1 Introduction

Quartz-enhanced photoacoustic spectroscopy (QEPAS) sensor technology is very attractive for trace gas detection of various simple molecules with narrow absorption spectra and larger molecules with broad, unresolved spectral absorption features [1, 2]. Three QEPAS spectrophone configurations have been usually used: “on beam” QEPAS (Fig. 1(a)) [1], “off beam” QEPAS (Fig. 1(b)) [3, 4] and “bare QTF” QEPAS (Fig. 1(d)) [5] (in which the spectrophone includes only a bare quartz tuning fork, QTF). The use of an optimized microresonator (mR) tube might increase the QEPAS sensor sensitivity by a factor of up to ∼30, compared to the “bare QTF” approach [5]. Therefore, it is very important to design an appropriate QEPAS spectrophone with optimal configuration and optimized mR parameters. Until now, almost all published papers mainly reported on instrumental developments and experimental applications. Only “bare QTF” QEPAS setup was successfully analyzed using a theoretical model by Petra and coworkers [6]. Firebaugh et al. [7] tried to use COMSOL Multiphysics to model QTF response and the dependence of the QEPAS signal upon the mR tube size, whereas their results showed that the used model is not well adapted in comparison with the conclusions acquired in [5]. In these previously reported works, the mR length l was roughly determined by λ/4<l<λ/2 (where λ is the acoustic wavelength inside the mR). In [5], the authors experimentally demonstrated that the optimum mR length was about 4.40 mm. Recently, Serebryakov et al. [8] reported a special mR called “a tube with a hole in the middle” to improve the sensitivity of the QEPAS sensor and the experimental results showed that the optimum length was about 9.3–9.4 mm for this special mR with an inner diameters of 0.40 mm. However, no suitable acoustic model has been suggested allowing one to theoretically determine optimum geometrical parameters of the mR.

Fig. 1
figure 1

Commonly used QEPAS spectrophone configurations: (a) on beam QEPAS; (b) off beam QEPAS; (c) half on beam QEPAS (introduced in the present work); (d) bare QTF QEPAS

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In this paper, we present an acoustic model referred to “orifice ended mR” (Figs. 2(a) and 2(b)) for describing the mR tube used in QEPAS. We introduce at first this model to analyze a single mR enhanced QEPAS setup noted as “half on beam QEPAS” (Fig. 1(c)) for theoretically determining the optimum mR geometrical parameters. We then applied the optimized mR to a conventional on-beam QEPAS approach. Comparison of the spectrophone performances between “on beam” and “half on beam” QEPAS approaches were carried out in terms of signal-to-noise ratio (SNR) gain through experimental investigation and theoretical analysis by monitoring water vapor absorption in the atmosphere. Using the present “orifice ended mR” model, an optimized mR allowed to achieve a SNR gain of up to ∼30 in a conventional on beam QEPAS setup. To our knowledge, this SNR gain is mostly close to the best results reported in [5] for the on-beam QEPAS.

Fig. 2
figure 2

“Orifice ended mR” model. (a) 3D map of an ideal “orifice ended mR” (left side: fully open; right side: partly open); (b) cross section profile along the axis of the real “orifice ended mR”; (c) setup seen from the axis of the mR; (d) setup observed from the side surface of the mR; (e) partly opening end area A 0 of the mR (seen from the axis of mR); (f) district area of the gap between the QTF and mR; (g) circular orifice with an equivalent area of A 1=A 0 of the real partly opening orifice

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2 Theoretical acoustic model for mR in QEPAS

In the case of using a single mR QEPAS approach (Figs. 1(c), 4), the mR can be modeled by an “orifice ended tube” originated in [9]. In such a QEPAS configuration, the mR is placed on one side of the QTF as close as possible to amplify the acoustic wave signal generated by the gas species in interaction with a modulated light inside the mR. The acoustic pressure antinode inside the mR is closed to the end of mR tube which is partially blocked by the QTF [5, 7]. The QTF was used to probe the amplified acoustic signal and transform the acoustic signal to electronic signal. As this QEPAS approach consists of only one single mR tube instead of two as in the case of conventional “on beam” QEPAS, we referred it to “half on beam” QEPAS (see Fig. 1(c)). In principle when the cross sectional dimensions of the resonator are much smaller than the acoustic wavelength, the excited acoustic wave can be described by a one-dimensional acoustic field along the length of the resonator [10]. In QEPAS, the mR (with an inner diameter ID≤1.00 mm in comparison with an acoustic wavelength of ∼10.4 mm in our experiment) can be thus treated as one-dimensional acoustic resonator, only longitudinal acoustic resonance occurs in the mR. In “half on beam” configuration, one end of the mR is fully open and another end close to the QTF is partially open. The gap (g) between the mR and QTF (see Figs. 2(b) and 2(d)) was set to g=20–50 µm, which was measured with a High-Definition USB scientific digital microscope (Supereyes A005). As the gap is very small compared to the dimension of the mR and QTF (see Fig. 2(d)), the mR in “half on beam” QEPAS can be considered as an “orifice ended tube” (see Fig. 2(a) for an ideal orifice ended mR and Fig. 2(b)), originally outlined in [9, 11]. In such a model, we assumed that the tube boundaries were perfectly sound hard.

According to [9], the natural resonant frequency f mR (that should be in principle equal to the resonant frequency f 0 of the QTF) of such an orifice ended mR tube can be determined by the inner diameter and length of the mR tube, the propagation velocity of the acoustic wave inside the mR, and the acoustic impedance (expressed usually through the acoustic conductivity) of the mR tube with the boundary condition. The following equation can be used to describe the natural frequencies of a mR tube that is fully open at one end and partly open at the other:

$$ \tan(kL_{\mathrm{eff}}) + kS/\sigma= 0$$
(1)

where k=2π/λ=2πf mR/υ is the wave number, υ (in m/s) is the propagation velocity of the acoustic wave inside the mR tube, f mR (in Hz) is the natural resonant frequency of mR. L eff (in mm) stands for the effective length of the mR tube, S (in mm2) is the cross section area of the mR tube (S=πR 2, R is the inner radius of the mR) and σ (in mm) is the acoustic conductivity.

When a constriction is formed by an orifice inside a tube having a larger inner diameter (see Fig. 3(a)), or a side hole is drilled on the tube side surface (see Fig. 3(b)), the acoustic conductivity of such an orifice or a side hole is defined as follows [11]:

$$ \sigma= A / t$$
(2)

where A=πr 2, r is the radius of the orifice or the side hole, and t is their corresponding thickness.

Fig. 3
figure 3

Definition of the acoustic conductivity used in the present theoretical model. (a) An orifice with a radius of r (its cross section area is thus A=πr 2) and a thickness of t inside a tube with a larger inner diameter (ID): its acoustic conductivity σ=A/t; (b) A side hole with a radius of r drilled on the tube (of thickness t) side surface: its acoustic conductivity σ=A/t; (c) An orifice forming a constriction and a termination at one end of the mR tube: its acoustic conductivity σ=d(1+d/ID)1.19; (d) The partially open mR end consisting of an orifice and a side hole: its acoustic conductivity σ=σ 0+σ g =d(1+d/ID)1.19+A g /t

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In the case where the orifice is located at one end of the tube and formed a termination to the tube (see Fig. 3(c)), meanwhile, the thickness of the orifice is infinitely thin (t→0, considered as an infinitely thin sheet), the acoustic conductivity of such an orifice is given by [12]:

$$ \sigma= d(1 + d/\mathrm{ID})^{1.19}$$
(3)

where d and ID (ID=2R) are the diameter (in mm) of the orifice and inner diameter (in mm) of the mR tube (Fig. 3(c)), respectively.

As the acoustic impedance for a fully open end equals to zero [11, 13] (the corresponding σ is thus equal to infinite), we shall only consider the total acoustic conductivity σ for the partly open end (see Fig. 3(d)). The acoustic conductivity of this part is attributed to two contributions: from the orifice formed by the QTF and from the gap area between the mR end and the QTF. The acoustic conductivity of the orifice (σ 0) formed by the QTF is given by Eq. (3). The gap area between the mR end and the QTF can be described with a model of a “tube with side hole” [11]. Here, the gap is equivalent to a narrow slit that can be further treated as an equivalent hole with 2πRg as its area surface A g , as presented by the dash area in Fig. 2(f). The acoustic conductivity of the gap area part can be then determined by Eq. (2). The total acoustic conductivity σ can be described as

$$ \sigma= \sigma_{0} + \sigma_{g} = d(1 + d/\mathrm{ID})^{1.19} + 2\pi Rg/t_{0}$$
(4)

where t 0=(OD−ID)/2 is the thickness of the mR tube and OD is the outer diameter of the mR tube.

In reality, the orifice is not circular (Fig. 2(e)), the diameter d of the orifice should be derived from the cross section A 0 that is equal to the cross section A 1 of an equivalent circular orifice (Fig. 2(g), dash area):

$$ A_{0} = A_{1} = 2R^{2}\arcsin \bigl(w/(2R) \bigr) + w \bigl(R^{2} - w^{2}/4\bigr)^{1/2}$$
(5)

and then

$$ d = 2(A_{0}/\pi)^{1/2},$$
(6)

where w is the gap width between the two QTF prongs (commonly w=0.30 mm). We can thus determine the effective length of the mR tube L eff through Eqs. (1)–(6).

By taking into account the end correction for both ends of the mR tube [9–12], the effective length L eff, described by Eq. (1), is larger than its physical length l. In fact, this can be understood as an effect of the mismatch between the one dimensional acoustic wave inside the mR and the three dimensional field (of spherical wave front) radiated from the opening end of the tube to free space [10].

In our case, the fully open end can be considered as an unflanged tube end [10, 11], the corresponding end correction can be approximated to be 0.6R. For the partly open end (treated as an infinitely thin orifice forming a termination to the mR tube combined with a side hole on the mR side surface), the end correction α can be determined as follows [9]:

$$ \mathrm{tan}(k\alpha) = kS/\sigma $$
(7)

where k, S, and σ are the same as defined in Eqs. (1) and (4), respectively.

Then the real length (l) of an orifice ended mR tube can be determined by the following equation:

$$ l = L_{\mathrm{eff}} - 0.6R - \alpha $$
(8)

Using Eqs. (1), (7)–(8), the optimum mR length can be expressed as follows:

$$ l = \frac{\upsilon}{2f_{\mathrm{mR}}} - \frac{\upsilon}{\pi f_{\mathrm{mR}}}\arctan \biggl(\frac{2\pi S}{\sigma} \frac{f_{\mathrm{mR}}}{\upsilon} \biggr) - 0.6R$$
(9)

Given that g=20–50 µm, υ=200–500 m/s, R=0.20–0.50 mm, d=0.37–0.61 mm, ID=0.40–1.00 mm, OD=0.70–1.30 mm, f 0=32–33 kHz, we can obtain 0.974 mm<σ<2.131 mm and so that 0<2πSf 0/(συ)<0.4 (f mR=f 0). In this case, Eq. (9) can be approximated to

$$ l = \frac{\upsilon}{2f_{0}} - \frac{2\pi R^{2}}{\sigma} - 0.6R$$
(10)

Finally, the real optimum length l of an orifice ended mR can be determined through Eqs. (4)–(6), (10). According to the results given in [10, 11, 13], the acoustic velocity υ (in m/s) of general gas, used in Eqs. (9)–(10), depends on the mol mass of molecule (M, kg/mol) and the absolute temperature T in Kelvin (K) of the gas, and the specific heat ratio of the gas constant γ:

$$ \upsilon= \sqrt{\gamma RT/M}$$
(11)

where R=8.3144 J/(mol K) is the universal gas constant. For ambient air, the acoustic velocity can be approximately determined in function of centigrade temperature T c (in °C) [13]:

$$ \upsilon(T_{c}) = 331.6 + 0.6T_{c}\ \mbox{(m/s)}$$
(12)

3 Experimental investigation for evaluating the acoustic model

QEPAS detection of water vapor absorption around 1.396 µm was performed to validate the presented acoustic model. The measurements have been carried out in ambient air at normal atmospheric pressure and temperature between 20–25 °C. The corresponding H2O vapor concentration was determined by direct absorption in a 45-cm-long optical path and used for QEPAS signal normalization. The “half on beam” QEPAS experimental arrangement is schematically displayed in Fig. 4. A fiber-coupled DFB diode laser (NLK1E5GAAA, NEL, Yokohama, Japan) tunable from 7154 to 7171 cm−1 with an optical power of ∼8 mW was chosen as excitation light source for generating photoacoustic signal. The target H2O absorption line has a strong absorption intensity of 1.174×10−20 cm−1/(mol cm−2) at 7161.41 cm−1. The diode current and temperature were controlled by a commercial diode laser controller (ILX Lightwave LDC-3724). Coarse and fine wavelength tuning was performed by changing the laser temperature and current, respectively. Wavelength modulation at half the QTF resonant frequency f 0/2 was achieved via laser current modulation by a sine-wave form from a function generator. The laser beam was collimated by a fiber-coupled collimator (f∼5 mm) in conjunction with a Grin lens of 50 mm focal length and subsequently focused into the mR. A QTF (DT-38, 32.768–12.5 pF, KDS, Daishinku Corp.) with a resonant frequency of f 0∼32.755 kHz was placed at one end of the mR for acoustic signal detection. The QTF-generated piezoelectric current was converted into voltage using a home-made transimpedance amplifier with a feedback resistor R g =10 MΩ. A preamplifier (EG&G, Model 5113, AMETEK Advanced Measurement Technology) with a gain of 100 combined with a 6 dB bandpass filter (10–100 kHz) was used for signal filtering and amplification. The QEPAS signal was demodulated at f 0 with second harmonic detection scheme by a lock-in amplifier (Stanford Research Systems, Model SR 830 DSP). The time constant of the lock-in amplifier was set to 1 s in association with an 18 dB/octave slope filter (Δf=0.094 Hz). The lock-in amplifier, the laser controller and the function generator were controlled by a personal computer with a GPIB card, RS232 communication port and a Labview program (Labview8.5/CVI).

Fig. 4
figure 4

Schematic diagram of the “half on beam” QEPAS experimental setup. RS232: communication port for remote control and data exchange; PC: personal computer; GPIB: General Purpose Interface Bus; DAQ card: data acquisition card

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Five mRs with different inner diameters and lengths (referred to as mR1, mR2, mR3, mR4, and mR5, respectively) were used to investigate the dependence of the “half on beam” QEPAS signal enhancement upon the mR parameters. Initially, each mR was cut to about 7 mm length. Then its length was gradually adjusted by cutting off the pieces from one end through a fine cutter. The mR lengths were measured using a vernier caliper with a precision of 0.02 mm. The corresponding geometrical parameters of each mR are listed in Table 1(a). In Fig. 5, the QEPAS signal enhancement factors (QSE) are plotted versus the mR lengths (with inner diameter as parameter) and the corresponding mR resonant frequencies (f mR). Based on these experimental results, the optimum mR lengths (as well as the corresponding natural resonant frequency f mR) were determined for each inner diameter.

Fig. 5
figure 5

“Half on beam” QEPAS signal enhancement factor as a function of mRs’ lengths (and the corresponding natural resonant frequency) with different inner diameters as parameters. Curves are Lorentzian profile fit. QSE factor: QEPAS signal enhancement factor. f mR: natural resonant frequency of the mR, calculated using l exp. in Table 1

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Table 1 The mR parameters used in the present work as well as the corresponding SNR gains measured (with an uncertainty of about 8 %) in “half on beam” QEPAS (a) and “on beam” QEPAS (b), respectively
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4 Results and discussions

The experimental results are summarized in Table 1. Using the specific QTF parameters (shown in Figs. 2(c) and 2(d): W=1.50 mm, H=3.8 mm, m=(W−w)/2=0.60 mm, T h =0.34 mm and w=0.30 mm, f 0∼32.755 kHz), theoretical optimum mR lengths l theor. are calculated and given in Table 1(a) for comparison with the experimentally determined values l exp. in “half on beam” QEPAS setup. As can be seen, the theoretical values closely matched the experimental data. The Q factor and the resonant frequency f 0 of the QTF were determined using a home-made “control and data processing electronics unit” described in [2, 4]. QEPAS enhancement factor (QSE factor) and signal to noise ratio gain (SNR Gain) [3, 4] were used to describe the characteristics of the mR used for improvement of the sensor sensitivity. The QSE factor and SNR Gain are defined as follows:

(13)
(14)

where S a and Q a are the normalized QEPAS signal and the corresponding QTF Q factor respectively for mR enhanced QEPAS. S b and Q b are the normalized QEPAS signal and the corresponding QTF Q factor for bare QTF-based QEPAS. Under the optimum conditions, the highest SNR gain of 21.6 was obtained by using mR1 (see Table 1) in a “half on beam” QEPAS configuration. A minimum detectable H2O concentration of 220 ppbv with a normalized noise equivalent absorption coefficient (NNEAC) of 5.3×10−9 cm−1 W/Hz1/2 was achieved.

We observed that the gap (g) between the mR and QTF is a critical parameter. A smaller g can more efficiently confine acoustic energy into the orifice ended mR. In addition, a smaller g makes a real “orifice end mR” (consisting of QTF prongs, mR and the gap, Fig. 2(b)) more close to the ideal “orifice end mR” model (Fig. 2(a)).

The same optimized mRs (mR1–mR5) were used in “on beam” QEPAS approach for performance evaluation in terms of SNR gain. The results were presented in Table 1(b). We found out that the optimal mR parameters obtained for the “half on beam” scheme is approximately valid for the “on beam” configuration (using the same mR parameters for both tubes). As a result, the use of optimum mR1 in “on beam” QEPAS permitted us to achieve a SNR gain of up to 32.2. A minimum detectable concentration of 150 ppbv for H2O and a NNEAC of 3.6×10−9 cm−1W/Hz1/2 were experimentally deduced. Taking into account the measurement uncertainly of ∼8 %, this SNR gain is very close to that reported in [5], which shows that the present model can be also applied to theoretical calculation of optimum mR parameters for conventional “on beam” QEPAS spectrophone configuration.

Performance comparison of different QEPAS spectrophone configurations (off beam, on beam, and half on beam) was then made in the present work. The results were summarized in Table 2. Although the sensitivity of the “half on beam QEPAS” setup is about 1.4 times lower than that in the on beam QEPAS arrangement, this approach provides a novel model for theoretical analysis of mR enhanced QEPAS. It is also very helpful for an optimum design of MEMS (Micro-ElectroMechanical Systems)-scale mR [14].

Table 2 Best NNEAC (cm−1W/Hz1/2) values achieved in off beam, on beam and half on beam QEPAS approaches
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It is clear that the mismatch between the QTF resonant frequency f 0 and that of the mR f mR will affect the QEPAS sensor sensitivity. In conventional broadband microphone based PAS [4], as the response frequency of microphone is in the range of 20–20 000 Hz, so that the natural resonant frequencies of a PA absorption cell can easily well match the frequency response of the microphone. In QEPAS, the QTF works as an acoustic transducer with a fixed resonant frequency located within a very narrow bandwidth (a few Hz). Maximum QEPAS signal strength can be obtained when f 0=f mR. The presented model shows that the optimum mR length depends not only on the QTF’s resonant frequency, but also on its inner diameter and the acoustic speed in the mR tube. As the acoustic speed is dependent on the medium through which the sound waves propagate, the mR parameters should be optimized for each target gas species. Two methods are currently employed to address this issue. One is to match ideally the mR length to the acoustic speed of the target species [3, 4], and the other method for practical field application is to match the sound speed to a settled mR length by introduction of a carrier gas in target species [5, 15].

5 Conclusions

An acoustic model of “orifice ended mR” was introduced and applied for the first time to determine optimum geometrical parameters of the mR used in on-beam QEPAS. The present work shows that the optimum tube length depends not only on the QTF’s resonant frequency, but also on its inner and outer diameter, the gap between the mR and QTF, and the acoustic speed in the mR tube. The subsequent results show that the analytical results agreed well with the experimental measurements. The presented acoustic theoretical model provides a well-adapted method for designing and optimizing QEPAS spectrophone configuration.