Abstract

We present a new numerical method for designing dispersion compensating optical fibers. The method is based on the solving of the Helmholtz wave equation with a finite-difference modesolver and uses topology optimization combined with a regularization filter for the design of the refractive index profile. We illustrate the applicability of the proposed method through numerical examples and, furthermore, address the problem of keeping the optimized design single moded by including a singlemode constraint in the optimization problem.

© 2008 Optical Society of America

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References

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  1. L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, "Dispersion compensating fibers," Opt. Fiber Technol. 6, 164-180 (2000).
    [CrossRef]
  2. G. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).
  3. A. Bjarklev, T. Rasmussen, O. Lumholt, K. Rottwitt, and M. Helmer, "Optimal design of single-cladded dispersion-compensating optical fibers," Opt. Lett. 19, 457-459 (1994).
    [CrossRef] [PubMed]
  4. J. L. Auguste, J. M. Blondy, J. Maury, J. Marcou, B. Dussardier, G. Monnom, R. Jindal, K. Thyagarajan, and B. P. Pal, "Conception, realization, and characterization of a very high negative chromatic dispersion fiber," Opt. Fiber Technol. 8, 89-105 (2002).
    [CrossRef]
  5. F. Gerome, J.-L. Auguste, J. Maury, J.-M. Blondy, and J. Marcou, "Theoretical and experimental analysis of a chromatic dispersion compensating module using a dual concentric core fiber," J. Lightwave Technol. 24, 442-448 (2006).
    [CrossRef]
  6. H. Subbaraman, T. Ling, Y. Jiang, M. Y. Chen, P. Cao, and R. T. Chen, "Design of a broadband highly dispersive pure silica photonic crystal fiber," Appl. Opt. 46, 3263-3268 (2007).
    [CrossRef] [PubMed]
  7. M. Bendsøe and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Comput. Methods Appl. Mech. Eng. 71, 197-224 (1988).
    [CrossRef]
  8. M. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).
  9. U. D. Larsen, O. Sigmund, and S. Bouwstra, "Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio," J. Microelectromech. Syst. 6, 99-106 (1997).
    [CrossRef]
  10. T. Borrvall and J. Petersson, "Topology optimisation of fluids in Stokes flow," Int. J. Numer. Methods Fluids 41, 77-107 (2003).
    [CrossRef]
  11. J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimisation: low-loss waveguide bends," Appl. Phys. Lett. 84, 2022-2024 (2004).
    [CrossRef]
  12. J. S. Jensen and O. Sigmund, "Topology optimization of photonic crystal structures: A high bandwidth low loss T-junction waveguide," J. Opt. Soc. Am. B 22, 69-71 (2005).
    [CrossRef]
  13. P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, J. S. Jensen, P. Shi, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express 12, 1996-2001 (2004).
    [CrossRef] [PubMed]
  14. A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibers (Kluwer, 2003).
    [CrossRef]
  15. T. Fujisawa, K. Saitoh, K. Wada, and M. Koshiba, "Chromatic dispersion profile optimization of dual-concentric-core photonic crystal fibers for broadband dispersion compensation," Opt. Express 14, 893-900 (2006).
    [CrossRef] [PubMed]
  16. F. Poletti, V. Finazzi, T. M. Monroe, N. G. R. Broderick, V. Tse, and D. J. Richardson, "Inverse design and fabrication tolerances of ultraflattened dispersion holey fibers," Opt. Express 13, 3728-3736 (2005).
    [CrossRef] [PubMed]
  17. O. Sigmund and J. Petersson, "Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima," Struct. Optim. 16, 68-75 (1998).
    [CrossRef]
  18. J. Riishede, N. A. Mortensen, and J. Lægsgaard, "A poor man's approach to modelling microstructured optical fibers," J. Opt. A, Pure Appl. Opt. 13, 534-538 (2003).
    [CrossRef]
  19. T. E. Bruns and D. A. Tortorelli, "Topology optimization of non-linear elastic structures and compliant mechanisms," Comput. Methods Appl. Mech. Eng. 190, 3443-3459 (2001).
    [CrossRef]
  20. B. Bourdin, "Filters in topology optimization," Int. J. Numer. Methods Eng. 50, 2143-2158 (2001).
    [CrossRef]
  21. K. Svanberg, "The method of moving assymptotes-a new method for structural optimization," Int. J. Numer. Methods Eng. 24, 359-373 (1987).
    [CrossRef]
  22. www.corning. com/opticalfiber
  23. A. P. Seyranian, E. Lund, and N. Olhoff, "Multiple eigenvalues in structural optimization problems," Struct. Optim. 8, 207-227 (1994).
    [CrossRef]
  24. J. S. Jensen and N. L. Pedersen, "On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases," J. Sound Vib. 289, 967-986 (2006).
    [CrossRef]

2007 (1)

2006 (3)

2005 (2)

J. S. Jensen and O. Sigmund, "Topology optimization of photonic crystal structures: A high bandwidth low loss T-junction waveguide," J. Opt. Soc. Am. B 22, 69-71 (2005).
[CrossRef]

F. Poletti, V. Finazzi, T. M. Monroe, N. G. R. Broderick, V. Tse, and D. J. Richardson, "Inverse design and fabrication tolerances of ultraflattened dispersion holey fibers," Opt. Express 13, 3728-3736 (2005).
[CrossRef] [PubMed]

2004 (2)

J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimisation: low-loss waveguide bends," Appl. Phys. Lett. 84, 2022-2024 (2004).
[CrossRef]

P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, J. S. Jensen, P. Shi, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express 12, 1996-2001 (2004).
[CrossRef] [PubMed]

2003 (2)

J. Riishede, N. A. Mortensen, and J. Lægsgaard, "A poor man's approach to modelling microstructured optical fibers," J. Opt. A, Pure Appl. Opt. 13, 534-538 (2003).
[CrossRef]

T. Borrvall and J. Petersson, "Topology optimisation of fluids in Stokes flow," Int. J. Numer. Methods Fluids 41, 77-107 (2003).
[CrossRef]

2002 (1)

J. L. Auguste, J. M. Blondy, J. Maury, J. Marcou, B. Dussardier, G. Monnom, R. Jindal, K. Thyagarajan, and B. P. Pal, "Conception, realization, and characterization of a very high negative chromatic dispersion fiber," Opt. Fiber Technol. 8, 89-105 (2002).
[CrossRef]

2001 (2)

T. E. Bruns and D. A. Tortorelli, "Topology optimization of non-linear elastic structures and compliant mechanisms," Comput. Methods Appl. Mech. Eng. 190, 3443-3459 (2001).
[CrossRef]

B. Bourdin, "Filters in topology optimization," Int. J. Numer. Methods Eng. 50, 2143-2158 (2001).
[CrossRef]

2000 (1)

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, "Dispersion compensating fibers," Opt. Fiber Technol. 6, 164-180 (2000).
[CrossRef]

1998 (1)

O. Sigmund and J. Petersson, "Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima," Struct. Optim. 16, 68-75 (1998).
[CrossRef]

1997 (1)

U. D. Larsen, O. Sigmund, and S. Bouwstra, "Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio," J. Microelectromech. Syst. 6, 99-106 (1997).
[CrossRef]

1994 (2)

A. P. Seyranian, E. Lund, and N. Olhoff, "Multiple eigenvalues in structural optimization problems," Struct. Optim. 8, 207-227 (1994).
[CrossRef]

A. Bjarklev, T. Rasmussen, O. Lumholt, K. Rottwitt, and M. Helmer, "Optimal design of single-cladded dispersion-compensating optical fibers," Opt. Lett. 19, 457-459 (1994).
[CrossRef] [PubMed]

1988 (1)

M. Bendsøe and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Comput. Methods Appl. Mech. Eng. 71, 197-224 (1988).
[CrossRef]

1987 (1)

K. Svanberg, "The method of moving assymptotes-a new method for structural optimization," Int. J. Numer. Methods Eng. 24, 359-373 (1987).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimisation: low-loss waveguide bends," Appl. Phys. Lett. 84, 2022-2024 (2004).
[CrossRef]

Comput. Methods Appl. Mech. Eng. (2)

M. Bendsøe and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Comput. Methods Appl. Mech. Eng. 71, 197-224 (1988).
[CrossRef]

T. E. Bruns and D. A. Tortorelli, "Topology optimization of non-linear elastic structures and compliant mechanisms," Comput. Methods Appl. Mech. Eng. 190, 3443-3459 (2001).
[CrossRef]

Int. J. Numer. Methods Eng. (2)

B. Bourdin, "Filters in topology optimization," Int. J. Numer. Methods Eng. 50, 2143-2158 (2001).
[CrossRef]

K. Svanberg, "The method of moving assymptotes-a new method for structural optimization," Int. J. Numer. Methods Eng. 24, 359-373 (1987).
[CrossRef]

Int. J. Numer. Methods Fluids (1)

T. Borrvall and J. Petersson, "Topology optimisation of fluids in Stokes flow," Int. J. Numer. Methods Fluids 41, 77-107 (2003).
[CrossRef]

J. Lightwave Technol. (1)

J. Microelectromech. Syst. (1)

U. D. Larsen, O. Sigmund, and S. Bouwstra, "Design and fabrication of compliant micromechanisms and structures with negative Poisson's ratio," J. Microelectromech. Syst. 6, 99-106 (1997).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

J. Riishede, N. A. Mortensen, and J. Lægsgaard, "A poor man's approach to modelling microstructured optical fibers," J. Opt. A, Pure Appl. Opt. 13, 534-538 (2003).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. S. Jensen and O. Sigmund, "Topology optimization of photonic crystal structures: A high bandwidth low loss T-junction waveguide," J. Opt. Soc. Am. B 22, 69-71 (2005).
[CrossRef]

J. Sound Vib. (1)

J. S. Jensen and N. L. Pedersen, "On maximal eigenfrequency separation in two-material structures: the 1D and 2D scalar cases," J. Sound Vib. 289, 967-986 (2006).
[CrossRef]

Opt. Express (3)

Opt. Fiber Technol. (2)

J. L. Auguste, J. M. Blondy, J. Maury, J. Marcou, B. Dussardier, G. Monnom, R. Jindal, K. Thyagarajan, and B. P. Pal, "Conception, realization, and characterization of a very high negative chromatic dispersion fiber," Opt. Fiber Technol. 8, 89-105 (2002).
[CrossRef]

L. Grüner-Nielsen, S. N. Knudsen, B. Edvold, T. Veng, D. Magnussen, C. C. Larsen, and H. Damsgaard, "Dispersion compensating fibers," Opt. Fiber Technol. 6, 164-180 (2000).
[CrossRef]

Opt. Lett. (1)

Struct. Optim. (2)

A. P. Seyranian, E. Lund, and N. Olhoff, "Multiple eigenvalues in structural optimization problems," Struct. Optim. 8, 207-227 (1994).
[CrossRef]

O. Sigmund and J. Petersson, "Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima," Struct. Optim. 16, 68-75 (1998).
[CrossRef]

Other (4)

G. Agrawal, Fiber-Optic Communication Systems, 2nd ed. (Wiley, 1997).

M. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer, 2003).

A. Bjarklev, J. Broeng, and A. S. Bjarklev, Photonic Crystal Fibers (Kluwer, 2003).
[CrossRef]

www.corning. com/opticalfiber

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Figures (7)

Fig. 1
Fig. 1

Schematic illustration of the different steps of the topology optimization method.

Fig. 2
Fig. 2

Geometry of the density filtering function s for increasing values of the filter width w. Only filter functions with an uneven number of grid points along the diameter are used.

Fig. 3
Fig. 3

Comparison of the dispersion of the optimized DCF design and the optimization target corresponding to 5 × the dispersion of a standard single-mode fiber.

Fig. 4
Fig. 4

Spatial distribution of the refractive index for the optimized DCF design.

Fig. 5
Fig. 5

(a) Cross section of the index profile along the diameter of the calculation domain. A density filter width of w = 5 has been applied. (b) The same calculation as in (a) but using a filter width of w = 1 , i.e., disabling the density filter.

Fig. 6
Fig. 6

Refractive index profile of an optimized DCF design obtained by using a single-mode constraint. A density filter width of w = 5 has been applied during the optimization.

Fig. 7
Fig. 7

Effective mode indices of the LP 01 , LP 11 , and LP 02 modes as a function of the iteration number. The single-mode constraint is seen to force the mode index of the higher-order modes below the background index thus keeping the optimized design single-moded.

Equations (26)

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D = λ c d 2 n eff d λ 2 .
n eff = β k
t 2 ψ ( x , y ) + n 2 ( x , y ) k 0 2 ψ ( x , y ) = β 2 ψ ( x , y ) .
2 ψ ( x , y ) ψ ( x Δ x , y ) 2 ψ ( x , y ) + ψ ( x + Δ x , y ) ( Δ x ) 2 + ψ ( x , y Δ y ) 2 ψ ( x , y ) + ψ ( x , y + Δ y ) ( Δ y ) 2 .
Φ ψ = β 2 ψ ,
Φ ( i , j ) = { 4 d 2 + n ( x i , y i ) 2 k 0 2 for i = j 1 d 2 for i , j nearest neighbors 0 otherwise } .
f = i = 1 N ( D i * D i ) 2 ( D i * ) 2 .
D λ c n eff , 1 2 n eff , 2 + n eff , 3 ( Δ λ ) 2 .
n i = n min + γ i ( n max n min ) ,
f γ i = 2 ( n max n min ) j = 1 N ( D j * D j ) ( D j * ) 2 D j n i .
n eff n i = ( β 2 k 0 ) n i = 1 k 0 2 1 2 n eff ( β 2 ) n i ,
ψ T n i Φ ψ + ψ T Φ n i ψ + ψ T Φ ψ n i = ( β 2 ) n i ψ T ψ + 2 β 2 ψ T ψ n i .
( β 2 ) n i = ψ T Φ n i ψ .
( β 2 ) n i = ψ T Φ n i ψ = 2 n i k 0 2 ψ i 2 ,
D n i = λ c n i ( Δ λ ) 2 ( ψ 1 , i 2 n eff , 1 2 ψ 2 , i 2 n eff , 2 + ψ 3 , i 2 n eff , 3 ) .
γ ̃ j = i = 1 N γ i s i ( j ) i = 1 N s i ( j ) ,
n j = n min + γ ̃ j ( n max n min ) .
f γ i = j = 1 N f γ ̃ j γ ̃ j γ i s j ( i ) = j = 1 N f γ ̃ j s j ( i ) M ( i ) .
min : f = i = 1 N ( D i * D i ) 2 ( D i * ) 2 ,
γ i : s.t. n eff L P 11 < n bg ,
s.t. n eff L P 01 > n bg .
n eff L P 01 n i = n i n eff ψ i 2 .
ψ = c 1 ψ 1 + c 2 ψ 2 ,
n eff n i = ( c 1 2 ψ 1 , i 2 + c 2 2 ψ 2 , i 2 + 2 c 1 c 2 ψ 1 , i ψ 2 , i ) n i n eff .
[ n i n eff ψ 1 , i 2 n i n eff ψ 1 , i ψ 2 , i n i n eff ψ 1 , i ψ 2 , i n i n eff ψ 2 , i 2 ] { c 1 c 2 } = { 0 0 } .
n eff L P 11 n i = n i n eff ( ψ 1 , i 2 + ψ 2 , i 2 ) .

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