Abstract

Inverse electromagnetic design has emerged as a way of efficiently designing active and passive electromagnetic devices. This maturing strategy involves optimizing the shape or topology of a device in order to improve a figure of merit–a process which typically requires that we compute the gradient of a figure of merit which describes device performance, potentially with respect to many design variables. In this paper, we introduce a new strategy based on smoothing abrupt material interfaces which enables us to efficiently compute these gradients with high accuracy irrespective of the resolution of the underlying simulation. This has advantages over previous approaches to shape and topology optimization in nanophotonics which are either prone to gradient errors or place important constraints on the shape of the device. As a demonstration of this new strategy, we optimize a non-adiabatic waveguide taper between a narrow and wide waveguide. This optimization leads to a non-intuitive design with a very low insertion loss of only 0.041 dB at 1550 nm.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (2)

2016 (3)

O. Sigmund, J. S. Jensen, and L. H. Frandsen, “On nanostructured silicon success,” Nat. Photon 10, 142–143 (2016).
[Crossref]

L. F. Frellsen, Y. Ding, O. Sigmund, and L. H. Frandsen, “Topology optimized mode multiplexing in silicon-on-insulator photonic wire waveguides,” Opt. Express 24, 16866–16873 (2016).
[Crossref] [PubMed]

W. Zhang, J. Yuan, J. Zhang, and X. Guo, “A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model,” Struct. Multidiscip. Optim. 53, 1243–1260 (2016).
[Crossref]

2015 (3)

P. Hansen and L. Hesselink, “Accurate adjoint design sensitivities for nano metal optics,” Opt. Express 23, 23899–23923 (2015).
[Crossref] [PubMed]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378 (2015).
[Crossref]

2014 (2)

2013 (3)

2012 (1)

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

2011 (1)

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

2009 (1)

2006 (2)

2005 (1)

B. Luyssaert, P. Bienstman, P. Vandersteegen, P. Dumon, and R. Baets, “Efficient Nonadiabatic Planar Waveguide Tapers,” J. Light. Technol. 23, 2462 (2005).
[Crossref]

2004 (5)

2003 (1)

M. H. Bakr and N. K. Nikolova, “An adjoint variable method for frequency domain TLM problems with conducting boundaries,” IEEE Microw. Wirel. Components Lett. 13, 408–410 (2003).
[Crossref]

2000 (1)

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Transactions on Microw. Theory Tech. 48, 2289–2296 (2000).
[Crossref]

1999 (1)

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Transactions on Microw. Theory Tech. 47, 1737–1739 (1999).
[Crossref]

1998 (1)

S. Dey and R. Mittra, “A modified locally conformal finite-difference time-domain algorithm for modeling three-dimensional perfectly conducting objects,” Microw. Opt. Technol. Lett. 17, 349–352 (1998).
[Crossref]

1997 (4)

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Transactions on Microw. Theory Tech. 45, 1645–1649 (1997).
[Crossref]

J. M. Johnson and V. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag. 39, 7–21 (1997).
[Crossref]

O. Sigmund, “On the Design of Compliant Mechanisms Using Topology Optimization,” Mech. Struct. Mach. 25, 493–524 (1997).
[Crossref]

H.-B. Lee and T. Itoh, “A systematic optimum design of waveguide-to-microstrip transition,” IEEE Transactions on Microw. Theory Tech. 45, 803–809 (1997).
[Crossref]

1992 (1)

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces [EM scattering],” IEEE Transactions on Antennas Propag. 40, 357–366 (1992).
[Crossref]

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]

1986 (1)

R. T. Haftka and R. V. Grandhi, “Structural shape optimization–A survey,” Comput. Methods Appl. Mech. Eng. 57, 91–106 (1986).
[Crossref]

Abrams, D.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” in International Society for Optics and Photonics, vol. 6283 (2006).

Acoleyen, K. V.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Allaire, G.

G. Allaire, F. Jouve, and A.-M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. 194, 363–393 (2004).
[Crossref]

Babinec, T. M.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

Baets, R.

G. Roelkens, D. V. Thourhout, and R. Baets, “High efficiency Silicon-on-Insulator grating coupler based on a poly-Silicon overlay,” Opt. Express 14, 11622–11630 (2006).
[Crossref] [PubMed]

B. Luyssaert, P. Bienstman, P. Vandersteegen, P. Dumon, and R. Baets, “Efficient Nonadiabatic Planar Waveguide Tapers,” J. Light. Technol. 23, 2462 (2005).
[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, D. V. Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref] [PubMed]

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Bakr, M. H.

E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “An adjoint variable method for sensitivity calculations of multiport devices,” IEEE Transactions on Microw. Theory Tech. 52589 (2004).
[Crossref]

M. H. Bakr and N. K. Nikolova, “An adjoint variable method for frequency domain TLM problems with conducting boundaries,” IEEE Microw. Wirel. Components Lett. 13, 408–410 (2003).
[Crossref]

Beckx, S.

Bendsøe, M. P.

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]

Bermel, P.

Bhargava, S.

Bienstman, P.

Bogaerts, W.

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, D. V. Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref] [PubMed]

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Borel, P. I.

Burr, G. W.

Campenhout, J. V.

Chen, C.-L.

C.-L. Chen, Foundations for Guided-Wave Optics (John Wiley & Sons, Inc., 2006), pp. 417–420.
[Crossref]

Cheon, C.

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Transactions on Microw. Theory Tech. 48, 2289–2296 (2000).
[Crossref]

Chu, T.

Y. Fu, T. Ye, W. Tang, and T. Chu, “Efficient adiabatic silicon-on-insulator waveguide taper,” Photonics Res. 2, A41–A44 (2014).
[Crossref]

Chung, Y.-S.

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Transactions on Microw. Theory Tech. 48, 2289–2296 (2000).
[Crossref]

De Cort, W.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

De Vos, K.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Debackere, P.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Dey, S.

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Transactions on Microw. Theory Tech. 47, 1737–1739 (1999).
[Crossref]

S. Dey and R. Mittra, “A modified locally conformal finite-difference time-domain algorithm for modeling three-dimensional perfectly conducting objects,” Microw. Opt. Technol. Lett. 17, 349–352 (1998).
[Crossref]

Ding, Y.

Dumon, P.

B. Luyssaert, P. Bienstman, P. Vandersteegen, P. Dumon, and R. Baets, “Efficient Nonadiabatic Planar Waveguide Tapers,” J. Light. Technol. 23, 2462 (2005).
[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, D. V. Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref] [PubMed]

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Dutton, R. W.

Elesin, Y.

L. H. Frandsen, Y. Elesin, L. F. Frellsen, M. Mitrovic, Y. Ding, O. Sigmund, and K. Yvind, “Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material,” Opt. Express 22, 8525–8532 (2014).
[Crossref] [PubMed]

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

Fan, S.

Farjadpour, A.

Frandsen, L. H.

Frellsen, L. F.

Fu, Y.

Y. Fu, T. Ye, W. Tang, and T. Chu, “Efficient adiabatic silicon-on-insulator waveguide taper,” Photonics Res. 2, A41–A44 (2014).
[Crossref]

Ghosh, S.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Grandhi, R. V.

R. T. Haftka and R. V. Grandhi, “Structural shape optimization–A survey,” Comput. Methods Appl. Mech. Eng. 57, 91–106 (1986).
[Crossref]

Guo, X.

W. Zhang, J. Yuan, J. Zhang, and X. Guo, “A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model,” Struct. Multidiscip. Optim. 53, 1243–1260 (2016).
[Crossref]

Haftka, R. T.

R. T. Haftka and R. V. Grandhi, “Structural shape optimization–A survey,” Comput. Methods Appl. Mech. Eng. 57, 91–106 (1986).
[Crossref]

Hahn, S.-Y.

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Transactions on Microw. Theory Tech. 48, 2289–2296 (2000).
[Crossref]

Hallynck, E.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Hansen, P.

Harpøth, A.

Hesselink, L.

Houshmand, B.

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Transactions on Microw. Theory Tech. 45, 1645–1649 (1997).
[Crossref]

Hughes, T.

Ibanescu, M.

Itoh, T.

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Transactions on Microw. Theory Tech. 45, 1645–1649 (1997).
[Crossref]

H.-B. Lee and T. Itoh, “A systematic optimum design of waveguide-to-microstrip transition,” IEEE Transactions on Microw. Theory Tech. 45, 803–809 (1997).
[Crossref]

Jensen, J.

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

Jensen, J. S.

O. Sigmund, J. S. Jensen, and L. H. Frandsen, “On nanostructured silicon success,” Nat. Photon 10, 142–143 (2016).
[Crossref]

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

P. I. Borel, A. Harpøth, 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]

Joannopoulos, J. D.

Johnson, J. M.

J. M. Johnson and V. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag. 39, 7–21 (1997).
[Crossref]

Johnson, S. G.

Jouve, F.

G. Allaire, F. Jouve, and A.-M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. 194, 363–393 (2004).
[Crossref]

Jurgens, T. G.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces [EM scattering],” IEEE Transactions on Antennas Propag. 40, 357–366 (1992).
[Crossref]

Kaneda, N.

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Transactions on Microw. Theory Tech. 45, 1645–1649 (1997).
[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]

Kottke, C.

Kristensen, M.

Lagoudakis, K. G.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

Lalau-Keraly, C. M.

Lazarov, B. S.

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

Lee, H.-B.

H.-B. Lee and T. Itoh, “A systematic optimum design of waveguide-to-microstrip transition,” IEEE Transactions on Microw. Theory Tech. 45, 803–809 (1997).
[Crossref]

Lin, Z.

Liu, Y.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” in International Society for Optics and Photonics, vol. 6283 (2006).

Loncar, M.

Lu, J.

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

J. Lu and J. Vuckovic, “Nanophotonic computational design,” Opt. Express 21, 13351–13367 (2013).
[Crossref] [PubMed]

Luyssaert, B.

Maute, K.

O. Sigmund and K. Maute, “Topology optimization approaches,” Struct. Multidiscip. Optim. 48, 1031–1055 (2013).
[Crossref]

Menon, R.

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378 (2015).
[Crossref]

Miller, O. D.

Mitrovic, M.

Mittra, R.

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Transactions on Microw. Theory Tech. 47, 1737–1739 (1999).
[Crossref]

S. Dey and R. Mittra, “A modified locally conformal finite-difference time-domain algorithm for modeling three-dimensional perfectly conducting objects,” Microw. Opt. Technol. Lett. 17, 349–352 (1998).
[Crossref]

Moore, T. G.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces [EM scattering],” IEEE Transactions on Antennas Propag. 40, 357–366 (1992).
[Crossref]

Nikolova, N. K.

E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “An adjoint variable method for sensitivity calculations of multiport devices,” IEEE Transactions on Microw. Theory Tech. 52589 (2004).
[Crossref]

M. H. Bakr and N. K. Nikolova, “An adjoint variable method for frequency domain TLM problems with conducting boundaries,” IEEE Microw. Wirel. Components Lett. 13, 408–410 (2003).
[Crossref]

Oskooi, A. F.

Pang, L.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” in International Society for Optics and Photonics, vol. 6283 (2006).

Park, I.-H.

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Transactions on Microw. Theory Tech. 48, 2289–2296 (2000).
[Crossref]

Petykiewicz, J.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Reports 7, 1786 (2017).
[Crossref]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

Piggott, A. Y.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Reports 7, 1786 (2017).
[Crossref]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

Polson, R.

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378 (2015).
[Crossref]

Preparata, F. P.

F. P. Preparata and M. Shamos, Computational Geometry: An Introduction, Monographs in Computer Science (Springer-Verlag, 1985).
[Crossref]

Rahmat-Samii, V.

J. M. Johnson and V. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag. 39, 7–21 (1997).
[Crossref]

Rodriguez, A.

Rodriguez, A. W.

Roelkens, G.

G. Roelkens, D. V. Thourhout, and R. Baets, “High efficiency Silicon-on-Insulator grating coupler based on a poly-Silicon overlay,” Opt. Express 14, 11622–11630 (2006).
[Crossref] [PubMed]

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Roundy, D.

Selvaraja, S.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Shamos, M.

F. P. Preparata and M. Shamos, Computational Geometry: An Introduction, Monographs in Computer Science (Springer-Verlag, 1985).
[Crossref]

Shen, B.

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378 (2015).
[Crossref]

Shi, P.

Shi, Y.

Sigmund, O.

L. F. Frellsen, Y. Ding, O. Sigmund, and L. H. Frandsen, “Topology optimized mode multiplexing in silicon-on-insulator photonic wire waveguides,” Opt. Express 24, 16866–16873 (2016).
[Crossref] [PubMed]

O. Sigmund, J. S. Jensen, and L. H. Frandsen, “On nanostructured silicon success,” Nat. Photon 10, 142–143 (2016).
[Crossref]

L. H. Frandsen, Y. Elesin, L. F. Frellsen, M. Mitrovic, Y. Ding, O. Sigmund, and K. Yvind, “Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material,” Opt. Express 22, 8525–8532 (2014).
[Crossref] [PubMed]

O. Sigmund and K. Maute, “Topology optimization approaches,” Struct. Multidiscip. Optim. 48, 1031–1055 (2013).
[Crossref]

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

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

P. I. Borel, A. Harpøth, 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, “On the Design of Compliant Mechanisms Using Topology Optimization,” Mech. Struct. Mach. 25, 493–524 (1997).
[Crossref]

Soliman, E. A.

E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “An adjoint variable method for sensitivity calculations of multiport devices,” IEEE Transactions on Microw. Theory Tech. 52589 (2004).
[Crossref]

Su, L.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Reports 7, 1786 (2017).
[Crossref]

Taflove, A.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces [EM scattering],” IEEE Transactions on Antennas Propag. 40, 357–366 (1992).
[Crossref]

Taillaert, D.

Tang, W.

Y. Fu, T. Ye, W. Tang, and T. Chu, “Efficient adiabatic silicon-on-insulator waveguide taper,” Photonics Res. 2, A41–A44 (2014).
[Crossref]

Thourhout, D. V.

Toader, A.-M.

G. Allaire, F. Jouve, and A.-M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. 194, 363–393 (2004).
[Crossref]

Umashankar, K.

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces [EM scattering],” IEEE Transactions on Antennas Propag. 40, 357–366 (1992).
[Crossref]

Van Thourhout, D.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Vandersteegen, P.

B. Luyssaert, P. Bienstman, P. Vandersteegen, P. Dumon, and R. Baets, “Efficient Nonadiabatic Planar Waveguide Tapers,” J. Light. Technol. 23, 2462 (2005).
[Crossref]

Vermeulen, D.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Veronis, G.

Vuckovic, J.

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Reports 7, 1786 (2017).
[Crossref]

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

J. Lu and J. Vuckovic, “Nanophotonic computational design,” Opt. Express 21, 13351–13367 (2013).
[Crossref] [PubMed]

Wang, J.

Wang, P.

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378 (2015).
[Crossref]

Wiaux, V.

Yablonovitch, E.

Ye, T.

Y. Fu, T. Ye, W. Tang, and T. Chu, “Efficient adiabatic silicon-on-insulator waveguide taper,” Photonics Res. 2, A41–A44 (2014).
[Crossref]

Yebo, N.

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

Yuan, J.

W. Zhang, J. Yuan, J. Zhang, and X. Guo, “A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model,” Struct. Multidiscip. Optim. 53, 1243–1260 (2016).
[Crossref]

Yvind, K.

Zhang, J.

W. Zhang, J. Yuan, J. Zhang, and X. Guo, “A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model,” Struct. Multidiscip. Optim. 53, 1243–1260 (2016).
[Crossref]

Zhang, W.

W. Zhang, J. Yuan, J. Zhang, and X. Guo, “A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model,” Struct. Multidiscip. Optim. 53, 1243–1260 (2016).
[Crossref]

Zhao, Z.

Comput. Methods Appl. Mech. Eng. (2)

R. T. Haftka and R. V. Grandhi, “Structural shape optimization–A survey,” Comput. Methods Appl. Mech. Eng. 57, 91–106 (1986).
[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]

IEEE Antennas Propag. Mag. (1)

J. M. Johnson and V. Rahmat-Samii, “Genetic algorithms in engineering electromagnetics,” IEEE Antennas Propag. Mag. 39, 7–21 (1997).
[Crossref]

IEEE Microw. Wirel. Components Lett. (1)

M. H. Bakr and N. K. Nikolova, “An adjoint variable method for frequency domain TLM problems with conducting boundaries,” IEEE Microw. Wirel. Components Lett. 13, 408–410 (2003).
[Crossref]

IEEE Transactions on Antennas Propag. (1)

T. G. Jurgens, A. Taflove, K. Umashankar, and T. G. Moore, “Finite-difference time-domain modeling of curved surfaces [EM scattering],” IEEE Transactions on Antennas Propag. 40, 357–366 (1992).
[Crossref]

IEEE Transactions on Microw. Theory Tech. (5)

E. A. Soliman, M. H. Bakr, and N. K. Nikolova, “An adjoint variable method for sensitivity calculations of multiport devices,” IEEE Transactions on Microw. Theory Tech. 52589 (2004).
[Crossref]

S. Dey and R. Mittra, “A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,” IEEE Transactions on Microw. Theory Tech. 47, 1737–1739 (1999).
[Crossref]

N. Kaneda, B. Houshmand, and T. Itoh, “FDTD analysis of dielectric resonators with curved surfaces,” IEEE Transactions on Microw. Theory Tech. 45, 1645–1649 (1997).
[Crossref]

H.-B. Lee and T. Itoh, “A systematic optimum design of waveguide-to-microstrip transition,” IEEE Transactions on Microw. Theory Tech. 45, 803–809 (1997).
[Crossref]

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using FDTD and design sensitivity analysis,” IEEE Transactions on Microw. Theory Tech. 48, 2289–2296 (2000).
[Crossref]

J. Comput. Phys. (1)

G. Allaire, F. Jouve, and A.-M. Toader, “Structural optimization using sensitivity analysis and a level-set method,” J. Comput. Phys. 194, 363–393 (2004).
[Crossref]

J. Light. Technol. (1)

B. Luyssaert, P. Bienstman, P. Vandersteegen, P. Dumon, and R. Baets, “Efficient Nonadiabatic Planar Waveguide Tapers,” J. Light. Technol. 23, 2462 (2005).
[Crossref]

Laser & Photon. Rev. (1)

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

Mech. Struct. Mach. (1)

O. Sigmund, “On the Design of Compliant Mechanisms Using Topology Optimization,” Mech. Struct. Mach. 25, 493–524 (1997).
[Crossref]

Microw. Opt. Technol. Lett. (1)

S. Dey and R. Mittra, “A modified locally conformal finite-difference time-domain algorithm for modeling three-dimensional perfectly conducting objects,” Microw. Opt. Technol. Lett. 17, 349–352 (1998).
[Crossref]

Nat. Photon (2)

A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photon 9, 374–377 (2015).
[Crossref]

O. Sigmund, J. S. Jensen, and L. H. Frandsen, “On nanostructured silicon success,” Nat. Photon 10, 142–143 (2016).
[Crossref]

Nat. Photonics (1)

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4 μm2 footprint,” Nat. Photonics 9, 378 (2015).
[Crossref]

Opt. Express (9)

L. F. Frellsen, Y. Ding, O. Sigmund, and L. H. Frandsen, “Topology optimized mode multiplexing in silicon-on-insulator photonic wire waveguides,” Opt. Express 24, 16866–16873 (2016).
[Crossref] [PubMed]

J. Wang, Y. Shi, T. Hughes, Z. Zhao, and S. Fan, “Adjoint-based optimization of active nanophotonic devices,” Opt. Express 26, 3236–3248 (2018).
[Crossref] [PubMed]

P. I. Borel, A. Harpøth, 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]

J. Lu and J. Vuckovic, “Nanophotonic computational design,” Opt. Express 21, 13351–13367 (2013).
[Crossref] [PubMed]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21, 21693–21701 (2013).
[Crossref] [PubMed]

L. H. Frandsen, Y. Elesin, L. F. Frellsen, M. Mitrovic, Y. Ding, O. Sigmund, and K. Yvind, “Topology optimized mode conversion in a photonic crystal waveguide fabricated in silicon-on-insulator material,” Opt. Express 22, 8525–8532 (2014).
[Crossref] [PubMed]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, D. V. Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).
[Crossref] [PubMed]

G. Roelkens, D. V. Thourhout, and R. Baets, “High efficiency Silicon-on-Insulator grating coupler based on a poly-Silicon overlay,” Opt. Express 14, 11622–11630 (2006).
[Crossref] [PubMed]

P. Hansen and L. Hesselink, “Accurate adjoint design sensitivities for nano metal optics,” Opt. Express 23, 23899–23923 (2015).
[Crossref] [PubMed]

Opt. Lett. (4)

Photonics Nanostructures - Fundamentals Appl. (1)

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

Photonics Res. (1)

Y. Fu, T. Ye, W. Tang, and T. Chu, “Efficient adiabatic silicon-on-insulator waveguide taper,” Photonics Res. 2, A41–A44 (2014).
[Crossref]

Sci. Reports (1)

A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Reports 7, 1786 (2017).
[Crossref]

Struct. Multidiscip. Optim. (2)

O. Sigmund and K. Maute, “Topology optimization approaches,” Struct. Multidiscip. Optim. 48, 1031–1055 (2013).
[Crossref]

W. Zhang, J. Yuan, J. Zhang, and X. Guo, “A new topology optimization approach based on moving morphable components (mmc) and the ersatz material model,” Struct. Multidiscip. Optim. 53, 1243–1260 (2016).
[Crossref]

Other (5)

C.-L. Chen, Foundations for Guided-Wave Optics (John Wiley & Sons, Inc., 2006), pp. 417–420.
[Crossref]

A. Michaels, “Emopt: Electromagnetic optimization toolbox,” https://github.com/anstmichaels/emopt (2018).

D. Vermeulen, K. V. Acoleyen, S. Ghosh, S. Selvaraja, W. De Cort, N. Yebo, E. Hallynck, K. De Vos, P. Debackere, P. Dumon, W. Bogaerts, G. Roelkens, D. Van Thourhout, and R. Baets, “Efficient tapering to the fundamental quasi-tm mode in asymmetrical waveguides,” in ECIO, (2010), p. paper WeP16.

F. P. Preparata and M. Shamos, Computational Geometry: An Introduction, Monographs in Computer Science (Springer-Verlag, 1985).
[Crossref]

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): What is the impact to the photomask industry?” in International Society for Optics and Photonics, vol. 6283 (2006).

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Figures (10)

Fig. 1
Fig. 1 Comparison of different strategies for representing abrupt material boundaries on a rectangular grid. On the far left, the desired structure is shown (a circle). The second image depicts a grayscale permittivity distribution which yields an approximation of the circle when a threshold is applied. The third image depicts a strictly binary grid (“Manhattan”) representation of the circle. The final picture depicts the circle represented using our own boundary smoothing method which best captures the circular boundary.
Fig. 2
Fig. 2 Visual depiction of boundary smoothing process. Internally, all material boundaries of the system are represented using polygons which are defined in a continuous domain. These shapes are then mapped onto a rectangular grid by computing the average value of permittivities and permeabilities which overlap with each cell in the grid. This mapping is achieved by computing the overlap area between grid cells and material domains.
Fig. 3
Fig. 3 Demonstration of boundary smoothing for a 0.5 μm diameter dielectric circle. (a) shows the relative permittivity of the smoothed grid computed for the circle on an intentionally coarse grid in order to clearly show the averaging which occurs at the circle’s boundary. (b) shows the difference in permittivity between the grid shown in (a) and the grid corresponding to the same circle which has been shifted in the y direction by 10−12 μm. The difference in permittivity is correspondingly small, highlighting the continuous nature of our boundary smoothing.
Fig. 4
Fig. 4 Demonstration of boundary smoothing applied to the calculation of the scattering cross section of an infinite dielectric cylinder (in 2D). The scattering cross sections computed using boundary smoothing (blue curve) and without using boundary smoothing (red curve) are compared to the theoretical value for a range of cylinder radii. On the left, the cylinder radius is varied by an amount equal to twice the grid spacing (30 nm). On the right, the radius is varied over a range much smaller than a single grid step. In both cases the cross-section of the grid-smoothed-cylinder evolves smoothly with changing radius while cross section of the cylinder represented on a strictly binary grid exhibits a step-like behavior. The smooth change in the cross section of the boundary-smoothed-cylinder is highly desirable when calculating sensitivities.
Fig. 5
Fig. 5 A short 18 μm taper from a 500 nm silicon waveguide to a 9 μm wide silicon waveguide. The taper is defined as a single polygon containing 200 vertices along its top and bottom diagonal edges. We choose the displacement of the x and y coordinates of these points (labeled δx and δy) as the design parameters of the system. This taper is used to validate the accuracy of gradients computed using boundary smoothing in conjunction with the adjoint method and serves as the starting point of an optimization to demonstrate application of these methods.
Fig. 6
Fig. 6 Accuracy of the gradient of the figure of merit computed using the adjoint method and boundary smoothing as a function of the size of the perturbation Δp used in the calculation (normalized to the wavelength λ) for a variety of different simulation resolutions. The error in this gradient is defined with respect to the “brute force” gradient which is computed by sequentially perturbing each design variable of the system by a very small amount, running a new simulation, and then computing the new figure of merit. Greater error is incurred as grid cells further from the original boundary of the shape contribute to the change in permittivity. The error saturates, meanwhile, when the perturbation is small enough such that only grid cells which straddle the boundary contribute to the derivative.
Fig. 7
Fig. 7 The initial structure overlaid with the real part of z-component of the magnetic field, Hz. Notice that the abrupt transition from the input to output waveguide results in significant curvature of the propagating wavefronts. This results in significant coupling to higher order modes in the output waveguide, and the fraction of power propagating in the fundamental mode of the output waveguide is only 51%.
Fig. 8
Fig. 8 The final optimized structure overlaid with the real part of Hz. The optimized structure is very successful in coupling the input power into the fundamental mode of the output waveguide, and no curving of the outgoing wavefronts is visible. This is reflected in the final efficiency of the structure of over 99% at the design wavelength.
Fig. 9
Fig. 9 Slices of the fields shown in Fig. 7 and Fig. 8 taken along a vertical line at the right edge of the simulation domain (where the figure of merit is computed). The top row shows the magnitude of Hz for the field at the beginning of the first iteration (left) and the field after the 97th iteration (right). The bottom row, meanwhile, shows the phase of Hz for the first (left) and 97th (right) iteration. The fields of the initial structure deviate significantly from the desired fields both in terms of amplitude and phase. The fields of the optimize solution, meanwhile, match very closely to the desired fundamental mode, with its phase deviating only at the edges where the amplitude of the field is exceptionally low (and thus incurring minimal loss).
Fig. 10
Fig. 10 (a) Plots of the final figure of merit versus wavelength and (b) the figure of merit versus iteration of the optimization. The optimization achieves a figure of merit of over 90% (−0.46 dB) in under 20 iterations, eventually reaching a final efficiency of just over 99% (−0.041 dB) at the design wavelength. Although the optimization is only performed for a single wavelength, the final structure shows excellent broadband performance with a −3 dB bandwidth of approximately 420 nm. The final structure can thus be used in conjunction with other components such as grating couplers without significantly reducing the overall bandwidth of the combined component.

Equations (36)

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× E i ω μ H = M
× H + i ω ε E = J
A x = b
A = ( i ω ε ( r ) × × i ω μ ( r ) ) , x = ( E H ) , b = ( J M )
p F = [ F p 1 , F p 2 , , F p M ]
F p i = 2 Re { F x x p i }
p i ( A x ) = b p i A x p i = b p i A p i x x p i = A 1 ( b p i A p i x ) .
x p i = A 1 A p i x .
F p i = 2 Re { F x A 1 A p i x }
y T = F x A 1
A T y = ( F x ) T
F p i = 2 Re { y T A p i x } .
F p k = 2 Re { j i ω ε j p k E j E j adj j i ω μ j p k H j H j adj } = 2 ω Im { j ε j p k E j E j adj j μ j p k H j H j adj }
ε ( i , j ) = 1 Δ x Δ y k C k ε k ( i , j )
A p i A ( p i + Δ p ) A ( p i ) Δ p
η = 1 4 P m P src | A d A E × H m * | 2
F ( E , H , p ) = η ( E , H ) f ROC ( p )
Error in F = | F FD F AM | | F FD |
E = E fwd + E back
= m ( a m e i k m z + b m e i k m z ) E m
E fwd = m a m e i k m z E m
E back = m b m e i k m z E m .
H = H fwd + H back
= m ( a m e i k m z b m e i k m z ) H m
A d A E m × H n * A d A E m × H m * = δ m n
a m e i k m z + b m e i k m z = A d A E × H m * S m
S m = A d A E m × H m * .
a m e i k m z b m e i k m z = A d A E m * × H S m * .
a m = 1 2 e i k m z ( A d A E × H m * S m + A d A E m * × H S m * )
η m fwd = P m P in = | a m | 2 Re { S m } Re { A d A E × H * } .
A d A E × H m * S m = A d A E m * × H S m *
η m = 1 2 Re { A d A E m × H m * } | A d A E × H m * | 2 1 2 Re { A d A E × H * } | A d A E m × H m * | 2
η m , guided = 1 4 P m P in | A d A E × H m * | 2
P m = 1 2 Re { A d A E m × H m * } P in = 1 2 Re { A d A E × H * }
R i = ( x 2 + y 2 ) 3 / 2 | x y y x |
f ROC ( p ) = α i 1 1 + exp [ κ ( R 0 R i ( p ) ) ]

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