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Abstract
We propose and demonstrate a half-circle interferometer using a hollow glass microsphere (HGM) resonator. The half-circle interference is induced by a mismatch between the fundamental mode in the HGM and the modes in the capillary wall. The theoretical model is verified by comparing the simulated and experimental results. The variation in capillary length induced by the axial pressure contributes the most to the half-circle interference, which features a device with a high hydrostatic pressure sensitivity of −1.099 nm/kPa. This device shows potential as a hydrostatic pressure sensor owing to its stability, high sensitivity, and robustness.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Pressure measurement is vital to geological research, chemical safety, and mineral extraction. Many concepts have been proposed and developed, of which most are based on fiber Bragg gratings (FBG) and microstructured optical fibers [1–5]. The typical pressure sensor based on Bragg gratings has a lower sensitivity of −3.04 pm/MPa than that of microstructured fiber [6–8]. The latter scheme offers enhanced sensitivity by facilitating mode coupling, which has been widely demonstrated [9,10]. The intermodal coupling of supermodes in a twin-core photonic crystal fiber can improve the sensitivity to −21 pm/MPa [10]. Additionally, the sensitivity can be improved by increasing the ratio of air region [11,12]. For example, the sensitivity of FBG written in grape microstructured fibers can be improved to −12.86 pm/MPa by enlarging the air hole [13]. By writing an FBG in a single-ring suspended fiber, which is composed of a suspended germanium-doped core accompanied by a supporting ring and an outer ring, the sensitivity can be enhanced to −43.6 pm/MPa [12]. In addition, the rocking filters realized by microstructured fibers can reach a sensitivity of −177 nm/MPa [14]. Overall, the hydrostatic pressure performance of the microstructured fiber is superior to that of Bragg gratings. However, the variation in fiber length induced by pressure is extremely small. Therefore, most schemes are based on changes in the refractive index.
The semi-enclosed structure enables the full utilization of pressure, particularly the axial pressure. The semi-enclosed structure is an in-fiber coupling structure that can improve the resonator performance in terms of robustness, stability, and ease of alignment [15–21]. It was investigated with one end opened such that the external environment can be sensed easily. Various schemes have been proposed, such as fiber pigtailed thin wall capillary couplers [22–25], etched photonics crystal fibers [26], suspended dual-core hollow fibers [27], and etched cavities on multimode fiber ends [28]. These structures were integrated with whispering gallery mode (WGM) microsphere resonators, and they excite the Fano resonance. However, the change in the microsphere radius under pressure was negligible. In addition, the fiber length variation only affects the shape of the Fano resonance, but did not significantly affect the resonant wavelength [29]. Therefore, these structures are typically applied in some sensing fields, such as temperature, relative humidity, and gas, which primarily change the effective refractive index of the microsphere resonators. Meanwhile, the performance is poor in other sensing fields, which primarily induce the deformation of the device, such as air pressure and hydrostatic pressure sensing [30].
In this study, we demonstrate a hydrostatic pressure sensor based on half-interference in a hollow glass microsphere (HGM) embedded in a capillary. First, the transmission principle and the corresponding model of the half-interference excited by the HGM are introduced and established. Second, the primary factor in hydrostatic sensing is derived, and the theoretical sensitivity versus the capillary length is predicted. To further prove the transmission principle, the fast Fourier transform (FFT) of the three samples with different initial lengths are calculated. The proposition of the existence of the reflection point at the coupling point is overturned by comparing the free spectral range (FSR) for the two cases. In addition, a hydrostatic pressure sensing experiment is designed to verify the sensing mechanism. Owing to its stability, high sensitivity, and robustness, a compact structure can be adopted as an efficient solution for hydrostatic pressure sensors.
2. Operation principle and device fabrication
Figure 1(a) depicts the structure of a hollow microsphere embedded in a capillary. The device is composed of a lead-in SMF (Corning, SMF-28) with a core refractive index n1 of 1.468, a short section of fused silica capillary (TSP075150, Polymicro Technologies, LLC), with an inner diameter of 75 µm and an outer diameter of 125 µm, and a HGM with a wall thickness of 1–2 µm. We denote the inner refractive index of the HGM as n0, which is the same as the air refractive index, the wall refractive index of the HGM as ns, the outer diameter of the HGM as R, the distance between the coupling point and the end of the capillary as L, and the cladding refractive index of the capillary as nc.
Fig. 1. Schematic illustration of (a) HGM resonator embedded in capillary and (b) optical pathways in structure with half-circle interference.
The input light E1 from the SMF couples into the capillary via the cone zone. A portion of the light traveling in the capillary couples into the HGM as E5 at the coupling point, whereas the other portion E2 travels along the capillary to the end and backs as E3. In the HGM, the light from the capillary travels along the microsphere wall until the other coupling point as E6. Subsequently, a portion of light E7 will couple out of the HGM, whereas the other portion travels around the microsphere. However, the light coupled from the capillary cannot match the conditions for the excitation of the fundamental WGM in the HGM. In addition, the HGM can only excite the fundamental mode and few higher order modes. Consequently, the portion of light that continues to travel around the HGM will not produce constructive interference. In other words, the WGM modes cannot be excited by the HGM. Owing to these factors, the reflection light only comprises two parts, the light coupled out of HGM E7 and the light reflected back from the end of capillary E4, as shown in Fig. 1(b). We denote the transmission coefficient and coupling coefficient as t and k, respectively, the round-trip transmission coefficient as ${\tau }$, and the halfway phase factor as p. The transfer matrix method is applied to calculate the reflection [31], and the normalized reflection can be calculated as
The fabrication process is based on the procedures shown in Fig. 2. First, a section of the capillary that has been cleaned and cut is fusion spliced with the SMF using a commercial fusion splicer (FITEL-S179). The discharge parameters of this fusion procedure are primarily the discharge intensity I = 165, prefuse time tp = 160 ms, discharge time td = 1000 ms, and Z-direction propulsion distance dz = 20 µm. Subsequently, the fusion splicer is required to discharge and pull the cone zone to decrease the cone angle for efficient coupling [32]. The discharge parameters of this step are modified as I = 6, tp = 50 ms, td = 1000 ms, and dz = 200 µm with the direction reversed. Finally, the microspheres can be placed in the device on a three-dimensional adjustment stage.
Fig. 2. Schematic diagrams of (a) splicing SMF with a capillary section, (b) tapering cone area, and (c) embedding process. (d) Micrograph of HGM resonator embedded in capillary.
Fiber taper coupling is the most effective method for exciting WGMs. To investigate the mode excitation experimentally, a HGM with a diameter of 64.92 µm is applied in the fiber taper coupling system. Figure 3(a) presents the schematic illustration. The input light is provided with a tunable laser, and the output light is collected by an optical detector. Meanwhile, the optical detector is followed by a digital oscilloscope to observe the transmission spectra, with the unit as voltage. The corresponding experimental result is shown in Fig. 3(c), where i is the order number and l is the mode number. It is clear that only the fundamental mode and few higher order modes in the form of WGMs can be excited because of the thin shell of the HGM. In the fiber taper coupling system, the phase-match condition can be fulfilled by adjusting the waist diameter of the fiber taper [33]. However, a capillary with a fixed wall thickness can only support modes with an effective refractive index of approximately 1.4430. The left part of Fig. 3(b) shows the simulations of the model field distributions of the fundamental mode in the HGM using COMSOL Multiphysics software, and the right part shows the model with the lowest effective refractive index in the capillary. Based on the calculation, the effective refractive index of the fundamental mode of the HGM is 1.3725, which is much smaller than that of the mode in the capillary, 1.4427. Only if the propagation constants are consistent (i.e., the effective refractive index of the modes is consistent) can the light coupled from the capillary form WGMs in the HGM [34]. This condition cannot be satisfied because of the mismatch between the effective refractive index of the WGMs in the HGM and the modes in the capillary. Therefore, WGMs cannot be excited in the HGM in this model. Light coupling always occurs between the HGM and the capillary. Therefore, the light coupled from the capillary on one side will couple out partially at the other side. Simultaneously, the other portion of light propagates along the wall of the HGM. Owing to the failure of WGM formation, this part of the light cannot yield constructive interference with the original light, and it will vanish gradually.
Fig. 3. (a) Schematic illustration of fiber taper coupling system. (b) Simulated mode field distributions in HGM (left) and capillary (right). (c) Experimental transmission spectra of coupling system with HGM.
In the following experiment, the reflection spectra of the device were obtained and observed using an optical sensing analyzer (SM125, Micron Optics, Inc.). The reflection spectrum features of a solid barium titanate glass microsphere and a HGM with capillaries of approximately the same length, as shown in Fig. 4. The Fano resonance can be observed for the barium titanate glass microspheres, whereas the interference spectra were obtained using the HGM. Barium titanate glass microspheres can easily excite WGMs and support multiple higher order modes. Therefore, the light reflected back from the end of the capillary will interfere with the WGMs to form Fano resonance. However, owing to the failure of mode matching, the WGMs cannot be excited in the HGM. Consequently, the light reflected back from the end of the capillary will interfere with the light coupled out of the HGM in the form of two-beam interference.
Fig. 4. Experimental reflection spectra of device with (a) solid barium titanate glass microsphere and (b) HGM.
3. Analysis of sensing characteristics
Based on Eq. (1), we can deduce that the wavelength shifts are determined by parameters nc, L, ns, and R. In particular, nc and ns barely change in the hydrostatic pressure sensing experiment, with the pressure-induced refractive index increasing on the order of 10−9 /kPa [35]. Therefore, the effects of nc and ns on the wavelength shifts are much smaller than that of L and R in the hydrostatic pressure range. In fact, the coupling point between the HGM and capillary is firm such that L can only be changed by the hydrostatic pressure, rather than the movement of the microsphere. This assumption is reasonable because the HGM is firmly locked near the cone zone. Hence, L and R are the most significant parameters affecting the wavelength shifts, and the effects of these two parameters on the sensing performance of the device will be investigated.
After performing the force analysis of the device, it is discovered that the sensing characteristics can be investigated based on three parameters, the pressure of the HGM from the surrounding PH, axial pressure of the capillary PCA, and radial pressure of the capillary PCR, as shown in Fig. 5. PH will affect the HGM and reduce R. PCA and PCR will affect the capillary, and PCA will diminish L and increase the inner diameter of the capillary rci, whereas PCR will increase L and diminish rci. Hence, the variation of the element in the device can be calculated as follows [36].
For PH,
where tH, GH, and vH are the thickness, modulus of elasticity, and Poisson’s ratio of the HGM, respectively.The pressures PCA and PCR, which are exerted on the capillary, affects L and rci. For PCA, the effect can be expressed as $\Delta $L = -PCAL /(Gctc) and $\; \Delta $rci = PCAvcrci /(Gctc). For PCR, the formula is modified as $\Delta $L = PCAvcrciL /(Gctc), $\Delta $rci = -PCAvcrci2 /(Gctc), where tc, Gc, and vc are the thickness, modulus of elasticity, and Poisson’s ratio of the capillary, respectively.
We assume that the capillary is a rigid body under hydrostatic pressure and that the capillary will not bend when the pressure is increased. However, elastic deformation will occur at low strains. Therefore, the pressure should not be high. We set PH, PCA, and PCR as 3.2 kPa, L as 300 µm, R as 35 µm, tH as 1 µm, and tc as 25 µm. The calculation results are presented in Table 1.
Table 1. Variation in theoretical parameters under hydrostatic pressure of 3.2 kPa
Based on the table, it can be concluded that the effect of PCR on L is negligible compared with that of PCA. rci can exclusively affect the coupling of the microspheres and capillaries. Moreover, the variation in rci is extremely small compared with the microsphere diameter. Therefore, the factors affecting the wavelength shifts can be simplified to PCA on L and PH on R. The effects of the two parameters on the wavelength shifts will be investigated. We set the center wavelength to 1550 nm, and the other parameters remain the same. When the variations in both L and R are considered, the wavelength shift is −3.07 nm. The wavelength shift is −3.07 nm when only the variation in L is considered, whereas it is almost 0 nm when only the variation in R is considered.
It is observed that the variation in R barely induced wavelength shifts, whereas the variation in L is the main factor that causes the wavelength shifts. This is because the optical path in the capillary is much longer than that in the HGM, and L varies more significantly compared to R.
Based on the analysis above, it can be concluded that PCA on L is the main factor that induces wavelength shifts. Therefore, $\Delta $L can be simplified as $\Delta $L≈ -PL/(Gctc). Based on $\Delta $λ/λ=$\; \Delta {\delta }$/$\; {\delta }$, we can easily deduce that the absolute value of sensitivity is inversely dependent on the initial length L.
4. Experiment
Figures 6(a), (b), and (c) show the experimental reflection spectra for different L values of 200, 325, and 575 µm, respectively, and their corresponding microsphere diameters are 61.92, 70.87, and 66.07 µm, respectively. Figure 6(b) shows the corresponding FFTs. To compare the FFTs, the intensity is normalized, as shown in Fig. 6(d). Hence, we can infer from it that, FSR is reducing with the increase in L, which is consistent with the previous analysis. A comparison of the theoretical and experimental optical paths, denoted as OPt and OPe, respectively, is shown in Table 2. The difference can be calculated as (OPt-OPe)/OPe $ \times \; 100\%$. The data statistics strongly support the discussion.
Fig. 6. Three experimental reflection spectra corresponding to three different value of L, (a) 200 µm, (b) 325 µm, and (c) 575 µm. (d) From bottom to top are the FFT of (a), (b), and (c). (e) FSRs from experiment (blue dots) and simulations with (red dots) and without (green dots) light passing through HGM.
Table 2. Comparison of theoretical and experimental OP
The interference spectra will be observed when the HGM are placed in the capillary, it was discovered that the interference spectrum formation is associated with the HGM. However, the coupling point between the HGM and capillary might serve as a reflection point of part of the incident light, which is more likely to be triggered in the experiments. Under this hypothesis, the light would reflect back from the capillary end interfere with the light reflected at the coupling point. To investigate the effects of the two interference principles in the experiment, their OPs are calculated using two different equations, as shown in Fig. 6(c). One is Eq. (1), and the other is ${\delta }$ = 2ncL, which is based on the existence of reflection at the coupling point between the HGM and capillary. The calculation result of the FSRs indicates that the reflection point had no contributions or contributed minimally to the interference spectra.
In the sensing experiment, the sensor is placed in a hydrostatic pressure chamber. The microsphere is stable because of the cone zone and van der Waals forces. Owing to the presence of air in the capillary, the microsphere will not touch the water. Therefore, the device can be safely placed in a hydrostatic pressure chamber without any encapsulation. The pressure in the chamber is increased from 0 to 3.2 kPa at an interval of 0.4 kPa with the room temperature unchanged. The pressure is fixed until the reflection spectra maintaining stable and then each data is recorded.
First, a solid barium titanate glass microsphere with a diameter of 71.57 µm is used to excite the WGM, and the sensing advantage of half-circle interference is verified by comparison. We set L to 310 µm and placed the device under the same hydrostatic pressure range, as previously mentioned. The corresponding experimental results are presented in Fig. 7(a). The result indicates that a sensitivity of 0.03 nm/kPa is obtained, which is not extremely high, as shown in Fig. 7(b). When the pressure is below 1 kPa, the wavelength valley shifts to the short wavelength. It is caused by the environment fluctuations and the measurement error under a small pressure range.
Fig. 7. (a) Experimental plots under different hydrostatic pressures with solid barium titanate glass microsphere. The microsphere diameter D and the capillary length L are 71.57 µm and 310 µm respectively. (b) Corresponding hydrostatic pressure response of (a).
The hydrostatic pressure performance of the HGM is presented in the following experiment. Figures 8(a) and 8(b) show the experimental reflection spectra without and with touching water for L = 200 µm and 270 µm, respectively. With touching water, FSR keeps the same. However, the intensity and contrast of the reflection spectra decreases and increases, respectively. This is due to the fact that the reduced reflectivity of the capillary end face changes the reflected light from the capillary end face [37].
Fig. 8. Experimental reflection spectra of samples without and with touching water when L is (a) 200 µm and (b) 270 µm.
Figures 9(a) and 9(c) show the experimental reflection spectra with hydrostatic pressure increasing for above samples, respectively. The center wavelength shifted from 1515.82 to 1512.45 nm with a sensitivity of −1.099 nm/kPa for L = 200 µm, which is approximately six times the highest sensitivity obtained based on microstructured fibers [14], as shown in Fig. 9(b). Furthermore, the center wavelength of the second sample shifted from 1562.04 to 1561.37 nm with a sensitivity of −0.214 nm/kPa for L = 270 µm, as shown in Fig. 9(d). The relationship between the sensitivity and L conformed to the previous analysis. In addition, compared with the solid barium titanate glass microsphere, the sensing experiment results of the HGMs highlighted the superiority of half-circle interference in hydrostatic sensing.
Fig. 9. Experimental plots of hydrostatic pressure with initial length L of (a) 200 µm and (c) 270 µm. (b) and (d) show corresponding hydrostatic pressure responses of (a) and (c), respectively.
5. Conclusions
We successfully demonstrated a hydrostatic pressure sensor based on half-circle interference with an HGM embedded in a capillary. Based on experimental and simulation proofs, we attributed the form of the half-circle interference to the few modes in the HGM and the mismatch between the WGMs in the HGM and the modes in the capillary. Subsequently, with the precondition that the capillary is a rigid body under hydrostatic pressure, it is discovered that the sensing characteristic is related to the pressure on the HGM and capillary. By calculating and comparing the effects of its sensing factors, it is simplified to the axial pressure on the capillary. In addition, we experimentally prove the transmission and sensing mechanisms. By comparing the FSRs, it is overturned that a reflection point existed at the coupling point. A sensitivity of −1.099 nm/kPa is obtained. The proposed device exhibits advantages of stability, robustness, and low cost, enabling it to be integrated extensively into hydrostatic pressure sensors.
Funding
National Natural Science Foundation of China (62022053, 61875116, 61675126); Natural Science Foundation of Shanghai (18ZR1415200); Open Project Program of Wuhan National Laboratory for Optoelectronics (2018WNLOKF014); 111 Project (D20031).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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