A New Method to Design Trifurcated Optical Fiber Displacement Sensors

1532 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024 A New Method to Design Trifurcated Optical Fiber Displacement Sensors Gorka Zubia , Graduate Student Member, IEEE, Joseba Zubia , Member, IEEE, Josu Amorebieta , Gotzon Aldabaldetreku , and Gaizka Durana Abstract—Trifurcated optical fiber displacement sensors (OFDSs) are increasingly used in industrial and aerospace applications. The critical element of the sensor is the fiber bundle. However, there is no straightforward method to calculate it from the working specifications of the sensor, the number of fibers needed, their size, and arrangement. This work presents a simple yet accurate method to design trifurcated OFDSs. The proposed method allows to derive the geometrical arrangement and size of the fibers from three simple equations, thus reducing significantly the difficulty and complexity of the OFDS design. Those three equations depend on the working point, working range, required sensitivity, and maximum size of the bundle. In this way, the proposed method will save valuable time for researchers and engineers who wish to design, fabricate, and use this type of OFDSs. The procedure is explained in detail with two examples. The results predicted by the model are compared with the experimental results of a bundle with identical fiber arrangement and dimensions. The results show good agreement with a deviation of less than 1% in the working range of the sensor. Index Terms—Design, fiber bundle, fiber optic sensors, intensity modulated optical sensor, method, optical fiber displacement sensor (OFDS), trifurcated. I. INTRODUCTION HIGH precision metrology is essential in Industry 4.0 to monitor and improve the efficiency and versatility of factory processes [1],[2]. As a result, distance sensors are used in a wide range of applications to measure several parameters of interest such as the thickness, height, roughness, deformaManuscript received 2 October 2023; revised 31 October 2023; accepted 25 November 2023. Date of publication 4 December 2023; date of current version 12 January 2024. This work was supported in part by the Grant I+D+i/PID2021-122505OB-C31, Grant TED2021129959B-C21, Grant PDC2022-133053-C21, and Grant RTC2019007194-4 through MCIN/AEI/10.13039/501100011033; in part by the “ERDF A way of making Europe”; in part by the “European Union Next Generation EU/PRTR”; in part by the Grant IT11452-22 and through the Basque Government; in part by the ELKARTEK 2023 under Grant µ4Smart-KK-2023/00016 and Grant Ekohegaz II-KK-2023/00051; and in part by the University of the Basque Country (UPV/EHU) Translight. The work of Gorka Zubia was supported by the Ph.D. fellowship from the Basque Government under Grant PRE_2022_2_0269. The associate editor coordinating the review of this article and approving it for publication was Dr. Sanjeev Raghuwanshi. (Corresponding author: Gorka Zubia.) Gorka Zubia and Josu Amorebieta are with the Department of Applied Mathematics, University of the Basque Country UPV/EHU, 48013 Bilbao, Spain (e-mail: [email protected]). Joseba Zubia, Gotzon Aldabaldetreku, and Gaizka Durana are with the Department of Communications Engineering, University of the Basque Country UPV/EHU, 48013 Bilbao, Spain. Digital Object Identifier 10.1109/JSEN.2023.3337311 tion, distortion, and gap between surfaces, for instance, [3], [4],[5]. To select the best-fitting device, we have to consider the application, the environment, and the required precision, as these factors are critical to determining the specifications of the sensor. A prime example of the latter can be acknowledged in the aeronautical turbine industry. In that field, noncontact solutions are practically compulsory in order to carry out any metrology measurements, as heavy metallic pieces are rotating at high speed significantly close to each other [6]. To that end, the most popular alternatives are capacitive [7], inductive [8], ultrasonic [9], draw-wire, and optical sensors [10]. Among them, optical fiber displacement sensors (OFDSs) offer many advantages such as electromagnetic immunity, high response, speed, small size, great versatility, and easy installation, for example, [11],[12]. Within this group of sensors, OFDSs based on the reflection of optical intensity add two significant advantages: Firstly, as an optical fiber is used to transmit light to the monitored surface, the sensing heads can be installed in remote locations where direct laser light cannot reach, such as inside an aircraft turbine [13]. Secondly, since these sensors are often used in a differential configuration [14], they minimize several potential error sources due to reflective surface roughness, temperature variations, fluctuations in the emitting light source, and so on [15]. This approach is particularly useful for dynamic © 2023 The Authors. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ ZUBIA et al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1533 turbine testing, as the blades rotate and the reflective surface is constantly changing [16],[17]. The working range and sensitivity of OFDSs depend on their geometric structure, i.e., the number and transverse arrangement of fibers in a bundle [18],[19],[20],[21]. The typical OFDS configuration uses two adjacent optical fibers to transmit and receive the light signal through reflection [22]. However, this arrangement limits the amount of reflected light, which reduces the working range. Other configurations have been proposed, e.g., a pair of bent-tip optical fibers [23], a hemispherical arrangement of fibers, and even randomly distributed transmitting and receiving fibers [24],[25], or more recently, an OFDS based on a multicore fiber with seven cores hexagonally arranged [26]. Some authors prefer to use fibers or bundles that form an angle with one another and with the normal to the reflecting surface [27]. It has been found that setting the angle between the transmitting and receiving fibers to 20◦ increases the sensitivity up to 30 times over the conventional parallel fiber configuration. Asymmetric approaches like this are well-suited for angle measurement. For example, Sagrario and Mead [28] designed an OFDS using a square structure to perform both axial and angular displacement measurements; Khiat et al. [29] developed a fiber sensor with four receiving fibers around the transmitting one, for long-distance, highresolution measurements. The drawback of such fiber bundles is that the sensor head increases proportionally with the angle. Additionally, many of these designs only allow the measurement of uniaxial rotation, that is, the ability to measure the tilt of the reflecting surface along just one axis [30]. In cases where the reflecting surface is neither homogeneous nor large and also vibrates, the angle is not maintained and changes rapidly. This applies to the measurement of tip clearance in aircraft turbines, where the reflecting surfaces are the blades and whose shape is twisted [31]. In these harsh environments, the angle measurement is not advantageous. Therefore, the designs used to measure displacements in turbines are based on azimuthally symmetric arrangements, specifically, bundles of fibers arranged in concentric rings around a central emitting fiber. The most common designs are the trifurcated OFDSs [32],[33]. These are based on a fiber bundle with an emitting fiber in the center surrounded by two fiber rings at different radial distances. Typically, fiber bundles are custom-designed to meet the working range or sensitivity required by the specific application [34],[35]. Those parameters vary based on the number of fibers in each receiving ring, the size and numerical aperture of each fiber, or the radii of the fiber rings [36]. Various methods have been proposed to model the response of OFDSs: Geometric approximations [25],[37], ray tracing, Monte Carlo calculations [38] or Gaussian beambased models. Of all these approaches, the one presented by Cao et al. [25] offers the widest range of possibilities, albeit it faces the drawback of requiring the implementation of 8 different formulas. The one based on a Gaussian or quasi-Gaussian beam has also excellent accuracy, but it is much simpler [39]. Anyway, designing the most suitable fiber bundle is a complex numerical task [25],[27],[38],[39] that the proposed approach in this article can greatly simplify. Fig. 1. Schematic of an OFDS. The constituent blocks are a lightemitting source, a fiber bundle to guide the light, and a receiver circuit to convert the collected light power into voltage. Unlike prior methods, the method introduced here allows us to compute the bundle design that best fits the working specifications. The proposed method enables the deduction of the geometrical arrangement and fiber size within a bundle from three simple equations, thereby significantly reducing the difficulty and complexity of OFDS design, which can be time-consuming and expensive. The three equations only depend on the specifications defined at each application, namely, the working point and range, the required sensitivity, and the maximum size of the bundle. As a result, the proposed method will save valuable time for researchers and engineers who wish to design, fabricate, and use this type of OFDSs. This method can trivially be extended to designs with four (tetrafurcated), five (pentafurcated), or more fiber rings. The only limitation of the method is that it is restricted to the design of azimuthally symmetric bundles. The article is organized into four sections. Firstly, the mathematical model is presented. Secondly, the results obtained are analyzed. Thirdly, the design equations are derived. Finally, the model is validated by comparing the theoretical results with experimental ones. II. MATHEMATICAL MODEL The schematic of an OFDS is shown in Fig. 1. It comprises a light source, a fiber bundle, and a receiver. The light emitted by the light source is guided through the transmitting fiber to the reflecting surface. After reflection, the light is collected by the receiving fibers of the bundle. Each of these is connected to separate photodiodes, where the optical power is finally converted into voltage. The latter is a function of the distance between the bundle-tip and the reflective surface. The design of the bundle is critical, since the response of the OFDS is highly dependent on its geometric arrangement, i.e., the number, size, and distribution of the transmitting and receiving fibers [15], [20],[22],[27],[31]. The aim of this work is to develop a novel approach to design a trifurcated OFDS, taking the desired working point and range as input variables. For this purpose, we have built a toy model of the response of the sensor to understand the influence of each of the geometric parameters in order to fine-tune its response and meet the requirements of the specific application. 1534 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024 Fig. 2. Structure of a trifurcated fiber bundle with radius R, composed of a transmitting fiber of radius rTlocated at the center, and two surrounding coaxial receiving rings. The positions of the rings are denoted as {ρ1, ρ2}, respectively. Their widths are labeled {∆ρ1,∆ρ2}. The diameters of the fibers {φ1, φ2}correspond to the widths of the rings in the toy model, and their position equal the position of the fibers {R1,R2}. Fig. 2illustrates the geometry of the problem. It consists of a circular structure composed of a transmitting fiber located at the center of the bundle and two coaxial rings of receiving fibers surrounding it. The toy model replaces the discrete fiber rings with continuous homogeneous rings. This greatly simplifies the calculations while preserving all the bundle geometry parameters. The positions of the rings and their widths are ρ1,ρ2,1ρ1, and 1ρ2, respectively. The diameters of the fibers {φ1, φ2} correspond to the widths of the rings in the toy model, and their position equals the positions of the fibers {R1,R2}. Other parameters are the radii of the transmitting fiber rTand the bundle R. The validity of the toy model is confirmed in Section V. We assume that the transmitting fiber is a single mode to reduce the modal noise at the output of the bundle [13]. Under this premise, we can approximate the irradiance at the output of the fiber as a Gaussian beam, using [40] I(ρ, z)=I0w0 w(z)2 exp−2ρ2 w2(z) =2P πw2(z)exp−2ρ2 w2(z)(1) where w(z)is the beamwidth, ρis the radial distance from the propagation axis zand I0is the intensity of the beam at the origin. Prefers to the total optical power transmitted by the beam and w0is the waist radius at its stretchiest point-where the intensity drops to 1/e2≈13.5% of its maximum value. The beamwidth w(z)spreads according to the formula w(z)=w0s1+z z02 .(2) It reaches its minimum value w0at the bundle tip (z=0). z0is the Rayleigh distance, defined as z0=πw2 0/λ, which is the distance over which a beam can propagate without Fig. 3. Evolution of the light intensity of a Gaussian beam with z, taking w0=2.4 µm and z0=27 µm. significantly diverging. The total light power emitted by the transmitting fiber, P, can easily be estimated as P=Z∞ 0 I(ρ)2πr dr =2πI0w0 w(z)2Z∞ 0 exp−2ρ2 w2(z)ρdρ P=π 2w2 0I0.(3) Since the light power remains constant at different zdistances from the transmitting fiber-tip, both the peak intensity and the intensity distribution profile vary with z. In fact, on the beam axis, the intensity I(0,z)drops to I(0,z)=I0 z2 0 z2+z2 0 .(4) Fig. 3shows the evolution of I(ρ, z)with z. For sufficiently large distances z≫z0H⇒ w(z)≈w0 z z0 w(z)≈ztan θ0=ztan(arcsin NA)(5) being NA the numerical aperture of the transmitting fiber. The divergence angle θ0and w0are parameters that solely rely on the light source and transmitting fiber characteristics. Thus, once both are selected, θ0and w0remain constant. At a certain distance of z, the light beam reaches a surface, reflects toward the receiver rings, and is collected after traveling a total distance of 2z. The intensity at an infinitesimal ring of width dρplaced at a distance ρfrom the fiber axis is d I (ρ, 2z)=20P πw2(2z)exp−2ρ2 w2(2z)2πρ dρ. (6) The ideal reflectance of the surface is 0. Thus, the calculation of the detected irradiance in the ring located at a radial distance ZUBIA et al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1535 ρ1with width 21ρ1is straightforward I(ρ1, 1ρ1,2z)=20P πw2(2z)Zρ1+1ρ1 ρ1−1ρ1 exp−2ρ2 w2(2z)2πρ dρ I(ρ1, 1ρ1,2z)=20Psinh4ρ11ρ1 w2(2z)exp−2ρ2 1+1ρ2 1 w2(2z). (7) Similarly, the measured irradiance in a second ring located at a distance ρ2and width 21ρ2is I(ρ2, 1ρ2,2z)=20Psinh4ρ21ρ2 w2(2z)exp−2ρ2 2+1ρ2 2 w2(2z). (8) Assuming that the two photodetectors collecting the light entering the two independent receiving rings have a typical quadratic response [6] Vi=ki20Psinh4ρi1ρi w2(2z)exp−2ρ2 i+1ρ2 i w2(2z)(9) where Viis the output voltage of photodetector i, and kiis a constant that includes the photodetector response and other factors such as noise, detector circuit gain, surface roughness, contamination, bends, etc. Consequently, the responsivity of the sensor, η(z), is defined as the ratio of the optical intensities of its two receiving rings, i.e., V2/V1 η(z)= k2sinh4ρ21ρ2 w2(2z)exp−2ρ2 2+1ρ2 2 w2(2z) k1sinh4ρ11ρ1 w2(2z)exp−2ρ2 1+1ρ2 1 w2(2z) η(z)=k sinh4ρ21ρ2 w2(2z) sinh4ρ11ρ1 w2(2z)exp−2ρ2 2+1ρ2 2−ρ2 1−1ρ2 1 w2(2z). (10) The constant kis defined as k=k2/k1. In the following (11), we assume that the response of both photodetectors is the same, so k=1. Although the responsivity of the OFDS depends on four geometric parameters, we encompass these into just three: {A1,A2,q}. Thereby, (10) with (5) compacts to η(z)= sinhA2 z2tan2θ0 sinhA1 z2tan2θ0exp−q z2(11) where          A1=ρ11ρ1 A2=ρ21ρ2 q=ρ2 2+1ρ2 2−ρ2 1−1ρ2 1/2 tan2θ0 w(z)≈ztan θ0      Geometrical parameters. In Fig. 4, the responsivity is plotted along with the optical power collected by the inner and outer rings, P1(z)and P2(z), respectively. We applied a scaling factor of 2.5 to P1(z)and P2(z)to visualize the three functions together. Still, one can observe that the response of each ring is significantly smaller than the combined output. Furthermore, according to Fig. 4, the inner ring (red curve) demonstrates excellent performance at short distances due to a narrow but steep linear range, which Fig. 4. Optical power collected by the inner-red, P1(z)-and outergreen, P2(z)-rings along with the responsivity-blue, η(z). A scale factor of 2.5 was applied to P1(z) and P2(z) to display the three curves together. The table shows the linear ranges and their slopes. In yellow, some typical points of interest of η(z) in mm: zTP =9.3, zHR =12, and zHM =13.7. makes the sensor highly sensitive. Conversely, the outer ring (green curve), has a more extensive linear range at the cost of a loss of sensitivity, i.e., a slower growth of its response. This is because the received light is broadened and varies radially smoother. For the same reason, the outer ring receives less power, in this case, five times less than the inner ring. However, we can also observe that taking the ratio between both curves results in a significantly improved response (blue curve). It is both greater in amplitude and larger in linear range. Thus, using a single-ring configuration has the only advantage of enabling shorter distance measurements. Nevertheless, as we have seen, working with a single ring has many disadvantages, mostly related to the inability to correct the emitting light source intensity fluctuations, ambient light, etc. Just as an example, if we define the linear range as the interval between 5% and 65% of the maximum value of the front slope of each curve, we obtain a linear range of 1z5,65 =1.13 mm when working with the inner ring, and 1z5,65 =3.08 mm for the outer. Finally, we can observe that the linear range of the responsivity-which combines the optical power of both rings-extends up to 1z5,65 =10.77 mm. The typical responsivity of the sensor is also presented in Fig. 4. It saturates at the limit value, when the distance to the reflector is sufficiently large lim z→∞η(z)= sinhA2 z2tan2θ0 sinhA1 z2tan2θ0exp−q z2=A2 A1=ρ21ρ2 ρ11ρ1 . (12) For convenience, let us refer to this limit as p p=lim z→∞η(z)=ρ21ρ2 ρ11ρ1 .(13) According to (13), the saturation value of the sensor increases with the ratio of the receiving areas, A2/A1. Fig. 5illustrates how each term of the responsivity function contributes to the result (11), revealing that the hyperbolic sine term (11) in orange is significant only for small z values. For larger zvalues, it just multiplies the exponential by the limit p. In fact, the responsivity given by (11) 1536 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024 Fig. 5. Sensor response η(z) in blue and S(z) in black for the following design specifications (mm): ρ1=0.5, ρ2=1.5, ∆ρ2=0.1, and ∆ρ2= 0.15. Also, θ0=5◦,q=130 mm2, and k=1. Contribution of the hyperbolic sines in orange; in red, the exponential one. is indistinguishable from η(z)=I(ρ2, 1ρ2,2z) I(ρ1, 1ρ1,2z)≈pexp−q z2.(14) A dashed vertical line has been added in Fig. 5to indicate where the responsivity (blue line) is no longer zero. From that point on, the hyperbolic-sine-term barely changes and resembles a constant. Another key parameter of the sensor is the sensitivity, §, defined as the slope of the responsivity S(z)=dη(z) dz =2pq z3exp−q z2.(15) The sensitivity is maximum when the second derivative equals zero, i.e., solving for the turning point,zTP, of the responsivity Smax =d2η(z) d2zz=zTP =0H⇒ zTP =r2 3q.(16) Thus, the value of the responsivity (14) at that point is η(zTP)=pexp−q z2 TP ≈0.22p.(17) An additional key point is the distance at which the responsivity reaches half its maximum value. We named it zHM η(zHM)=1 2p.(18) Solving for zHM yields to zHM =rq ln 2 ≈1.2√q.(19) Moreover, substituting (16) in (19) revealed that the relationship between zTP and zHM can be expressed as zHM =r3 2 ln 2 zTP =1.47zTP.(20) In previous Fig. 4, we plotted the responsivity along with the proposed key parameters, namely: zTP,zHM, and zHR. The latter is the distance point at half of the working range of the sensor. We assumed that the working range of the sensor is between the [5,65]% of ηmax, which corresponds to the most linear region of the responsivity. So, zHR will be at 35% of the maximum responsivity ηmax. As we will see in Section V, the working range may also be defined by fitting the responsivity to a straight line using Pearson’s correlation coefficient. III. RESULTS AND DISCUSSION After introducing the mathematical model in Section II, the next step is to analyze the results. Equations (11) and (14) characterize the responsivity of the sensor. Notice that it depends on the geometric parameters through two expressions ρ2 2+1ρ2 2−ρ2 1−1ρ2 1and ρ21ρ2 ρ11ρ1 . The roles of {ρ2, 1ρ2}and {ρ1, 1ρ1}are symmetrical with respect to the responsivity. Also, changes in {ρ2, 1ρ2}are equivalent, and the same is true for {ρ1, 1ρ1}. Therefore, we conducted our initial analysis of the responsivity through global changes in qand p. The dependence of the responsivity on those parameters is better exemplified in Fig. 6. Fig. 6(a) shows the responsivity as a function of qand z for different values of p, from 1 to 5 in steps of 1. The green surface corresponds to p=5, and the blue at the bottom corresponds to p=1. Inspecting Fig. 6(a) leads us to two outcomes. 1) First, the shape of the curve does not change as the value of pincreases. This implies that the position of the working point is independent of p, as evidenced in (16) and (19). However, it does change the scale and slope of the curve and, therefore, the sensitivity in the neighborhood of the working point. 2) Second, the analysis of the responsivity reveals two simultaneous effects: on the one hand, the reduction of the parameter qshifts the turning point to smaller values; thus if we want to work at short distances, the parameter qmust be small. This fact is supported by (16). On the other hand, it shows that this decrement of qis associated with an increment in the slope of the responsivity; thus, the sensitivity and qare inversely proportional. When one increases, the other decreases. In short, qis a trade-off parameter between the position of the working point and the sensitivity of the sensor q↓ H⇒ zTP ↓and q↓ H⇒ Smax ↑. As previously stated, the working range is defined as 1z5,65 =z65 −z5, which equals to 0.94√q. This can easily be deduced by following the same procedure that led to (16) and (20). Thus, a larger working range of the sensor implies a larger value of q; which, in turn, decreases the sensitivity. In conclusion, the range of distances at which the responsivity is linear is inversely proportional to the sensitivity q↑ H⇒ 1z5,65 ↑so, 1z5,65 ↑ H⇒ S(z)↓. This loss of sensitivity can be corrected, at least partially, in two ways. a) Externally, increasing the gain of the photodetectors. b) Increasing pas it is proportional to the sensitivity (15). ZUBIA et al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1537 Fig. 6. Responsivity as a function of qand p.(a) Responsivity as a function of {q,z}for different p, from p=1 to 5 in steps of 1. The top green surface corresponds to p=5 and the blue at the bottom to p=1. (b) Responsivity as a function of pand zfor different q, from q=0.1 mm2to 10.1 mm2in steps of 2 mm2. The top blue surface corresponds to q=0.1 mm2and the cyan at the bottom to q=10.1 mm2. The red markers indicate the same point. The maximum and minimum values of pare given by the size of the bundle, R≥ρ2+1ρ2, and the size of the transmitting fiber, rT≤ρ1−1ρ1. For that purpose, to increase p, the outer receiving ring should either be wider, located further away, or both. From (13), widening the area of the outer ring or reducing the area of the inner ring leads to a larger p. This is possible while leaving qunchanged since they are independent. The only practical limitations of this procedure are the maximum gain of the photodetectors [ki,(10)] and the maximum ratio of the areas of the receiving rings p=A2/A1, which, in summary, depend on both the size of the bundle Rand the size of the emitting fiber rT. Identical conclusions can be drawn from Fig. 6(b), which shows the sensitivity as a function of pand zfor different values of q. In summary, to achieve a wider working range and higher sensitivity, we need to increase {p,q}in the sensor design. Moreover, a higher responsivity can be obtained by increasing p, see (13). In a second analysis, we conducted a set of simulations to understand better the influence of each parameter by varying one while the rest remained fixed. Fig. 7shows the results of the toy model for the responsivity η(z)and sensitivity, S(z). The reference parameters are ρ1=0.5 mm ρ2=1.0 mm 1ρ1=0.2 mm 1ρ2=0.2 mm θ0=5◦. A. Variation of the Position of the Inner Ring ρ1 This variation is equivalent to changing the position of the inner ring fibers. Looking at Fig. 7(a), we can hardly observe any change in the position of the sensitivity. This is because the range of values over which it can vary is small since the rest of the bundle limits the inner ring. However, the responsivity value drops sharply when we increase the value of ρ1, bringing the two rings closer. This decrease of η(z)is accompanied by a substantial drop in S(z), see Fig. 7(b), and a variation of its maximum toward smaller values of z. The maximum value of the sensitivity is given at zTP distance, Smax =S(zTP), substituting (16) in (15) S(zTP)=3√3 exp−3 2ρ21ρ2tan θ0 ρ11ρ1qρ2 2+1ρ2 2−ρ2 1−1ρ2 1 .(21) Also, the working range decreases slightly with an increase of ρ1. So, as predicted by (16) and (21) ρ1↑ H⇒ zTP ↓,S(zTP)↓, 1z5,65 ↓. B. Variation of the Inner Ring Width 1ρ1 The responsivity value is very sensitive to the width of the inner ring, as shown in Fig. 7(c). Increasing the width of the inner ring causes a significant decrease in the responsivity. When 1ρ1goes from [0.1,3]mm, the responsivity divides by four. Fig. 7(d) depicts that the same is true for the sensitivity, which becomes three times smaller. In short, the responsivity decreases and flattens when the radius of the inner fibers increases. The position of S(zTP), though, remains constant, meaning that qis less sensitive to changes in 1ρ1. Likewise, the working range is independent of 1ρ1. Summarizing 1ρ1↑ H⇒ η(z)↓,S(z)↓. C. Variation of the Position of the Outer Radius ρ2 If we modify the outer radius of the fiber ring, increasing the distance ρ2, as shown in Fig. 7(e), we observe two effects on the responsivity. On the one hand, the curve shifts toward larger values of z; on the other hand, both its amplitude and working range increase. However, based on Fig. 7(f), there is a small decrease in sensitivity as ρ2increases. The position of its maximum value zTP grows linearly with ρ2. So ρ2↑ H⇒ q↑,p↑, η(z)↑,zTP ↑S(z)↓. This can be understood from (13) and (16). An increase of ρ2raises the value of qand p, and, therefore, zTP. Hence, the maximum value of the responsivity is also augmented. (21) explains the slight decay of sensitivity as ρ2increases. 1538 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024 Fig. 7. All the values from the legends are in millimeters. Vertical lines show the zTP points for every curve. (a) through (j) show the responsivity and sensitivity for five different geometrical parameter configurations: {ρ1,∆ρ1, ρ2,∆ρ2, θ0}. Data were normalized with respect to the maximum value. D. Variation of the Outer Ring Width 1ρ2 If we now look at the variation of the sensor response with the width of the outer ring, Fig. 7(g) and (h), we observe the opposite behavior. This is due to the minus sign in the qfactor. By increasing the value of 1ρ2in the range [0.1,3]mm, we triple the value of both responsivity and sensitivity. As with 1ρ1(Section III-B), the position of the turning point zTP hardly varies and can be considered constant for ZUBIA et al.: NEW METHOD TO DESIGN TRIFURCATED OPTICAL FIBER DISPLACEMENT SENSORS 1539 TABLE I LIST OF POSSIBLE WORKING POINTS OF THE SENSOR TOGETHER WITH THE VALUE OF THEIR RESPECTIVE RESPONSIVITIES this 1ρ2range. The same happens with the working range. Then 1ρ2↑ H⇒ η(z)↑,S(z)↑. E. Variation of the Numerical Aperture NA =sin(θ0) The last parameter to vary is the numerical aperture of the source. In Fig. 7(i) and (j) we vary the NA between [0.1,3], which corresponds to the acceptance angles θ0= [5,15]◦. If we increase the NA, the responsivity increases in slope and shifts toward smaller zvalues, see Fig. 7(j). Its value is almost quadrupled and the position of the maximum, i.e., the value of zTP, is halved. It is noteworthy the huge dependence of the working range on NA; the working range is indeed divided by three. Consequently NA ↑ H⇒ η(z)↑,zTP ↓,S(z)↑. F. Dead Zone Some authors define the dead [39] or blind [20] zone of an OFDS as the range of distances in which the responsivity is very small or zero. In Table Iwe show that the value z5 for which the responsivity is 5% of its maximum value is z5=0.58√q. This parameter indicates the point at which the responsivity starts to rise. It is evident that, in order to minimize the dead zone, qmust be reduced. As previously discussed, there are two approaches to address this issue: either by moving the receiver fiber rings, increasing ρ1, and decreasing ρ2; or by increasing the NA of the source. This is easy to understand in both cases; if we decrease the distance between the fiber rings to the center of the bundle, the reflected light reaches the two rings significantly due to their proximity to the transmitting fibers. Conversely, increasing the NA causes the light to scatter rapidly, reaching the receiving rings even at a short distance. This can easily be observed for the case of two equal fiber rings 1ρ1=1ρ2 q=ρ2 2−ρ2 1 2 tan2θ0=(ρ2−ρ1)(ρ2+ρ1) 2 tan2(arcsin NA).(22) So NA ↑∨(ρ2−ρ1)↓∨(ρ2+ρ1)↑ H⇒ Dead zone ↓. The individual inspection of the parameters yields the following design tips. 1) To operate at small distances, we should either increase the NA of the source, bring the fiber rings closer, or decrease the fiber rings to bundle center distance. 2) Distances zTP,zHM,and zHR give us information about the position of the responsivity curve (see Table I). 3) Sensitivity can be improved by decreasing the width of the inner ring (the radius of the inner fibers), or equivalently, increasing the width of the outer ring (the radius of the outer fibers). 4) The position working point is independent of the width of the rings, i.e., the diameter of the fibers. IV. DESIGN PROCEDURE Although the fiber bundle sensor has a simple principle and many outstanding advantages, most of the research has focused on the model and calculation of the response of a specific bundle, i.e., modeling the response of the bundle knowing a priori its geometric parameters and configuration [13],[14],[15], [16],[17],[18]. However, there is no research on the design of such bundles from given specifications. In other words, no procedure that relates the target distance and the working range to the fiber parameters and bundle configuration. The ultimate goal is to determine {ρ1, ρ2, 1ρ1, 1ρ2}from the design requirements, which are: the size of the bundle (outer radius R) and the working point, zWP, the sensitivity, S(z), and working range, 1z, of the OFDS Input     R zWP S(zWP) 1z     | {z } Design requirements −→    Solve design equations   −→ Output Toy model Experimental     ρ1 21ρ1 ρ2 21ρ2    =    R1 φ1 R2 φ2     |{z } Geometrical parameters . The procedure can be summarized in the following steps. 1) First, for practical reasons, it is usually necessary to set a maximum radius Rof the bundle R=ρ2+1ρ2(first design equation).(23) 1540 IEEE SENSORS JOURNAL, VOL. 24, NO. 2, 15 JANUARY 2024 2) Then, qis determined from the working point. If we consider the turning point of the responsivity as the working point, i.e., zWP =zTP =√2q/3, then the parameter qbecomes fixed q=3 2z2 TP (second design equation).(24) Any alternative working point also sets the qparameter. In general, and as we have previously indicated, the relationship between the working point zWP and the parameter qis of the form zWP =h√q, being ha numerical constant, see Table I. These two conditions impose restrictions on the slope of the responsivity. Indeed 1ρ2=R−ρ2(25) 1ρ1=sρ2 2+(R−ρ2)2−ρ2 1−z2 WP tan2θ0 h2(26) and then, by substituting (25) and (26) in (13) p=ρ2(R−ρ2) ρ1qρ2 2+(R−ρ2)2−ρ2 1−z2 WP tan2θ0 h2 .(27) 3) We could set either the sensitivity, S(zWP), or the maximum value of the responsivity of the sensor. It is equivalent to fixing the value of p. This is the third design equation. With the choice of {p,q}the responsivity of the sensor becomes fixed η(z)=pexp−z2 WP h2z2(third design equation).(28) All that remains is to determine from the three design equations the geometric values of the two rings, i.e., the sizes and placement of the fibers. Since we have four parameters {ρ1, ρ2, 1ρ1, 1ρ2}and three conditions, the problem is underdetermined. We can leave the design based on a free parameter we can take at will, e.g., ρ2. Expressing the three design equations as a function of ρ2, we obtain BUNDLE DESIGN COOKBOOK             1ρ2=R−ρ2(DE1) 1ρ1=sρ2 2+(R−ρ2)2−ρ2 1−z2 WP tan2θ0 h2(DE2) ρ1=ρ21ρ2 p1ρ1 (DE3)             . A simple inspection of the latter brings two clues about the bundle design process. The smaller the radius of the inner ring ρ1, the greater the intrinsic slope pof the responsivity, i.e., the greater the sensitivity. This would recommend placing the first ring as close as possible to the emitter fiber, establishing the new condition rT=ρ1−1ρ1(29) ρ1=rT 2±sR2 2−r2 T 4−z2 WP tan2θ0 2h2−ρ2 2−Rρ2.(30) However, this choice is not always compatible with the selection of the working point or q. Notice that we can easily raise the chosen slope at the working point, S(zWP), by just tuning the asymmetric gain factor k, which is determined from the selected photodetectors gains as demonstrated in (11) dη(z) dz zWP =S(zWP)=2kp h2zWP exp−1 h2.(31) Two additional constraints must always be satisfied for the design to be physically feasible, namely rT≤ρ1−1ρ1(32a) ρ2≥ρ1+1ρ1+1ρ2.(32b) These inequalities only reflect that the rings in the toy model and the fibers in the experimental fiber bundle cannot overlap. Another design option is to set a known linear response, f(z)=z0+mz, in a specified range, [zi,zf]and find the responsivity that best approximates it within that range. Then, we look for the responsivity that minimizes the error in the working range of the sensor [zi,zf]. For the sake of ease of reading, the design equations for this case have been developed in the Appendix. We shall illustrate the preceding procedure with two realworld examples, both of which are carefully discussed in Sections V-A and V-B. V. EXPERIMENTAL VALIDATION The experimental setup has been described in detail previously in [34] and [35]. Regarding the light source, a laser module from Frankfurt Components (HSML-0660-20-FC, Frankfurt Laser Company, Friedrichsdorf, Germany) was employed. It had a nominal output power of 20 mW at 660 nm. An optical isolator (IOF-660, Thorlabs, Newton, NJ, US) was placed between the laser and the bundle to avoid reflections that could destabilize the light source. For the opto-electrical conversion, two Thorlabs PDA100A-EC photodetectors were used. Finally, the responsitivity is calculated as the quotient of the two obtained voltage signals, {V1,V2}for Section V-A, and {V1,V3}for Section V-B. To validate our model, a tetrafurcated bundle was designed and fabricated, consisting of a single-mode transmitting fiber at its center and three concentric rings of multimode receiving fibers. The chosen emitting fiber was a single-mode fiber as reasoned in Section II. The fibers of the first two rings have the same diameter, 0.2 mm. However, the outer ring fibers are 0.34 mm in diameter. The total diameter of the bundle is 1.12 mm. A microscopic image of its cross section is shown in Fig. 8(b). The values of the position and radii of the fibers are presented in Table II. This bundle allows one to validate twice the model. Firstly, we compare the responsivity of two rings (first and second)