1. Introduction

2. Ultrasonic sensing model of the DBR fiber laser

2.1. Response of the DBR fiber laser to ultrasonic wave

A DBR fiber laser consists of a pair of fiber Bragg gratings (FBGs) and a piece of Er-doped fiber in between works as a sensing unit. The process of a soundwave applying to the DBR fiber laser changing the cavity length and in turn the laser wavelength, which involves two physical effects, elastic deformation and photoelastic effect of the DBR fiber laser, so the induced strain of the DBR fiber laser under ultrasonic wave needs to be figured out before sensor modelling. In this work, the displacement of the DBR fiber laser is obtained by solving partial differential equation of one-dimensional vibration of the optical fiber, and numerically proved by the finite element simulation.

Figure 1 illustrates the gas detection scheme proposed in this paper. Transmitting transducers generate ultrasonic waves of several specific frequencies, which are received by the DBR fiber laser.

As a kind of mechanical wave, ultrasonic wave acting on the DBR fiber laser follows the wave theory of a string and generates longitudinal stress waves propagating along the fiber axis, which produces axial strain on the DBR fiber laser. Considering the DBR fiber laser as a thin rod consisting of several segments, longitudinal displacement of each segment leads to compression or elongation of its adjacent segments.

Assuming the ultrasonic wave as a plane wave emitted from the transducer, the acoustic sound pressure of z direction (perpendicular to the fiber axis) p(z) can be written as the following.

where p₀ is the sound pressure at the transmitter, $\alpha$ is the attenuation coefficient.

Assuming the DBR fiber laser is an uniform bar, and u(x, t) is the displacement perpendicular to the fiber axis at position x and time t of the DBR fiber laser. The vibration equation has an expression as:

In Equation (2), v is the velocity of sound in the optical fiber. The displacement on the DBR fiber laser caused by the continuous force of ultrasonic waves presents as u(x, t), and the sound pressure p modulates the sensor’s geometry, which leads to wavelength shift of the DBR fiber laser.

Equation (3) is one dimensional vibration equation of the DBR fiber laser, in which, m is a positive integer, d is the fiber segment length under ultrasonic wave, L is the length of the DBR fiber laser, v₀ is the initial velocity of the fiber vibration, p_m$\left(\tau \right)$ is the sound pressure acting on DBR fiber laser, which is proportional to p(z) in Equation(1).

Figure 2 illustrates the displacement distribution of the DBR fiber laser under ultrasonic sound pressure of 2 Pa and 25 kHz according to Equation (3).

25 kHz ultrasonic wave is launches perpendicularly to the axis of the DBR fiber laser, the two ends of which are fixed. The ultrasonic wave acting on the DBR fiber laser sets to 2 Pa, and covers fiber segment length of 20 mm.

Figure 2a plots the displacement distribution of the DBR fiber laser by solving the vibration equation as well as the finite element simulation results (Figure 2b). The center node has the maximum displacement about 1.5×10^-17 m for the DBR fiber laser length of 48 mm.

The wavelength shift of the DBR fiber laser caused by ultrasonic wave can be deduced from the obtained displacement. Since p is a function of the ultrasonic frequency, the wavelength shifts of the DBR fiber laser under 25 kHz (black square), 40 kHz (red dot), 112 kHz (blue up triangle), 200 kHz (green down triangle) and 300 kHz (purple diamond) ultrasonic wave are compared in Figure 3. In Figure 3, the maximum wavelength shift of the DBR fiber laser increases with the ultrasonic source strength which is proportional to the sound pressure applied to the DBR fiber laser at a specific frequency and decreases while the ultrasonic frequency increases.

The DBR fiber laser is essentially the sensing unit and subjected to sound pressure. The cavity length changes with the applied ultrasonic wave, then results in the wavelength shift of the DBR fiber laser. In Figure 3, the intensity of the ultrasonic wave is indicated by the source strength. For ultrasonic waves of a certain frequency, the wavelength shift of the DBR fiber laser is proportional to the intensity of the sound source, and decreases with the frequency of the sound source increase, which results from the faster attenuation of the ultrasonic wave at higher frequency. Since better linearity of the DBR fiber lasers sensing system brings with smaller systematic error in gas detection, the response linearity of the DBR fiber lasers sensing system is experimentally demonstrated in section 2.2.

The DBR fiber laser has two specific frequencies to be determined, the resonance frequency and the theoretical maximum response frequency that limited by the geometry of the DBR fiber laser. Using the equation of string vibration, resonant frequency $f_m=mv/2L,m=1,2,3\dots$ When the density and Young's modulus of the communication fiber are chosen, the sound velocity in the DBR fiber laser is about 5570 m/s. So the frequency response of the fundamental mode is around 58 kHz. The fundamental vibration mode has the maximum amplitude, which is inversely proportional to m square (m is the order of harmonic mode). On the other hand, only the axial strain needs to be considered when the ultrasonic wavelength is much larger than the diameter of the DBR fiber laser (125m). Therefore, the frequency response of the DBR fiber laser in CO2 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.

The output light frequency f_light of the DBR fiber laser depends on its cavity geometry written as follow:

where Q is the positive integer, c is the velocity of light, n is the refractive index of the fiber, L is length of the DBR fiber laser.

The relation of output frequency, cavity length and refractive index of the DBR fiber laser has an expression as the derivative of Equation (4):

ΔL, Δn, Δf_light, Δ_light are the variation of cavity length, refractive index change caused by photoelastic effect, the frequency shift and the wavelength shift of the DBR fiber laser. For convenience, frequency variation rewrites in the form of relative wavelength shift as Equation (6).

According to Hooke's law, the strain tensor of the DBR fiber laser in an uniform sound field can be expressed as:

Where E is the Young's modulus of the DBR fiber laser, μ is the poisson ratio. By substituting Equation (7) into Equation (6), the relative sound pressure sensitivity K_DBR of the DBR fiber laser can be obtained:

As can be seen from equation (8), the sound pressure sensitivity of the DBR fiber laser depends on the effective elastic coefficient p₁₁ and p_12, 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.

Assuming the operating wavelength of the DBR fiber laser is ₀, and the wavelength shift is Δ(t). The corresponding phase difference introduced by the NE-MZI has an expression as:

Where L_OPD is optical path difference of the NE-MZI, and Δ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 ultrasonic transducer under the same frequency and circumstance pressure. 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 onto the DBR fiber laser. The demonstration scheme and results are plotted in Figure 4.

Figure 4 a is the experimental scheme of the demonstration experiment, Figure 4b shows the influences of the ambient pressure on the ultrasonic noise power of the DBR fiber laser output. The noise power increases with the ambient pressure, and is much smaller than the ultrasonic signal power. Noise will affect the accuracy of ultrasonic measurement, so the noise signal is subtracted during ultrasonic measurement. Figure 4 c and d are the sample experimental spectrums of phase difference to transducer output voltages for 25 kHz and 300 kHz ultrasonic wave. Figure 4 shows the phase difference of the DBR fiber laser changes linearly with the ultrasonic pressure, and the linear correlation coefficients of the experimental results are over 0.99.

2.2.2. Calibration of the ultrasonic response of the DBR fiber laser.

Since it is the first time to measure the ultrasonic sound pressure quantitively using a DBR fiber laser, the ultrasonic response of the proposed DBR fiber laser system needs tested and compared with their corresponding transducers at five ultrasonic frequencies of our applications.

Figure 5 shows the optical responses of the DBR fiber laser to five sampling ultrasonic waves between 20 to 300 kHz at 25, 40, 112, 200 and 300 kHz, where the ultrasonic waves launched at five frequencies are tested simultaneously. And the inlet plots the background noise without ultrasonic signal applied.

The background noise power is relative large at lower frequency (for example, raising to -50dB/Hz^1/2 below 1 kHz), decreases with the frequency increase, and 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 raises to -60 dB/Hz^1/2 at 20 kHz as the ultrasonic wave launched. Therefore, calibration is necessary for the DBR fiber laser before sensing application.

Five piezoelectric transducers of 25 kHz, 40 kHz, 112 kHz, 200 kHz and 300 kHz are chosen for the sensitivity calibration of the DBR fiber laser. The results are shown in Figure 6.

Figure 6 shows the sensitivity of the DBR fiber laser and transducers over the frequency range from 25 to 300 kHz. The average sensitivity of the DBR fiber laser is -45.4 dB, and relatively flat over the ultrasonic range. By now, the DBR fiber laser system is ready for ultrasonic detection and gas sensing.

3. Measurement of the acoustic relaxation absorption and concentration of CO₂ with the DBR fiber laser ultrasonic sensor.

3.1. Measurement of the acoustic relaxation absorption at a specific frequency.

Relaxation frequency and the corresponding intensity of the absorption curve are the unique characteristics of a specific gas and generally used to identify the unknown composition of a gas mixture. The composition or concentration of a certain gas in the mixture can be identified with the sound velocity spectrum or the absorption spectrum, which depends on the number of components in the mixture. Characteristic frequency, absorption coefficient and sound velocity of the measured gas can be evaluated from the comparison of the reconstruction spectrum with the theoretical or published experimental ultrasonic spectral data of gases [17]. Since the transmitting transducers find no substitutes at present, the experiment chooses a decompression gas chamber to simulate the main molecular relaxation process.

Figure 7 illustrates the experimental system proposed in this paper. A pressure tunable gas chamber ranging from 0.01 to 1 atm is designed for the acoustic absorption spectrum measurement. Transmitting transducers embedded in a plate fixed on a step motor emit ultrasonic waves of 25 and 40kHz, which are received by the DBR fiber laser. The output signal of the DBR fiber laser feeds into a NE-MZI through a wavelength division multiplexer (WDM), and interrogates the phase difference and corresponding wavelength shift of the DBR fiber laser under applied ultrasonic wave using the 3×3 coupler method, which effectively improves the sensitivity of the system.

In the gas chamber, the DBR fiber laser is placed on the path and axially perpendicular to the direction of the ultrasonic emission. Since the diameter of optical fiber is much smaller than the ultrasonic wavelength, reflection from the fiber is no concern of our experiments. An aluminum plate places next to the DBR fiber laser on the opposite side of the transmitting transducer to form an effective standing wave.

CO₂ is a popular validation gas in reduction studies of acoustic relaxation absorption spectrum. Acoustic absorption measures first for the reconstruction of the acoustic relaxation absorption curve of CO₂. According to 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% CO₂ locates at 40 kHz, an ultrasonic transducer of 25 kHz is a better choice to satisfy the measurement of the relaxation absorption spectrum of CO₂, 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.

The incidence ultrasonic wave is reflected on the receiving transducer/aluminum plate and interferes with its reflected counterpart, thus sound field is a superposition of sound absorption and interference. According to the classical definition of sound absorption coefficient in Equation (10), acoustic pressure need to be obtained along the direction of ultrasonic emission at various distances, where the transmitting transducer moves on a linear guide rail controlled by a step-motor from 30 mm (distance to the DBR fiber laser) to 100 mm, and records the output phase difference of the DBR fiber laser sensor every 1 mm.

Where α is the absorption coefficient, U1 and U2 are the output voltages of the receiving transducer obtained in the positions L1 and L2 respectively. The output voltage U is also defined as a product of sound pressure sensitivity kt and intensity p. Equation (10) clears that the absorption coefficient α is independent of the sensitivity of the sensor kt, for a specific ultrasonic frequency and only determined by the gas characteristics. Therefore, the absorption coefficient α keeps the same no matter either a transducer or a DBR fiber laser sensor is used for testing as long as the f/p keeps the same. The sensitivity determines the lowest detectable acoustic pressure or the noise equivalent sound pressure, so the measurement precision of acoustic attenuation coefficient improves with the sensitivity.

Testing results of the 25kHz sound field distribution recovered by the DBR fiber laser sensor is plotted in Figure 8a. Based on the linear relation of phase difference and acoustic pressure and Equation (10), a diagram of peak pressures level in Figure 8a to the distance and its linear fitting are shown in Figure 8b, and the slope of which is divided by 20 is equal to the absorption coefficient of the gas, where the phase difference variation limits to a linear zone because of the weak ultrasonic absorption.

From Figure 8a, the standing sound wave field produced by the superposition of the incident and reflected ultrasonic waves shows a tendency of oscillation decline. According to Equation (10) and the direct conversion relationship between sound pressure and sound pressure level, the slope of the fitting line in Figure 8b divided by 20 is acoustic absorption coefficient α, and equals to 0.01 dB/mm. The ultrasonic wavelength  is 11.3 mm, the acoustic relaxation absorption coefficient  is 0.113. Following the same process to get the acoustic relaxation absorption coefficients at various frequencies by tuning the gas chamber pressure, the acoustic relaxation absorption spectrum is recovered.

4. Discussion and conclusion

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