Abstract

A compact highly sensitive microfiber coupler based reflective micro-force sensor is presented. The device is fabricated by fusing two twisted optical fibers and then connecting two of the pigtails to form a Sagnac loop. The sensor has a high force sensitivity of ~3754 nm/N which is three orders of magnitude larger than traditional optical fiber force sensors, and a low detection limit of ~1.6 µN. The good repeatability is also shown in this paper.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
    [CrossRef]
  5. B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
    [CrossRef]
  6. J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2014 (1)

M. Ding, P. Wang, J. Wang, G. Brambilla, “FIB-milled gold-coated singlemode-multimode-singlemode fiber tip refractometer,” IEEE Photonics Technol. Lett. 26(3), 239–241 (2014).
[CrossRef]

2013 (2)

2012 (2)

M. Ding, P. Wang, G. Brambilla, “A microfiber coupler tip thermometer,” Opt. Express 20(5), 5402–5408 (2012).
[CrossRef] [PubMed]

W. Luo, J. L. Kou, Y. Chen, F. Xu, Y. Q. Lu, “Ultra-highly sensitive surface-corrugated microfiber Bragg grating force sensor,” Appl. Phys. Lett. 101(13), 133502 (2012).
[CrossRef]

2009 (1)

2008 (1)

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

2007 (2)

2006 (4)

J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
[CrossRef]

M. Sumetsky, Y. Dulashko, J. M. Fini, A. Hale, D. J. DiGiovanni, “The microfiber loop resonator: theory, experiment, and application,” J. Lightwave Technol. 24(1), 242–250 (2006).
[CrossRef]

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

2004 (1)

2003 (1)

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

2001 (1)

W. Zhang, X. Dong, Q. Zhao, G. Kai, S. Yuan, “FBG-type sensor for simultaneous measurement of force (or displacement) and temperature based on bilateral cantilever beam,” IEEE Photonics Technol. Lett. 13(12), 1340–1342 (2001).
[CrossRef]

1999 (1)

1994 (1)

S. Huang, F. Luo, Y. Pan, “A fiber optic sensor for measuring distributed forces,” J. Intell. Mater. Syst. Struct. 5(3), 427–431 (1994).
[CrossRef]

1985 (3)

M. S. Yataki, D. N. Payne, M. P. Varnham, “All-fibre polarising beamsplitter,” Electron. Lett. 21(6), 249–251 (1985).
[CrossRef]

J. D. Love, M. Hall, “Polarisation modulation in long couplers,” Electron. Lett. 21, 519–521 (1985).

F. P. Payne, C. D. Hussey, M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett. 21(13), 561–563 (1985).
[CrossRef]

Alam, S.-U.

Araújo, F. M.

Ashcom, J. B.

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

Brambilla, G.

Cai, Z.

J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
[CrossRef]

Chan, T. L.

Chen, R.

Chen, Y.

W. Luo, J. L. Kou, Y. Chen, F. Xu, Y. Q. Lu, “Ultra-highly sensitive surface-corrugated microfiber Bragg grating force sensor,” Appl. Phys. Lett. 101(13), 133502 (2012).
[CrossRef]

DiGiovanni, D. J.

Ding, M.

M. Ding, P. Wang, J. Wang, G. Brambilla, “FIB-milled gold-coated singlemode-multimode-singlemode fiber tip refractometer,” IEEE Photonics Technol. Lett. 26(3), 239–241 (2014).
[CrossRef]

M. Ding, P. Wang, G. Brambilla, “A microfiber coupler tip thermometer,” Opt. Express 20(5), 5402–5408 (2012).
[CrossRef] [PubMed]

Dong, B.

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

Dong, X.

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

W. Zhang, X. Dong, Q. Zhao, G. Kai, S. Yuan, “FBG-type sensor for simultaneous measurement of force (or displacement) and temperature based on bilateral cantilever beam,” IEEE Photonics Technol. Lett. 13(12), 1340–1342 (2001).
[CrossRef]

Dulashko, Y.

Ferreira, L. A.

Finazzi, V.

Fini, J. M.

Frazão, O.

Gattass, R. R.

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

Giles, I. P.

Gong, Y.

J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
[CrossRef]

Guerreiro, A.

Guo, T.

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

Guo, X.

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

Hale, A.

Hall, M.

J. D. Love, M. Hall, “Polarisation modulation in long couplers,” Electron. Lett. 21, 519–521 (1985).

Hao, J.

J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
[CrossRef]

He, S. L.

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

Huang, G.

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

Huang, S.

S. Huang, F. Luo, Y. Pan, “A fiber optic sensor for measuring distributed forces,” J. Intell. Mater. Syst. Struct. 5(3), 427–431 (1994).
[CrossRef]

Hussey, C. D.

F. P. Payne, C. D. Hussey, M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett. 21(13), 561–563 (1985).
[CrossRef]

Ismaeel, R.

Jiang, X. S.

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

Jin, L.

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

Jung, Y.

Kai, G.

W. Zhang, X. Dong, Q. Zhao, G. Kai, S. Yuan, “FBG-type sensor for simultaneous measurement of force (or displacement) and temperature based on bilateral cantilever beam,” IEEE Photonics Technol. Lett. 13(12), 1340–1342 (2001).
[CrossRef]

Kou, J. L.

W. Luo, J. L. Kou, Y. Chen, F. Xu, Y. Q. Lu, “Ultra-highly sensitive surface-corrugated microfiber Bragg grating force sensor,” Appl. Phys. Lett. 101(13), 133502 (2012).
[CrossRef]

Lau, K. S.

Liao, C. R.

Liao, T.

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

Liu, J.

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

Lou, J. Y.

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

Love, J. D.

J. D. Love, M. Hall, “Polarisation modulation in long couplers,” Electron. Lett. 21, 519–521 (1985).

Lu, Y. Q.

W. Luo, J. L. Kou, Y. Chen, F. Xu, Y. Q. Lu, “Ultra-highly sensitive surface-corrugated microfiber Bragg grating force sensor,” Appl. Phys. Lett. 101(13), 133502 (2012).
[CrossRef]

Luo, F.

S. Huang, F. Luo, Y. Pan, “A fiber optic sensor for measuring distributed forces,” J. Intell. Mater. Syst. Struct. 5(3), 427–431 (1994).
[CrossRef]

Luo, W.

W. Luo, J. L. Kou, Y. Chen, F. Xu, Y. Q. Lu, “Ultra-highly sensitive surface-corrugated microfiber Bragg grating force sensor,” Appl. Phys. Lett. 101(13), 133502 (2012).
[CrossRef]

Maxwell, I.

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

Mazur, E.

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

Miao, Y.

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

Ng, J. H.

J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
[CrossRef]

Pan, Y.

S. Huang, F. Luo, Y. Pan, “A fiber optic sensor for measuring distributed forces,” J. Intell. Mater. Syst. Struct. 5(3), 427–431 (1994).
[CrossRef]

Payne, D. N.

M. S. Yataki, D. N. Payne, M. P. Varnham, “All-fibre polarising beamsplitter,” Electron. Lett. 21(6), 249–251 (1985).
[CrossRef]

Payne, F. P.

F. P. Payne, C. D. Hussey, M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett. 21(13), 561–563 (1985).
[CrossRef]

Richardson, D. J.

Santos, J. L.

Shen, M. Y.

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

Silva, S. F. O.

Sumetsky, M.

Tong, L. M.

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

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

Tsao, A.

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

Varghese, P.

J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
[CrossRef]

Varnham, M. P.

M. S. Yataki, D. N. Payne, M. P. Varnham, “All-fibre polarising beamsplitter,” Electron. Lett. 21(6), 249–251 (1985).
[CrossRef]

Vienne, G.

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

Wang, D. N.

Wang, J.

M. Ding, P. Wang, J. Wang, G. Brambilla, “FIB-milled gold-coated singlemode-multimode-singlemode fiber tip refractometer,” IEEE Photonics Technol. Lett. 26(3), 239–241 (2014).
[CrossRef]

Wang, P.

M. Ding, P. Wang, J. Wang, G. Brambilla, “FIB-milled gold-coated singlemode-multimode-singlemode fiber tip refractometer,” IEEE Photonics Technol. Lett. 26(3), 239–241 (2014).
[CrossRef]

M. Ding, P. Wang, G. Brambilla, “A microfiber coupler tip thermometer,” Opt. Express 20(5), 5402–5408 (2012).
[CrossRef] [PubMed]

Wang, Y.

Wong, K. H.

Xu, F.

W. Luo, J. L. Kou, Y. Chen, F. Xu, Y. Q. Lu, “Ultra-highly sensitive surface-corrugated microfiber Bragg grating force sensor,” Appl. Phys. Lett. 101(13), 133502 (2012).
[CrossRef]

F. Xu, G. Brambilla, “Manufacture of 3-D microfiber coil resonators,” IEEE Photonics Technol. Lett. 19(19), 1481–1483 (2007).
[CrossRef]

Xue, L.

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

Yang, D. R.

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

Yang, Q.

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

Yataki, M. S.

F. P. Payne, C. D. Hussey, M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett. 21(13), 561–563 (1985).
[CrossRef]

M. S. Yataki, D. N. Payne, M. P. Varnham, “All-fibre polarising beamsplitter,” Electron. Lett. 21(6), 249–251 (1985).
[CrossRef]

Yuan, S.

W. Zhang, X. Dong, Q. Zhao, G. Kai, S. Yuan, “FBG-type sensor for simultaneous measurement of force (or displacement) and temperature based on bilateral cantilever beam,” IEEE Photonics Technol. Lett. 13(12), 1340–1342 (2001).
[CrossRef]

Zeng, X.

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

Zhang, W.

W. Zhang, X. Dong, Q. Zhao, G. Kai, S. Yuan, “FBG-type sensor for simultaneous measurement of force (or displacement) and temperature based on bilateral cantilever beam,” IEEE Photonics Technol. Lett. 13(12), 1340–1342 (2001).
[CrossRef]

Zhao, L.

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

Zhao, Q.

B. Dong, Q. Zhao, L. Zhao, L. Jin, Y. Miao, T. Liao, X. Zeng, “Simultaneous measurement of temperature and force based on a special-strain-function-chirped FBG,” Sens. Actuators A Phys. 147(1), 169–172 (2008).
[CrossRef]

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

W. Zhang, X. Dong, Q. Zhao, G. Kai, S. Yuan, “FBG-type sensor for simultaneous measurement of force (or displacement) and temperature based on bilateral cantilever beam,” IEEE Photonics Technol. Lett. 13(12), 1340–1342 (2001).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

W. Luo, J. L. Kou, Y. Chen, F. Xu, Y. Q. Lu, “Ultra-highly sensitive surface-corrugated microfiber Bragg grating force sensor,” Appl. Phys. Lett. 101(13), 133502 (2012).
[CrossRef]

X. S. Jiang, L. M. Tong, G. Vienne, X. Guo, A. Tsao, Q. Yang, D. R. Yang, “Demonstration of optical microfiber knot resonator,” Appl. Phys. Lett. 88(22), 223501 (2006).
[CrossRef]

Electron. Lett. (4)

J. Hao, Z. Cai, J. H. Ng, Y. Gong, P. Varghese, “Simultaneous temperature and lateral force measurement using simple arc-shaped FBG sensor module,” Electron. Lett. 42(25), 1446–1447 (2006).
[CrossRef]

F. P. Payne, C. D. Hussey, M. S. Yataki, “Polarisation analysis of strongly fused and weakly fused tapered couplers,” Electron. Lett. 21(13), 561–563 (1985).
[CrossRef]

M. S. Yataki, D. N. Payne, M. P. Varnham, “All-fibre polarising beamsplitter,” Electron. Lett. 21(6), 249–251 (1985).
[CrossRef]

J. D. Love, M. Hall, “Polarisation modulation in long couplers,” Electron. Lett. 21, 519–521 (1985).

IEEE Photonics Technol. Lett. (4)

W. Zhang, X. Dong, Q. Zhao, G. Kai, S. Yuan, “FBG-type sensor for simultaneous measurement of force (or displacement) and temperature based on bilateral cantilever beam,” IEEE Photonics Technol. Lett. 13(12), 1340–1342 (2001).
[CrossRef]

L. Xue, Q. Zhao, J. Liu, G. Huang, T. Guo, X. Dong, “Force sensing with temperature self-compensated based on a loop thin-wall section beam,” IEEE Photonics Technol. Lett. 18(1), 271–273 (2006).
[CrossRef]

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[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic of a MFC.

Fig. 2
Fig. 2

Structure of the miniature MFC Sagnac loop.

Fig. 3
Fig. 3

The calculated output power from port P2 under three different mechanical forces.

Fig. 4
Fig. 4

The calculated force sensitivity as a function of the microfiber diameter a. A smaller diameter will cause a higher sensitivity. Inset: cross section of the MFC coupling region in the “weakly fusing” approximation.

Fig. 5
Fig. 5

Microscope images of (a) the transition region and (b) coupling region of the MFC, the diameter of each coupled microfiber is ~1.6 µm; (c) Output spectrum at port P2.

Fig. 6
Fig. 6

Schematic of the measurement setup.

Fig. 7
Fig. 7

Output spectrum of the device around the peak wavelength 1553 nm under different forces.

Fig. 8
Fig. 8

Wavelength shift dependence on the stretched length. The red and blue curves refer to increasing and decreasing forces, respectively.

Equations (9)

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C x = 2 3/2 ( n 1 2 n 0 2 ) 1/2 U 2 ( 2 n 1 2 V+1 ) n 1 3 a( π ) V 7/2
C y = 2 3/2 ( n 1 2 n 0 2 ) 1/2 U 2 ( 2 n 1 2 V1 ) n 1 3 a( π ) V 7/2
P 2 = 1 2 { 1+cos[2( C x ¯ + C y ¯ ) L coupler ]cos[2( C x ¯ C y ¯ ) L coupler ] }
Δ L coupler L coupler = ε coupler = f E A coupler
Δ n 1 n 1 = f E A coupler ×P
P= n 1 2 2 [ p 12 ν( p 11 + p 12 )]
L coupler + L pigtail = L fiber
Δ L coupler +Δ L Pigtail = L stretch
ε coupler E A coupler = ε pigtail E A pigtail

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