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

Optics Express, Vol. 20, Issue 8, pp. 9187-9196 (2012)

http://dx.doi.org/10.1364/OE.20.009187

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### Abstract

A novel spectrophone sensor prototype consisting of a T-shaped acoustic microresonator (T-mR) in off-beam quartz-enhanced photoacoustic spectroscopy (T-mR QEPAS) is introduced for the first time. Its performance was evaluated and optimized through an acoustic model and experimental investigation via detection of water vapor in the atmosphere. The present work shows that the use of T-mR in QEPAS based sensor can improve the detection sensitivity by a factor of up to ~30, compared with that using only a bare QTF. This value is as high as that obtained in a conventional “on-beam” QEPAS, while keeping the advantages of “off-beam” QEPAS configuration: it is no longer necessary to couple excitation light beam through the narrow gap between the QTF prongs. In addition, the T-mR is really suitable for mass production with high precision.

© 2012 OSA

## 1. Introduction

_{1})). 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.

## 2. Theoretical background

9. B. Baumann, B. Kost, H. Groninga, and M. Wolff, “Eigenmode analysis of photoacoustic sensors via finite element method,” Rev. Sci. Instrum. **77**(4), 044901 (2006). [CrossRef]

10. A. Elia, V. Spagnolo, C. D. Franco, P. M. Lugarà, and G. Scamarcio, “Trace gas sensing using quantum cascade lasers and a fiber-coupled optoacoustic sensor: application to formaldehyde,” 15th International Conference on Photoacoustic and Photothermal Phenomena (ICPPP15),” J. Phys.: Conference Series **214**, 012037 (2010). [CrossRef]

_{3})) consists of a long main pipe of length

*l*(diameter

*D)*and a short branch pipe of length

*l*

_{1}(diameter

*D*

_{1}). 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(a

_{4}) and 1(a

_{5})). The gap

*g*between the QTF and the branch pipe end (Fig. 1(a

_{5})) 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(a

_{2})). A drilling with a diameter

*D*

_{1}and a real length

*l*

_{1}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

*T*

_{1}, 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(a

_{2}). Optimum parameters of the T-mR can be determined via an acoustic model, originated from references [12

12. P. Merkli, “Acoustic resonance frequencies for a T-tube,” Z. Angew. Math. Phys. **29**(3), 486–498 (1978) (ZAMP). [CrossRef]

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

14. A. Miklós, P. Hess, and Z. Bozóki, “Application of acoustic resonators in photoacoustic trace gas analysis and metrology,” Rev. Sci. Instrum. **72**(4), 1937–1955 (2001). [CrossRef]

*Z*inside the resonator at acoustic boundary condition [12

12. P. Merkli, “Acoustic resonance frequencies for a T-tube,” Z. Angew. Math. Phys. **29**(3), 486–498 (1978) (ZAMP). [CrossRef]

*p/(uS)*[12

12. P. Merkli, “Acoustic resonance frequencies for a T-tube,” Z. Angew. Math. Phys. **29**(3), 486–498 (1978) (ZAMP). [CrossRef]

*p*,

*u*are the acoustic pressure and fluid velocity inside the pipe, respectively,

*S*is the cross section area of the pipe).

*L*

_{0}rather than its real physical length

*l*

_{0}. 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

*r*

_{0}, the input acoustic impedances

*Z*at x

*= 0*(one end of the resonator) and x

*= L*

_{0}(the other end of the resonator), noted as

*Z(0)*and

*Z(L*

_{0}

*)*respectively, are expressed by Eq. (1) [11, 12

**29**(3), 486–498 (1978) (ZAMP). [CrossRef]

_{1}): x

_{1}and x

_{2}stand for position coordinate of the branch pipe and the main pipe, respectively;

*ρ*is the density of the fluid inside the pipe;

*S*the cross section area of the pipe. The axis origin is noted as 0.

_{0}*k = ω/υ = 2πf/υ*is the wave number,

*f*is the acoustic frequency (that should be equal to the QTF resonant frequency

*f*

_{0}= 32.740 kHz, in our case),

*υ*the acoustic wave velocity and

*j*.

^{2}= −1_{2}= -

*L*

_{2}), E (x

_{2}= 0) for branch 2, at F (x

_{2}= 0), B (x

_{2}=

*L*

_{3}) for branch 3, at 0 (x

_{1}= 0), C (x

_{1}= -

*L*

_{1}) for branch 1 are noted as Z

_{2A}(

*-L*

_{2}), Z

_{2E}(

*0*), Z

_{3F}(

*0*), Z

_{3B}(

*L*

_{3}), Z

_{10}(

*0*), Z

_{1C}(

*-L*

_{1}) (shown in Fig. 2(b

_{1})), 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(b

_{1})), respectively.

*L*

_{1}is the effective length of the branch 1 (its corresponding physical length is

*l*

_{1}).

_{1}), Z

_{2E}(

*0*), Z

_{3F}(

*0*) and Z

_{10}(

*0*) satisfy the following relation deduced by continuity equations [12

**29**(3), 486–498 (1978) (ZAMP). [CrossRef]

13. D. Li and J. S. Vipperman, “On the design of long T-shaped acoustic resonators,” J. Acoust. Soc. Am. **116**(5), 2785–2792 (2004). [CrossRef]

*S*

_{2}

*= S*

_{3}

*= S = πR*, with

^{2}*R = D/2*, the radius of the main pipe branches 2 and 3;

*S*

_{1}

*= πR*

_{1}

*, with*

^{2}*R*

_{1}

*= D*

_{1}

*/2*, the radius of the short branch 1.

_{2}

*= -L*) and B (x

_{2}_{2}

*= L*) satisfies the following relationship [11, 12

_{3}**29**(3), 486–498 (1978) (ZAMP). [CrossRef]

*L*

_{2}=

*L*

_{3},

*L*

_{2}and

*L*

_{3}are the effective length of the main pipe branches 2 and 3, respectively (their corresponding physical length are

*l*

_{2}and

*l*

_{3}, respectively).

_{4}) and Fig. 2(b

_{2})). The acoustic impedance of the partially masked end at x

_{1}

*=*-

*L*

_{1}was denoted as

*Z*

_{1}

*-*

_{C}(*L*

_{1}

*) = jρω/α*[13

13. D. Li and J. S. Vipperman, “On the design of long T-shaped acoustic resonators,” J. Acoust. Soc. Am. **116**(5), 2785–2792 (2004). [CrossRef]

*Z*

_{o}

*= jρω/α*

_{ο}(see Fig. 2(b

_{1})) and the acoustic impedance of the gap between the QTF and the end of the branch pipe

*Z*(see Fig. 2(b

_{s}= jρω/α_{s}_{1})) [8

8. H. Yi, W. Chen, X. Guo, S. Sun, K. Liu, T. Tan, W. Zhang, and X. Gao, “An acoustic model for microresonator in on beam quartz-enhanced photoacoustic spectroscopy,” Appl. Phys. B (2012). DOI . [CrossRef] [PubMed]

*Ω*

_{0}

*= πD*

_{1}

*g*and a thickness of

*T*

_{1}[8

8. H. Yi, W. Chen, X. Guo, S. Sun, K. Liu, T. Tan, W. Zhang, and X. Gao, “An acoustic model for microresonator in on beam quartz-enhanced photoacoustic spectroscopy,” Appl. Phys. B (2012). DOI . [CrossRef] [PubMed]

*Z*

_{1C}

*(0)*,

*Z*

_{o}and

*Z*satisfy the following equation at the junction of x

_{s}_{1}= -

*L*

_{1}: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(b

_{2}) and 2(b

_{4}), dash area) and the acoustic conductivity of the gap

*α*[8

_{s}8. H. Yi, W. Chen, X. Guo, S. Sun, K. Liu, T. Tan, W. Zhang, and X. Gao, “An acoustic model for microresonator in on beam quartz-enhanced photoacoustic spectroscopy,” Appl. Phys. B (2012). DOI . [CrossRef] [PubMed]

_{3}), dash area):

*α*

_{ο}and

*α*

_{s}can be calculated using the following equations: 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(b

_{4}), dash area) equivalent to the real orifice area

*Ω*

_{1}(Fig. 2(b

_{2}), dash area).

### 2.2. End correction - determination of the resonator physical length

14. A. Miklós, P. Hess, and Z. Bozóki, “Application of acoustic resonators in photoacoustic trace gas analysis and metrology,” Rev. Sci. Instrum. **72**(4), 1937–1955 (2001). [CrossRef]

_{1})) 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(b

_{1})) 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

13. D. Li and J. S. Vipperman, “On the design of long T-shaped acoustic resonators,” J. Acoust. Soc. Am. **116**(5), 2785–2792 (2004). [CrossRef]

*l*of circular tubes based on the flange width T (T is the wall thickness of the mR tube here):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 Δ

*l≈*0.6

*r*[15, 16]. When T becomes thicker (T≥3

*r*), the tube end forms a flange, the end correction for such a flanged end can be expressed as Δ

*l≈*0.85

*r*[15, 16].

^{2}(see Fig. 1(a

_{2})), 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 Δ

*l*

_{2A}and Δ

*l*

_{3B}can be approximately calculated by:

*Δl*

_{1C}can be determined by the following equation [15]:

_{1}and D have the same order of magnitude, the interior end of the branch pipe at junction 0 (shown in Fig. 2(b

_{1})) can be seen as unflanged [16], the end corrections, denoted as Δ

*l*

_{10}, may be determined by Hybrid Rayleigh’s end corrections outlined in [13

**116**(5), 2785–2792 (2004). [CrossRef]

### 2.3. General consideration of T-mR design

*D*

_{1}is usually limited by the value of w = 0.30 mm and

*l*

_{1}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:

*D*

_{1}= 0.34-0.50 mm,

*T*

_{1}≈0.44-0.52 mm and

*l*

_{1}= 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 mm

^{2}(not circular, see Fig. 1(a

_{2})) 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

_{2}O 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 H

_{2}O 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 H

_{2}O in volume mixing ratio is about 3%. As the concentration of the ambient H

_{2}O varied over time, all measured QEPAS signals were normalized to the H

_{2}O vapor concentration for instrument performance comparison.

6. K. Liu, X. Guo, H. Yi, W. Chen, W. Zhang, and X. Gao, “Off-beam quartz-enhanced photoacoustic spectroscopy,” Opt. Lett. **34**(10), 1594–1596 (2009). [CrossRef] [PubMed]

7. K. Liu, H. Yi, A. A. Kosterev, W. Chen, L. Dong, L. Wang, T. Tan, W. Zhang, F. K. Tittel, and X. Gao, “Trace gas detection based on off-beam quartz enhanced photoacoustic spectroscopy: optimization and performance evaluation,” Rev. Sci. Instrum. **81**(10), 103103 (2010). [CrossRef] [PubMed]

5. L. Dong, A. A. Kosterev, D. Thomazy, and F. K. Tittel, “QEPAS spectrophones: design, optimization, and performance,” Appl. Phys. B **100**(3), 627–635 (2010). [CrossRef]

7. K. Liu, H. Yi, A. A. Kosterev, W. Chen, L. Dong, L. Wang, T. Tan, W. Zhang, F. K. Tittel, and X. Gao, “Trace gas detection based on off-beam quartz enhanced photoacoustic spectroscopy: optimization and performance evaluation,” Rev. Sci. Instrum. **81**(10), 103103 (2010). [CrossRef] [PubMed]

_{a}and

*Q*

_{a}are the QEPAS signal normalized to the H

_{2}O vapor concentration and the corresponding QTF

*Q*factor, respectively for T-mR enhanced QEPAS. S

_{b}and

*Q*

_{b}are the H

_{2}O 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

*t*is the centigrade temperature (°C).

*Q*QTF and the low-

*Q*mR resonator [5

5. L. Dong, A. A. Kosterev, D. Thomazy, and F. K. Tittel, “QEPAS spectrophones: design, optimization, and performance,” Appl. Phys. B **100**(3), 627–635 (2010). [CrossRef]

*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

5. L. Dong, A. A. Kosterev, D. Thomazy, and F. K. Tittel, “QEPAS spectrophones: design, optimization, and performance,” Appl. Phys. B **100**(3), 627–635 (2010). [CrossRef]

*f*signal of a 1.816% of H

_{2}O vapor in air at 1 atm and 25.6°C was acquired using T-mR1 based QEPAS spectrophone (see Fig. 4 ).

**100**(3), 627–635 (2010). [CrossRef]

7. K. Liu, H. Yi, A. A. Kosterev, W. Chen, L. Dong, L. Wang, T. Tan, W. Zhang, F. K. Tittel, and X. Gao, “Trace gas detection based on off-beam quartz enhanced photoacoustic spectroscopy: optimization and performance evaluation,” Rev. Sci. Instrum. **81**(10), 103103 (2010). [CrossRef] [PubMed]

*V*(rms, root mean square) can be determined using the following equation:where

*R*= 246 kΩ is the QTF equivalent dynamic resistance, Δ

*f =*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 H

_{2}O was achieved which led to a normalized noise equivalent absorption coefficient (NNEAC) of 3.9 × 10

^{−9}cm

^{−1}W/Hz

^{1/2}.

## 4. Conclusions

## Acknowledgments

## References and links

1. | A. A. Kosterev, Y. A. Bakhirkin, R. F. Curl, and F. K. Tittel, “Quartz-enhanced photoacoustic spectroscopy,” Opt. Lett. |

2. | F. K. Tittel, G. Wysocki, A. A. Kosterev, and Y. Bakhirkin, “Semiconductor laser based trace gas sensor technology: recent advances and applications,” in |

3. | H. Yi, K. Liu, W. Chen, T. Tan, L. Wang, and X. Gao, “Application of a broadband blue laser diode to trace NO |

4. | S. Böttger, M. Angelmahr, and W. Schade, “Photoacoustic Gas Detection with LED QEPAS,” in CLEO/Europe and EQEC 2011 Conference Digest, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CH_P14. |

5. | L. Dong, A. A. Kosterev, D. Thomazy, and F. K. Tittel, “QEPAS spectrophones: design, optimization, and performance,” Appl. Phys. B |

6. | K. Liu, X. Guo, H. Yi, W. Chen, W. Zhang, and X. Gao, “Off-beam quartz-enhanced photoacoustic spectroscopy,” Opt. Lett. |

7. | K. Liu, H. Yi, A. A. Kosterev, W. Chen, L. Dong, L. Wang, T. Tan, W. Zhang, F. K. Tittel, and X. Gao, “Trace gas detection based on off-beam quartz enhanced photoacoustic spectroscopy: optimization and performance evaluation,” Rev. Sci. Instrum. |

8. | H. Yi, W. Chen, X. Guo, S. Sun, K. Liu, T. Tan, W. Zhang, and X. Gao, “An acoustic model for microresonator in on beam quartz-enhanced photoacoustic spectroscopy,” Appl. Phys. B (2012). DOI . [CrossRef] [PubMed] |

9. | B. Baumann, B. Kost, H. Groninga, and M. Wolff, “Eigenmode analysis of photoacoustic sensors via finite element method,” Rev. Sci. Instrum. |

10. | A. Elia, V. Spagnolo, C. D. Franco, P. M. Lugarà, and G. Scamarcio, “Trace gas sensing using quantum cascade lasers and a fiber-coupled optoacoustic sensor: application to formaldehyde,” 15th International Conference on Photoacoustic and Photothermal Phenomena (ICPPP15),” J. Phys.: Conference Series |

11. | S. L. Firebaugh, F. Roignant, and E. A. Terray, “Enhancing sensitivity in tuning fork photoacoustic spectroscopy systems,” in |

12. | P. Merkli, “Acoustic resonance frequencies for a T-tube,” Z. Angew. Math. Phys. |

13. | D. Li and J. S. Vipperman, “On the design of long T-shaped acoustic resonators,” J. Acoust. Soc. Am. |

14. | A. Miklós, P. Hess, and Z. Bozóki, “Application of acoustic resonators in photoacoustic trace gas analysis and metrology,” Rev. Sci. Instrum. |

15. | E. G. Richardson, |

16. | L. E. Kinsler and A. R. Frey, |

**OCIS Codes**

(230.0230) Optical devices : Optical devices

(280.3420) Remote sensing and sensors : Laser sensors

(300.6260) Spectroscopy : Spectroscopy, diode lasers

(300.6430) Spectroscopy : Spectroscopy, photothermal

**ToC Category:**

Spectroscopy

**History**

Original Manuscript: February 9, 2012

Revised Manuscript: March 28, 2012

Manuscript Accepted: March 28, 2012

Published: April 5, 2012

**Virtual Issues**

Vol. 7, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Hongming Yi, Weidong Chen, Shanwen Sun, Kun Liu, Tu Tan, and Xiaoming Gao, "T-shape microresonator-based high sensitivity quartz-enhanced photoacoustic spectroscopy sensor," Opt. Express **20**, 9187-9196 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-8-9187

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### References

- A. A. Kosterev, Y. A. Bakhirkin, R. F. Curl, and F. K. Tittel, “Quartz-enhanced photoacoustic spectroscopy,” Opt. Lett. 27(21), 1902–1904 (2002). [CrossRef] [PubMed]
- F. K. Tittel, G. Wysocki, A. A. Kosterev, and Y. Bakhirkin, “Semiconductor laser based trace gas sensor technology: recent advances and applications,” in Mid-Infrared Coherent Sources and Applications, Ebrahim-Zadeh, M., Sorokina, I.T., Eds. (Springer, 2007), 467–493.
- H. Yi, K. Liu, W. Chen, T. Tan, L. Wang, and X. Gao, “Application of a broadband blue laser diode to trace NO2 detection using off-beam quartz-enhanced photoacoustic spectroscopy,” Opt. Lett. 36(4), 481–483 (2011). [CrossRef] [PubMed]
- S. Böttger, M. Angelmahr, and W. Schade, “Photoacoustic Gas Detection with LED QEPAS,” in CLEO/Europe and EQEC 2011 Conference Digest, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CH_P14.
- L. Dong, A. A. Kosterev, D. Thomazy, and F. K. Tittel, “QEPAS spectrophones: design, optimization, and performance,” Appl. Phys. B 100(3), 627–635 (2010). [CrossRef]
- K. Liu, X. Guo, H. Yi, W. Chen, W. Zhang, and X. Gao, “Off-beam quartz-enhanced photoacoustic spectroscopy,” Opt. Lett. 34(10), 1594–1596 (2009). [CrossRef] [PubMed]
- K. Liu, H. Yi, A. A. Kosterev, W. Chen, L. Dong, L. Wang, T. Tan, W. Zhang, F. K. Tittel, and X. Gao, “Trace gas detection based on off-beam quartz enhanced photoacoustic spectroscopy: optimization and performance evaluation,” Rev. Sci. Instrum. 81(10), 103103 (2010). [CrossRef] [PubMed]
- H. Yi, W. Chen, X. Guo, S. Sun, K. Liu, T. Tan, W. Zhang, and X. Gao, “An acoustic model for microresonator in on beam quartz-enhanced photoacoustic spectroscopy,” Appl. Phys. B (2012). DOI 10.1007/s00340-012-4988-7. [CrossRef] [PubMed]
- B. Baumann, B. Kost, H. Groninga, and M. Wolff, “Eigenmode analysis of photoacoustic sensors via finite element method,” Rev. Sci. Instrum. 77(4), 044901 (2006). [CrossRef]
- A. Elia, V. Spagnolo, C. D. Franco, P. M. Lugarà, and G. Scamarcio, “Trace gas sensing using quantum cascade lasers and a fiber-coupled optoacoustic sensor: application to formaldehyde,” 15th International Conference on Photoacoustic and Photothermal Phenomena (ICPPP15),” J. Phys.: Conference Series 214, 012037 (2010). [CrossRef]
- S. L. Firebaugh, F. Roignant, and E. A. Terray, “Enhancing sensitivity in tuning fork photoacoustic spectroscopy systems,” in Sensor Application Symposium (SAS), 23–25 Feb. (IEEE Press, New York 2010), 30–35.
- P. Merkli, “Acoustic resonance frequencies for a T-tube,” Z. Angew. Math. Phys. 29(3), 486–498 (1978) (ZAMP). [CrossRef]
- D. Li and J. S. Vipperman, “On the design of long T-shaped acoustic resonators,” J. Acoust. Soc. Am. 116(5), 2785–2792 (2004). [CrossRef]
- A. Miklós, P. Hess, and Z. Bozóki, “Application of acoustic resonators in photoacoustic trace gas analysis and metrology,” Rev. Sci. Instrum. 72(4), 1937–1955 (2001). [CrossRef]
- E. G. Richardson, Technical Aspects of Sound: Sonic Range and Airborne Sound (Elsevier Pub. Co., 1957), pp. 12–13 and 487–496.
- L. E. Kinsler and A. R. Frey, Fundamentals of Acoustics (John Wiley& Sons, Inc., 1962), pp. 116–118 and 186–213.

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