T-shape microresonator-based high sensitivity quartz-enhanced photoacoustic spectroscopy sensor

Hongming Yi; Weidong Chen; Shanwen Sun; Kun Liu; Tu Tan; Xiaoming Gao

doi:10.1364/OE.20.009187

1. Introduction

Quartz-enhanced photoacoustic spectroscopy (QEPAS) technology [1, 2] has been increasingly applied to selective and sensitive detection of trace gases since its introduction in 2002. Using a commercially available very low cost quartz tuning fork (QTF) as sharply resonant acoustic energy transducer, QEPAS technology is a photo-detector free highly sensitive spectroscopic technique: in a small size (~mm) light-gas interaction as well as acoustic detection take place together. Distributed feedback (DFB) diode laser [2], quantum cascade laser (QCL) [2], low cost broadband laser diode (BBLD) [3] and light emitting diode (LED) [4] have been successfully used as light exciting sources in QEPAS-based sensors. In order to obtain high detection sensitivity, microresonators (mR) are usually used in “on beam” [2, 5] or “off beam” configured QEPAS [3, 4, 6, 7]. The mR is in general made of glass or stainless steel tube and its length was set so that the first longitudinal mode resonance of the acoustic wave corresponds to the resonant frequency of the QTF. To date, reported results indicated that the “on beam” QEPAS with optimum mR yielded the highest signal-to-noise ratio (SNR) gain: about 30 times higher than the QEPAS setup using only a “bare QTF” [5, 8]. In “on beam” configuration, two mRs are placed on each side of the QTF with a gap of about 20-50 μm. As the gap between the QTF prongs is only 0.2-0.3 mm wide, it constrained the inner diameter of the mR and hence the size of the laser excitation beam passing through the mR tubes and the gap between the QTF prongs. As a result, optical alignment for “on beam” scheme is very difficult and the system assembling is inconvenient. It is also difficult to use excitation light sources with low spatial radiation quality in such “on beam” QEPAS. In “off beam” approach, the excitation light beam is coupled through the mR, while the QTF is placed outside the mR for “off beam” detection of the resonant photoacoustic signal through a small slit made in the middle of the mR tube. It is no longer necessary to couple the light beam through the narrow gap between the QTF prongs, the shortages of optical alignment related to the “on beam” configuration were offset. However, the sensitivity of this off beam QEPAS setup was ~1.6 times lower [5] than that obtained in on-beam scheme.

In this paper, we introduce a novel T-shaped mR (T-mR) for use, for the first time, in “off-beam” QEPAS sensor (Fig. 1 (a₁)). As shown in the present work, the use of T-mR in QEPAS sensor can improve the detection sensitivity by a factor of up to ~30, compared with that using only a bare QTF. This signal-to-noise ratio (SNR) gain is as high as that obtained in a conventional “on-beam” QEPAS, while keeping the advantages of “off-beam” QEPAS configuration: it is easier for optical alignment and mechanical assembling. Performance of the T-mR based QEPAS sensor is evaluated and optimized through an acoustic model and experimental investigation via detection of water vapor in the atmosphere.

Fig. 1 T-shaped mR based QEPAS spectrophone configuration. (a₁) 3D map of an ideal T-shaped mR based QEPAS approach; (a₂) 3D map of a T-mR made with a cubic aluminum block in the present work; (a₃) cross section profile along axis of the main pipe of an ideal T-shaped mR; (a₄) orifice formed by a QTF placed as close as possible to the branch pipe end of an ideal T-shaped mR; (a₅) setup consisting of an ideal T-shaped mR and a QTF observed from the cross section along axis of the branch pipe.

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2. Theoretical background

In fact, T-shaped photoacoustic (PA) cell has been successfully used in conventional microphone-based photoacoustic spectroscopy (PAS) [9, 10]. Similar to the T-shaped PA cell, a T-mR for use in QEPAS (Fig. 1(a₃)) consists of a long main pipe of length l (diameter D) and a short branch pipe of length l₁ (diameter D₁). The branch pipe is perpendicularly intersected the main pipe in the middle of the main pipe. The QTF is placed at the end of the branch pipe to “off-beam” probe the PA signal excited inside the main pipe (Fig. 1(a₄) and 1(a₅)). The gap g between the QTF and the branch pipe end (Fig. 1(a₅)) is set as small as possible (commonly, the gap is g ~10 μm, measured with a High-Definition USB scientific digital microscope). Because the dimension of the T-shaped mR is in the range of submillimeter to millimeter, it is difficult to acquire a tube with such dimension in real condition. Instead of using a stainless steel or glass tube, the T-mR is made of a cubic aluminum block with dimension of 2 × 3 × 9 mm in the present work (Fig. 1(a₂)). A drilling with a diameter D₁ and a real length l₁ was made to form the short branch pipe, and another drilling of diameter D and a real length l formed the main pipe. The averaged wall thickness of the main pipe and branch pipe are noted as T and T₁, respectively. In order to reduce the viscous drag [11] between the branch pipe and the QTF, the side surface of the branch pipe end was shaped as shown in a 3D map in Fig. 1(a₂). Optimum parameters of the T-mR can be determined via an acoustic model, originated from references [12–16].

2.1. Acoustic impedance - Determination of the main pipe’s effective length

Based on the theoretical analysis of acoustic resonator given in [14], a mR tube used in QEPAS can be treated as one-dimensional acoustic resonator, and only longitudinal acoustic resonance occurs inside the mR. Optimum parameters of a T-mR should match the T-mR resonant frequency to the QTF resonant frequency. These parameters can be determined through the resonant condition of the T-mR. Acoustic impedance Z inside the resonator at acoustic boundary condition [12–15] is usually used to describe the characteristics of a T-mR, including its resonant condition which allows determining the effective length of the main pipe. It is usually expressed as Z = p/(uS) [12–16] (where p, u are the acoustic pressure and fluid velocity inside the pipe, respectively, S is the cross section area of the pipe).

In acoustic resonator modeling, it is convenient to use effective length L₀ rather than its real physical length l₀. In the following analysis, the main pipe is divided into two parts: branches 2 and 3, respectively; branch pipe denoted as branch 1. For an acoustic resonator pipe with a length of L ₀ and an inner radius of r₀, the input acoustic impedances Z at x = 0 (one end of the resonator) and x = L₀ (the other end of the resonator), noted as Z(0) and Z(L₀) respectively, are expressed by Eq. (1) [11, 12]:

Z (L_{0}) = (Z (0) - j \frac{ρ υ}{S_{0}} \tan (k L_{0})) / (1 + \frac{S_{0}}{j ρ υ} Z (0) \tan (k L_{0}))

where x is the position coordinate, as shown in Fig. 2 (b₁): x₁ and x₂ stand for position coordinate of the branch pipe and the main pipe, respectively; ρ is the density of the fluid inside the pipe; S₀ the cross section area of the pipe. The axis origin is noted as 0. k = ω/υ = 2πf/υ is the wave number, f is the acoustic frequency (that should be equal to the QTF resonant frequency f₀ = 32.740 kHz, in our case), υ the acoustic wave velocity and j² = −1.

Fig. 2 Theoretical model for calculation of the optimum T-mR parameters. (b₁) T-shaped mR and coordinate system; (b₂) orifice area Ω₁ of the branch pipe end close to the QTF (seen from the axis of the branch pipe and the gap between QTF prongs); (b₃) gap (between the QTF and the branch pipe end) district surface area Ω₀; (b₄) equivalent of a circular orifice (Fig. 2(b₄)) to the real non-circular orifice (b₂) with an effective area Ω = Ω₁.

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The acoustic impedance at A (x₂ = -L₂), E (x₂ = 0) for branch 2, at F (x₂ = 0), B (x₂ = L₃) for branch 3, at 0 (x₁ = 0), C (x₁ = -L₁) for branch 1 are noted as Z_2A(-L₂), Z_2E(0), Z_3F(0), Z_3B(L₃), Z₁₀(0), Z_1C(-L₁) (shown in Fig. 2(b₁)), respectively. Z_o, Z_s are the input acoustic impedances of the orifice and the gap between the QTF and the end of the branch pipe (Fig. 2(b₁)), respectively. L₁ is the effective length of the branch 1 (its corresponding physical length is l₁).

At the junction of the three branches as shown in Fig. 2(b₁), Z_2E(0), Z_3F(0) and Z₁₀(0) satisfy the following relation deduced by continuity equations [12, 13, 15]:

\frac{1}{Z_{10} (0)} = \frac{1}{Z_{2 E} (0)} + \frac{1}{Z_{3 F} (0)}

For the T-shaped mR used in the present work, S₂ = S₃ = S = πR², with R = D/2, the radius of the main pipe branches 2 and 3; S₁ = πR₁², with R₁ = D₁/2, the radius of the short branch 1.

According to Eq. (1), Z₁₀(0), Z_2A(-L₂) and Z_3B(-L₃) can be given by:

Z_{10} (0) = (Z_{1 C} (- L_{1}) - j \frac{ρ υ}{S_{0}} \tan (k L_{1})) / (1 + \frac{S_{0}}{j ρ υ} Z_{1 C} (- L_{1}) \tan (k L_{1}))

Z_{2 A} (- L_{2}) = (Z_{2 E} (0) - j \frac{ρ υ}{S_{2}} \tan (k L_{2})) / (1 + \frac{S_{2}}{j ρ υ} Z_{2 E} (0) \tan (k L_{2}))

Z_{3 B} (L_{3}) = (Z_{3 F} (0) - j \frac{ρ υ}{S_{3}} \tan (k L_{3})) / (1 + \frac{S_{3}}{j ρ υ} Z_{3 F} (0) \tan (k L_{3}))

When acoustic resonance occurs in the T-mR, the acoustic boundary condition at A (x₂ = -L₂) and B (x₂ = L₃) satisfies the following relationship [11, 12]:

Z_{2 A} (- L_{2}) = Z_{3 B} (L_{3}) = 0

where L₂ = L₃, L₂ and L₃ are the effective length of the main pipe branches 2 and 3, respectively (their corresponding physical length are l₂ and l₃, respectively).

Based on Eqs. (4)–(6), Z₂_E (x₂ = 0) and Z₃_F (x₂ = 0) can be given as follows:

Z_{2 E} (0) = j ρ υ \tan (k L_{2}) / S_{2}

Z_{3 F} (0) = j ρ υ \tan (k L_{3}) / S_{3}

We will now consider the acoustic impedance of the gap between the QTF and the end of the branch pipe. For the branch pipe (branch 1), one end is connected to the middle of the main pipe (inserted with branches 2 and 3), and the other end close to the QTF is partially masked which can be considered as a terminal orifice (Fig. 1(a₄) and Fig. 2(b₂)). The acoustic impedance of the partially masked end at x₁ = -L₁ was denoted as Z₁_C(-L₁) = jρω/α [13], and it is determined by the acoustic impedance of the orifice Z_o = jρω/α_ο (see Fig. 2(b₁)) and the acoustic impedance of the gap between the QTF and the end of the branch pipe Z_s = jρω/α_s (see Fig. 2(b₁)) [8]. In fact, this gap part can be considered as a narrow side slit of branch pipe with a surface area of Ω₀ = πD₁g and a thickness of T₁ [8]. Analogous to the boundary condition at the junction 0, Z_1C(0), Z_o and Z_s satisfy the following equation at the junction of x₁ = -L₁:

\frac{1}{Z_{1 C} (0)} = \frac{α}{j ρ υ k} = \frac{1}{Z_{o}} + \frac{1}{Z_{s}} = \frac{α_{o}}{j ρ υ k} + \frac{α_{s}}{j ρ υ k} = \frac{α_{o} + α_{s}}{j ρ υ k}

where α is the total acoustic conductivity of the partially masked end that is the sum of the acoustic conductivity of the orifice α_ο (Fig. 2(b₂) and 2(b₄), dash area) and the acoustic conductivity of the gap α_s [8] (Fig. 2(b₃), dash area):

α = α_{o} + α_{s}

Using the theory outlined in [8, 15], α_ο and α_s can be calculated using the following equations:

α_{o} = d(1 + {d/D}_{1})^{1.19}

α_{s} = Ω_{0} / T_{1} = π D_{1} g / T_{1}

α = α_{o} + α_{s} =d {(1 {+d/D}_{1})}^{1.19} + π D_{1} g / T_{1}

d = 2 {(Ω / π)}^{1 / 2}

Ω = Ω_{1} = 2 R_{1}^{2} \arcsin (w / (2 R_{1})) + w {(R_{1}^{2} - w^{2} / 4)}^{1 / 2}

where w is the gap between two QTF prongs, w = 0.30 mm, and d is the equivalent diameter of the orifice. In reality, the orifice is not circular; the diameter d may be derived from a circular orifice with a cross section Ω (Fig. 2(b₄), dash area) equivalent to the real orifice area Ω₁ (Fig. 2(b₂), dash area).

Substituting the acoustic impedances Z₂_E(0), Z₃_F(0) and Z₁₀(0) (Eqs. (3), (7)–(9)) into Eq. (2) with L₂ = L₃ = L/2, we can deduce the following equation for the main pipe effective length L:

\tan (k \frac{L}{2}) / 2 S = (\frac{k}{α} - \frac{\tan (k L_{1})}{S_{1}}) / (1 + \frac{k}{α} S_{1} \tan (k L_{1}))

2.2. End correction - determination of the resonator physical length

Due to the fact 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 mR tube to free space [14], end (ends at A, B and C, shown in Fig. 2(b₁)) corrections for the main and branch pipes should be performed. On the other hand, interior end corrections at the junction of the branch pipe (at 0 as shown in Fig. 2(b₁)) should be taken into account too. The end corrections result in a longer effective length than its real physical length. The theories of end corrections given in [13–16] were adopted to calculate effective lengths.

(1) Wirz suggested [15] an approximate formula for end-correction Δl of circular tubes based on the flange width T (T is the wall thickness of the mR tube here):

Δ l = (0.60 + 0.22 \exp (- k T / r)) \times r

where k is the acoustic wave number, r is the radius of the tube. In real application, if T is rather thin, the tube end can be treated as unflanged, the end correction is Δl≈0.6r [15, 16]. When T becomes thicker (T≥3r), the tube end forms a flange, the end correction for such a flanged end can be expressed as Δl≈0.85r [15, 16].

In our designed T-mR, the main pipe was formed by drilling a hole on a flat surface with a cross section area of 2 × 3 mm² (see Fig. 1(a₂)), two ends of the main pipe can be treated as flanged [16]. As a result, for the ends at A and B of the main pipe, the end corrections Δl_2A and Δl_3B can be approximately calculated by:

Δ l_{2A} = Δ l_{3B} \approx 0. 85R

(2) For the outer end of the branch pipe, which is partially masked (treated as an orifice), the end correction Δl_1C can be determined by the following equation [15]:

tan (k Δ l_{1 C}) = k S_{1} / α

(3) Because D₁ and D have the same order of magnitude, the interior end of the branch pipe at junction 0 (shown in Fig. 2(b₁)) can be seen as unflanged [16], the end corrections, denoted as Δl₁₀, may be determined by Hybrid Rayleigh’s end corrections outlined in [13–16]:

Δ l_{10} \approx 0.60 R_{1}

As a result, the effective length of the main and branch pipe are then calculated as follows:

L_{1} = l_{1} + Δ l_{10} + Δ l_{1 C}

for the branch pipe, and

L = l + Δ l_{2 A} + Δ l_{3 B}

for the main pipe with both ends treated as flanged.

The real physical length of the main pipe can thus be deduced by:

l \approx L - Δ l_{2 A} - Δ l_{3 B} = L - 1.70 R

2.3. General consideration of T-mR design

For a T-mR used in QEPAS, D₁ is usually limited by the value of w = 0.30 mm and l₁ is taken as short as possible to avoid coupling losses of sound wave. D would be a choice depending on the light beam diameter that we can obtain, with such a pre-determination of these 3 parameters, we can determine the parameters l of the main pipe using Eqs. (11)–(22). The wall thickness can be approximately estimated as:

T \approx {(6 / π - R^{2})}^{1 / 2} - R

T_{1} \approx {(1.5 / π)}^{1 / 2} - R_{1}

In the present work, we evaluated the performances of the T-mR based QEPAS sensor with different geometrical dimension: D₁ = 0.34-0.50 mm, T₁≈0.44-0.52 mm and l₁ = 0.10-0.85 mm for the branch pipe, D between 0.46 and 1.50 mm for the main pipe with a cross section of 2 × 3 mm² (not circular, see Fig. 1(a₂)) and an approximate wall thickness T ≈0.63-1.15 mm. The corresponding real lengths of the main pipe were calculated based on the acoustic model described above and compared with the optimum experimental values.

3. Experiment and results

Performances of the T-mR based QEPAS spectrophone were evaluated using water vapor detection in ambient air at normal atmospheric pressure and temperature between 24.2 and 27.2 °C. The used H₂O line intensity is 1.174 × 10⁻²⁰ cm⁻¹/ (mol.cm⁻²) at 7161.41 cm⁻¹. Output power of the used fiber-coupled DFB laser (NLK1E5GAAA, NEL) was ~8 mW. The ambient H₂O vapor relative humidity (RH) was determined with a hygrometer (humidity sensor, SHT75) with an uncertainty of 2% and the temperature of the ambient air was measured by a temperature sensor (PT100) with a precision of 0.1°C. The absolute concentration of water vapor was deduced from the measured RH value and the temperature. The uncertainty of the deduced absolute concentration of H₂O in volume mixing ratio is about 3%. As the concentration of the ambient H₂O varied over time, all measured QEPAS signals were normalized to the H₂O vapor concentration for instrument performance comparison.

The experimental arrangement of the T-mR QEPAS, similar to that used in [6, 7], is schematically presented in Fig. 3 .

Fig. 3 Experimental set up of a T-mR based QEPAS. 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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The resonant frequency of the used QTF was f₀~32.740 kHz. The QTF-generated piezoelectric current was converted into voltage by a home-made transimpedance amplifier with a feedback resistor R_g = 10 MΩ. A pre-amplifier ΠEG&G, Model 5113, AMETEK Advanced Measurement Technology| was used with a gain of 100 and an 6 dB bandpass filter Π10–100 kHz| for signal filtering and further amplification prior to demodulation at f0 by a lock-in amplifier (Stanford Research Systems, Model SR 830 DSP|. The time constant of the lock-in amplifier was set to 1s in combination with an 18 dB/octave slope filter (the corresponding bandwidth is Δf = 0.094 Hz). Ten T-shape mRs with different diameters and lengths (referred to as T-mR 1 to T-mR 10, respectively) were used for sensor performance evaluation. Initially, the length l of the main pipe was cut to 9 mm long, and then its length was gradually adjusted by symmetrically cutting off the pieces from both ends. The optimum lengths of the main pipe versus its inner diameters D and the parameters D₁ and L₁ of the branch pipe can be experimentally obtained. The geometrical parameters of each T-mR are listed in Table 1 .

Table 1. Parameters of the T-shaped mR tested in the experiment, as well as the related experimental results

View Table

SNR Gain is used to describe the sensitivity enhancement factor of QEPAS spectrophone. The definition of SNR Gain is given as follow [5, 7, 8]:

SNR {Gain=(S}_{a} / S_{b}) \times {(Q_{b} / Q_{a})}^{1 / 2} = (QSE factor) \times {(Q_{b} / Q_{a})}^{1 / 2}

where S_a and Q_a are the QEPAS signal normalized to the H₂O vapor concentration and the corresponding QTF Q factor, respectively for T-mR enhanced QEPAS. S_b and Q_b are the H₂O vapor concentration normalized QEPAS signal and the corresponding QTF Q factor for bare QTF-QEPAS. QSE factor, standing for QEPAS signal enhancement factor, is defined as S_a/S_b. Acoustic velocity υ (m/s) in ambient air at normal atmospheric pressure was calculated using the following expression [8, 16]:

υ (t) = 331.6 + 0. 6 t

where t is the centigrade temperature (°C).

As can be seen in Table 1, the theoretically calculated T-mR parameters closely match the experimental results. With the optimum parameters, the highest SNR gain of ~30 was obtained by using T-mR1 and T-mR5. Due to the acoustic coupling between the high-Q QTF and the low-Q mR resonator [5], the QTF Q factor in the T-mR QEPAS setup changes from Q_b~9000 to Q_a~5000 (see Table 1). It is worth noting that the change of QTF’s Q factor in the T-mR QEPAS versus the bare QTF QEPAS is smaller than that in the “on beam” QEPAS, in which Q changes from more than 10000 (bare QTF QEPAS) to even below 2500 (mR enhanced “on beam” QEPAS) [5]. This results show that T-mR possesses smaller energy accumulation wastage inside the tube compared with the mR tube used in “on beam” configuration. For sensitivity evaluation, a photoacoustic 2f signal of a 1.816% of H₂O vapor in air at 1 atm and 25.6°C was acquired using T-mR1 based QEPAS spectrophone (see Fig. 4 ).

Fig. 4 Second harmonic QEPAS signal of H₂O vapor absorption at 7161.41cm⁻¹ using T-mR1 based QEPAS sensor.

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Based on the experimentally measured noise in the QEPAS baseline, a noise level (1σ) of ~1µV was deduced at the output of the transimpedance amplifier which included the noise resulting from the transimpedance amplifier and the fundamental thermal noise of the used QTF [2, 5–7]. This thermal noise $\sqrt{< V_{_{N}}^{2} >}$ ≈0.80 μV (rms, root mean square) can be determined using the following equation:

\sqrt{< V_{_{N}}^{2} >} = \sqrt{Δ f} R_{g} \sqrt{(4 k_{B} T) / R}

where R = 246 kΩ is the QTF equivalent dynamic resistance, Δf = 0.094 Hz is the detection bandwidth and k_B is the Boltzmann constant. The corresponding minimum detectable concentration (MDC) of 165 ppbv (parts per billion by volume) for H₂O was achieved which led to a normalized noise equivalent absorption coefficient (NNEAC) of 3.9 × 10⁻⁹ cm⁻¹W/Hz^1/2.

4. Conclusions

In comparison with the currently used QEPAS spectrophone configurations, T-shaped mR based QEPAS combines the advantages from “on beam” and “off beam” QEPAS schemes while offsetting their shortages: high sensitivity (equivalent to that in the “on beam” QEPAS) and easy for optical alignment and suited for sensor applications using low cost exciting light sources (like the “off beam” QEPAS approach). In addition, the T-shaped mR design permits for precise manufacture. All these characteristics would make “T-mR QEPAS” an optimal choice for QEPAS sensor applications.

Acknowledgments

This research was supported by National Natural Science Foundation of China (Foundation No. 41175036). The support of the Groupement de Recherche International SAMIA between CNRS (France), RFBR (Russia) and CAS (China) is acknowledged.

References and links

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Type of T-mR	Parameters of branch pipe, mm		Parameters of main pipe, mm			QSE factor	Q_b	Q_a	SNR Gain	υ, m/s	C_H2O
Type of T-mR	D₁	l₁	D	l_theo.	l_exp.	QSE factor	Q_b	Q_a	SNR Gain	υ, m/s	C_H2O
T-mR1	0.34	0.18	0.46	8.58	8.50	22.5	8756	5457	28.5	347	1.816%
T-mR2	0.40	0.85	0.50	6.40	6.30	22.1	8700	5710	27.3	346	2.012%
T-mR3	0.40	0.61	0.50	6.89	6.80	21.7	9406	5608	28.1	346	2.621%
T-mR4	0.40	0.64	0.80	5.45	5.34	12.8	8860	4557	17.8	346	1.812%
T-mR5	0.50	0.66	0.50	7.47	7.35	23.0	8707	5257	29.6	347	2.821%
T-mR6	0.50	0.80	0.50	7.03	7.00	21.0	9056	5200	27.7	346	1.810%
T-mR7	0.50	0.74	0.80	5.69	5.60	12.9	9413	6331	15.7	346	1.732%
T-mR8	0.40	0.43	1.00	5.23	5.14	9.8	8736	7632	10.5	347	2.612%
T-mR9	0.40	0.23	1.20	5.13	5.00	7.4	8840	7280	8.2	346	2.523%
T-mR10	0.40	0.10	1.50	4.92	4.90	5.7	8834	7632	6.1	347	2.256%

T-shape microresonator-based high sensitivity quartz-enhanced photoacoustic spectroscopy sensor

Abstract

1. Introduction

2. Theoretical background

2.1. Acoustic impedance - Determination of the main pipe’s effective length

2.2. End correction - determination of the resonator physical length

2.3. General consideration of T-mR design

3. Experiment and results

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (28)

Optics Express