Abstract

Modern nanoscale manufacturing techniques allow for a high degree of flexibility in designing surface microstructures and nanostructures. Injection molding of nanosized features allows for mass production of plastic components with a tailored nanostructure producing specific optical effects depending on the purpose. This work details the use of topology optimization for designing periodic polymer grating surfaces with complex optical properties. A method based on robust topology optimization is formulated for designing the nanostructure of plastic surfaces with extreme reflection or transmission properties. Topology optimization allows for free distribution of material but a mechanical constraint based on the fundamental free mechanical vibration frequency ensures connected structures. Several examples are given to illustrate the efficiency of the method.

© 2012 Optical Society of America

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  1. L. D. Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” CIRP Ann. 52, 561–577 (2003).
    [CrossRef]
  2. L. Alting, F. Kimura, H. N. Hansen, and G. Bissacco, “Micro engineering,” CIRP Ann. 52, 635–657 (2003).
    [CrossRef]
  3. P. Utko, F. Persson, A. Kristensen, and N. B. Larsen, “Injection molded nanofluidic chips: fabrication method and functional tests using single-molecule DNA experiments,” Lab Chip 11, 303–308 (2011).
    [CrossRef]
  4. K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
    [CrossRef]
  5. R. T. Lee and G. S. Smith, “Detailed electromagnetic simulation for the structural color of butterfly wings,” Appl. Opt. 48, 4177–4190 (2009).
    [CrossRef]
  6. S. Banerjee, J. B. Cole, and T. Yatagai, “Colour characterization of a morpho butterfly wing-scale using a high accuracy nonstandard finite-difference time-domain method,” Micron 38, 97–103 (2007).
    [CrossRef]
  7. K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
    [CrossRef]
  8. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
    [CrossRef]
  9. D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–339 (1993).
    [CrossRef]
  10. D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
    [CrossRef]
  11. K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
    [CrossRef]
  12. M. P. Bendsoe and O. Sigmund, Topology Optimisation–Theory, Methods and Applications, 2nd ed. (Springer-Verlag, 2003).
  13. 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, 1191–1198 (2005).
    [CrossRef]
  14. J. Jin, The Finite Element Method in Electromagnetics, 2nd ed. (Wiley-IEEE, 2002).
  15. F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B 28, 387–397 (2011).
    [CrossRef]
  16. J. S. Jensen, “A note on sensitivity analysis of linear dynamic systems with harmonic excitation,” Tech. rep. (Technical University of Denmark, Department of Mechanical Engineering, 2009).
  17. O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidisc. Optim. 33, 401–424 (2007).
    [CrossRef]
  18. J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Num. Methods Eng. 61, 238–254 (2004).
    [CrossRef]
  19. F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
    [CrossRef]
  20. O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sin. 25, 227–239 (2009).
    [CrossRef]
  21. K. Svanberg, “Method of moving asymptotes–a new method for structural optimization,” Int. J. Num. Methods Eng. 24, 359–373 (1987).
    [CrossRef]
  22. M. Schevenels, B. S. Lazarov, and O. Sigmund, “Robust topology optimization accounting for spatially varying manufacturing errors,” Comput. Methods Appl. Mech. Eng. 200, 3613–3627 (2011).
    [CrossRef]
  23. E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
    [CrossRef]
  24. J. S. Jensen, “Topology optimization of dynamics problems with Padé approximants,” Int. J. Num. Methods Eng. 72, 1605–1630 (2007).
    [CrossRef]

2012 (1)

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

2011 (6)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

M. Schevenels, B. S. Lazarov, and O. Sigmund, “Robust topology optimization accounting for spatially varying manufacturing errors,” Comput. Methods Appl. Mech. Eng. 200, 3613–3627 (2011).
[CrossRef]

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
[CrossRef]

P. Utko, F. Persson, A. Kristensen, and N. B. Larsen, “Injection molded nanofluidic chips: fabrication method and functional tests using single-molecule DNA experiments,” Lab Chip 11, 303–308 (2011).
[CrossRef]

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B 28, 387–397 (2011).
[CrossRef]

2010 (2)

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
[CrossRef]

2009 (2)

2007 (3)

J. S. Jensen, “Topology optimization of dynamics problems with Padé approximants,” Int. J. Num. Methods Eng. 72, 1605–1630 (2007).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidisc. Optim. 33, 401–424 (2007).
[CrossRef]

S. Banerjee, J. B. Cole, and T. Yatagai, “Colour characterization of a morpho butterfly wing-scale using a high accuracy nonstandard finite-difference time-domain method,” Micron 38, 97–103 (2007).
[CrossRef]

2005 (1)

2004 (1)

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Num. Methods Eng. 61, 238–254 (2004).
[CrossRef]

2003 (2)

L. D. Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” CIRP Ann. 52, 561–577 (2003).
[CrossRef]

L. Alting, F. Kimura, H. N. Hansen, and G. Bissacco, “Micro engineering,” CIRP Ann. 52, 635–657 (2003).
[CrossRef]

1999 (1)

D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
[CrossRef]

1993 (1)

D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–339 (1993).
[CrossRef]

1987 (1)

K. Svanberg, “Method of moving asymptotes–a new method for structural optimization,” Int. J. Num. Methods Eng. 24, 359–373 (1987).
[CrossRef]

Alting, L.

L. Alting, F. Kimura, H. N. Hansen, and G. Bissacco, “Micro engineering,” CIRP Ann. 52, 635–657 (2003).
[CrossRef]

Andreassen, E.

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
[CrossRef]

Andresen, K. O.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Banerjee, S.

S. Banerjee, J. B. Cole, and T. Yatagai, “Colour characterization of a morpho butterfly wing-scale using a high accuracy nonstandard finite-difference time-domain method,” Micron 38, 97–103 (2007).
[CrossRef]

Belytschko, T.

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Num. Methods Eng. 61, 238–254 (2004).
[CrossRef]

Bendsoe, M. P.

M. P. Bendsoe and O. Sigmund, Topology Optimisation–Theory, Methods and Applications, 2nd ed. (Springer-Verlag, 2003).

Bissacco, G.

L. Alting, F. Kimura, H. N. Hansen, and G. Bissacco, “Micro engineering,” CIRP Ann. 52, 635–657 (2003).
[CrossRef]

Chiffre, L. D.

L. D. Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” CIRP Ann. 52, 561–577 (2003).
[CrossRef]

Chung, K.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Clausen, A.

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
[CrossRef]

Cole, J. B.

S. Banerjee, J. B. Cole, and T. Yatagai, “Colour characterization of a morpho butterfly wing-scale using a high accuracy nonstandard finite-difference time-domain method,” Micron 38, 97–103 (2007).
[CrossRef]

Diaz, A. R.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
[CrossRef]

Dobson, D. C.

D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
[CrossRef]

D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–339 (1993).
[CrossRef]

Fuchi, K.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
[CrossRef]

Guest, J.

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Num. Methods Eng. 61, 238–254 (2004).
[CrossRef]

Han, M. G.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Hansen, H. N.

L. Alting, F. Kimura, H. N. Hansen, and G. Bissacco, “Micro engineering,” CIRP Ann. 52, 635–657 (2003).
[CrossRef]

Hansen, M.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Hansen, T. S.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Heo, C.-J.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Jensen, J. S.

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B 28, 387–397 (2011).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[CrossRef]

J. S. Jensen, “Topology optimization of dynamics problems with Padé approximants,” Int. J. Num. Methods Eng. 72, 1605–1630 (2007).
[CrossRef]

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, 1191–1198 (2005).
[CrossRef]

J. S. Jensen, “A note on sensitivity analysis of linear dynamic systems with harmonic excitation,” Tech. rep. (Technical University of Denmark, Department of Mechanical Engineering, 2009).

Jepsen, S. T.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Jin, J.

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

Jin, Y.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Kimura, F.

L. Alting, F. Kimura, H. N. Hansen, and G. Bissacco, “Micro engineering,” CIRP Ann. 52, 635–657 (2003).
[CrossRef]

Kristensen, A.

P. Utko, F. Persson, A. Kristensen, and N. B. Larsen, “Injection molded nanofluidic chips: fabrication method and functional tests using single-molecule DNA experiments,” Lab Chip 11, 303–308 (2011).
[CrossRef]

Kunzmann, H.

L. D. Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” CIRP Ann. 52, 561–577 (2003).
[CrossRef]

Larsen, N. B.

P. Utko, F. Persson, A. Kristensen, and N. B. Larsen, “Injection molded nanofluidic chips: fabrication method and functional tests using single-molecule DNA experiments,” Lab Chip 11, 303–308 (2011).
[CrossRef]

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Lazarov, B. S.

M. Schevenels, B. S. Lazarov, and O. Sigmund, “Robust topology optimization accounting for spatially varying manufacturing errors,” Comput. Methods Appl. Mech. Eng. 200, 3613–3627 (2011).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
[CrossRef]

Lee, H.-S.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Lee, R. T.

Lee, S. Y.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Lucca, D. A.

L. D. Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” CIRP Ann. 52, 561–577 (2003).
[CrossRef]

Matschuk, M.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Ouedraogo, R.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
[CrossRef]

Park, N.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Peggs, G. N.

L. D. Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” CIRP Ann. 52, 561–577 (2003).
[CrossRef]

Persson, F.

P. Utko, F. Persson, A. Kristensen, and N. B. Larsen, “Injection molded nanofluidic chips: fabrication method and functional tests using single-molecule DNA experiments,” Lab Chip 11, 303–308 (2011).
[CrossRef]

Prevost, J.

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Num. Methods Eng. 61, 238–254 (2004).
[CrossRef]

Rothwell, E.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
[CrossRef]

Rozlosnik, N.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Schevenels, M.

M. Schevenels, B. S. Lazarov, and O. Sigmund, “Robust topology optimization accounting for spatially varying manufacturing errors,” Comput. Methods Appl. Mech. Eng. 200, 3613–3627 (2011).
[CrossRef]

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
[CrossRef]

Selmeczi, D.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Shim, J. W.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Shin, J. H.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Sigmund, O.

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B 28, 387–397 (2011).
[CrossRef]

M. Schevenels, B. S. Lazarov, and O. Sigmund, “Robust topology optimization accounting for spatially varying manufacturing errors,” Comput. Methods Appl. Mech. Eng. 200, 3613–3627 (2011).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[CrossRef]

O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sin. 25, 227–239 (2009).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidisc. Optim. 33, 401–424 (2007).
[CrossRef]

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, 1191–1198 (2005).
[CrossRef]

M. P. Bendsoe and O. Sigmund, Topology Optimisation–Theory, Methods and Applications, 2nd ed. (Springer-Verlag, 2003).

Smith, G. S.

Sorensen, H. S.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Svanberg, K.

K. Svanberg, “Method of moving asymptotes–a new method for structural optimization,” Int. J. Num. Methods Eng. 24, 359–373 (1987).
[CrossRef]

Taboryski, R.

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Temme, A.

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
[CrossRef]

Utko, P.

P. Utko, F. Persson, A. Kristensen, and N. B. Larsen, “Injection molded nanofluidic chips: fabrication method and functional tests using single-molecule DNA experiments,” Lab Chip 11, 303–308 (2011).
[CrossRef]

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

Wang, F.

F. Wang, J. S. Jensen, and O. Sigmund, “Robust topology optimization of photonic crystal waveguides with tailored dispersion properties,” J. Opt. Soc. Am. B 28, 387–397 (2011).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

Yang, S.-M.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Yatagai, T.

S. Banerjee, J. B. Cole, and T. Yatagai, “Colour characterization of a morpho butterfly wing-scale using a high accuracy nonstandard finite-difference time-domain method,” Micron 38, 97–103 (2007).
[CrossRef]

Yu, S.

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Acta Mech. Sin. (1)

O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sin. 25, 227–239 (2009).
[CrossRef]

Adv. Mater. (1)

K. Chung, S. Yu, C.-J. Heo, J. W. Shim, S.-M. Yang, M. G. Han, H.-S. Lee, Y. Jin, S. Y. Lee, N. Park, and J. H. Shin, “Flexible, angle-independent, structural color reflectors inspired by morpho butterfly wings,” Adv. Mater. 24, 2375–2379 (2012).
[CrossRef]

Appl. Math. Optim. (1)

D. C. Dobson, “Optimal shape design of blazed diffraction gratings,” Appl. Math. Optim. 40, 61–78 (1999).
[CrossRef]

Appl. Opt. (1)

CIRP Ann. (2)

L. D. Chiffre, H. Kunzmann, G. N. Peggs, and D. A. Lucca, “Surfaces in precision engineering, microengineering and nanotechnology,” CIRP Ann. 52, 561–577 (2003).
[CrossRef]

L. Alting, F. Kimura, H. N. Hansen, and G. Bissacco, “Micro engineering,” CIRP Ann. 52, 635–657 (2003).
[CrossRef]

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

M. Schevenels, B. S. Lazarov, and O. Sigmund, “Robust topology optimization accounting for spatially varying manufacturing errors,” Comput. Methods Appl. Mech. Eng. 200, 3613–3627 (2011).
[CrossRef]

Eur. J. Appl. Math. (1)

D. C. Dobson, “Optimal design of periodic antireflective structures for the Helmholtz equation,” Eur. J. Appl. Math. 4, 321–339 (1993).
[CrossRef]

Int. J. Num. Methods Eng. (3)

J. S. Jensen, “Topology optimization of dynamics problems with Padé approximants,” Int. J. Num. Methods Eng. 72, 1605–1630 (2007).
[CrossRef]

K. Svanberg, “Method of moving asymptotes–a new method for structural optimization,” Int. J. Num. Methods Eng. 24, 359–373 (1987).
[CrossRef]

J. Guest, J. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Num. Methods Eng. 61, 238–254 (2004).
[CrossRef]

J. Micromech. Microeng. (1)

K. O. Andresen, M. Hansen, M. Matschuk, S. T. Jepsen, H. S. Sorensen, P. Utko, D. Selmeczi, T. S. Hansen, N. B. Larsen, N. Rozlosnik, and R. Taboryski, “Injection molded chips with integrated conducting polymer electrodes for electroporation of cells,” J. Micromech. Microeng. 20, 055010 (2010).
[CrossRef]

J. Opt. Soc. Am. B (2)

Lab Chip (1)

P. Utko, F. Persson, A. Kristensen, and N. B. Larsen, “Injection molded nanofluidic chips: fabrication method and functional tests using single-molecule DNA experiments,” Lab Chip 11, 303–308 (2011).
[CrossRef]

Laser Photon. Rev. (1)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photon. Rev. 5, 308–321 (2011).
[CrossRef]

Micron (1)

S. Banerjee, J. B. Cole, and T. Yatagai, “Colour characterization of a morpho butterfly wing-scale using a high accuracy nonstandard finite-difference time-domain method,” Micron 38, 97–103 (2007).
[CrossRef]

Struct. Multidisc. Optim. (4)

K. Fuchi, A. R. Diaz, E. Rothwell, R. Ouedraogo, and A. Temme, “Topology optimization of periodic layouts of dielectric materials,” Struct. Multidisc. Optim. 42, CP11–493 (2010).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784 (2011).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidisc. Optim. 33, 401–424 (2007).
[CrossRef]

E. Andreassen, A. Clausen, M. Schevenels, B. S. Lazarov, and O. Sigmund, “Efficient topology optimization in MATLAB using 88 lines of code,” Struct. Multidisc. Optim. 43, 1–16 (2011).
[CrossRef]

Other (3)

M. P. Bendsoe and O. Sigmund, Topology Optimisation–Theory, Methods and Applications, 2nd ed. (Springer-Verlag, 2003).

J. S. Jensen, “A note on sensitivity analysis of linear dynamic systems with harmonic excitation,” Tech. rep. (Technical University of Denmark, Department of Mechanical Engineering, 2009).

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

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Figures (11)

Fig. 1.
Fig. 1.

Computational domain composed of a fixed air region ΩA, a fixed bulk region ΩB, and an active optimization region ΩD where material is distributed to form the surface structure.

Fig. 2.
Fig. 2.

Optimized robust design of antireflective surface for seven different input angles. (a) Dilated design, (b) intermediate design, (c) eroded design, and (d) overlaid contour plot comparing the surface contours of the three projections.

Fig. 3.
Fig. 3.

Angle sweep of robust optimized designs for λ=600nm and maximized transmittance.

Fig. 4.
Fig. 4.

Threshold value sweep of optimized designs for λ=600nm and maximized transmittance.

Fig. 5.
Fig. 5.

Optimized robust design with mechanical constraint for incident angle [0, 10, 20, 30, 40, 50, 60]° and E- and H-polarized light. (a) Dilated design, (b) intermediate design, (c) eroded design, and (d) overlaid contour plot.

Fig. 6.
Fig. 6.

Responses for the robust optimization with mechanical constraint for incident angles [0, 10, 20, 30, 40, 50, 60]° and E- and H-polarized light. (a) Angle sweep of robust optimized intermediate projection design. (b) Threshold value sweep for optimized design.

Fig. 7.
Fig. 7.

(a) Intermediate projection of unconstrained optimization and (b) contour plot of reflective surface design with mechanical constraint using unconstrained optimization design as initial guess.

Fig. 8.
Fig. 8.

Wavelength sweep of reflective surface design optimized with objectives of wavelengths [500, 550, 600, 650, 700] nm using unconstrained optimized design as initial guess.

Fig. 9.
Fig. 9.

(a) Initial guess and (b) contour plot of optimized angle selective surface design.

Fig. 10.
Fig. 10.

Angular sweep for optimized angle selective surface design.

Fig. 11.
Fig. 11.

Surface microstructure constituted by optimized intermediate angle selective surface design.

Equations (29)

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·(A(x)u(x))+ω2B(x)u(x)=0,
u0(x)=Uexp(jωA1Bk^·x).
n^u=jωAB(1n^k^)u0=jωAB(1+cos(θ))u0.
u(x,d)=u(x,0)exp(jωdsin(θ)BA1).
x(sysxAux)+y(sxsyAuy)+ω2sxsyBu=0,
sx(x)=1jσ(x)ωϵ,
Ω(Au)·(T)dΩΩTω2BudΩ+ΩPML(sysxAuxTx+sxsyAuyTy)dΩΩPMLTω2sxsyBudΩ=ΓITjωAB(1+cos(θ))u0dΓ.
(Kω2M)u=f,
K=eΩAe(Kxe+Kye)+eΩPMLAe[(sysx)Kxe+(sxsy)Kye],Kxe=NTxNxdV,Kye=NTyNydVM=eΩBeMe+eΩPMLBe(sxsy)eMe,Me=NTNdV,f=jω(1+cos(θ))eΓIAeBefe,fe=NTu0dS.
0ρe1,eΩD.
Ae=A1+ρep(A2A1),Be=B1+ρep(B2B11),
T=PTP0R=PRP0,
Φ0T=PoutP0Φ0R=P0PinP0=1PinP0.
V(x)=12ω(jCu¯u),
(Px)e=Ce2ω(i(ue)TQeu¯e),
Qe=NTxNdy|x=xΓ.
Pin=eein(Px)ePout=eeout(Px)e,
dΦdρi=2(λTSρiu),
STλ=12(Φ0uRiΦ0uI).
STλ=eeiniCe4ω(Qe(Qe)T)Tu¯e.
ρ˜i=jNe,iw(xj)vjρjjNe,iw(xj)vj,
w(xj)=R|xjxi|,
ρ¯i=tanh(βη)+tanh(β(ρ˜iη))tanh(βη)+tanh(β(1η)).
Φρj=iNe,jΦρ¯iρ¯iρ˜iρ˜iρj,
minρj:maxi{hi(ρ¯jd),hi(ρ¯ji),hi(ρ¯je)},s.t.:(Kiqωi2Miq)u=fi,0<ρj1i=1,k,j=1,N,q={e,i,d}.
u(,y)=0,u(x,0)=u(x,d),
Eeq=Emin+(E1Emin)(ρ¯eq)pmeq=mmin+(m1mmin)ρ¯eq.
minρj:maxq:maxi:{hi((ρ˜j)q)},s.t.:(Kiqω2Miq)u=fi,(K^iλ1qM^i)x=0,λ1q>δ,0<ρj1,i=1,k,j=1,N,q={e,i,d}.
δ=δmin+(δmaxδmin)(ββ0βmaxβ0).

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