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Measurement of the Acoustic Relaxation Absorption Spectrum of CO_{2} Using a Distributed Bragg Reflector Fiber Laser

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## Abstract

**:**

_{2}using a decompression gas chamber between 0.1 and 1 atm to accommodate the main molecular relaxation processes, and interrogates with a non-equilibrium Mach-Zehnder interferometer (NE-MZI) to gain a sound pressure sensitivity of −45.4 dB. The measurement error of the acoustic relaxation absorption spectrum is less than 1.32%.

## 1. Introduction

^{8}Hz at room temperature and pressure [7]. Current consumer ultrasonic transducers are only launched at one frequency, aside from a few with kilohertz bandwidth. A large number of acoustic transducers at different frequencies are required to cover the entire frequency spectral line to be measured [8]. Therefore, a wide-band ultrasonic sensor with a flat response would be preferred. Since the ultrasonic transmitter is not able to be replaced at present, an alternative approach to adjusting the ambient pressure, which is inversely proportional to the relaxation frequency of gas molecules [9], to obtain the whole sound propagation spectrum with a transmitter and a DBR fiber laser is proposed in this paper.

^{1/2}and a measurement range from 5 to 25 MHz [16]. As a highly sensitive candidate, the DBR fiber laser is also promising for marine applications such as pressure, tsunami, and earth dynamics [17].

_{2}using the DBR fiber laser ultrasonic sensor.

## 2. Ultrasonic Sensing Model of the DBR Fiber Laser

#### 2.1. Response of the DBR Fiber Laser to Ultrasonic Wave

_{0}is the sound pressure at the transmitter and $\alpha $ is the attenuation coefficient.

_{0}is the initial velocity of the fiber vibration, and P

_{m}$\left(\tau \right)$ is the sound pressure acting on the DBR fiber laser, which is proportional to P(z) in Equation (1).

^{−17}m for the DBR fiber laser length of 48 mm.

_{2}reaches above 2 MHz (the wavelength of the sound wave and the response bandwidth increase while the molar mass of the gas decreases), which fully meets the experimental requirements.

_{light}of the DBR fiber laser depends on its cavity geometry, written as follows:

_{light}, and Δλ

_{light}are the variations of cavity length, refractive index change caused by the photoelastic effect, the frequency shift, and the wavelength shift of the DBR fiber laser. For convenience, frequency variation is rewritten in the form of a relative wavelength shift as Equation (6).

_{DBR}of the DBR fiber laser can be obtained:

_{11}and P

_{12}. The Young’s modulus and Poisson ratio of the DBR fiber laser can be effectively improved by selecting appropriate materials for packaging.

#### 2.2. Ultrasonic Wave Sensing Model of the DBR Fiber Laser

#### 2.2.1. The Optical Interrogation Signal of the DBR Fiber Laser

_{0}and the wavelength shift is Δλ(t). The corresponding phase difference introduced by the NE-MZI has an expression as follows:

_{OPD}is the optical path difference of the NE-MZI, and Δφ (t) is the phase difference caused by the wavelength shift of the DBR fiber laser, i.e., the demodulation signal recorded in the experiments. The ultrasonic response comparison of the DBR fiber laser with the ultrasonic transducer takes advantage of the proportional relation between the applied sound pressure and the output voltage of the ultrasonic transducer under the same frequency and pressure conditions. The DBR fiber laser is placed right in front of the receiving transducer, facing the transmitting transducer. The phase difference proves to be proportional to the sound pressure applied to the DBR fiber laser. The demonstration scheme and results are plotted in Figure 4.

#### 2.2.2. Calibration of the Ultrasonic Response of the DBR Fiber Laser

^{1/2}below 1 kHz), decreases with frequency increase, and drops below −75 dB/Hz

^{1/2}after 15 kHz, where the noise equivalent sound pressure (NESP) reaches about −80 dB/Hz

^{1/2}. However, the noise power rises to −60 dB/Hz

^{1/2}at 20 kHz as the ultrasonic wave launches. Therefore, calibration is necessary for the DBR fiber laser before sensing applications.

## 3. Measurement of the Acoustic Relaxation Absorption and Concentration of CO_{2} with the DBR Fiber Laser Ultrasonic Sensor

#### 3.1. Measurement of the Acoustic Relaxation Absorption at a Specific Frequency

_{2}is a popular validation gas in reduction studies of the acoustic relaxation absorption spectrum. Acoustic absorption measures are first used for the reconstruction of the acoustic relaxation absorption curve of CO

_{2}. According to the Clapeyron equation, a decreasing chamber pressure is equivalent to an increasing ultrasonic frequency. Since the effective relaxation frequency of the acoustic relaxation absorption curve of 100% CO

_{2}is located at 40 kHz, an ultrasonic transducer of 25 kHz is a better choice to satisfy the measurement of the relaxation absorption spectrum of CO

_{2}, where f/p is controlled by adjusting the chamber pressure. The distance between the transmitting transducer and the DBR fiber laser varies between 30~100 mm to satisfy the far-field prerequisite for ultrasonic attenuation coefficient measurement.

_{1}and U

_{2}are the output voltages of the receiving transducer obtained in positions L

_{1}and L

_{2}, respectively. The output voltage U is also defined as a product of sound pressure sensitivity k

_{t}and intensity P. Equation (10) clarifies that the absorption coefficient α is independent of the sensitivity of the sensor k

_{t}for a specific ultrasonic frequency and only determined by the gas characteristics. Therefore, the absorption coefficient α stays the same no matter whether a transducer or a DBR fiber laser sensor is used for testing, as long as the f/p stays the same. The sensitivity determines the lowest detectable acoustic pressure, or the noise equivalent sound pressure, so the measurement precision of the acoustic attenuation coefficient improves with the sensitivity.

#### 3.2. Measurement Results of the Absorption Spectrum of CO_{2}

_{2}concentrations of 100%, 80%, 50%, and 20%. Measurements use two ultrasonic transducers of 25 kHz and 40 kHz as the emitting sources and the DBR fiber laser as the sensing probe. Repeating the test steps in Section 3.1, acoustic relaxation absorption coefficients are obtained corresponding to a series of f/p around the effective relaxation frequency. The acoustic relaxation absorption spectrum of 100% CO

_{2}is restored by the fitting spectrum of 25 kHz (red square) and 40 kHz (pink round), as shown in Figure 9a. Similar fitting curves are plotted in Figure 9b–d for CO

_{2}concentrations of 80%, 50%, and 20%, respectively. The black solid curves are theoretical curves corresponding to the four CO

_{2}concentrations.

_{2}, which leads to an error of 0.82% of concentration. In Figure 9b, the deviation of the experimental spectrum from the theoretical spectrum leads to an error of 0.84%. For 50% and 20% CO

_{2}, the errors are 1.26% and 1.32%, respectively. While the relaxation absorption coefficient decreases with the concentration of CO

_{2}, the measurement error of the corresponding acoustic relaxation absorption spectra increases. On the other side, it shows that measurements taken as close as possible to the absorption peak frequency will give better precision.

## 4. Discussion

#### 4.1. Systematic Error Results from Variation of f/p

#### 4.2. The Temperature and Pressure Instability of the Gas Chamber and Compensation

## 5. Conclusions

_{2}with various concentrations. With a relative wide and flat frequency response, the DBR fiber laser sensor measures and restores a full acoustic relaxation absorption spectrum of CO

_{2}using a decompression gas chamber between 0.1 and 1 atm to accommodate the main molecular relaxation processes and interrogates with a NE-MZI to gain a sound pressure sensitivity of −45.4 dB. The measurement error of the acoustic relaxation absorption spectrum is less than 1.32%. Combined with the gas sensing approach based on ultrasonic velocity, the DBR fiber laser sensor is a promising candidate for the analysis of ternary or quaternary gases.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Displacement distribution along the DBR fiber laser under 25 kHz and a 2 Pa ultrasonic wave: (

**a**) solution of the vibration equation and (

**b**) finite element simulation.

**Figure 3.**Corresponding wavelength shift of the DBR fiber laser to the source strength of ultrasonic waves at various frequencies.

**Figure 4.**(

**a**) Schematic diagram of linearity demonstration with a DBR fiber laser and a pair of transducers; (

**b**) experimental results of the noise power to ambient pressure in the gas chamber; (

**c**,

**d**) are sample ultrasonic calibration results at 25 kHz and 300 kHz using the DBR fiber laser and a receiving transducer.

**Figure 8.**The testing results of the DBR fiber laser sensor. (

**a**) The ultrasonic sound field distribution of 25 kHz in 100% CO

_{2}and (

**b**) the acoustic pressure to distance curve.

**Figure 9.**Restored curves of the acoustic absorption relaxation (pink fitting curve) with the DBR fiber laser and the theoretical curve (black solid curve) [18] of CO

_{2}, concentrations of (

**a**) 100%, (

**b**) 80%, (

**c**) 50%, and (

**d**) 20%. Using both 25 kHz and 40 kHz ultrasonic transducers, testing in the gas chamber over a pressure range of 0.1 to 1 atm.

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**MDPI and ACS Style**

Shen, K.; Yuan, J.; Li, M.; Wen, X.; Lu, H.
Measurement of the Acoustic Relaxation Absorption Spectrum of CO_{2} Using a Distributed Bragg Reflector Fiber Laser. *Sensors* **2023**, *23*, 4740.
https://doi.org/10.3390/s23104740

**AMA Style**

Shen K, Yuan J, Li M, Wen X, Lu H.
Measurement of the Acoustic Relaxation Absorption Spectrum of CO_{2} Using a Distributed Bragg Reflector Fiber Laser. *Sensors*. 2023; 23(10):4740.
https://doi.org/10.3390/s23104740

**Chicago/Turabian Style**

Shen, Kun, Jixian Yuan, Min Li, Xiaoyan Wen, and Haifei Lu.
2023. "Measurement of the Acoustic Relaxation Absorption Spectrum of CO_{2} Using a Distributed Bragg Reflector Fiber Laser" *Sensors* 23, no. 10: 4740.
https://doi.org/10.3390/s23104740