1. Introduction

Curvature is an important physical parameter and should be monitored in various applications, such as security monitoring and civil engineering. Recently, fiber sensors based on in-fiber Mach-Zehnder interferometer (MZI) have attracted many attentions [1–5]. Compared to the conventional MZIs, the in-fiber MZI has an in-line fiber structure, and the interference arm difference can be easily controlled. Consequently, it has been successfully applied as curvature sensors. For instance, Frazão [6] proposed a fiber multimode interference structure and a long period grating (LPG) based MZI bending sensor. Zhou [7] used a photonic crystal fiber (PCF) based MZI with a fiber Bragg grating (FBG) for simultaneous curvature and temperature measurement. Wei [8] proposed an all-fiber MZI curvature sensor fabricated by collapsing splicing both ends of a PCF with single mode fibers (SMFs). A highly sensitive MZI based bending sensor by mismatch fusion splicing a PCF between two sections of SMFs was also proposed [9]. However, all these curvature or bending sensors were based on curvature-induced interference wavelength shift. Expensive wavelength demodulation method and external temperature compensation part were needed. In order to decrease the production cost, some curvature sensors based on intensity demodulation method were also proposed. Hernandez [10] proposed compact optical fiber curvature sensor based on concatenating two fiber tapers. Silva [11] used a suspended multi-core fiber for simultaneous curvature and strain measurement. Dong [12] proposed a temperature insensitive curvature sensor with a core-offset polarization maintaining PCF based MZI. These curvature sensors were carried out by detecting the interference fringe visibility variation. However, the measurement accuracy is easily disturbed by the light source power fluctuations with the use of light intensity demodulation method. Thus, the curvature sensor with the characteristics of intensity demodulation while independent of light power fluctuations was desirable.

In this work, a simple polarization-dependent curvature sensor based on in-fiber MZI with difference arithmetic demodulation method is reported. The proposed in-fiber MZI is formed by core-offset fusion splicing a polarization maintaining fiber (PMF) with a length of 2 cm between two SMFs. The core mode in the PMF has different effective indices corresponding to the two orthogonal polarization modes because of the birefringence of the PMF. Therefore, the interference patterns of the MZI corresponding to the two orthogonal polarization modes have different resonant dip wavelengths. By detecting the fringe visibility of the interference patterns, the curvature variation of the MZI can be obtained. And with the use of the difference arithmetic demodulation method, the measurement errors caused by the temperature and light source power fluctuations can be decreased effectively.

2. Materials and methods

The schematic diagram of the proposed curvature sensor is shown in Fig. 1 . A gain-flattened erbium-doped fiber amplified spontaneous emission (ASE) source with wavelength range of 1450 to 1650 nm is used as the light source. And the polarization state of the output light from the ASE source is a non-polarized light. The output spectrum is detected with an optical spectrum analyzer (OSA, AQ6370, Japan). The maximum resolution of the OSA is 20 pm. The MZI is formed by inserting a PMF (PANDA, 1017-C, YOFC) with a length of 2 cm between two single mode fibers (SMF-28), the birefringence index of the PMF is 7.7024 × 10⁻⁴. Both ends of the PMF are mismatch fusion spliced [13] (using a commercial fusion splicer (Fujikura FSM-40S)) with the lead-in SMF1 and lead-out SMF2, respectively. The core offset size is about 4.5 μm. The core and cladding diameters of the used SMF are 9 μm and 125 μm, respectively. The PMF inserted between the two SMFs has the same core and cladding diameters as the SMF. A polarization controller (PC) is used to adjust the polarization states of the input light in order to obtain a high fringe visibility.

 

Fig. 1 Schematic diagram of the proposed curvature sensor. The inset picture shows the partially enlarged drawing of the sensing head

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As shown in the inset of Fig. 1, the left end of the PMF is mismatch fusion spliced with the SMF1. And the core mode of the SMF1 is partly coupled into the cladding of the PMF. The right end of the PMF is mismatch fusion spliced with the SMF2. The excited cladding modes are recoupled back to the core of SMF2 after propagating throught the PMF. Therefore, a kind of MZI system is formed. The curvature on the MZI part is applied by decreasing the separation distance of the translation stages. The bent fiber is normally approximated as an arc of circle. The curvature is given by [14, 15],

The resonant dip wavelength satisfies the equation of ${\Phi }^m=(2k+1)\pi$, where k is natural number. Therefore, the resonant dip wavelength corresponding to slow axis and fast axis polarization modes, λ_s and λ_f can be described as,

As the SMF1 and SMF2 are mismatch fusion spliced with the PMF, the misalignment degree is controllable. To control the degree of the mismatch, during the fusion splicing process, the manual splicing mode of the fusion splicer is used. The fiber core offset size is about 4.5 μm in order to ensure the splicing loss is approximately 3 dB. And during the manual operation, the mismatch degree can be adjusted by the splicing loss showing on the screen of the fusion splicer. As the SMF is mismatch fusion spliced along the fast axis direction of the right end of the PMF, the light corresponding to the fast axis polarization mode is dominant. On the other hand, as the SMF is mismatch fusion spliced along the slow axis direction of the right end of the PMF, the light corresponding to the slow axis polarization mode is dominant. In our design, the SMF is mismatch fusion spliced along the direction between the slow and fast axis of the PMF (the optimization direction is the 45° deflecting to slow/fast axis), as shown in Fig. 2(a) . The cladding modes corresponding to the fast axis polarization mode and slow axis polarization mode are excited simultaneously. In experiment, the polymer coating of the fiber at the sensing head is removed. The core-offset size of the two fibers (SMF and PMF) is only about 4.5 μm. Compare to the diameter of the fiber cladding (125 μm). The small core-offset size of the fiber has little influence to the mechanical stability when it is bent. Besides, the MZI part is symmetrical bent along with the lead-in and lead-out SMFs and the curvature is not applied directly on the core-offset part of the fiber. In a word, the core-offset fiber is strength enough for our experiment. If it is used in practical applications, we should consider to finding a special re-coating material to strengthen the mechanical stability of the sensing head.

 

Fig. 2 (a) Schematic diagram of the SMF is mismatched fusion spliced along the direction between the slow and fast axis of the PMF. Initial interference pattern of the polarization-dependent MZI as the SMF is offset along the (b) fast axis, (c) slow axis and (d) between the fast and slow axis of the PMF.

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The initial interference patterns during the fabrication process of the in-fiber MZI are recorded. As expected, owing to the difference of the effective refractive index between the two orthogonal core modes and the excited cladding modes, the resonant dip wavelengths of the interference patterns corresponding to slow axis and fast axis polarization modes are different. The effective refractive index difference between the core and the excited strong cladding modes on the fast axis and the slow axis of the PMF are 0.03829 and 0.03906, respectively. Through the theoretical arithmetic, the maximum fringe visibility of the interference corresponding to the fast and slow axis polarization modes are occurred at the wavelengths of 1531.6 nm and 1562.4 nm, respectively. And during the experiment, the lead-in and lead-out SMFs are mismatch fusion spliced along the fast axis and slow axis of the PMF, respectively. The measured values are 1526 nm and 1562.5 nm, as shown in Fig. 2(b) and (c). There is a little deviation between theoretical and experimental values corresponding to the fast axis polarization mode. We think this is mainly owing to the mismatch direction is not strictly along the fast axis of the PMF during the fusion splicing process. The wavelength of 1526 nm is closer to the theoretical value. So the dip wavelength of 1526 nm is selected as the experimental dip wavelength. As the SMFs are mismatch fusion spliced between the fast and slow axis of the PMF, the theoretical arithmetic value of the resonant dip wavelengths difference corresponding to the fast and slow polarization modes is 30.81nm. As shown in Fig. 2(d), the measured two maximum resonant dip wavelengths are occurred at the wavelengths of 1536.6 nm and 1568.4 nm, respectively. The wavelength difference is very close to the theoretical arithmetic value. So we thought that the resonant dips wavelengths of λ_f and λ_s occurred at Fig. 2(d) are the fast axis and slow polarization modes dominant, respectively.

From Fig. 2(b) and Fig. 2 (c), it can be seen that the interference patterns have a little asymmetry since there are more than two modes involved in the interference patterns. In our experiment, it can be assumed that only one exited cladding mode is dominant. In order to examine the assumption, we obtained the spatial frequency spectra of the interference patterns of Fig. 2 (b) and Fig. 2(c) by using the fast Fourier transform method. As shown in Fig. (3) , it can be seen that the power is mainly distributed in the core mode and a strong cladding mode. Other excited higher order cladding modes are weak. The main interference pattern is mainly formed by the interference of the dominant strong cladding mode with the core mode. Other weak cladding modes should also interference with the core mode, and the interference between the core mode and the weak cladding modes will slightly modulate the main interference pattern. However, the modulation effect is very weak.

 

Fig. 3 Spatial frequency spectra of the proposed sensor.

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3. Results and discussion

Figure 4 represents the interference patterns of the proposed in-fiber MZI corresponding to the curvature variation. The guiding core mode in the PMF has different effective refractive index corresponding to slow axis and fast axis polarization modes of the PMF. The maximum resonant dip wavelength corresponding to fast axis and slow axis polarization mode are 1536.6 nm and 1568.4 nm, respectively. The experiment is carried out at room temperature (25 °C). As the curvature on the PMF part is increased from 0 to 0.982m⁻¹, the fringe visibility of the interference patterns corresponding to the slow polarization mode and fast polarization mode are decreased simultaneously. At the same time, λ_s is shifted towards smaller wavelengths direction as the curvature is increased. The fringe visibility evaluated for a fixed λ_s will bring a big error to the experiment results. During the experiment, we don’t measure the fixed λ_s. As the λ_s shift, the dip wavelength of the interference patterns around the initial λ_s is chosen as the new one. And the error can be decreased effectively. The proposed in-fiber MZI has different sensitivities to external fluctuations corresponding to the two orthogonal polarization modes. As shown in Fig. 3, the order of the excited cladding modes corresponding to the slow axis and fast axis polarization mode are different. This result in the exited cladding modes corresponding to the slow axis and fast axis polarization mode have different response to the bending of the MZI.

 

Fig. 4 The fringe visibility variation of the interference patterns corresponding to fast polarization mode and slow polarization mode as the curvature variation.

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The misalignment degree of the fiber core controls the initial fringe visibility of the interference patterns. The direction of curvature variation on the fiber is mainly influence the order of the weak cladding modes while the power of the strong cladding mode will keep almost unchanged. In experiment, the main interference pattern is mainly formed by the interference of the dominant strong cladding mode with the core mode. Other weak cladding modes should also interferes with the core mode, and the interference between the core mode and the weak cladding modes will slightly modulate the main interference pattern. However, the modulation effect is very weak. So, even the sensing head in our design is bent along the different directions, the two dominant resonant dips wavelength in the interference patterns have the same response.

In our experiment, the pattern fringe visibility K of the interference pattern can be described as

Equation (6) shows that the fringe visibility of the interference patterns greatly depends on the light intensity ratio of the I₁ and I₂. For the two polarization directions of slow axis and fast axis, I₁ and I₂ are the intensities of the light propagating along the fiber core and cladding, respectively. The fringe visibility gets its maximum value at the ratio of 1:1. The variation of curvature is supposed to change the couple efficiency between the core mode and the cladding mode of the SMF and PMF. Thus the power distribution between I₁ and I₂ will be modified, which induces a change on the visibility of the interference patterns [8]. Curvature variation on the PMF part of the MZI can be obtained by monitoring the fringe visibility variation of the interference patterns. As shown in Fig. 5 , the fringes visibility of the interference patterns corresponding to slow axis polarization mode and fast axis polarization mode are both decreased with the increasing of the curvature of the PMF. The curvature sensitivity corresponding to the fast and slow axis polarization modes are −8.268dB/m⁻¹ and −10.264dB/m⁻¹, respectively. K_s and K_f are defined as the fringe visibility of the interference patterns corresponds to the slow and fast axis polarization mode, respectively.

 

Fig. 5 Fringes visibility variation of the interference patterns for the two orthogonal polarization modes as a function of the curvature.

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During the experiment, the mismatch splice method is used to form the MZI system. And the big insertion loss can’t be ignored. However, to obtain the value of the curvature variation, we only need to measure the relative power variations of the resonance dips. So the accuracy of the experiment results will not be influenced by the insertion loss. The accuracy of the experiment results is mainly influenced by the temperature and light source power fluctuation. As shown in Fig. 6(a) , the light source power fluctuations will lead to a big variation of the interference patterns’ fringe visibility. In a variation range of 10% of the light source power fluctuations, the fringe visibility of the interference patterns are decreased by 5.5 dB and 6.5 dB corresponding to the fast and slow axis polarization mode, respectively. During the temperature response test, the total MZI part is placed into a temperature controlled container. The temperature of the container is set to increase from 10 to 80 °C with a step of 10 °C. As shown in Fig. 6(b), the resonant dip wavelengths of the interference patterns corresponding to slow axis polarization mode and fast axis polarization mode have a little fluctuation as the temperature is increased. The resonant dip wavelengths have a red shift as the temperature is increased mainly owing to the thermo-optic effect. The temperature variation will also influence the mismatch degree of the fusion splice point, resulting in the decreasing of the coupling efficiency between the core and cladding modes. The fringe visibility of the interference patterns also has a little variation as the temperature is increased.

 

Fig. 6 Interference patterns variation as function of (a) the light source power fluctuations and (b) the temperature response.

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Each resonant dip wavelength will be affected by the temperature and light source power fluctuations. The precise experimental results can’t be obtained if only one resonant dip wavelength is chosen as the experimental parameters. Under this condition, to obtain precise experimental results, Wang [17] proposed a difference arithmetic demodulation method, and Li [18, 19] proposed a similar method, too. Because the temperature and the light source power fluctuations result in the same variation tendency for the interference patterns of the two orthogonal polarization modes. During the measurement process, the interference patterns corresponding to the two orthogonal polarization modes are detected simultaneously. A difference arithmetic demodulation method is used to decrease the measurement error caused by the temperature and light source power fluctuations. The temperature and light source power fluctuations contribute the same effects to the K_s and K_f, (K_s and K_f are the fringe visibility of the interference patterns corresponds to the slow and fast axis polarization mode, respectively.). ΔK is the difference value between the fringes visibility corresponding to the slow polarization mode and fast polarization mode. The noise induced by the two external factors can be eliminated effectively by detecting the difference value of ΔK = (K_s-K_f)/(K_s + K_f). As shown in Fig. 7 (a) , the maximal error caused by the light source power fluctuations is 3.10 dB/μW and 3.37 dB/μW for fast axis and slow axis polarization mode, respectively. After using the difference arithmetic demodulation method, the error is decreased to 0.307 dB/μW. As shown in Fig. 7(b), the maximum temperature error of the fast axis polarization mode and the slow axis polarization mode are 0.050 dB/°C and 0.067 dB/°C, respectively. The temperature error is decreased to 0.012 dB/°C after the difference arithmetic demodulation method is used. The main interference pattern is mainly formed by the interference of the dominant strong cladding mode with the core mode. Other weak cladding modes should also interferes with the core mode, and the interference between the core mode and the weak cladding modes will slightly modulate the main interference pattern. The orders of the weak cladding modes corresponding to the slow and fast axis polarization mode are different. Each order of weak cladding has the different temperature response. Thus, the resonant dip wavelength corresponding to the two arithmetic polarization modes has the different temperature response.

 

Fig. 7 (a) Light source power fluctuations and (b) temperature response of the proposed curvature sensor before and after using difference arithmetic demodulation method.

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After using difference arithmetic demodulation method, the relationship between the fringe visibility and the curvature is shown in Fig. 8 . These experimental data are fitted by the Boltzmann function. The curvature sensitivity is determined by calculating the first-order derivative of the Boltzmann curves. The difference value of ΔK = (K_s-K_f)/(K_s + K_f) cannot be obtained at the curvature value of 0 m⁻¹. Owing to this drawback, the measurement range of the proposed sensor has to start at the value of 0.1 m⁻¹. As shown in Fig. 8, the experimental measurement range is 0.1 to 0.982m⁻¹. But a linear response is only within the measurement range of 0.1 to 0.35 m⁻¹. Based on the above reason, accurate to say that the measurement range of the proposed sensor is 0.1 to 0.35 m⁻¹ after a difference arithmetic demodulation method is used. The curvature sensitivity in the measurement range of 0 to 0.1 m⁻¹ can be obtained from the Fig. 5. After using the difference arithmetic demodulation method, the maximum sensitivity of −0.882 dB/m⁻¹ is obtained. The accuracy of the experiment results is mainly influenced by the temperature and light source power fluctuation. Compared to the intensity demodulation based curvature sensor which have been already published [10–12]. Here, a difference arithmetic demodulation method is used. Through our experimental demonstration, the measurement error caused by that two factors are decreased by 94% and 91%, respectively. Compare to the PCF based curvature sensor (reference [7]), the curvature variation is measured by detecting the wavelength shift of the interference patterns. It used a wavelength demodulation method and the OSA is necessary. In our design, in order to detecting the fringe visibility variation, the OSA is still necessary. But we have proposed an alternative intensity demodulation method. If the fusion splicing quality at the mismatch point of the fiber is excellent enough. The resonant dip wavelengths both corresponding to the slow and fast axis polarization mode will not shift with the curvature variation. We can use photoelectric detector (PD) combined with a controllable band pass filter to replace the OSA. The maximal fringe visibility of the resonant dip wavelength corresponding to the slow and fast axis polarization modes can be detected by the PD.

 

Fig. 8 Relationship between the fringe visibility and the curvature after using difference arithmetic demodulation method.

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4. Conclusion

In conclusion, a compact polarization-dependent curvature sensor based on in-fiber MZI is proposed. The MZI is fabricated by core-offset fusion splicing one section of PMF between two SMFs. The core mode in the PMF has different effective indices corresponding to the two orthogonal polarization modes because of the birefringence of the PMF. Therefore, the interference patterns of the MZI corresponding to the two orthogonal polarization modes have different resonant dip wavelengths. The curvature variation on the MZI part can be measured by detecting the fringe visibilities variation of the interference patterns, which depends on the light intensity ratio of the core and cladding of the PMF. Experiment results show that high sensitivity of −0.882 dB/m⁻¹ is obtained within the measurement range of 0.1 to 0.35 m⁻¹. Benefited from the using of difference arithmetic demodulation method, the measurement error caused by the temperature and light source power fluctuations are decreased effectively. The alternative cost-efficient demodulation method and simple fabrication process shows the proposed sensor has a great potential for many sensing applications.