length, centered at 85% of the prong length, which is the best position for this QTF, and found an optimum for a radius of 6.5 mm. The sensitivity was only increased by a factor 2 compared to the bare QTF, mainly because of the non adapted prong width.

To conclude this section, we demonstrated that coupling between the QTF and the acoustic resonator was primordial to achieve good sensitivities. The new radial resonator we designed is well adapted to its QTF and the other way around. Last but not least, it allows more straightforward operation as verified in the next section.

4 Experimental study

To confirm the previous results, the radial resonator was realised in duraluminium and we carried out the experimental characterisation of the system. In order to tune the radial mode eigenfrequency to the QTF one, we used cylinders of different radii that we inserted in a larger one as shown in Fig. 6. The rotation of the inner cylinder was blocked, and walls were used to close the resonator and avoid important acoustic losses. A 2 mm diameter hole was drilled in these walls to let the laser beam pass. This hole dimension was not critical and it could easily be replaced by optical windows. For the smallest radii we evaluated the distance between the QTF prong and the cavity wall to be around 200 µm. For all measurements, we operated the sensor at atmospheric temperature and pressure.

Fig. 6

Pictures of the radial resonator of different radii (Left) and placed around the QTF (Right)

4.1 Photoacoustic setup

We used the setup depicted in Fig. 7 to carry out photoacoustic measurements. A telecom laser diode, emitting around 17 mW, was set to 6,490.02 cm$^{-1}$ (i.e., 1,540.83 nm) to target a rovibrational line of C₂H₂. The laser beam was focused into the cell by a lens with a focal length of 200 mm, resulting in a waist of 45 µm, enabling an easy pathway between the QTF prongs. We filled the cell with a certified mixture of 1,000 ppmv C₂H₂ in nitrogen, yielding to a spectral line with a peak absorption of $1.84 \times 10^{-4}$ cm$^{-1}$ and a full width at half maximum (FWHM) of 0.12 cm$^{-1}$.

The average laser wavelength was tuned by temperature variation with a thermoelectric controller (ITC 510, Thorlabs) and measured with a Fizeau lambdameter (WS6 785, HighFinesse). The laser wavelength was modulated around the absorption line by current modulation, according to the wavelength modulation scheme (WM), at half the resonator frequency, $f_{{0}}/2$. The sine modulation was provided by a function generator (33000B, Agilent Technologies). The modulation depth was adapted to the FWHM of the broadened absorption line by maximising the photoacoustic signal. We found an optimal modulation depth amplitude of 0.132 cm$^{-1}$ which is in agreement with the theoretical value of $1.1 \times \mathrm {FWHM}$ [30]. The generated heat amplitude introduced in the medium H was derived to be $1.04 \,\times 10^{-6} \, {\text{W cm}}^{-1}$ as detailed in Appendix A.

The QTF piezoelectric signal was amplified by means of a conventional amplification circuit comprising a transimpedance amplifier. Thereafter, the signal was fed to a lock-in amplifier (SR530, Stanford Research Systems) and was demodulated at the reference frequency $f_{{0}}/2$. We measured the signal in a noise equivalent bandwidth of 1/8 Hz (set by cascading two 6 dB/octave filters with time constants of 1 s each), while the noise was measured by the lock-in amplifier in a 1 Hz bandwidth. The SNR was then computed by deriving the signal and noise in the same bandwidth, assuming white noise.

Fig. 7

Scheme of the experimental setup

4.2 Limiting noise

To assess that we will be able to reach the sensitivity determined by FEM, we performed a preliminary study of the system noise. Indeed, as we have seen, the coupling of an acoustic resonator to the QTF leads to a decrease of the system quality factor Q^*, which in turn modifies the level of the Brownian noise. To verify that the Brownian noise was still limiting compared to other system noise sources, we measured the system noise spectral density for different resonator radii. We used a spectrum analyser (35670A, Agilent Technologies) for the noise measurement and an impedance analyser (4294A, Keysight) to derive the quality factor values. The setup was placed in a bulk metallic cell to shield from environmental sounds and vibrations. The results are presented in Fig. 8.

Fig. 8

Noise spectral density of the system for different resonator radii. The noise of the lowest quality factor system is plotted in the upper right hand corner

We observe that the system noise is actually limited by the Brownian noise in every case. Indeed, the noise spectral density presents a resonance shape with a quality factor corresponding to the one derived thanks to the impedance analyser. However, we also observe a noise peak at 21,130 Hz that may be due to the electronic circuit. This parasite did not alter the photoacoustic measurements since we integrated the noise in a smaller bandwidth. We obtained the best coupling between the QTF and the radial resonator, i.e. the lowest quality factor of 2,106, for a resonator radius of 10 mm.

4.3 Photoacoustic measurements

We then performed photoacoustic measurements with our system for the different resonator radii. We obtained the results presented in Fig. 9.

Fig. 9

Experimental frequency, coupled quality factor and SNR when varying the radial resonator radius

For each radius, we scanned the response over the modulation frequency and derived the resonance frequency, quality factor and SNR. The best coupling is obtained when the radial resonator radius is 10 mm, which is consistent with FEM calculation and previous noise measurements. For this value, the system quality factor reaches 704. The discrepancy between this value and the ones provided by FEM of 335 or derived from noise measurements of 2,106 might come from an experimental positioning which is different from FEM modelling. In addition, the shape of the resulting curves is similar to the ones obtained by FEM modelling. The frequency presents an antiresonance around the radius yielding the best coupling. The quality factor increases with the radius, and reaches a quality factor of 59,000 for a 16 mm radius. Also, despite the small distance between the QTF prong and the cavity of around 200 µm, we did not observe the effect of squeeze damping.

Eventually, we observe a discrepancy for the SNR value at 10 mm. Indeed, the best SNR is not obtained for the lowest quality factor as expected. Instead, the value is reduced compared to the two adjacent radii, passing from 777 to 591. Despite repeating this particular measurement, we always obtained the same result. We still did not manage to find a definite explanation for this discrepancy. It might come from the positioning of the radial resonator around the QTF which could not have been perfectly glued and could present an angle, thus an asymmetry between the resonators. We plan to investigate this potential issue in future work through improved adjustment of the QTF gluing and modification of the radial resonator. At this point, we however cannot rule out that this discrepancy can also originate from an unexpected physical phenomenon.

Finally, we derived the system NNEA for the best SNR. We obtained a sensitivity of $3.9 \times 10^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$ that is close to the one derived through FEM of 3 $\times$ 10$^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$. The slight discrepancy might be due to the SNR dropout we have discussed above. Compared to the bare QTF, this represents a SNR improvement of a factor 5 which is inferior to SNR improvements obtained using classic configurations such as on-beam. This originates from various phenomena, such as a reduced pressure generation compared to microtubes because of the bigger cell volume [23].

To summarise, these results are presented in Table 1 and compared to the ones obtained with watch QTFs reported in the literature in on-beam configuration.

Table 1 Comparison of our radial resonator with classic on-beam QEPAS systems from the literature. The SNR increase is normalised compared to the corresponding bare QTF SNR

5 Conclusion

We have presented the first QEPAS system using a radial resonator to enhance the acoustic signal. The QTF was designed to provide a high quality factor and a good coupling with acoustic waves. In turn, the acoustic resonator was designed to maximise energy transfer with the QTF, by choosing a radial mode of resonance. The radial resonator increased the photoacoustic SNR by a factor of 5 compared to the bare QTF. This resulted in an experimental sensitivity of 3.9 $\times$ 10$^{-9}$ W cm$^{-1}$ Hz$^{-1/2}$, close to the QEPAS state-of-the-art sensitivity [10].

Furthermore, this new resonator presents several benefits over traditional microtube resonators used in on-beam QEPAS. First, the laser beam alignment is relaxed since it has to be focused in a $2 \times 2$ mm area, to compare to 1 cm long tubes with typical diameters of 500 µm. Second, machining of the resonator is easy and so is the positioning relatively to the QTF. For the same reason, this system would be more suitable for use in laser intracavity setups that may enable to enhance the sensitivity several hundred times as already demonstrated [31, 32].

The use of such radial acoustic resonators well coupled to the QTF paves the way towards new resonators for QEPAS, possibly allowing to enhance the state-of-the-art sensitivity or presenting unique practical advantages compared to traditional setups. In the future, we aim at studying new configurations and to further adapt our radial resonator to improve the sensitivity.

6 Note

After the submission of this paper, the use of a radial resonator for QEPAS was reported in [33]. An acoustic radial resonator of length 6.9 mm and radius 6.4 mm was placed around a standard watch tuning fork vibrating at 32,768 Hz. The SNR was reported to be enhanced by one order of magnitude compared to the bare QTF.

Appendix A: Computation of the heat and NNEA

In photoacoustics, the sensor sensitivity is often characterised by computing the NNEA. In most cases this figure of merit is computed from the gas absorption $\alpha _{\mathrm {G}}$ and laser mean power $P_{\mathrm {L}}$ as follows:

$$\begin{aligned} \mathrm {NNEA} = \frac{\alpha _{\mathrm {G}} \times P_{\mathrm {L}}}{\mathrm {SNR} \times \sqrt{{\Delta } f}}, \end{aligned}$$

(2)

with SNR the sensor signal-to-noise ratio and ${\Delta } f$ the measurement bandwidth. This computation method allows to quantify the sensor sensitivity in its actual mode of operation. Also, we observed that the resulting values presented discrepancies for a same detector, mainly depending on the gas used or laser modulation scheme.

In order to compare the detectors on an equal footing, we propose to use a generalised definition of the NNEA so that a given detector will always present the same NNEA. To do so, instead of using directly the gas absorption $\alpha _{\mathrm {G}}$ and the laser mean power $P_{\mathrm {L}}$ we use the generated heat inside the medium H. This heat is directly proportional to the acoustic pressure later seen by the detector. It takes into account the gas absorption, the laser power, the overlap between the two and also the gas relaxation rate and detection frequency. The NNEA is now defined as follows:

$$\begin{aligned} \mathrm {NNEA} = \frac{H}{\mathrm {SNR} \times \, \sqrt{ {\Delta } f}}, \end{aligned}$$

(3)

with H the modulus of its complex form $\widehat{H}$ defined as:

$$\begin{aligned} { \left\{ \begin{array}{lll} \widehat{H}(\omega ) = { \frac{2\times \mathrm{e}^{-j\times \,\tan ^{-1}(\omega \,\times \tau _{\mathrm {V-T}})}}{\sqrt{1 + (\omega \,\times \tau _{\mathrm {V-T}})^2}} \,} \widehat{c}(\omega )\\ \widehat{c}(\omega ) = \! {\frac{\omega }{2\pi }\!\times \! \int _{-\pi /\omega }^{+\pi /\omega } \!\alpha _{\mathrm {G}}(t)\!\times \! P_{\mathrm {L}}(t) \!\times \! \mathrm{e}^{-j \times \omega \times t} \times \! \mathrm{d}t}\\ \end{array} \right. }, \end{aligned}$$

(4)

with $\tau _{\mathrm {V-T}}$ the gas relaxation rate and $\omega$ the angular modulation frequency. The first term takes into account gas relaxation rate effect on the signal. As a consequence, the NNEA does not depend on the detected gas, gas matrix and detection wavelength modifying the gas relaxation rate. The second term accounts for the overlap between laser emission and gas absorption, making the NNEA independent of the modulation scheme.

When the laser is modulated in amplitude with a 50% duty cycle one can easily derive $H=\alpha _{\mathrm {G}} \times P_{\mathrm {L}} /2$, with $P_{\mathrm {L}}$ the laser mean power before modulation. When using a wavelength modulation, the result is more complex and the overlap should be computed. However, we can derive an approximate value when using a sine modulation and targeting a Lorentzian-like profile of absorption. In this particular case we can derive $H \thickapprox \alpha _{\mathrm {G}} \times P_{\mathrm {L}} /3$.

We used this approximated expression to compute H in this article. Here, the maximal gas absorption $\alpha _{\mathrm {G}}$ is equal to 1.84 $\times \!$ 10$^{-4}$ cm$^{-1}$, the laser mean power $P_{\mathrm {L}}$ is equal to 17 mW, resulting in a heat deposition H of 1.04 $\times \!$ 10$^{-6}$ W cm$^{-1}$.