Abstract

A robust topology optimization method is formulated to tailor dispersion properties of photonic crystal waveguides, with consideration of manufacturing uncertainties. Slightly dilated and eroded realizations are considered as well as the real structure, and by worst-case optimization, we also ensure a satisfactory performance in the case of an under- or overetching scenario in the manufacturing process. Two photonic crystal waveguides facilitating slow light with group indexes of ng=25 and ng=100 and bandwidths of Δω/ω=2.3% and 0.3%, respectively, are obtained through the proposed robust design procedure. In addition, a novel waveguide design with two different constant group index waveguide regions is demonstrated. The numerical examples illustrate the efficiency of the robust optimization formulation and indicate that the topology optimization procedure can provide a useful tool for designing waveguides that are robust to manufacturing uncertainties such as under or overetching.

© 2011 Optical Society of America

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References

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  1. T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
    [CrossRef]
  2. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals—Molding the Flow of Light, 2nd ed.(Princeton University Press, 2008).
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    [CrossRef] [PubMed]
  4. D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007).
    [CrossRef] [PubMed]
  5. L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450(2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  7. S. Kubo, D. Mori, and T. Baba, “Low-group-velocity and low-dispersion slow light in photonic crystal waveguides,” Opt. Lett. 32, 2981–2983 (2007).
    [CrossRef] [PubMed]
  8. M. P. Bendsøe and O. Sigmund, Topology Optimization—Theory, Methods and Applications (Springer Verlag, 2004).
  9. 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]
  10. 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]
  11. 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]
  12. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics—a review,” Laser Photon. Rev. (to be published).
    [CrossRef]
  13. R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves in Random and Complex Media 17, 477–489 (2007).
    [CrossRef]
  14. O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mechan. Struct. Mach. 25, 493–524(1997).
    [CrossRef]
  15. O. Sigmund and J. Petersson, “Numerical instabilities in topology optimization: a survey on procedures dealing with checkerboards, mesh-dependencies and local minima,” Struct. Opt. 16, 68–75 (1998).
    [CrossRef]
  16. 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]
  17. B. Bourdin, “Filters in topology optimization,” Int. J. Numer. Methods Eng. 50, 2143–2158 (2001).
    [CrossRef]
  18. J. K. Guest, J. H. Prevost, and T. Belytschko, “Achieving minimum length scale in topology optimization using nodal design variables and projection functions,” Int. J. Numer. Methods Eng. 61, 238–254 (2004).
    [CrossRef]
  19. M. Y. Wang and S. Wang, “Bilateral filtering for structural topology optimization,” Int. J. Numer. Methods Eng. 63, 1911–1938(2005).
    [CrossRef]
  20. O. Sigmund, “Morphology-based black and white filters,” Struct. Multidisc. Optim. 33, 401–424 (2007).
    [CrossRef]
  21. J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
    [CrossRef]
  22. S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2009).
    [CrossRef]
  23. O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sin. 25, 227–239 (2009).
    [CrossRef]
  24. F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. (to be published).
    [CrossRef]
  25. A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Multidisc. Optim. 8, 207–227 (1994).
  26. N. L. Pedersen and A. K. Nielsen, “Optimization of practical trusses with constraints on eigenfrequencies, displacements,” Struct. Multidisc. Optim. 25, 436–445 (2003).
    [CrossRef]
  27. K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
    [CrossRef]

2009 (3)

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2009).
[CrossRef]

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

2008 (2)

2007 (4)

S. Kubo, D. Mori, and T. Baba, “Low-group-velocity and low-dispersion slow light in photonic crystal waveguides,” Opt. Lett. 32, 2981–2983 (2007).
[CrossRef] [PubMed]

D. Mori, S. Kubo, H. Sasaki, and T. Baba, “Experimental demonstration of wideband dispersion-compensated slow light by a chirped photonic crystal directional coupler,” Opt. Express 15, 5264–5270 (2007).
[CrossRef] [PubMed]

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves in Random and Complex Media 17, 477–489 (2007).
[CrossRef]

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

2006 (1)

2005 (3)

2004 (3)

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

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]

2003 (1)

N. L. Pedersen and A. K. Nielsen, “Optimization of practical trusses with constraints on eigenfrequencies, displacements,” Struct. Multidisc. Optim. 25, 436–445 (2003).
[CrossRef]

2002 (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (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]

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. Opt. 16, 68–75 (1998).
[CrossRef]

1997 (1)

O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mechan. Struct. Mach. 25, 493–524(1997).
[CrossRef]

1994 (1)

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Multidisc. Optim. 8, 207–227 (1994).

Baba, T.

Belytschko, T.

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

Bendsøe, M. P.

M. P. Bendsøe and O. Sigmund, Topology Optimization—Theory, Methods and Applications (Springer Verlag, 2004).

Borel, P. I.

Bourdin, B.

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

Bruns, T. E.

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]

Cai, Y.

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2009).
[CrossRef]

Cheng, G.

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2009).
[CrossRef]

Fage-Pedersen, J.

Frandsen, L. H.

Gomez-Iglesias, A.

Guest, J. K.

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

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

Harpøth, A.

Jensen, J. S.

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 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, J. S. Jensen, P. Shi, 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, “Topology optimization for nano-photonics—a review,” Laser Photon. Rev. (to be published).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals—Molding the Flow of Light, 2nd ed.(Princeton University Press, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals—Molding the Flow of Light, 2nd ed.(Princeton University Press, 2008).

Kraus, T. F.

Kristensen, M.

Kubo, S.

Lavrinenko, A. V.

Lazarov, B. S.

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. (to be published).
[CrossRef]

Li, J.

Lund, E.

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Multidisc. Optim. 8, 207–227 (1994).

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals—Molding the Flow of Light, 2nd ed.(Princeton University Press, 2008).

Mori, D.

Nielsen, A. K.

N. L. Pedersen and A. K. Nielsen, “Optimization of practical trusses with constraints on eigenfrequencies, displacements,” Struct. Multidisc. Optim. 25, 436–445 (2003).
[CrossRef]

O’Faolain, L.

Olhoff, N.

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Multidisc. Optim. 8, 207–227 (1994).

Pedersen, N. L.

N. L. Pedersen and A. K. Nielsen, “Optimization of practical trusses with constraints on eigenfrequencies, displacements,” Struct. Multidisc. Optim. 25, 436–445 (2003).
[CrossRef]

Petersson, J.

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

Prevost, J. H.

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

Sasaki, H.

Seyranian, A. P.

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Multidisc. Optim. 8, 207–227 (1994).

Shi, P.

Sigmund, O.

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

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves in Random and Complex Media 17, 477–489 (2007).
[CrossRef]

O. Sigmund, “Morphology-based black and white filters,” 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]

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, J. S. Jensen, P. Shi, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[CrossRef] [PubMed]

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

O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mechan. Struct. Mach. 25, 493–524(1997).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics—a review,” Laser Photon. Rev. (to be published).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. (to be published).
[CrossRef]

M. P. Bendsøe and O. Sigmund, Topology Optimization—Theory, Methods and Applications (Springer Verlag, 2004).

Stainko, R.

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves in Random and Complex Media 17, 477–489 (2007).
[CrossRef]

Svanberg, K.

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

Tortorelli, D. A.

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]

Wang, F.

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. (to be published).
[CrossRef]

Wang, M. Y.

M. Y. Wang and S. Wang, “Bilateral filtering for structural topology optimization,” Int. J. Numer. Methods Eng. 63, 1911–1938(2005).
[CrossRef]

Wang, S.

M. Y. Wang and S. Wang, “Bilateral filtering for structural topology optimization,” Int. J. Numer. Methods Eng. 63, 1911–1938(2005).
[CrossRef]

White, T. P.

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals—Molding the Flow of Light, 2nd ed.(Princeton University Press, 2008).

Xu, S.

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2009).
[CrossRef]

Acta Mech. Sin. (1)

O. Sigmund, “Manufacturing tolerant topology optimization,” Acta Mech. Sin. 25, 227–239 (2009).
[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. (2)

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[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. (3)

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

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

M. Y. Wang and S. Wang, “Bilateral filtering for structural topology optimization,” Int. J. Numer. Methods Eng. 63, 1911–1938(2005).
[CrossRef]

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

Mechan. Struct. Mach. (1)

O. Sigmund, “On the design of compliant mechanisms using topology optimization,” Mechan. Struct. Mach. 25, 493–524(1997).
[CrossRef]

Nat. Photon. (1)

T. Baba, “Slow light in photonic crystals,” Nat. Photon. 2, 465–473 (2008).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

SIAM J. Optim. (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

Struct. Multidisc. Optim. (4)

A. P. Seyranian, E. Lund, and N. Olhoff, “Multiple eigenvalues in structural optimization problems,” Struct. Multidisc. Optim. 8, 207–227 (1994).

N. L. Pedersen and A. K. Nielsen, “Optimization of practical trusses with constraints on eigenfrequencies, displacements,” Struct. Multidisc. Optim. 25, 436–445 (2003).
[CrossRef]

S. Xu, Y. Cai, and G. Cheng, “Volume preserving nonlinear density filter based on heaviside functions,” Struct. Multidisc. Optim. 41, 495–505 (2009).
[CrossRef]

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

Struct. Opt. (1)

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

Waves in Random and Complex Media (1)

R. Stainko and O. Sigmund, “Tailoring dispersion properties of photonic crystal waveguides by topology optimization,” Waves in Random and Complex Media 17, 477–489 (2007).
[CrossRef]

Other (4)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics—a review,” Laser Photon. Rev. (to be published).
[CrossRef]

F. Wang, B. S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. (to be published).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals—Molding the Flow of Light, 2nd ed.(Princeton University Press, 2008).

M. P. Bendsøe and O. Sigmund, Topology Optimization—Theory, Methods and Applications (Springer Verlag, 2004).

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Figures (13)

Fig. 1
Fig. 1

(a) Schematic illustration of the supercell of a triangular lattice PhCW. (b) Corresponding band structure. Gray region indicates slab mode region, the dotted curve represents the light line, the solid curves denote the even guided modes, and the dashed curves denote the odd guided modes. (c) Group index of the even guided modes in the bandgap.

Fig. 2
Fig. 2

Schematic illustration of objective and constraints in the robust formulation. The crosses denote the prescribed properties, upward arrow indicates pushing upward, downward arrow indicates pushing downward, and the inset shows the prescribed group index and actual group index versus wavenumber.

Fig. 3
Fig. 3

Illustration of the smoothed threshold projection for η = 0.5 and different values of β.

Fig. 4
Fig. 4

Illustration of design domain. The dash-dot curves denote the symmetric axes of design domain.

Fig. 5
Fig. 5

Robust design of PhCWs. (a) Dilated design. (b) Intermediate design. (c) Eroded design. (d) Band structure of dilated design. The crosses indicate the design range and the bold curve in the inset denotes the bandwidth of prescribed group index. (e) Band structure of intermediate design. (f) Band structure of eroded design. (g) Group index of the different design realizations and prescribed group index. (h) PhCW constituted by intermediate design.

Fig. 6
Fig. 6

Amplitudes of the magnetic field at k i = 0.425 2 π / a . (a) Amplitude of the magnetic field for dilated design. (b) Amplitude of the magnetic field for intermediate design. (c) Amplitude of the magnetic field for eroded design. (d) Amplitude of the lower odd magnetic mode for intermediate design. (e) Amplitude of the lower even mode for intermediate design.

Fig. 7
Fig. 7

Optimized design of PhCW without considering robustness for η = 0.5 . (a) Optimized design. (b) Group index of different designs and prescribed group index.

Fig. 8
Fig. 8

Maximum error between actual group index and prescribed group index in the design wavenumber range versus η for normal and robust formulation.

Fig. 9
Fig. 9

Robust design of PhCWs with small GVD. (a) Dilated design. (b) Intermediate design. (c) Eroded design. (d) PhCW constituted by intermediate design. (e) Amplitude of the magnetic field at k i = 0.425 2 π / a for dilated design. (f) Amplitude of the magnetic field at k i = 0.425 2 π / a for intermediate design. (g) Amplitude of the magnetic field at k i = 0.425 2 π / a for eroded design. (h) Band structure of intermediate design. (i) Groups indexes of the different design realizations and prescribed group index.

Fig. 10
Fig. 10

Robust design of PhCWs with large bandwidth. (a) Dilated design. (b) Intermediate design. (c) Eroded design. (d) PhCW constituted by intermediate design. (e) Amplitude of the magnetic field at k i = 0.376 2 π / a for dilated design. (f) Amplitude of the magnetic field at k i = 0.376 2 π / a for intermediate design. (g) Amplitude of the magnetic field at k i = 0.376 2 π / a for eroded design. (h) Band structure of intermediate design. (i) Groups indexes of the different design realizations and prescribed group index.

Fig. 11
Fig. 11

Robust design of PhCWs with two constant group indexes. (a) Dilated design. (b) Intermediate design. (c) Eroded design. (d) PhCW constituted by intermediate design. (e) Amplitude of the magnetic field at k i = 0.345 2 π / a for dilated design. (f) Amplitude of the magnetic field at k i = 0.345 2 π / a for intermediate design. (g) Amplitude of the magnetic field at k i = 0.345 2 π / a for eroded design. (h) Band structure of intermediate design (f) Groups indexes of the different design realizations and prescribed group index.

Fig. 12
Fig. 12

Robust design of PhCWs with fundamental free vibration frequency constraint. (a) Dilated design. (b) Intermediate design. (c) Eroded design. (d) Amplitude of the magnetic field at k i = 0.425 2 π / a for dilated design. (e) Amplitude of the magnetic field at k i = 0.425 2 π / a for intermediate design. (f) Amplitude of the magnetic field at k i = 0.425 2 π / a for eroded design. (g) Group indexes of the different design realizations and prescribed group index.

Fig. 13
Fig. 13

Robust design of PhCWs with n g = 100 . (a) Dilated design. (b) Intermediate design. (c) Eroded design. (d) Amplitude of the magnetic field at k i = 0.425 2 π / a for dilated design. (e) Amplitude of the magnetic field at k i = 0.425 2 π / a for intermediate design. (f) Amplitude of the magnetic field at k i = 0.425 2 π / a for eroded design. (g) Group indexes of the different design realizations and prescribed group index.

Tables (5)

Tables Icon

Table 1 Performance of Different Designs in Fig. 5

Tables Icon

Table 2 Performance of Different Designs in Fig. 9

Tables Icon

Table 3 Performance of Different Designs in Fig. 10

Tables Icon

Table 4 Performance of Different Designs in Fig. 11

Tables Icon

Table 5 Performance of Different Designs in Fig. 11

Equations (18)

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

· ( 1 ε r h ) + ( ω c ) 2 h = 0 ,
h ( x , a ) = exp ( i k a ) h ( x , 0 ) h ( 0 , y ) = h ( b , y ) ,
( K k ω 2 M ) h = 0 ,
v g = ω k .
n g = c k ω .
n g ( ω n ( k i ) ) = c ( k i k ) ω n ( k ) ω n ( k i ) ,
min ρ j     max q     max k i     f ( ρ ¯ q ) = ( c ( k i k i 1 ) ω n q ( k i 1 ) ω n q ( k i ) n g * ) 2 s.t. [ K k q ( ω q ) 2 M q ] h q = 0 max k i i     ω n 1 q ( k i i ) a 1 min k i     ω n q ( k i ) ω n q ( 0 ) a 2     max k i     ω n q ( k i ) min k i i     ω n + 1 q ( k i i ) a 2     max k i     ω n q ( k i ) f v = j ρ ¯ j d v j j v j f v * 0 ρ j 1 j = 1 , , N , i = 2 , , m , a 1 < 1 , a 2 > 1 , q = { d , i , e }
ρ ˜ e = j N e w ( x j ) v j ρ j j N e w ( x j ) v j ,
N e = { j | x j x e r } ,
ρ ¯ e = { η { exp [ β ( 1 ρ ˜ e / η ) ] ( 1 ρ ˜ e / η ) exp ( β ) } 0 ρ ˜ e η ( 1 η ) { 1 exp [ β ( ρ ˜ e η ) / ( 1 η ) ] + ( ρ ˜ e η ) / ( 1 η ) exp ( β ) } + η η < ρ ˜ e 1 .
1 ε e q = ( 1 ρ ¯ e q ) 1 ε 1 + ρ ¯ e q 1 ε 2 ,
F ρ j = e N j F ρ ¯ e q ρ ¯ e q ρ ˜ e ρ ˜ e ρ j ,
f ( ρ ¯ q ) ρ ¯ e q = 2 ( c ( k i k i 1 ) ω n q ( k i 1 ) ω n q ( k i ) n g * ) c ( k i k i 1 ) ( ω n q ( k i 1 ) ω n q ( k i ) ) 2 ( ω n q ( k i 1 ) ρ ¯ e q ω n q ( k i ) ρ ¯ e q ) .
ρ ¯ e q ρ ˜ e = { β exp [ β ( 1 ρ ˜ e / η q ) ] + exp ( β ) 0 ρ ˜ e η q β exp [ β ( ρ ˜ e η q ) / ( 1 η q ) ] + exp ( β ) η q < ρ ˜ e 1 .
ρ ˜ e ρ j = w ( x j ) v j i N e w ( x i ) v i .
u ( x , a ) = u ( x , 0 ) u ( 0 , y ) = u ( b , y ) = 0.
E e q = E min + ( E 1 E min ) ( ρ ¯ e q ) p , m e q = m min + ( m 1 m min ) ρ ¯ e q ,
min ρ j     max q     max k i     f ( ρ ¯ q ) = ( c ( k i k i 1 ) ω n q ( k i 1 ) ω n q ( k i ) n g * ) 2 s.t. [ K k q ( ω q ) 2 M q ] h q = 0 ( K ˜ q λ 1 q M ˜ q ) u q = 0 max k i i     ω n 1 q ( k i i ) a 1 min k i     ω n q ( k i ) ω n q ( 0 ) a 2     max k i     ω n q ( k i ) min k i i     ω n + 1 q ( k i i ) a 2     max k i     ω n q ( k i ) λ 1 q λ 0 δ , λ 0 = E 1 m 1 f v = j ρ ¯ j d v j j v j f v * 0 ρ j 1 j = 1 , , N , i = 2 , , m , a 1 < 1 , a 2 > 1 , q = { d , i , e } .

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