Abstract

A compact high temperature sensor utilizing a multipath Michelson interferometer (MI) structure based on weak coupling multicore fiber (MCF) is proposed and experimentally demonstrated. The device is fabricated by program-controlled tapering the spliced region between single mode fiber (SMF) and a segment of MCF. After that, a spherical reflective structure is formed by arc-fusion splicing the end face of MCF. Theoretical analysis has been implemented for this specific multipath MI structure; beam propagation method based simulation and corresponding experiments were performed to investigate the effect of taper and spherical end face on system’s performance. Benefiting from the multipath interferences and heterogeneous structure between the center core and surrounding cores of the all-solid MCF, an enhanced temperature sensitivity of 165 pm/°C up to 900°C and a high-quality interference spectrum with 25 dB fringe visibility were achieved.

© 2016 Optical Society of America

1. Introduction

Silica optical fiber sensors have attracted intense interests for temperature sensing due to the advantages of miniaturization, electromagnetic isolation, high stability, reliability and so on. All of the characteristics make them desirable for applications in harsh environments, for instance in high temperature or strong magnetic fields. Over the years, a variety of fiber based sensing microstructures have been extensively studied, among which the grating based structures, such as fiber Bragg grating (FBG) [1,2] and long period grating (LPG) [3], always suffer from degradation on optical performances and thermal drifts of the resonance peaks when being placed in high-temperature environments. As an alternative, interferometer structures are introduced in conventional or speciaty optical fibers, such as Fabry–Perot interferometer (FPI) [4–6], Mach-Zehnder interferometer (MZI) [7,8] and fiber Michelson interferometer (MI) [9–11]. Recently, the multicore fiber (MCF) based interferometer structure has attracted researchers’ attentions [12–15]. Due to the multipath evolutions of light in different cores, sophisticated interference pattern and super-modes are supported.

In 2012, a MI structured high temperature sensor based on twin-core fiber (TCF) was proposed [13]. The high temperature-response characteristic was analyzed and the annealing process was discussed in detail. Whereas the maximum temperature measured with this sensor was just 700°C, and the relatively flat dips in spectrum leads to large uncertainty in reading the wavelengths. Subsequently, a fiber optic MCF-MZI sensor fabricated by splicing a strong coupling MCF between two SMFs was proposed in 2014 [14,15]. A super-mode interference was used in this structure. This MCF sensor shows some improvements in signal strength, accuracy, reproducibility, polarization insensitivity, polarization-independent, etc. However, its temperature sensitivity is relatively low and the measurement range is limited in one free-spectrum range (FSR). Typically, the MZI structure works in the transmission direction, which is not suitable to act as probe sensors. On the contrary, the MI works in the reflection mode and it is more compact in practical use and installation [11,16]. Therefore, the MI structure based on MCF needs to be investigated and improved sensing performance could be anticipated. Noted that the established MCF based high temperature fiber sensors utilize the strongly coupled MCF thus the super-mode induced interference pattern is used for measurement. It has been witnessed that the weak-coupling MCF has been developed for ultra-high capacity spatial-division multiplexed (SDM) optical fiber communication system with excellent optical properties [17]. The low loss telecom grade MCFs exhibit indispensable applications in preliminary sensing experiments [18,19] and it will pave the way for future high performance optical fiber sensing system with the compatible fan-in/fan-out devices [20]. This kind of structures can realize a multi-path interference with the help of isolated cores. And the large volume manufactured weak coupling MCF can be used to reduce the cost and promote the application of MCF based sensing.

In this paper, with our in-house developed weak-coupling trench-assisted all-solid heterogeneous seven-core MCF, we implement a multipath MI based high temperature sensor probe through advanced programmable tapering technology and a special tailored spherical end face. The tapering process was conducted at the splice point between the lead-in SMF and MCF in order to enhance the coupling between cores of MCF, thus facilitating the generation of multipath propagation. Moreover, even distribution of optical power among cores has been achieved with the help of spherical structure fabricated at the MCF end face. Theoretical analysis assisted by experiments have been carried out to figure out the principle of this specific multipath MI structure; beam propagation method based simulation and corresponding experiments were performed to investigate the effect of taper and spherical end face on system’s performance. Inter-core interference has been found dominates the fringe pattern and the carefully controlled inter-core refractive index difference leads to a large FSR (166.34 nm with a MCF length of 10 mm). The large FSR helps to increase the measurement accuracy especially in point sensing applications and, in the meantime, keeps large measurement temperature range without overlapping reading problem [11]. It has been demonstrated that a multipath interferometer has higher sensitivity to phase change in comparison with a two-path interferometer [18,21]. As we know, higher phase sensitivity will be helpful to improve the reading accuracy of measurement, which stems from the fact that the slopes of resulted interference peaks are steeper than the sinusoidal output of a two-path interference. The multipath MI device shows a large extinction ratio of 25 dB and a sensitivity of approximately 165 pm/°C at high temperature region. Thanks to the all solid fiber structure, this device is stable in high temperature environment. Taking advantages of the high sensitivity, high extinction ratio, wide FSR and steady performance, our proposed MCF-MI high temperature technique will be very useful in practical applications.

2. Sensor structure and operation principle

Figure 1(a) shows the cross section of the fabricated MCF used in our experiment. The MCF has seven Ge-doped cores surrounded by low refractive index (RI) trench and pure silica cladding. Differing from the homogeneous structures, the MCF used in this paper has six identical ambient cores and a different center core with a little lower RI. The surrounding cores distributed in the vertices of the regular hexagon. The core diameter, the overall fiber cladding diameters and the pitch size (Λ) between the cores are measured to be 8.8μm, 150μm, and 42μm, respectively. The RI difference between central core and the ambient cores is at 10−4 orders of magnitude, and the core differs from cladding in RI with the difference of about 5.9×103. The trench and large core pitch can efficiently avoid the spatial crosstalk between cores, so single mode guiding can be guaranteed in all the cores of this MCF, which matches well to the mode field of the conventional SMF.

 figure: Fig. 1

Fig. 1 (a) The cross section image of the MCF. (b) Schematic diagram and operation principle of the Michelson-type multipath interferometer. (c) MCF-MI structure under microscope. (d) Spherical end face of MCF under microscope.

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In this work, the designed MCF-based fiber sensor structure, as sketched in Fig. 1(b), was fabricated by splicing a segment of MCF to a single mode fiber (SMF) using a fusion splicer (Fujikura, FSM-100P + ) with a taper region, and then cleaving the MCF with a desired length (Furukawa, S325). The microscope photo of this structure is given in Fig. 1(c). The lengths of waist area, transition area and the waist diameter are measured to be 150μm, 250μm, and146μm respectively. The splice point is 146μmaway from the taper region. The taper was manufactured with the arc discharge technique of the fusion splicer with programmable control such that the whole process is repeatable and stable. MCF and SMF without coating layer are fixed into the fusion splicer by two clamps to prevent relative displacement. Firstly, a standard program is taken to splice two segments together. Then, the arc discharges are applied locally to heat the fusion point. At mean time, the splicer starts to pull the fiber from one side and feed with different speed from the other side to taper the heating area [19,20]. In our work, the power of discharge process was set to be [275] bit. The speed of tapering process was about 0.75μm/ms. The parameters of taper region is optimized to realize a low splicing loss. However, the trench structure within the MCF results in weak couplings from central core to the ambient cores. Therefore, the optical power in the central core will be much stronger than those in surrounding cores if we use the flat end face as the reflector. According to the principle, the contrast of the fringe is determined by the power distribution among all the involved beams. The imbalance power distribution will lead to a low contrast interference fringe and hinder its sensing application. To achieve a stable and high extinction ratio interference spectrum, a spherical end face was adopted by arc discharge, as shown in Fig. 1(d), which results in the focus and divergent of light and a second coupling between modes. Through adjusting the relevant parameters by the program of fusion splicer, an expected interference spectrum can be realized.

When light propagates from the SMF to the tapered region, a coupling between center core and surrounding cores of MCF will appear due to the decreased distance between cores, and the optical fields distribute in every core. After the taper region, the large pitch size and trenches ensure mode isolation between the adjacent cores. The light transmit independently in different cores until being reflected by the spherical end face, where the second coupling occurs. The function of spherical end face is to balance the power between central core and surrounding cores and realize a high extinction ratio. When the back reflected modes reach the tapered region again, surrounding core modes interference with center core mode due to the optical path difference caused by spherical end face and different refractive index between cores. The interfered optical field in the center core is then collected by the SMF. In particular, multipath interference happened as the optical waves propagated back and forth along different paths with distinct RI [18]. Therefore, an obvious interference phenomenon will be observed in the reflected light. Both simulation and experiment have been implemented to verify the benefits of the taper and spherical end face structure. Figures 2(a) and 2(b) show distribution of transmitted optical field when white light is launched from the SMF on the other side, and the simulated electric field intensity distribution of MCF without and with tapering. It shows clearly that light in the center core can be coupled to surrounding cores with tapering structure. In addition, the FFT-spatial frequency spectra of the reflection spectrum with or without spherical end face are compared. It is demonstrated that the spherical end face can enhance the inter-core interference, as shown in Fig. 2(c). After fabrication of the spherical end face, we actually observed a total power reduction of the reflected light beam, as can be illustrated in the inset of Fig. 2(c). Generally, 3-4 dB further loss is induced by the spherical end face. However, the key and important parameter for sensing is the extinction ratio of interfered spectrum. Although larger return loss could be introduced by the spherical end face, the power re-distribution function of the end face is essential to balance the optical power among cores thus much better contrast can be obtained. As shown by the inset of Fig. 2(c), clear interference pattern can be obtained with the help of spherical end face. The additional loss is negligible for the sensing application. Through optimizing the parameters of arc discharge, for instance, discharge number and power, we can strengthen the inter-core coupling to improve the extinction ratio of interference spectrum.

 figure: Fig. 2

Fig. 2 Light distributions at MCF end face and simulated electric field intensity distribution with different processing methods of fusion point. (a) Splicing only. (b) Splicing and tapering. (c). FFT-spatial frequency spectrum of the reflectivity spectrum with different processing methods at MCF end face. Inset: The blue curve and red curve show the results of spatial frequency spectra with and without spherical end face, respectively.

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Figure 3(a) shows a typical reflective spectrum of the MCF-MI, with a 10mm MCF segment. A high extinction ratio of 20-30 dB and a wide FSR of about 166nm have been obtained. To understand the principle of this device, fast Fourier Transform (FFT) of the reflection spectra with different MCF length are taken to obtain the spatial frequency spectra, as exhibited in Fig. 3(b). It is obvious that for the length 10mm, the FFT-spatial frequency spectrum has two dominant peaks at 0.006nm1 and 0.012nm1. According to the relationship between effective refractive index difference and the spatial frequency ζ [9]:

Δneff=ζλ02/2L,
where λ0 is the center wavelength, and L is the length of MCF, the value of Δneff can be easily obtained. The Δneff of peak at 0.006nm1 is calculated to be at 10−4 orders of magnitude, which is close to the refractive index difference of center core and surrounding cores. Similarly, the Δneff of peak at 0.012nm1 is calculated to be at 10−3 orders of magnitude, corresponding to the refractive index difference of core and cladding. The red curve and dark yellow curve show the spatial frequency analyses with MCF length of 20mm and 30mm, which are consistent with the situation for 10mm MCF. Therefore, the results suggest that the interference spectrum is dominated by the inter-core interference, during which the optical fields propagating along seven cores interfered each other at the splicing point. The other interference patterns have little influence on the quality of reflectance spectrum.

 figure: Fig. 3

Fig. 3 (a) Typical reflectance spectrum of MCF- MI device, the length of MCF is 10mm. (b) FFT-spatial frequency spectrum of the reflectance spectrums with different MCF length.

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To explain the operation principle, phase-amplitude vector and vector addition are used to analyze the superposition of the light which propagate through center core and surrounding cores. In general, the electric field after superposition can be calculated by vector addition. Considering phase of the first complex amplitude as zero, for simplicity, thus the amplitude and phase of resultant vector can be expressed as

E0=(n=1NEn0cosδn)2+(n=1NEn0sinδn)2,
δ=arctan(n=1NEn0sinδn/n=1NEn0cosδn),
where En0and δnis the amplitude and phase of one vector participating in this superposition. In our work, assuming the light field distribute symmetrically among all cores, the Eq. (2) and Eq. (3) can be simplified as
I=Ic+36Is+12IcIscosΔΦ,
ΔΦ=δcδs,
δ=arctan(Ecsinδc+6EssinδsEccosδc+6Escosδs),
where subscript c and s denote the physical quantity of center core and surrounding core mode. As is well known, for MI structure, the phase difference ΔΦ can be given by:
ΔΦ=[4πL(neff,cneff,s)]/λ,
where λ is the wavelength, neff,c and neff,s are the effective refractive index of central core and surrounding core, respectively. When ΔΦ=(2m+1)π,(m=0,±1,±2,...), the intensity takes minimum value. Combining it with Eq. (7), the relationship between dip wavelength and temperature can be expressed by
λT=4[(neff,cTneff,sT)L+(neff,cneff,s)LT]/(2m+1),
where LT is related to thermal expansion coefficient, the neff,cT and neff,sT are related to the thermo-optic coefficients. For 2m+1=[4L(neff,cneff,s)]/λ, the Eq. (8) can be derived as
λT=(ΔneffT/Δneff+LT/L)λ,
where Δneff=neff,cneff,s. As shown in Eq. (8) and Eq. (9), it is evident that the changes in temperature affect the RI and the length of the MCF section, leading to the shift of resonance wavelength. More accurately, the differential thermal expansion coefficient and the differential thermo-optic coefficients determine the sensitivity of the sensor. Since the thermo-optic coefficient of center core of MCF is different from that of the surrounding core, such a kind of MI has much higher temperature sensitivity [22]. The difference is multiplied with the length of fiber sensor thus the sensitivity is actually enhanced by this heterogeneous all-solid multicore fiber structure. Furthermore, an even higher temperature sensitivity can be anticipated through optimizing the parameters of MCF, such as the doping content and refractive index difference between central core and ambient cores. Monitoring the shift in reflection spectrum allows for accurate temperature measurement in real time.

3. Experimental setup and measurement results

With our proposed and fabricated MCF based MI structure, we conducted the high temperature sensing experiments as depicted in Fig. 4. Light emitting from a supercontinuum source (SCS, from YSL Photonics) was launched into the MCF-MI sensor head through an isolator and a 3dB coupler. When reaching the spherical surface, the light was reflected and finally back-propagated into the optical spectrum analyzer (OSA, YOKOGAWA, AQ6370C) to monitor the output.

 figure: Fig. 4

Fig. 4 Experimental setup for the MCF-MI for temperature sensing.

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Firstly, the reflection spectrum of the MCF-MI structure with different MCF segment length was examined as the spectral periodicity of the interfered signal depends on it. The FSR can be approximated as:

ΔλFSRλ02/2ΔneffL,

The experiment result is shown in Fig. 5(a). When the length of MCF was set to be 10mm, 20mm and 30mm, the FSR was calculated to be 166.34 nm, 87.61 nm and 57.69 nm, respectively. A small amount of error comes from the inaccuracy of MCF length. When L is doubled, the FSR falls in half. It clearly indicates that FSR is inversely proportional to the length of MCF, which is in consistent with the theoretical prediction given by Eq. (10). For the above mentioned interference mechanism and small refractive index difference between center and surrounding cores of MCF, the length of the sensing fiber can be greatly reduced in our sensor under the same FSR, comparing to the sensors reported before [12–15]. It may have some potential applications in a wide range of temperature measurement, especially when the point sensing is required.

 figure: Fig. 5

Fig. 5 (a) Measured interference pattern for the proposed device for different MCF lengths. (b) Spectral response to temperature for the MCF based Michelson interferometer.

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After tradeoff between temperature measurement range and extinction ratio of spectrum, the sensor with 10 mm length of MCF was chosen to perform the temperature measurement experiment. The MCF-MI sensor probe was put into the temperature controlled high-temperature furnace, as exhibited in Fig. 4. At first, a long annealing procedure was needed to eliminate all the residual stress in the MCF that has been accumulated during the fiber drawing process. Thus the sensor head was heated up to 900°C and maintained for about 4 hours. Then it was cooled down to the room temperature, under which this device features deep reflection notches. As the temperature rising during the annealing process, an expected red shift was observed up to 900°C. At 900°C, the shift was still ongoing during the first few minutes, and then the reflection spectrum stabilized and remained stable throughout the final 2.5 hours.

To figure out the temperature response of the stabilized sensor, a continuous test was performed by heating up and cooling down the furnace (resolution ± 1°C). The temperature was changed from 50 °C to 900 °C with 100°C increment changed in 35-minute intervals (15-min heating, 20-min holding), and the wavelength dips shift to longer wavelengths as we expected. Figure 5(b) shows the spectral response of the MCF-MI sensor, in which the spectrum at each temperature step during the heating process are compared and a monotonic spectral shift with temperature is obtained. The relationship between temperature and the chosen dip wavelength is shown in Fig. 6(a) and the red solid line shows a linear fitting to the position of the interference minimum. Owing to the wide FSR, all the data can be recorded in one period.

 figure: Fig. 6

Fig. 6 (a) Dip wavelength shift with temperature increasing and its linear fit. (b) Repeatability: the dip wavelength at different temperatures during the first and second heating cycles.

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According to the experiment result, we divide the detection region into two parts, and a piecewise linear fitting is introduced. Hence the sensor response can be approximated by two linear regions for two temperature ranges. For the temperature range of 50°C-250°C, a sensitivity of ∼40 pm/°C is obtained. For the higher temperature range from 250°C to 900°C, the sensitivity is approximately 165 pm/°C. The R-Square of each linear region is 0.99813 and 0.98943, respectively. The achieved temperature sensitivity of this MCF-MI device is much higher than the similar structure based on TCF (∼44.6 pm/°C) [13] and the strong coupling MCF based MZI structure (∼51.7 pm/°C) [14,15]. Therefore the environment temperature can be calculated by the slope and the wavelength shift of the spectrum dip in one FSR.

For the purpose of demonstrating the reproducibility of the MCF temperature sensor, the heating cycle has been repeated twice with the same experimental procedures and under the same conditions, and the result is shown in Fig. 6(b). The blue circles and the red triangle points represent the measured data during the first and second heating cycle, respectively. As expected, data during different cycle matched well and no hysteresis has been observed. This stable and accurate performance should be attributed to the annealing process, which eliminates residual stress and thermal memory.

4. Conclusion

In conclusion, we have designed and demonstrated a single-taper MI structured high temperature sensor with good performance based on the low-loss telecom grade all-solid weak-coupling seven-core fiber. Enhanced inter-core interference has been obtained through programmable tapering at the SMF-MCF splice point. With the contribution of the second coupling at the spherical end face of MCF, the extinction ratio of MI spectrum is significantly improved (~25dB). With the wide FSR (166.34 nm with a MCF length of 10 mm) and distinguished wavelength dip in the reflection spectrum with high extinction ratio, accurate temperature measurement in a large range can be guaranteed. Between 250°C and 900°C, a sensitivity of 165 pm/°C has been achieved. Through optimizing the parameters of MCF, such as the doping content and the refractive index difference between central core and ambient cores, an even higher temperature sensitivity can be anticipated. This type of sensor is very promising due to its high sensitivity, high extinction ratio, wide detection range and simple fabrication process. It's worth noting that by taking advantage of the MCF’s characteristics of spatial-division multiplexing, we can measure multiple parameters simultaneously with the compatible fan-in/fan-out devices.

Founding

National Natural Science Foundation of China (Grant No. 61331010, 61290311), the 863 High Technology Plan (Grant No. 2013AA013402), and the Program for New Century Excellent Talents in University (NCET-13-0235).

References and links

1. B. Zhang and M. Kahrizi, “High-temperature resistance fiber Bragg grating temperature sensor fabrication,” IEEE Sens. J. 7(4), 586–591 (2007). [CrossRef]  

2. X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015). [CrossRef]  

3. J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

4. W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015). [CrossRef]   [PubMed]  

5. P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015). [CrossRef]  

6. T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010). [CrossRef]  

7. Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014). [CrossRef]  

8. L. V. Nguyen, D. Hwang, S. Moon, D. S. Moon, and Y. Chung, “High temperature fiber sensor with high sensitivity based on core diameter mismatch,” Opt. Express 16(15), 11369–11375 (2008). [CrossRef]   [PubMed]  

9. N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015). [CrossRef]  

10. J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015). [CrossRef]  

11. J. Yang, Y. Zheng, L. H. Chen, C. C. Chan, X. Dong, P. P. Shum, and H. Su, “Miniature temperature sensor with germania-core optical fiber,” Opt. Express 23(14), 17687–17692 (2015). [CrossRef]   [PubMed]  

12. M. S. Yoon, S. B. Lee, and Y. G. Han, “In-line interferometer based on intermodal coupling of a multicore fiber,” Opt. Express 23(14), 18316–18322 (2015). [CrossRef]   [PubMed]  

13. P. Rugeland and W. Margulis, “Revisiting twin-core fiber sensors for high-temperature measurements,” Appl. Opt. 51(25), 6227–6232 (2012). [CrossRef]   [PubMed]  

14. are A. Van Newkirk, Z. Sanjabi Eznaveh, E. Antonio-Lopez, G. Salceda-Delgado, A. Schulzgen, and R. Amezcua-Correa, “High temperature sensor based on supermode interference in multicore fiber,” in CLEO: 2014, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM2N.7.

15. J. E. Antonio-Lopez, Z. S. Eznaveh, P. L. Kamwa, A. Schulzgen, and R. A. Correa, “Multicore fiber sensor for high-temperature applications up to 1000 degrees C,” Opt. Lett. 39(15), 4309–4312 (2014). [CrossRef]   [PubMed]  

16. T. Zhu, D. Wu, M. Liu, and D. W. Duan, “In-line fiber optic interferometric sensors in single-mode fibers,” Sensors (Basel) 12(8), 10430–10449 (2012). [CrossRef]   [PubMed]  

17. B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015). [CrossRef]   [PubMed]  

18. Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013). [CrossRef]  

19. R. Wang, L. Duan, M. Tang, S. Fu, P. Zang, Z. Feng, B. Li, W. Tong, D. Liu, and P. P. Shum, “Long period grating in multicore fiber and its application,” in Asia Communications and Photonics Conference 2015, OSA Technical Digest (online) (Optical Society of America, 2015), AM1D. 5.

20. L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

21. G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, “All-fiber three-path Mach-Zehnder interferometer,” Opt. Lett. 21(4), 302–304 (1996). [CrossRef]   [PubMed]  

22. D. Wu, T. Zhu, K. S. Chiang, and M. Deng, “All single-mode fiber Mach–Zehnder interferometer based on two Peanut-Shape structures,” J. Lightwave Technol. 30(5), 805–810 (2012). [CrossRef]  

References

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  • |
  • |

  1. B. Zhang and M. Kahrizi, “High-temperature resistance fiber Bragg grating temperature sensor fabrication,” IEEE Sens. J. 7(4), 586–591 (2007).
    [Crossref]
  2. X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
    [Crossref]
  3. J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).
  4. W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015).
    [Crossref] [PubMed]
  5. P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
    [Crossref]
  6. T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
    [Crossref]
  7. Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
    [Crossref]
  8. L. V. Nguyen, D. Hwang, S. Moon, D. S. Moon, and Y. Chung, “High temperature fiber sensor with high sensitivity based on core diameter mismatch,” Opt. Express 16(15), 11369–11375 (2008).
    [Crossref] [PubMed]
  9. N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
    [Crossref]
  10. J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
    [Crossref]
  11. J. Yang, Y. Zheng, L. H. Chen, C. C. Chan, X. Dong, P. P. Shum, and H. Su, “Miniature temperature sensor with germania-core optical fiber,” Opt. Express 23(14), 17687–17692 (2015).
    [Crossref] [PubMed]
  12. M. S. Yoon, S. B. Lee, and Y. G. Han, “In-line interferometer based on intermodal coupling of a multicore fiber,” Opt. Express 23(14), 18316–18322 (2015).
    [Crossref] [PubMed]
  13. P. Rugeland and W. Margulis, “Revisiting twin-core fiber sensors for high-temperature measurements,” Appl. Opt. 51(25), 6227–6232 (2012).
    [Crossref] [PubMed]
  14. are A. Van Newkirk, Z. Sanjabi Eznaveh, E. Antonio-Lopez, G. Salceda-Delgado, A. Schulzgen, and R. Amezcua-Correa, “High temperature sensor based on supermode interference in multicore fiber,” in CLEO: 2014, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM2N.7.
  15. J. E. Antonio-Lopez, Z. S. Eznaveh, P. L. Kamwa, A. Schulzgen, and R. A. Correa, “Multicore fiber sensor for high-temperature applications up to 1000 degrees C,” Opt. Lett. 39(15), 4309–4312 (2014).
    [Crossref] [PubMed]
  16. T. Zhu, D. Wu, M. Liu, and D. W. Duan, “In-line fiber optic interferometric sensors in single-mode fibers,” Sensors (Basel) 12(8), 10430–10449 (2012).
    [Crossref] [PubMed]
  17. B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
    [Crossref] [PubMed]
  18. Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
    [Crossref]
  19. R. Wang, L. Duan, M. Tang, S. Fu, P. Zang, Z. Feng, B. Li, W. Tong, D. Liu, and P. P. Shum, “Long period grating in multicore fiber and its application,” in Asia Communications and Photonics Conference 2015, OSA Technical Digest (online) (Optical Society of America, 2015), AM1D. 5.
  20. L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).
  21. G. Weihs, M. Reck, H. Weinfurter, and A. Zeilinger, “All-fiber three-path Mach-Zehnder interferometer,” Opt. Lett. 21(4), 302–304 (1996).
    [Crossref] [PubMed]
  22. D. Wu, T. Zhu, K. S. Chiang, and M. Deng, “All single-mode fiber Mach–Zehnder interferometer based on two Peanut-Shape structures,” J. Lightwave Technol. 30(5), 805–810 (2012).
    [Crossref]

2015 (9)

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015).
[Crossref] [PubMed]

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

J. Yang, Y. Zheng, L. H. Chen, C. C. Chan, X. Dong, P. P. Shum, and H. Su, “Miniature temperature sensor with germania-core optical fiber,” Opt. Express 23(14), 17687–17692 (2015).
[Crossref] [PubMed]

M. S. Yoon, S. B. Lee, and Y. G. Han, “In-line interferometer based on intermodal coupling of a multicore fiber,” Opt. Express 23(14), 18316–18322 (2015).
[Crossref] [PubMed]

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

2014 (2)

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

J. E. Antonio-Lopez, Z. S. Eznaveh, P. L. Kamwa, A. Schulzgen, and R. A. Correa, “Multicore fiber sensor for high-temperature applications up to 1000 degrees C,” Opt. Lett. 39(15), 4309–4312 (2014).
[Crossref] [PubMed]

2013 (1)

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

2012 (3)

2010 (1)

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

2009 (1)

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

2008 (1)

2007 (1)

B. Zhang and M. Kahrizi, “High-temperature resistance fiber Bragg grating temperature sensor fabrication,” IEEE Sens. J. 7(4), 586–591 (2007).
[Crossref]

1996 (1)

Ahmad, H.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Antonio-Lopez, J. E.

Cao, Z.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Chan, C. C.

Chen, L. H.

Cheng, Y.

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Chiang, K. S.

D. Wu, T. Zhu, K. S. Chiang, and M. Deng, “All single-mode fiber Mach–Zehnder interferometer based on two Peanut-Shape structures,” J. Lightwave Technol. 30(5), 805–810 (2012).
[Crossref]

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

Chung, Y.

Correa, R. A.

Deng, L.

Deng, M.

Ding, W.

W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015).
[Crossref] [PubMed]

Dong, X.

Duan, D. W.

T. Zhu, D. Wu, M. Liu, and D. W. Duan, “In-line fiber optic interferometric sensors in single-mode fibers,” Sensors (Basel) 12(8), 10430–10449 (2012).
[Crossref] [PubMed]

Duan, L.

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

Eznaveh, Z. S.

Feng, Z.

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Fu, H.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Fu, S.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Galindez, C.

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Gan, L.

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Gao, F.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

Gao, H.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Gao, R.

W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015).
[Crossref] [PubMed]

Garcia-Allende, P. B.

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Guo, T.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Han, Y. G.

Hua, L.

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

Huang, J.

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

Hwang, D.

Ji, X.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Jiang, Y.

W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015).
[Crossref] [PubMed]

Jones, J. D. C.

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Kahrizi, M.

B. Zhang and M. Kahrizi, “High-temperature resistance fiber Bragg grating temperature sensor fabrication,” IEEE Sens. J. 7(4), 586–591 (2007).
[Crossref]

Kamwa, P. L.

Ke, T.

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

Lan, X.

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

Lazaro, J. M.

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Lee, S. B.

Li, B.

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Li, H.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Li, L.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Li, Y.

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

Lim, K. S.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Liu, D.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Liu, M.

T. Zhu, D. Wu, M. Liu, and D. W. Duan, “In-line fiber optic interferometric sensors in single-mode fibers,” Sensors (Basel) 12(8), 10430–10449 (2012).
[Crossref] [PubMed]

Liu, Q.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Liu, S.

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Liu, Y.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015).
[Crossref] [PubMed]

Lopez-Higuera, J. M.

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Lu, L.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Margulis, W.

Mirapeix, J.

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Moon, D. S.

Moon, S.

Nguyen, L. V.

Ouyang, J.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

Qiao, X.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Quintela, A.

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Rao, Y.

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

Reck, M.

Rong, Q.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Rugeland, P.

Schulzgen, A.

Shao, M.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Shui, T.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Shum, P. P.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

J. Yang, Y. Zheng, L. H. Chen, C. C. Chan, X. Dong, P. P. Shum, and H. Su, “Miniature temperature sensor with germania-core optical fiber,” Opt. Express 23(14), 17687–17692 (2015).
[Crossref] [PubMed]

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Song, Y.

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

Su, D.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Su, H.

Tang, M.

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Tong, W.

B. Li, Z. Feng, M. Tang, Z. Xu, S. Fu, Q. Wu, L. Deng, W. Tong, S. Liu, and P. P. Shum, “Experimental demonstration of large capacity WSDM optical access network with multicore fibers and advanced modulation formats,” Opt. Express 23(9), 10997–11006 (2015).
[Crossref] [PubMed]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Wang, R.

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Wang, Y.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Wei, H.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Weihs, G.

Weinfurter, H.

Wu, D.

D. Wu, T. Zhu, K. S. Chiang, and M. Deng, “All single-mode fiber Mach–Zehnder interferometer based on two Peanut-Shape structures,” J. Lightwave Technol. 30(5), 805–810 (2012).
[Crossref]

T. Zhu, D. Wu, M. Liu, and D. W. Duan, “In-line fiber optic interferometric sensors in single-mode fibers,” Sensors (Basel) 12(8), 10430–10449 (2012).
[Crossref] [PubMed]

Wu, Q.

Xiao, H.

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

Xu, Z.

Yan, X.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Yang, H.

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

Yang, J.

Yin, C.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Yoon, M. S.

Yu, B.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Zeilinger, A.

Zhang, B.

B. Zhang and M. Kahrizi, “High-temperature resistance fiber Bragg grating temperature sensor fabrication,” IEEE Sens. J. 7(4), 586–591 (2007).
[Crossref]

Zhang, P.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

Zhang, Z.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Zhao, N.

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Zhao, Z.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

Zhen, S.

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Zheng, Y.

Zhu, B.

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

Zhu, T.

T. Zhu, D. Wu, M. Liu, and D. W. Duan, “In-line fiber optic interferometric sensors in single-mode fibers,” Sensors (Basel) 12(8), 10430–10449 (2012).
[Crossref] [PubMed]

D. Wu, T. Zhu, K. S. Chiang, and M. Deng, “All single-mode fiber Mach–Zehnder interferometer based on two Peanut-Shape structures,” J. Lightwave Technol. 30(5), 805–810 (2012).
[Crossref]

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

Appl. Opt. (1)

Appl. Phys. B (1)

Z. Zhao, M. Tang, S. Fu, S. Liu, H. Wei, Y. Cheng, W. Tong, P. P. Shum, and D. Liu, “All-solid multi-core fiber-based multipath Mach-Zehnder interferometer for temperature sensing,” Appl. Phys. B 112(4), 491–497 (2013).
[Crossref]

IEEE Photonics J. (2)

P. Zhang, M. Tang, F. Gao, B. Zhu, Z. Zhao, L. Duan, S. Fu, J. Ouyang, H. Wei, P. P. Shum, and D. Liu, “Simplified hollow-core fiber-based Fabry-Perot interferometer with modified vernier effect for highly sensitive high-temperature measurement,” IEEE Photonics J. 7(1), 7100210 (2015).
[Crossref]

L. Gan, R. Wang, D. Liu, L. Duan, S. Liu, S. Fu, B. Li, Z. Feng, H. Wei, W. Tong, P. P. Shum, and M. Tang, “Spatial-division multiplexed Mach-Zehnder interferometers in heterogeneous multicore fiber for multi-parameter measurement,” IEEE Photonics J. 8(1), 7800908 (2015).

IEEE Photonics Technol. Lett. (2)

X. Qiao, Y. Wang, H. Yang, T. Guo, Q. Rong, L. Li, D. Su, K. S. Lim, and H. Ahmad, “Ultra-high temperature chirped fiber Bragg grating through thermal activation,” IEEE Photonics Technol. Lett. 27(12), 1305–1308 (2015).
[Crossref]

J. Huang, X. Lan, Y. Song, Y. Li, L. Hua, and H. Xiao, “Microwave interrogated sapphire fiber Michelson interferometer for high temperature sensing,” IEEE Photonics Technol. Lett. 27(13), 1398–1401 (2015).
[Crossref]

IEEE Sens. J. (1)

B. Zhang and M. Kahrizi, “High-temperature resistance fiber Bragg grating temperature sensor fabrication,” IEEE Sens. J. 7(4), 586–591 (2007).
[Crossref]

J. Lightwave Technol. (1)

Opt. Commun. (2)

T. Zhu, T. Ke, Y. Rao, and K. S. Chiang, “Fabry-Perot optical fiber tip sensor for high temperature measurement,” Opt. Commun. 283(19), 3683–3685 (2010).
[Crossref]

N. Zhao, H. Fu, M. Shao, X. Yan, H. Li, Q. Liu, H. Gao, Y. Liu, and X. Qiao, “High temperature probe sensor with high sensitivity based on Michelson interferometer,” Opt. Commun. 343, 131–134 (2015).
[Crossref]

Opt. Express (4)

Opt. Fiber Technol. (1)

Z. Cao, Z. Zhang, X. Ji, T. Shui, R. Wang, C. Yin, S. Zhen, L. Lu, and B. Yu, “Strain-insensitive and high temperature fiber sensor based on a Mach-Zehnder modal interferometer,” Opt. Fiber Technol. 20(1), 24–27 (2014).
[Crossref]

Opt. Lett. (2)

Proc. SPIE (1)

J. D. C. Jones, J. M. Lazaro, A. Quintela, P. B. Garcia-Allende, J. Mirapeix, C. Galindez, and J. M. Lopez-Higuera, “High temperature fiber sensor based on a thermo-mechanical written LPG,” Proc. SPIE 7503, 75033W (2009).

Rev. Sci. Instrum. (1)

W. Ding, Y. Jiang, R. Gao, and Y. Liu, “High-temperature fiber-optic Fabry-Perot interferometric sensors,” Rev. Sci. Instrum. 86(5), 055001 (2015).
[Crossref] [PubMed]

Sensors (Basel) (1)

T. Zhu, D. Wu, M. Liu, and D. W. Duan, “In-line fiber optic interferometric sensors in single-mode fibers,” Sensors (Basel) 12(8), 10430–10449 (2012).
[Crossref] [PubMed]

Other (2)

are A. Van Newkirk, Z. Sanjabi Eznaveh, E. Antonio-Lopez, G. Salceda-Delgado, A. Schulzgen, and R. Amezcua-Correa, “High temperature sensor based on supermode interference in multicore fiber,” in CLEO: 2014, OSA Technical Digest (online) (Optical Society of America, 2014), paper SM2N.7.

R. Wang, L. Duan, M. Tang, S. Fu, P. Zang, Z. Feng, B. Li, W. Tong, D. Liu, and P. P. Shum, “Long period grating in multicore fiber and its application,” in Asia Communications and Photonics Conference 2015, OSA Technical Digest (online) (Optical Society of America, 2015), AM1D. 5.

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Figures (6)

Fig. 1
Fig. 1 (a) The cross section image of the MCF. (b) Schematic diagram and operation principle of the Michelson-type multipath interferometer. (c) MCF-MI structure under microscope. (d) Spherical end face of MCF under microscope.
Fig. 2
Fig. 2 Light distributions at MCF end face and simulated electric field intensity distribution with different processing methods of fusion point. (a) Splicing only. (b) Splicing and tapering. (c). FFT-spatial frequency spectrum of the reflectivity spectrum with different processing methods at MCF end face. Inset: The blue curve and red curve show the results of spatial frequency spectra with and without spherical end face, respectively.
Fig. 3
Fig. 3 (a) Typical reflectance spectrum of MCF- MI device, the length of MCF is 10mm. (b) FFT-spatial frequency spectrum of the reflectance spectrums with different MCF length.
Fig. 4
Fig. 4 Experimental setup for the MCF-MI for temperature sensing.
Fig. 5
Fig. 5 (a) Measured interference pattern for the proposed device for different MCF lengths. (b) Spectral response to temperature for the MCF based Michelson interferometer.
Fig. 6
Fig. 6 (a) Dip wavelength shift with temperature increasing and its linear fit. (b) Repeatability: the dip wavelength at different temperatures during the first and second heating cycles.

Equations (10)

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Δ n e f f = ζ λ 0 2 / 2 L ,
E 0 = ( n = 1 N E n 0 cos δ n ) 2 + ( n = 1 N E n 0 sin δ n ) 2 ,
δ = arc tan ( n = 1 N E n 0 sin δ n / n = 1 N E n 0 cos δ n ) ,
I = I c + 36 I s + 12 I c I s cos Δ Φ ,
Δ Φ = δ c δ s ,
δ = arc tan ( E c sin δ c + 6 E s sin δ s E c cos δ c + 6 E s cos δ s ) ,
Δ Φ = [ 4 π L ( n e f f , c n e f f , s ) ] / λ ,
λ T = 4 [ ( n e f f , c T n e f f , s T ) L + ( n e f f , c n e f f , s ) L T ] / ( 2 m + 1 ) ,
λ T = ( Δ n e f f T / Δ n e f f + L T / L ) λ ,
Δ λ F S R λ 0 2 / 2 Δ n e f f L ,

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