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Photonics
, Volume 8 (8) – Jul 30, 2021

/lp/multidisciplinary-digital-publishing-institute/study-of-the-vernier-effect-based-on-the-fabry-perot-interferometer-kk7U7j38ur

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hv photonics Article Study of the Vernier Effect Based on the Fabry–Perot Interferometer: Methodology and Application 1 , 2 1 , 2 1 , 2 , 1 , 2 1 , 2 1 , 2 Haitao Gao , Junxian Wang , Jian Shen *, Shubing Zhang , Danping Xu , Yanan Zhang and 1 , 2 , Chaoyang Li * School of Information and Communication Engineering, Hainan University, Haikou 570228, China; gaohaitao@hainanu.edu.cn (H.G.); 20181685310051@hainanu.edu.cn (J.W.); 18085208210034@hainanu.edu.cn (S.Z.); xudanping@hainanu.edu.cn (D.X.); zhangyanan@hainanu.edu.cn (Y.Z.) State Key Laboratory of Marine Resource Utilization in South China Sea, Hainan University, Haikou 570228, China * Correspondence: shenjian@hainanu.edu.cn (J.S.); lichaoyang@hainanu.edu.cn (C.L.) Abstract: The optical Vernier effect is a powerful tool for improving the sensitivity of an optical sensor, which relies on the use of two sensor units with slightly detuned frequencies. However, an improper amount of detuning can easily cause the Vernier effect to be unusable. In this work, the effective generation range of the Vernier effect and the corresponding interferometer conﬁguration are suggested and experimentally demonstrated through a tunable cascaded Fabry–Perot interferometer structure. We further demonstrate a practical method to increase the magniﬁcation factor of the Vernier effect based on the device bandwidth. Only the optical path length of an interferometer probe and the sensitivity of the measurement parameters are needed to design this practical interferometer based on the Vernier effect. Our results provide potential insights for the sensing applications of the Vernier effect. Keywords: vernier effect; envelope; Fabry–Perot interferometer; optical path length Citation: Gao, H.; Wang, J.; Shen, J.; Zhang, S.; Xu, D.; Zhang, Y.; Li, C. Study of the Vernier Effect Based on the Fabry–Perot Interferometer: 1. Introduction Methodology and Application. The ﬁber Fabry–Perot interferometer (FPI) has been widely used in many ﬁelds due to Photonics 2021, 8, 304. https:// its stable interference fringes, easy demodulation, and compact structure [1–4]. In recent doi.org/10.3390/photonics8080304 decades, ﬁber optic FPI sensors have been studied for their excellent performance beneﬁts, such as low cost, fast response, anti-electromagnetic interference ability, and durability Received: 1 July 2021 in harsh environments [5–8]. The structure of the ﬁber optic FPI sensor can be grouped Accepted: 27 July 2021 into two categories: (i) an intrinsic Fabry–Perot interferometer (IFPI), where the light is Published: 30 July 2021 reﬂected by the reﬂector inside the ﬁber, a typical example is how the IFPI can be formed by a pair of Bragg gratings separated by a small gap [9], and (ii) an extrinsic Fabry–Perot Publisher’s Note: MDPI stays neutral interferometer (EFPI), where the light exits the ﬁber and propagates inside the external with regard to jurisdictional claims in cavity. Such an external cavity can be made of polymer or an air cavity encapsulated by published maps and institutional afﬁl- a diaphragm [10,11]. The development of science and technology puts forward higher iations. requirements for the performance of sensors, such as the need for a high resolution for biomedical applications [12]. Improving sensitivity is an effective way to realize high- resolution sensors [13]. Therefore, it is of great signiﬁcance to study how to improve the sensitivity of optical ﬁber sensors. Copyright: © 2021 by the authors. The Vernier effect was applied to Vernier calipers for the ﬁrst time. By comparing Licensee MDPI, Basel, Switzerland. the difference in the smallest resolution of two calipers, the detection resolution was This article is an open access article improved. Recently, the Vernier effect has also been applied to the ﬁeld of ﬁber optic distributed under the terms and sensors. Regarding the interference signal as the Vernier scale, the Vernier effect relies conditions of the Creative Commons on the overlap between two interference signals with similar frequencies. The envelope Attribution (CC BY) license (https:// of overlapping signals exhibits the ability to amplify wavelength shifts compared to an creativecommons.org/licenses/by/ individual sensing interferometer. 4.0/). Photonics 2021, 8, 304. https://doi.org/10.3390/photonics8080304 https://www.mdpi.com/journal/photonics Photonics 2021, 8, 304 2 of 15 In 2014, Zhang et al. proposed the application of the Vernier effect to optical ﬁber sen- sors for the ﬁrst time. In this application, two hollow photonic crystal ﬁber (PCF) columns were used for the sensing FPI and the reference FPI, respectively, and were connected in se- ries on the single-mode ﬁber (SMF). The sensitivity of the sensor structure to the axial strain and the magnetic ﬁeld was ampliﬁed simultaneously [14]. Since then, optical ﬁber sensors based on the Vernier effect have aroused widespread interest. Optical ﬁber sensors based on the Vernier effect can be categorized into two approaches: (i) Compact structures, which consists of two interferometers connected in series that interact with each other; however, the structure is relatively compact, such as that of cascaded FPIs [15–19], Mach–Zehnder interferometers (MZI) [20], Sagnac loops [21], and hybrid cascaded conﬁgurations [22]; and (ii) separated structures, which consists of two independent interferometers that are physi- cally separated and do not interfere with each other, for example, the parallel combination of two interferometers of the same type [23–25] or a mixed conﬁguration of different types of interferometers [26,27]. The introduction of the optical Vernier effect using FPIs is quite popular, corresponding to almost half of the publications on this topic [28]. In 2015, Quan et al. proposed an ultra-high sensitivity open-cavity FPI gas refractive index sensor based on PCF and the Vernier effect and using a compact probe conﬁguration with a refractive index sensitivity of 30899 nm/RIU [15]. However, a sensor probe with similar compact structure cannot completely ignore the reference interferometer in the application. A series conﬁguration with physical separation seems to be a good solution to this problem [13]. In 2021, Yi, G et al. reported a high-temperature sensor based on the parallel double FPI structure. This separate conﬁguration prevents the reference interferometer from being affected by temperature [29]. However, there are still some problems to be solved in the application of the Vernier effect. In practical applications, a large ampliﬁcation factor causes the envelope to exceed the bandwidth of the device, which may become undetectable. Deng and D. N. Wang proposed the precise control the magniﬁcation by controlling the length of two FPIs so that the envelope is in the range of the light source [30]. A. D. Gomes et al. innovatively proposed the concept of the harmonic Vernier effect, which further improved the sensitivity by tracking the nodes of the inner envelope [18,31]. In another report by the author, the use of a modal interference combined with extreme optical Vernier effects to produce a measurable envelope while maintaining extremely high magniﬁcation is proposed. This method demonstrated the ultra-sensitive ﬁber refraction with a sensitivity of 500 m/RIU, and the calculated magniﬁcation was higher than 850 times that of normal magniﬁcation [32]. Recently, people have also begun to consider signal processing methods to solve the envelope problem. Zuo et al. proposed the use of Hilbert–Huang transform to extract the square envelope of the Vernier for order calibration so that the measurement range is increased several times [33]. However, the sensitivity of the sensor and the measurement range are always in competition. For the application of the Vernier effect, we still need to consider the sensitivity of a single sensor to design the magniﬁcation and measurement range. In this paper, a cascaded interferometer structure of a tunable air cavity and a SiO cavity made only with commercial welding technology was proposed. First, the charac- teristic spectrum of the Vernier effect was obtained by simulating a theoretical model of the proposed device. The accordion phenomenon was then found by employing dynamics simulations to calculate the length of the air cavity. When the length of the air cavity grad- ually increases, the envelope of the reﬂection spectrum expands and contracts regularly. After that, we studied the relationship between the optical path length (OPL) ratio of two interferometers and the accordion phenomenon, and it is found that the Vernier effect can be stably generated in the ratio range of 0.67 to 1.5. Moreover, considering the limitations of the experimental equipment, a design method to increase the magniﬁcation factor of the Vernier effect as much as possible within the bandwidth of the experimental equipment was proposed and veriﬁed by the experiments. At the same time, the relationship between the sensor sensitivity and the measurement range can be ﬂexibly adjusted. Our research results provide a reference for the practical application of the Vernier effect. Photonics 2021, 8, x FOR PEER REVIEW 3 of 16 the Vernier effect as much as possible within the bandwidth of the experimental equip- ment was proposed and verified by the experiments. At the same time, the relationship between the sensor sensitivity and the measurement range can be flexibly adjusted. Our Photonics 2021, 8, 304 3 of 15 research results provide a reference for the practical application of the Vernier effect. 2. Device Fabrication and Theoretical Principle 2. Device The fabr Fabrication ication process d and Theoretical iagram of t Principle he proposed interferometer device is illustrated in Figure 1. First, a section of SMF (Corning SMF28e+) and hollow fiber (HF, TSP075150, The fabrication process diagram of the proposed interferometer device is illustrated with a core diameter of 75 µm and a cladding diameter of 150 µm) was cleaved, and a in Figure 1. First, a section of SMF (Corning SMF28e+) and hollow ﬁber (HF, TSP075150, commercial welding machine (Fujikura, Tokyo, Japan, 80C+) was then used to splice them with a core diameter of 75 m and a cladding diameter of 150 m) was cleaved, and a together, as shown in Figure 1a. In order to obtain a silica microcavity, as seen in Figure commercial welding machine (Fujikura, Tokyo, Japan, 80C+) was then used to splice them 1b, the SMF was cleaved to obtain a SiO2 cavity with a length of 200 µm. Second, as shown together, as shown in Figure 1a. In order to obtain a silica microcavity, as seen in Figure 1b, in Figure 1c, a large diameter hollow fiber (LDHF, TSP0150375, with a core of 150 µm and the SMF was cleaved to obtain a SiO cavity with a length of 200 m. Second, as shown in a cladding diameter of 375 µm) with a length of 5 cm was prepared, the SiO2 cavity was Figure 1c, a large diameter hollow ﬁber (LDHF, TSP0150375, with a core of 150 m and complet a cladding ely diameter inserted int ofo 375 it wit m) h part with of t a length he HF outsid of 5 cme, and the place of contact was prepared, the SiO cavity betwee was n the LDHF completely an inserted d the HF into was itfixed with with A part of Bthe adhe HF sive outside, . The excess HF and the place at the end w of contact as rbetween emoved after the AB adhesive solidified. Finally, the optical guide fiber was inserted from the the LDHF and the HF was ﬁxed with AB adhesive. The excess HF at the end was removed other end of after the AB adhesive the LDHF without being fixed wi solidiﬁed. Finally, the optical th guide adhesive, ﬁberand was a inserted n air cafr vit om y c the ould b other e formed betw end of the LDHF een the optic withoutabeing l guide ﬁxed fiber with and the end adhesive,face and of the SiO an air cavity 2 cavity, as shown could be formed in between the optical guide ﬁber and the end face of the SiO cavity, as shown in Figure 1d. Figure 1d. Figure 1. Figure 1. The schematic diagr The schematic diagram am of the fabrication process. of the fabrication process. ((a a,,b b) Fabricat ) Fabrication ion process of S process of SiO iO2 cavity cavity; ; (c,d) fabrication process of air cavity. (c,d) fabrication process of air cavity. To analyze the principle of cascaded interferometer, we ﬁrst deﬁned the interface of the To analyze the principle of cascaded interferometer, we first defined the interface of structure. The schematic diagram of the device is shown in Figure 2. The Fresnel reﬂection the structure. The schematic diagram of the device is shown in Figure 2. The Fresnel re- was generated at the surface (M , M and M ) marked in Figure 2, which is formed by the flection was generated at the sur1face2 (M1, M23 and M3) marked in Figure 2, which is formed refractive index (RI) mismatch between different media. The ﬁrst interferometer (FPI ) by the refractive index (RI) mismatch between different media. The first interferometer 1 consists of an air chamber located between the mirror interfaces M and M , while the 1 2 (FPI1) consists of an air chamber located between the mirror interfaces M1 and M2, while second interferometer (FPI ) is an SiO chamber located between the mirror interfaces M 2 2 2 the second interferometer (FPI2) is an SiO2 chamber located between the mirror interfaces and M . The light intensity reﬂectivity of the three interfaces is R , R and R , which can 3 1 2, 3 M2 and M3. The light intensity reflectivity of the three interfaces is R1, R2, and R3, which be calculated by the Fresnel reﬂection equation as follows: can be calculated by the Fresnel reflection equation as follows: n n 1 2 R = R = R = ( ) (1) 1 2 3 n + n 1 2 where n and n are the RI of air and SiO , respectively. According to Equation (1), the 1 2 2 reﬂectivity of each interface was very small (less than 3.6%, the RI of SiO and air at 1550 nm are, 1.4682 and 1, respectively), which can be treated as double-beam interference. Therefore, the reﬂected electric ﬁeld E of the cascaded FPI can be expressed as follows [34]: jj j(j +j ) 1 1 2 E = E ( R + Ae + Be ) r i 1 (2) A = (1 a )(1 b )(1 R ) R 1 1 1 1 B = A(1 a )(1 b )(1 R ) R /R 2 2 2 3 1 Photonics 2021, 8, 304 4 of 15 where E is the input electric ﬁeld, a , a , b , and b are the transmission loss factors at M i 1 2 1 2 1 and M and the loss factor of FPI and FPI , respectively. j and j represent the phase 2 1 2 1 2 difference of the FPI and FPI , which can be calculated as 1 2 4pn L 4pn L 1 1 2 2 j = , j = (3) 1 2 l l where L and L are the length of FPI and FPI , respectively. l is the propagation light 1 2 1 2 wavelength. Equation (3) shows that when the refractive index or the length of the interfer- ometer changes, the corresponding phase will also change, so the OPL can be used as a reference for the phase change of the interferometer. The reﬂected intensity can be derived as p p E E 2 2 I = = R + A + B + 2 R A cos j + 2AB cos j + 2 R B cos(j + j ) (4) r 2 2 1 1 1 1 1 where I is the total reﬂected light intensity, and E is the conjugate complex number of E . r r Half-wave loss occurs when light is reﬂected from a light-sparse medium to a light-dense medium. Since the RI of the FPI in the proposed sensor is smaller than FPI , there is a 1 2 half-wave loss at the M interface. The total reﬂected intensity was corrected to be p p E E r 2 2 I = = R + A + B 2 R A cos j 2AB cos j + 2 R B cos(j + j ) (5) Photonics 2021, 8, x FOR PEER REVIEW r 1 1 1 2 1 1 24 of 16 Figure 2. Schematic diagram of theoretical analysis of cascade FPI. Figure 2. Schematic diagram of theoretical analysis of cascade FPI. From Equation (5), one could ﬁnd that the change of the total reﬂected light intensity is mainly determined by three cosine terms, so the change of phase is the dominant factor nn - 12 2 RR==R =() for the change of total reﬂected light intensity. To show the characteristic spectrum of(1 the ) 12 3 nn + Vernier effect, we used Equation (5) to calculate the reﬂection spectra of the theoretical model. where n1 and n2 are the RI of air and SiO2, respectively. According to Equation (1), the The cavity length of FPI was set from 270 m to 320 m (which makes the OPL of reflectivity of each interface wa 1 s very small (less than 3.6%, the RI of SiO2 and air at 1550 FPI approximate to that of SiO ) with a step size of 10 m, and the cavity length of FPI 1 2 2 nm are, 1.4682 and 1, respectively), which can be treated as double-beam interference. was 200 m. The wavelength of the light source ranged from 1418 nm to 1718 nm, and the Therefore, the reflected electric field Er of the cascaded FPI can be expressed as follows calculation parameters used in the simulation are summarized in Table 1 [35,36]. [34]: −−jjϕϕ()+ϕ Table 1. Parameters for theoreticalEE=+ calculation() R andAe simulation.+Be ri 1 AR =− (1 a )(1− b )(1− )R (2) Cavity Transmission Loss Interface11 Transmission1 Loss1 RI a = 0.02 BA=− (1 a )(1 b−= b0.4 )(1−R ) R /R n = 1 1 1 1 22 2 3 1 a = 0.05 b = 0.4 n = 1.4682 2 2 2 where Ei is the input electric field, a1, a2, b1, and b2 are the transmission loss factors at M1 and M2 and the loss factor of FPI1 and FPI2, respectively. φ1 and φ2 represent the phase The calculated reﬂection spectra are shown in Figure 3. The upper envelope connected difference of the FPI1 and FPI2, which can be calculated as by the valleys of the high-frequency fringes can be observed in Figure 3a–f, which is a 44 ππ nL n L 12 1 2 ϕϕ== , (3) 1 2 λλ where L1 and L2 are the length of FPI1 and FPI2, respectively. λ is the propagation light wavelength. Equation (3) shows that when the refractive index or the length of the inter- ferometer changes, the corresponding phase will also change, so the OPL can be used as a reference for the phase change of the interferometer. The reflected intensity can be de- rived as EE rr IR == +A+B+ 2RA cosϕϕ + 2AB cos + 2RB cos(ϕ+ϕ ) (4) r 11 1 2 1 12 where Ir is the total reflected light intensity, and E is the conjugate complex number of Er. Half-wave loss occurs when light is reflected from a light-sparse medium to a light- dense medium. Since the RI of the FPI1 in the proposed sensor is smaller than FPI2, there is a half-wave loss at the M2 interface. The total reflected intensity was corrected to be EE rr IR == +A+B − 2RA cosϕϕ − 2AB cos + 2RB cos(ϕ+ϕ ) (5) r 2 11 1 2 1 12 From Equation (5), one could find that the change of the total reflected light intensity is mainly determined by three cosine terms, so the change of phase is the dominant factor for the change of total reflected light intensity. To show the characteristic spectrum of the Vernier effect, we used Equation (5) to calculate the reflection spectra of the theoretical model. Photonics 2021, 8, x FOR PEER REVIEW 5 of 16 Photonics 2021, 8, 304 5 of 15 The cavity length of FPI1 was set from 270 µm to 320 µm (which makes the OPL of FPI1 approximate to that of SiO2) with a step size of 10 µm, and the cavity length of FPI2 was 200 µm. The wavelength of the light source ranged from 1418 nm to 1718 nm, and the calculation parameters used in the simulation are summarized in Table 1 [35,36]. typical feature of the Vernier effect. The Vernier envelope can be described by the FSR of FPI and FPI [36]: 1 2 FSR FSR Table 1. Parameters for theoretical calculation and simulation. 1 2 FSR = (6) envelope jFSR FSR j 1 2 Cavity Transmission Loss Interface Transmission Loss RI l l m m+1 a1 = 0.02 b1 = 0.4 n1 = 1 FSR = (OPL = 2n L , i = 1, 2) (7) i i i i OPL a2 = 0.05 b2 = 0.4 n2 = 1.4682 where l and l are the adjacent peak wavelengths of the reﬂection spectrum and m m +1 FSR The calc is the ulate FSR d reflection spectra are of the envelope. Figur sh eow 3a–c n in Fig shows urthat e 3. The upp FSR er envelope c expands as o the n- envelope envelope length of the FPI increases, while Figure 3d–f shows the opposite. Equations (6) and (7) nected by the valleys of the high-frequency fringes can be observed in Figure 3a–f, which describe the relationship between L and FSR , and it can be found that the envelope is a typical feature of the Vernier effect. The Vernier envelope can be described by the FSR 1 envelope decays rapidly around 295 m, as shown in Figure 4. of FPI1 and FPI2 [36]: (a) (b) -10 -10 -20 -20 L =270μm L =280μm 1 1 -30 -30 Upper envelope Upper envelope (c) (d) -10 -10 -20 -20 L =290μm L =300μm 1 1 -30 -30 Upper envelope Upper envelope (e) (f) -10 -10 -20 -20 L =310μm L =320μm 1 1 -30 -30 Upper envelope Upper envelope 1450 1525 1600 1675 1450 1525 1600 1675 Wavelength/nm Wavelength/nm Photonics 2021, 8, x FOR PEER REVIEW 6 of 16 Figure 3. The simulated reflection spectrum and upper envelope of different length L1. (a−f) L1 Figure 3. The simulated reﬂection spectrum and upper envelope of different length L . (af) L 1 1 changing from 270 µm to 320 µm. changing from 270 m to 320 m. FSR FSR FSR = envelope (6) FSR − FSR λλ mm +1 FSR==(2 OPL n L,i=1,2) (7) ii ii 600 OPL where λm and λm+1 are the adjacent peak wavelengths of the reflection spectrum and FSRen- 0 100 200 300 400 500 600 velope is the FSR of the envelope. Figure 3a–c shows that FSRenvelope expands as the length of the FPI1 increases, while Figure 3d–f shows the opposite. Equations (6) and (7) describe the rela 200 tionship between L1 and FSRenvelope, and it can be found that the envelope decays rapidly around 295 µm, as shown in Figure 4. 0 200 400 600 800 1000 1200 L /μm Figure 4. The theoretical relationship between L and FSR . Figure 4. The theoretical relationship between L1 and FSRenvelope. 1 envelope We guessed that the trend of this change has boundaries. To ﬁnd the boundary We guessed that the trend of this change has boundaries. To find the boundary con- conditions of the change of the Vernier effect, the length of FPI was set in the range ditions of the change of the Vernier effect, the length of FPI1 was set in the range between between 10 m and 1200 m with a step of 0.1 m, and the length of FPI was set to 169 m. 10 µm and 1200 µm with a step of 0.1 µm, and the length of FPI2 was set to 169 µm. The The other parameters were the same. For the convenience of comparison, the ratio of OPL other parameters were the same. For the convenience of comparison, the ratio of OPL1 to to OPL is deﬁned as OPL2 is defined as OPL D = (8) OPL OPL D = (8) OPL With L1 increasing from 10 µm to 1200 µm, the continuous changes of the simulated reflectance spectrum and D are recorded. The difference from the description in Figure 4 is that we discovered a peculiar phenomenon in the continuous changing spectrum. As L1 increases, the envelope of the spectrum will be repeatedly compressed and stretched, sim- ilar to a musician playing accordion; thus, we called it the accordion phenomenon. The D values of several reflection spectra in the stretched state are recorded as shown in Figure 5. Among them, as shown in Figure 5b, D = 1 means that the frequencies of the two inter- ferometers are completely equal, and the corresponding envelope is stretched infinitely. The same situation appears in Figure 5a,c, and their common feature is that the envelope is fully stretched, which means that they cannot be tracked. Moreover, in the adjacent stretched state, the envelope of the spectrum has undergone compression and expansion stages, which indicates that these changes have turning points. Therefore, the effective range of the Vernier effect can be found from these turning points during the accordion phenomenon. Intensity/dBm FSRenvelope/nm Intensity/dBm Photonics 2021, 8, 304 6 of 15 With L increasing from 10 m to 1200 m, the continuous changes of the simulated reﬂectance spectrum and D are recorded. The difference from the description in Figure 4 is that we discovered a peculiar phenomenon in the continuous changing spectrum. As L increases, the envelope of the spectrum will be repeatedly compressed and stretched, similar to a musician playing accordion; thus, we called it the accordion phenomenon. The D values of several reﬂection spectra in the stretched state are recorded as shown in Figure 5. Among them, as shown in Figure 5b, D = 1 means that the frequencies of the two interferometers are completely equal, and the corresponding envelope is stretched inﬁnitely. The same situation appears in Figure 5a,c, and their common feature is that the envelope is fully stretched, which means that they cannot be tracked. Moreover, in the adjacent stretched state, the envelope of the spectrum has undergone compression and expansion stages, which indicates that these changes have turning points. Therefore, the Photonics 2021, 8, x FOR PEER REVIEW 7 of 16 effective range of the Vernier effect can be found from these turning points during the accordion phenomenon. -5 (a) -10 -15 -20 -25 D = 0.50 -30 Envelope -35 (b) -10 -15 -20 -25 D = 1.00 -30 Envelope -35 (c) -10 -15 -20 -25 D = 2.00 -30 Envelope -35 1450 1500 1550 1600 1650 1700 Wavelength/nm Figure 5. Figure 5. The r The ratio atio D D and andits its ccorr orresponding esponding reflect reﬂection ion spe spectr ctruum m in the stret in the stretched ched state: state: (a () a) DD = 0 =.0.50, 50, (b) D = 1.00, (c) D = 2.00. The red line represents the upper envelope. (b) D = 1.00, (c) D = 2.00. The red line represents the upper envelope. The generant condition of the Vernier effect is that the frequencies of the two inter- The generant condition of the Vernier effect is that the frequencies of the two inter- ferometers are close enough, so we chose a ratio near D = 1 for simulation, as shown in ferometers are close enough, so we chose a ratio near D = 1 for simulation, as shown in Figure 6. Figure 6g,l shows the reﬂectance spectra at different ratio D. Figure 6a–f shows Figure 6. Figure 6g,l shows the reflectance spectra at different ratio D. Figure 6a–f shows the fast Fourier transform (FFT) of the corresponding reﬂectance spectra, where the red the fast Fourier transform (FFT) of the corresponding reflectance spectra, where the red dot represents the OPL of the FPI , and the blue dot represents the sum of the OPL of dot represents the OPL of the FPI1, 1and the blue dot represents the sum of the OPL of the the two interferometers. With L changing, the simulated signal and its frequency change two interferometers. With L1 changing, the simulated signal and its frequency change syn- synchronously. Figure 6g is the spectrum in the stretched state, and the corresponding chronously. Figure 6g is the spectrum in the stretched state, and the corresponding FFT is FFT is shown in Figure 6a, so we can calculate that the frequency of FPI is twice that of shown in Figure 6a, so we can calculate that the frequency of FPI2 is twic2e that of FPI1. In FPI . In other words, when D is 0.5, twice the frequency of FPI resonates with that of FPI . 1 1 2 other words, when D is 0.5, twice the frequency of FPI1 resonates with that of FPI2. When When D increases from 0.5 to 0.67, the envelope of the reﬂection spectrum is continuously D increases from 0.5 to 0.67, the envelope of the reflection spectrum is continuously com- compressed from the stretched state until it is as it is shown in Figure 6h. In contrast, pressed from the stretched state until it is as it is shown in Figure 6h. In contrast, when D when D increases from 0.67 to 1, the envelope continues to expand until D = 1 (refer to increases from 0.67 to 1, the envelope continues to expand until D = 1 (refer to Figure 5b). Figure 5b). Additionally, the envelope then starts the next stage of compression and ex- Additionally, the envelope then starts the next stage of compression and expansion. Fig- pansion. Figure 6i,j shows samples taken during the expansion stage and the compression ure 6i,j shows samples taken during the expansion stage and the compression stage near stage near D = 1, respectively, and the red envelope represents the typical characteristics of D = 1, respectively, and the red envelope represents the typical characteristics of the Ver- the Vernier effect. It can be seen that the change trend of the envelope near D = 0.67 (here is nier effect. It can be seen that the change trend of the envelope near D = 0.67 (here is an an approximate value, the exact value is two-thirds) is opposite, so it is an inﬂection point. approximate value, the exact value is two-thirds) is opposite, so it is an inflection point. Meanwhile, we can find another inflection point at D = 1.5 in the next cycle of envelope change. As shown in Figure 6h,k, their common feature is that the spectrum exhibits pe- riodicity and the inner envelope formed by the blue discontinuous line connecting the secondary peaks are all in a stretched state. The reason for the appearance of these inflec- tion point features can be explained as follows: in Figure 6b, three times the FPI1 frequency and two times the FPI2 frequency are equal (the frequency multiple in Figure 6e is the opposite), so resonance occurs between them. Here, we verified our conjecture that the boundary of the envelope change exists through simulation. Intensity/dBm Photonics 2021, 8, 304 7 of 15 Meanwhile, we can ﬁnd another inﬂection point at D = 1.5 in the next cycle of envelope change. As shown in Figure 6h,k, their common feature is that the spectrum exhibits periodicity and the inner envelope formed by the blue discontinuous line connecting the secondary peaks are all in a stretched state. The reason for the appearance of these inﬂection point features can be explained as follows: in Figure 6b, three times the FPI frequency and two times the FPI frequency are equal (the frequency multiple in Figure 6e is the opposite), Photonics 2021, 8, x FOR PEER REVIEW 8 of 16 so resonance occurs between them. Here, we veriﬁed our conjecture that the boundary of the envelope change exists through simulation. OPL (a) (g) -10 OPL +OPL 1 2 251.70μm -20 746.99μm -30 D = 0.5 OPL (b) (h) -10 OPL +OPL 332.90μm 1 2 -20 828.18μm -30 D = 0.67 OPL (c) (i) -10 438.45μm OPL +OPL 1 2 -20 933.73μm -30 D = 0.88 OPL (d) (j) -10 6 584.60μm OPL +OPL 1 2 -20 1079.88μm -30 D = 1.18 OPL (e) (k) -10 746.99μm OPL +OPL 1 2 -20 1242.27μm -30 D = 1.50 OPL (f) 868.78μm (l) -10 OPL +OPL 1 2 1364.06μm -20 -30 D = 1.75 314 942 1571 2199 2827 1540 1560 1580 1600 OPL/μm Wavelength/nm Figure 6. Figure 6. Simulated Simulated refle reﬂection ctionspectra spectra and FFT spatial and FFT spatial spectra spectra of FPI of FPI with 1 wdif ithfer dient fferlengths. ent lengt (h a s. ( f)athe −f) the FFT spatial frequency spectrum of the cascaded FPI. The red dot represents the OPL of the FPI1, FFT spatial frequency spectrum of the cascaded FPI. The red dot represents the OPL of the FPI , and and the blue dot marks the sum of the OPL of the FPI1 and the FPI2. (g−l) Simulated reflection spectra the blue dot marks the sum of the OPL of the FPI and the FPI . (gl) Simulated reﬂection spectra 1 2 with different ratio D. The red line represents the upper envelope, and the blue discontinuous line with different ratio D. The red line represents the upper envelope, and the blue discontinuous line represents the inner envelope connected by the secondary peaks. represents the inner envelope connected by the secondary peaks. According to the turning point obtained from the simulation, we can use the Vernier According to the turning point obtained from the simulation, we can use the Vernier effect in the effective range D from 0.67 to 1.5. However, in practical applications, we need effect in the effective range D from 0.67 to 1.5. However, in practical applications, we need to try our best to make the spectra envelope of the Vernier effect within the observable to try our best to make the spectra envelope of the Vernier effect within the observable range of the device. If the maximum bandwidth supported by the device is expressed as range of the device. If the maximum bandwidth supported by the device is expressed as Wspan, the Vernier envelope needs to meet the following equation: W , the Vernier envelope needs to meet the following equation: span FSR FSR FSR =≤ W envelope span (9) FSR = W (9) envelope D −1 span jD 1j The appropriate ratio D can be selected according to the device bandwidth Wspan to The appropriate ratio D can be selected according to the device bandwidth W to span produce the Vernier effect. In addition, the sensitivity is also a very important issue for produce the Vernier effect. In addition, the sensitivity is also a very important issue for sensing applications. The magnification M of the Vernier effect on the sensitivity of the sensing applications. The magniﬁcation M of the Vernier effect on the sensitivity of the sensor can be calculated as sensor can be calculated as FSR W envelope span FSR W envelope span M=≈ (10) M = (10) FSR (1 N−⋅) S FSR (N 1) S 2s sening 2 sen sin g where Ssensing is the sensitivity of the sensing probe, and N is the number of measuring points with step length as the unit sensitivity (generally, in order to fit the measurement results well, N needs to be no less than 5). According to Equation (10), the D value of the maximum magnification Vernier effect supported by the device is calculated as (1NS −⋅) sen sin g D =± 1 (11) span Therefore, we can accurately design the interferometer to generate the Vernier effect, and the sensitivity of the sensor can also be enhanced. The following three steps explain how this method works: Amplitude/a.u. Intensity/dBm Photonics 2021, 8, 304 8 of 15 where S is the sensitivity of the sensing probe, and N is the number of measuring sensing points with step length as the unit sensitivity (generally, in order to ﬁt the measurement results well, N needs to be no less than 5). According to Equation (10), the D value of the maximum magniﬁcation Vernier effect supported by the device is calculated as (N 1) S sen sin g D = 1 (11) span Photonics 2021, 8, x FOR PEER REVIEW 9 of 16 Therefore, we can accurately design the interferometer to generate the Vernier effect, and the sensitivity of the sensor can also be enhanced. The following three steps explain how this method works: (1) Parameter acquisition: the material and length parameters of the sensing interferom- (1) eter are Parameter desig acquisition: ned accordthe ing to the material requirements and lengthand parameters its OPL, of and the Ssensing sensing can be me interfer as- - ured ometer through are designed sensing accor expe ding riments. The to the requir appropriate ements and Wspan its can OPL, the and n be S selected can de be- sensing pending on t measured thr h ough e equipment sensing condit experiments. ions. ToThe obta appr in a mo opriate re acc Wuratcan e linear thenfit be , tselected he num- span ber of depending sampli on ng point the equipment s per unitconditions. sensitivity N T o isobtain generaa lly mor set to 5. e accurate linear ﬁt, the number of sampling points per unit sensitivity N is generally set to 5. (2) Parameter comparison: by substituting the parameters (Wspan, N, Ssensing) obtained in (2) Parameter comparison: by substituting the parameters (W , N, S ) obtained in step (1) into Equation (11), the ratio D of the reference interfer span ometer to the sensing sensing step (1) into Equation (11), the ratio D of the reference interferometer to the sensing interferometer can be calculated. If the D is between 0.67 and 1.5, it is within the ef- interferometer can be calculated. If the D is between 0.67 and 1.5, it is within the fective range of the Vernier effect. In addition, the inequality of equation (9) must be effective range of the Vernier effect. In addition, the inequality of equation (9) must satisfied to ensure that the envelope can be observed within the bandwidth of the be satisﬁed to ensure that the envelope can be observed within the bandwidth of the device. Otherwise, we must choose a nearby value that satisfies the condition. device. Otherwise, we must choose a nearby value that satisﬁes the condition. (3) The production of the reference interferometer: the OPL of the reference interferom- (3) The production of the reference interferometer: the OPL of the reference interfer- eter can be calculated by substituting the D obtained in step 2 into Equation (8). ometer can be calculated by substituting the D obtained in step 2 into Equation (8). Therefore, we can freely choose the material and length parameters of the reference Therefore, we can freely choose the material and length parameters of the reference interferometer for production. interferometer for production. Finally, we can obtain the reflection spectrum with the Vernier effect within the band- Finally, we can obtain the reﬂection spectrum with the Vernier effect within the width of the experimental equipment, and the amplification factor is also as increased as bandwidth of the experimental equipment, and the ampliﬁcation factor is also as increased much as possible. as much as possible. 3. Experiments and Discussion 3. Experiments and Discussion The device shown in Figure 7a was used to verify the inflection point of the accordion The device shown in Figure 7a was used to verify the inﬂection point of the accor- phenomenon in the simulation. An amplified spontaneous emission (ASE, with a wave- dion phenomenon in the simulation. An ampliﬁed spontaneous emission (ASE, with a length ranging from 1525 nm to 1605 nm) broadband light source supplies light to the wavelength ranging from 1525 nm to 1605 nm) broadband light source supplies light to proposed interferometer via a circulator, and the reflected light is then collected by an the proposed interferometer via a circulator, and the reﬂected light is then collected by an optical spectrum analyzer (OSA, MS9740A with a resolution of 0.1 nm). In the experiment, optical spectrum analyzer (OSA, MS9740A with a resolution of 0.1 nm). In the experiment, two samples with different lengths are fabricated, as shown in Figure 7b. The probe is two samples with different lengths are fabricated, as shown in Figure 7b. The probe is fixed on the optical fiber coupling platform, on which the length of the air chamber can ﬁxed on the optical ﬁber coupling platform, on which the length of the air chamber can be be adjusted, and the computer display screen can assist us in inserting the light guide fiber adjusted, and the computer display screen can assist us in inserting the light guide ﬁber into the LDHF. into the LDHF. OSA S1 Screen Fiber coupled system 169 μm HF Coupler S2 204μm HF ASE Displacement stage CCD (a) (b) Figure Figure 7. 7. ((a a)) Schematic Schematic d diagram iagram of experi of experiment ment d device evice bas based ed on the on the V Vernier effect. ( ernier effect. (b b)) M Micr icroscopic oscopic image of FPI2. image of FPI . Sample S1, which had a length of 169 µm, was inserted into the LDHF, and the length of the air cavity was adjusted with a step size of 1 µm. At the same time, the real-time spectral changes could be observed on the OSA. Figure 8 presents the reflected intensity spectra for the Vernier effect together with the corresponding FFT. Figure 8g shows the measured reflectance spectrum from when the OPL ratio between interferometers was 0.5, which can be calculated from Figure 8a. The red dots in Figure 8a–f represent the OPL of the air cavity, and the blue dots represent the sum of the OPL of the air cavity and the silica cavity. Therefore, the ratio D corresponding to the spectrum can be calculated. Figure 8h,k represent the inflection point measured by the experiment, which are similar Photonics 2021, 8, 304 9 of 15 Sample S1, which had a length of 169 m, was inserted into the LDHF, and the length of the air cavity was adjusted with a step size of 1 m. At the same time, the real-time spectral changes could be observed on the OSA. Figure 8 presents the reﬂected intensity spectra for the Vernier effect together with the corresponding FFT. Figure 8g shows the measured reﬂectance spectrum from when the OPL ratio between interferometers was 0.5, which can be calculated from Figure 8a. The red dots in Figure 8a–f represent the Photonics 2021, 8, x FOR PEER REVIEW 10 of 16 OPL of the air cavity, and the blue dots represent the sum of the OPL of the air cavity and the silica cavity. Therefore, the ratio D corresponding to the spectrum can be calculated. Figure 8h,k represent the inﬂection point measured by the experiment, which are similar to to the simulation in Figure 6h,k. In order to prove the reliability of the experimental re- the simulation in Figure 6h,k. In order to prove the reliability of the experimental results, sults, sample S2, which had a length of 204 µm, was used to repeat the experiment. As a sample S2, which had a length of 204 m, was used to repeat the experiment. As a result, result, we also found the inflections seen in Figure 9. Among them, Figure 9d represents we also found the inﬂections seen in Figure 9. Among them, Figure 9d represents the the interference spectrum of S2 when air cavity length is 0, and the images in Figure 9e,f interference spectrum of S2 when air cavity length is 0, and the images in Figure 9e,f are are the inflection points of sample S2. There are some differences in the reflection intensity the inﬂection points of sample S2. There are some differences in the reﬂection intensity between the experimental spectrum and the simulation spectrum, which is mainly caused between the experimental spectrum and the simulation spectrum, which is mainly caused by the loss in the transmission process and the fluctuations in the power of the light by the loss in the transmission process and the ﬂuctuations in the power of the light source. source. Nevertheless, the change trend of the interference signal was still the same. In Nevertheless, the change trend of the interference signal was still the same. In addition, addition, the longer the length of the cavity, the more peaks in the spectrum, which easily the longer the length of the cavity, the more peaks in the spectrum, which easily leads to leads to the loss of peak information. As shown in Figure 9f, some secondary peaks have the loss of peak information. As shown in Figure 9f, some secondary peaks have almost almost disappeared. The turning points calculated in the experiment have slight errors, disappeared. The turning points calculated in the experiment have slight errors, which which are caused by the low resolution of the FFT and the accumulation displacement are caused by the low resolution of the FFT and the accumulation displacement platform platform errors. errors. -9 OPL (a) 1 (g) 246.21μm OPL +OPL 1 2 -18 738.63μm -27 D = 0.50 OPL (b) 1 (h) 338.51μm -18 OPL +OPL 1 2 830.96μm -24 -30 D = 0.69 OPL (c) 1 (i) 4 -12 430.87μm OPL +OPL 1 2 923.29μm -17 D = 0.88 -22 OPL -11 (d) 1 (j) 4 584.75μm OPL +OPL 1 2 -15 1077.17μm -19 D = 1.18 OPL (e) 1 (k) -13 738.63μm OPL +OPL 1 2 -16 1231.05μm -19 D = 1.50 OPL (f) 1 (l) -13 861.73μm OPL +OPL 1 2 -16 1354.15μm D = 1.75 -19 314 942 1571 2199 2827 1540 1560 1580 1600 OPL/μm Wavelength/nm Figure 8. The measured reflection spectrum of FPI2 with a length of 169 µm and the corresponding Figure 8. The measured reﬂection spectrum of FPI with a length of 169 m and the corresponding cal- calculated FFT spectrum. (a–f) the FFT spatial frequency spectrum of the cascaded FPI. (g–l) Simu- culated FFT spectrum. (a–f) the FFT spatial frequency spectrum of the cascaded FPI. (g–l) Simulated lated reflection spectra with different ratio D. reﬂection spectra with different ratio D. OPL (a) 2 (d) -8 599.92μm -12 -16 D = 0 -20 OPL (b) 1 (e) -16 6 409.03μm OPL +OPL 1 2 -20 1008.95μm -24 D = 0.68 -28 OPL (c) (f) 3 1 899.87μm -19 OPL +OPL 1 2 -21 1499.79μm -23 D = 1.50 314 942 1571 2199 2827 1540 1560 1580 1600 Wavelength/nm OPL/μm Figure 9. The measured reflection spectrum of FPI2 with a length of 204 µm and the corresponding calculated FFT spectrum. (a–c) the FFT spatial frequency spectrum of the cascaded FPI. (d–f) Simu- lated reflection spectra with different ratio D. Amplitude/a.u. Amplitude/a.u. Intensity/dBm Intensity/dBm Photonics 2021, 8, x FOR PEER REVIEW 10 of 16 to the simulation in Figure 6h,k. In order to prove the reliability of the experimental re- sults, sample S2, which had a length of 204 µm, was used to repeat the experiment. As a result, we also found the inflections seen in Figure 9. Among them, Figure 9d represents the interference spectrum of S2 when air cavity length is 0, and the images in Figure 9e,f are the inflection points of sample S2. There are some differences in the reflection intensity between the experimental spectrum and the simulation spectrum, which is mainly caused by the loss in the transmission process and the fluctuations in the power of the light source. Nevertheless, the change trend of the interference signal was still the same. In addition, the longer the length of the cavity, the more peaks in the spectrum, which easily leads to the loss of peak information. As shown in Figure 9f, some secondary peaks have almost disappeared. The turning points calculated in the experiment have slight errors, which are caused by the low resolution of the FFT and the accumulation displacement platform errors. -9 OPL (a) 1 (g) 246.21μm OPL +OPL 1 2 -18 738.63μm -27 D = 0.50 OPL (b) 1 (h) 338.51μm -18 OPL +OPL 1 2 830.96μm -24 -30 D = 0.69 OPL (c) 1 (i) 4 -12 430.87μm OPL +OPL 1 2 923.29μm -17 D = 0.88 -22 OPL -11 (d) 1 (j) 584.75μm OPL +OPL 1 2 -15 1077.17μm -19 D = 1.18 OPL (e) 1 (k) -13 738.63μm OPL +OPL 1 2 -16 1231.05μm -19 D = 1.50 OPL (f) (l) -13 861.73μm OPL +OPL 1 2 -16 1354.15μm D = 1.75 -19 314 942 1571 2199 2827 1540 1560 1580 1600 OPL/μm Wavelength/nm Figure 8. The measured reflection spectrum of FPI2 with a length of 169 µm and the corresponding Photonics 2021, 8, 304 10 of 15 calculated FFT spectrum. (a–f) the FFT spatial frequency spectrum of the cascaded FPI. (g–l) Simu- lated reflection spectra with different ratio D. OPL (a) 2 (d) -8 599.92μm -12 -16 D = 0 -20 OPL (b) 1 (e) -16 6 409.03μm OPL +OPL 1 2 -20 1008.95μm -24 D = 0.68 -28 OPL (c) (f) 3 1 -19 899.87μm OPL +OPL 1 2 -21 1499.79μm -23 D = 1.50 314 942 1571 2199 2827 1540 1560 1580 1600 Wavelength/nm OPL/μm Figure 9. The measured reflection spectrum of FPI2 with a length of 204 µm and the corresponding Figure 9. The measured reﬂection spectrum of FPI with a length of 204 m and the corresponding cal- Photonics 2021, 8, x FOR PEER REVIEW calcu lated FFT spectrum. (a–c) the FFT spatial frequency spectrum of the cascaded FPI. (d–f11 of ) Simu- 16 culated FFT spectrum. (a–c) the FFT spatial frequency spectrum of the cascaded FPI. (d–f) Simulated lated reflection spectra with different ratio D. reﬂection spectra with different ratio D. To further validate the properties of our proposed method, a poly (dimethylsiloxane) To further validate the properties of our proposed method, a poly (dimethylsiloxane) (PDMS)-ﬁlled FPI temperature sensor probe was fabricated and tested. The microscopic (PDMS)-filled FPI temperature sensor probe was fabricated and tested. The microscopic image and experimental setup for temperature are illustrated in Figure 10. The sensing image and experimental setup for temperature are illustrated in Figure 10. The sensing probe was fabricated by splicing a SMF with HF ﬁlled with PDMS, where the length of probe was fabricated by splicing a SMF with HF filled with PDMS, where the length of HF was 130 m and the thickness of the PDMS was 108 m. The PDMS was composed HF was 130 µm and the thickness of the PDMS was 108 µm. The PDMS was composed of of a mixture of Sylgard 184-A and the hardener Sylgard 184-B in a ratio of 10:1. The FPI a mixture of Sylgard 184-A and the hardener Sylgard 184-B in a ratio of 10:1. The FPI structure ﬁlled with PDMS was baked on a heating table (with an accuracy of 0.1 C) at structure filled with PDMS was baked on a heating table (with an accuracy of 0.1 °C) at 80 80 C for 3 h to complete the sensor. The measured reﬂection spectrum of the sensor and °C for 3 h to complete the sensor. The measured reflection spectrum of the sensor and its its FFT spectrum at 25 C are shown in Figure 11a, and its FSR and OPL were 8.01 nm FFT spectrum at 25 °C are shown in Figure 11a, and its FSR and OPL were 8.01 nm and and 307.76 m, respectively. In the experiment, the heating platform was adjusted from 25 307.76 µm, respectively. In the experiment, the heating platform was adjusted from 25 to to 65 C and from 65 to 25 C, respectively, collecting data at 3 C intervals. Figure 11c,d 65 °C and from 65 to 25 °C, respectively, collecting data at 3 °C intervals. Figure 11c,d displays the reﬂection spectrum of the temperature response of the sensor. The peak displays the reflection spectrum of the temperature response of the sensor. The peak with with a wavelength of 1534 nm moves to longer wavelengths as the temperature rises, a wavelength of 1534 nm moves to longer wavelengths as the temperature rises, while the while the cooling process moves to shorter wavelengths, where the total wavelengths shift cooling process moves to shorter wavelengths, where the total wavelengths shift from 25 from 25 to 61 C is about 74 nm. The experiment was repeated for two cycles, and the to 61 °C is about 74 nm. The experiment was repeated for two cycles, and the average average response of the probe along with error bars are shown in Figure 11b. The linear response of the probe along with error bars are show n in Figure 11b. The linear fit tem- ﬁt temperature sensitivity of the sensor is 2.056 nm/ C and maintains a good linearity of perature sensitivity of the sensor is 2.056 nm/°C and maintains a good linearity of 99.98%. 99.98%. Figure 10. Experimental system and micrographs of the proposed PDMS-filled FPI temperature sen- Figure 10. Experimental system and micrographs of the proposed PDMS-ﬁlled FPI temperature sor. sensor. In order to generate the Vernier effect, we need to design the reference cavity. There- fore, the parameters (Wspan = 80 nm, N = 5 (1 °C), Ssensing = 2.056 nm/°C) can be substituted into Equation (11) to calculate the ratio D as 1.103 and 0.897, respectively. The comparison shows that the D is close to the inequality (D ≥ 1.1 and D ≤ 0.9), which can be solved by Equation (9). Therefore, the parameters of the reference interferometer can be selected ac- cording to Equation (8). Here D = 0.9 was selected as the configuration parameters of the reference interferometer. To facilitate the calculation, air is used as the material RI of the reference interferometer. In the system configuration of the Vernier effect, the parallel configuration scheme was applied to prove the universality of our proposed method. The schematic diagram of the experimental system and the micrograph of the reference interferometer are shown in Figure 12. The difference from the system scheme in Figure 10 is that we have added a ring coupler to obtain the reference interferometer signal. The reference interferometer is composed of an HF with a length of 138 µm sandwiched by two SMFs. The total measured reflection spectrum of the sensor and its FFT spectrum at 25 °C are shown in Figure 13a, and the OPL of the reference FPI is 276.99 µm and the double frequencies of sensing FPI is 615.52 µm. Therefore, the ratio D can be calculated by Equation (8) to be 0.9, which is consistent with the designed value. Amplitude/a.u. Amplitude/a.u. Intensity/dBm Intensity/dBm Photonics 2021, 8, 304 11 of 15 Photonics 2021, 8, x FOR PEER REVIEW 12 of 16 -30 -33 (a) (c) 4 8.01nm -30 25℃ -36 -32 -32 -34 -35 34℃ -36 307.76μm -32 1540 1560 1580 2 43℃ -36 Wavelength/nm -33 52℃ 1 -39 -36 615.52μm -42 61℃ 0 -48 188 440 691 942 1194 1540 1560 1580 1600 Wavelength/nm OPL/μm -34 (b) (d) 1605 -40 61℃ -46 -32 Sensitivity = 2.056nm/℃ 52℃ -38 R = 0.9998 -32 43℃ -36 -31 -35 34℃ Experiment data -31 Linear fit -34 25℃ -37 25 31 37 43 49 55 61 1540 1560 1580 1600 Temperature/℃ Wavelength/nm Figure Figure 11. 11. ((a a)) T The he measured re measured reﬂection flection spe spectr ctrum of the sensor um of the sensor and and its FFT spe its FFT spectr ctrum. ( um. (b b)) Linear Linear fitting curve ﬁtting curvessof of temperatur temperature e sensitivities of the sensor, and error bars indicate the measurement standard deviation for two measurement cycles. (c) sensitivities of the sensor, and error bars indicate the measurement standard deviation for two measurement cycles. (c) and and (d): Reflected spectrums at different temperatures; (c) temperature-rise process (d) temperature-drop process. (d): Reﬂected spectrums at different temperatures; (c) temperature-rise process (d) temperature-drop process. In order to generate the Vernier effect, we need to design the reference cavity. Therefore, the parameters (W = 80 nm, N = 5 (1 C), S = 2.056 nm/ C) can be substituted span sensing into Equation (11) to calculate the ratio D as 1.103 and 0.897, respectively. The comparison shows that the D is close to the inequality (D 1.1 and D 0.9), which can be solved by Equation (9). Therefore, the parameters of the reference interferometer can be selected according to Equation (8). Here D = 0.9 was selected as the conﬁguration parameters of the reference interferometer. To facilitate the calculation, air is used as the material RI of the reference interferometer. In the system conﬁguration of the Vernier effect, the parallel conﬁguration scheme was applied to prove the universality of our proposed method. The schematic diagram Figure 12. Experimental system and reference FPI micrographs of the proposed PDMS-filled FPI of the experimental system and the micrograph of the reference interferometer are shown temperature sensor based on the Vernier effect. in Figure 12. The difference from the system scheme in Figure 10 is that we have added a ring coupler to obtain the reference interferometer signal. The reference interferometer is composed of an HF with a length of 138 m sandwiched by two SMFs. The total measured reﬂection spectrum of the sensor and its FFT spectrum at 25 C are shown in Figure 13a, and the OPL of the reference FPI is 276.99 m and the double frequencies of sensing FPI is 615.52 m. Therefore, the ratio D can be calculated by Equation (8) to be 0.9, which is consistent with the designed value. Wavelength/nm Amplitude/a.u. Intensity/dBm Intensity/dBm Intensity/dBm Photonics 2021, 8, x FOR PEER REVIEW 12 of 16 -30 -33 4 (a) (c) -30 8.01nm 25℃ -36 -32 -32 -34 34℃ -35 -36 307.76μm -32 1540 1560 1580 43℃ -36 Wavelength/nm -33 52℃ 1 -39 -36 615.52μm -42 61℃ 0 -48 188 440 691 942 1194 1540 1560 1580 1600 Wavelength/nm OPL/μm -34 (b) (d) -40 61℃ -46 Sensitivity = 2.056nm/℃ -32 52℃ -38 R = 0.9998 -32 43℃ -36 -31 34℃ -35 Experiment data -31 Linear fit -34 1530 25℃ -37 1540 1560 1580 1600 25 31 37 43 49 55 61 Temperature/℃ Wavelength/nm Figure 11. (a) The measured reflection spectrum of the sensor and its FFT spectrum. (b) Linear fitting curves of temperature Photonics 2021, 8, 304 12 of 15 sensitivities of the sensor, and error bars indicate the measurement standard deviation for two measurement cycles. (c) and (d): Reflected spectrums at different temperatures; (c) temperature-rise process (d) temperature-drop process. Photonics 2021, 8, x FOR PEER REVIEW 13 of 16 Figure 12. Experimental system and reference FPI micrographs of the proposed PDMS-ﬁlled FPI Figure 12. Experimental system and reference FPI micrographs of the proposed PDMS-filled FPI temperature sensor based on the Vernier effect. temperature sensor based on the Vernier effect. -38 (a) (c) -44 -50 41℃ -36 -38 -44 6 -42 42℃ -50 276.99μm -48 -38 -44 4 -54 -50 43℃ 1540 1570 1600 -38 Wavelength/nm -44 44℃ -50 -38 615.52μm -44 -50 45℃ 188 440 691 942 1194 1540 1560 1580 1600 OPL/μm Wavelength/nm (b) (d) -38 -44 Sensitivity Sensitivity -50 45℃ = 18.18nm/℃ = -18.22nm/℃ -38 -44 2 2 R = 0.99978 R = 0.99979 -50 44℃ -38 1570 -44 43℃ -50 Temperature-rise -38 -44 Temperature-drop -50 42℃ Linear fit -38 Linear fit -44 41℃ -50 41 42 43 44 45 44 43 42 41 1540 1560 1580 1600 Temperature/℃ Wavelength/nm Figure 13. Figure 13.(( aa ) T ) The he total measured refle total measured reﬂection ction spectrum of the sens spectrum of the sensor or with the with theV V e ernier rnier effect a effect and nd its F its FFT FT spe spectr ctru um. m. (( b b ) Linea ) Linear r fitting curves of temperature sensitivities of the sensor; reflected spectrums at different temperatures: (c)temperature-rise ﬁtting curves of temperature sensitivities of the sensor; reﬂected spectrums at different temperatures: (c)temperature-rise process (d) temperature-drop process. process (d) temperature-drop process. The temperat The temperatur ure ch e characteristics aracteristics o offthe thecombination combination of of the thre efer refer ence ence and and thetsensing he sensiFPI ng are tested by placing the sensors on a heating table with a temperature range from 41 C to FPI are tested by placing the sensors on a heating table with a temperature range from 41 °C to 45 C. 45 °C. T To clearly o cle compar arly com e the pare interfer the inter ence ference spectra spectr under a dif under ferent ditemperatur fferent temperat e, Figur ure, F e 13ic,d g- shows the total reﬂectance spectra based on Vernier effect when the temperature rises and ure 13c,d shows the total reflectance spectra based on Vernier effect when the temperature drops. The red discontinuous line represents the wavelength red shift as the temperature rises and drops. The red discontinuous line represents the wavelength red shift as the increases, and the blue discontinuous line represents the blue shift of the wavelength temperature increases, and the blue discontinuous line represents the blue shift of the during the cooling process. To investigate the sensing characteristics of this sensor, the wavelength during the cooling process. To investigate the sensing characteristics of this linear ﬁtting results of the heating and cooling processes are demonstrated in Figure 13b. sensor, the linear fitting results of the heating and cooling processes are demonstrated in The linear ﬁtting sensitivity of the heating process is 18.18 nm/ C, and the linearity is Figure 13b. The linear fitting sensitivity of the heating process is 18.18 nm/°C, and the 0.99978. In the temperature cooling process, the sensitivity is 18.22 nm/ C and the linearity is 0.99978. In the temperature cooling process, the sensitivity is −18.22 nm/°C and linearity is 0.99979. This shows that the sensor has good repeatability. The sensitivity of the the linearity is 0.99979. This shows that the sensor has good repeatability. The sensitivity cooling process with better linearity was selected as a reference. Thus, the sensitivity is of the cooling process with better linearity was selected as a reference. Thus, the sensitivity enhanced 8.86 times, more than the single sensor without Vernier effect, which is largely is enhanced 8.86 times, more than the single sensor without Vernier effect, which is largely consistent with the theory (M = 9.72). consistent with the theory (M = 9.72). In this work, we studied the conditions for generating the Vernier effect and its ap- plications in practice. Consistent with previous studies of Vernier effect [28,37], the enve- lope of the Vernier effect is regenerated when the OPL of the interferometer changes pro- portionally. However, we found that there is a regular expansion and contraction of the envelope during the regeneration process, which means that there is a turning point in the trend of change. Using this phenomenon, we demonstrated the effective range of the Vernier effect determined by inflection points, which provides a reference range for the generation of the Vernier effect. In addition, in practical applications, when the envelope of the Vernier effect exceeds the bandwidth of the device, the sensing signal cannot be monitored. The researchers pro- posed controlling the length of the interferometer with the design of the magnification to solve this problem [30]. However, if the sensitivity of the sensing probe is too high, the offset of the envelope will still be unmeasured. Therefore, within the effective range of the Wavelength/nm Wavelength/nm Amplitude/a.u. Amplitude/a.u. Intensity/dBm Intensity/dBm Intensity/dBm Intensity/dBm Intensity/dBm Intensity/dBm Photonics 2021, 8, 304 13 of 15 In this work, we studied the conditions for generating the Vernier effect and its applications in practice. Consistent with previous studies of Vernier effect [28,37], the envelope of the Vernier effect is regenerated when the OPL of the interferometer changes proportionally. However, we found that there is a regular expansion and contraction of the envelope during the regeneration process, which means that there is a turning point in the trend of change. Using this phenomenon, we demonstrated the effective range of the Vernier effect determined by inﬂection points, which provides a reference range for the generation of the Vernier effect. In addition, in practical applications, when the envelope of the Vernier effect exceeds the bandwidth of the device, the sensing signal cannot be monitored. The researchers proposed controlling the length of the interferometer with the design of the magniﬁcation to solve this problem [30]. However, if the sensitivity of the sensing probe is too high, the offset of the envelope will still be unmeasured. Therefore, within the effective range of the Vernier effect, we proposed a practical design method, considering the sensitivity of the sensing probe and the bandwidth of the device, which ensures that we can generate and track the Vernier signal. The experiment proves that the temperature sensing performance of the sensor made using the proposed method, which demonstrates a sensitivity of –18.22 nm/ C and an excellent linear ﬁt in the range of 41–45 C. The sensing performance of the proposed FPI temperature probe was compared with the recently proposed FPI temperature ﬁber sensors, as shown in Table 2. The temperature sensitivity of the proposed sensor is the highest when compared to other sensors. Among them, compared to the sensor proposed in [4,8,38], the sensor we proposed has higher sensitivity and occupies a smaller bandwidth. The measurement range of the proposed sensor is smaller than the sensor proposed in [4,38,39], but we can change the design parameters to expand the measurement range, and this is at the expense of sensitivity. Table 2. Sensing performance comparison for recently proposed FPI temperature sensors. Type Range Sensitivity W References span Capillary/PDMS 46–50 C 17.758 nm/ C 110 nm [8] Microﬁber/Microsphere/PDMS 30–40 C 3.90 nm/ C 40 nm [39] Microﬁber/PDMS 43–50 C 11.86 nm/ C 90 nm [4] Microﬁber/Capillary/PDMS 42–54 C 6.386 nm/ C 90 nm [38] Capillary/PDMS 41–45 C 18.22 nm/ C 80 nm This work To further improve the sensitivity of the Vernier effect, a light source with a larger bandwidth and a higher resolution device are required. However, a larger magniﬁcation means that the frequencies between the interferometers are close enough, which requires the accurate production of the corresponding reference sensor. However, all of these factors come at a huge cost. The combination with new technology is a way to improve the Vernier effect. For example, an effective way to improve the detection range of the Vernier effect is to use signal processing technology to expand the envelope periodically [33]. The development of higher sensitivity also can be combined with new modes, such as the use of a low-mode interferometer to modulate the large envelope of the Vernier effect, which allows the excellent wavelength offset performance of the large envelope to be retained [32]. 4. Conclusions In this paper, the Vernier effect was studied in a cascaded adjustable air cavity and a ﬁxed SiO cavity interferometer device. In the simulation calculation, the continuous expansion and contraction of the reﬂectance spectrum envelope was deﬁned as an accordion phenomenon. The effective range of the Vernier effect was proven in this phenomenon when the ratio of OPL to OPL is from 0.67 to 1.5. In addition, based only on the optical 1 2 path length of an interferometer probe and the sensitivity of the measurement parameters, a method was demonstrated to increase the magniﬁcation factor of the Vernier effect that can be easily measured by equipment. Using this method, equipment resources can not Photonics 2021, 8, 304 14 of 15 only be more fully utilized to improve the performance of the interference sensor but can also provide a new method of sensor conﬁguration based on the Vernier effect. The results are instructive and universal for the practical application of the Vernier effect. Author Contributions: Conceptualization, H.G.; methodology, H.G.; formal analysis, H.G. and S.Z.; writing—original draft preparation, H.G.; writing—review and editing, H.G., J.W., D.X., and Y.Z.; supervision, J.S. and C.L.; project administration, J.S. and C.L. All authors have read and agreed to the published version of the manuscript. 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Photonics – Multidisciplinary Digital Publishing Institute

**Published: ** Jul 30, 2021

**Keywords: **vernier effect; envelope; Fabry–Perot interferometer; optical path length

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