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Topology optimization and fabrication of photonic crystal structures

P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, P. Shi, J. S. Jensen, and O. Sigmund

Open Access Open Access

Abstract

Topology optimization is used to design a planar photonic crystal waveguide component resulting in significantly enhanced functionality. Exceptional transmission through a photonic crystal waveguide Z-bend is obtained using this inverse design strategy. The design has been realized in a silicon-on-insulator based photonic crystal waveguide. A large low loss bandwidth of more than 200 nm for the TE polarization is experimentally confirmed.

©2004 Optical Society of America

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References

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  • Publication
  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. T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996).
    [Crossref]
  4. M. Thorhauge, L. H. Frandsen, and P. I. Borel, “Efficient Photonic Crystal Directional Couplers,” Opt. Lett. 28, 1525–1527 (2003).
    [Crossref] [PubMed]
  5. L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).
  6. D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
    [Crossref]
  7. M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. Appl. Mech. Eng. 71, 197–224 (1988).
    [Crossref]
  8. M. P. Bendsøe and O. Sigmund, Topology optimization — Theory, Methods and Applications (Springer-Verlag, 2003).
  9. T. P. Felici and D. F. G. Gallagher, “Improved waveguide structures derived from new rapid optimization techniques,” Proc. SPIE 4986, 375–385 (2003).
    [Crossref]
  10. J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11, 1378–1384 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1378.
    [Crossref] [PubMed]
  11. 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]
  12. M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
    [Crossref]
  13. T. Uusitupa, K. Kärkkäinen, and K. Nikoskinen, “Studying 120° PBG waveguide bend using FDTD,” Microwave Opt. Technol. Lett. 39, 326–333 (2003).
    [Crossref]
  14. It should be emphasized that the method can readily be implemented in a 3D finite element model where the computational requirements naturally will be significantly higher.
  15. K. Svanberg, “The method of moving asymptotes: a new method for structural optimization,” Int. J. Numer. Meth. Engng. 24, 359–373 (1987).
    [Crossref]
  16. 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]
  17. O. Sigmund and J. S. Jensen, “Systematic design of phononic band gap materials and structures by topology optimization,” Phil. Trans. R. Soc. Lond. A 361, 1001–1019 (2003).
    [Crossref]
  18. P.I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, Y. X. Zhuang, M. Kristensen, and H. M. H. Chong, “Efficient propagation of TM polarized light in photonic crystal components exhibiting band gaps for TE polarized light,” Opt. Express 11, 1757–1762 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1757.
    [Crossref] [PubMed]
  19. A. Lavrinenko, P. I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. M. H. Chong, “Comprehensive FDTD modelling of photonic crystal waveguide components,” Opt. Express 12, 234–248 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-2-234.
    [Crossref] [PubMed]

2004 (3)

2003 (7)

O. Sigmund and J. S. Jensen, “Systematic design of phononic band gap materials and structures by topology optimization,” Phil. Trans. R. Soc. Lond. A 361, 1001–1019 (2003).
[Crossref]

P.I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, Y. X. Zhuang, M. Kristensen, and H. M. H. Chong, “Efficient propagation of TM polarized light in photonic crystal components exhibiting band gaps for TE polarized light,” Opt. Express 11, 1757–1762 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1757.
[Crossref] [PubMed]

T. Uusitupa, K. Kärkkäinen, and K. Nikoskinen, “Studying 120° PBG waveguide bend using FDTD,” Microwave Opt. Technol. Lett. 39, 326–333 (2003).
[Crossref]

T. P. Felici and D. F. G. Gallagher, “Improved waveguide structures derived from new rapid optimization techniques,” Proc. SPIE 4986, 375–385 (2003).
[Crossref]

J. Smajic, C. Hafner, and D. Erni, “Design and optimization of an achromatic photonic crystal bend,” Opt. Express 11, 1378–1384 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-12-1378.
[Crossref] [PubMed]

M. Thorhauge, L. H. Frandsen, and P. I. Borel, “Efficient Photonic Crystal Directional Couplers,” Opt. Lett. 28, 1525–1527 (2003).
[Crossref] [PubMed]

D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
[Crossref]

2000 (1)

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[Crossref]

1996 (1)

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996).
[Crossref]

1988 (1)

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. 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. Meth. Engng. 24, 359–373 (1987).
[Crossref]

Baets, R.

D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
[Crossref]

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

Beckx, S.

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

Bendsøe, M. P.

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. Appl. Mech. Eng. 71, 197–224 (1988).
[Crossref]

M. P. Bendsøe and O. Sigmund, Topology optimization — Theory, Methods and Applications (Springer-Verlag, 2003).

Bogaerts, W.

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

Borel, P. I.

Borel, P.I.

Brand, S.

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996).
[Crossref]

Chong, H.

D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
[Crossref]

Chong, H. M. H.

De La Rue, R. M.

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996).
[Crossref]

De La Rue, R.M.

D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
[Crossref]

Dumon, P.

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

Erni, D.

Felici, T. P.

T. P. Felici and D. F. G. Gallagher, “Improved waveguide structures derived from new rapid optimization techniques,” Proc. SPIE 4986, 375–385 (2003).
[Crossref]

Frandsen, L. H.

Frandsen, L.H.

D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
[Crossref]

Gallagher, D. F. G.

T. P. Felici and D. F. G. Gallagher, “Improved waveguide structures derived from new rapid optimization techniques,” Proc. SPIE 4986, 375–385 (2003).
[Crossref]

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]

O. Sigmund and J. S. Jensen, “Systematic design of phononic band gap materials and structures by topology optimization,” Phil. Trans. R. Soc. Lond. A 361, 1001–1019 (2003).
[Crossref]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[Crossref] [PubMed]

Kärkkäinen, K.

T. Uusitupa, K. Kärkkäinen, and K. Nikoskinen, “Studying 120° PBG waveguide bend using FDTD,” Microwave Opt. Technol. Lett. 39, 326–333 (2003).
[Crossref]

Kikuchi, N.

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. Appl. Mech. Eng. 71, 197–224 (1988).
[Crossref]

Kim, W. J.

Kosaka, H.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[Crossref]

Krauss, T. F.

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996).
[Crossref]

Kristensen, M.

Lavrinenko, A.

Niemi, T.

Nikoskinen, K.

T. Uusitupa, K. Kärkkäinen, and K. Nikoskinen, “Studying 120° PBG waveguide bend using FDTD,” Microwave Opt. Technol. Lett. 39, 326–333 (2003).
[Crossref]

O’Brien, J. D.

Sigmund, O.

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,” Phil. Trans. R. Soc. Lond. A 361, 1001–1019 (2003).
[Crossref]

M. P. Bendsøe and O. Sigmund, Topology optimization — Theory, Methods and Applications (Springer-Verlag, 2003).

Smajic, J.

Svanberg, K.

K. Svanberg, “The method of moving asymptotes: a new method for structural optimization,” Int. J. Numer. Meth. Engng. 24, 359–373 (1987).
[Crossref]

Taillaert, D.

D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
[Crossref]

Thorhauge, M.

Tokushima, M.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[Crossref]

Tomita, A.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[Crossref]

Uusitupa, T.

T. Uusitupa, K. Kärkkäinen, and K. Nikoskinen, “Studying 120° PBG waveguide bend using FDTD,” Microwave Opt. Technol. Lett. 39, 326–333 (2003).
[Crossref]

Wiaux, V.

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

Wouters, J.

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

Yablonovitch, E.

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

Yamada, H.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[Crossref]

Zhuang, Y. X.

P.I. Borel, L. H. Frandsen, M. Thorhauge, A. Harpøth, Y. X. Zhuang, M. Kristensen, and H. M. H. Chong, “Efficient propagation of TM polarized light in photonic crystal components exhibiting band gaps for TE polarized light,” Opt. Express 11, 1757–1762 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-15-1757.
[Crossref] [PubMed]

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

Appl. Phys. Lett. (2)

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952–954 (2000).
[Crossref]

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. Meth. Appl. Mech. Eng. (1)

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. Appl. Mech. Eng. 71, 197–224 (1988).
[Crossref]

IEEE Photon. Technol. Lett. (1)

D. Taillaert, H. Chong, P.I. Borel, L.H. Frandsen, R.M. De La Rue, and R. Baets, “A Compact Two-dimensional Grating Coupler used as a Polarization Splitter,” IEEE Photon. Technol. Lett. 15, 1249–1251 (2003).
[Crossref]

Int. J. Numer. Meth. Engng. (1)

K. Svanberg, “The method of moving asymptotes: a new method for structural optimization,” Int. J. Numer. Meth. Engng. 24, 359–373 (1987).
[Crossref]

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

Microwave Opt. Technol. Lett. (1)

T. Uusitupa, K. Kärkkäinen, and K. Nikoskinen, “Studying 120° PBG waveguide bend using FDTD,” Microwave Opt. Technol. Lett. 39, 326–333 (2003).
[Crossref]

Nature (1)

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic-bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996).
[Crossref]

Opt. Express (3)

Opt. Lett. (1)

Phil. Trans. R. Soc. Lond. A (1)

O. Sigmund and J. S. Jensen, “Systematic design of phononic band gap materials and structures by topology optimization,” Phil. Trans. R. Soc. Lond. A 361, 1001–1019 (2003).
[Crossref]

Phys. Rev. Lett. (2)

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]

Proc. SPIE (1)

T. P. Felici and D. F. G. Gallagher, “Improved waveguide structures derived from new rapid optimization techniques,” Proc. SPIE 4986, 375–385 (2003).
[Crossref]

Other (3)

M. P. Bendsøe and O. Sigmund, Topology optimization — Theory, Methods and Applications (Springer-Verlag, 2003).

L. H. Frandsen, P. I. Borel, Y. X. Zhuang, A. Harpøth, M. Thorhauge, M. Kristensen, W. Bogaerts, P. Dumon, R. Baets, V. Wiaux, J. Wouters, and S. Beckx, “Ultra-low-loss 3-dB Photonic Crystal Waveguide Splitter,” Opt. Lett. (to be published).

It should be emphasized that the method can readily be implemented in a 3D finite element model where the computational requirements naturally will be significantly higher.

Supplementary Material (2)

Media 1: MOV (150 KB)     
Media 2: MOV (482 KB)     

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Figures (6)
Fig. 1.
Fig. 1. Top: Standard and two modified Z-bend waveguides. Bottom: Transmission through the bends calculated using a 2D frequency domain finite element model.

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Fig. 2.
Fig. 2. Left: Schematic illustration of the topology optimization procedure. The yellow area sketches the design domain of one bend. Middle: (149 kB) Movie of how the material is redistributed in the design domain in the optimization procedure. In about 600 iteration steps a final design is obtained that has optimized transmission properties. Right: (482 kB) Movie of TE polarized light propagating through the topology optimized Z-bend.

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3. The transmission for TE polarized light through the un-optimized (standard) design (black) and the optimized design (blue). The transmission spectra are based on a 2D frequency domain finite element model.

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4. Scanning electron micrograph of the fabricated Z-bend. The number, shape and size of the holes at each bend are designed using topology optimization. The inset shows a magnified view of the optimized holes as designed (white contour) and actually fabricated.

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Fig. 5.
Fig. 5. Experimental setup used to characterize the waveguide samples.

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Fig. 6.
Fig. 6. The measured (gray) and 3D FDTD calculated (red) loss per bend for TE polarized light in the fabricated structure. Also shown is the 3D FDTD calculated bend loss for the un-optimized (black) Z-bend.

Download Full Size | PPT Slide | PDF

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