Abstract

A pulse-delaying optimization scheme based on topology optimization for transient response of photonic crystal structures (PhCs) is formulated to obtain slow-light devices. The optimization process is started from a qualified W1 PhC waveguide design with group index ng40 obtained from a simple Edisonian parameter search. Based on this, the proposed pulse delaying and subsequent pulse restoring strategies yield a design that increases the group index by 75% to ng70±10% for an operational full-width at half-maximum (FWHM) bandwidth BFWHM=6nm, and simultaneously minimizes interface penalty losses between the access ridge and the W1 PhC waveguide. To retain periodicity and symmetry, the active design set is limited to the in-/outlet region and a distributed supercell, and manufacturability is further enhanced by density filtering techniques combined with material phase projections.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  46. K. Inoue, N. Kawai, Y. Sugimoto, N. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65, 121308 (2002).
    [CrossRef]
  47. R. S. Jacobsen, A. V. Lavrinenko, L. H. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. I. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express 13, 7861–7871 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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2011 (4)

B. Lazarov, R. Matzen, and Y. Elesin, “Topology optimization of pulse shaping filters using the Hilbert transform envelope extraction,” Struct. Multidisc. Optim. , published online, doi:10.1007/s00158-011-0642-y (2011).
[CrossRef]

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

L. Dai, T. Li, and C. Jiang, “Wideband ultralow high-order-dispersion photonic crystal slow-light waveguide,” J. Opt. Soc. Am. B 28, 1622–1626 (2011).
[CrossRef]

2010 (3)

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

R. Matzen, J. S. Jensen, and O. Sigmund, “Topology optimization for transient response of photonic crystal structures,” J. Opt. Soc. Am. B 27, 2040–2050 (2010).
[CrossRef]

2009 (2)

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

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

2008 (2)

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension: design of filters and pulse modulators,” Struct. Multidisc. Optim. 36, 585–595 (2008).
[CrossRef]

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

2007 (4)

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
[CrossRef]

L. O’Faolain, T. P. White, D. O’Brien, X. Yuan, M. D. Settle, and T. F. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express 15, 13129–13138(2007).
[CrossRef] [PubMed]

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

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

2006 (1)

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” 14, 9444–9450 (2006).
[CrossRef]

2005 (8)

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71(2005).
[CrossRef]

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90° photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

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]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114(2005).
[CrossRef]

R. S. Tucker, P. C. Ku, and C. J. Chang-Hasnain, “Slow-light optical buffers: capabilities and fundamental limitations,” J. Lightwave Technol. 23, 4046–4066 (2005).
[CrossRef]

R. S. Jacobsen, A. V. Lavrinenko, L. H. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. I. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express 13, 7861–7871 (2005).
[CrossRef] [PubMed]

2004 (6)

T. J. Karle, Y. J. Chai, C. N. Morgan, I. H. White, and T. F. Krauss, “Observation of pulse compression in photonic crystal coupled cavity waveguides,” J. Lightwave Technol. 22 , 514–519 (2004).
[CrossRef]

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[CrossRef] [PubMed]

P. I. Borel, A. Harpoth, 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]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. E87C, 258–265 (2004).

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 (2)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

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

2002 (4)

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

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, and T. F. Krauss, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002).
[CrossRef]

K. Inoue, N. Kawai, Y. Sugimoto, N. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65, 121308 (2002).
[CrossRef]

M. Soljacic, S. G. Johnson, S. H. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052–2059 (2002).
[CrossRef]

2001 (4)

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. dYerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on Inp membranes,” Appl. Phys. Lett. 79, 2312–2314 (2001).
[CrossRef]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276(2001).
[CrossRef] [PubMed]

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[CrossRef]

1999 (3)

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

1996 (2)

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996).
[CrossRef]

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]

1994 (1)

D. A. Tortorelli and P. Michaleris, “Design sensitivity analysis: overview and review,” Inverse Probl. Eng. 1, 71–105 (1994).
[CrossRef]

1987 (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]

1979 (1)

K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B 19, 5057–5067 (1979).
[CrossRef]

Akahane, Y.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Asakawa, K.

K. Inoue, N. Kawai, Y. Sugimoto, N. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65, 121308 (2002).
[CrossRef]

Asano, T.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Baba, T.

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

Beggs, D. M.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

Bendsøe, M. P.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, 2nd ed. (Springer Verlag, 2004).

Benisty, H.

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, and T. F. Krauss, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Borel, P. I.

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” 14, 9444–9450 (2006).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71(2005).
[CrossRef]

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90° photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

R. S. Jacobsen, A. V. Lavrinenko, L. H. Frandsen, C. Peucheret, B. Zsigri, G. Moulin, J. Fage-Pedersen, and P. I. Borel, “Direct experimental and numerical determination of extremely high group indices in photonic crystal waveguides,” Opt. Express 13, 7861–7871 (2005).
[CrossRef] [PubMed]

P. I. Borel, A. Harpoth, 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).
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Peucheret, C.

Rattier, M.

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, and T. F. Krauss, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002).
[CrossRef]

Riley, D. J.

J. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays (Wiley & Sons, 2007).

Rojo-Romeo, P.

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. dYerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on Inp membranes,” Appl. Phys. Lett. 79, 2312–2314 (2001).
[CrossRef]

Rostovtsev, Y.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Sautenkov, V. A.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Schulz, S. A.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

Scully, M. O.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Seassal, C.

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. dYerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on Inp membranes,” Appl. Phys. Lett. 79, 2312–2314 (2001).
[CrossRef]

Settle, M. D.

Shi, P.

Shinya, A.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Sigmund, O.

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

R. Matzen, J. S. Jensen, and O. Sigmund, “Topology optimization for transient response of photonic crystal structures,” J. Opt. Soc. Am. B 27, 2040–2050 (2010).
[CrossRef]

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

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

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension: design of filters and pulse modulators,” Struct. Multidisc. Optim. 36, 585–595 (2008).
[CrossRef]

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

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

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90° photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[CrossRef]

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71(2005).
[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]

L. H. Frandsen, A. Harpoth, P. I. Borel, M. Kristensen, J. S. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60° bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[CrossRef] [PubMed]

P. I. Borel, A. Harpoth, 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,” Phil. Trans. Roy. Soc. London A 361, 1001–1019 (2003).
[CrossRef]

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784, doi:10.1007/s00158-010-0602-y (2010).
[CrossRef]

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, 2nd ed. (Springer Verlag, 2004).

Slusher, R. E.

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[CrossRef]

B. J. Eggleton, C. M. de Sterke, and R. E. Slusher, “Bragg solitons in the nonlinear Schrödinger limit: experiment and theory,” J. Opt. Soc. Am. B 16, 587–599 (1999).
[CrossRef]

Smith, C. J. M.

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, and T. F. Krauss, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002).
[CrossRef]

Soljacic, M.

Song, B. S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Stainko, R.

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

Sugimoto, Y.

K. Inoue, N. Kawai, Y. Sugimoto, N. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65, 121308 (2002).
[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]

Takahashi, C.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, J.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Tortorelli, D. A.

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114(2005).
[CrossRef]

D. A. Tortorelli and P. Michaleris, “Design sensitivity analysis: overview and review,” Inverse Probl. Eng. 1, 71–105 (1994).
[CrossRef]

Tucker, R. S.

van Hulst, N. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

Viktorovitch, P.

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. dYerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on Inp membranes,” Appl. Phys. Lett. 79, 2312–2314 (2001).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

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. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784, doi:10.1007/s00158-010-0602-y (2010).
[CrossRef]

Weisbuch, C.

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, and T. F. Krauss, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002).
[CrossRef]

Welch, G. R.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

White, I. H.

White, T. P.

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

L. O’Faolain, T. P. White, D. O’Brien, X. Yuan, M. D. Settle, and T. F. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express 15, 13129–13138(2007).
[CrossRef] [PubMed]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 2008).

Yablonovitch, E.

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. E87C, 258–265 (2004).

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

Yamada, K.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yang, L. R.

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

Yokohama, I.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yuan, X.

Zibrov, A. S.

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

Zsigri, B.

Acta Mech. Sinica (1)

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

Appl. Phys. Lett. (4)

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. dYerville, D. Cassagne, and C. Jouanin, “Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on Inp membranes,” Appl. Phys. Lett. 79, 2312–2314 (2001).
[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]

W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Topology optimization of a photonic crystal waveguide termination to maximize directional emission,” Appl. Phys. Lett. 86, 111114(2005).
[CrossRef]

L. R. Yang, A. V. Lavrinenko, J. M. Hvam, and O. Sigmund, “Design of one-dimensional optical pulse-shaping filters by time-domain topology optimization,” Appl. Phys. Lett. 95, 261101 (2009).
[CrossRef]

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Electron. Lett. (1)

P. I. Borel, L. H. Frandsen, A. Harpoth, M. Kristensen, J. S. Jensen, and O. Sigmund, “Topology optimised broadband photonic crystal Y-splitter,” Electron. Lett. 41, 69–71(2005).
[CrossRef]

IEEE J. Quantum. Electron. (1)

G. Lenz, B. J. Eggleton, C. K. Madsen, and R. E. Slusher, “Optical delay lines based on optical filters,” IEEE J. Quantum. Electron. 37, 525–532 (2001).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. S. Jensen, O. Sigmund, L. H. Frandsen, P. I. Borel, A. Harpoth, and M. Kristensen, “Topology design and fabrication of an efficient double 90° photonic crystal waveguide bend,” IEEE Photon. Technol. Lett. 17, 1202–1204 (2005).
[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. E87C, 258–265 (2004).

Inverse Probl. Eng. (1)

D. A. Tortorelli and P. Michaleris, “Design sensitivity analysis: overview and review,” Inverse Probl. Eng. 1, 71–105 (1994).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. (1)

S. A. Schulz, L. O’Faolain, D. M. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. 12, 104004 (2010).
[CrossRef]

J. Opt. Soc. Am. B (6)

J. Phys. D (1)

T. F. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D 40, 2666–2670 (2007).
[CrossRef]

Laser Photon. Rev. (1)

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

Nat. Mater. (1)

M. Soljacic and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211–219 (2004).
[CrossRef] [PubMed]

Nat. Photon. (1)

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

Nature (5)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[CrossRef]

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]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

M. D. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276(2001).
[CrossRef] [PubMed]

Y. A. Vlasov, M. O’Boyle, H. F. Hamann, and S. J. McNab, “Active control of slow light on a chip with photonic crystal waveguides,” Nature 438, 65–69 (2005).
[CrossRef] [PubMed]

Opt. Express (4)

Opt. Quantum Electron. (1)

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, and T. F. Krauss, “Transmission properties of two-dimensional photonic crystal channel waveguides,” Opt. Quantum Electron. 34, 171–181 (2002).
[CrossRef]

Phil. Trans. Roy. Soc. London A (1)

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

Phys. Lett. A (1)

G. Diener, “Superluminal group velocities and information transfer,” Phys. Lett. A 223, 327–331 (1996).
[CrossRef]

Phys. Rev. B (2)

K. Ohtaka, “Energy band of photons and low-energy photon diffraction,” Phys. Rev. B 19, 5057–5067 (1979).
[CrossRef]

K. Inoue, N. Kawai, Y. Sugimoto, N. Carlsson, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B 65, 121308 (2002).
[CrossRef]

Phys. Rev. Lett. (5)

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]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs,” Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005).
[CrossRef] [PubMed]

M. M. Kash, V. A. Sautenkov, A. S. Zibrov, L. Hollberg, G. R. Welch, M. D. Lukin, Y. Rostovtsev, E. S. Fry, and M. O. Scully, “Ultraslow group velocity and enhanced nonlinear optical effects in a coherently driven hot atomic gas,” Phys. Rev. Lett. 82, 5229–5232 (1999).
[CrossRef]

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. (3)

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

J. Dahl, J. S. Jensen, and O. Sigmund, “Topology optimization for transient wave propagation problems in one dimension: design of filters and pulse modulators,” Struct. Multidisc. Optim. 36, 585–595 (2008).
[CrossRef]

B. Lazarov, R. Matzen, and Y. Elesin, “Topology optimization of pulse shaping filters using the Hilbert transform envelope extraction,” Struct. Multidisc. Optim. , published online, doi:10.1007/s00158-011-0642-y (2011).
[CrossRef]

Waves Random Complex Media (1)

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

Other (7)

L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” 14, 9444–9450 (2006).
[CrossRef]

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods, and Applications, 2nd ed. (Springer Verlag, 2004).

http://www.esf.org/euriy.

http://www.topopt.dtu.dk.

F. Wang, B. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Struct. Multidisc. Optim. 43, 767–784, doi:10.1007/s00158-010-0602-y (2010).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 2008).

J. Jin and D. J. Riley, Finite Element Analysis of Antennas and Arrays (Wiley & Sons, 2007).

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Figures (11)

Fig. 1
Fig. 1

Ridge waveguide optimization setup. The computational domain contains a solution region Ω S , PML region Ω PML , and design region Ω D encapsulated by the dashed boundary. The “active” design set consists of a y mirrored in-/outlet region Ω IO and a x y mirrored supercell Ω SC that is a subset of Ω D . Whenever an optimized design is presented, we only show the material distribution in Ω IO and Ω SC . The waveguide mode is excited at Γ inc with an analytically given amplitude profile Ψ ( y ) . The objective is measured at (multiple) point(s) on Ω E .

Fig. 2
Fig. 2

(a) Rotated supercell used in the eigenvalue computation to obtain the (b) band diagram. This shows the normalized frequencies versus normalized wave vectors for an even (solid) and odd (dashed) PhC-WG mode in the bandgap, and the dashed line indicates the light line above which leaky modes live. The dielectric and the air band correspond to the lower and upper gray dense mode regions, respectively. (c) Group index n g versus normalized frequency.

Fig. 3
Fig. 3

Transmitted— | T | 2 , reflected energy | R | 2 , and energy balance | T | 2 + | R | 2 through in- and outlets as a function of the wavelength for the initial solution (and thus the sum of the reflection and transmission does not equal 1). The interval between the dashed lines indicates the bandgap region. The crystal lattice is assumed to be a = 370 nm .

Fig. 4
Fig. 4

Pulse delay strategy for pulses with a group delay 0 and η. The parameter η is introduced as a design variable. The g * parameter controls the temporal spreading of the output pulse envelope. The α parameter mainly specifies the transmitted energy, i.e., the amplitude decrease.

Fig. 5
Fig. 5

Symmetry conditions for the “active” design set. (a) x symmetry is imposed for Ω IO . (b)  x y symmetry is imposed for Ω SC . This reduces the number of active design variables further.

Fig. 6
Fig. 6

Pulse-delayed slow-light designs. (a) Transient intensity response of the ridge waveguide pulse peaking at t = 0 ps and time- delayed pulse for g * = 0.001 , 0.005, and 0.05 peaking at t = 1.5 ps , 2.0 ps , and 3.0 ps , respectively. In/-outlet and supercell design for (b)  g * = 0.001 , (c)  g * = 0.005 , and (d)  g * = 0.05 .

Fig. 7
Fig. 7

Group index versus normalized frequency. (a) Optimized design obtained from the delay formulation in Eq. (8) for three dif ferent values for the pulse relaxation: g * = 0.001 , 0.005, and 0.05. (b) A comparison between the pulse delayed design for g * = 0.001 and the pulse delayed-shaped design.

Fig. 8
Fig. 8

Pulse-delayed-shaped slow-light design. (a) Intensity response of the initial envelope, peaking at t = 0 ps , and the optimized envelope peaking at t = 1.5 ps with and without pulse shaping. (b) Band diagram for the optimized supercell structure and the transmission spectrum for the structure of finite length without optimized in-/outlets. The dashed line is the light line, and the solid and dashed dispersion curves represent the even and odd modes, respectively. The inset magnifies the anticrossing curves in the square with α and β indicating the fundamental and higher order modes, respectively. (c) The optimized in-/outlet region (left) and supercell (right).

Fig. 9
Fig. 9

Optimized supercell design without in-/outlet design. Transmitted— | T | 2 , reflected energy | R | 2 , and energy balance | T | 2 + | R | 2 as a function of the wavelength. The interval between the dashed lines indicates the bandgap region.

Fig. 10
Fig. 10

In-/outlet and supercell design in Fig. 11a. Transmitted— | T | 2 , reflected— | R | 2 , and energy balance | T | 2 + | R | 2 as a function of the wavelength. The interval between the dashed lines indicates the bandgap region.

Fig. 11
Fig. 11

H z field pattern for slow-light device. (a) Optimized structure at a / λ = 0.2163 , obtained from delaying and subsequently shaping the pulse. (b) Initial geometry at a / λ = 0.2163 . The material distribution is shown with a 0.6 threshold.

Equations (12)

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x ( A Ψ x ) + Ψ y ( A Ψ y ) B c 2 2 Ψ t 2 = 0 Ψ = H z ( r , t ) , A = 1 / ε r ( r ) , B = 1 ,
e = 1 M ( T e u ¨ + R e u ˙ + S e u + g e f e ) = 0 ,
v g d ω d k = c n g ,
v g = c 2 L c Δ λ λ 2 ,
v g = η L ,
f = 0 T i Ω E [ r i ( t ) α p i ( t η ) ] 2 d t 0 T i Ω E α 2 p i ( t ) 2 d t ,
ε r 1 ( ρ e ) = ( 1 ρ e ) ( ε r I ) 1 + ρ e ( ε r II ) 1 ,
min ρ R M , η R η s.t.:   governing Eq.   ( 2 ) g ( ρ , η ) = f ( ρ , η ) / g * 1 0 0 ρ e 1 , e Ω D 0 η η max ,
g η = 0 T i Ω E 2 [ r i ( t ) α p i ( t s η ) ] α p i ( t s η ) η d t g * 0 T i Ω E α 2 p i ( t ) 2 d t ,
min ρ R M f ( ρ , η ) s.t.:  governing Eq.   ( 2 ) 0 ρ e 1 , e Ω D .
Ψ inc ( x , t ) = Ψ ( y ) sin [ k x ω c ( t t 0 ) ] e ( t t 0 x / v p ) 2 T 0 2 ,
T 0 = 4 ln 2 π B FWHM .

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