Abstract

We study the effects of macroscopic bends and twists in an optical waveguide and how they influence the transmission capabilities of a waveguide. These mechanical stresses and strains distort the optical indicatrix of the medium, producing optical anistropy. The spatially varying refractive indices are incorporated into the full-wave Maxwell’s equations. The governing equations are discretized by using a vector finite-element method cast in a high-order finite element approximation. This approach allows us to study the complexities of the mechanical deformation within a framework of a high-order formulation that can, in turn, reduce the computational requirement without degrading its performance. The optical activities generated, total energy produced, and power loss due to the mechanical stresses and strains are reported and discussed.

© 2006 Optical Society of America

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  1. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 1, 302-307 (1966).
  2. A. Taflove and M. E. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
    [CrossRef]
  3. K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).
  4. V. Shankar, A. Mohammadia, and W. Hall, "A time-domain finite-volume treatment for the Maxwell equations," Electromagnetics 10, 127-145 (1990).
    [CrossRef]
  5. R. W. Noack and D. A. Anderson, "Time-domain solutions of Maxwell's equations using a finite-volume formulation," Presented at the AIAA 30th. Aerospace Sciences Meeting, Reno, Nev., Jan. 6-9 1992, paper 92-0451.
  6. S. Brandon and P. Rambo, "Stability of the DSI electromagnetic update algorithm on a chevron grid," in Proceedings of the 22nd IEEE International Conference on Plasma Science, (IEEE, 1995).
  7. D. J. Riley and C. D. Turner, "VOLMAX: a solid model based transient volumetric Maxwell solver using hybrid grids," IEEE Trans. Antennas Propag. 39, 20-23 (1997).
  8. J. C. Nedelec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
    [CrossRef]
  9. J. C. Nedelec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
    [CrossRef]
  10. A. Bossavit, "Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism," IEE Proc. A: Sci., Meas. Technol. 135, 493-500 (1988).
  11. A. Konrad, "Vector variational formulations of electromagnetic fields in anisotropic media," IEEE Trans. Microwave Theory Tech. 24, 533-559 (1976).
    [CrossRef]
  12. A. Bossavit, "Solving Maxwell equations in a closed cavity, and the question of spurious modes," IEEE Trans. Magn. 26, 702-705 (1990).
    [CrossRef]
  13. R. Rieben, D. White, and G. Rodrigue, "High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations," IEEE Trans. Antennas Propag. 52, 2190-2195 (2004).
    [CrossRef]
  14. R. Rieben, "A novel high order time domain vector finite element for the simulation of electronic device," Ph.D. dissertation, UCRL-TH-205466 (University of California, Davis, 2004).
  15. M. Li and X. Chen, "Fiber spinning for reducing polarization mode dispersion in single mode fibers: theory and applications," Opt. Fiber Technol. 8, 162-169 (2002).
  16. R. Ulrich and A. Simon "Polarization optics of twisted single-mode fibers," Appl. Opt. 18, 2241-2251 (1979).
    [CrossRef] [PubMed]
  17. R. T. Deck, M. Mirkov, and B. G. Bagley, "Determination of bending losses in rectangular waveguides," J. Lightwave Technol. 16, 1703-1714 (1998).
    [CrossRef]
  18. Z. Menachem, "Wave propagation in a curved waveguide with arbitrary dielectric transverse profiles," Electromagn. Waves 42, 173-192 (2003).
    [CrossRef]
  19. H. Tai and R. Rogowski, "Optical anisotropy induced by torsion and bending in an optical fiber," Opt. Fiber Technol. 8, 162-169 (2002).
    [CrossRef]
  20. G. Durana, J. Zubia, J. Arrue, G. Aldabaldetreku, and J. Mateo, "Dependence of bending losses on cladding thickness in plastic optical fibers," Appl. Opt. 42, 997-1002 (2003).
    [CrossRef] [PubMed]
  21. XYZ Scientific Applications Inc., TrueGrid home page, http://www.truegrid.com(2002).

2004 (1)

R. Rieben, D. White, and G. Rodrigue, "High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations," IEEE Trans. Antennas Propag. 52, 2190-2195 (2004).
[CrossRef]

2003 (2)

Z. Menachem, "Wave propagation in a curved waveguide with arbitrary dielectric transverse profiles," Electromagn. Waves 42, 173-192 (2003).
[CrossRef]

G. Durana, J. Zubia, J. Arrue, G. Aldabaldetreku, and J. Mateo, "Dependence of bending losses on cladding thickness in plastic optical fibers," Appl. Opt. 42, 997-1002 (2003).
[CrossRef] [PubMed]

2002 (2)

H. Tai and R. Rogowski, "Optical anisotropy induced by torsion and bending in an optical fiber," Opt. Fiber Technol. 8, 162-169 (2002).
[CrossRef]

M. Li and X. Chen, "Fiber spinning for reducing polarization mode dispersion in single mode fibers: theory and applications," Opt. Fiber Technol. 8, 162-169 (2002).

1998 (1)

1997 (1)

D. J. Riley and C. D. Turner, "VOLMAX: a solid model based transient volumetric Maxwell solver using hybrid grids," IEEE Trans. Antennas Propag. 39, 20-23 (1997).

1990 (2)

V. Shankar, A. Mohammadia, and W. Hall, "A time-domain finite-volume treatment for the Maxwell equations," Electromagnetics 10, 127-145 (1990).
[CrossRef]

A. Bossavit, "Solving Maxwell equations in a closed cavity, and the question of spurious modes," IEEE Trans. Magn. 26, 702-705 (1990).
[CrossRef]

1988 (1)

A. Bossavit, "Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism," IEE Proc. A: Sci., Meas. Technol. 135, 493-500 (1988).

1986 (1)

J. C. Nedelec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
[CrossRef]

1980 (1)

J. C. Nedelec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
[CrossRef]

1979 (1)

1976 (1)

A. Konrad, "Vector variational formulations of electromagnetic fields in anisotropic media," IEEE Trans. Microwave Theory Tech. 24, 533-559 (1976).
[CrossRef]

1975 (1)

A. Taflove and M. E. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 1, 302-307 (1966).

Aldabaldetreku, G.

Anderson, D. A.

R. W. Noack and D. A. Anderson, "Time-domain solutions of Maxwell's equations using a finite-volume formulation," Presented at the AIAA 30th. Aerospace Sciences Meeting, Reno, Nev., Jan. 6-9 1992, paper 92-0451.

Arrue, J.

Bagley, B. G.

Bossavit, A.

A. Bossavit, "Solving Maxwell equations in a closed cavity, and the question of spurious modes," IEEE Trans. Magn. 26, 702-705 (1990).
[CrossRef]

A. Bossavit, "Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism," IEE Proc. A: Sci., Meas. Technol. 135, 493-500 (1988).

Brandon, S.

S. Brandon and P. Rambo, "Stability of the DSI electromagnetic update algorithm on a chevron grid," in Proceedings of the 22nd IEEE International Conference on Plasma Science, (IEEE, 1995).

Brodwin, M. E.

A. Taflove and M. E. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
[CrossRef]

Chen, X.

M. Li and X. Chen, "Fiber spinning for reducing polarization mode dispersion in single mode fibers: theory and applications," Opt. Fiber Technol. 8, 162-169 (2002).

Deck, R. T.

Durana, G.

Hall, W.

V. Shankar, A. Mohammadia, and W. Hall, "A time-domain finite-volume treatment for the Maxwell equations," Electromagnetics 10, 127-145 (1990).
[CrossRef]

Konrad, A.

A. Konrad, "Vector variational formulations of electromagnetic fields in anisotropic media," IEEE Trans. Microwave Theory Tech. 24, 533-559 (1976).
[CrossRef]

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Li, M.

M. Li and X. Chen, "Fiber spinning for reducing polarization mode dispersion in single mode fibers: theory and applications," Opt. Fiber Technol. 8, 162-169 (2002).

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

Mateo, J.

Menachem, Z.

Z. Menachem, "Wave propagation in a curved waveguide with arbitrary dielectric transverse profiles," Electromagn. Waves 42, 173-192 (2003).
[CrossRef]

Mirkov, M.

Mohammadia, A.

V. Shankar, A. Mohammadia, and W. Hall, "A time-domain finite-volume treatment for the Maxwell equations," Electromagnetics 10, 127-145 (1990).
[CrossRef]

Nedelec, J. C.

J. C. Nedelec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
[CrossRef]

J. C. Nedelec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
[CrossRef]

Noack, R. W.

R. W. Noack and D. A. Anderson, "Time-domain solutions of Maxwell's equations using a finite-volume formulation," Presented at the AIAA 30th. Aerospace Sciences Meeting, Reno, Nev., Jan. 6-9 1992, paper 92-0451.

Rambo, P.

S. Brandon and P. Rambo, "Stability of the DSI electromagnetic update algorithm on a chevron grid," in Proceedings of the 22nd IEEE International Conference on Plasma Science, (IEEE, 1995).

Rieben, R.

R. Rieben, D. White, and G. Rodrigue, "High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations," IEEE Trans. Antennas Propag. 52, 2190-2195 (2004).
[CrossRef]

R. Rieben, "A novel high order time domain vector finite element for the simulation of electronic device," Ph.D. dissertation, UCRL-TH-205466 (University of California, Davis, 2004).

Riley, D. J.

D. J. Riley and C. D. Turner, "VOLMAX: a solid model based transient volumetric Maxwell solver using hybrid grids," IEEE Trans. Antennas Propag. 39, 20-23 (1997).

Rodrigue, G.

R. Rieben, D. White, and G. Rodrigue, "High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations," IEEE Trans. Antennas Propag. 52, 2190-2195 (2004).
[CrossRef]

Rogowski, R.

H. Tai and R. Rogowski, "Optical anisotropy induced by torsion and bending in an optical fiber," Opt. Fiber Technol. 8, 162-169 (2002).
[CrossRef]

Shankar, V.

V. Shankar, A. Mohammadia, and W. Hall, "A time-domain finite-volume treatment for the Maxwell equations," Electromagnetics 10, 127-145 (1990).
[CrossRef]

Simon, A.

Taflove, A.

A. Taflove and M. E. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
[CrossRef]

Tai, H.

H. Tai and R. Rogowski, "Optical anisotropy induced by torsion and bending in an optical fiber," Opt. Fiber Technol. 8, 162-169 (2002).
[CrossRef]

Turner, C. D.

D. J. Riley and C. D. Turner, "VOLMAX: a solid model based transient volumetric Maxwell solver using hybrid grids," IEEE Trans. Antennas Propag. 39, 20-23 (1997).

Ulrich, R.

White, D.

R. Rieben, D. White, and G. Rodrigue, "High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations," IEEE Trans. Antennas Propag. 52, 2190-2195 (2004).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 1, 302-307 (1966).

Zubia, J.

Appl. Opt. (2)

Electromagn. Waves (1)

Z. Menachem, "Wave propagation in a curved waveguide with arbitrary dielectric transverse profiles," Electromagn. Waves 42, 173-192 (2003).
[CrossRef]

Electromagnetics (1)

V. Shankar, A. Mohammadia, and W. Hall, "A time-domain finite-volume treatment for the Maxwell equations," Electromagnetics 10, 127-145 (1990).
[CrossRef]

IEE Proc. A: Sci., Meas. Technol. (1)

A. Bossavit, "Whitney forms: A class of finite elements for three-dimensional computations in electromagnetism," IEE Proc. A: Sci., Meas. Technol. 135, 493-500 (1988).

IEEE Trans. Antennas Propag. (3)

D. J. Riley and C. D. Turner, "VOLMAX: a solid model based transient volumetric Maxwell solver using hybrid grids," IEEE Trans. Antennas Propag. 39, 20-23 (1997).

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media," IEEE Trans. Antennas Propag. 1, 302-307 (1966).

R. Rieben, D. White, and G. Rodrigue, "High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations," IEEE Trans. Antennas Propag. 52, 2190-2195 (2004).
[CrossRef]

IEEE Trans. Magn. (1)

A. Bossavit, "Solving Maxwell equations in a closed cavity, and the question of spurious modes," IEEE Trans. Magn. 26, 702-705 (1990).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

A. Taflove and M. E. Brodwin, "Numerical solution of steady-state electromagnetic scattering problems using the time-dependent Maxwell's equations," IEEE Trans. Microwave Theory Tech. 23, 623-630 (1975).
[CrossRef]

A. Konrad, "Vector variational formulations of electromagnetic fields in anisotropic media," IEEE Trans. Microwave Theory Tech. 24, 533-559 (1976).
[CrossRef]

J. Lightwave Technol. (1)

Numer. Math. (2)

J. C. Nedelec, "Mixed finite elements in R3," Numer. Math. 35, 315-341 (1980).
[CrossRef]

J. C. Nedelec, "A new family of mixed finite elements in R3," Numer. Math. 50, 57-81 (1986).
[CrossRef]

Opt. Fiber Technol. (2)

H. Tai and R. Rogowski, "Optical anisotropy induced by torsion and bending in an optical fiber," Opt. Fiber Technol. 8, 162-169 (2002).
[CrossRef]

M. Li and X. Chen, "Fiber spinning for reducing polarization mode dispersion in single mode fibers: theory and applications," Opt. Fiber Technol. 8, 162-169 (2002).

Other (5)

R. Rieben, "A novel high order time domain vector finite element for the simulation of electronic device," Ph.D. dissertation, UCRL-TH-205466 (University of California, Davis, 2004).

XYZ Scientific Applications Inc., TrueGrid home page, http://www.truegrid.com(2002).

K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

R. W. Noack and D. A. Anderson, "Time-domain solutions of Maxwell's equations using a finite-volume formulation," Presented at the AIAA 30th. Aerospace Sciences Meeting, Reno, Nev., Jan. 6-9 1992, paper 92-0451.

S. Brandon and P. Rambo, "Stability of the DSI electromagnetic update algorithm on a chevron grid," in Proceedings of the 22nd IEEE International Conference on Plasma Science, (IEEE, 1995).

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Figures (15)

Fig. 1
Fig. 1

Computational mesh for a bent fiber.

Fig. 2
Fig. 2

Computational mesh for a twisted fiber.

Fig. 3
Fig. 3

Power flow in kW μ m 2 inside straight fiber.

Fig. 4
Fig. 4

Power flow in kW μ m 2 for a 15 ° bend.

Fig. 5
Fig. 5

Power flow in kW μ m 2 for a 60 ° bend.

Fig. 6
Fig. 6

Power flow in kW μ m 2 for a 15 ° , 30 ° , 45 ° , and 6 ° bend at t = 0.128 ps .

Fig. 7
Fig. 7

Electric field propagation in kV μ m for a 15 ° bend.

Fig. 8
Fig. 8

Comparison of time evolution of total energy for a bent fiber.

Fig. 9
Fig. 9

Power flow in kW μ m 2 for a 20 π twist fiber.

Fig. 10
Fig. 10

Electric field propagation in kV μ m 2 for a twisted fiber.

Fig. 11
Fig. 11

Comparison of time evolution of total energy for a twisted fiber.

Fig. 12
Fig. 12

Power loss in the core as a function of time for bent fibers.

Fig. 13
Fig. 13

Power loss in the core as a function of time for twisted fibers.

Fig. 14
Fig. 14

Electric field propagation in kV μ m 2 for 20 π twist.

Fig. 15
Fig. 15

Comparison of time evolution of total energy in the core among a bent fiber, a twisted fiber, and a combined bent and twisted fiber.

Tables (2)

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Table 1 Computational Requirement

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Table 2 Computational Experiments

Equations (33)

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

ϵ E t = × ( μ 1 B ) σ E J ( t ) ,
B t = × E σ * ( μ 1 B ) ,
M ϵ e t = K T M μ b M σ e M ϵ j ,
b t = K e M σ * ,
E ̃ = e T M ϵ e + b T M μ b .
k η i k ϵ k j = δ i j .
i j η i j x i x j = 1 ,
i j η ̃ i j x i x j = 1
i j Δ i j x i x j = 0 .
Δ ( k η i k ϵ k j ) k η i k ϵ k j = k ( η i k + Δ η i k ) ( ϵ k j + Δ ϵ k j ) k η i k ϵ k j = k Δ η i k ϵ k j + k η i k Δ ϵ k j = 0 ,
Δ ϵ k l = i Δ η i k k ϵ k j .
[ Δ η 11 Δ η 22 Δ η 33 Δ η 23 Δ η 31 Δ η 12 ] = [ p 11 p 12 p 13 p 14 p 15 p 16 p 11 p 12 p 13 p 14 p 15 p 16 p 11 p 12 p 13 p 14 p 15 p 16 p 11 p 12 p 13 p 14 p 15 p 16 p 11 p 12 p 13 p 14 p 15 p 16 p 11 p 12 p 13 p 14 p 15 p 16 ] [ S 11 S 22 S 33 S 23 S 31 S 12 ] .
S i j = 1 2 ( u i x j + u j x i ) ,
P = [ p 11 p 12 p 12 0 0 0 p 12 p 11 p 12 0 0 0 p 12 p 12 p 11 0 0 0 0 0 0 ϕ 0 0 0 0 0 0 ϕ 0 0 0 0 0 0 ϕ ] ,
u r = 0 , u θ = α z r , u z = 0 ,
u x = α z y , u y = α z x , u z = 0 .
S 12 = 0 , S 13 = 12 α y , S 23 = 12 α x .
[ Δ η 11 Δ η 22 Δ η 33 Δ η 23 Δ η 31 Δ η 12 ] = [ 0 0 0 ϕ α x ϕ α y 0 ] .
Δ ϵ = ϵ ( Δ η ) ϵ = ϵ 2 [ 0 0 ϕ α y 0 0 ϕ α x ϕ α y ϕ α x 0 ] .
n x = n co ,
n y = n co [ 1 + n co 2 4 ( p 11 p 12 ) τ 2 r 2 4 ] ,
n z = n co [ 1 n co 2 4 ( p 11 p 12 ) τ 2 r 2 4 ] .
[ Δ η 11 Δ η 22 Δ η 33 Δ η 23 Δ η 31 Δ η 12 ] = [ p 12 S 33 p 12 S 33 p 11 S 33 0 0 0 ] ,
Δ ϵ = ϵ ( Δ η ) ϵ ,
Δ ϵ = ϵ 2 [ p 12 S 33 0 0 0 p 12 S 33 0 0 0 p 11 S 33 ] .
n x = n co ( 1 + n co 2 p 12 x 2 R ) ,
n y = n co [ 1 + n co 2 ( p 11 + p 12 ) x 4 R + n co 2 4 ( p 11 p 12 ) x 2 R 2 ] ,
n z = n co [ 1 + n co 2 ( p 11 + p 12 ) x 4 R n co 2 4 ( p 11 p 12 ) x 2 R 2 ] .
Δ ϵ = ϵ ( Δ η ) ϵ = ϵ 2 [ p 12 S 33 0 ϕ α y 0 p 12 S 33 ϕ α x ϕ α y ϕ α x p 11 S 33 ] .
n x = n co ( 1 + n co 2 p 12 x 2 R ) ,
n y = n co [ 1 + n co 2 ( p 11 + p 12 ) x 4 R + n co 2 4 ( p 11 p 12 ) x 2 R 2 + τ 2 r 2 4 ] ,
n z = n co [ 1 + n co 2 ( p 11 + p 12 ) x 4 R n co 2 4 ( p 11 p 12 ) x 2 R 2 + τ 2 r 2 4 ] .
[ h ] α = g τ ,

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