1 Introduction

The curvature and deformation measurement have attracted considerable interest nowadays due to the importance, it plays in modern engineering such as earthquake prediction, volcano and structure health monitoring [1–3]. More and more optical fiber-based devices have been utilized for curvature sensing since they offer the advantages of compact size, convenient interrogation, and immunity to the electromagnetic field [4–7].

However, most of these curvature or bending sensors were based on the measurement of curvature-induced interference wavelength shift, which increases the measuring difficulty due to the demand of precise wavelength interrogator. To overcome this disadvantage, several approaches based on intensity measurement are proposed and demonstrated. For instance, Silva et al. [8] measured the amplitude change of the spatial frequency spectrum to achieve simultaneous curvature and strain measurement; Wang et al. [9] demonstrated a curvature sensor by measuring the spectrum intensity of a periodically tapered soft glass fiber; Monzon-Hernandez et al. [10] proposed a curvature sensor using two concatenated fiber tapers. However, most of them offer relatively low curvature measuring sensitivities. Another big challenge for bending or curvature sensor focuses on the temperature or optical light source power fluctuation introduced measuring error, which requires special consideration in practical applications. Although various schemes have been used to diminish the error including special types of fiber Bragg gratings [11–13], the sensing elements of them are relatively difficult to fabricate.

Recently, methods utilizing the interference between guided modes in fiber for surrounding parameters measurement have attracted extensive attentions [14, 15]. Compared to the conventional interferometers, the in-fiber interferometers are with compact dimensions and their interference arms can be easily controlled. Consequently, they have been successfully applied as curvature sensors as well [16–18]. In this paper, we propose and demonstrate a novel modal interferometer using single biconically tapered singlemode fiber (SMF) spliced between two multimode fibers (MMF) for curvature sensing. Due to the modes power redistribution happened at the tapered section of the SMF, the interference fringe visibility is greatly enhanced and directly related to the bending radius. Therefore, the bending curvature could be simply determined by monitoring the fringe visibility variation. The curvature measuring sensitivities before and after tapering the SMF are compared to study the modes coupling enhancement effect of the SMF taper. To obtain the hysteresis of the sensing performance, fringe visibility changes of the interference spectrum in both increasing and decreasing the bending curvature are also recorded. Meanwhile, we investigate the measuring range shift of the sensing structures with different taper diameters. Furthermore, the temperature response demonstrates that the fringe visibility is almost independent on the environmental temperature change.

2 Working principle and experimental setup

The configuration of the curvature sensor is illustrated in Fig. 1a. An interrogation system (Micron Optics, Inc., sm125–500), including an optical spectrometer and an extremely low-noise tunable laser source ranging from 1,510 to 1,590 nm with the wavelength scanning step of 5 pm and accuracy of 1 pm, is employed to monitor the interference spectrum. Light emitted from the tunable laser source is launched to illuminate the sensing element through the transmission link from port 2 to port 3 of the circulator, and then, the modulated optical signal is re-coupled into the interrogator for detection from port 1 to port 2. Consequently, the final optical transmission spectrum is collected through the signal processing system for analysis.

Fig. 1
figure 1

Schematic diagram of the proposed sensing system based on tapered MSM fiber structure

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Figure 1b shows the structure of the sensing element, which is fabricated with the following steps. First, a section of standard SMF is spliced between two MMF to construct an ordinary multimode–singlemode–multimode (MSM) fiber structure. Then, the central part of the SMF is tapered down with flame-heated adiabatically drawing technique. By carefully controlling the drawing length and velocity, a biconical taper with desired waist diameter and length could be obtained. The inset in Fig. 1b shows the image of the tapered section of the central SMF under an optical microscope. Here, the two MMFs act as the multimode generator and collector, respectively, while the tapered SMF works as a mode power converter. When light transmits from the lead-in SMF to the first MMF, high-order modes will be excited due to the fiber core mismatch. Then, the light will propagate into the cladding layer of the sensing SMF, resulting in the power exchange from core modes to multiple cladding modes. The cladding and core mode will be collected by the second MMF and interfere with each other due to the different optical length. Since the optical field distribution is very sensitive to the surrounding change on account of the existence of the fiber taper with thin diameter, the interferometric structure could be used for sensing applications.

Due to the different optical path of the core and the cladding modes propagated in the sensing tapered SMF, an intrinsic fiber Mach–Zehnder interferometer based on modal interference is formed.

Then, the fringe visibility of the interference spectrum, according to the interference intensity equation, could be illustrated as:

$$\begin{aligned} F & = {{(I_{\hbox{max} } - I_{\hbox{min} } )} \mathord{\left/ {\vphantom {{(I_{\hbox{max} } - I_{\hbox{min} } )} {(I_{\hbox{max} } + I_{\hbox{min} } )}}} \right. \kern-0pt} {(I_{\hbox{max} } + I_{\hbox{min} } )}} \\ & \; = \frac{2}{{\left( {\sqrt {{{I_{\text{core}} } \mathord{\left/ {\vphantom {{I_{\text{core}} } {I_{\text{clad}} }}} \right. \kern-0pt} {I_{\text{clad}} }}} + \sqrt {{{I_{\text{clad}} } \mathord{\left/ {\vphantom {{I_{\text{clad}} } {I_{\text{core}} }}} \right. \kern-0pt} {I_{\text{core}} }}} } \right)}}. \\ \end{aligned}$$
(1)

where I max and I min represent the maximum and minimum interference intensities, I core and I clad are the intensities of the core and cladding modes, respectively.

From Eq. (1), the fringe visibility of the interference spectrum is largely dependent on the power ratio of the core and cladding modes.

After the tapering process, the radius of the fiber goes down. When the radius of the core reaches a low value, light cannot be confined in the core region and it will be controlled by the cladding–air boundary [19]. It has been demonstrated both theoretically and experimentally in Ref. [20, 21] that bent taper in SMF will lead the power interchange between core and cladding modes (as shown in Fig. 1b), resulting in the oscillatory nature of the optical intensity arisen from the lossy cladding modes that is largely dependent on the bending curvature. However, in this paper, by using the MMF as the cladding modes collector, the power of cladding modes will be reserved and interfere with the core mode.

Accordingly, when the fiber is bent, part of the core mode will couple to the cladding modes in the tapered section of the central SMF, resulting in an increasing power proportion of the cladding mode [20]. In this case, the power distribution among fiber core and cladding modes will be more uniform, and consequently, the fringe visibility and regularity of the interference spectrum will be great improved. In addition, because of the strong cladding modes in the taper region, the fringe visibility of the interference spectrum will be much sensitive to the bending. By monitoring the fringe visibility of the interference spectrum, bending characteristic could be simply demodulated. Meanwhile, since the optical power of the interference mode is temperature independent, the sensor will be capable of overcoming the environment temperature influence. Moreover, by measuring the fringe visibility that is only relevant to the relative power difference between the maximum and minimum interference power, power fluctuation effect of the light source could be also eliminated, which could be meaningful for practical applications.

Figure 2 shows the experimental setup for applying the bending effect onto the sensing element. The sensing interferometer is fixed by two fiber holders that are mounted on two translation stages. By moving one of the translation stages toward the other one, the sensing fiber bends naturally, as depicted by the dotted curves.

Fig. 2
figure 2

Experimental setup used for applying the bending onto the sensing element

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The introduced bending radius could be calculated in terms of the distance d that the translation stage moves and the initial separation L between two stages. The exact relationship between bending radius and these two parameters could be described as:

$$L - d = 2 \times R\sin (L/2R).$$
(2)

where R is the bending radius. In the experiment, the initial separation of the two stages is set to be 370 mm.

3 Sensor structure and experimental research

In order to demonstrate the mode coupling enhancement induced by the SMF taper, the curvature responses of the sensing element before and after the tapering process are tested as contrast. The fiber used for fabricating the sensing element are standard SMF and MMF with the core and cladding diameter of 9/125 and 100/125 μm, which are supported by Yangtze Optical Fibre and Cable Company Ltd. First, an ordinary MSM fiber structure is fabricated, and its curvature response is investigated for comparison, as presented in Fig. 3a. The central SMF is 50 mm long, and both of the surrounding MMF pieces are 5 mm in length. It can be noticed that when the fiber is kept straight, interference spectrum with the highest extinction ratio of about 8 dB in the whole wavelength range is observed, as presented by the blue curve. However, as the fiber is bent, the optical extinction ratio begins to drop firstly and then increases. The decrease in the extinction ratio is caused by the bending introduced loss. Although the bending effect will enhance the coupling between core and cladding modes, cladding mode loss also depends on the bending radius [22, 23]. So, when the bending radius is large, the influence of the bending induced cladding mode loss is dominating since the coupling from core mode to cladding mode is very weak. Thus, the total power of cladding mode is getting lower, and the extinction ratio drops accordingly. The red curves in Fig. 3 are the spectrum at the curvature where lowest extinction ratio is observed. As the bending radius becomes even smaller, more and more optical power of core mode couples to and propagates as the cladding mode, which makes a significant enlargement of the extinction ratio of the interference spectrum. The brown spectrums in Fig. 3 depict the highest fringe visibility observed during the measurement. As verified in Fig. 3a, the extinction ratio reaches as high as 13 dB at the wavelength around 1524 nm when the fiber radius is bent to 2.6246 m (with the curvature of 0.3810 m−1).

Fig. 3
figure 3

Optical spectrum of the sensing element in different bending curvature. a Before tapering process, b after tapering process

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Then, the fiber is tapered at the central part of the SMF utilizing flame-heated adiabatically drawing technique. The taper is 5 mm in length with the waist diameter of 55 μm. After tapering the SMF, the interference spectrum in response to curvature change is recorded again for comparison. As depicted in Fig. 3b, the optical spectrum exhibits a similar response with the fiber curvature changing from 0 (no bending effect) to 0.5378 m−1. Nevertheless, the extinction ratio of the interference spectrum changes more prominent. A stronger interference spectrum with higher extinction ratio of over 20 dB is observed at the curvature of 0.5378 m−1, which is attributed to the more efficient mode power coupling effect in the tapered SMF. Furthermore, the spectral uniformity is much enhanced after tapering the fiber. The spectrum with improved quality will make the measurement easier and more accurate. In addition, owing to the optical length increase in the cladding modes caused by the taper [24, 25], the resonance wavelength gets a regular blue shift in response to the increasing bending curvature. Although the standard SMF also becomes asymmetric under bending effect, the effective refractive index change of the core and cladding modes is not apparent since the bending radius under test is large. However, the effective refractive index change of cladding modes induced by the fiber asymmetry becomes more significant for tapered fibers, resulting in an increased resonance wavelength shift, according to Ref. [26].

We can also find that the transmission loss of the fiber structure becomes larger as the bending radius increases, but the measuring accuracy will not be influenced because we only need to measure the relative power difference between the upper and lower power of the transmission spectrum.

In the experiment, the peak and dip power of certain period of the transmission spectrum with the most prominent extinction ratio change are selected to record and calculate the fringe visibility, according to Eq. (1). In order to test the curvature response of the sensor, the fringe visibility is recorded by decreasing the translation stage separation with an interval of 0.05 mm.

Figure 4a, b shows the fringe visibility changes with the bending curvature before and after tapering process, respectively. Before tapering, it is obvious that the measuring fringe visibility first decreases along with the bending radius until it reaches the lowest point (with the radius of R = 2.6246 m), which is determined by the cladding mode loss caused by bending effect. After further reducing the bending radius, the fringe visibility increases almost linearly with a responsivity of 3.8685 m−1 and correlative coefficient of 0.9982 within the curvature region from 0.2487 to 0.33297 m−1, as shown in the section between the two dotted lines in Fig. 4a. The linear fit here could largely simplify the measuring difficulty since it could be more convenient for calibration, which could be favorable for practical applications.

Fig. 4
figure 4

Measured fringe visibility changes with the bending curvature. a Before tapering process, b after tapering process

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After tapering the SMF, an increased measuring sensitivity of 4.6164 m−1 with a correlative coefficient of 0.9964 is achieved, as displayed in Fig. 4b. However, the linear measuring region shifts to the curvature region from 0.3487 to 0.4541 m−1. Based on the optical power detecting resolution of about 0.01 dB of the power detecting system, the resolution of the sensor could be calculated as at least 0.0092 m−1 in the linear region, according to Eq. (1). When the fiber bending increases any further, the fringe visibility drops correspondingly. This may arises from the oscillatory power interchange distribution between core and cladding modes.

Since multiple cladding modes excitation will make the spectrum slightly modulated, resulting in the measuring error afterward, the transmission spectrums of the tapered MSM sensing structure before and after bending are Fourier transformed to analyze the exact number of propagating modes contributing in the interference procedure. As presented in Fig. 5, whether with or without the bending effect, it is obvious that the transmission spectrum is primarily composed of the interference between the core and one dominating cladding mode, as seen from the marked dominating frequency peaks. Although other cladding modes are still excited as well, shown by several other frequency peaks with low intensities, they exhibit very weakly contribution to the interference pattern and could be neglected. Hence, the transmission spectrum is mainly carried on the information of one interference spectrum, which may largely diminish the measurement error.

Fig. 5
figure 5

Spatial frequency spectrum of the tapered MSM sensing structure before and after bending

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Moreover, as fiber bending changes in two directions, measuring the curvature increase and decrease to investigate the hysteresis of the sensor performance is particularly essential. As presented in Fig. 6, the red downward-pointing triangle indicates the process of increasing the fiber bending curvature, while the blue dot illustrates the reverse process. It is obvious that the recorded data in two directions are almost coincident within the whole curvature region. The inset of Fig. 6 shows the specified characteristic of the linear measuring region. Half the maximum fringe visibility deviation is 0.01, which is less than 0.5 % comparing with the highest fringe visibility. The slight error may be caused by the deviation of the translation stage. The result demonstrates that the system is of great repeatability and stability.

Fig. 6
figure 6

Measured fringe visibility change in response to the curvature increase and decrease after tapering process; the inset shows the enlarged fragment of the linear measuring region between two dotted lines

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Note that the optical power distribution between core and cladding modes depends on the physical dimension of the fiber taper [7, 20], the curvature measuring sensitivity, as well as the whole measurement range could be adjusted by carefully optimizing the waist diameter of the tapered SMF. In order to investigate the variation tendency of the linear measurement range and sensitivity on the taper diameter, another two sensing structures with taper diameters of 65 and 51 μm are fabricated and tested. To make the measurement results meaningful and persuasive, all the three fiber structures are fabricated with the same SMF and MMF length and diameter. The experimental results for all the three sensing structures are depicted in Fig. 7.

Fig. 7
figure 7

Measured fringe visibility variation of the interference patterns as the curvature increases corresponding to different taper diameters. a Taper diameter of 65 μm, b taper diameter of 55 μm, and c taper diameter of 51 μm

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It is obvious that the linear measurement range shifts to the larger curvature as the taper diameter reduces. In Fig. 7a, for the fiber taper diameter of 65 μm, the fringe visibility changes linearly within the curvature range from 0.2487 to 0.3487 m−1 with a sensitivity and coefficient of 4.1074 and 0.9994 m−1, respectively. When the taper diameter decreases to 51 μm, the linear measuring region shifts to the curvature range from 0.3811 to 0.4784 m−1, and the sensitivity increases to 4.7049 m−1. In this case, the linear operation range of the sensor could be adjusted by proposing the fiber structures with certain taper diameters. The results also demonstrate that curvature sensitivity could be further improved by employing fiber taper with smaller diameter, which could offer stronger mode coupling effect.

However, as the fiber taper is thinner than 50 μm, its mechanical strength drops promptly. To overcome the problem, the fiber could be embedded in a flexible material with high strength and low refractive index coefficient for practical applications, as presented in Ref. [17].

Moreover, the stability of the sensor is also investigated. Figure 8 shows the repeatedly scanned spectrums of the sensing structure with the taper diameter of 55 μm at room temperature, within 24 min at the time interval of 4 min. According to the experimental results depicted in Fig. 8b, the biggest extinction ratio fluctuation of the interference spectrum is about 0.15 dB, indicating a biggest fringe visibility deviation of ~2.5 % that may arise from the environmental disturbance. Therefore, the measuring accuracy of the sensor can be calculated as 0.1154 m−1.

Fig. 8
figure 8

a Repeatedly scanned interference spectrum of the sensor with the taper diameter of 55 μm at the time interval of 4 min and b extinction ratio fluctuation calculated from a

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As we all know, the performance of most fiber bending sensor is also influenced by the temperature fluctuation. In order to discriminate the temperature cross-sensitive effect, the fringe visibility as well as the resonance wavelength shift of the sensing structure with the taper diameter of 55 μm in response to the temperature change is investigated. Temperature measurement is employed by heating the sensing head in air with low thermal refractive index coefficient. During the temperature response test, the sensing element is kept straight and placed into a commercial temperature controllable oven with the resolution of 1 °C. A little tension is applied onto the sensing element to eliminate any bending effect. The temperature is set to increase from 20 to 65 °C with 5 °C increment. As demonstrated in Fig. 9, although the resonant dip wavelengths shifts to the longer wavelength with a coefficient of 33.93 pm/°C owing to the thermo-optic effect, the profile of the interference pattern keeps almost stable, and the maximum fluctuation of fringe visibility is less than 3 %. The experimental result demonstrates that the sensor has the intrinsic ability to overcome the temperature cross-sensitive effect. Moreover, due to the relative large fiber diameter, the sensing structure can also be immune to the slight refractive index change of the environment [10].

Fig. 9
figure 9

a Optical spectra of the sensing element in different temperature, b partial enlarged optical spectra of a certain interference period in a, c resonance wavelength shift in response to temperature change, and d fringe visibility variation in response to temperature change

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4 Conclusion

We have investigated an optical fiber bending sensor based on MSM fiber structure with a biconical taper in the center. Owing to the thin dimension of the tapered SMF, the bending applied onto the sensing element will enhance the power redistribution between core and cladding modes, resulting in the fringe visibility change of the interference spectrum. The experimental result shows the fringe visibility changes almost linearly in certain curvature region. An improved high sensitivity of 4.6164 m−1 in curvature range from 0.3487 to 0.4541 m−1 is achieved after tapering the SMF into a waist diameter of 55 μm. In addition, the linear measurement range shifts to the larger curvature along with higher sensitivity when the fiber taper diameter decreases. Meanwhile, the increasing and decreasing curvature are applied to investigate the stability and hysteresis of the sensor. Half the maximum fringe visibility deviation of 0.01 is obtained, which is less than 0.5 % comparing with the highest fringe visibility. Furthermore, the temperature response of the sensor demonstrates that the fringe visibility is almost insensitive to the temperature change. The features of compact size, easy fabrication, and high sensitivity with low hysteresis make the device very attractive in civil and mechanical engineering applications where measurement of large curvatures with high sensitivity is often required.