Abstract

A stress optical fiber sensor was manufactured and tested. It uses light coupling between two parallel and laterally fused, all-silica multimode optical fibers along a cladding length of a few centimeters. This sensor is dedicated to the measurement of high values of stress. A theoretical model was developed using the mode coupling and the perturbation theory to calculate the global coupling coefficient of light. A serial optical fiber sensor network interrogated by the time-division multiplexing method was realized and tested. The major applications of this sensor are control and monitoring of civil engineering structures and concretes.

© 1998 Optical Society of America

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References

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  1. W. B. Spillman, J. S. Sirkis, P. T. Gardiner, “Smart materials and structures: What are they?” Smart Mater. Struct. 5, 247–254 (1996).
    [CrossRef]
  2. S. K. Sheem, J. H. Cole, “Acoustic sensitivity of single-mode optical power dividers,” Opt. Lett. 4, 322–324 (1979).
    [CrossRef] [PubMed]
  3. R. A. Zangaro, L. Silveira, R. Silva Barreto, “Optical fiber sensor for the measurement of stress in concrete structures,” Measurement 16, 103–105 (1995).
    [CrossRef]
  4. W. J. Bock, W. Urbanczyk, J. Wojcik, M. Beaulieu, “White-light interferometric fiber-optic pressure sensor,” IEEE Trans. Instrum. Meas. 44, 694–697 (1995).
    [CrossRef]
  5. J. Radhakrishman, M. A. El-Sherif, “Analysis on spatial modulation for fiber-optic sensor applications,” Opt. Fiber Technol. 2, 114–126 (1996).
    [CrossRef]
  6. A. Tardy, M. Jurczyszyn, J. M. Caussignac, G. Morel, G. Briant, “High sensitivity transducer for fibre-optic pressure sensing applied to dynamic technical testing and vehicle detection on roads,” in Optical Fiber Sensors, H. J. Ardity, J. P. Dakin, R. T. Kersten, eds., Proc. SPIE40, 215–220 (1989).
    [CrossRef]
  7. Y. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photon. Technol. Lett. 12, 1471–1473 (1995).
    [CrossRef]
  8. E. Bernabeu, M. C. Navarette, “Pressure sensor based on a 3-branch fibre optic interferometrical device,” Meas. Control 28, 164–166 (1995).
  9. M. Ketata, “Capteur de contraintes et de déformations à fibres optiques,” Ph.D. dissertation (Ecole Centrale Paris, Paris, 1989).
  10. P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
    [CrossRef]
  11. J. Dakin, B. Culshaw, Optical Fiber Sensors: Principle and Components (Artech House, Boston, 1988), Chap. 2, pp. 9–24.
  12. S. Lacroix, F. Gonthier, J. Bures, “Modeling of symmetric 2 × 2 fused-fiber couplers,” Appl. Opt. 33, 8361–8369 (1994).
    [CrossRef] [PubMed]
  13. J. S. Sirkis, “Optical and mechanical isotropies in embedded fiber optic sensors,” Smart Mater. Struct. 2, 255–259 (1993).
    [CrossRef]
  14. M. Frocht, Photoelasticity, Vol. 2 (Wiley, New York, 1948), pp. 125–129.
  15. R. Gafsi, A. Malki, F. Ahdad, P. Lecoy, J. Bures, “Static stress optical fiber sensor,” in Proceedings of Eurosensors X, Tenth Conference on Solid-State Transducers, Leuven, Belgium (Eurosensors, Leuven, Belgium, 1996), pp. 889–892.
  16. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 18, pp. 374–406;Chap. 29, pp. 567–585.
  17. P. McIntyre, A. W. Snyder, “Power transfer between optical fibers,” J. Opt. Soc. Am. A 63, 1518–1527 (1973).
    [CrossRef]
  18. D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–841 (1973).
  19. D. Marcuse, “The coupling of degenerate modes in two parallel dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1831 (1971).
  20. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1973).

1996 (2)

J. Radhakrishman, M. A. El-Sherif, “Analysis on spatial modulation for fiber-optic sensor applications,” Opt. Fiber Technol. 2, 114–126 (1996).
[CrossRef]

W. B. Spillman, J. S. Sirkis, P. T. Gardiner, “Smart materials and structures: What are they?” Smart Mater. Struct. 5, 247–254 (1996).
[CrossRef]

1995 (4)

R. A. Zangaro, L. Silveira, R. Silva Barreto, “Optical fiber sensor for the measurement of stress in concrete structures,” Measurement 16, 103–105 (1995).
[CrossRef]

W. J. Bock, W. Urbanczyk, J. Wojcik, M. Beaulieu, “White-light interferometric fiber-optic pressure sensor,” IEEE Trans. Instrum. Meas. 44, 694–697 (1995).
[CrossRef]

Y. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photon. Technol. Lett. 12, 1471–1473 (1995).
[CrossRef]

E. Bernabeu, M. C. Navarette, “Pressure sensor based on a 3-branch fibre optic interferometrical device,” Meas. Control 28, 164–166 (1995).

1994 (1)

1993 (1)

J. S. Sirkis, “Optical and mechanical isotropies in embedded fiber optic sensors,” Smart Mater. Struct. 2, 255–259 (1993).
[CrossRef]

1979 (1)

1973 (3)

P. McIntyre, A. W. Snyder, “Power transfer between optical fibers,” J. Opt. Soc. Am. A 63, 1518–1527 (1973).
[CrossRef]

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–841 (1973).

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1973).

1971 (1)

D. Marcuse, “The coupling of degenerate modes in two parallel dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1831 (1971).

Ahdad, F.

R. Gafsi, A. Malki, F. Ahdad, P. Lecoy, J. Bures, “Static stress optical fiber sensor,” in Proceedings of Eurosensors X, Tenth Conference on Solid-State Transducers, Leuven, Belgium (Eurosensors, Leuven, Belgium, 1996), pp. 889–892.

Beaulieu, M.

W. J. Bock, W. Urbanczyk, J. Wojcik, M. Beaulieu, “White-light interferometric fiber-optic pressure sensor,” IEEE Trans. Instrum. Meas. 44, 694–697 (1995).
[CrossRef]

Bernabeu, E.

E. Bernabeu, M. C. Navarette, “Pressure sensor based on a 3-branch fibre optic interferometrical device,” Meas. Control 28, 164–166 (1995).

Bock, W. J.

W. J. Bock, W. Urbanczyk, J. Wojcik, M. Beaulieu, “White-light interferometric fiber-optic pressure sensor,” IEEE Trans. Instrum. Meas. 44, 694–697 (1995).
[CrossRef]

Briant, G.

A. Tardy, M. Jurczyszyn, J. M. Caussignac, G. Morel, G. Briant, “High sensitivity transducer for fibre-optic pressure sensing applied to dynamic technical testing and vehicle detection on roads,” in Optical Fiber Sensors, H. J. Ardity, J. P. Dakin, R. T. Kersten, eds., Proc. SPIE40, 215–220 (1989).
[CrossRef]

Bures, J.

S. Lacroix, F. Gonthier, J. Bures, “Modeling of symmetric 2 × 2 fused-fiber couplers,” Appl. Opt. 33, 8361–8369 (1994).
[CrossRef] [PubMed]

R. Gafsi, A. Malki, F. Ahdad, P. Lecoy, J. Bures, “Static stress optical fiber sensor,” in Proceedings of Eurosensors X, Tenth Conference on Solid-State Transducers, Leuven, Belgium (Eurosensors, Leuven, Belgium, 1996), pp. 889–892.

Caussignac, J. M.

A. Tardy, M. Jurczyszyn, J. M. Caussignac, G. Morel, G. Briant, “High sensitivity transducer for fibre-optic pressure sensing applied to dynamic technical testing and vehicle detection on roads,” in Optical Fiber Sensors, H. J. Ardity, J. P. Dakin, R. T. Kersten, eds., Proc. SPIE40, 215–220 (1989).
[CrossRef]

Cole, J. H.

Culshaw, B.

J. Dakin, B. Culshaw, Optical Fiber Sensors: Principle and Components (Artech House, Boston, 1988), Chap. 2, pp. 9–24.

Dakin, J.

J. Dakin, B. Culshaw, Optical Fiber Sensors: Principle and Components (Artech House, Boston, 1988), Chap. 2, pp. 9–24.

Dammak, H.

P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
[CrossRef]

El-Sherif, M. A.

J. Radhakrishman, M. A. El-Sherif, “Analysis on spatial modulation for fiber-optic sensor applications,” Opt. Fiber Technol. 2, 114–126 (1996).
[CrossRef]

Frocht, M.

M. Frocht, Photoelasticity, Vol. 2 (Wiley, New York, 1948), pp. 125–129.

Gafsi, R.

R. Gafsi, A. Malki, F. Ahdad, P. Lecoy, J. Bures, “Static stress optical fiber sensor,” in Proceedings of Eurosensors X, Tenth Conference on Solid-State Transducers, Leuven, Belgium (Eurosensors, Leuven, Belgium, 1996), pp. 889–892.

Gardiner, P. T.

W. B. Spillman, J. S. Sirkis, P. T. Gardiner, “Smart materials and structures: What are they?” Smart Mater. Struct. 5, 247–254 (1996).
[CrossRef]

Gonthier, F.

Jurczyszyn, M.

A. Tardy, M. Jurczyszyn, J. M. Caussignac, G. Morel, G. Briant, “High sensitivity transducer for fibre-optic pressure sensing applied to dynamic technical testing and vehicle detection on roads,” in Optical Fiber Sensors, H. J. Ardity, J. P. Dakin, R. T. Kersten, eds., Proc. SPIE40, 215–220 (1989).
[CrossRef]

Ketata, M.

P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
[CrossRef]

M. Ketata, “Capteur de contraintes et de déformations à fibres optiques,” Ph.D. dissertation (Ecole Centrale Paris, Paris, 1989).

Kim, Y.

Y. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photon. Technol. Lett. 12, 1471–1473 (1995).
[CrossRef]

Lacroix, S.

Latry, O.

P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
[CrossRef]

Lecoy, P.

P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
[CrossRef]

R. Gafsi, A. Malki, F. Ahdad, P. Lecoy, J. Bures, “Static stress optical fiber sensor,” in Proceedings of Eurosensors X, Tenth Conference on Solid-State Transducers, Leuven, Belgium (Eurosensors, Leuven, Belgium, 1996), pp. 889–892.

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 18, pp. 374–406;Chap. 29, pp. 567–585.

Malki, A.

R. Gafsi, A. Malki, F. Ahdad, P. Lecoy, J. Bures, “Static stress optical fiber sensor,” in Proceedings of Eurosensors X, Tenth Conference on Solid-State Transducers, Leuven, Belgium (Eurosensors, Leuven, Belgium, 1996), pp. 889–892.

P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
[CrossRef]

Marcuse, D.

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–841 (1973).

D. Marcuse, “The coupling of degenerate modes in two parallel dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1831 (1971).

McIntyre, P.

P. McIntyre, A. W. Snyder, “Power transfer between optical fibers,” J. Opt. Soc. Am. A 63, 1518–1527 (1973).
[CrossRef]

Miry, R.

P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
[CrossRef]

Morel, G.

A. Tardy, M. Jurczyszyn, J. M. Caussignac, G. Morel, G. Briant, “High sensitivity transducer for fibre-optic pressure sensing applied to dynamic technical testing and vehicle detection on roads,” in Optical Fiber Sensors, H. J. Ardity, J. P. Dakin, R. T. Kersten, eds., Proc. SPIE40, 215–220 (1989).
[CrossRef]

Navarette, M. C.

E. Bernabeu, M. C. Navarette, “Pressure sensor based on a 3-branch fibre optic interferometrical device,” Meas. Control 28, 164–166 (1995).

Neikirk, D. P.

Y. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photon. Technol. Lett. 12, 1471–1473 (1995).
[CrossRef]

Radhakrishman, J.

J. Radhakrishman, M. A. El-Sherif, “Analysis on spatial modulation for fiber-optic sensor applications,” Opt. Fiber Technol. 2, 114–126 (1996).
[CrossRef]

Sheem, S. K.

Silva Barreto, R.

R. A. Zangaro, L. Silveira, R. Silva Barreto, “Optical fiber sensor for the measurement of stress in concrete structures,” Measurement 16, 103–105 (1995).
[CrossRef]

Silveira, L.

R. A. Zangaro, L. Silveira, R. Silva Barreto, “Optical fiber sensor for the measurement of stress in concrete structures,” Measurement 16, 103–105 (1995).
[CrossRef]

Sirkis, J. S.

W. B. Spillman, J. S. Sirkis, P. T. Gardiner, “Smart materials and structures: What are they?” Smart Mater. Struct. 5, 247–254 (1996).
[CrossRef]

J. S. Sirkis, “Optical and mechanical isotropies in embedded fiber optic sensors,” Smart Mater. Struct. 2, 255–259 (1993).
[CrossRef]

Snyder, A. W.

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1973).

P. McIntyre, A. W. Snyder, “Power transfer between optical fibers,” J. Opt. Soc. Am. A 63, 1518–1527 (1973).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 18, pp. 374–406;Chap. 29, pp. 567–585.

Spillman, W. B.

W. B. Spillman, J. S. Sirkis, P. T. Gardiner, “Smart materials and structures: What are they?” Smart Mater. Struct. 5, 247–254 (1996).
[CrossRef]

Tardy, A.

A. Tardy, M. Jurczyszyn, J. M. Caussignac, G. Morel, G. Briant, “High sensitivity transducer for fibre-optic pressure sensing applied to dynamic technical testing and vehicle detection on roads,” in Optical Fiber Sensors, H. J. Ardity, J. P. Dakin, R. T. Kersten, eds., Proc. SPIE40, 215–220 (1989).
[CrossRef]

Urbanczyk, W.

W. J. Bock, W. Urbanczyk, J. Wojcik, M. Beaulieu, “White-light interferometric fiber-optic pressure sensor,” IEEE Trans. Instrum. Meas. 44, 694–697 (1995).
[CrossRef]

Wojcik, J.

W. J. Bock, W. Urbanczyk, J. Wojcik, M. Beaulieu, “White-light interferometric fiber-optic pressure sensor,” IEEE Trans. Instrum. Meas. 44, 694–697 (1995).
[CrossRef]

Zangaro, R. A.

R. A. Zangaro, L. Silveira, R. Silva Barreto, “Optical fiber sensor for the measurement of stress in concrete structures,” Measurement 16, 103–105 (1995).
[CrossRef]

Appl. Opt. (1)

Bell Syst. Tech. J. (2)

D. Marcuse, “Coupled mode theory of round optical fibers,” Bell Syst. Tech. J. 52, 817–841 (1973).

D. Marcuse, “The coupling of degenerate modes in two parallel dielectric waveguides,” Bell Syst. Tech. J. 50, 1791–1831 (1971).

IEEE Photon. Technol. Lett. (1)

Y. Kim, D. P. Neikirk, “Micromachined Fabry–Perot cavity pressure transducer,” IEEE Photon. Technol. Lett. 12, 1471–1473 (1995).
[CrossRef]

IEEE Trans. Instrum. Meas. (1)

W. J. Bock, W. Urbanczyk, J. Wojcik, M. Beaulieu, “White-light interferometric fiber-optic pressure sensor,” IEEE Trans. Instrum. Meas. 44, 694–697 (1995).
[CrossRef]

J. Opt. Soc. Am. A (2)

P. McIntyre, A. W. Snyder, “Power transfer between optical fibers,” J. Opt. Soc. Am. A 63, 1518–1527 (1973).
[CrossRef]

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1973).

Meas. Control (1)

E. Bernabeu, M. C. Navarette, “Pressure sensor based on a 3-branch fibre optic interferometrical device,” Meas. Control 28, 164–166 (1995).

Measurement (1)

R. A. Zangaro, L. Silveira, R. Silva Barreto, “Optical fiber sensor for the measurement of stress in concrete structures,” Measurement 16, 103–105 (1995).
[CrossRef]

Opt. Fiber Technol. (1)

J. Radhakrishman, M. A. El-Sherif, “Analysis on spatial modulation for fiber-optic sensor applications,” Opt. Fiber Technol. 2, 114–126 (1996).
[CrossRef]

Opt. Lett. (1)

Smart Mater. Struct. (2)

W. B. Spillman, J. S. Sirkis, P. T. Gardiner, “Smart materials and structures: What are they?” Smart Mater. Struct. 5, 247–254 (1996).
[CrossRef]

J. S. Sirkis, “Optical and mechanical isotropies in embedded fiber optic sensors,” Smart Mater. Struct. 2, 255–259 (1993).
[CrossRef]

Other (7)

M. Frocht, Photoelasticity, Vol. 2 (Wiley, New York, 1948), pp. 125–129.

R. Gafsi, A. Malki, F. Ahdad, P. Lecoy, J. Bures, “Static stress optical fiber sensor,” in Proceedings of Eurosensors X, Tenth Conference on Solid-State Transducers, Leuven, Belgium (Eurosensors, Leuven, Belgium, 1996), pp. 889–892.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman and Hall, London, 1983), Chap. 18, pp. 374–406;Chap. 29, pp. 567–585.

M. Ketata, “Capteur de contraintes et de déformations à fibres optiques,” Ph.D. dissertation (Ecole Centrale Paris, Paris, 1989).

P. Lecoy, A. Malki, H. Dammak, M. Ketata, O. Latry, R. Miry, “A new optical sensor for distributed sensing,” in Distributed and Multiplexing Fiber Optic Sensors, A. D. Kersey, J. P. Dakin, eds., Proc. SPIE1586, 96–106 (1991).
[CrossRef]

J. Dakin, B. Culshaw, Optical Fiber Sensors: Principle and Components (Artech House, Boston, 1988), Chap. 2, pp. 9–24.

A. Tardy, M. Jurczyszyn, J. M. Caussignac, G. Morel, G. Briant, “High sensitivity transducer for fibre-optic pressure sensing applied to dynamic technical testing and vehicle detection on roads,” in Optical Fiber Sensors, H. J. Ardity, J. P. Dakin, R. T. Kersten, eds., Proc. SPIE40, 215–220 (1989).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

Stress optical fiber sensor principle: (a) schematic design of the stress optical fiber sensor; (b) cut view AA′; (c) longitudinal cut view of the sensing part of the sensor.

Fig. 2
Fig. 2

Geometric model of the fused optical fibers: (a) Longitudinal view of the fused optical fibers; (b) cut view AA′.

Fig. 3
Fig. 3

Experimental setup for manufacturing fused optical fibers.

Fig. 4
Fig. 4

Longitudinal photograph view of the two fused optical fibers.

Fig. 5
Fig. 5

Distribution of stresses in the optical fiber when the sensor is submitted to an external stress of 1 kPa applied along the x axis. (a) Distribution of σ x in the optical fiber when the sensor is submitted to an external stress of 1 kPa applied along the x axis. (b) Distribution of σ y in the optical fiber when the sensor is submitted to an external stress of 1 kPa applied along the x axis.

Fig. 6
Fig. 6

Distribution of the stresses in the optical fiber when the sensor is submitted to an external stress of 1 kPa applied along the y axis. (a) Distribution of σ x in the optical fiber when the sensor is submitted to an external stress of 1 kPa applied along the y axis. (b) Distribution of σ y in the optical fiber when the sensor is submitted to an external stress of 1 kPa applied along the y axis.

Fig. 7
Fig. 7

Computed clad’s refractive index along the y axis when the sensor is submitted to an external stress of 1 kPa applied along the y direction.

Fig. 8
Fig. 8

Computed clad’s refractive index along the y axis when the sensor is submitted to an external stress of 1 kPa applied along the x direction.

Fig. 9
Fig. 9

Schema of the fused zone of the two optical fibers. (a) Transverse cut view. (b) Longitudinal cut view.

Fig. 10
Fig. 10

Calibration of the optical fiber sensor.

Fig. 11
Fig. 11

Measurement of the global coupling coefficient versus temperature variation and when the sensor is submitted to a stress of 42.3 kPa applied along the y axis.

Fig. 12
Fig. 12

Evolution of the global coupling coefficient versus time.

Fig. 13
Fig. 13

Experimental setup of the TDM network for the stress optical fiber sensors.

Fig. 14
Fig. 14

Temporal responses of the network of optical fiber stress sensors interrogated by the TDM method.

Tables (6)

Tables Icon

Table 1 Calculated Initial Coupling Coefficient between Two Step-Index Fibers (50–125 μm) with NA = 0.22, Δ = 11 × 10-3, V = 40.656, and λ = 850 nm

Tables Icon

Table 2 Calculated Initial Coupling Coefficient between Two Step-Index Fibers (62.5–125 μm) with NA = 0.22, Δ = 11 × 10-3, V = 50.82, and λ = 850 nm

Tables Icon

Table 3 Calculated Initial Coupling Coefficient between Two Step-Index Fibers (110/125 μm) with NA = 0.37, Δ = 3 × 10-2, V = 150.427, and λ = 850 nm

Tables Icon

Table 4 Calculated Initial Coupling Coefficient between Two Graded-Index Fibers (62.5–125 μm) with V = 50.82, NA = 0.22; Δ = 16 × 10-3, and λ = 850 nm

Tables Icon

Table 5 Calculated Results of Fiber-Optic Sensor Sensitivities

Tables Icon

Table 6 Experimental Results of Fiber-Optic Sensor Sensitivities

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

f = 1 - l x l x 0 1 - 1 2 .
Δ n x x ,   y = - n 0 3 x ,   y 2 E A σ x x ,   y + B σ y x ,   y ;
Δ n y x ,   y = - n 0 3 x ,   y 2 E B σ x x ,   y + A σ y x ,   y .
C pq 1 2 = 1 2 k ε 0 μ 0 1 / 2 A 2 n 1,0 2 - n 2,0 2 × e p 1 r ,   θ N p e q 2 r ,   θ N q d A ,
r 1   sin   θ 1 = r 2   sin   θ 2 , r 1 2 = d 2 + r 2 2 + 2 r 2 d   cos   θ 2 .
κ pq = aC pq 1 2 Δ = aC qp 2 1 Δ = aC pq Δ ,
κ pq = - 1 l q u l p m p u l q m q 2 V 3 K l p + l q w l p m p d a cos l p + l q α + K l p - l q w l p m p d a cos l p - l q α K l p - 1 w l p m p K l p + 1 w l p m p K l q - 1 w l q m q K l q + 1 w l q m q 1 / 2 ,
V = 2 π a λ n 1,0 2 - n 2,0 2 1 / 2 ,
κ 0 m p , 0 m q = u 0 m p u 0 m q V 3 K 0 w 0 m p d a K 1 w 0 m p K 1 w 0 m q .
κ pq = κ pq 1 2 = κ qp 2 1 = 1 2 ka Δ ε 0 μ 0 1 / 2 0 a 0 2 π × n 1,0 2 - n 2,0 2 e p 1 r 1 ,   θ 1 N p e q 2 r 2 ,   θ 2 N q   r 2 d r 2 d θ 2 ,
e p 1 r 1 ,   θ 1 = K l p w l p m p r 1 a K l p w l p m p cos l p θ 1 - α u x
e q 2 r 2 ,   θ 2 = r 2 a l q L m q - 1 l q × V r 2 a 2 exp - V 2 r 2 a 2 cos l q θ 2 - α u x
n 1,0 r = n 1,0 1 - 2 Δ r / a 2 1 / 2
C pq = 1 2   k   ε 0 μ 0 1 / 2 A n 2 - n 0 2 e p r ,   θ e ¯ q * r ,   θ d A N p + N q ,
η pq = aC pq Δ = aC pq 1 2 Δ = aC qp 2 1 Δ .
η pq = ka N p + N q Δ ε 0 μ 0 1 / 2 Fiber 2   Core   n 1,0 r Δ n 1 r ,   θ × e ¯ p r ,   θ e ¯ q * r ,   θ d A + n 2,0 Fiber 2   Cladding   Δ n 2 r ,   θ e ¯ p r ,   θ e ¯ q * r ,   θ d A .
η pq = - ka 2 E N p + N q Δ ε 0 μ 0 1 / 2 0 a 0 2 π   n 1,0 4 r ,   θ A σ x r ,   θ + B σ y r ,   θ e ¯ p r ,   θ e ¯ q * r ,   θ r d r d θ + n 2,0 4 a b 0 2 π A σ x r ,   θ + B σ y r ,   θ e ¯ p r ,   θ e ¯ q * r ,   θ r d r d θ .
η pq = - ka 2 E N p + N q Δ ε 0 μ 0 1 / 2 0 a 0 2 π   n 1,0 4 r ,   θ B σ x r ,   θ + A σ y r ,   θ e ¯ p r ,   θ e ¯ q * r ,   θ r d r d θ + n 2,0 4 a b 0 2 π B σ x r ,   θ + A σ y r ,   θ e ¯ p r ,   θ e ¯ q * r ,   θ r d r d θ .

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