Abstract

The shape of a multi-core optical fiber is calculated by numerically solving a set of Frenet-Serret equations describing the path of the fiber in three dimensions. Included in the Frenet-Serret equations are curvature and bending direction functions derived from distributed fiber Bragg grating strain measurements in each core. The method offers advantages over prior art in that it determines complex three-dimensional fiber shape as a continuous parametric solution rather than an integrated series of discrete planar bends. Results and error analysis of the method using a tri-core optical fiber is presented. Maximum error expressed as a percentage of fiber length was found to be 7.2%.

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References

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  1. R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Elsevier, 2010).
  2. M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
    [CrossRef]
  3. G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003).
    [CrossRef] [PubMed]
  4. R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
    [CrossRef]
  5. S. M. Klute, R. G. Duncan, R. S. Fielder, G. W. Butler, J. H. Mabe, A. K. Sang, R. J. Seeley, and M. T. Raum, “Fiber-optic shape sensing and distributed strain measurements on a morphing chevron,” in Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit (Reno, Nevada, Jan. 9–12, 2006).
  6. A. Gray, Modern Differential Geometry of Curves and Surfaces (CRC Press, 1993), Chap. 7.
  7. J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. 38(4), 605–618 (1996).
    [CrossRef]
  8. C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001), Chap. 14.
  9. B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
    [CrossRef]

2007

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

2003

2001

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

2000

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

1996

J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. 38(4), 605–618 (1996).
[CrossRef]

Allison, S. G.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

Barton, J. S.

Batten, C. F.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

Bennion, I.

G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003).
[CrossRef] [PubMed]

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Blanchard, P. M.

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Burnett, J. G.

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Childers, B. A.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

Duncan, R. G.

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

Flockhart, G. M. H.

Froggatt, M. E.

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

Gander, M. J.

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Gifford, D. K.

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

Greenaway, A. H.

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Hare, D. A.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

Jegley, D. C.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

Jones, J. D. C.

G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003).
[CrossRef] [PubMed]

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Kreger, S. T.

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

Langer, J.

J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. 38(4), 605–618 (1996).
[CrossRef]

MacPherson, W. N.

G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003).
[CrossRef] [PubMed]

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

McBride, R.

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Moore, T. C.

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

Sang, A. K.

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

Seeley, R. J.

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

Singer, D. A.

J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. 38(4), 605–618 (1996).
[CrossRef]

Wolfe, M. S.

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

Zhang, L.

G. M. H. Flockhart, W. N. MacPherson, J. S. Barton, J. D. C. Jones, L. Zhang, and I. Bennion, “Two-axis bend measurement with Bragg gratings in multicore optical fiber,” Opt. Lett. 28(6), 387–389 (2003).
[CrossRef] [PubMed]

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Electron. Lett.

M. J. Gander, W. N. MacPherson, R. McBride, J. D. C. Jones, L. Zhang, I. Bennion, P. M. Blanchard, J. G. Burnett, and A. H. Greenaway, “Bend measurement using Bragg gratings in multi-core fiber,” Electron. Lett. 36(2), 120–121 (2000).
[CrossRef]

Opt. Lett.

Proc. SPIE

R. G. Duncan, M. E. Froggatt, S. T. Kreger, R. J. Seeley, D. K. Gifford, A. K. Sang, and M. S. Wolfe, “High-accuracy fiber-optic shape sensing,” Proc. SPIE 6530, 65301S, 65301S-11 (2007).
[CrossRef]

B. A. Childers, M. E. Froggatt, S. G. Allison, T. C. Moore, D. A. Hare, C. F. Batten, and D. C. Jegley, “Use of 3000 Bragg grating strain sensors distributed on four eight-meter optical fibers during static load tests of a composite structure,” Proc. SPIE 4332, 133–142 (2001).
[CrossRef]

SIAM Rev.

J. Langer and D. A. Singer, “Lagrangian aspects of the Kirchhoff elastic rod,” SIAM Rev. 38(4), 605–618 (1996).
[CrossRef]

Other

C. de Boor, A Practical Guide to Splines (Springer-Verlag, 2001), Chap. 14.

R. Kashyap, Fiber Bragg Gratings, 2nd ed. (Elsevier, 2010).

S. M. Klute, R. G. Duncan, R. S. Fielder, G. W. Butler, J. H. Mabe, A. K. Sang, R. J. Seeley, and M. T. Raum, “Fiber-optic shape sensing and distributed strain measurements on a morphing chevron,” in Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit (Reno, Nevada, Jan. 9–12, 2006).

A. Gray, Modern Differential Geometry of Curves and Surfaces (CRC Press, 1993), Chap. 7.

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Figures (4)

Fig. 1
Fig. 1

Typical symmetrical tri-core geometry and parameters for determining strain due to bending.

Fig. 2
Fig. 2

(a) Y vs. X for curves P1 - P5. (b) Z vs. θ of curves C1-C3.

Fig. 3
Fig. 3

(a) Absolute measurement error and (b) error per unit length versus arc length for planar curves P1-P5.

Fig. 4
Fig. 4

(a) Absolute measurement error and (b) error per unit length versus arc length for three-dimensional curves C1-C3.

Tables (1)

Tables Icon

Table 1 Measurement Error Statistics

Equations (17)

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r( s )=x( s )i+y( s )j+z( s )k
T( s )= dr( s ) ds = x ( s )i+ y ( s )j+ z ( s )k,
N( s )= T ( s )= x ( s )i+ y ( s )j+ z ( s )k ( x ( s ) ) 2 + ( y ( s ) ) 2 + ( z ( s ) ) 2 ,
B( s )=T( s )×N( s ).
T ( s )=κ( s )N( s ) ; N ( s )=κ( s )T( s )+τ( s )B( s ) ; B ( s )=τ( s )N( s ),
κ( s )= T ( s ) = ( x ( s ) ) 2 + ( y ( s ) ) 2 + ( z ( s ) ) 2
τ( s )= ( r ( s )× r ( s ) ) r ( s ) r ( s )× r ( s ) 2
r(s)= T(s)ds+ r 0 ,
Δε= ε 2 ε 1 = d R = d 1 + d 2 R ,
ε i =κ r i cos( θ b 3π 2 θ i ),
κ app,i = ε i r i ( cos θ i j+sin θ i k ),
κ app = i=1 N ε i r i cos θ i j i=1 N ε i r i sin θ i k .
κ= 2| κ app | N = 2 ( i=1 N ε i cos θ i ) 2 + ( i=1 N ε i sin θ i ) 2 Nr .
θ=angle( κ app ).
τ( s )= θ ( s ).
r 0 =( x 0 , y 0 , z 0 )=( 0,0,0 ).
N 0 =cos( θ( 0 ) )j+sin( θ( 0 ) )k.

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