Abstract

We investigate the refractive index profile of the cross-section of fused type fiber-optic couplers by solving the convective diffusion equation. We assume the refractive index to be a linear function of the dopant concentration. The viscous sintering of the optical fibers is considered as the motion of an incompressible Newtonian fluid which is driven by the surface tension acting at the free boundary. The internal velocity field is obtained using conformal mapping methods. We present numerical solutions of the resulting equations and compare them with experimental observations.

© 2004 Optical Society of America

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References

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  1. S. Lacroix, F. Gonthier, and J. Bures, “Modeling of symmetric 2×2 fused-fiber couplers,” Appl. Opt. 33, 8361–8369 (1994).
    [Crossref] [PubMed]
  2. C. Farget, J.P. Meunier, and P.E. Bonneau, “An Efficient Taper Shape Model for Fused Optical Fiber Components,” Int. Conf. Fiber Opt. Photon. Photonics-96” 1141–1146 (1996).
  3. L. J. Cummings and P. D. Howell, “On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity,” J. Fluid Mech. 389, 361–389 (1999).
    [Crossref]
  4. R. W. Hopper, “Plane Stokes flow driven by capillarity on a free surface,” J. Fluid Mech. 213, 349–375 (1990).
    [Crossref]
  5. S. Richardson, “Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 3, 193–207 (1992).
    [Crossref]
  6. S. Richardson, “Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 8, 311–329 (1997).
  7. S. Richardson, “Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region,” Eur. J. Appl. Math. 11, 249–269 (2000).
    [Crossref]
  8. Glass: Science and Technology (Academic Press, 1986)

2000 (1)

S. Richardson, “Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region,” Eur. J. Appl. Math. 11, 249–269 (2000).
[Crossref]

1999 (1)

L. J. Cummings and P. D. Howell, “On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity,” J. Fluid Mech. 389, 361–389 (1999).
[Crossref]

1997 (1)

S. Richardson, “Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 8, 311–329 (1997).

1994 (1)

1992 (1)

S. Richardson, “Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 3, 193–207 (1992).
[Crossref]

1990 (1)

R. W. Hopper, “Plane Stokes flow driven by capillarity on a free surface,” J. Fluid Mech. 213, 349–375 (1990).
[Crossref]

Bonneau, P.E.

C. Farget, J.P. Meunier, and P.E. Bonneau, “An Efficient Taper Shape Model for Fused Optical Fiber Components,” Int. Conf. Fiber Opt. Photon. Photonics-96” 1141–1146 (1996).

Bures, J.

Cummings, L. J.

L. J. Cummings and P. D. Howell, “On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity,” J. Fluid Mech. 389, 361–389 (1999).
[Crossref]

Farget, C.

C. Farget, J.P. Meunier, and P.E. Bonneau, “An Efficient Taper Shape Model for Fused Optical Fiber Components,” Int. Conf. Fiber Opt. Photon. Photonics-96” 1141–1146 (1996).

Gonthier, F.

Hopper, R. W.

R. W. Hopper, “Plane Stokes flow driven by capillarity on a free surface,” J. Fluid Mech. 213, 349–375 (1990).
[Crossref]

Howell, P. D.

L. J. Cummings and P. D. Howell, “On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity,” J. Fluid Mech. 389, 361–389 (1999).
[Crossref]

Lacroix, S.

Meunier, J.P.

C. Farget, J.P. Meunier, and P.E. Bonneau, “An Efficient Taper Shape Model for Fused Optical Fiber Components,” Int. Conf. Fiber Opt. Photon. Photonics-96” 1141–1146 (1996).

Richardson, S.

S. Richardson, “Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region,” Eur. J. Appl. Math. 11, 249–269 (2000).
[Crossref]

S. Richardson, “Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 8, 311–329 (1997).

S. Richardson, “Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 3, 193–207 (1992).
[Crossref]

Appl. Opt. (1)

Eur. J. Appl. Math. (3)

S. Richardson, “Two-dimensional slow viscous flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 3, 193–207 (1992).
[Crossref]

S. Richardson, “Two-dimensional Stokes flows with time-dependent free boundaries driven by surface tension,” Eur. J. Appl. Math. 8, 311–329 (1997).

S. Richardson, “Plane Stokes flows with time-dependent free boundaries in which the fluid occupies a doubly-connected region,” Eur. J. Appl. Math. 11, 249–269 (2000).
[Crossref]

J. Fluid Mech. (2)

L. J. Cummings and P. D. Howell, “On the evolution of non-axisymmetric viscous fibres with surface tension, inertia and gravity,” J. Fluid Mech. 389, 361–389 (1999).
[Crossref]

R. W. Hopper, “Plane Stokes flow driven by capillarity on a free surface,” J. Fluid Mech. 213, 349–375 (1990).
[Crossref]

Other (2)

C. Farget, J.P. Meunier, and P.E. Bonneau, “An Efficient Taper Shape Model for Fused Optical Fiber Components,” Int. Conf. Fiber Opt. Photon. Photonics-96” 1141–1146 (1996).

Glass: Science and Technology (Academic Press, 1986)

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Figures (5)

Fig. 1.
Fig. 1.

Schematic diagram of the conformal mapping

Fig. 2.
Fig. 2.

Cross-section evolution without diffusion

Fig. 3.
Fig. 3.

cross-section at f=0.355

Fig. 4.
Fig. 4.

cross-section at f=0.6

Fig. 5.
Fig. 5.

cross-section at f=0.92

Equations (28)

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

c t = u · c + D 2 c
p = μ 2 u
u = 0
u = μ u * γ t = γ t * μ R p = p * R γ
p = 2 u
u = 0
D = D * μ γ R
2 [ u x ( ζ ) + i u y ( ζ ) ] = ϕ ( ζ ) Ω ( ζ ) Ω ( ζ ) ϕ ( ζ ) ¯ ψ ( ζ ) ¯ ζ 1
ϕ ( ζ , t ) = ζ Ω ( ζ , t ) F ( ζ , t ) + t Ω ( ζ , t ) ζ 1
F ( ζ , t ) = 1 2 π i 1 2 Ω ( σ , t ) σ + ζ σ ( σ ζ ) d σ
β j Ω ( λ j ) = r j 2 d λ j d t = λ j F ( λ j )
Ω ( ζ ) = β 1 ζ + β 2 ζ 1 λ ζ
β 1 ( β 1 + β 2 ) = r 1 2
β 2 [ β 1 + β 2 1 ( 1 λ 2 ) 2 ] = r 2 2
x ( θ ) = β 1 cos θ + β 2 cos θ λ 1 2 λ cos θ + λ 2
y ( θ ) = β 1 sin θ + β 2 sin θ 1 2 λ cos θ + λ 2
W = x ( 0 ) x ( π ) = 2 β 1 + β 2 [ 1 1 λ + 1 1 + λ ]
f = W ( t 0 ) W W ( t 0 ) W ( t )
ψ ( σ , t ) = Ω ( σ , t ) ¯ [ ( 1 + σ Ω " ( σ , t ) Ω ( σ , t ) ) F ( σ , t ) + σ F ( σ , t )
t Ω ( σ , t ) Ω ( σ , t ) ] σ Ω ( σ , t ) ¯ F ( σ , t ) t Ω ( σ , t ) ¯
Ω 1 ( ζ ) = β 1 ζ + β 2 ζ λ
2 [ u x ( ζ , t ) + i u y ( ζ , t ) ] = ζ Ω ( ζ , t ) F ( ζ , t ) + t Ω ( ζ , t )
+ Ω ( ζ , t ) Ω 1 ( ζ , t ) ¯ Ω ( ζ , t ) ¯ [ Ω ( ζ , t ) F ( ζ , t ) + ζ Ω " ( ζ , t ) F ( ζ , t ) ¯
+ ζ Ω ( ζ , t ) F ( ζ , t ) t Ω ( ζ , t ) ] ¯
ζ Ω 1 ( ζ , t ) F ( ζ , t ) ¯ + t Ω 1 ( ζ , t ) ¯
d λ d t = λ F ( λ )
t = μ γ λ 0.9999 d x x F ( x )
γ = 0.28 0.00004 ( T 1200 ) N m

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