Abstract

We perform full 3D topology optimization (in which “every voxel” of the unit cell is a degree of freedom) of photonic-crystal structures in order to find optimal omnidirectional band gaps for various symmetry groups, including fcc (including diamond), bcc, and simple-cubic lattices. Even without imposing the constraints of any fabrication process, the resulting optimal gaps are only slightly larger than previous hand designs, suggesting that current photonic crystals are nearly optimal in this respect. However, optimization can discover new structures, e.g. a new fcc structure with the same symmetry but slightly larger gap than the well known inverse opal, which may offer new degrees of freedom to future fabrication technologies. Furthermore, our band-gap optimization is an illustration of a computational approach to 3D dispersion engineering which is applicable to many other problems in optics, based on a novel semidefinite-program formulation for nonconvex eigenvalue optimization combined with other techniques such as a simple approach to impose symmetry constraints. We also demonstrate a technique for robust topology optimization, in which some uncertainty is included in each voxel and we optimize the worst-case gap, and we show that the resulting band gaps have increased robustness to systematic fabrication errors.

© 2014 Optical Society of America

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2014 (1)

H. Men, R. M. Freund, N. C. Nguyen, J. Saa-Seoane, and J. Peraire, “Fabrication-adaptive optimization with an application to photonic crystal design,” Oper. Res. 62, 418–434 (2014).
[Crossref]

2013 (1)

2012 (2)

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express 20, 21558–21575 (2012).
[Crossref] [PubMed]

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. 10, 153–165 (2012).
[Crossref]

2011 (4)

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. Multidiscip. 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]

D. Bertsimas, D. B. Brown, and C. Caramanis, “Theory and applications of robust optimization,” SIAM Rev. 53, 464–501 (2011).
[Crossref]

2010 (2)

H. Men, N. C. Nguyen, R. M. Freund, P. A. Parrilo, and J. Peraire, “Bandgap optimization of two-dimensional photonic crystals using semidefinite programming and subspace methods,” J. Comput. Phys. 229, 3706–3725 (2010).
[Crossref]

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res. 58, 161–178 (2010).
[Crossref]

2009 (2)

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (2009).
[Crossref]

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

2008 (1)

O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. 100, 153904 (2008).
[Crossref] [PubMed]

2007 (4)

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[Crossref]

L. He, C. Y. Kao, and S. J. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comput. Phys. 225, 891–909 (2007).
[Crossref]

J. Harrison, P. Kuchment, A. Sobolev, and B. Winn, “On occurrence of spectral edges for periodic operators inside the brillouin zone,” J. Phys. A 40, 7597–7618 (2007).
[Crossref]

H. G. Beyer and B. Sendhoff, “Robust optimization–a comprehensive survey,” Comput. Methods Appl. Mech. Eng. 196, 3190–3218 (2007).
[Crossref]

2006 (5)

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[Crossref] [PubMed]

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5, 93–96 (2006).
[Crossref] [PubMed]

Y. Watanabe, Y. Sugimoto, N. Ikeda, N. Ozaki, A. Mizutani, Y. Takata, Y. Kitagawa, and K. Asakawa, “Broadband waveguide intersection with low crosstalk in two-dimensional photonic crystal circuits by using topology optimization,” Opt. Express 14, 9502–9507 (2006).
[Crossref] [PubMed]

S. Halkjær, O. Sigmund, and J. S. Jensen, “Maximizing band gaps in plate structures,” Struct. Multidiscip. Optim. 32, 263–275 (2006).
[Crossref]

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, and B. Miao, “Photonic crystal structures and applications: Perspective, overview, and development,” IEEE J. Sel. Top. Quantum Electron. 12, 1416–1437 (2006).
[Crossref]

2005 (7)

C. Y. Kao, S. J. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B 81, 235–244 (2005).
[Crossref]

M. Burger and S. J. Osher, “A survey on level set methods for inverse problems and optimal design,” Eur. J. Appl. Math. 16, 263–301 (2005).
[Crossref]

M. Maldovan and E. Thomas, “Photonic crystals: six connected dielectric networks with simple cubic symmetry,” J. Opt. Soc. Am. B 22, 466–473 (2005).
[Crossref]

J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett. 30, 2397–2399 (2005).
[Crossref] [PubMed]

J. Serbin and M. Gu, “Superprism phenomena in polymeric woodpile structures,” J. Appl. Phys. 98, 123101 (2005).
[Crossref]

A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, and S. G. Johnson, “Disorder-immune confinement of light in photonic-crystal cavities,” Opt. Lett. 30, 3192–3194 (2005).
[Crossref] [PubMed]

T. Bruns, “A reevaluation of the simp method with filtering and an alternative formulation for solid–void topology optimization,” Struct. Multidiscip. Optim. 30, 428–436 (2005).
[Crossref]

2004 (7)

C. Luo, M. Soljačić, and J. Joannopoulos, “Superprism effect based on phase velocities,” Opt. Lett. 29, 745–747 (2004).
[Crossref] [PubMed]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[Crossref] [PubMed]

Y. Ni, L. Zhang, L. An, J. Peng, and C. Fan, “Dual-core photonic crystal fiber for dispersion compensation,” IEEE Photon. Technol. Lett. 16, 1516–1518 (2004).
[Crossref]

B. Zsigri, J. Lægsgaard, and A. Bjarklev, “A novel photonic crystal fibre design for dispersion compensation,” J. Opt. A Pure Appl. Opt. 6, 717 (2004).
[Crossref]

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. 87, 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]

M. Maldovan and E. L. Thomas, “Diamond-structured photonic crystals,” Nat. Mater. 3, 593–600 (2004).
[Crossref] [PubMed]

2003 (7)

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

O. Toader, M. Berciu, and S. John, “Photonic band gaps based on tetragonal lattices of slanted pores,” Phys. Rev. Lett. 90, 233901 (2003).
[Crossref] [PubMed]

K. Michielsen and J. Kole, “Photonic band gaps in materials with triply periodic surfaces and related tubular structures,” Phys. Rev. B 68, 115107 (2003).
[Crossref]

M. Maldovan, C. K. Ullal, W. C. Carter, and E. L. Thomas, “Exploring for 3d photonic bandgap structures in the 11 fcc space groups,” Nat. Mater. 2, 664–667 (2003).
[Crossref] [PubMed]

L. P. Shen, W. P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540–542 (2003).
[Crossref]

L. Wu, M. Mazilu, and T. F. Krauss, “Beam steering in planar-photonic crystals: from superprism to supercollimator,” J. Lightwave Technol. 21, 561–566 (2003).
[Crossref]

P. V. Parimi, W. T. Lu, P. Vodo, and S. Sridhar, “Photonic crystals: Imaging by flat lens using negative refraction,” Nature 426, 404 (2003).
[Crossref] [PubMed]

2002 (6)

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quant. 8, 1246–1257 (2002).
[Crossref]

C. Luo, S. G. Johnson, and J. Joannopoulos, “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

L. Wu, M. Mazilu, T. Karle, and T. F. Krauss, “Superprism phenomena in planar photonic crystals,” IEEE J. Quantum. Elect. 38, 915–918 (2002).
[Crossref]

R. Biswas, M. Sigalas, K. Ho, and S. Lin, “Three-dimensional photonic band gaps in modified simple cubic lattices,” Phys. Rev. B 65, 205121 (2002).
[Crossref]

M. Maldovan, A. Urbas, N. Yufa, W. Carter, and E. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123 (2002).
[Crossref]

L. F. Shen, S. L. He, and S. H. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B 66, 165315 (2002).
[Crossref]

2001 (2)

M. Stolpe and K. Svanberg, “An alternative interpolation scheme for minimum compliance topology optimization,” Struct. Multidiscip. Optim. 22, 116–124 (2001).
[Crossref]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[Crossref] [PubMed]

2000 (5)

S. G. Johnson and J. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77, 3490–3492 (2000).
[Crossref]

S. Saltiel and Y. S. Kivshar, “Phase matching in nonlinear χ(2) photonic crystals,” Opt. Lett. 25, 1204–1206 (2000).
[Crossref]

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,” Phys. Rev. B. 62, 10696–10705 (2000).
[Crossref]

S. J. Cox and D. C. Dobson, “Band structure optimization of two-dimensional photonic crystals in H-polarization,” J. Comput. Phys. 158, 214–224 (2000).
[Crossref]

M. Doosje, B. J. Hoenders, and J. Knoester, “Photonic bandgap optimization in inverted fcc photonic crystals,” J. Opt. Soc. Am. B 17, 600–606 (2000).
[Crossref]

1999 (5)

M. P. Bendsøe and O. Sigmund, “Material interpolation schemes in topology optimization,” Arch. Appl. Mech. 69, 635–654 (1999).
[Crossref]

D. C. Dobson and S. J. Cox, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

T. Birks, D. Mogilevtsev, J. Knight, and P. S. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: Toward microscale lightwave circuits,” J. Lightwave Technol. 17, 2032–2038 (1999).
[Crossref]

1998 (5)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

A. Chutinan and S. Noda, “Spiral three-dimensional photonic-band-gap structure,” Phys. Rev. B 57, R2006–R2008 (1998).
[Crossref]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896–3903 (1998).
[Crossref]

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463–466 (1998).
[Crossref]

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

1996 (1)

1995 (1)

S. J. Cox, “The generalized gradient at a multiple eigenvalue,” J. Funct. Anal. 133, 30–40 (1995).
[Crossref]

1994 (2)

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

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, “Design of three-dimensional photonic crystals at submicron lengthscales,” Appl. Phys. Lett. 65, 1466–1468 (1994).
[Crossref]

1992 (1)

H. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

1990 (1)

K. Ho, C. Chan, and C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

1988 (1)

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

An, L.

Y. Ni, L. Zhang, L. An, J. Peng, and C. Fan, “Dual-core photonic crystal fiber for dispersion compensation,” IEEE Photon. Technol. Lett. 16, 1516–1518 (2004).
[Crossref]

Asakawa, K.

Avniel, Y.

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (2009).
[Crossref]

Bendsøe, M. P.

M. P. Bendsøe and O. Sigmund, “Material interpolation schemes in topology optimization,” Arch. Appl. Mech. 69, 635–654 (1999).
[Crossref]

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

M. P. Bendsøe, Optimization of Structural Topology, Shape, and Material (Springer, 1995).
[Crossref]

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

Ben-Tal, A.

A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization (Princeton University, 2009).

Berciu, M.

O. Toader, M. Berciu, and S. John, “Photonic band gaps based on tetragonal lattices of slanted pores,” Phys. Rev. Lett. 90, 233901 (2003).
[Crossref] [PubMed]

Berger, V.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463–466 (1998).
[Crossref]

V. Berger, “Nonlinear photonic crystals,” Phys. Rev. Lett. 81, 4136–4139 (1998).
[Crossref]

Bertsimas, D.

D. Bertsimas, D. B. Brown, and C. Caramanis, “Theory and applications of robust optimization,” SIAM Rev. 53, 464–501 (2011).
[Crossref]

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization for unconstrained simulation-based problems,” Oper. Res. 58, 161–178 (2010).
[Crossref]

D. Bertsimas, O. Nohadani, and K. M. Teo, “Robust optimization in electromagnetic scattering problems,” J. Appl. Phys. 101, 074507 (2007).
[Crossref]

Beyer, H. G.

H. G. Beyer and B. Sendhoff, “Robust optimization–a comprehensive survey,” Comput. Methods Appl. Mech. Eng. 196, 3190–3218 (2007).
[Crossref]

Birks, T.

T. Birks, D. Mogilevtsev, J. Knight, and P. S. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[Crossref]

Biswas, R.

R. Biswas, M. Sigalas, K. Ho, and S. Lin, “Three-dimensional photonic band gaps in modified simple cubic lattices,” Phys. Rev. B 65, 205121 (2002).
[Crossref]

Bjarklev, A.

B. Zsigri, J. Lægsgaard, and A. Bjarklev, “A novel photonic crystal fibre design for dispersion compensation,” J. Opt. A Pure Appl. Opt. 6, 717 (2004).
[Crossref]

Boyd, S.

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (2009).
[Crossref]

Boyd, S. P.

Bravetti, P.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463–466 (1998).
[Crossref]

Brown, D. B.

D. Bertsimas, D. B. Brown, and C. Caramanis, “Theory and applications of robust optimization,” SIAM Rev. 53, 464–501 (2011).
[Crossref]

Bruns, T.

T. Bruns, “A reevaluation of the simp method with filtering and an alternative formulation for solid–void topology optimization,” Struct. Multidiscip. Optim. 30, 428–436 (2005).
[Crossref]

Burger, M.

M. Burger and S. J. Osher, “A survey on level set methods for inverse problems and optimal design,” Eur. J. Appl. Math. 16, 263–301 (2005).
[Crossref]

M. Burger, S. J. Osher, and E. Yablonovitch, “Inverse problem techniques for the design of photonic crystals,” IEICE Trans. Electron. 87, 258–265 (2004).

Busch, K.

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896–3903 (1998).
[Crossref]

Caramanis, C.

D. Bertsimas, D. B. Brown, and C. Caramanis, “Theory and applications of robust optimization,” SIAM Rev. 53, 464–501 (2011).
[Crossref]

Carter, W.

M. Maldovan, A. Urbas, N. Yufa, W. Carter, and E. Thomas, “Photonic properties of bicontinuous cubic microphases,” Phys. Rev. B 65, 165123 (2002).
[Crossref]

Carter, W. C.

M. Maldovan, C. K. Ullal, W. C. Carter, and E. L. Thomas, “Exploring for 3d photonic bandgap structures in the 11 fcc space groups,” Nat. Mater. 2, 664–667 (2003).
[Crossref] [PubMed]

Chan, C.

K. Ho, C. Chan, and C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Chen, C.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, and B. Miao, “Photonic crystal structures and applications: Perspective, overview, and development,” IEEE J. Sel. Top. Quantum Electron. 12, 1416–1437 (2006).
[Crossref]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[Crossref] [PubMed]

Chen, G. X.

L. P. Shen, W. P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540–542 (2003).
[Crossref]

Chutinan, A.

A. Chutinan and S. Noda, “Spiral three-dimensional photonic-band-gap structure,” Phys. Rev. B 57, R2006–R2008 (1998).
[Crossref]

Cox, S. J.

S. J. Cox and D. C. Dobson, “Band structure optimization of two-dimensional photonic crystals in H-polarization,” J. Comput. Phys. 158, 214–224 (2000).
[Crossref]

D. C. Dobson and S. J. Cox, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
[Crossref]

S. J. Cox, “The generalized gradient at a multiple eigenvalue,” J. Funct. Anal. 133, 30–40 (1995).
[Crossref]

Dahlem, M. S.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5, 93–96 (2006).
[Crossref] [PubMed]

Dobson, D. C.

S. J. Cox and D. C. Dobson, “Band structure optimization of two-dimensional photonic crystals in H-polarization,” J. Comput. Phys. 158, 214–224 (2000).
[Crossref]

D. C. Dobson and S. J. Cox, “Maximizing band gaps in two-dimensional photonic crystals,” SIAM J. Appl. Math. 59, 2108–2120 (1999).
[Crossref]

Doosje, M.

El Ghaoui, L.

A. Ben-Tal, L. El Ghaoui, and A. Nemirovski, Robust Optimization (Princeton University, 2009).

Elesin, Y.

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. 10, 153–165 (2012).
[Crossref]

Fan, C.

Y. Ni, L. Zhang, L. An, J. Peng, and C. Fan, “Dual-core photonic crystal fiber for dispersion compensation,” IEEE Photon. Technol. Lett. 16, 1516–1518 (2004).
[Crossref]

Fan, S.

J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett. 30, 2397–2399 (2005).
[Crossref] [PubMed]

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, “Design of three-dimensional photonic crystals at submicron lengthscales,” Appl. Phys. Lett. 65, 1466–1468 (1994).
[Crossref]

Farjadpour, A.

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (2009).
[Crossref]

Fiore, A.

A. Fiore, V. Berger, E. Rosencher, P. Bravetti, and J. Nagle, “Phase matching using an isotropic nonlinear optical material,” Nature 391, 463–466 (1998).
[Crossref]

Freund, R. M.

H. Men, R. M. Freund, N. C. Nguyen, J. Saa-Seoane, and J. Peraire, “Fabrication-adaptive optimization with an application to photonic crystal design,” Oper. Res. 62, 418–434 (2014).
[Crossref]

H. Men, N. C. Nguyen, R. M. Freund, P. A. Parrilo, and J. Peraire, “Bandgap optimization of two-dimensional photonic crystals using semidefinite programming and subspace methods,” J. Comput. Phys. 229, 3706–3725 (2010).
[Crossref]

Gu, M.

J. Serbin and M. Gu, “Superprism phenomena in polymeric woodpile structures,” J. Appl. Phys. 98, 123101 (2005).
[Crossref]

Halkjær, S.

S. Halkjær, O. Sigmund, and J. S. Jensen, “Maximizing band gaps in plate structures,” Struct. Multidiscip. Optim. 32, 263–275 (2006).
[Crossref]

Harrison, J.

J. Harrison, P. Kuchment, A. Sobolev, and B. Winn, “On occurrence of spectral edges for periodic operators inside the brillouin zone,” J. Phys. A 40, 7597–7618 (2007).
[Crossref]

Haus, J.

H. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

He, L.

L. He, C. Y. Kao, and S. J. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comput. Phys. 225, 891–909 (2007).
[Crossref]

He, S. L.

L. F. Shen, S. L. He, and S. H. Xiao, “Large absolute band gaps in two-dimensional photonic crystals formed by large dielectric pixels,” Phys. Rev. B 66, 165315 (2002).
[Crossref]

Hietala, V.

Ho, K.

R. Biswas, M. Sigalas, K. Ho, and S. Lin, “Three-dimensional photonic band gaps in modified simple cubic lattices,” Phys. Rev. B 65, 205121 (2002).
[Crossref]

K. Ho, C. Chan, and C. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[Crossref] [PubMed]

Hoenders, B. J.

Hougaard, K.

O. Sigmund and K. Hougaard, “Geometric properties of optimal photonic crystals,” Phys. Rev. Lett. 100, 153904 (2008).
[Crossref] [PubMed]

Huang, W. P.

L. P. Shen, W. P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540–542 (2003).
[Crossref]

Ibanescu, M.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5, 93–96 (2006).
[Crossref] [PubMed]

A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, and S. G. Johnson, “Disorder-immune confinement of light in photonic-crystal cavities,” Opt. Lett. 30, 3192–3194 (2005).
[Crossref] [PubMed]

Ikeda, N.

Inguva, R.

H. Sözüer, J. Haus, and R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[Crossref]

Ippen, E. P.

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5, 93–96 (2006).
[Crossref] [PubMed]

Jensen, J. S.

Y. Elesin, B. S. Lazarov, J. S. Jensen, and O. Sigmund, “Design of robust and efficient photonic switches using topology optimization,” Photon. Nanostruct. 10, 153–165 (2012).
[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]

S. Halkjær, O. Sigmund, and J. S. Jensen, “Maximizing band gaps in plate structures,” Struct. Multidiscip. Optim. 32, 263–275 (2006).
[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]

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

Jian, S. S.

L. P. Shen, W. P. Huang, G. X. Chen, and S. S. Jian, “Design and optimization of photonic crystal fibers for broad-band dispersion compensation,” IEEE Photon. Technol. Lett. 15, 540–542 (2003).
[Crossref]

Joannopoulos, J.

C. Luo, M. Soljačić, and J. Joannopoulos, “Superprism effect based on phase velocities,” Opt. Lett. 29, 745–747 (2004).
[Crossref] [PubMed]

C. Luo, S. G. Johnson, and J. Joannopoulos, “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

S. G. Johnson and J. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77, 3490–3492 (2000).
[Crossref]

Joannopoulos, J. D.

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express 20, 21558–21575 (2012).
[Crossref] [PubMed]

P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljačić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5, 93–96 (2006).
[Crossref] [PubMed]

A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, and S. G. Johnson, “Disorder-immune confinement of light in photonic-crystal cavities,” Opt. Lett. 30, 3192–3194 (2005).
[Crossref] [PubMed]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[Crossref] [PubMed]

S. Fan, P. R. Villeneuve, R. D. Meade, and J. D. Joannopoulos, “Design of three-dimensional photonic crystals at submicron lengthscales,” Appl. Phys. Lett. 65, 1466–1468 (1994).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

John, S.

O. Toader, M. Berciu, and S. John, “Photonic band gaps based on tetragonal lattices of slanted pores,” Phys. Rev. Lett. 90, 233901 (2003).
[Crossref] [PubMed]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896–3903 (1998).
[Crossref]

Johnson, S. G.

X. Liang and S. G. Johnson, “Formulation for scalable optimization of microcavities via the frequency-averaged local density of states,” Opt. Express 21, 30812–30841 (2013).
[Crossref]

A. Oskooi, A. Mutapcic, S. Noda, J. D. Joannopoulos, S. P. Boyd, and S. G. Johnson, “Robust optimization of adiabatic tapers for coupling to slow-light photonic-crystal waveguides,” Opt. Express 20, 21558–21575 (2012).
[Crossref] [PubMed]

A. Mutapcic, S. Boyd, A. Farjadpour, S. G. Johnson, and Y. Avniel, “Robust design of slow-light tapers in periodic waveguides,” Eng. Optim. 41, 365–384 (2009).
[Crossref]

A. Rodriguez, M. Ibanescu, J. D. Joannopoulos, and S. G. Johnson, “Disorder-immune confinement of light in photonic-crystal cavities,” Opt. Lett. 30, 3192–3194 (2005).
[Crossref] [PubMed]

C. Luo, S. G. Johnson, and J. Joannopoulos, “All-angle negative refraction in a three-dimensionally periodic photonic crystal,” Appl. Phys. Lett. 81, 2352–2354 (2002).
[Crossref]

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
[Crossref] [PubMed]

S. G. Johnson and J. Joannopoulos, “Three-dimensionally periodic dielectric layered structure with omnidirectional photonic band gap,” Appl. Phys. Lett. 77, 3490–3492 (2000).
[Crossref]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light (Princeton University, 2011).

Jones, E.

Kao, C. Y.

L. He, C. Y. Kao, and S. J. Osher, “Incorporating topological derivatives into shape derivatives based level set methods,” J. Comput. Phys. 225, 891–909 (2007).
[Crossref]

C. Y. Kao, S. J. Osher, and E. Yablonovitch, “Maximizing band gaps in two-dimensional photonic crystals by using level set methods,” Appl. Phys. B 81, 235–244 (2005).
[Crossref]

Karle, T.

L. Wu, M. Mazilu, T. Karle, and T. F. Krauss, “Superprism phenomena in planar photonic crystals,” IEEE J. Quantum. Elect. 38, 915–918 (2002).
[Crossref]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: Toward microscale lightwave circuits,” J. Lightwave Technol. 17, 2032–2038 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: Toward microscale lightwave circuits,” J. Lightwave Technol. 17, 2032–2038 (1999).
[Crossref]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, 10096–10099 (1998).
[Crossref]

Kikuchi, N.

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

Kitagawa, Y.

Kivshar, Y. S.

Knight, J.

T. Birks, D. Mogilevtsev, J. Knight, and P. S. J. Russell, “Dispersion compensation using single-material fibers,” IEEE Photon. Technol. Lett. 11, 674–676 (1999).
[Crossref]

Knoester, J.

Kole, J.

K. Michielsen and J. Kole, “Photonic band gaps in materials with triply periodic surfaces and related tubular structures,” Phys. Rev. B 68, 115107 (2003).
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Figures (9)

Fig. 1
Fig. 1

Schematic of optimization process of photonic-crystal band-gap structures. Computation from (a) to (b) is often known as the forward problem, in which given the photonic crystal dielectric function (or in a discrete representation εi), one computes the band structure by solving the Maxwell equations. The optimal design problem (PN), or the inverse problem, seeks to compute the optimal dielectric function ε* that maximizes the frequency band gap. The robust optimal design problem (PR) seeks to compute a more robust optimal solution ε ˜ δ * by solving a maxmin optimization problem, in case the nominal optimum ε* is not easily fabricable.

Fig. 2
Fig. 2

Material structure and the optimization design region in a diamond lattice. The optimization design region is limited to the rectangular parallelepiped, denoted by U in (1), and is only 1/64 of the cube. Through necessary symmetric operations, e.g., (2), ..., (48), the material ε(u) (top) is reconstructed.

Fig. 3
Fig. 3

Optimized photonic crystals of prescribed symmetries (space groups no. 214, 221, 225, 227) with complete gaps between consecutive bands.

Fig. 4
Fig. 4

Band structures of initial and optimized photonic crystals of a face-centered cubic (fcc) lattice (no. 225). The two initial solutions and their band structures are shown on the left, and their corresponding optimal solutions and the band structures are shown on the right.

Fig. 5
Fig. 5

(a) Geometry characterization of SC5 with three parameters (r1, r2, r3). (b) Reconstructed photonic crystal with ( r 1 * , r 2 * , r 3 * ) = ( 0.14 a , 0.36 a , 0.105 a ) , and a frequency gap of 16.7%.

Fig. 6
Fig. 6

(a) Geometry characterization of Diamond2 with one parameters r. (b) Reconstructued photonic crystal with r* = 0.1a, and a frequency gap of 30.7%.

Fig. 7
Fig. 7

(a) Geometry characterization of FCC8 with three parameters (r1, r2, r3). (b) Reconstructued photonic crystal with ( r 1 * , r 2 * , r 3 * ) = ( 0.12 a , 0.19 a , 0.08 a ) , and a frequency gap of 17.8%.

Fig. 8
Fig. 8

Gap vs. refractive-index contrast nhi/nlo

Fig. 9
Fig. 9

Robust optimal designs for Diamond2, where u ˜ δ * solves the FA problem (4) with uncertainty δ in u, and u ˜ 0 * is the nominal optimum. Top curve: gap size g ( u ˜ δ * ) of FA structure versus δ, showing tradeoff between gap size and robustness to greater uncertainties. Middle curve: worst-case gap min d g ( u ˜ δ * + d ) of the FA structure plus errors, which degrades the gap size. Bottom curve: much greater degradation of gap size is found if we add uncertainty to the nominal (non-robust) structure u ˜ 0 * . (In this case, we show the mean gap size for random perturbations d, with the standard deviation shown as error bars; this is an upper bound on the worst-case degradation.)

Equations (4)

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k × ( 1 ε ( u ) k × H ) = ( ω c ) 2 H : = λ H ,
g ( u ) : = min k ω m + 1 ( ε ( u ) , k ) max k ω m ( ε ( u ) , k ) 1 2 ( min k ω m + 1 ( ε ( u ) , k ) + max k ω m ( ε ( u ) , k ) ) .
f ( u ) : = min k λ m + 1 ( ε ( u ) , k ) max k λ m ( ε ( u ) , k ) 1 2 ( min k λ m + 1 ( ε ( u ) , k ) + max k λ m ( ε ( u ) , k ) )
u ˜ δ * = arg max u [ 0 , 1 ] N u min u u 1 / N u δ , u [ 0 , 1 ] N u f ( u ) ,

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