High Sensitivity Fiber Refractive Index Sensors Based on Asymmetric Supermodes Interference in Tapered Four Core Fiber

: We demonstrate high sensitivity ﬁber refractive index (RI) sensor based on asymmetric supermode interferences in tapered four core ﬁber (TFCF). To make TFCF-based RI sensors, the whitelight was launched into any one of the cores to deﬁne the excitation orientation and is called a vertex-core excitation scheme. When the four-core ﬁber (FCF) was gradually tapered, the four cores gathered closer and closer. Originally, the power coupling occurred between its two neighboring cores ﬁrst and these three cores are grouped to produce supermodes. Subsequently, the fourth diagonal core enters the evanescent ﬁeld overlapping region to excite asymmetric supermodes interferences. The output spectral responses of the two cores next to the excitation core are mutually in phase whereas the spectral responses of the diagonal core are in phase and out of phase to that of the excitation core at the shorter and longer wavelengths, respectively. Due to the lowest limitation of the available refractive index of liquids, the best sensitivity can be achieved when the tapered diameter is 10 µ m and the best RI sensitivity S is 3249 nm/RIU over the indices ranging from 1.41–1.42. This is several times higher than that at other RI ranges due to the asymmetric supermodes.


Introduction
In-line fiber interferometric sensors are featured with high accuracy, high bandwidth, high signal capacity, high environmental stability, in situ and real-time monitoring for measuring various kinds of physical, chemical, and biological measurands [1][2][3][4][5]. To satisfy the giant demands of the bandwidth for telecommunications or sensing purposes, many different kinds of active and passive multicore fibers (MCF) have been extensively investigated over the past few years to enlarge the signal capacity based on space division multiplexing [6][7][8][9][10][11]. In addition, the MCF can also provide multi-parameter sensing [12][13][14] or high power lasing based on coherent beam combination [15][16][17], ascribing to the multicore structures. However, most of the MCFs are very different from the traditional singlemode fibers (SMF), not only in the physical core distributions but also in their fundamental optical properties. For example, the MCFs are incompatible with the standard SMF in their core sizes, the numerical apertures (NA), and the core configurations. Therefore, the fan-in/fanout couplers are always required to conquer the power/signal transmission issues for linking up the MCF and SMF. Besides, the evanescent power coupling may occur when the MCF is heavily bent [18]. Moreover, it is noted that the supermodes are excited to give rise to interferences when the MCF is substantially tapered [19][20][21][22]. With this interesting phenomenon, it brings new physical findings and applications for MCF. Except for the

Fabrication, Experimental Set-Up, and Working Principle
In fabrication, a FCF (Chiral Photonics: SM-4C1500) with the 4 Ge-doped cores arranged in a square shape, shown in Figure 1a. Typically, the NA for each core is 0.157 and the core diameter is 8 μm. The shortest distances between two neighboring cores and the diagonal cores are 50 μm and 74 μm, respectively. The insertion loss and crosstalk for the individual 4 cores are ranging over 1.1~1.7 dB and −46.8~−55 dB, respectively, due to weak power coupling. In order to combine or to separate the four core channels to individually link up to the SMF, the fan-in/fan-out couplers, made from the same FCF, are essential components. However, a small core misalignment between the fan-in/fan-out couplers and the FCF due to core dislocations from different batches of the manufactured FCF may occur to excite some unwanted higher order modes to give rise to noises in spectral responses. To make the TFCF RI sensors, a segment of the uncoated FCF was fixed on the fiber taper station, shown in Figure 2a. The FCF was heated using hydrogen flame and then elongated to thin down the D. During tapering, the boundary of the 4 Ge-doped cores gradually becomes blurred under a 1000× CCD microscope, as can be seen from Figure  2b-e with D = 37, 35.5, 33, 24 μm, respectively, due to the Ge ions diffusion. In addition, the mode field distributions for different cores gradually expand to overlap with each other. The SLD lights were launched into the core c#1 and an optical spectrum analyzer (OSA) was used to individually monitor and record the spectral responses at the output ends of cores for the power coupling c#1-#4. When D is less than 30 μm, the directthrough, namely from the input core c#1 to output core c#1, spectral responses start to vary, which reflects that the supermodes based on the tri-core structure were excited to produce interferences. The extinction ratios (ER) of the spectral responses increase with a decreasing D. Moreover, the excited supermodes based on the tri-core structure also rapidly transit to the quadruple-core structure with decreasing D. The corresponding crosscoupled, namely from the input core c#1 to output core c#2/3/4, spectral responses were also recorded by an OSA. To make RI sensors, the liquids with different indices were prepared using DI water and glycerin at different mixing ratios. The indices of the liquid samples were measured by an Abbe refractometer working at the nD wavelength (589.3 nm). From all of the output spectra, the direct-through type were found to be the most sensitive route to the external index variations since the mode field of the excited asymmetric supermodes is mainly distributed surrounding the excitation core, as can be observed in Figure 2g-i, especially when the ambient index is close to the effective index of TFCF.

Fabrication, Experimental Set-Up and Working Principle
In fabrication, a FCF (Chiral Photonics: SM-4C1500) with the 4 Ge-doped cores arranged in a square shape, shown in Figure 1a. Typically, the NA for each core is 0.157 and the core diameter is 8 µm. The shortest distances between two neighboring cores and the diagonal cores are 50 µm and 74 µm, respectively. The insertion loss and crosstalk for the individual 4 cores are ranging over 1.1~1.7 dB and −46.8~−55 dB, respectively, due to weak power coupling. In order to combine or to separate the four core channels to individually link up to the SMF, the fan-in/fan-out couplers, made from the same FCF, are essential components. However, a small core misalignment between the fan-in/fan-out couplers and the FCF due to core dislocations from different batches of the manufactured FCF may occur to excite some unwanted higher order modes to give rise to noises in spectral responses. To make the TFCF RI sensors, a segment of the uncoated FCF was fixed on the fiber taper station, shown in Figure 2a. The FCF was heated using hydrogen flame and then elongated to thin down the D. During tapering, the boundary of the 4 Ge-doped cores gradually becomes blurred under a 1000× CCD microscope, as can be seen from Figure 2b-e with D = 37, 35.5, 33, 24 µm, respectively, due to the Ge ions diffusion. In addition, the mode field distributions for different cores gradually expand to overlap with each other. The SLD lights were launched into the core c#1 and an optical spectrum analyzer (OSA) was used to individually monitor and record the spectral responses at the output ends of cores for the power coupling c#1-#4. When D is less than 30 µm, the direct-through, namely from the input core c#1 to output core c#1, spectral responses start to vary, which reflects that the supermodes based on the tri-core structure were excited to produce interferences. The extinction ratios (ER) of the spectral responses increase with a decreasing D. Moreover, the excited supermodes based on the tri-core structure also rapidly transit to the quadruplecore structure with decreasing D. The corresponding cross-coupled, namely from the input core c#1 to output core c#2/3/4, spectral responses were also recorded by an OSA. To make RI sensors, the liquids with different indices were prepared using DI water and glycerin at different mixing ratios. The indices of the liquid samples were measured by an Abbe refractometer working at the n D wavelength (589.3 nm). From all of the output spectra, the direct-through type were found to be the most sensitive route to the external index variations since the mode field of the excited asymmetric supermodes is mainly distributed surrounding the excitation core, as can be observed in Figure 2g-i, especially when the ambient index is close to the effective index of TFCF. The working principle for the excited supermodes in the MCF with differen configurations has been extensively studied [26,[32][33][34][35][36]. Due to the vertex-core exci the laser lights are well-confined to propagate in the core right after the splicing When FCF was gradually tapered, the four cores gathered closer and the evanescen of the excitation core, c#1, increasingly overlap with the two neighboring cores, c c#3, first, shown in Figure 2f, and then came along with the supermodes, shown in 2g, based on this three-core structure, c#1-#3. With continuous tapering, the fourt #4, at the diagonal position with respect to the excitation core gradually enters the e cent fields overlapping region to excite new supermodes, Figure 2h,i, based on th core structure, c#1-#4. Figure 2f-i were the far field mode patterns using whiteligh ning 1250-1650 nm from superluminescent diodes (SLD) under a 1000× CCD micro Partial guiding lights from SLDs near the 1310 nm wavelengths can be observed CCD microscope [23]. From the three-core to four-core structure, the excited super are highly determined by the tapered diameter D of the TFCF and are intrinsically metric due to the vertex-core excitation scheme. Consequently, the output interf fringes will be in phase when the output spectral responses are individually me from the two cores, c#2 and c#3, next to the excitation core, after using a fan-out co Due to the supermode interferences, the output spectral characteristics for the exc core become highly wavelength dependent. However, the output spectral response core c#2 and c#3 are very different from the c#1. Ambiguously, the output spect sponses from the diagonal core c#4 are somehow in phase and out of phase relative The working principle for the excited supermodes in the MCF with different core configurations has been extensively studied [26,[32][33][34][35][36]. Due to the vertex-core excitation, the laser lights are well-confined to propagate in the core right after the splicing point. When FCF was gradually tapered, the four cores gathered closer and the evanescent fields of the excitation core, c#1, increasingly overlap with the two neighboring cores, c#2 and c#3, first, shown in Figure 2f, and then came along with the supermodes, shown in Figure 2g, based on this three-core structure, c#1-#3. With continuous tapering, the fourth core, #4, at the diagonal position with respect to the excitation core gradually enters the evanescent fields overlapping region to excite new supermodes, Figure 2h,i, based on the four-core structure, c#1-#4. Figure 2f-i were the far field mode patterns using whitelight spanning 1250-1650 nm from superluminescent diodes (SLD) under a 1000× CCD microscope. Partial guiding lights from SLDs near the 1310 nm wavelengths can be observed by the CCD microscope [23]. From the three-core to four-core structure, the excited supermodes are highly determined by the tapered diameter D of the TFCF and are intrinsically asymmetric due to the vertex-core excitation scheme. Consequently, the output interference fringes will be in phase when the output spectral responses are individually measured from the two cores, c#2 and c#3, next to the excitation core, after using a fan-out coupler. Due to the supermode interferences, the output spectral characteristics for the excitation core become highly wavelength dependent. However, the output spectral responses from core c#2 and c#3 are very different from the c#1. Ambiguously, the output spectral responses from the diagonal core c#4 are somehow in phase and out of phase relative to the spectral Photonics 2022, 9, 45 5 of 11 characteristics from the core c#1 at the shorter wavelength and longer wavelength side of the 1450 nm, respectively.
In this work, the evanescent power overlapping was experimentally observed to begin when D is approaching 30 µm where the supermodes are produced based on a tri-core structure, namely the grouped cores c#1-#3. However, it is found that when D = 26 µm, the diagonal core, c#4, the supermodes at the wavelengths longer than 1450 nm are initially generated. The details about the supermodes evolutions will be further discussed later in this work. In accordance with the couple mode theory for the strong coupling MCF, the supermodes are excited when the distance between cores is small enough to start the effective evanescent power coupling. The superposed electric-field intensity for the interferences can be expressed as follows: the I i and I j are the intensity of the ith and jth supermodes, the n i and n j are the effective index for ith and jth supermodes, l is the length of MCF, and λ is the operating wavelength.
The OPD between two supermodes is defined as and the wavelengths for the interference peaks can be expressed as: the ∆ n e f f is the effective index difference between the cores and m is the order of interferences. From Formula (3), the FSR can be written as: Thus, from Formula (4), it is known that the FSR increases with a decreasing l or ∆ n e f f whereas the FSR decreases with a decreasing λ [39]. These formulas can help explain the working principles for RI sensors.

Measurements and Discussions
In measurement, four TFCF samples, A 1 , A 2 , A 3 , A 4 with the D of 26 µm, 15 µm, 13 µm, and 10 µm, respectively, were prepared to investigate their spectral characteristics. The optical resolution (RES) of OSA was set at 0.05 nm or 0.5 nm, contingent upon the spectral spanning range. To investigate the spectral responses for the power coupling ports between input/output cores, denoted as c#-#, the power coupling condition from input core 1 to output core 1, namely c#1-#1, is called the direct-through state. On the other hand, the conditions from input core 1 to output core 2/3/4, namely c#1-#2/3/4, are defined as the cross-coupled states. The spectral responses of the samples A 1 , A 2 , A 3 , A 4 with the states of c#1-#1 in air can be found in Figure 3a. From which, the normalized transmission loss was observed with oscillating power variations at the very beginning tapering stage at D = 26 µm. When D continuously decreases, the spectral curves come along with more and more oscillations and the free spectral range (FSR) decreases accordingly. In Figure 3b, the spectral responses of the sample A 1 with its all power coupling states c#1-#1/2/3/4 in air were recorded and shown there. Explicitly, it is very interesting to note that the oscillating spectral curves for c#1-#2 and c#1-#3 are highly similar to each other since core #2 and #3 are neighboring cores with the same distance with respect to the excitation core #1. On the contrary, the spectral curve for c#1-#4 is somehow in phase and out of phase relative to the shorter and longer wavelength side of 1450 nm of c#1-#1. In fact, D = 30 µm was experimentally found to be close to the critical point to start the excitation of the supermodes. Thus, for D = 26 µm, the core c#1,2,3 are grouped together to produce the tri-core-based supermodes first, whereas the evanescent power overlapping only occurs at the longer wavelengths for core c#4 to generate the quadruple-core-based supermodes. This is the reason why the oscillating spectral curve for c#1-#4 is partially in phase to that of c#1-#1. However, this phenomenon disappears rapidly soon after D increasingly decreases. The spectral curves for all power coupling start when D = 15 µm, 13 µm, and 10 µm are respectively shown in Figure 3c-e. As predicted, along with the decreasing D, the FSR decreases. The relationships between different c#-# are quite chaotic and no strong regular manners can be found due to the asymmetric supermodes based on the vertex-core excitation scheme. However, when D = 13 µm and 10 µm, the power oscillations become highly overlap with each other for the curves c#1-#1 and c#1-#4 at the longer wavelengths, shown in Figure 3d,e. supermodes. Thus, for D = 26 μm, the core c#1,2,3 are grouped together to produce the tricore-based supermodes first, whereas the evanescent power overlapping only occurs at the longer wavelengths for core c#4 to generate the quadruple-core-based supermodes. This is the reason why the oscillating spectral curve for c#1-#4 is partially in phase to that of c#1-#1. However, this phenomenon disappears rapidly soon after D increasingly decreases. The spectral curves for all power coupling start when D = 15 μm, 13 μm, and 10 μm are respectively shown in Figure 3c-e. As predicted, along with the decreasing D, the FSR decreases. The relationships between different c#-# are quite chaotic and no strong regular manners can be found due to the asymmetric supermodes based on the vertexcore excitation scheme. However, when D = 13 μm and 10 μm, the power oscillations become highly overlap with each other for the curves c#1-#1 and c#1-#4 at the longer wavelengths, shown in Figure 3d,e.  For RI sensing, the two TFCFs with D of 13 μm and 10 μm were selected due to their better performances. The prepared liquids were ranging over the index 1.334-1.342, 1.37-1.38, 1.41-1.42 [40]. Each of them was individually applied into a U-groove and then lifted to a right position by a precision translation stage to make the TFCF entirely immersed. After each time of measurement at room temperature, the samples were thoroughly cleaned using DI water and alcohol to complete the experiments. The refractive indices of the liquids were accurately measured using an Abbe refractometer and the step of the index increment was from 0.0003 to 0.001 contingent upon the fiber samples. At the beginning, the spectral responses with the best interferences at some specific wavelengths were intentionally searched and observed from OSA. Subsequently, the refractive indices of the liquids were finely tuned around there. When the RI gradually increases, the resonant wavelengths red-shift. However, the extinction ratio downgrades slowly with increasing RI since the phase-matching conditions are no longer well-satisfied. Hence, the RI sensitivity was measured at some specifically selected wavelengths and at some RI intervals over 1.334-1.42. The spectral responses of the samples A3 and A4 for the direct-  For RI sensing, the two TFCFs with D of 13 µm and 10 µm were selected due to their better performances. The prepared liquids were ranging over the index 1.334-1.342, 1.37-1.38, 1.41-1.42 [40]. Each of them was individually applied into a U-groove and then lifted to a right position by a precision translation stage to make the TFCF entirely immersed. After each time of measurement at room temperature, the samples were thoroughly cleaned using DI water and alcohol to complete the experiments. The refractive indices of the liquids were accurately measured using an Abbe refractometer and the step of the index increment was from 0.0003 to 0.001 contingent upon the fiber samples. At the beginning, the spectral responses with the best interferences at some specific wavelengths were intentionally searched and observed from OSA. Subsequently, the refractive indices of the liquids were finely tuned around there. When the RI gradually increases, the resonant wavelengths red-shift. However, the extinction ratio downgrades slowly with increasing RI since the phase-matching conditions are no longer well-satisfied. Hence, the RI sensitivity was measured at some specifically selected wavelengths and at some RI intervals over 1.334-1.42. The spectral responses of the samples A 3 and A 4 for the direct-through and cross-coupled states were all recorded at the wavelengths around 1500-1600 nm as well as the S are shown in Figure 4a-l. In Figure 4e,g, the D, L, maximum wavelength shift ∆λ, index variations, and c#-# are (13 µm, 1.8 mm, 12.49 nm, 1.41-1.4139, cross-coupled c#1-#2) and (10 µm, 2.1 mm, 4.62 nm, 1.334-1.342, direct-through c#1-#1), respectively. Figure 4g,i,k are the spectral responses of the sample A 4 on the c#1-#1 states over different index ranges while Figure 4h,j,l are their corresponding sensitivity. Explicitly, the different index liquids give rise to different resonant wavelengths and the sensitivity gradually increases with an increasing external index, shown in Figure 4g,i,k. This is because an external index closing to the effective index of the TFCF can expand the mode fields much more to extend to the outside of TFCF to increase the index sensitivity. From Figure 4e,k, the resonant wavelengths red-shift with an increasing index and the maximum S can be found to be 2911 nm/RIU, shown in Figure 4f, and 3249 nm/RIU, shown in Figure 4l, respectively. It is worthy to note that the S has been shown to be significantly improved compared with the reported fiber interferometers using FCF [26,32,33]. In addition, the coefficients of determination R 2 of linear fitting are 0.976 and 0.9885, respectively, and which reflects the 0394λ exhibits high linearity. In Figure 4k, it is interesting to note that when the ambient index gradually goes up, the resonant wavelength red-shifts. On the other hand, when the tapered diameter is as small as 10 µm and the ambient index is higher than 1.41, two resonant wavelengths can be generated and clearly observed around the refractive index range 1.41-1.417. This is because when a silica tapered fiber is surrounded by the liquid with a refractive index closing to its effective refractive index, more high order supermodes can be excited and survived. This is why the two resonant wavelengths can be observed. However, when the ambient refractive index continuously increases, the phase-matching conditions are no longer well-satisfied. Thus, one of the resonant wavelengths disappears. To simplify the comparisons among samples A 2 , A 3 , A 4 , the best measured working parameters including D, L, ER, FSR, ∆λ, S, and c#-# are listed in Table 1. Obviously, a smaller D can lead to the larger ∆λ and S since the higher order supermodes were excited when the four cores are tightly gathered. Moreover, by selecting the proper tapered diameter and RI liquids, the desired spectral responses as well as the S can be achieved. However, in terms of the best S, the D as small as 10 µm would be the best option over the low RI range near 1.334. increases, the phase-matching conditions are no longer well-satisfied. Thus, one of the resonant wavelengths disappears. To simplify the comparisons among samples A2, A3, A4, the best measured working parameters including D, L, ER, FSR, Δλ, S, and c#-# are listed in Table 1. Obviously, a smaller D can lead to the larger Δλ and S since the higher order supermodes were excited when the four cores are tightly gathered. Moreover, by selecting the proper tapered diameter and RI liquids, the desired spectral responses as well as the S can be achieved. However, in terms of the best S, the D as small as 10 μm would be the best option over the low RI range near 1.334.

Conclusions
In this work, the asymmetric supermode interferences based on the TFCF in the vertexcore excitation scheme were demonstrated to achieve high sensitivity RI sensors. The best sensitivity can be as high as 3249 nm/RIU over the index range 1.41-1.42 and the best extinction ratio is 15.5 dB while the tapered diameter is 10 µm. When the tapered diameter is approaching 30 µm, the supermodes excitations were started to produce. Initially, the supermodes are exited based on a tri-core structure and then gradually transitioned to a quadruple-core structure. This can be verified to explain that the spectral responses of the diagonal core c#4 are in-phase and out-of-phase relative to that of the excitation core c#1 at the short-and long-wavelength side of 1450 nm, respectively. When the tapered diameter gradually decreases, the phases of the cross-coupled and the direct-through spectra become less correlated due to over-coupling. The best sensitivity is achieved when the tapered diameter and tapered length are 10 µm and 2.1 mm, respectively. The measured index sensitivity for fiber sample A 4 is 535 nm/RIU, 884 nm/RIU, and 3249 nm/RIU over the indices ranging from 1.334-1.342, 1.37-1.38, and 1.41-1.42, respectively. The best sensitivity of the TFCF-based RI sensor is 3249 nm/RIU. The excited asymmetric supermodes in TFCF based on a vertex-core excitation are very different from the mode characteristics of the symmetric supermodes in tapered MCF. This RI sensor can also be used to measure the variations of temperature, humidity, blood glucose concentration, and many other applications in biochemical and biomedical fields. It is thus highly promising for developing high sensitivity fiber interferometric sensors.

Data Availability Statement:
The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.