Abstract

We present a distributed refractive index (RI) sensor using tapered optical fibers in optical frequency domain reflectometry (OFDR). RI of the external medium surrounding the tapered optical fibers is measured by the optical frequency shifts of the local back-reflection spectra in OFDR. By a spectrum interpolation, we can increase the resolution of RI measurements without decreasing the sensing spatial resolution. In our experiments, we realize a truly distributed RI sensing with a 4.25 mm spatial resolution and 2.1 cm measurement distance. We calibrate the relationship between the optical frequency shifts of the local back-reflection spectra and RI variation. RI ranges from 1.3574 to 1.3686 and the sensitivity is about 8565 GHz/RIU (68.52 nm/RIU) in the presented sensor. We also measure RI variation in a glycerol solution diffusion to verify the capability of distributed RI sensing by the presented sensor.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (3)

T. Wang, K. Liu, J. Jiang, M. Xue, P. Chang, and T. Liu, “Temperature-insensitive refractive index sensor based on tilted moiré FBG with high resolution,” Opt. Express 25(13), 14900–14909 (2017).
[Crossref] [PubMed]

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Y. Du, S. Jothibasu, Y. Zhuang, C. Zhu, and J. Huang, “Rayleigh backscattering based macrobending single mode fiber for distributed refractive index sensing,” Sens. Actuators B Chem. 248, 346–350 (2017).
[Crossref]

2015 (5)

Y. Chen, Q. Han, T. Liu, and X. Lü, “Self-temperature-compensative refractometer based on single mode–multimode–single mode fiber structure,” Sens. Actuators B Chem. 212, 107–111 (2015).
[Crossref]

Z. Chen, G. Hefferman, and T. Wei, “Multiplexed oil level meter using a thin core fiber cladding mode exciter,” IEEE Photonics Technol. Lett. 27(21), 2215–2218 (2015).
[Crossref]

J. Harris, P. Lu, H. Larocque, L. Chen, and X. Bao, “In-fiber Mach–Zehnder interferometric refractive index sensors with guided and leaky modes,” Sens. Actuators B Chem. 206, 246–251 (2015).
[Crossref]

X. Zhang and W. Peng, “Fiber optic refractometer based on leaky-mode interference of bent fiber,” IEEE Photonics Technol. Lett. 27(1), 11–14 (2015).
[Crossref]

Y. Chen, Q. Han, T. Liu, F. Liu, and Y. Yao, “Simultaneous measurement of refractive index and temperature using a cascaded FBG/droplet-like fiber structure,” IEEE Sens. J. 15(11), 6432–6436 (2015).
[Crossref]

2014 (2)

F. De-Jun, L. Guan-Xiu, L. Xi-Lu, J. Ming-Shun, and S. Qing-Mei, “Refractive index sensor based on plastic optical fiber with tapered structure,” Appl. Opt. 53(10), 2007–2011 (2014).
[Crossref] [PubMed]

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

2013 (2)

2012 (4)

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

J. Zhang, Q. Sun, R. Liang, J. Wo, D. Liu, and P. Shum, “Microfiber Fabry-Perot interferometer fabricated by taper-drawing technique and its application as a radio frequency interrogated refractive index sensor,” Opt. Lett. 37(14), 2925–2927 (2012).
[Crossref] [PubMed]

J. Wo, G. Wang, Y. Cui, Q. Sun, R. Liang, P. P. Shum, P. P. Shum, and D. Liu, “Refractive index sensor using microfiber-based Mach-Zehnder interferometer,” Opt. Lett. 37(1), 67–69 (2012).
[Crossref] [PubMed]

W. B. Ji, H. H. Liu, S. C. Tjin, K. K. Chow, and A. Lim, “Ultrahigh sensitivity refractive index sensor based on optical microfiber,” IEEE Photonics Technol. Lett. 24(20), 1872–1874 (2012).
[Crossref]

2011 (2)

Y. Tian, W. Wang, N. Wu, X. Zou, and X. Wang, “Tapered optical fiber sensor for label-free detection of biomolecules,” Sensors (Basel) 11(4), 3780–3790 (2011).
[Crossref] [PubMed]

Q. Wu, Y. Semenova, P. Wang, and G. Farrell, “High sensitivity SMS fiber structure based refractometer--analysis and experiment,” Opt. Express 19(9), 7937–7944 (2011).
[Crossref] [PubMed]

2010 (5)

2009 (4)

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach–Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

D. K. Wu, B. T. Kuhlmey, and B. J. Eggleton, “Ultrasensitive photonic crystal fiber refractive index sensor,” Opt. Lett. 34(3), 322–324 (2009).
[Crossref] [PubMed]

O. Frazão, P. Caldas, J. L. Santos, P. V. S. Marques, C. Turck, D. J. Lougnot, and O. Soppera, “Fabry-Perot refractometer based on an end-of-fiber polymer tip,” Opt. Lett. 34(16), 2474–2476 (2009).
[Crossref] [PubMed]

R. St. Gelais, J. Masson, and Y. A. Peter, “All-silicon integrated Fabry–Pérot cavity for volume refractive index measurement in microfluidic systems,” Appl. Phys. Lett. 94(24), 243905 (2009).
[Crossref]

2008 (2)

2007 (1)

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79(3), 930–937 (2007).
[Crossref] [PubMed]

2006 (1)

D. Monzón-Hernández and J. Villatoro, “High-resolution refractive index sensing by means of a multiple-peak surface plasmon resonance optical fiber sensor,” Sens. Actuators B Chem. 115(1), 227–231 (2006).
[Crossref]

2005 (3)

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(20), 4057 (2005).
[Crossref]

D. Monzón-Hernández, J. Villatoro, and D. Luna-Moreno, “Miniature optical fiber refractometer using cladded multimode tapered fiber tips,” Sens. Actuators B Chem. 110(1), 36–40 (2005).
[Crossref]

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor,” IEEE Photonics Technol. Lett. 17(6), 1247–1249 (2005).
[Crossref]

2004 (2)

J. Villatoro, D. Monzon-Hernandez, and D. Talavera, “High resolution refractive index sensing with cladded multimode tapered optical fibre,” Electron. Lett. 40(2), 106–107 (2004).
[Crossref]

G. Brambilla, V. Finazzi, and D. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express 12(10), 2258–2263 (2004).
[Crossref] [PubMed]

2003 (1)

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

1998 (1)

Ashcom, J. B.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Bao, X.

J. Harris, P. Lu, H. Larocque, L. Chen, and X. Bao, “In-fiber Mach–Zehnder interferometric refractive index sensors with guided and leaky modes,” Sens. Actuators B Chem. 206, 246–251 (2015).
[Crossref]

Brambilla, G.

G. Brambilla, “Optical fibre nanowires and microwires: a review,” J. Opt. 12(4), 043001 (2010).
[Crossref]

F. Xu and G. Brambilla, “Demonstration of a refractometric sensor based on optical microfiber coil resonator,” Appl. Phys. Lett. 92(10), 101126 (2008).
[Crossref]

G. Brambilla, V. Finazzi, and D. Richardson, “Ultra-low-loss optical fiber nanotapers,” Opt. Express 12(10), 2258–2263 (2004).
[Crossref] [PubMed]

Caldas, P.

Chang, P.

Chen, L.

J. Harris, P. Lu, H. Larocque, L. Chen, and X. Bao, “In-fiber Mach–Zehnder interferometric refractive index sensors with guided and leaky modes,” Sens. Actuators B Chem. 206, 246–251 (2015).
[Crossref]

Chen, Q.

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach–Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

Chen, Y.

Y. Chen, Q. Han, T. Liu, and X. Lü, “Self-temperature-compensative refractometer based on single mode–multimode–single mode fiber structure,” Sens. Actuators B Chem. 212, 107–111 (2015).
[Crossref]

Y. Chen, Q. Han, T. Liu, F. Liu, and Y. Yao, “Simultaneous measurement of refractive index and temperature using a cascaded FBG/droplet-like fiber structure,” IEEE Sens. J. 15(11), 6432–6436 (2015).
[Crossref]

Chen, Z.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Z. Chen, G. Hefferman, and T. Wei, “Multiplexed oil level meter using a thin core fiber cladding mode exciter,” IEEE Photonics Technol. Lett. 27(21), 2215–2218 (2015).
[Crossref]

Chow, K. K.

W. B. Ji, H. H. Liu, S. C. Tjin, K. K. Chow, and A. Lim, “Ultrahigh sensitivity refractive index sensor based on optical microfiber,” IEEE Photonics Technol. Lett. 24(20), 1872–1874 (2012).
[Crossref]

Cui, Y.

De-Jun, F.

Ding, J. F.

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor,” IEEE Photonics Technol. Lett. 17(6), 1247–1249 (2005).
[Crossref]

Du, Y.

Y. Du, S. Jothibasu, Y. Zhuang, C. Zhu, and J. Huang, “Rayleigh backscattering based macrobending single mode fiber for distributed refractive index sensing,” Sens. Actuators B Chem. 248, 346–350 (2017).
[Crossref]

Eggleton, B. J.

Fan, X.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79(3), 930–937 (2007).
[Crossref] [PubMed]

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(20), 4057 (2005).
[Crossref]

Farrell, G.

Finazzi, V.

Frazão, O.

Froggatt, M.

Gao, S.

Gao, Z.

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

Gattass, R. R.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Guan, B. O.

Guan, H.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Guan, J.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Guan-Xiu, L.

Guo, F.

Han, M.

Han, Q.

Y. Chen, Q. Han, T. Liu, and X. Lü, “Self-temperature-compensative refractometer based on single mode–multimode–single mode fiber structure,” Sens. Actuators B Chem. 212, 107–111 (2015).
[Crossref]

Y. Chen, Q. Han, T. Liu, F. Liu, and Y. Yao, “Simultaneous measurement of refractive index and temperature using a cascaded FBG/droplet-like fiber structure,” IEEE Sens. J. 15(11), 6432–6436 (2015).
[Crossref]

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

Han, Y.

Hanumegowda, N. M.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(20), 4057 (2005).
[Crossref]

Harris, J.

J. Harris, P. Lu, H. Larocque, L. Chen, and X. Bao, “In-fiber Mach–Zehnder interferometric refractive index sensors with guided and leaky modes,” Sens. Actuators B Chem. 206, 246–251 (2015).
[Crossref]

He, S.

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor,” IEEE Photonics Technol. Lett. 17(6), 1247–1249 (2005).
[Crossref]

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Hefferman, G.

Z. Chen, G. Hefferman, and T. Wei, “Multiplexed oil level meter using a thin core fiber cladding mode exciter,” IEEE Photonics Technol. Lett. 27(21), 2215–2218 (2015).
[Crossref]

Huang, J.

Y. Du, S. Jothibasu, Y. Zhuang, C. Zhu, and J. Huang, “Rayleigh backscattering based macrobending single mode fiber for distributed refractive index sensing,” Sens. Actuators B Chem. 248, 346–350 (2017).
[Crossref]

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

Ji, W. B.

W. B. Ji, H. H. Liu, S. C. Tjin, K. K. Chow, and A. Lim, “Ultrahigh sensitivity refractive index sensor based on optical microfiber,” IEEE Photonics Technol. Lett. 24(20), 1872–1874 (2012).
[Crossref]

Jiang, J.

Jin, L.

Jin, S.

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Jothibasu, S.

Y. Du, S. Jothibasu, Y. Zhuang, C. Zhu, and J. Huang, “Rayleigh backscattering based macrobending single mode fiber for distributed refractive index sensing,” Sens. Actuators B Chem. 248, 346–350 (2017).
[Crossref]

Kaur, A.

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

Kuhlmey, B. T.

Lan, X.

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

Larocque, H.

J. Harris, P. Lu, H. Larocque, L. Chen, and X. Bao, “In-fiber Mach–Zehnder interferometric refractive index sensors with guided and leaky modes,” Sens. Actuators B Chem. 206, 246–251 (2015).
[Crossref]

Li, J.

Li, Y.

P. Wang, Y. Semenova, Y. Li, Q. Wu, and G. Farrell, “A macrobending single mode fiber refractive index sensor for low refractive index liquids,” Photonics Lett. Pol. 2(2), 67–69 (2010).
[Crossref]

T. Wei, Y. Han, Y. Li, H. L. Tsai, and H. Xiao, “Temperature-insensitive miniaturized fiber inline Fabry-Perot interferometer for highly sensitive refractive index measurement,” Opt. Express 16(8), 5764–5769 (2008).
[Crossref] [PubMed]

Liang, R.

Lim, A.

W. B. Ji, H. H. Liu, S. C. Tjin, K. K. Chow, and A. Lim, “Ultrahigh sensitivity refractive index sensor based on optical microfiber,” IEEE Photonics Technol. Lett. 24(20), 1872–1874 (2012).
[Crossref]

Liu, D.

Liu, F.

Y. Chen, Q. Han, T. Liu, F. Liu, and Y. Yao, “Simultaneous measurement of refractive index and temperature using a cascaded FBG/droplet-like fiber structure,” IEEE Sens. J. 15(11), 6432–6436 (2015).
[Crossref]

Liu, H. H.

W. B. Ji, H. H. Liu, S. C. Tjin, K. K. Chow, and A. Lim, “Ultrahigh sensitivity refractive index sensor based on optical microfiber,” IEEE Photonics Technol. Lett. 24(20), 1872–1874 (2012).
[Crossref]

Liu, K.

Liu, S.

Liu, T.

T. Wang, K. Liu, J. Jiang, M. Xue, P. Chang, and T. Liu, “Temperature-insensitive refractive index sensor based on tilted moiré FBG with high resolution,” Opt. Express 25(13), 14900–14909 (2017).
[Crossref] [PubMed]

Y. Chen, Q. Han, T. Liu, and X. Lü, “Self-temperature-compensative refractometer based on single mode–multimode–single mode fiber structure,” Sens. Actuators B Chem. 212, 107–111 (2015).
[Crossref]

Y. Chen, Q. Han, T. Liu, F. Liu, and Y. Yao, “Simultaneous measurement of refractive index and temperature using a cascaded FBG/droplet-like fiber structure,” IEEE Sens. J. 15(11), 6432–6436 (2015).
[Crossref]

Long, S.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Lou, J.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Lougnot, D. J.

Lu, H.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Lu, P.

J. Harris, P. Lu, H. Larocque, L. Chen, and X. Bao, “In-fiber Mach–Zehnder interferometric refractive index sensors with guided and leaky modes,” Sens. Actuators B Chem. 206, 246–251 (2015).
[Crossref]

Y. Wang, M. Yang, D. N. Wang, S. Liu, and P. Lu, “Fiber in-line Mach-Zehnder interferometer fabricated by femtosecond laser micromachining for refractive index measurement with high sensitivity,” J. Opt. Soc. Am. B 27(3), 370–374 (2010).
[Crossref]

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach–Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

Lu, Y.

Lü, X.

Y. Chen, Q. Han, T. Liu, and X. Lü, “Self-temperature-compensative refractometer based on single mode–multimode–single mode fiber structure,” Sens. Actuators B Chem. 212, 107–111 (2015).
[Crossref]

Luna-Moreno, D.

D. Monzón-Hernández, J. Villatoro, and D. Luna-Moreno, “Miniature optical fiber refractometer using cladded multimode tapered fiber tips,” Sens. Actuators B Chem. 110(1), 36–40 (2005).
[Crossref]

Luo, Y.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Marques, P. V. S.

Masson, J.

R. St. Gelais, J. Masson, and Y. A. Peter, “All-silicon integrated Fabry–Pérot cavity for volume refractive index measurement in microfluidic systems,” Appl. Phys. Lett. 94(24), 243905 (2009).
[Crossref]

Maxwell, I.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Mazur, E.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Men, L.

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach–Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

Ming-Shun, J.

Monzon-Hernandez, D.

J. Villatoro, D. Monzon-Hernandez, and D. Talavera, “High resolution refractive index sensing with cladded multimode tapered optical fibre,” Electron. Lett. 40(2), 106–107 (2004).
[Crossref]

Monzón-Hernández, D.

D. Monzón-Hernández and J. Villatoro, “High-resolution refractive index sensing by means of a multiple-peak surface plasmon resonance optical fiber sensor,” Sens. Actuators B Chem. 115(1), 227–231 (2006).
[Crossref]

D. Monzón-Hernández, J. Villatoro, and D. Luna-Moreno, “Miniature optical fiber refractometer using cladded multimode tapered fiber tips,” Sens. Actuators B Chem. 110(1), 36–40 (2005).
[Crossref]

Moore, J.

Patel, B. C.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(20), 4057 (2005).
[Crossref]

Peng, W.

X. Zhang and W. Peng, “Fiber optic refractometer based on leaky-mode interference of bent fiber,” IEEE Photonics Technol. Lett. 27(1), 11–14 (2015).
[Crossref]

Peter, Y. A.

R. St. Gelais, J. Masson, and Y. A. Peter, “All-silicon integrated Fabry–Pérot cavity for volume refractive index measurement in microfluidic systems,” Appl. Phys. Lett. 94(24), 243905 (2009).
[Crossref]

Qi, L.

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Qing-Mei, S.

Richardson, D.

Santos, J. L.

Semenova, Y.

Shao, L. Y.

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor,” IEEE Photonics Technol. Lett. 17(6), 1247–1249 (2005).
[Crossref]

Shen, M.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Shum, P.

Shum, P. P.

Sooley, K.

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach–Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

Soppera, O.

St. Gelais, R.

R. St. Gelais, J. Masson, and Y. A. Peter, “All-silicon integrated Fabry–Pérot cavity for volume refractive index measurement in microfluidic systems,” Appl. Phys. Lett. 94(24), 243905 (2009).
[Crossref]

Stica, C. J.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(20), 4057 (2005).
[Crossref]

Sun, L. P.

Sun, Q.

Suter, J. D.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79(3), 930–937 (2007).
[Crossref] [PubMed]

Talavera, D.

J. Villatoro, D. Monzon-Hernandez, and D. Talavera, “High resolution refractive index sensing with cladded multimode tapered optical fibre,” Electron. Lett. 40(2), 106–107 (2004).
[Crossref]

Tan, Y.

Tang, J.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Tian, Y.

Y. Tian, W. Wang, N. Wu, X. Zou, and X. Wang, “Tapered optical fiber sensor for label-free detection of biomolecules,” Sensors (Basel) 11(4), 3780–3790 (2011).
[Crossref] [PubMed]

Tjin, S. C.

W. B. Ji, H. H. Liu, S. C. Tjin, K. K. Chow, and A. Lim, “Ultrahigh sensitivity refractive index sensor based on optical microfiber,” IEEE Photonics Technol. Lett. 24(20), 1872–1874 (2012).
[Crossref]

Tong, L.

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Tsai, H. L.

Turck, C.

Villatoro, J.

D. Monzón-Hernández and J. Villatoro, “High-resolution refractive index sensing by means of a multiple-peak surface plasmon resonance optical fiber sensor,” Sens. Actuators B Chem. 115(1), 227–231 (2006).
[Crossref]

D. Monzón-Hernández, J. Villatoro, and D. Luna-Moreno, “Miniature optical fiber refractometer using cladded multimode tapered fiber tips,” Sens. Actuators B Chem. 110(1), 36–40 (2005).
[Crossref]

J. Villatoro, D. Monzon-Hernandez, and D. Talavera, “High resolution refractive index sensing with cladded multimode tapered optical fibre,” Electron. Lett. 40(2), 106–107 (2004).
[Crossref]

Wang, D. N.

Wang, G.

Wang, J.

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Wang, P.

Wang, T.

Wang, W.

Y. Tian, W. Wang, N. Wu, X. Zou, and X. Wang, “Tapered optical fiber sensor for label-free detection of biomolecules,” Sensors (Basel) 11(4), 3780–3790 (2011).
[Crossref] [PubMed]

Wang, X.

Y. Tian, W. Wang, N. Wu, X. Zou, and X. Wang, “Tapered optical fiber sensor for label-free detection of biomolecules,” Sensors (Basel) 11(4), 3780–3790 (2011).
[Crossref] [PubMed]

Wang, X. D.

X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
[Crossref] [PubMed]

Wang, Y.

Wei, T.

Z. Chen, G. Hefferman, and T. Wei, “Multiplexed oil level meter using a thin core fiber cladding mode exciter,” IEEE Photonics Technol. Lett. 27(21), 2215–2218 (2015).
[Crossref]

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

T. Wei, Y. Han, Y. Li, H. L. Tsai, and H. Xiao, “Temperature-insensitive miniaturized fiber inline Fabry-Perot interferometer for highly sensitive refractive index measurement,” Opt. Express 16(8), 5764–5769 (2008).
[Crossref] [PubMed]

White, I.

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(20), 4057 (2005).
[Crossref]

White, I. M.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79(3), 930–937 (2007).
[Crossref] [PubMed]

Wo, J.

Wolfbeis, O. S.

X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
[Crossref] [PubMed]

Wu, D. K.

Wu, N.

Y. Tian, W. Wang, N. Wu, X. Zou, and X. Wang, “Tapered optical fiber sensor for label-free detection of biomolecules,” Sensors (Basel) 11(4), 3780–3790 (2011).
[Crossref] [PubMed]

Wu, Q.

Xiao, H.

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

T. Wei, Y. Han, Y. Li, H. L. Tsai, and H. Xiao, “Temperature-insensitive miniaturized fiber inline Fabry-Perot interferometer for highly sensitive refractive index measurement,” Opt. Express 16(8), 5764–5769 (2008).
[Crossref] [PubMed]

Xi-Lu, L.

Xu, F.

F. Xu and G. Brambilla, “Demonstration of a refractometric sensor based on optical microfiber coil resonator,” Appl. Phys. Lett. 92(10), 101126 (2008).
[Crossref]

Xue, M.

Yan, J. H.

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor,” IEEE Photonics Technol. Lett. 17(6), 1247–1249 (2005).
[Crossref]

Yang, M.

Yao, Y.

Y. Chen, Q. Han, T. Liu, F. Liu, and Y. Yao, “Simultaneous measurement of refractive index and temperature using a cascaded FBG/droplet-like fiber structure,” IEEE Sens. J. 15(11), 6432–6436 (2015).
[Crossref]

Ye, M.

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Yu, J.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Yuan, J.

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Zhang, A. P.

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor,” IEEE Photonics Technol. Lett. 17(6), 1247–1249 (2005).
[Crossref]

Zhang, J.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

J. Zhang, Q. Sun, R. Liang, J. Wo, D. Liu, and P. Shum, “Microfiber Fabry-Perot interferometer fabricated by taper-drawing technique and its application as a radio frequency interrogated refractive index sensor,” Opt. Lett. 37(14), 2925–2927 (2012).
[Crossref] [PubMed]

Zhang, X.

X. Zhang and W. Peng, “Fiber optic refractometer based on leaky-mode interference of bent fiber,” IEEE Photonics Technol. Lett. 27(1), 11–14 (2015).
[Crossref]

Zhang, Z.

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Zhao, C. L.

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Zheng, J.

Zhou, J.

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

Zhu, C.

Y. Du, S. Jothibasu, Y. Zhuang, C. Zhu, and J. Huang, “Rayleigh backscattering based macrobending single mode fiber for distributed refractive index sensing,” Sens. Actuators B Chem. 248, 346–350 (2017).
[Crossref]

Zhu, H.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79(3), 930–937 (2007).
[Crossref] [PubMed]

Zhuang, Y.

Y. Du, S. Jothibasu, Y. Zhuang, C. Zhu, and J. Huang, “Rayleigh backscattering based macrobending single mode fiber for distributed refractive index sensing,” Sens. Actuators B Chem. 248, 346–350 (2017).
[Crossref]

Zou, X.

Y. Tian, W. Wang, N. Wu, X. Zou, and X. Wang, “Tapered optical fiber sensor for label-free detection of biomolecules,” Sensors (Basel) 11(4), 3780–3790 (2011).
[Crossref] [PubMed]

Zourob, M.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79(3), 930–937 (2007).
[Crossref] [PubMed]

Anal. Chem. (2)

X. D. Wang and O. S. Wolfbeis, “Fiber-optic chemical sensors and biosensors (2008-2012),” Anal. Chem. 85(2), 487–508 (2013).
[Crossref] [PubMed]

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, “Integrated refractive index optical ring resonator detector for capillary electrophoresis,” Anal. Chem. 79(3), 930–937 (2007).
[Crossref] [PubMed]

Appl. Opt. (3)

Appl. Phys. Lett. (4)

R. St. Gelais, J. Masson, and Y. A. Peter, “All-silicon integrated Fabry–Pérot cavity for volume refractive index measurement in microfluidic systems,” Appl. Phys. Lett. 94(24), 243905 (2009).
[Crossref]

F. Xu and G. Brambilla, “Demonstration of a refractometric sensor based on optical microfiber coil resonator,” Appl. Phys. Lett. 92(10), 101126 (2008).
[Crossref]

P. Lu, L. Men, K. Sooley, and Q. Chen, “Tapered fiber Mach–Zehnder interferometer for simultaneous measurement of refractive index and temperature,” Appl. Phys. Lett. 94(13), 131110 (2009).
[Crossref]

N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(20), 4057 (2005).
[Crossref]

Electron. Lett. (1)

J. Villatoro, D. Monzon-Hernandez, and D. Talavera, “High resolution refractive index sensing with cladded multimode tapered optical fibre,” Electron. Lett. 40(2), 106–107 (2004).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Tang, J. Zhou, J. Guan, S. Long, J. Yu, H. Guan, H. Lu, Y. Luo, J. Zhang, and Z. Chen, “Fabrication of side-polished single mode-multimode-single mode fiber and its characteristics of refractive index sensing,” IEEE J. Sel. Top. Quantum Electron. 23(2), 1–8 (2017).
[Crossref]

IEEE Photonics Technol. Lett. (5)

J. F. Ding, A. P. Zhang, L. Y. Shao, J. H. Yan, and S. He, “Fiber-taper seeded long-period grating pair as a highly sensitive refractive-index sensor,” IEEE Photonics Technol. Lett. 17(6), 1247–1249 (2005).
[Crossref]

W. B. Ji, H. H. Liu, S. C. Tjin, K. K. Chow, and A. Lim, “Ultrahigh sensitivity refractive index sensor based on optical microfiber,” IEEE Photonics Technol. Lett. 24(20), 1872–1874 (2012).
[Crossref]

Z. Chen, G. Hefferman, and T. Wei, “Multiplexed oil level meter using a thin core fiber cladding mode exciter,” IEEE Photonics Technol. Lett. 27(21), 2215–2218 (2015).
[Crossref]

Q. Han, X. Lan, J. Huang, A. Kaur, T. Wei, Z. Gao, and H. Xiao, “Long-period grating inscribed on concatenated double-clad and single-clad fiber for simultaneous measurement of temperature and refractive index,” IEEE Photonics Technol. Lett. 24(13), 1130–1132 (2012).
[Crossref]

X. Zhang and W. Peng, “Fiber optic refractometer based on leaky-mode interference of bent fiber,” IEEE Photonics Technol. Lett. 27(1), 11–14 (2015).
[Crossref]

IEEE Sens. J. (1)

Y. Chen, Q. Han, T. Liu, F. Liu, and Y. Yao, “Simultaneous measurement of refractive index and temperature using a cascaded FBG/droplet-like fiber structure,” IEEE Sens. J. 15(11), 6432–6436 (2015).
[Crossref]

J. Opt. (1)

G. Brambilla, “Optical fibre nanowires and microwires: a review,” J. Opt. 12(4), 043001 (2010).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (1)

L. Tong, R. R. Gattass, J. B. Ashcom, S. He, J. Lou, M. Shen, I. Maxwell, and E. Mazur, “Subwavelength-diameter silica wires for low-loss optical wave guiding,” Nature 426(6968), 816–819 (2003).
[Crossref] [PubMed]

Opt. Express (5)

Opt. Lett. (5)

Photonics Lett. Pol. (1)

P. Wang, Y. Semenova, Y. Li, Q. Wu, and G. Farrell, “A macrobending single mode fiber refractive index sensor for low refractive index liquids,” Photonics Lett. Pol. 2(2), 67–69 (2010).
[Crossref]

Sens. Actuators B Chem. (6)

D. Monzón-Hernández and J. Villatoro, “High-resolution refractive index sensing by means of a multiple-peak surface plasmon resonance optical fiber sensor,” Sens. Actuators B Chem. 115(1), 227–231 (2006).
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J. Harris, P. Lu, H. Larocque, L. Chen, and X. Bao, “In-fiber Mach–Zehnder interferometric refractive index sensors with guided and leaky modes,” Sens. Actuators B Chem. 206, 246–251 (2015).
[Crossref]

L. Qi, C. L. Zhao, J. Yuan, M. Ye, J. Wang, Z. Zhang, and S. Jin, “Highly reflective long period fiber grating sensor and its application in refractive index sensing,” Sens. Actuators B Chem. 193(31), 185–189 (2014).
[Crossref]

Y. Chen, Q. Han, T. Liu, and X. Lü, “Self-temperature-compensative refractometer based on single mode–multimode–single mode fiber structure,” Sens. Actuators B Chem. 212, 107–111 (2015).
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D. Monzón-Hernández, J. Villatoro, and D. Luna-Moreno, “Miniature optical fiber refractometer using cladded multimode tapered fiber tips,” Sens. Actuators B Chem. 110(1), 36–40 (2005).
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Y. Du, S. Jothibasu, Y. Zhuang, C. Zhu, and J. Huang, “Rayleigh backscattering based macrobending single mode fiber for distributed refractive index sensing,” Sens. Actuators B Chem. 248, 346–350 (2017).
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Sensors (Basel) (1)

Y. Tian, W. Wang, N. Wu, X. Zou, and X. Wang, “Tapered optical fiber sensor for label-free detection of biomolecules,” Sensors (Basel) 11(4), 3780–3790 (2011).
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S. T. Kreger, D. K. Gifford, M. E. Froggatt, B. J. Soller, and M. S. Wolfe, “High resolution distributed strain or temperature measurements in single-and multi-mode fiber using swept-wavelength interferometry,” Optical Fiber Sensors, Optical Society of America, 2006, p. ThE42.

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Figures (7)

Fig. 1
Fig. 1 (a) The simulated results of the relationship between the refractive index of environment medium and the local back-reflection spectra shift when the diameters of the taper waist is about 4.3 μm. RI sensitivities can be obtained from the slope of linear fitting curve. (b) The simulated results of RI sensitivities in different diameters of the taper waist.
Fig. 2
Fig. 2 Experimental setup for distributed RI sensor using tapered fibers in OFDR. TLS is tunable laser source, FRM is Faraday rotating mirror; PC is polarization controller; BPD is balanced photo detector; DAQ is data acquisition card; FUT is fiber under test. The tapered fiber is immersed in a tank with glycerol solution.
Fig. 3
Fig. 3 Configuration of small-scale fiber-tapering device. The RC SMF was fixed and clamp the fiber on the platform and they are pulled simultaneously to both sides during the heating process. The oxyhydrogen flame is also driven to move back and forth. The diameter of the taper waist section in the tapered fiber is 4.3 μm measured by a microscopy shown in the photography above.
Fig. 4
Fig. 4 Measured back-reflected light signals in the spatial domain for a FUT in the different RI of the environment medium. The tapered fiber is from 3.51 m to 3.42 m. (a) RI of the environment medium is 1.3574. (b) is local zoom of (a). (c) RI of the environment medium is 1.3686. (d) is local zoom of (c).
Fig. 5
Fig. 5 RI measurement based on the optical frequency shifts of the back-reflection spectra in the tapered fiber. Cross-correlation of measurement and reference of the back-reflection spectra with Δn = 0.0009 (RI ranges between 1.3574 and 1.3686). (a) Raw data with the length of N = 250. (b) Padding M = 500 zeros in the local back-reflected light segment in the spatial domain. (c) Local zoom area of (a) and (b). The cross-correlation peak shift one point (8 GHz) after a spectrum interpolation.
Fig. 6
Fig. 6 (a) Measured optical frequency shifts of the back-reflection spectra in the tapered fiber as a function of distance at various RIs. The optical frequency shifts of the tapered section increase along with RI increasing. (b) Measured optical frequency shifts of the back-reflection spectra as function of RI. The slope of the fitting line is the RI measurement sensitivity of this method.
Fig. 7
Fig. 7 Distance-time mapping trace of RI variation in a diffusion process of a glycerol solution. The glycerol will diffuse from one end to the other end in the tank over time. The RI variation also occurred from the one end to the other end of the tapered fiber over time.

Equations (17)

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Φ=ΔβL,
Δβ= c( U 2 2 U 1 2 ) 4π n c ρ 2 ν exp( 2 V ),
V= 2πρν c n c 2 n e 2 ,
Φ( ν 0 , n e0 )=Φ( ν 0 +δν, n e0 +δ n e ).
n e =δ n e + n e0 = n c 2 [ c( ν 0 δν) πρ ν 2 ln( ν 0 δν ν 0 )+ c ν 0 n c 2 n e0 2 ] 2 .
ΔX=NΔZ,
ΔZ=c/2nΔν.
δ n min =Δv/M+N,
δ n min =δ ν min /RES= Δν/ [RES(M+N)] .
δ n min = c 2nΔXRES( M N +1) .
δ n min = c 2nΔXRES .
ARESηRES,
Φ( ν 0 , n e0 )=Δβ( ν 0 , n e0 )L= c( U 2 2 U 1 2 ) 4π n c ρ 2 ν 0 exp( 2 V )L,
Φ( ν 0 +δν, n e0 +δ n e0 )=Δβ( ν 0 +δν, n e0 +δ n e0 )L= c( ν 0 δν)( U 2 2 U 1 2 ) 4π n c ρ 2 ν 0 2 exp( 2 V 1 )L,
V= 2πρ ν 0 c n c 2 n e0 2 ,
V 1 = 2πρ ν 0 2 c( ν 0 δν) n c 2 ( n e0 +δ n e ) 2 .
c( U 2 2 U 1 2 ) 4π n c ρ 2 ν 0 exp( 2 V )L= c( ν 0 δν)( U 2 2 U 1 2 ) 4π n c ρ 2 ν 0 2 exp( 2 V 1 )L.

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