Abstract

A T junction in a photonic crystal waveguide is designed with the topology-optimization method. The gradient-based optimization tool is used to modify the material distribution in the junction area so that the power transmission in the output ports is maximized. To obtain high transmission in a large frequency range, we use an active-set strategy by using a number of target frequencies that are updated repeatedly in the optimization procedure. We apply a continuation method based on artificial damping to avoid undesired local maxima and also introduce artificial damping in a penalization scheme to avoid nondiscrete properties in the design domain.

© 2005 Optical Society of America

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References

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. M. Burger, S. J. Osher, and E. Yablonovitch, "Inverse problem techniques for the design of photonic crystals," IEICE Trans. Electron. E87-C, 258-265 (2004).
  4. J. S. Jensen and O. Sigmund, "Systematic design of photonic crystal structures using topology optimization: low-loss waveguide bends," Appl. Phys. Lett. 84, 2022-2024 (2004).
    [CrossRef]
  5. P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund, "Topology optimization and fabrication of photonic crystal structures," Opt. Express 12, 1996-2001 (2004).
    [CrossRef] [PubMed]
  6. K. B. Chung, J. S. Yoon, and G. H. Song, "Analysis of optical splitters in photonic crystals," in Photonic Bandgap Material and Devices, A.Adibi, A.Scherer, and S.-Yu.Lin, eds., Proc. SPIE 4655, 349-355 (2002).
  7. J. Smajic, C. Hafner, and D. Erni, "Optimization of photonic crystal structures," J. Opt. Soc. Am. A 21, 2223-2232 (2004).
    [CrossRef]
  8. W. J. Kim and J. D. O'Brien, "Optimization of a two-dimensional photonic-crystal waveguide branch by simulated annealing and the finite-element method," J. Opt. Soc. Am. B 21, 289-295 (2004).
    [CrossRef]
  9. T. Felici and T. F. G. Gallagher, "Improved waveguide structures derived from new rapid optimization techniques," in Physics and Simulation of Optoelectronic Devices XI, M.Osinski, H.Amano, and P.Blood, eds., Proc. SPIE 4986, 375-385 (2003).
  10. M. P. Bendsøe and N. Kikuchi, "Generating optimal topologies in structural design using a homogenization method," Comput. Methods Appl. Mech. Eng. 71, 197-224 (1988).
    [CrossRef]
  11. H. L. Thomas, M. Zhou, and U. Schramm, "Issues of commercial optimization software development," Struct. Multidiscip. Optim. 23, 97-110 (2002).
    [CrossRef]
  12. M. P. Bendsøe and O. Sigmund, Topology Optimization--Theory, Methods and Applications (Springer, Berlin, 2003).
  13. O. Sigmund and J. S. Jensen, "Systematic design of phononic band-gap materials and structures by topology optimization," Proc. R. Soc. London, Ser. A 361, 1001-1019 (2003).
  14. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
  15. M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performanceabsorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microw. Wirel. Compon. Lett. 11, 152-154 (2001).
    [CrossRef]
  16. R. D. Cook, D. S. Malkus, M. E. Plesha, and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed. (Wiley, New York, 2002).
  17. K. Svanberg, "The method of moving asymptotes--a new method for structural optimization," Int. J. Numer. Methods Eng. 24, 359-373 (1987).
    [CrossRef]
  18. J. S. Jensen, "Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures, J. Sound Vib. 266, 1053-1078 (2003).
    [CrossRef]
  19. G. Allaire and G. A. Francfort, "A numerical algorithm for topology and shape optimization," in Topology Optimization of Structures, M.P.Bendsøe and C.A. M.Soares, eds. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 239-248.
  20. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, New York, 2002).

2004 (5)

2003 (2)

O. Sigmund and J. S. Jensen, "Systematic design of phononic band-gap materials and structures by topology optimization," Proc. R. Soc. London, Ser. A 361, 1001-1019 (2003).

J. S. Jensen, "Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures, J. Sound Vib. 266, 1053-1078 (2003).
[CrossRef]

2002 (1)

H. L. Thomas, M. Zhou, and U. Schramm, "Issues of commercial optimization software development," Struct. Multidiscip. Optim. 23, 97-110 (2002).
[CrossRef]

2001 (1)

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performanceabsorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microw. Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

1996 (1)

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

1988 (1)

M. P. 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 (3)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

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

Allaire, G.

G. Allaire and G. A. Francfort, "A numerical algorithm for topology and shape optimization," in Topology Optimization of Structures, M.P.Bendsøe and C.A. M.Soares, eds. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 239-248.

Bendsøe, M. P.

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

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

Borel, P. I.

Burger, M.

M. Burger, S. J. Osher, and E. Yablonovitch, "Inverse problem techniques for the design of photonic crystals," IEICE Trans. Electron. E87-C, 258-265 (2004).

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Chung, K. B.

K. B. Chung, J. S. Yoon, and G. H. Song, "Analysis of optical splitters in photonic crystals," in Photonic Bandgap Material and Devices, A.Adibi, A.Scherer, and S.-Yu.Lin, eds., Proc. SPIE 4655, 349-355 (2002).

Cook, R. D.

R. D. Cook, D. S. Malkus, M. E. Plesha, and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed. (Wiley, New York, 2002).

Erni, D.

Fan, S.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Felici, T.

T. Felici and T. F. G. Gallagher, "Improved waveguide structures derived from new rapid optimization techniques," in Physics and Simulation of Optoelectronic Devices XI, M.Osinski, H.Amano, and P.Blood, eds., Proc. SPIE 4986, 375-385 (2003).

Francfort, G. A.

G. Allaire and G. A. Francfort, "A numerical algorithm for topology and shape optimization," in Topology Optimization of Structures, M.P.Bendsøe and C.A. M.Soares, eds. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 239-248.

Frandsen, L. H.

Gallagher, T. F. G.

T. Felici and T. F. G. Gallagher, "Improved waveguide structures derived from new rapid optimization techniques," in Physics and Simulation of Optoelectronic Devices XI, M.Osinski, H.Amano, and P.Blood, eds., Proc. SPIE 4986, 375-385 (2003).

Hafner, C.

Harpøth, A.

Jensen, J. S.

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

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

O. Sigmund and J. S. Jensen, "Systematic design of phononic band-gap materials and structures by topology optimization," Proc. R. Soc. London, Ser. A 361, 1001-1019 (2003).

J. S. Jensen, "Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures, J. Sound Vib. 266, 1053-1078 (2003).
[CrossRef]

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, New York, 2002).

Joannopoulos, J. D.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

John, S.

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Kikuchi, N.

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

Kim, W. J.

Koshiba, M.

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performanceabsorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microw. Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

Kristensen, M.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Malkus, D. S.

R. D. Cook, D. S. Malkus, M. E. Plesha, and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed. (Wiley, New York, 2002).

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

O'Brien, J. D.

Osher, S. J.

M. Burger, S. J. Osher, and E. Yablonovitch, "Inverse problem techniques for the design of photonic crystals," IEICE Trans. Electron. E87-C, 258-265 (2004).

Plesha, M. E.

R. D. Cook, D. S. Malkus, M. E. Plesha, and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed. (Wiley, New York, 2002).

Sasaki, S.

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performanceabsorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microw. Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

Schramm, U.

H. L. Thomas, M. Zhou, and U. Schramm, "Issues of commercial optimization software development," Struct. Multidiscip. Optim. 23, 97-110 (2002).
[CrossRef]

Shi, P.

Sigmund, O.

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

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

O. Sigmund and J. S. Jensen, "Systematic design of phononic band-gap materials and structures by topology optimization," Proc. R. Soc. London, Ser. A 361, 1001-1019 (2003).

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

Smajic, J.

Song, G. H.

K. B. Chung, J. S. Yoon, and G. H. Song, "Analysis of optical splitters in photonic crystals," in Photonic Bandgap Material and Devices, A.Adibi, A.Scherer, and S.-Yu.Lin, eds., Proc. SPIE 4655, 349-355 (2002).

Svanberg, K.

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

Thomas, H. L.

H. L. Thomas, M. Zhou, and U. Schramm, "Issues of commercial optimization software development," Struct. Multidiscip. Optim. 23, 97-110 (2002).
[CrossRef]

Tsuji, Y.

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performanceabsorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microw. Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Witt, R. J.

R. D. Cook, D. S. Malkus, M. E. Plesha, and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed. (Wiley, New York, 2002).

Yablonovitch, E.

M. Burger, S. J. Osher, and E. Yablonovitch, "Inverse problem techniques for the design of photonic crystals," IEICE Trans. Electron. E87-C, 258-265 (2004).

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yoon, J. S.

K. B. Chung, J. S. Yoon, and G. H. Song, "Analysis of optical splitters in photonic crystals," in Photonic Bandgap Material and Devices, A.Adibi, A.Scherer, and S.-Yu.Lin, eds., Proc. SPIE 4655, 349-355 (2002).

Zhou, M.

H. L. Thomas, M. Zhou, and U. Schramm, "Issues of commercial optimization software development," Struct. Multidiscip. Optim. 23, 97-110 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

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

Comput. Methods Appl. Mech. Eng. (1)

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

IEEE Microw. Wirel. Compon. Lett. (1)

M. Koshiba, Y. Tsuji, and S. Sasaki, "High-performanceabsorbing boundary conditions for photonic crystal waveguide simulations," IEEE Microw. Wirel. Compon. Lett. 11, 152-154 (2001).
[CrossRef]

IEICE Trans. Electron. (1)

M. Burger, S. J. Osher, and E. Yablonovitch, "Inverse problem techniques for the design of photonic crystals," IEICE Trans. Electron. E87-C, 258-265 (2004).

Int. J. Numer. Methods Eng. (1)

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

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

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

J. Sound Vib. (1)

J. S. Jensen, "Phononic band gaps and vibrations in one- and two-dimensional mass-spring structures, J. Sound Vib. 266, 1053-1078 (2003).
[CrossRef]

Opt. Express (1)

Phys. Rev. Lett. (3)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "High transmission through sharp bends in photonic crystal waveguides," Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Proc. R. Soc. London, Ser. A (1)

O. Sigmund and J. S. Jensen, "Systematic design of phononic band-gap materials and structures by topology optimization," Proc. R. Soc. London, Ser. A 361, 1001-1019 (2003).

Struct. Multidiscip. Optim. (1)

H. L. Thomas, M. Zhou, and U. Schramm, "Issues of commercial optimization software development," Struct. Multidiscip. Optim. 23, 97-110 (2002).
[CrossRef]

Other (6)

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

T. Felici and T. F. G. Gallagher, "Improved waveguide structures derived from new rapid optimization techniques," in Physics and Simulation of Optoelectronic Devices XI, M.Osinski, H.Amano, and P.Blood, eds., Proc. SPIE 4986, 375-385 (2003).

K. B. Chung, J. S. Yoon, and G. H. Song, "Analysis of optical splitters in photonic crystals," in Photonic Bandgap Material and Devices, A.Adibi, A.Scherer, and S.-Yu.Lin, eds., Proc. SPIE 4655, 349-355 (2002).

R. D. Cook, D. S. Malkus, M. E. Plesha, and R. J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed. (Wiley, New York, 2002).

G. Allaire and G. A. Francfort, "A numerical algorithm for topology and shape optimization," in Topology Optimization of Structures, M.P.Bendsøe and C.A. M.Soares, eds. (Kluwer, Dordrecht, The Netherlands, 1993), pp. 239-248.

J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley, New York, 2002).

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Figures (10)

Fig. 1
Fig. 1

Computational model consisting of 15 × 15 unit cells in Ω with centrally placed circular rods of diameter 0.36 a . A wave input is provided on Γ inp , and absorbing boundaries are specified on Γ abs . Perfectly matching layers Ω PML are added to avoid reflections from the input and output waveguide ports. The objective is to maximize the power transmission in the two subdomains near the output ports Ω J 1 and Ω J 2 .

Fig. 2
Fig. 2

Design domain Ω D and initial material distribution.

Fig. 3
Fig. 3

Optimized T-junction topology for target frequency ω ̃ = 0.32 and the corresponding field distribution.

Fig. 4
Fig. 4

Optimized T-junction topology for target frequency ω ̃ = 0.38 and the corresponding field distribution.

Fig. 5
Fig. 5

Optimized T-junction topology for target frequency ω ̃ = 0.44 and the corresponding field distribution.

Fig. 6
Fig. 6

Transmission through the top and bottom output ports. Results for the standard junction is shown along with three optimized designs for three different frequencies.

Fig. 7
Fig. 7

Illustration of update scheme for target frequencies. The frequency curve is computed with Padé approximations to obtain high frequency resolution at low computational cost. Discrete markers indicate transmission values computed directly.

Fig. 8
Fig. 8

Design optimized for frequency range ω ̃ = 0.32 0.44 . The field is computed for ω ̃ = 0.38 .

Fig. 9
Fig. 9

Design optimized for frequency range ω ̃ = 0.32 0.44 with an enlarged design domain. The field is computed for ω ̃ = 0.38 .

Fig. 10
Fig. 10

Transmission spectra for the design optimized for the entire frequency range ω ̃ = 0.32 0.44 . Shown also is the spectrum for the design optimized for ω ̃ = 0.38 .

Equations (29)

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[ A ( x ) u ( x ) ] + ω 2 B ( x ) u ( x ) = 0 in Ω .
n ( A u ) = 2 i ω A B U on Γ inp ,
n ( A u ) + i ω A B u = 0 on Γ abs ,
x ( s y s x A u x ) + y ( s x s y A u y ) + ω 2 s x s y B u = 0 in Ω PML ,
( ω 2 M + i ω C + K ) u = f ,
M = e Ω B e M e + e Ω PML B e s x e s y e M e , M e = N T N d V ,
C = C abs = e Γ abs A e B e C e , C e = N T N d S ,
K = e Ω A e K e + e Ω PML A e [ ( s y e s x e ) K x e + ( s x e s y e ) K y e ] ,
K e = K x e + K y e = N T x N x d V + N T y N y d V ,
f = 2 i ω e Γ inp A e B e f e , f e = N T d S ,
0 x e 1 , e Ω D .
A e = A 1 + x e ( A 2 A 1 ) ,
B e = B 1 + x e ( B 2 B 1 ) ,
P ( x ) = { P x P y } T = 1 2 ω A R ( i u ¯ u ) .
J 1 = 1 N y e Ω J 1 P y e ,
J 2 = 1 N y e Ω J 2 P y e ,
P y e = 1 2 ω A e R [ i ( u e ) T Q y e u ¯ e ] ,
Q y e = N T y N d S y = 0 .
max x e J ( ω ) ,
J ( ω ) = J 1 ( ω ) + J 2 ( ω ) ,
d J d x e = 2 R [ λ T d ( ω 2 M + i ω C + K ) d x e u ] ,
( ω 2 M + i ω C + K ) T λ = i ω 4 e Ω J A e [ Q y e ( Q y e ) T ] T u ¯ e ,
d J ̂ d x e = ( H e ̂ d J d x e ) H e ̂ ,
H e ̂ = r min dist ( k , e ) , { e Ω D dist ( k , e ) r min } , k = 1 , , N Ω D .
C = C abs + C art ,
C art = β ω M ,
C pen e = 4 ϵ x e ( 1 x e ) ω M e , e Ω D ,
max x e ω 1 , , ω N min ω i I i [ J ( ω i ) J * ( ω i ) ] ,
I 1 = [ ω ̃ 1 ; ω ̃ 2 [ , , I n = ] ω ̃ N ; ω ̃ N + 1 ] ,

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