1. Introduction

Over the last few decades, distributed optical fiber sensors (DOFSs) have been widely incorporated into new-generation long-haul sensing applications [1,2]. DOFSs have numerous advantages, such as high sensitivity to various external perturbations, multipoint sensing (i.e., the fiber is equivalent to sensor networks), low transmission losses, high resistivity in harsh environments, low cost, small size, and compact and flexible geometry. Currently, DOFSs are remarkable candidates for various sensing applications in both smart materials and smart structures [3]. Thus, a portfolio of measured parameter changes have been recorded and investigated using DOFSs, including strain [4], temperature [5], bending [6], gas [7], magnetic and electric fields [8], refractive index [9], and acoustic sensing [10]. More recently, the performance of Brillouin fiber sensors have been extensively demonstrated in the structural health monitoring of risky civil structures, such as gas pipelines and chemical plants [11,12]. In short, Brillouin fiber sensors are based on the monitoring of the frequency shift of a backward optical signal in optical fibers. This back-scattering signal is attributed to the interaction (i.e., spatial overlap) between guided optical modes and higher-order acoustic modes under a resonance condition. The trace of this interaction is a light peak located at a well-known Brillouin frequency, which is sensitive to external disturbances, such as temperature and strain. Thus, by monitoring the Brillouin frequency shift (BFS), it is possible to discriminate temperature and strain responses [13].

Most Brillouin fiber sensors (e.g., in a Brillouin optical time-domain analyzer [BOTDA]) utilize the G-family of standard single-mode fibers (SMFs; e.g., G$_{652}$, G$_{655}$, and G$_{657}$) [14]. However, the joint effects of multiparameter crosstalk may degrade the sensing performance of SMF-based Brillouin sensors. Hence, much research has been devoted to specialty Brillouin fiber sensors with the aim of addressing the joint effects of various environmental parameters and explicitly quantifying each response [15,16]. This is based on discriminative multiparameter measurements performed by tracking the BFSs of distinguishable multiple high-gain Brillouin traces (i.e., peaks) that emerge in the fiber’s Brillouin gain spectrum (BGS). This requires the presence of multiple higher-order acoustic modes that strongly interact with the guided optical modes. In principle, the temperature coefficient $C_{(i/j)}^T$ and strain coefficient $C_{(i/j)}^\varepsilon$, both of which quantify the sensitivities of the fiber, can be determined using the BFSs of at least two Brillouin scattering peaks ($i$ and $j$). These coefficients are key features of a Brillouin fiber sensor that are required to be high. In addition, the errors associated with the simultaneous measurement of temperature and strain ($\delta T$ and $\delta \varepsilon$, respectively) are considered potential critical challenges that must be reduced in specialty Brillouin fiber sensors.

Targeting the aforementioned metrics, the authors in [15] proposed and numerically investigated an M-shaped SMF (M-SMF) in multiple stimulated Brillouin scattering (i.e., two peaks were used from the three peaks that emerged in the spectrum) to discriminate temperature and strain. The reported results outperformed those of standard SMFs in terms of Brillouin gain (BG) and temperature/strain coefficients (maximum $C_{(i/j)}^T$= 1.5187 MHz/$^\circ$C, maximum $C_{(i/j)}^\varepsilon$= 0.0664 MHz/$\mu \varepsilon$). The achieved errors were 0.47$^\circ$C for temperature and 12.3 $\mu \varepsilon$ for strain.

Photonic crystal fibers (PCFs) [16,17], polarization-maintaining fibers (PMFs) [18], and large-effective-area ($A_{eff}$) fibers (LEAFs) [19] have been proposed and demonstrated for stimulated Brillouin scattering sensing, with promising results reported in terms of high strain and temperature coefficients. However, due to the high cost of manufacturing PCFs, the complexity of the sensing system (e.g., the fibers require splicing with SMFs), and large discriminative measurement errors (especially in LEAFs), sensing lengths are fairly limited when using PCFs, PMFs, or LEAFs.

Moreover, space-division multiplexing (SDM) fibers, such as multicore fibers [20,21], have been demonstrated in multiplexed Brillouin distributed sensing. One study achieved temperature and strain coefficients of $C_{(i/j)}^T$= 0.971 MHz/$^\circ$C and $C_{(i/j)}^\varepsilon$=0.0729 MHz/$\mu \varepsilon$, respectively [20]. In addition, few-mode fibers (FMFs), another subset of SDM fibers, have recently been used as sensing elements in Brillouin scattering [22,23]. Compared with the aforementioned fibers, FMFs have acceptable costs similar to those of standard SMFs; thus, they tend to be deployed in long-distance sensing applications. However, sensing systems that incorporate FMF sensors are more complex due to the presence of multiple optical modes sharing the same core.

Intuitively, ring core fibers (RCFs) are specialty fibers that have recently demonstrated high potential in orbital angular momentum–mode division multiplexing (OAM-MDM) optical telecommunication networks [24,25]. A few recent theoretical and empirical demonstrations have harnessed these specialty fibers for distributed Brillouin fiber sensors, producing promising results. In [26,27] an inverse-parabolic graded-index fiber (IPGIF) has been demonstrated for multiparameter discrimination, with the experimental results revealing high strain and temperature coefficients (maximum $C_{(i/j)}^T$= 0.99386 MHz/$^\circ$C and maximum $C_{(i/j)}^\varepsilon$= 0.04202 MHz/$\mu \varepsilon$, respectively). These promising results in high responses are attributed to the unique sharp index profile of the IPGIF. Recently, In 2022, the authors in [28] performed the first measurement of Brillouin characterization in an OAM fiber (which has a refractive index very similar to that of IPGIF) operating at 1.55 $\mu$m. The reported $C_{(i/j)}^T$ and $C_{(i/j)}^\varepsilon$ were quite competitive (maximum $C_{(i/j)}^T$= 0.886 MHz/$^\circ$C and maximum $C_{(i/j)}^\varepsilon$= 0.04217 MHz/$\mu \varepsilon$, respectively).

The cited works have demonstrated that the fiber structure (i.e., material composition and profile shapes) could pave the way for sensing system simplification and performance improvement in terms of high gain, high response, and low measurement errors. Noteworthily, based on recently reported results, the ring shape featuring sharp rise and fall of the index profile (i.e., unique sharp variation in [[26], [28]] /multiple sharp variations in [15]) could lead to large improvements through unleashing fiber sensitivities and reducing measurement errors.

In fact, in [15], the author did not conduct a design optimization in order to enhance the sensitivities and to decrease the measurement errors in M-shape fiber. Optimization efforts were focused to improve the Brillouin gain (in /mW) of initially obtained peaks in the absence of any disturbances on the fiber. Moreover, the designed fiber has a high inner-doped region (i.e., undepressed inner core), which may complicate the sensor characterization and tend to degrade the acoustic properties and performances. Furthermore, the promising results in [26–28] using only inner-graded shape (interior graded index shape and outer sharp index shape) confirm that graded index variation collocated with both ring structure and high index contrast may contribute to high responses. Hence, investigating a graded index ring core fiber in Brillouin sensing is reasonable and may provide further design guidelines or recommendations for further improving simultaneously the performance of temperature and strain sensing. In addition, the recently cited works ([15], [26], and [28]) reported small sensitivity coefficient differences in the strain or temperature dependence for different Brillouin peaks, which resulted in a large amplification factor of measurement error. Therefore, the race is ongoing, and it remains challenging to find appropriate fiber designs associated with high responses for discriminative measurements and small measurement errors.

The present study proposed and numerically investigated a novel RCF sensor, which we refer to as ring hyperbolic tangent (R-HTAN) fiber, for stimulated Brillouin scattering sensing. although the graded shape, the designed R-HTAN-based Brillouin sensor achieved discriminative measurements of temperature and strain with high sensitivity and low measurements errors thanks to the high index contrast. The designed R-HTAN fiber sensor was inspired by the hyperbolic tangent function, which has demonstrated prestigious merits in handling OAM modes with high inter-channel separation [29,30].

The proposed R-HTAN fiber’s refractive index and acoustic velocity profile have a shape parameter $\alpha$ that controls variation in the refractive index and longitudinal acoustic velocity profiles (i.e., the smoothness parameter). The parameter $\alpha$ controls the smooth/abrupt rise and fall of the index profile between the center and the edge of the core (inner/outer core) and between the edge of the core and the cladding region. In principle, during the manufacturing process, the heating, the temperature and imperfections could change the fiber parameters especially the shape parameter $\alpha$ which subsequently changes fiber sensor performances afterwards. Hence, we deal with the fiber sensor performance metrics at varying $\alpha$ in order to explore the limits of potentials of the proposed R-HTAN fiber and cover much scenarios in case of manufacturing error. Furthermore, this provides high flexibility for deriving appropriate designs aimed at supporting optical/acoustic modes with optimized performance.

The remainder of this paper is organized as follows. Section 2 presents the key generic parameters of the proposed R-HTAN fiber’s refractive index and longitudinal acoustic velocity profile. Section 3 describes an optical and acoustical modal analysis that we numerically performed to discover the supported optical/longitudinal acoustic modes in the designed R-HTAN fiber. Next, Section 4 characterizes the multiple Brillouin peaks that emerged in the R-HTAN fiber’s BGS. Then, Section 5 demonstrates the discriminative properties of temperature and strain sensing using the R-HTAN fiber by measuring the BFS of multiple Brillouin peaks. The temperature and strain coefficients are reported and compared with those obtained in recent fiber-based Brillouin sensors. To the best of our knowledge, our designed R-HTAN fiber sensor possesses the highest temperature and strain coefficients. Section 6 discusses the respective errors related to the measurement operation and peaks selection. Lastly, Section 7 provides the study’s conclusions and ends with some future research perspectives.

2. Proposed fiber: refractive index and longitudinal acoustic profiles

The proposed R-HTAN fiber is a ring fiber with an optical refractive index profile inspired by the HTAN function. Ring fibers are OAM fibers that have been demonstrated to handle robust OAM modes for next-generation OAM-MDM communication systems [31,32]. Such fibers exhibit a high-doped region (i.e., ring) that is centered at the fiber cross-section. The HTAN function was borrowed from other fields and harnessed as a refractive index profile due to its capability to enhance intermodal separation and to suppress the induced crosstalk between supported optical modes. The proposed optical refractive index profile n(r) of the proposed R-HTAN fiber is as follows:

Given the direct dependencies of both indices (i.e., the refractive index and longitudinal acoustic velocity) on the GeO$_2$/F dopant concentration, strain variation ($\Delta S$), and temperature variation ($\Delta T$), the parameters $n$ and $V_L$ are expressed as follows [15]:

The designed R-HTAN fiber has a high refractive index contrast between the core (i.e., the ring layer) and the cladding. We selected $n_1$ = 1.494 and $n_2$ = 1.444 (i.e., undoped silica) as the refractive indices of the ring and cladding, respectively, at a wavelength $\lambda = 1.55 \mu$m. Using (2) and assuming an ideal fiber ($\Delta T = \Delta S = 0$), the concentration of GeO$_2$ was calculated as 34.6 $wt\%$. Since in this study we perform a purely numerical analysis (generalized approach), we assume that the power launched inside the fiber is below a specific threshold that may induce nonlinear effects caused by the high GeO$_2$ doping concentration. In addition, various experimental demonstrations have used approximately similar GeO$_2$ doping concentration such as the 33.9 $wt\%$ in [26–28].

In addition, using Eq. (2), the equivalent acoustic velocity was calculated as 4463.2 m/s. The refractive index profile and the longitudinal acoustic velocity profile are depicted in Fig. 1 for different values of shape parameter $\alpha$. Considering the effect of the smoothness parameter $\alpha$, when $\alpha =0$, the R-HTAN fiber corresponds to the well-known step index inner and outer RCF, whereas when $\alpha =1$, the R-HTAN fiber corresponds to the graded index RCF. Both profiles are next used in the simulations and characterization of optical and acoustic modes.

 

Fig. 1. (a) Refractive index profile of the proposed R-HTAN fiber, and (b) Equivalent longitudinal acoustic profile of the proposed R-HTAN fiber; $n_2$ = 1.444, $w_{GeO_2}$ = 34.6 $wt\%$, and $a =3 \mu$m.

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The R-HTAN fiber parameters require precise adjustments throughout the manufacturing process. Although it is well known that MCVD (modified chemical vapor deposition) could be a powerful fiber fabrication method that guarantees the exactitude of the manufactured optical fiber parameters while keeping high-performance levels, it is quite challenging to guarantee that, especially the shape parameter $\alpha$. In order to deal with these challenges, in our numerical analysis, we deal with various structures (i.e., varying $\alpha$) in order to cover all the possible scenarios of fiber structures.

3. Optical and acoustic modal analysis

This section numerically deals with the linearly polarized optical modes supported in the designed R-HTAN fiber. These modes are obtainable by solving the optical propagation equation (i.e., the Helmotz equation). After we obtained the optical modes (the propagation constants and fields), we could obtain the acoustic modes through resolving the scalar equation of acoustic displacement field distribution. For our numerical analysis, we used the finite-element method integrated in the COMSOL Multiphysics software package.

3.1 Optical modes in the R-HTAN fiber

Using the parameters highlighted in Fig. 1, the designed R-HTAN fiber supports six eigenmodes at $\lambda = 1.55 \mu$m. These eigenmodes are the true solutions of the vectorial version of the propagation equation that results from Maxwell equations [26]. These modes are HE$_{1,1}$, TE$_{0,1}$, HE$_{2,1}$, and TM$_{0,1}$. Each HE mode is twofold-degenerate, containing odd and even modes with the same propagation constant $\beta$. The linear combination of HE$_{1,1}$ (odd and even) gives the fundamental linearly polarized mode LP$_{0,1}$, whereas the combination of TE$_{0,1}$, HE$_{2,1}$ (odd and even), and TM$_{0,1}$ gives the mode LP$_{1,1}$.

Figure 2(a) depicts the propagation constant ($\beta$) of supported modes, while Fig. 2(b) depicts the electric field distributions of supported modes in the R-HTAN fiber at $\alpha$ = 1. In Fig. 2(b), the rainbow scale indicates the normalized electric field, while the red arrows indicate the direction of the transverse electric field. In regard to the effect of parameter $\alpha$, the number of supported modes is unchangeable across $\alpha$. Only their associated propagation constant $\beta$ changes (also the effective index $n_{eff}$), which decreases across $\beta$. In case of bent R-HTAN fiber, the confinement loss induced by bending is calculated for bend radius of 0.02 m and at $\lambda = 1.55 \mu$m. The maximum bending losses are 1 dB/km and 0.6 dB/km for $\alpha$=0 and $\alpha$=1, respectively. The designed R-HTAN fiber especially at $\alpha$=1 outperforms the M-shape fiber [15] (0.1262 dB/turn) in terms of resistivity against practical condition such as bending.

 

Fig. 2. (a) Propagation constant of supported modes in the R-HTAN fiber versus shape parameter $\alpha$, and (b) Supported optical modes in the R-HTAN fiber. The rainbow scale indicates the normalized electric field (i.e., amplitude), while the red arrows indicate the direction of the transverse electric field. The parameters are maintained as $n_2$ = 1.444, $w_{GeO_2}$ = 34.6 $wt\%$, $a =3 \mu$m, and $\alpha =1$.

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3.2 Longitudinal acoustic modes in the R-HTAN fiber

To calculate the longitudinal acoustic modes supported in the R-HTAN fiber, the eigenvalue equation of the acoustic displacement field distribution needed to be solved. This scalar eigenvalue equation in optical fiber is presented as follows [15,26]:

By substituting $\beta$, obtained from the aforementioned optical modal analysis into Eq. (4), the considered equation could be numerically solved, including the determination of $\Omega _j$ and $\xi _j$ of guided modes. Subsequently, the frequency downshift experienced by scattered light (Stokes) can be expressed as $\nu _b = \Omega _j/2\pi$. Moreover, the effective velocity $V_{eff}$ of each longitudinal acoustic mode was obtained using $V_{eff}= \Omega _j /\beta _{ac}$.

Using our designed R-HTAN fiber, Eq. (4) was solved for different values of shape parameter $\alpha$ and using the propagation constant of the fundamental optical mode ($\beta _{LP_{0,1}}$ in Fig. 2(a)). Should note here that due to the co-existence of multiple co-propagating optical modes, higher optical modes (TE$_{0,1}$, HE$_{2,1}$, and TM$_{0,1}$) forming LP$_{1,1}$ mode could also contribute to photo-acoustic interactions. In principle, these modes are enough separated to consider them as independent channels each of which will stimulate acoustic modes that contribute to multiple Brillouin peaks exploited later in optical sensing [27]. However, most of multi-parameter Brillouin-based sensing systems incorporating FMFs only accommodate LP$_{0,1}$ since higher optical vector modes require delicate spectral and spatial control in order to be detected. Therefore, for the purpose of clarity and system simplification, it is more conducive to study photo-acoustic interaction in ring FMFs with only harnessing LP$_{0,1}$ mode for sensing. The results of mode effective velocity ($V_{eff}$) across acoustic mode numbers as well as for the BFS ($\nu _b$) across effective velocities are presented in Fig. 3(a) and 3(b). As indicated in Fig. 3(a), the number of supported longitudinal acoustic modes decreased as the shape parameter $\alpha$ increased. This is explained by the fact that $\alpha$ tended to decrease the propagation constant $\beta _{op}$ value of the LP$_{0,1}$ mode before its intercalation into Eq. (4). This tended to degrade the number of acoustic eigenvalues (i.e., the effect of the decrease of the absorption coefficient). On the other hand, it is well known that dominant axisymmetric acoustic modes (i.e., L$_{0,m}$) are the major actors responsible for Brillouin scattering (i.e., strong coupling with optical modes).

Figure 4 depicts the 2D mode profiles of the longitudinal acoustic displacement and field pattern of the supported axisymmetric acoustic modes in the R-HTAN fiber with shape parameter $\alpha =1$. The R-HTAN fiber supports six higher-order axially symmetric acoustic modes, namely L$_{0,1}$, L$_{0,2}$, L$_{0,3}$, L$_{0,4}$, L$_{0,5}$, and L$_{0,6}$. Based on Fig. 4, the higher-order acoustic modes are more concentrated toward the center of the core compared with the lower one (the same could be deduced from the field patterns [inset]). In principle, this behavior is due to the high effective acoustic velocities of higher acoustic modes compared with those of lower modes. In addition, the concentric circles formatting the mode patterns (i.e., oscillations) are located in the ring region, which is explained by the high concentration of GeO$_2$ in the considered region. The interaction between the characterized optical and acoustic modes are responsible for Brillouin scattering.

 

Fig. 3. (a) Effective acoustic velocity versus acoustic mode number for different values of shape parameter $\alpha$, and (b) Brillouin frequency versus mode effective velocities for different values of shape parameter $\alpha$. The R-HTAN fiber parameters are maintained as $n_2$ = 1.444, $w_{GeO_2}$ = 34.6 $wt\%$, and $a =3 \mu$m

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Fig. 4. 2D mode profiles and field patterns (inset) of the six high acoustic modes: (a) L$_{0,1}$, (b) L$_{0,2}$, (c) L$_{0,3}$, (d) L$_{0,4}$, (e) L$_{0,5}$, and (f) L$_{0,6}$. Dashed dark lines indicate the refractive index profile and vertical dashed lines delineate the core region. The fiber parameters are maintained as $n_2$ = 1.444, $w_{GeO_2}$ = 34.6 $wt\%$, $a =3 \mu$m, and $\alpha =1$.

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4. Brillouin gain spectrum of the R-HTAN fiber

Brillouin scattering is defined as the interaction between optical modes and acoustic modes. The BG associated with each $j^{th}$ acoustic mode is calculated as $g_0/A_{ao}^{j}$, where $g_0$ is the BG coefficient and $A_{ao}^{j}$ is the acousto-optic effective area [15]. The latter evaluates the spatial overlap between acoustic mode and optical mode field distributions. The BG coefficient is expressed as follows [15]:

 

Fig. 5. Numerical simulations of the Brillouin gain spectrum of the designed R-HTAN fiber. The resulting multiple peaks were fitted using the Lorentz function. Dashed lines are individual peaks and solid dark lines are accumulated peaks (i.e., the spectrum). (a) The R-HTAN fiber at $\alpha$ = 0 and (b) the R-HTAN fiber at $\alpha$ = 1. The fiber parameters are maintained as $n_2$ = 1.444, $w_{GeO_2}$ = 34.6 wt%, and $a$ = 3 $\mu$m.

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For purposes of comparison, the number of Brillouin back-scattering peaks in the case of $\alpha$ = 0 is five, whereas the number is six in the case of $\alpha$ = 1 for the BGS of the R-HTAN fiber. This is explained by the fact that the acoustic mode $L_{0,6}$ is unexcited in case of $\alpha$ = 0 (i.e., cutoff), despite the large number of excited acoustic modes compared with those supported in the case of $\alpha$ = 1 (Fig. 3(a)). Based on the aforementioned analysis, the BGS of the R-HTAN fiber contains more peaks than that of recently investigated fibers, namely three in [15, 26, 28], which were equivalent to just three higher-order acoustic modes ($L_{\text {0,m}}$, m = 1, 2, and 3). To maximize the resulting BG, a few dominant acoustic modes seem desirable. However, there are cases where multiple peaks are highly desirable, such as in situations where more than two parameters are required to be monitored and discriminated.

Table 1 recapitulates several attributes of both BGSs in the designed R-HTAN fiber. Due to the distinct mode field distributions, P$_1$ had the highest BG compared with other peaks that emerged in the spectrum. This is because the acoustic mode L$_{0,1}$ (responsible for P$_1$) had the highest overlap with the optical mode LP$_{0,1}$. By contrast, higher-order acoustic modes ($L_{\text {0,m}}$ m = 2,…,6) were more oscillatory (visualized in Fig. 4), leading to a steady decrease in the overlap with the LP$_0,1$ mode and subsequently reducing the BG.

Table 1. Characteristics of the Brillouin gain spectrum (BGS) in the R-HTAN fiber for $\alpha$ = 0 and $\alpha$ = 1.

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The minimum BFS difference ($\delta \nu _b$) between two successive peaks was 281.9 MHz (P$_1$–P$_2$) in case of $\alpha$ = 0, whereas it was approximately 362.77 MHz (P$_5$–P$_6$) for $\alpha$ = 1. These values enhanced the distinguishability of the peaks and made them independently detected. In addition, the BG of the obtained peaks (e.g., peak P$_1$) was larger than those in recently proposed fibers, namely 0.1139 /mW in M-SMF and 0.2704 /mW in Step-SMF [15]. Considering the effect of the smoothness parameter $\alpha$ on the BG, $\alpha$ tended to decrease the BG of the two first peaks, whereas it slowly raised the BG of the other peaks. Moreover, $\alpha$ slowly shifted the BFS of each peak (except P$_5$) by tens of MHz to the right (i.e., the peaks moved toward to a higher frequency).

5. Discriminative temperature and strain sensing using the R-HTAN fiber

This section deals with the multiple peaks’ responses to temperature and strain variations. These traces are utilized in the monitoring of both environmental variables (i.e., temperature and strain). According to (2) and (3), any variation in strain or temperature automatically causes a change in the refractive index as well as in the longitudinal acoustic velocity profiles of an optical fiber. These changes lead to differences in the effective index of LP$_{0,1}$ and in the effective velocities of the supported longitudinal acoustic modes. In turn, these variations in strain or temperature lead to changes of the BGS. In principle, these changes have two types: (1) vertical changes, where the BG of peaks changes (i.e., peak amplitude), and (2) longitudinal changes, where the BFs of peaks (i.e., locations) shift toward higher frequencies. The strength of these shifts depends on the sensitivity of the fiber and/or the strength of the perturbation (i.e., high variation).

5.1 Brillouin gain variation in R-HTAN Fiber

Figure 6(a–d) highlights the impact of strain and temperature variations on the BG of each peak. Values for BG <10$^{-4}$ (/mW) were assumed to be too low to be detected in the spectrum. Thus, the associated acoustic mode was considered a cut mode. Considering the impact of strain (Fig. 6(a), 6(c)), the BG of P$_1$ clearly increased as the strain increased for both cases of shape parameter $\alpha$: from 1.05 /mW to 1.15 /mW for $\alpha$ = 0 and from 0.91 /mW to 1.05 /mW for $\alpha$ = 1, when the strain ranged from 0 to 1500 $\mu \varepsilon$. For all other peaks except P$_5$, the associated BGs slowly decreased across the strain (approximately 2.2 $\times$ 10$^{-7}$/mW/$\mu \varepsilon$ for $\alpha$ = 0 and 1.6 $\times$ 10$^{-7}$ /mW/$\mu \varepsilon$ for $\alpha$ = 1). The associated longitudinal acoustic mode (L$_0,5$) was cut at strain values exceeding 600 $\mu \varepsilon$ for $\alpha$ = 0, whereas it resisted until 900 $\mu \varepsilon$ in the case of $\alpha$ = 1. In addition, the extra peak for $\alpha$ = 1 (i.e., P$_6$) was cut at 150 $\mu \varepsilon$. These disparities were caused by the different sensitivities of supported acoustic modes due to their different inherent features (e.g., distinct mode field distributions). Considering the impact of temperature variation (Fig. 6(b), 6(d)), the BG of P$_1$ visibly increased as the temperature increased: from 1.01 /mW to 1.16 /mW for $\alpha$ = 0 and from 0.85 /mW to 1.09 /mW for $\alpha$ = 1 when the temperature increased from -20$^{\circ }$C to 100$^{\circ }$C. For all other peaks except P$_5$, their associated BGs slowly decreased across the temperature range (approximately 5.9 $\times$ 10$^{-6}$/mW/$^{\circ }$C for $\alpha$ = 0 and 5.3 $\times$ 10$^{-6}$/mW/$^{\circ }$C for $\alpha$ = 1). The associated longitudinal acoustic mode (L$_{0,5}$) was cut at a temperature before 20$^{\circ }$C for $\alpha$ = 0 and before 40$^{\circ }$C for $\alpha$ = 1, while P$_6$ was cut after 0$^{\circ }$C.

 

Fig. 6. Numerical simulations of Brillouin gain variation of the designed R-HTAN fiber versus strain and temperature: (a) and (b) for the R-HTAN fiber at $\alpha$ = 0, and (c) and (d) for the R-HTAN fiber at $\alpha$ = 1.

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5.2 Brillouin Gain frequency shift in R-HTAN fiber

As an arbitrary choice, Fig. 7 presents the numerical results of strain and temperature impacts on the BFS of the first peak (Fig. 7(a) and 7(b)) and on the BFS of the third peak (Fig. 7(c) and 7(d)) for the designed R-HTAN fiber at both $\alpha$ = 0 and $\alpha$ = 1. All peaks’ amplitudes were normalized to their maximum. From Fig. 7, one can see that as the applied temperature or strain increased, the BGSs of the peaks moved toward to a higher frequency. By tracking these frequency shifts (i.e., locations shifts) in the spectrum, it was possible to monitor the strain and temperature variations of the cable (i.e., the R-HTAN fiber).

 

Fig. 7. Numerical simulations of normalized Brillouin gain of the first and third peaks versus strain and temperature (Lorentz fitting): (a) and (c) for the R-HTAN fiber at $\alpha$ = 0, and (b) and (d) for the R-HTAN fiber at $\alpha$ = 1.

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5.3 Temperature and strain coefficients in R-HTAN Fiber

The influences of strain and temperature on the BFS (given in MHz) of each peak were numerically calculated and are depicted in Fig. 8(a) and 8(b) for $\alpha$ = 0 and in Fig. 8(c) and 8(d) for $\alpha$ = 1. All of the relations between the BFSs of peaks and both parameters (i.e., strain and temperature) exhibited excellent linear dependencies. The slope of each curve yielded the temperature or strain coefficient of each peak (equivalent to each $L_{0,m}$ mode). Thus, using linear fitting, the temperature and strain coefficients in case of $\alpha$ = 0 and $\alpha$ = 1 are highlighted in the same figure (Fig. 8). In principle, considering only two sensing parameters (e.g., temperature and strain), the sensing performance of the peaks in the Brillouin spectrum are given by the following dependencies [26]:

 

Fig. 8. Calculated Brillouin frequency shifts versus strain and temperature: (a) and (b) for the R-HTAN fiber at $\alpha$ = 0, and (c) and (d) for the R-HTAN fiber at $\alpha$ = 1.

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Fig. 9. (a) Variation of the strain coefficients of the R-HTAN fiber versus parameter $\alpha$. (b) Variation of the temperature coefficients of the R-HTAN fiber versus parameter $\alpha$.

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In principle, due to the large refractive index contrast at either the inner interface of the core (inner layer / ring layer) or at the border between the core and the cladding, high difference in acoustic properties will arise in these layers. The extreme coupling between shear and longitudinal acoustic modes at the layers interfaces causes this discrepancy in acoustic properties. In addition, the abrupt/jump of the refractive index ($\alpha$ = 0) between the center and the edge of the core as well as between the core and the cladding will create strong coupling between shear and longitudinal acoustic modes compared to the case of the smooth transition of the refractive index ($\alpha \neq$ 1).

Considering P$_1$, P$_2$, and P$_3$ (since they have the highest $C^{T}$ and $C^{\varepsilon }$) in our designed fiber, the temperature and strain coefficients for $\alpha$ = 0 were $C^{T}_{P1}$ = 1.922 MHz/$^\circ$C, $C^{T}_{P2}$ = 1.928 MHz/$^\circ$C, $C^{T}_{P3}$ = 1.917 MHz/$^\circ$C, $C^{\varepsilon }_{P1}$ = 0.0863 MHz/$\mu \varepsilon$, $C^{\varepsilon }_{P2}$ = 0.087 MHz/$\mu \varepsilon$, and $C^{\varepsilon }_{P3}$ = 0.0877 MHz/$\mu \varepsilon$, respectively. Considering $\alpha$ = 1, the temperature and strain coefficients were $C^{T}_{P1}$ = 1.872 MHz/$^\circ$C, $C^{T}_{P2}$ = 1.782 [MHz/$^\circ$C, $C^{T}_{P3}$ = 1.623 MHz/$^\circ$C, $C^{\varepsilon }_{P1}$ = 0.0842 MHz/$\mu \varepsilon$, $C^{\varepsilon }_{P2}$ = 0.0808 MHz/$\mu \varepsilon$, and $C^{\varepsilon }_{P3}$ = 0.075 MHz/$\mu \varepsilon$, respectively. To the best of our knowledge, the obtained temperature and strain coefficients outperform those demonstrated in recently designed fibers (see Table 2).

Table 2. Temperature and strain coefficients of the designed R-HTAN fiber for $\alpha$ = 0 and $\alpha$ = 1 compared with the coefficients of other types of fibers. The maximum achievable $C^{T}_{i,j}$ and $C^{\varepsilon }_{i,j}$ were selected.

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Although the case of $\alpha =1$ is corresponding to a graded index RCF, the obtained temperature and strain coefficients outperform those in recently reported fibers. This is attributed to the high index contrast (inner/outer core, core/cladding), which tends to subsequently change acoustic properties at different interfaces. In addition, to exploit the multipeak feature yielded by the designed R-HTAN fiber, it is possible to utilize all of the spectrum peaks in multiparameter sensing. By selecting more than two peaks in the Brillouin spectrum, the designed R-HTAN fiber would potentially be applicable in situations where more than two environmental parameters are required to be discriminated. Moreover, the harnessing of independent optical parameters rather than the BFS only (such as the BFS and the Birefringence in [18]) in order to get independent responses, could be an alternative solution to discriminate the effects of the temperature and the strain. This could be a viable solution especially in case of low difference between the temperature and the strain coefficients (i.e. high measurement errors).

6. Errors measurements and peaks selection

The errors associated with the measurement process using optical fiber sensors are considered potential challenges due to their influence on the credibility and trust of both the sensor and the measurement operation equally. Therefore, by using two peaks $i$ and $j$ from the BGS, the error analysis for the measurement of temperature and strain is expressed as follows [26]:

We consider all peaks combinations since all peaks have large Brillouin gain (BG $\geq$ 0.001 /mW, according to Fig. 5) to be detected in the spectrum. Table 3(a-d) highlights errors measurements of temperature and strain for all peaks combinations (10 combinations ($\alpha$= 0) and 15 combinations ($\alpha$= 1)) in case of $\alpha$= 0 (a, b) and $\alpha$= 1 (c, d), respectively.

Table 3. (a) Temperature errors for all peaks combinations for $\alpha =0$, (b) strain errors for all peaks combinations for $\alpha =0$, (c) Temperature errors for all peaks combinations for $\alpha =1$, (d) strain errors for all peaks combinations for $\alpha =1$.

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Because, peaks P$_{4}$, P$_{5}$ and P$_{6}$ are cut at varying either the temperature or the strain (Table 3), and are not detectable at the entire ranges of both parameters, we don’t take them into consideration (although they observe low errors measurements). Table 3 reveals that the respective errors increase with the smoothness parameter $\alpha$. Eventhought, in some cases the event of cutting a peak in the BG spectrumm could be a sensing information. This may give preference to $\alpha$=1 against $\alpha$= 0 since the case of $\alpha$=1 has much acoustic modes that are stimulated and cut. In addition, considering distributed sensing realm, bend loss would be a potential issue of system stability in case of $\alpha$=0.

If we consider P$_{1}$ and P$_{3}$ (P$_{1}$ and P$_{3}$ have the highest BG which tend to eliminate the influence of noise and enhance the measurement accuracy) in the temperature and strain sensing, the calculated accuracies in simultaneous measurements, are 0.045$^\circ$C and 0.16 $\mu \varepsilon$ for $\alpha$=0 and 0.25$^\circ$C and 6.6517 $\mu \varepsilon$ for $\alpha$=1. On the other side, $P_{2}$ has higher sensitivity compared to $P_{3}$ (except for the strain at $\alpha$=0), selecting P$_{1}$ and P$_{2}$ would be justifiable. The respective calculated accuracies of temperature and strain in simultaneous measurements are 0.0846$^\circ$C and 0.7250 for $\alpha$= 0 and 0.2803$^\circ$C and 7.4184$\mu \varepsilon$ for $\alpha$= 1. Considering the case of P$_{2}$-P$_{3}$ combination (P$_{2}$ and P$_{3}$ have high BG), the respective errors are 0.03$^\circ$C and 0.477$\mu \varepsilon$ for $\alpha$= 0 and 0.23$^\circ$C and 6.3317$\mu \varepsilon$ for $\alpha$= 1. As discussed above, a large discrepancy between temperature and strain coefficients tends to obtain low measurement errors. Taking into consideration the requirement of high BG, in our work, P$_{1}$-P$_{3}$ offers optimal discrimination capability.

The obtained measurements errors outperform those reported in M-shape FMF [15] (0.47$^\circ$C and 12.3$\mu \varepsilon$ ), in IPGI-FMF [26] (0.85$^\circ$C and 17.4$\mu \varepsilon$ ), and in OAM-fiber [28] (0.93$^\circ$C and 18.2$\mu \varepsilon$).

At last, in view of the direct influence of the manufacturing process on the fiber structure, especially on the fluctuation of the shape parameter $\alpha$ and on the sensing performances (i.e., temperature and strain coefficients and measurements errors), it is inevitable that some manufacturing errors will arise. In this study, it is worthy to discuss that we took into consideration the two limits of the shape parameter $\alpha$ (0 and 1). This helps to estimate the fabricating error effects on fiber performances especially on the discrimination capability of the designed fiber. Both selected values of $\alpha$ prove the outperformance of the proposed R-HTAN fiber compared to state-of-the -art fibers. Hence, the designed fiber could be a promising candidate for next-generation Brillouin sensing systems, enabling temperature and strain discrimination.

7. Conclusion

In summary, we proposed and numerically demonstrated a novel multiparameter Brillouin sensor based on the R-HTAN fiber. The R-HTAN fiber has a shape factor ($\alpha$) that controls the optical refractive index and longitudinal acoustic velocity profiles. Using the finite-element method, the optical and longitudinal acoustic modes were investigated through numerical modal analysis. The BGS of the designed fiber was numerically characterized and revealed the presence of multiple separate peaks (min $\Delta \nu _b$ = 281.90 MHz), which originated from the interaction of the LP$_{0,1}$ optical mode and various longitudinal acoustic modes. Furthermore, by analyzing the BFS of two peaks, across the temperature and strain variations, the responses of the designed R-HTAN fiber against both parameters were yielded for different values of the shape parameter $\alpha$. The achieved temperature and strain coefficients outperformed those obtained in the recent specialty fiber-based Brillouin sensors. In addition, the analysis of temperature and strain measurement errors produced results that were consistent with the literature. Thus, the proposed fiber could be a promising candidate for next-generation Brillouin sensing systems, enabling temperature and strain discrimination.

In future work, we aim to deeply examine the new R-HTAN fiber, including the Brillouin scattering of higher optical or vector modes (i.e., LP$_{1,1}$ as a pump), the effects of further profile parameters on BGs and BFSs, and the fiber’s capability in distributed sensing. The expected results of such investigations may motivate the industry to prototype the proposed R-HTAN sensor.