Abstract

For the fused tapering optical fibers, a physics–mathematics equation of dynamic-shape curves is deduced. The closed-form solution of the equation can be acquired by use of its initial, boundary, and volume-conserved conditions. The shape curves in different tapering processes can be described by the solution conveniently, accurately, and integratively. A large quantity of experimental data has been provided and shown to have a coincidence with the theoretical results properly. Especially, the initial length of the fused zone is known, and the solution can predict the whole tapering shape curve with an arbitrary elongating length.

© 2006 Optical Society of America

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References

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  1. M. Eisenmann and E. Weidel, "Single-mode fused biconical couplers for wavelength division multuplexing with channel spacing between 100-nm and 300-nm," J. Lightwave Technol. 6, 113-119 (1988).
    [CrossRef]
  2. S. Lacroix, F. Gonthier, and J. Bures, "Modeling of symmetrical 2 × 2 fused-fiber couplers," Appl. Opt. 33, 8361-8369 (1994).
    [CrossRef] [PubMed]
  3. W. K. Burns, M. Abebe, and C. A. Villarruel, "Parabolic model for shape of fiber taper," Appl. Opt. 24, 2753-2755 (1985).
    [CrossRef] [PubMed]
  4. W. K. Burns and M. Abebe, "Coupling model for fused fiber couplers with parabolic taper shape," Appl. Opt. 26, 4190-4192 (1987).
    [CrossRef] [PubMed]
  5. J. M. P. Rodrigues, T. S. M. Maclean, B. K. Gazey, and J. F. Miller, "Parabolic shape of a tapered fused coupler--comparison with experiment," Appl. Opt. 26, 1578-1581 (1987).
    [CrossRef] [PubMed]
  6. J. Dewynne, J. R. Ockendon, and P. Wilmott, "On a mathematical model for fiber tapering," SIAM J. Appl. Math. 49, 983-990 (1989).
    [CrossRef]
  7. R. P. Kenny, T. A. Birks, and K. P. Oakley, "Control of optical fibre taper shape," Electron. Lett. 27, 1654-1656 (1991).
    [CrossRef]
  8. T. A. Birks and Y. W. Li, "The shape of fiber tapers," J. Lightwave Technol. 10, 432-438 (1992).
    [CrossRef]

1994

1992

T. A. Birks and Y. W. Li, "The shape of fiber tapers," J. Lightwave Technol. 10, 432-438 (1992).
[CrossRef]

1991

R. P. Kenny, T. A. Birks, and K. P. Oakley, "Control of optical fibre taper shape," Electron. Lett. 27, 1654-1656 (1991).
[CrossRef]

1989

J. Dewynne, J. R. Ockendon, and P. Wilmott, "On a mathematical model for fiber tapering," SIAM J. Appl. Math. 49, 983-990 (1989).
[CrossRef]

1988

M. Eisenmann and E. Weidel, "Single-mode fused biconical couplers for wavelength division multuplexing with channel spacing between 100-nm and 300-nm," J. Lightwave Technol. 6, 113-119 (1988).
[CrossRef]

1987

1985

Abebe, M.

Birks, T. A.

T. A. Birks and Y. W. Li, "The shape of fiber tapers," J. Lightwave Technol. 10, 432-438 (1992).
[CrossRef]

R. P. Kenny, T. A. Birks, and K. P. Oakley, "Control of optical fibre taper shape," Electron. Lett. 27, 1654-1656 (1991).
[CrossRef]

Bures, J.

Burns, W. K.

Dewynne, J.

J. Dewynne, J. R. Ockendon, and P. Wilmott, "On a mathematical model for fiber tapering," SIAM J. Appl. Math. 49, 983-990 (1989).
[CrossRef]

Eisenmann, M.

M. Eisenmann and E. Weidel, "Single-mode fused biconical couplers for wavelength division multuplexing with channel spacing between 100-nm and 300-nm," J. Lightwave Technol. 6, 113-119 (1988).
[CrossRef]

Gazey, B. K.

Gonthier, F.

Kenny, R. P.

R. P. Kenny, T. A. Birks, and K. P. Oakley, "Control of optical fibre taper shape," Electron. Lett. 27, 1654-1656 (1991).
[CrossRef]

Lacroix, S.

Li, Y. W.

T. A. Birks and Y. W. Li, "The shape of fiber tapers," J. Lightwave Technol. 10, 432-438 (1992).
[CrossRef]

Maclean, T. S. M.

Miller, J. F.

Oakley, K. P.

R. P. Kenny, T. A. Birks, and K. P. Oakley, "Control of optical fibre taper shape," Electron. Lett. 27, 1654-1656 (1991).
[CrossRef]

Ockendon, J. R.

J. Dewynne, J. R. Ockendon, and P. Wilmott, "On a mathematical model for fiber tapering," SIAM J. Appl. Math. 49, 983-990 (1989).
[CrossRef]

Rodrigues, J. M. P.

Villarruel, C. A.

Weidel, E.

M. Eisenmann and E. Weidel, "Single-mode fused biconical couplers for wavelength division multuplexing with channel spacing between 100-nm and 300-nm," J. Lightwave Technol. 6, 113-119 (1988).
[CrossRef]

Wilmott, P.

J. Dewynne, J. R. Ockendon, and P. Wilmott, "On a mathematical model for fiber tapering," SIAM J. Appl. Math. 49, 983-990 (1989).
[CrossRef]

Appl. Opt.

Electron. Lett.

R. P. Kenny, T. A. Birks, and K. P. Oakley, "Control of optical fibre taper shape," Electron. Lett. 27, 1654-1656 (1991).
[CrossRef]

J. Lightwave Technol.

T. A. Birks and Y. W. Li, "The shape of fiber tapers," J. Lightwave Technol. 10, 432-438 (1992).
[CrossRef]

M. Eisenmann and E. Weidel, "Single-mode fused biconical couplers for wavelength division multuplexing with channel spacing between 100-nm and 300-nm," J. Lightwave Technol. 6, 113-119 (1988).
[CrossRef]

SIAM J. Appl. Math.

J. Dewynne, J. R. Ockendon, and P. Wilmott, "On a mathematical model for fiber tapering," SIAM J. Appl. Math. 49, 983-990 (1989).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Diagram of varying curves for the fused fiber as tapering.

Fig. 2
Fig. 2

The enlarged sketch of abcd area in Fig. 1.

Fig. 3
Fig. 3

The reciprocal of the degenerated factor l / d ( L s ) vs elongating length L s for samples titled for c# to k# ( L f 0 = 6.293 mm , L t = L f 0 + L s ) .

Fig. 4
Fig. 4

The taper-shape comparison between theory and experiment for samples titled from c# to k#. (a) For samples of d# ( L s = 22.02 mm ) , g# ( L s = 16 mm ) and j# ( L s = 10 mm ) . (b) For samples of e# ( L s = 20 mm ) , h# ( L s = 14 mm ) and k# ( L s = 8 mm ) . (c) For samples of c# ( L s = 24.02 mm ) , f# ( L s = 18 mm ) and i# ( L s = 12 mm ) .

Equations (106)

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R 0
L f 0
L f
L s
L t
L t = L f 0 + L s
R ( x )
R ( x )
L f
L s
L f = d ( L s ) L f 0 . d ( L s )
x = L f / 2
p q
R ( x )
x = L f / 2
x ( x > L f / 2 )
Δ V R
b efg
x
x + Δ x
Δ V f
Δ V R
Δ V R = π { [ R 2 ( x + Δ x ) - R 2 ( x ) ] / 2 + R 2 ( x ) } Δ x .
Δ x 2
Δ V R = π R 2 ( x ) Δ x .
Δ x
Δ V R
R ( x )
Δ x
R ( x ) > R ( x ) - Δ R ( x )
Δ R ( x ) > 0
0 x L f / 2
Δ V f
Δ V f = π L f { R 2 ( x ) - [ R ( x ) - Δ R ( x ) ] 2 } / 2 ;
Δ R ( x )
Δ R ( x ) 2
Δ R ( x ) = R ( x ) Δ x
Δ V f = π L f R ( x ) R ( x ) Δ x .
Δ V R
Δ V f
Δ x
Δ x
R 2 ( x ) = L f R ( x ) R ( x ) .
R ( x )
x = L f / 2
R ( x )
R ( x ) = R ( x ) exp [ i ( a x + b ) ] ;
R ( x ) R ( x ) = 1 L f exp [ 2 i ( a x + b ) ] .
R ( x ) | x = L t / 2 = R 0 ,
R ( x ) | x = L t / 2 = 0 ,
R ( x ) | x = 0 = 0 ,
2 π 0 L t / 2 R 2 ( x ) d x = π R 0 2 L f 0 .
R ( x )
π R 0 2 L f 0
R ( x ) = c exp { - i 2 a d ( L s ) L f 0 exp [ 2 i ( a x + b ) ] } ;
R ( x ) = c exp [ 1 2 a d ( L s ) L f 0 sin 2 ( a x + b ) ] .
c = R 0 exp [ - 1 2 a d ( L s ) L f 0 sin ( a L t + 2 b ) ] ,
a = π L t , b = π 4 + n π , n = 0 , ± 1 , ± 2 ,   .
n = 0
b = - π / 4
R ( x ) = R 0 exp { - L t 2 π d ( L s ) L f 0 [ 1 - sin ( 2 π x L t - π 2 ) ] } .
L t
R 0
L t
L s
L f
d ( L s )
d ( L s )
L f
d ( L s )
d ( L s )
L f 0
L s
1 / d ( L s )
d ( L s )
L s
L f 0
L s
l / d ( L s )
L f 0
L t
L s
L t
H 2
O 2
5.5 mm
0.05 mm / s
L f 0 = 6.293 mm
R ct
R k t
R c e
R k e
L t
L t
l / d ( L s )
L s
( L f 0 = 6.293 mm , L t = L f 0 + L s )
d# ( L s = 22.02 mm )
g# ( L s = 16 mm )
j# ( L s = 10 mm )
e# ( L s = 20 mm )
h# ( L s = 14 mm )
k# ( L s = 8 mm )
c# ( L s = 24.02 mm )
f# ( L s = 18 mm )
i# ( L s = 12 mm ) .

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