Numerical and Experiment Analysis of Sapphire Sandwich-Structure Fabry–Perot Pressure Sensor through Fast Fourier Transform and Mean Square Error Demodulation Algorithm

Pressure sensors prepared from sapphire exhibit excellent characteristics, including high-temperature resistance, high hardness, and resistance to electromagnetic interference. A Fast Fourier Transform and Mean Square Error (FFT-MSE) demodulation algorithm was employed to demodulate a sapphire sandwich-structure Fabry–Perot (F-P) pressure sensor. Through simulation analysis, the experimental results indicated that the demodulation error of the air cavity length in the range of 206 μm to 216 μm was less than 0.0008%. Compared to single demodulation methods and combined demodulation methods based on FFT or Minimum Mean Square Error (MMSE), the method proposed in this work reduced the demodulation error by more than three times and increased accuracy by more than six times. The algorithm was utilized to demodulate the sapphire sandwich-structure F-P pressure sensor, and the test results indicated that the fitting error of the sensor was less than 0.025% within the pressure range of 0 MPa to 10 MPa. The repeatability error was less than 0.066%, the zero-point deviation was 1.26%, and the maximum stability deviation was 0.0063% per 30 min. The algorithm effectively demodulated the actual cavity length variation in the sapphire sandwich-structure F-P pressure sensor, providing a solution for the performance evaluation of the sapphire sandwich-structure F-P pressure sensor.


Introduction
The measurement of pressure values in the operating states of key components in high-hypervelocity vehicles and advanced aerospace engines is essential for assessing their operating states, structural optimization, and safety performance [1].For example, during aero-engine testing, pressure measurements are required for intakes, combustion chambers, tail nozzles, and exhaust outlets [2][3][4].The melting point of sapphire material is as high as 2040 • C [5], and the pressure sensors made from it possess superior properties such as high-temperature resistance, high hardness, anti-electromagnetic interference, and high-temperature stability [6][7][8].It is reported that pressure sensors based on sapphire Materials 2024, 17, 3649 2 of 10 material are mainly F-P pressure sensors [9][10][11][12][13][14]. Sapphire Fabry-Perot (F-P) pressure sensors have high accuracy, high stability, and high-temperature environment adaptability, making them more suitable for pressure measurement in extreme aerospace environments.The main demodulation algorithms for sapphire F-P pressure sensors include bimodal methods [15][16][17], Fourier transform methods [18][19][20], phase quadrature methods [21][22][23], and mutual correlation methods [24][25][26].Zhiqiang Shao et al. achieved the demodulation of the sapphire F-P cavity length ranging from 103 µm to 283 µm by using the mutual correlation function method with a pressure range of 0 MPa to 5 MPa [27].Zilong Guo et al. achieved the demodulation of the sapphire F-P cavity with a length range of 60 µm to 95 µm using the white light interference method with a pressure range of 0 MPa to 2.9 MPa and accuracy of up to 0.04% [28].Yang Cui et al. achieved the demodulation of the quartz F-P cavity with a length range of 614 µm to 630 µm using the white light interference method with a pressure range of 0 MPa to 10 MPa [29].
This work combines the Fast Fourier Transform (FFT) and Mean Square Error (MSE) demodulation methods, interrelating the computational results to achieve air cavity length demodulation in a sapphire sandwich-structure F-P cavity.The objective was to optimize the low accuracy of FFT and the instability of MSE.Simulation results indicated that the maximum deviation is less than 0.1 nm, and demodulation accuracy reaches 0.0008% using the FFT-MSE demodulation method.During actual testing, the sapphire sandwichstructure pressure sensor undergoes multiple large-range pressure tests from 0-10 MPa.The fitting error of this sensor is less than 0.016%, the repeatability error is less than 0.066%, and no mode-jumping problem arises, demonstrating that the method has good reliability and stability.This method provides a reliable solution for determining the air cavity length of sapphire sandwich-structure F-P cavity.

Test System
Figure 1 illustrates the sapphire sandwich-structure F-P pressure sensor test system, which comprises a optical spectrum analyzer (DAQ6370, Yokogawa Measurement Technology Co., Ltd., Tokyo, Japan), a super-continuum spectral light source (SuperK COMPACT, NKT Photonics A/S, Copenhagen, Denmark), a pressure controller (860-25M, Beijing ConST Instruments Technology Inc., Beijing, China), an optical isolator, an optical coupler, a computer, a pressure connection table, a gas cylinder, and a pressure reducing valve.The super-continuum spectral light source emits laser light with a wavelength of 400 nm to 2400 nm. Figure S1 illustrates the coupling efficiency between the signal emitted by the super-continuum spectral light source and the single-mode fiber, which ranges from 17.71% to 23.8%, within the wavelength range of 1520 nm to 1570 nm.The laser light passes through the optical isolator and optical coupler before entering the sapphire sandwich-structure F-P pressure sensor.Its reflection spectrum then passes through the optical coupler, and the optical spectrum analyzer collects the reflection spectrum signals (sampling wavelength range: 1520 nm to 1570 nm, number of sampling points: 1001, scanning speed: 1×).The collected reflectance spectral signals are transferred to a computer with demodulation system software written in MATLAB (version number: 9.6.0.1047502), which demodulates the changes in the air cavity length of the sapphire sandwich-structure F-P cavity.The pressure in the gas cylinder is approximately 15 MPa, reduced to about 12 MPa by a pressure reducing valve.The pressure controller connects the gas cylinder to the pressure connection table, regulating the pressure within it.The sapphire sandwich-structure F-P pressure sensor is mounted onto the pressure connection table.

Sensor Structure
Figure 2 illustrates the sensitive core of the sapphire sandwich-structure F-P ca which comprises a basal cavity, an air cavity, and a pressure-sensitive diaphragm constructed from sapphire.The optical fiber consists of quartz.The super-contin spectral light source emits a beam that reflects off the upper surface of the basal c (reflectivity R1), the lower surface of the basal cavity (reflectivity R2), and the upper su of the pressure-sensitive diaphragm (reflectivity R3), resulting in an interferom reflectance spectrum.The sapphire basal cavity diaphragm thickness (d1) is 600 µm, a diameter of 8 mm.The sapphire pressure-sensing diaphragm thickness (d3) is 200 with a diameter of 8 mm.The sapphire air cavity diaphragm thickness (d2) is 216 with an outer diameter of 8 mm and an inner diameter of 4 mm.The quartz optical is a single-mode fiber with a diameter of 125 µm.Figure 3a shows the upper structure of the sensitive core of the sapphire sandw structure F-P pressure sensor.The diameter of the sensitive core is approximately 8 Figure 3b illustrates the physical diagram of the sapphire sandwich-structure F-P pre sensor.The sensor is encased in 304 stainless steel and features a thread diameter mm with a pitch of 1.5 mm.The scanning electron microscope (SEM) manufacture Carl Zeiss AG, model SUPRA55 SAPPHIRE, combined with energy-dispersive X spectroscopy (EDS), is used to analyze the micro-structure and element distribution o sensitive core of the sapphire sandwich-structure F-P pressure sensor.The micro-stru in A-A cross section of the sapphire sandwich-structure F-P cavity appears in Figu

Sensor Structure
Figure 2 illustrates the sensitive core of the sapphire sandwich-structure F-P cavity, which comprises a basal cavity, an air cavity, and a pressure-sensitive diaphragm, all constructed from sapphire.The optical fiber consists of quartz.The super-continuum spectral light source emits a beam that reflects off the upper surface of the basal cavity (reflectivity R 1 ), the lower surface of the basal cavity (reflectivity R 2 ), and the upper surface of the pressure-sensitive diaphragm (reflectivity R 3 ), resulting in an interferometric reflectance spectrum.The sapphire basal cavity diaphragm thickness (d1) is 600 µm, with a diameter of 8 mm.The sapphire pressure-sensing diaphragm thickness (d3) is 200 µm, with a diameter of 8 mm.The sapphire air cavity diaphragm thickness (d2) is 216 µm, with an outer diameter of 8 mm and an inner diameter of 4 mm.The quartz optical fiber is a single-mode fiber with a diameter of 125 µm.

Sensor Structure
Figure 2 illustrates the sensitive core of the sapphire sandwich-structure F-P ca which comprises a basal cavity, an air cavity, and a pressure-sensitive diaphragm constructed from sapphire.The optical fiber consists of quartz.The super-contin spectral light source emits a beam that reflects off the upper surface of the basal c (reflectivity R1), the lower surface of the basal cavity (reflectivity R2), and the upper su of the pressure-sensitive diaphragm (reflectivity R3), resulting in an interferom reflectance spectrum.The sapphire basal cavity diaphragm thickness (d1) is 600 µm, a diameter of 8 mm.The sapphire pressure-sensing diaphragm thickness (d3) is 200 with a diameter of 8 mm.The sapphire air cavity diaphragm thickness (d2) is 216 with an outer diameter of 8 mm and an inner diameter of 4 mm.The quartz optical is a single-mode fiber with a diameter of 125 µm.Figure 3a shows the upper structure of the sensitive core of the sapphire sandw structure F-P pressure sensor.The diameter of the sensitive core is approximately 8 Figure 3b illustrates the physical diagram of the sapphire sandwich-structure F-P pres sensor.The sensor is encased in 304 stainless steel and features a thread diameter mm with a pitch of 1.5 mm.The scanning electron microscope (SEM) manufacture Carl Zeiss AG, model SUPRA55 SAPPHIRE, combined with energy-dispersive X spectroscopy (EDS), is used to analyze the micro-structure and element distribution o sensitive core of the sapphire sandwich-structure F-P pressure sensor.The micro-stru in A-A cross section of the sapphire sandwich-structure F-P cavity appears in Figur Figure 3a shows the upper structure of the sensitive core of the sapphire sandwichstructure F-P pressure sensor.The diameter of the sensitive core is approximately 8 mm. Figure 3b illustrates the physical diagram of the sapphire sandwich-structure F-P pressure sensor.The sensor is encased in 304 stainless steel and features a thread diameter of 20 mm with a pitch of 1.5 mm.The scanning electron microscope (SEM) manufactured by Carl Zeiss AG, model SUPRA55 SAPPHIRE, combined with energy-dispersive X-ray spectroscopy (EDS), is used to analyze the micro-structure and element distribution of the sensitive core of the sapphire sandwich-structure F-P pressure sensor.The micro-structure in A-A cross section of the sapphire sandwich-structure F-P cavity appears in Figure 3c,

Simulation and Analysis
To effectively simulate the reflectance spectrum of the sapphire sandwich-stru F-P cavity, it is necessary to construct the reflectance spectrum using basal c reflectance spectrum expression, air cavity reflectance spectrum expression, interferometric reflectance spectrum expression for both the basal and air cav [28,30,31].The basal cavity reflectance spectrum expression is as follows: n1 denotes the refractive index of the basal cavity, L1 represents the basal cavity le I0 indicates the incident spectrum, I1 denotes the basal cavity reflectance spectru represents the speed of light, and v indicates the incident light frequency.The air ca reflectance spectrum expression is as follows: n2 denotes the refractive index of the air cavity, L2 represents the air cavity len and I2 indicates the air cavity reflectance spectrum.The interferometric reflect spectrum expression for the basal and air cavities is as follows: The reflectance spectrum expression for the sapphire sandwich-structure F-P c is as follows: The flowchart of the FFT-MSE demodulation algorithm is shown in Figure 4. reflectance spectrum of the sapphire sandwich-structure F-P cavity (Figure S2) is plo using Equation ( 4), with a wavelength range from 1520 nm to 1570 nm.To simulat reflectance spectrum of the sapphire sandwich-structure F-P cavity more efficie normal distribution noise is introduced.The reflectance spectrum with added no distribution noise is shown in Figure 5a.The signal-to-noise ratio (SNR) distributio

Simulation and Analysis
To effectively simulate the reflectance spectrum of the sapphire sandwich-structure F-P cavity, it is necessary to construct the reflectance spectrum using basal cavity reflectance spectrum expression, air cavity reflectance spectrum expression, and interferometric reflectance spectrum expression for both the basal and air cavities [28,30,31].The basal cavity reflectance spectrum expression is as follows: n 1 denotes the refractive index of the basal cavity, L 1 represents the basal cavity length, I 0 indicates the incident spectrum, I 1 denotes the basal cavity reflectance spectrum, c represents the speed of light, and v indicates the incident light frequency.The air cavity reflectance spectrum expression is as follows: n 2 denotes the refractive index of the air cavity, L 2 represents the air cavity length, and I 2 indicates the air cavity reflectance spectrum.The interferometric reflectance spectrum expression for the basal and air cavities is as follows: The reflectance spectrum expression for the sapphire sandwich-structure F-P cavity is as follows: The flowchart of the FFT-MSE demodulation algorithm is shown in Figure 4.The reflectance spectrum of the sapphire sandwich-structure F-P cavity (Figure S2) is plotted using Equation ( 4), with a wavelength range from 1520 nm to 1570 nm.To simulate the reflectance spectrum of the sapphire sandwich-structure F-P cavity more efficiently, normal distribution noise is introduced.The reflectance spectrum with added normal distribution noise is shown in Figure 5a.The signal-to-noise ratio (SNR) distribution in the wavelength range from 1520 nm to 1570 nm is shown in Figure S3, with the SNR greater than 35.1.The sapphire sandwich-structure F-P cavity reflectance spectrum is subjected to cubic spline interpolation.The frequency domain spectrum of the sapphire sandwich-structure F-P cavity is obtained by using the FFT (shown in Figure 5b).The frequency corresponding to the first peak is the air cavity frequency, the second peak corresponds to the basal cavity frequency, and the third peak results from the combined effect of the air cavity and basal cavity.The frequency domain spectrum of the sapphire sandwich-structure F-P cavity is low-pass filtered with a cut-off frequency of 20 Hz.The air cavity reflection spectrum is obtained by Fourier inverse transform and is then normalized as shown in Figure 5c.The normalization calculation formula is as follows: Materials 2024, 17, x FOR PEER REVIEW 6  I 4 denotes the normalized air cavity reflectance spectrum, S down1 represents the lower envelope of the air cavity reflectance spectrum, and S up1 indicates the upper envelope of the air cavity reflectance spectrum.The rough air cavity length is then calculated using the bimodal method, which employs the wavelengths (λ 1 and λ 2 ) corresponding to the two adjacent wave peaks of the air cavity spectrum signal to determine the rough air cavity length (L 4 ).The calculation formula is as follows [32]: The calculated L 4 is used to construct the reference air cavity length (L 5 ), which ranges from L 4 − ∆L to L 4 + ∆L, with a step size (∆d) of 10 nm, and ∆L is 10,000 nm.L 5 is then substituted into Equation ( 2) to construct the reference air cavity reflectance spectrum.The normalized reference air cavity reflectance spectrum (I 5 ) calculated using Equation (5).I 4 and I 5 are substituted into the MSE calculation formula; the result is shown in Figure S4.The MSE calculation formula is as follows: where N denotes the number of sampling points of the sapphire sandwich-structure F-P cavity reflectance spectrum after three iterations of spline interpolation, and n represents the sample points of L 5 .The minimum value of MSE corresponds to the reference air cavity length, identified as the first air cavity length.As shown in Figure 5d, after calculating the second MSE, the next step is to determine which of the reference air cavity lengths corresponding to " 1 ⃝", " 2 ⃝", and " 3 ⃝" is closest to the first air cavity length and then use this value as the air cavity length.Each subsequent air cavity length calculation references the immediately preceding air cavity length.The results of the simulation calculations for the variation in air cavity length from 206 µm to 216 µm are shown in Figure 5e.The simulation error curve is shown in Figure 5f, with the simulation error being less than 0.0008%.As shown in Table 1, compared with the dual peak method [15], dual/mult wavelength method [33,34], phase demodulation method [26,28,35], MMSE method [36-38], and FFT method [18,19,39], the method proposed in this work achieves a demodulation error that is more than three times smaller and accuracy that is more than six times higher.The simulation parameters not explicitly stated above are listed in Table 2.

Experiments
To effectively evaluate the feasibility of the demodulation algorithm, it is necessary to test the sapphire sandwich-structure F-P pressure sensor.As shown in Figure 6a, the sapphire sandwich-structure F-P pressure sensor is pressurized from 0 MPa to 10 MPa in a single round, with a boost rate of the pressure controller at 0.003 MPa/s.Secondorder fitting equations are used for modeling, resulting in an R 2 value of 0.99998.The fitting error curve is shown in Figure 6b, and the fitting error (e f ) is calculated using the following formula: Y denotes the air cavity length curve, Y f represents the fitting curve of the air cavity length, and Y F.S. indicates the variation range of the air cavity length.According to Equation ( 8), the absolute value of e f is calculated to be less than 0.016%.To evaluate the repeatability of the sapphire sandwich-structure F-P pressure sensor, two rounds of 0 MPa to 10 MPa pressurization are performed, with the pressure controller's boost rate at 0.003 MPa/s.The test results are shown in Figure 6c, and the repeatability error (δ R ) is calculated as follows: ∆Y denotes the difference between the change in curves of the air cavity lengths of the two rounds, and Y F.S. represents the variation range of the air cavity length.According to Equation ( 9), the calculated δ R is less than 0.066%, indicating that the sapphire sandwichstructure F-P pressure sensor has excellent repeatability.The stability of the sensor is a key indicator for evaluating its performance.Stability tests are conducted on the sapphire sandwich-structure F-P pressure sensor for 1 MPa to 10 MPa.The test results, shown in Figure 6d,e, indicate stability values of 0.0063%/30 min at 1 MPa, 0.0047%/30 min at 2 MPa, 0.0047%/30 min at 3 Mpa, 0.0050%/30 min at 4 MPa, 0.0053%/30 min at 5 MPa, 0.0039%/30 min at 6 MPa, 0.0052%/30 min at 7 MPa, 0.0048%/30 min at 8 MPa, 0.0051%/30 min at 9 MPa, and 0.0057%/30 min at 10 MPa.Notably, no "mode jumping" occurs during the 8 h continuous demodulation process.Figure 6f shows the 0 MPa to 10 MPa boost and buck curves, and the deviation (δ D ) of the zero point is calculated as follows: occurs during the 8 h continuous demodulation process.Figure 6f shows the 0 MPa to 10 MPa boost and buck curves, and the deviation (δ ) of the zero point is calculated as follows: ∆Y丨 the deviation value of the zero point, and YF.S. indicates the variation range of the air cavity length.The pressure controller has a ramp-up and ramp-down rate of 0.003 MPa/s.After one round of ramp-up and ramp-down, the air cavity length offset of the sapphire sandwich-structure F-P pressure sensor at atmospheric pressure is 125.87 nm, and according to Equation (10), the δ is calculated to be 1.26%.The performance of the sapphire sandwich-structure F-P pressure sensor is accurately evaluated using the FFT-MSE algorithm, which provides a generalized method for calculating the cavity length of sandwich-structure F-P cavity.

Conclusions
In conclusion, through the simulation analysis of the sapphire sandwich-structure F-P cavity, experimental results show that the demodulation error of the air cavity length in the range of 206 µm to 216 µm is less than 0.0008%, indicating that the algorithm has excellent accuracy.Compared to single demodulation methods and combined demodulation methods based on FFT or MMSE, the method proposed in this work reduces demodulation errors by more than three times and increases accuracy by more than six times.Testing the application of the sapphire sandwich-structure F-P pressure ∆Y| P=0 the deviation value of the zero point, and Y F.S. indicates the variation range of the air cavity length.The pressure controller has a ramp-up and ramp-down rate of 0.003 MPa/s.After one round of ramp-up and ramp-down, the air cavity length offset of the sapphire sandwich-structure F-P pressure sensor at atmospheric pressure is 125.87 nm, and according to Equation (10), the δ D is calculated to be 1.26%.The performance of the sapphire sandwich-structure F-P pressure sensor is accurately evaluated using the FFT-MSE algorithm, which provides a generalized method for calculating the cavity length of sandwich-structure F-P cavity.

Conclusions
In conclusion, through the simulation analysis of the sapphire sandwich-structure F-P cavity, experimental results show that the demodulation error of the air cavity length in the range of 206 µm to 216 µm is less than 0.0008%, indicating that the algorithm has excellent accuracy.Compared to single demodulation methods and combined demodulation methods based on FFT or MMSE, the method proposed in this work reduces demodulation errors by more than three times and increases accuracy by more than six times.Testing the application of the sapphire sandwich-structure F-P pressure sensor using the algorithm reveals a fitting error of less than 0.025%, a repeatability error of less than 0.066%, a deviation of the zero point of 1.26%, and a maximum stability deviation of 0.0063%/30 min in the pressure range of 0 MPa to 10 MPa.The test results show that the algorithm effectively demodulates the actual cavity length variation in the sapphire sandwich-structure F-P pressure sensor, providing a robust solution for its performance evaluation.The performance of the sapphire sandwich-structure F-P pressure sensor is accurately evaluated using the FFT-MSE algorithm.

Figure 1 .
Figure 1.Diagram of the sapphire sandwich-structure F-P pressure sensor test system.

Figure 2 .
Figure 2. The sensitive core of the sapphire sandwich-structure F-P cavity.

Figure 1 .
Figure 1.Diagram of the sapphire sandwich-structure F-P pressure sensor test system.

Figure 1 .
Figure 1.Diagram of the sapphire sandwich-structure F-P pressure sensor test system.

Figure 2 .
Figure 2. The sensitive core of the sapphire sandwich-structure F-P cavity.

Figure 2 .
Figure 2. The sensitive core of the sapphire sandwich-structure F-P cavity.
and the corresponding surface elements are Al and O.This is due to the main material composition of sapphire being Al 2 O 3 .From top to bottom, the layers are the sapphire pressure-sensitive diaphragm ( 1 ⃝), sapphire air cavity diaphragm ( 2 ⃝), and sapphire basal cavity diaphragm( 3  ⃝).Materials 2024, 17, x FOR PEERREVIEW  4    and the corresponding surface elements are Al and O.This is due to the main mat composition of sapphire being Al2O3.From top to bottom, the layers are the sapp pressure-sensitive diaphragm (①), sapphire air cavity diaphragm (②), and sapp basal cavity diaphragm (③).

Figure 3 .
Figure 3. (a) The upper structure of the sensitive core of the sapphire sandwich-structur pressure sensor.(b) Physical diagram of sapphire sandwich-structure F-P pressure sensor.(c micro-structure in the A-A cross section of the sensitive core of the sapphire sandwich-structu P pressure sensor and its corresponding surface element distribution (Al, O).

Figure 3 .
Figure 3. (a) The upper structure of the sensitive core of the sapphire sandwich-structure F-P pressure sensor.(b) Physical diagram of sapphire sandwich-structure F-P pressure sensor.(c) The microstructure in the A-A cross section of the sensitive core of the sapphire sandwich-structure F-P pressure sensor and its corresponding surface element distribution (Al, O).

Figure 5 .
Figure 5. (a)Sapphire sandwich-structure F-P cavity reflectance spectrum after adding noise.(b) Sapphire sandwich-structure F-P cavity frequency domain spectrum.(c) Air cavity spectrum and normalized air cavity spectrum.(d) Result of MSE.(e) Result of demodulated air cavity length and theoretical air cavity length.(f) Simulation error curve.

Figure 5 .
Figure 5. (a) Sapphire sandwich-structure F-P cavity reflectance spectrum after adding noise.(b) Sapphire sandwich-structure F-P cavity frequency domain spectrum.(c) Air cavity spectrum and normalized air cavity spectrum.(d) Result of MSE.(e) Result of demodulated air cavity length and theoretical air cavity length.(f) Simulation error curve.

Table 1 .
Comparison of the simulation performance of different algorithms.