A High-Sensitivity Dual-Axis Accelerometer with Two FP Cavities Assembled on Single Optical Fiber

In this paper, a dual-axis Fabry–Pérot (FP) accelerometer assembled on single optical fiber is proposed. The sensor is equipped with a special beam-splitting prism to split the light into two perpendicular directions (the X- and Y-axes); the prism surface coated with semi-permeable film and the reflective sheet on the corresponding Be-Cu vibration-sensitive spring form two sets of FP cavities of different sizes. When the Be-Cu spring with a proof mass (PM) is subjected to the vibration signal, the cavity length of the corresponding FP cavity is changed and the interference signal returns to the collimator through the original path of the prism. After bandpass filtering and demodulation, the two cavity lengths are obtained, and the acceleration measurement in dual-axis directions is completed. The resonant frequency of the proposed dual-axis fiber optic accelerometer is around 280 Hz. The results of the spectral measurements show 3.93 μm/g (g = 9.8 m/s2: gravity constant) and 4.19 μm/g for the applied acceleration along the X- and Y-axes, respectively, and the cross-axis sensitivity is below 5.1%. Within the angle range of 180°, the maximum error of measured acceleration is less than 3.77%. The proposed fiber optic dual-axis FP accelerometer has high sensitivity and strong immunity to electromagnetic interference. The size of the sensor mainly depends on the size of the prism, which is easy to reduce and mass produce. Moreover, this FP construction method has high flexibility and development potential.


Introduction
With the popularity of the Internet of Everything and the increased demand for structural health monitoring, accelerometry is of great importance in the aerospace, automotive, and robotics industries, and has a wide range of applications in sensing and monitoring scenarios, such as bridges and railroads [1][2][3]. Accelerometers are generally divided into electrical accelerometers and optical accelerometers. Among them, fiber optic accelerometers have gained considerable attention and undergone rapid development due to their high sensitivity, corrosion resistance, anti-electromagnetic interference, and intrinsic safety [4]. Fiber optic accelerometers mainly contain two types: the wavelength type and the interferometric type. Fiber Bragg grating (FBG)-type accelerometers, as the classical type of wavelength type accelerometers, have been continuously improved since their first demonstration by Berkoff et al. [5][6][7]; however, their sensitivity is still lower when compared with interferometric accelerometers. Interferometric accelerometers exist in various types, such as Fabry-Pérot, Mach-Zehnder [8,9], Michelson [10,11], and Sagnac [12,13], among which fiber optic Fabry-Pérot sensors have the advantages of simple fabrication of FP cavity sensing head, good reliability, flexible fabrication, and many other advantages such as high sensitivity [14].
A typical fiber optic FP accelerometer structure is formed by connecting the spring and the PM together, and then pasting the reflective film on the PM to form an FP cavity with a fiber end face. When the external acceleration acts, the relative displacement between the PM and the fiber changes the length of the FP cavity due to the inertia effect, and the acceleration value of the measured object is obtained through the demodulation of the cavity length [15][16][17][18]. However, the number of FP cavities that can be constructed by a single fiber is extremely limited. In 2019, Xiaoying Liu et al. [19] proposed a compact fiber optic Fabry-Pérot (FP) sensor for the simultaneous measurement of acoustics and temperature, which consists of a silicon-glass-silica sandwich structure formed by anodic bonding, with a circular shape. The air FP cavity with circular holes and grooves utilizes the deformation of the silicon film for acoustic measurements, and the silicon FP cavity utilizes the temperature sensitivity of the silicon refractive index for temperature measurements. Using a multilayer reflective structure is a common method used to construct multiple FP cavities, so multiple parameters can be measured on the basis of a single fiber. However, multiple FP cavities are parallel to each other, which makes it difficult to expand to different directions in acceleration measurement. In 2021, Majid Taghavi et al. [20] used two optical fibers to construct separate FP cavities in the vertical direction to complete dual-axis acceleration measurements, which is the most common method of multi-dimensional acceleration measurements [21,22]. This is the most common method used to construct an FP cavity with an optical fiber end face alone in each axis direction, or directly combine multiple one-dimensional accelerometers into a whole to complete multi-dimensional measurement, but this method not only makes the sensor more vulnerable to the influence of fiber bending performance in multiple directions, but also makes the accelerometer require more fiber interfaces, which increase the complexity of the system. This paper proposes a single fiber optic dual-axis FP accelerometer. The light source of the sensor enters the beam-splitting prism through the collimator, and the output surface of the beam-splitting prism is coated with semi-transmissive and semi-reflective film, which forms an FP cavity with the reflector at the spring. This method of forming an FP cavity provides a good platform for the design and installation of springs. The results of this paper show that the accelerometer we proposed can successfully accomplish the measurement of dual-axis acceleration; the resonant frequency of the accelerometer is about 280 Hz, and has a flat frequency response in the range of 0-120 Hz. Moreover, the X-axis and Y-axis sensitivities of the accelerometer reach 3.93 µm/g and 4.19 µm/g, respectively, at a vibration frequency of 100 Hz, and the error of the measured acceleration value does not exceed 3.77% within the measurement range of 180 • . According to the results, this dual-axis accelerometer only needs one interface, has high sensitivity, and completes the two-dimensional direction measurement well. The structure of the sensor is simple and efficient, with the prism-splitting surface providing the role of vertical beam splitting. The output surface involved in the construction of the FP cavity while providing a platform for the construction of the spring. The FP cavity-forming method used in the accelerometer we proposed has great potential for mass production in the multi-dimensional sensing field. Figure 1a shows a schematic diagram of the structure of the dual-axis FP accelerometer (sensor housing not included). The sensor consists of a collimator, a special optical prism, fan-shaped shims, Be-Cu springs, and PMs. The collimator is coated with a transmissionenhancing film. The incident surface of the prism is coated with a transmission-enhancing film, the spectroscope of the prism can divide the light with a wavelength range of 1500 nm to 1600 nm equally into two vertical directions, and the output side is coated with a 30% reflection and 70% transmission film. The fan-shaped shims are fixed onto the output side of the prism to provide a platform for the installation of the spring, and the thickness of the fan-shaped shim is used to control the cavity length of the FP cavity, so that the two FP cavities in the vertical direction have unequal cavity lengths. In Figure 1b, the collimator output surface is coated with a transmission-enhancing film, marked as R 1 ; the two transmission-enhancing films found on the prism are marked as R 2 and R 8 ; the 50:50 splitting surface of the prism is marked as R 3 ; and the two 30% reflection and 70% the collimator output surface is coated with a transmission-enhancing film, marked as R1; the two transmission-enhancing films found on the prism are marked as R2 and R8; the 50:50 splitting surface of the prism is marked as R3; and the two 30% reflection and 70% transmission films of the prism are marked as R4 and R5. The centers of the springs in two vertical directions are equipped with reflective sheets, which are marked as R6 and R7. The prism's semi-permeable film R4 and the corresponding directional spring reflective sheet R6 form FP1 with a cavity length of L1, and the prism's semi-permeable film R5 and the corresponding directional spring reflective sheet R7 form FP2 with a cavity length of L2, where L1 is significantly smaller than L2. Since the prism is a cubic structure, the distances R1 to R4 and R5 are almost equal, which makes the optical path difference (OPD) of the interference signal between R4 and R5 close to 0, which can be ignored. There is Michelson interference between R6 and R7, and its arm difference is the difference between L1 and L2. The lights reflected from R4, R5, R6, and R7 interfere with each other and the total light intensity [20] returned to the collimator is Ι λ = I I I 2 I I cos ϕ 2 I I cos ϕ 2 I I cos ϕ ϕ

Structure and Principle
The intensity of I1, I2, and I3 refers to the light intensities of the three reflected lights, where I1 is the sum of the reflected light from R4 and R5, I2 is the reflected light from R6, and I3 is the reflected light from R7, all of which are only related to the wavelength of the incident light λ; thus, they can be considered as a constant, approximately. From Equation (1), it can be seen that the reflection spectrum of the sensor is mainly a linear superposition The prism's semi-permeable film R 4 and the corresponding directional spring reflective sheet R 6 form FP 1 with a cavity length of L 1 , and the prism's semi-permeable film R 5 and the corresponding directional spring reflective sheet R 7 form FP 2 with a cavity length of L 2 , where L 1 is significantly smaller than L 2 . Since the prism is a cubic structure, the distances R 1 to R 4 and R 5 are almost equal, which makes the optical path difference (OPD) of the interference signal between R 4 and R 5 close to 0, which can be ignored. There is Michelson interference between R 6 and R 7 , and its arm difference is the difference between L 1 and L 2 . The lights reflected from R4, R 5 , R 6 , and R 7 interfere with each other and the total light intensity [20] returned to the collimator is I(λ) = I 1 + I 2 + I 3 − 2 I 1 I 2 cos(φ 1 ) − 2 I 1 I 3 cos(φ 2 ) + 2 I 2 I 3 cos(φ 2 − φ 1 ) The intensity of I 1 , I 2 , and I 3 refers to the light intensities of the three reflected lights, where I 1 is the sum of the reflected light from R 4 and R 5 , I 2 is the reflected light from R 6 , and I 3 is the reflected light from R 7, all of which are only related to the wavelength of the incident light λ; thus, they can be considered as a constant, approximately. From Equation (1), it can be seen that the reflection spectrum of the sensor is mainly a linear superposition of three groups of cosine functions, where the interference spectra of FP 1 and FP 2 correspond to the fourth and fifth terms of Equation (1). φ 1 and φ 2 are the phase shifts of FP 1 and FP 2 , respectively, which can be expressed as where n air is the effective refractive index of air and is defaulted to 1. From Equation (2), it can be seen that φ 1 and φ 2 are positively correlated with cavity length, so in the process of making the sensor, the unequal size of the cavity lengths L 1 and L 2 can lead to unequal frequencies of the corresponding spectrum. The spectra of different frequencies correspond to different FP cavity lengths, so they can be separated by appropriate band-pass filtering to obtain the cavity lengths L 1 and L 2 for FP 1 and FP 2 , respectively. The cavity lengths of FP 1 and FP 2 are L 1 and L 2 when the sensor remains stationary. When excited by acceleration in the dual-axis plane, the springs with PMs produce axial movement which will cause the cavity length to change correspondingly; the cavity lengths of FP 1 and FP 2 change ∆L 1 (t) and ∆L 2 (t). Existing literature studies have shown that there is a linear relationship between acceleration and the change in cavity length of the FP cavity of optical fiber [23]. The acceleration value is linearly related to the cavity length of the fiber optic FP cavity as follows: where S 1 and S 2 are the sensitivities of the FP 1 and FP 2 spring, that is, the amount of change in cavity length after receiving an acceleration of 1 g. a 1 (t) and a 2 (t) correspond to the acceleration of the X-axis and Y-axis, respectively, and the relationship between a 1 (t) and a 2 (t) is shown in Figure 2. The following sections will take θ, which is the angle between the X-axis of the sensor and the ground, as the angle reference. of three groups of cosine functions, where the interference spectra of FP1 and FP2 correspond to the fourth and fifth terms of Equation (1). ϕ and ϕ are the phase shifts of FP1 and FP2, respectively, which can be expressed as where nair is the effective refractive index of air and is defaulted to 1. From Equation (2), it can be seen that ϕ and ϕ are positively correlated with cavity length, so in the process of making the sensor, the unequal size of the cavity lengths L1 and L2 can lead to unequal frequencies of the corresponding spectrum. The spectra of different frequencies correspond to different FP cavity lengths, so they can be separated by appropriate band-pass filtering to obtain the cavity lengths L1 and L2 for FP1 and FP2, respectively. The cavity lengths of FP1 and FP2 are L1 and L2 when the sensor remains stationary. When excited by acceleration in the dual-axis plane, the springs with PMs produce axial movement which will cause the cavity length to change correspondingly; the cavity lengths of FP1 and FP2 change ΔL t and ΔL t . Existing literature studies have shown that there is a linear relationship between acceleration and the change in cavity length of the FP cavity of optical fiber [23]. The acceleration value is linearly related to the cavity length of the fiber optic FP cavity as follows: where S and S are the sensitivities of the FP1 and FP2 spring, that is, the amount of change in cavity length after receiving an acceleration of 1 g. a1(t) and a2(t) correspond to the acceleration of the X-axis and Y-axis, respectively, and the relationship between a1(t) and a2(t) is shown in Figure 2. The following sections will take θ, which is the angle between the X-axis of the sensor and the ground, as the angle reference. The directions of a1(t) and a2(t) are perpendicular to each other, and thus the total acceleration values a(t) can be expressed as It is worth noting that in the later experiment, the direction of the total acceleration a(t) is perpendicular to the ground. Therefore, the following equation can be deduced: a t = a t sin θ , a t = a t cos θ The prism is a 10 mm × 10 mm × 10 mm sized cube, the incident surfaces R2 and R8 are coated with a transmission-enhancing film to avoid crosstalk on the interference signal, and the beam-splitting surface R3 uses 50:50 vertical beam splitting to divide the incident light equally into two vertical directions. In order to make the reflected light intensity of different reflective surfaces approximately equal, so that the interference spectrum has good contrast, the two output surfaces R4 and R5 are coated with a 30% reflection and 70% transmission film. The Be-Cu springs are square with a thickness of 0.05 mm and side The directions of a 1 (t) and a 2 (t) are perpendicular to each other, and thus the total acceleration values a(t) can be expressed as It is worth noting that in the later experiment, the direction of the total acceleration a(t) is perpendicular to the ground. Therefore, the following equation can be deduced: The prism is a 10 mm × 10 mm × 10 mm sized cube, the incident surfaces R 2 and R 8 are coated with a transmission-enhancing film to avoid crosstalk on the interference signal, and the beam-splitting surface R 3 uses 50:50 vertical beam splitting to divide the incident light equally into two vertical directions. In order to make the reflected light intensity of different reflective surfaces approximately equal, so that the interference spectrum has good contrast, the two output surfaces R 4 and R 5 are coated with a 30% reflection and 70% transmission film. The Be-Cu springs are square with a thickness of 0.05 mm and side length of 10 mm, which matches the size of the prism. The spring is designed symmetrically to increase its stability against different angles. A 0.1 g PM is glued to one side of the spring to increase its sensitivity, and a reflective sheet is glued to the other side of the spring to form the FP cavity with the prism's corresponding output surface; the two directional springs are made in the same way.
The sensor is fabricated as follows: First, using the sensor housing as a medium, the relative positions of the collimator and prism are fixed to ensure that the reflected light from the prisms R 4 and R 5 can return to the collimator. The four equally thick fan-shaped shims are glued to the four corners of the prism's output side, which not only help to control the length of the FP cavity, but also help to maintain the parallelism between the prism's output side and the reflective sheet of the spring, so that the desired FP and interference spectrum can be obtained. To ensure the interference spectra of the two FP cavities can be separated, it is important that the thickness of the shims in the two vertical directions is different, which means that L 1 , L 2 , and L 2 − L 1 are not equal. Specifically, the thickness of the shims is about 650 for FP 1 and 1750 for FP 2 . Finally, the completed springs are glued on the fan-shaped shims, and the internal structure of the sensor is shown in Figure 1c. After packaging, the appearance of the sensor is as shown in Figure 1d.

Experiment and Results
Figure 3a shows a diagram of the sensor test experiment system, and the structure in the red wireframe is shown in Figure 3b. The vibration experiment uses a Danish B&K vibration test platform with a frequency excitation range of 5-1000 kHz. The reference sensor uses a piezoelectric accelerometer from B&K with a resonant frequency of 16 kHz, a sensitivity of 10 pC/ms −2 , and a measurement range of 0.1-4800 Hz. The broadband light generated by the broadband light source (BBS) reaches the optical sensor through the circulator, and the optical sensor is fixed on the rotating platform, which has an adjustment range of 90 • . The frequency and amplitude of the excitation signal are controlled by the PC and power amplifier, respectively, so that the vibration exciter produces the expected vibration excitation. Meanwhile, the vibration signal is recorded by the electrical accelerometer on the vibration platform. The optical accelerometer, that is also excited, reflects the signal light back to the circulator and reaches the demodulator, which sends the spectral information of 1530 nm-1560 nm to the PC for storage and subsequent demodulation at an acquisition frequency of 2000 Hz. length of 10 mm, which matches the size of the prism. The spring is designed symme cally to increase its stability against different angles. A 0.1 g PM is glued to one side of spring to increase its sensitivity, and a reflective sheet is glued to the other side of spring to form the FP cavity with the prism's corresponding output surface; the two rectional springs are made in the same way. The sensor is fabricated as follows: First, using the sensor housing as a medium, relative positions of the collimator and prism are fixed to ensure that the reflected li from the prisms R4 and R5 can return to the collimator. The four equally thick fan-shap shims are glued to the four corners of the prism's output side, which not only help control the length of the FP cavity, but also help to maintain the parallelism between prism's output side and the reflective sheet of the spring, so that the desired FP and in ference spectrum can be obtained. To ensure the interference spectra of the two FP cavi can be separated, it is important that the thickness of the shims in the two vertical dir tions is different, which means that L1, L2, and L2 − L1 are not equal. Specifically, the th ness of the shims is about 650 for FP1 and 1750 for FP2. Finally, the completed springs glued on the fan-shaped shims, and the internal structure of the sensor is shown in Fig  1c. After packaging, the appearance of the sensor is as shown in Figure 1d.

Experiment and Results
Figure 3a shows a diagram of the sensor test experiment system, and the structur the red wireframe is shown in Figure 3b. The vibration experiment uses a Danish B vibration test platform with a frequency excitation range of 5-1000 kHz. The refere sensor uses a piezoelectric accelerometer from B&K with a resonant frequency of 16 k a sensitivity of 10 pC/ms −2 , and a measurement range of 0.1-4800 Hz. The broadband li generated by the broadband light source (BBS) reaches the optical sensor through the culator, and the optical sensor is fixed on the rotating platform, which has an adjustm range of 90°. The frequency and amplitude of the excitation signal are controlled by PC and power amplifier, respectively, so that the vibration exciter produces the expec vibration excitation. Meanwhile, the vibration signal is recorded by the electrical ac erometer on the vibration platform. The optical accelerometer, that is also excited, refle the signal light back to the circulator and reaches the demodulator, which sends the sp tral information of 1530 nm-1560 nm to the PC for storage and subsequent demodulat at an acquisition frequency of 2000 Hz.
(a)  The reflection spectra of the optical fiber FP sensor is shown in Figure 4a, which is the result of mixing multiple groups of interference spectra. Figure 4b shows the result of the FFT transformation of the reflection spectra, where there are three distinct frequency components corresponding to peaks 1, 2, and 3, respectively, and half of the OPDs corresponding to the three peaks are demodulated with values of 613, 1152, and 1765, respectively. Combined with the structure of the sensor, we can draw the conclusion that peak 1 corresponds to FP1 and its cavity length is L1, and peak 3 corresponds to FP2 and its cavity length is L2. The incident light is divided into two beams by the beam-splitter and reflected back by the corresponding reflective surfaces R6 and R7, forming a Michelson interference. According to Figure 1b, peak 2 corresponds to the Michelson interference phenomenon formed between R6 and R7, with the value of the arm length difference L3 equal to the difference between the cavity length L2 and L1. Figure 4c,d show the interference spectra corresponding to FP1 and FP2 alone after bandpass filtering. Due to the influence of the filter performance, the initial position amplitude of the spectrum is attenuated. However, this has little impact on the subsequent demodulation process, because the demodulation principle in this paper is based on the phase of the spectrum rather than the amplitude. The reflection spectra of the optical fiber FP sensor is shown in Figure 4a, which is the result of mixing multiple groups of interference spectra. Figure 4b shows the result of the FFT transformation of the reflection spectra, where there are three distinct frequency components corresponding to peaks 1, 2, and 3, respectively, and half of the OPDs corresponding to the three peaks are demodulated with values of 613, 1152, and 1765, respectively. Combined with the structure of the sensor, we can draw the conclusion that peak 1 corresponds to FP 1 and its cavity length is L 1 , and peak 3 corresponds to FP 2 and its cavity length is L 2 . The incident light is divided into two beams by the beam-splitter and reflected back by the corresponding reflective surfaces R6 and R7, forming a Michelson interference. According to Figure 1b, peak 2 corresponds to the Michelson interference phenomenon formed between R 6 and R 7 , with the value of the arm length difference L 3 equal to the difference between the cavity length L 2 and L 1 . Figure 4c,d show the interference spectra corresponding to FP 1 and FP 2 alone after bandpass filtering. Due to the influence of the filter performance, the initial position amplitude of the spectrum is attenuated. However, this has little impact on the subsequent demodulation process, because the demodulation principle in this paper is based on the phase of the spectrum rather than the amplitude. The reflection spectra of the optical fiber FP sensor is shown in Figure 4a, which is the result of mixing multiple groups of interference spectra. Figure 4b shows the result of the FFT transformation of the reflection spectra, where there are three distinct frequency components corresponding to peaks 1, 2, and 3, respectively, and half of the OPDs corresponding to the three peaks are demodulated with values of 613, 1152, and 1765, respectively. Combined with the structure of the sensor, we can draw the conclusion that peak 1 corresponds to FP1 and its cavity length is L1, and peak 3 corresponds to FP2 and its cavity length is L2. The incident light is divided into two beams by the beam-splitter and reflected back by the corresponding reflective surfaces R6 and R7, forming a Michelson interference. According to Figure 1b, peak 2 corresponds to the Michelson interference phenomenon formed between R6 and R7, with the value of the arm length difference L3 equal to the difference between the cavity length L2 and L1. Figure 4c,d show the interference spectra corresponding to FP1 and FP2 alone after bandpass filtering. Due to the influence of the filter performance, the initial position amplitude of the spectrum is attenuated. However, this has little impact on the subsequent demodulation process, because the demodulation principle in this paper is based on the phase of the spectrum rather than the amplitude.

Frequency Response
In order to determine the resonant frequency and operating frequency range of the fiber optic accelerometer, the response of the sensor at different vibration frequencies needs to be measured. Adjust θ to 90° by rotating the platform, apply the vibration signal to FP1, adjust the vibration amplitude of the exciter to keep it constant and scan the frequency from 20 Hz to 480 Hz in steps of 20 Hz, record the response of FP1, and ignore the response of FP2. Then, adjust θ to 180° and repeat the same operation to obtain the response of FP2.
The frequency response of the optical sensor is shown in Figure 5, and the experimental results show that the resonance frequencies of the optical sensor were obtained as about 280 Hz for both the X-and Y-axes. These values in the two directions are slightly different. This difference can be attributed to a slight weight difference between the PMs and a machining error of springs. Considering the results of frequency response in both directions, we can see that the sensor has a fairly flat frequency response when the frequency is lower than 100 Hz; thus, the flat region 0~100 Hz is taken as the operating frequency range.  (c) independent interference spectra of FP 1 after filter; (d) independent interference spectra of FP 2 after filter.

Frequency Response
In order to determine the resonant frequency and operating frequency range of the fiber optic accelerometer, the response of the sensor at different vibration frequencies needs to be measured. Adjust θ to 90 • by rotating the platform, apply the vibration signal to FP 1 , adjust the vibration amplitude of the exciter to keep it constant and scan the frequency from 20 Hz to 480 Hz in steps of 20 Hz, record the response of FP 1 , and ignore the response of FP 2 . Then, adjust θ to 180 • and repeat the same operation to obtain the response of FP 2 .
The frequency response of the optical sensor is shown in Figure 5, and the experimental results show that the resonance frequencies of the optical sensor were obtained as about 280 Hz for both the X-and Y-axes. These values in the two directions are slightly different. This difference can be attributed to a slight weight difference between the PMs and a machining error of springs. Considering the results of frequency response in both directions, we can see that the sensor has a fairly flat frequency response when the frequency is lower than 100 Hz; thus, the flat region 0~100 Hz is taken as the operating frequency range.  Figure 4. (a) Reflection spectra of the sensor; (b) spatial frequency spectra of the reflection light; (c) independent interference spectra of FP1 after filter; (d) independent interference spectra of FP2 after filter.

Frequency Response
In order to determine the resonant frequency and operating frequency range of the fiber optic accelerometer, the response of the sensor at different vibration frequencies needs to be measured. Adjust θ to 90° by rotating the platform, apply the vibration signal to FP1, adjust the vibration amplitude of the exciter to keep it constant and scan the frequency from 20 Hz to 480 Hz in steps of 20 Hz, record the response of FP1, and ignore the response of FP2. Then, adjust θ to 180° and repeat the same operation to obtain the response of FP2.
The frequency response of the optical sensor is shown in Figure 5, and the experimental results show that the resonance frequencies of the optical sensor were obtained as about 280 Hz for both the X-and Y-axes. These values in the two directions are slightly different. This difference can be attributed to a slight weight difference between the PMs and a machining error of springs. Considering the results of frequency response in both directions, we can see that the sensor has a fairly flat frequency response when the frequency is lower than 100 Hz; thus, the flat region 0~100 Hz is taken as the operating frequency range.

Sensitivity Calibration
To determine the correspondence between the cavity length change and the acceleration value in two directions of the dual-axis FP accelerometer, the sensitivity of the springs on both axes was calibrated separately by adjusting the angle of the sensor. In the same way as described above, adjust θ to 90 • to calibrate FP 1 and adjust θ to 180 • to calibrate FP 2 . The PC controls the vibration exciter, applies a sinusoidal vibration signal of a specified frequency to the sensor, adjusts different vibration amplitudes through the power amplifier knob, records and demodulates the FP cavity length change in the corresponding direction, compares the amplitude with the results recorded by the electrical accelerometer, and then obtains the single-sided sensitivity of the optical fiber accelerometer at the corresponding frequency. Figure 6 shows the linear response of the sensor to the X-axis and Y-axis at 20 Hz and 100 Hz. It can be seen that the linearity of the sensor's response to the vibration signal is very good, and is higher than 99.9%. The corresponding results of amplitude-acceleration at 100 Hz are shown in Figure 6b, and the sensitivity of FP 1 and FP 2 are 3.93 µm/g and 4.19 µm/g, respectively. Compared to the value 3.22 nm/g reported in [20], the sensor shows a much higher sensitivity. The sensitivity of the two axes is slightly different, which may be due to the error in the spring processing.

Sensitivity Calibration
To determine the correspondence between the cavity length change and the acceleration value in two directions of the dual-axis FP accelerometer, the sensitivity of the springs on both axes was calibrated separately by adjusting the angle of the sensor. In the same way as described above, adjust θ to 90° to calibrate FP1 and adjust θ to 180° to calibrate FP2. The PC controls the vibration exciter, applies a sinusoidal vibration signal of a specified frequency to the sensor, adjusts different vibration amplitudes through the power amplifier knob, records and demodulates the FP cavity length change in the corresponding direction, compares the amplitude with the results recorded by the electrical accelerometer, and then obtains the single-sided sensitivity of the optical fiber accelerometer at the corresponding frequency. Figure 6 shows the linear response of the sensor to the X-axis and Y-axis at 20 Hz and 100 Hz. It can be seen that the linearity of the sensor's response to the vibration signal is very good, and is higher than 99.9%. The corresponding results of amplitude-acceleration at 100 Hz are shown in Figure 6b, and the sensitivity of FP1 and FP2 are 3.93 μm/g and 4.19 μm/g, respectively. Compared to the value 3.22 nm/g reported in [20], the sensor shows a much higher sensitivity. The sensitivity of the two axes is slightly different, which may be due to the error in the spring processing.

Cross-Axis Sensitivity
For accelerometers, especially multi-dimensional accelerometers, it is necessary to consider not only the response of the sensor in the sensitive direction, but also the effect of the vibration component in the non-sensitive direction on the measurement in the sensitive direction. Adjust θ to 90° and 180°, respectively, and record and demodulate the sizes of the optical sensor's FP cavities L1 and L2 in both directions. The cross-sensitivity of the sensor can be obtained by comparing the peak amplitude of two FP cavity length changes. Figure 6 shows the response of the sensor to the vibration signal in the sensitive direction and the non-sensitive direction at 100 Hz vibration frequency. Figure 7a shows the change in cavity length of FP1 and FP2 when the X-axis of the sensor is vibrated. Average the wave crest and wave trough, respectively, calculate the difference between them to obtain the maximum displacement of FP cavity, and compare the maximum displacement of FP1 and FP2 to obtain the cross-sensitivity. The results show that the cross-axis sensitivity of the sensor is 4.9%. Figure 7b shows the change in cavity length of FP1 and FP2 when the Y-axis of the sensor is vibrated, and the results show that the cross-axis sensitivity of the sensor is 5.1%.
Considering the structure of the spring, the main reason why the cross-sensitivity of the sensor is slightly high is that when the PM and the spring are bonded as a whole, their

Cross-Axis Sensitivity
For accelerometers, especially multi-dimensional accelerometers, it is necessary to consider not only the response of the sensor in the sensitive direction, but also the effect of the vibration component in the non-sensitive direction on the measurement in the sensitive direction. Adjust θ to 90 • and 180 • , respectively, and record and demodulate the sizes of the optical sensor's FP cavities L 1 and L 2 in both directions. The cross-sensitivity of the sensor can be obtained by comparing the peak amplitude of two FP cavity length changes. Figure 6 shows the response of the sensor to the vibration signal in the sensitive direction and the non-sensitive direction at 100 Hz vibration frequency. Figure 7a shows the change in cavity length of FP 1 and FP 2 when the X-axis of the sensor is vibrated. Average the wave crest and wave trough, respectively, calculate the difference between them to obtain the maximum displacement of FP cavity, and compare the maximum displacement of FP 1 and FP 2 to obtain the cross-sensitivity. The results show that the cross-axis sensitivity of the sensor is 4.9%. Figure 7b shows the change in cavity length of FP 1 and FP 2 when the Y-axis of the sensor is vibrated, and the results show that the cross-axis sensitivity of the sensor is 5.1%.
Considering the structure of the spring, the main reason why the cross-sensitivity of the sensor is slightly high is that when the PM and the spring are bonded as a whole, their center of gravity is on the PM instead of the spring. This structural feature makes it difficult for the spring to keep stable when it is subjected to vibration in the cross-direction, thus producing unexpected shaking, which leads to the increase in cross-sensitivity. center of gravity is on the PM instead of the spring. This structural feature makes it difficult for the spring to keep stable when it is subjected to vibration in the cross-direction, thus producing unexpected shaking, which leads to the increase in cross-sensitivity.

Multi-Angle Measurement
When adjusting the PC and power amplifier, keep the vibration frequency at 100 Hz, refer to the value of the electrical accelerometer, and keep the maximum span of the sine wave crest and wave trough at 0.697 g. Take the angle between the X-axis of the sensor and the ground as a reference, rotate the optical accelerometer from 45° to 225° in sequence by adjusting the rotating platform in steps of 5°, and record spectral signals in turn. As the rotation range of the rotating platform is 90°, the sensor needs to be re-fixed when rotating from 45° to 135° to test the 135° to 225° range.
It can be seen from Figure 6c,d that the sensitivities of FP1 and FP2 are known to be 3.93 μm/g and 4.19 μm/g, respectively. Based on the variation in the cavity lengths of the two FP cavities and their respective sensitivities, the accelerations of a1(t) and a2(t) at different angles are calculated and the total acceleration a(t) is finally obtained based on Equation (4). The span of the acceleration amplitude is calculated according to the difference between the wave crest and wave trough of the measured acceleration. The acceleration of the X-axis, Y-axis and the total acceleration at different measuring angles are shown in Figure 8. It should be noted that the angle range from 45° to 225° are measured data, and the other half are mirror data for better observation. At 90°, the acceleration measured by FP1 is close to the maximum value, and the acceleration measured by FP2 is close to 0, which is reversed at 180°. The acceleration amplitudes measured by FP1 and FP2 are approximately equal at around 45°, 135°, and 225°. The experimental results are consistent with equation (5) and show that the dual-axis FP accelerometer responds effectively in the angle range of 45° to 225°, and the maximum error of the measured acceleration value does not exceed 3.77% when comparing its measured acceleration with that of the electrical accelerometer. When the angle is between 130° and 135°, FP1 and FP2 showed a poor continuity of response due to the disassembly and re-fixing of the sensor, as they were rotated to that angle.

Multi-Angle Measurement
When adjusting the PC and power amplifier, keep the vibration frequency at 100 Hz, refer to the value of the electrical accelerometer, and keep the maximum span of the sine wave crest and wave trough at 0.697 g. Take the angle between the X-axis of the sensor and the ground as a reference, rotate the optical accelerometer from 45 • to 225 • in sequence by adjusting the rotating platform in steps of 5 • , and record spectral signals in turn. As the rotation range of the rotating platform is 90 • , the sensor needs to be re-fixed when rotating from 45 • to 135 • to test the 135 • to 225 • range.
It can be seen from Figure 6c,d that the sensitivities of FP 1 and FP 2 are known to be 3.93 µm/g and 4.19 µm/g, respectively. Based on the variation in the cavity lengths of the two FP cavities and their respective sensitivities, the accelerations of a 1 (t) and a 2 (t) at different angles are calculated and the total acceleration a(t) is finally obtained based on Equation (4). The span of the acceleration amplitude is calculated according to the difference between the wave crest and wave trough of the measured acceleration. The acceleration of the X-axis, Y-axis and the total acceleration at different measuring angles are shown in Figure 8. It should be noted that the angle range from 45 • to 225 • are measured data, and the other half are mirror data for better observation. At 90 • , the acceleration measured by FP 1 is close to the maximum value, and the acceleration measured by FP 2 is close to 0, which is reversed at 180 • . The acceleration amplitudes measured by FP 1 and FP 2 are approximately equal at around 45 • , 135 • , and 225 • . The experimental results are consistent with Equation (5) and show that the dual-axis FP accelerometer responds effectively in the angle range of 45 • to 225 • , and the maximum error of the measured acceleration value does not exceed 3.77% when comparing its measured acceleration with that of the electrical accelerometer. When the angle is between 130 • and 135 • , FP 1 and FP 2 showed a poor continuity of response due to the disassembly and re-fixing of the sensor, as they were rotated to that angle.