Abstract

Here, we introduce an efficient method to perform pixel-by-pixel optimization of photonic devices, based on the Dyson equation in the Green’s function formalism. Unlike continuous optimization techniques widely used for photonic device optimization, pixel-by-pixel optimization automatically results in structures consisting of a discrete set of permittivities, and it incorporates the constraint of a minimum feature size throughout the optimization process. Thus pixel-by-pixel optimizations automatically result in structures amenable to lithographic nanofabrication. We show that the use of a Green’s function formalism enables an efficient pixel-by-pixel update of the structure, where each update is guaranteed to improve the performance of the structure.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Inverse design of digital nanophotonic devices using the adjoint method

Kaiyuan Wang, Xinshu Ren, Weijie Chang, Longhui Lu, Deming Liu, and Minming Zhang
Photon. Res. 8(4) 528-533 (2020)

References

  • View by:
  • |
  • |
  • |

  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, 378–382 (2015).
    [Crossref]
  2. A. Y. Piggott, J. Lu, K. G. Lagoudakis, J. Petykiewicz, T. M. Babinec, and J. Vučković, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9, 374–377 (2015).
    [Crossref]
  3. J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20, 7221–7236 (2012).
    [Crossref]
  4. D. Pavesi, Silicon Fundamentals for Photonics Applications. Silicon photonics (2004).
  5. M. Soltani, S. Yegnanarayanan, and A. Adibi, “Ultra-high Q planar silicon microdisk resonators for chip-scale silicon photonics,” Opt. Express 15, 4694–4704 (2007).
    [Crossref]
  6. P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett. 34, 2760–2762 (2009).
    [Crossref]
  7. M. P. Bendsøe and O. Sigmund, Topology Optimization Theory, Methods and Applications (Springer, 2003).
  8. N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
    [Crossref]
  9. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. 29, 2288–2290 (2004).
    [Crossref]
  10. 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]
  11. V. Liu, Y. Jiao, D. A. Miller, and S. Fan, “Design methodology for compact photonic-crystal-based wavelength division multiplexers,” Opt. Lett. 36, 591–593 (2011).
    [Crossref]
  12. O. D. Miller, “Photonic design: from fundamental solar cell physics to computational inverse design,” arXiv:1308.0212 (2013).
  13. A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučković, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7, 1786(2017).
    [Crossref]
  14. S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
    [Crossref]
  15. X. Guo, W. Zhang, and W. Zhong, “Explicit feature control in structural topology optimization via level set method,” Comput. Methods Appl. Mech. Engrg. 272, 354–378 (2014).
    [Crossref]
  16. P. D. Dunning and H. A. Kim, “A new hole insertion method for level set based structural topology optimization,” Int. J. Numer. Methods Eng. 93(1), 118–134 (2013).
    [Crossref]
  17. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [Crossref]
  18. E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer-Verlag, 1990).
  19. E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
    [Crossref]
  20. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
    [Crossref]
  21. O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909 (1998).
    [Crossref]
  22. O. J. Martin, A. Dereux, and C. Girard, “Iterative scheme for computing exactly the total field propagating in dielectric structures of arbitrary shape,” J. Opt. Soc. Am. A 11, 1073–1080 (1994).
    [Crossref]
  23. S. Boutami and S. Fan, “Efficient pixel-by-pixel optimization of photonic devices utilizing the Dyson’s equation in a Green function formalism: Part II. implementation using standard electromagnetic solvers,” J. Opt. Soc. Am. B 36, 2387–2394 (2019).
    [Crossref]
  24. W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency–domain Maxwell’s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
    [Crossref]
  25. S. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12, 1267–1272 (1995).
    [Crossref]
  26. G. F. Oster, N. Shubin, J. D. Murray, and P. Alberch, “Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny,” Evolution 42, 862–884 (1988).
    [Crossref]

2019 (1)

2017 (1)

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

2015 (2)

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4  μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

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

2014 (1)

X. Guo, W. Zhang, and W. Zhong, “Explicit feature control in structural topology optimization via level set method,” Comput. Methods Appl. Mech. Engrg. 272, 354–378 (2014).
[Crossref]

2013 (2)

P. D. Dunning and H. A. Kim, “A new hole insertion method for level set based structural topology optimization,” Int. J. Numer. Methods Eng. 93(1), 118–134 (2013).
[Crossref]

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]

2012 (2)

J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20, 7221–7236 (2012).
[Crossref]

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency–domain Maxwell’s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
[Crossref]

2011 (1)

2009 (1)

2007 (1)

2004 (1)

2002 (1)

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

1998 (1)

O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909 (1998).
[Crossref]

1995 (1)

1994 (1)

1988 (3)

G. F. Oster, N. Shubin, J. D. Murray, and P. Alberch, “Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny,” Evolution 42, 862–884 (1988).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[Crossref]

1987 (1)

1973 (1)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
[Crossref]

Adibi, A.

Alberch, P.

G. F. Oster, N. Shubin, J. D. Murray, and P. Alberch, “Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny,” Evolution 42, 862–884 (1988).
[Crossref]

Allebach, J. P.

Babinec, T. M.

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

Bakr, M. H.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Bandler, J. W.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Bendsøe, M. P.

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

Bhargava, S.

Boutami, S.

Brimont, A.

Chen, J. C.

Cuesta, F.

Dereux, A.

Devenyi, A.

Draine, B. T.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

Dunning, P. D.

P. D. Dunning and H. A. Kim, “A new hole insertion method for level set based structural topology optimization,” Int. J. Numer. Methods Eng. 93(1), 118–134 (2013).
[Crossref]

Dutton, R. W.

Economou, E. N.

E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer-Verlag, 1990).

Fan, S.

Galán, J. V.

Georgieva, N. K.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Girard, C.

Glavic, S.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Griol, A.

Guo, X.

X. Guo, W. Zhang, and W. Zhong, “Explicit feature control in structural topology optimization via level set method,” Comput. Methods Appl. Mech. Engrg. 272, 354–378 (2014).
[Crossref]

Håkansson, A.

Jiao, Y.

Joannopoulos, J. D.

Kim, H. A.

P. D. Dunning and H. A. Kim, “A new hole insertion method for level set based structural topology optimization,” Int. J. Numer. Methods Eng. 93(1), 118–134 (2013).
[Crossref]

Lagoudakis, K. G.

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

Lalau-Keraly, C. M.

Liu, V.

Lu, J.

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

J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20, 7221–7236 (2012).
[Crossref]

Martí, J.

Martin, O. J.

Meade, R. D.

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, 378–382 (2015).
[Crossref]

Miller, D. A.

Miller, O. D.

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]

O. D. Miller, “Photonic design: from fundamental solar cell physics to computational inverse design,” arXiv:1308.0212 (2013).

Murray, J. D.

G. F. Oster, N. Shubin, J. D. Murray, and P. Alberch, “Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny,” Evolution 42, 862–884 (1988).
[Crossref]

Osher, S.

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[Crossref]

Oster, G. F.

G. F. Oster, N. Shubin, J. D. Murray, and P. Alberch, “Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny,” Evolution 42, 862–884 (1988).
[Crossref]

Pavesi, D.

D. Pavesi, Silicon Fundamentals for Photonics Applications. Silicon photonics (2004).

Pennypacker, C. R.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
[Crossref]

Petykiewicz, J.

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

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

Piggott, A. Y.

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

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

Piller, N. B.

O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909 (1998).
[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, 378–382 (2015).
[Crossref]

Purcell, E. M.

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
[Crossref]

Sanchis, P.

Seldowitz, M. A.

Sethian, J. A.

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[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, 378–382 (2015).
[Crossref]

Shin, W.

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency–domain Maxwell’s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
[Crossref]

Shubin, N.

G. F. Oster, N. Shubin, J. D. Murray, and P. Alberch, “Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny,” Evolution 42, 862–884 (1988).
[Crossref]

Sigmund, O.

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

Soltani, M.

Su, L.

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

Sweeney, D. W.

Veronis, G.

Villalba, P.

Vuckovic, J.

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

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

J. Lu and J. Vučković, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20, 7221–7236 (2012).
[Crossref]

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, 378–382 (2015).
[Crossref]

Winn, J. N.

Yablonovitch, E.

Yegnanarayanan, S.

Zhang, W.

X. Guo, W. Zhang, and W. Zhong, “Explicit feature control in structural topology optimization via level set method,” Comput. Methods Appl. Mech. Engrg. 272, 354–378 (2014).
[Crossref]

Zhong, W.

X. Guo, W. Zhang, and W. Zhong, “Explicit feature control in structural topology optimization via level set method,” Comput. Methods Appl. Mech. Engrg. 272, 354–378 (2014).
[Crossref]

Appl. Opt. (1)

Astrophys. J. (2)

E. M. Purcell and C. R. Pennypacker, “Scattering and absorption of light by nonspherical dielectric grains,” Astrophys. J. 186, 705 (1973).
[Crossref]

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[Crossref]

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

X. Guo, W. Zhang, and W. Zhong, “Explicit feature control in structural topology optimization via level set method,” Comput. Methods Appl. Mech. Engrg. 272, 354–378 (2014).
[Crossref]

Evolution (1)

G. F. Oster, N. Shubin, J. D. Murray, and P. Alberch, “Evolution and morphogenetic rules: the shape of the vertebrate limb in ontogeny and phylogeny,” Evolution 42, 862–884 (1988).
[Crossref]

IEEE Trans. Microwave Theory Tech. (1)

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microwave Theory Tech. 50, 2751–2758 (2002).
[Crossref]

Int. J. Numer. Methods Eng. (1)

P. D. Dunning and H. A. Kim, “A new hole insertion method for level set based structural topology optimization,” Int. J. Numer. Methods Eng. 93(1), 118–134 (2013).
[Crossref]

J. Comput. Phys. (2)

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[Crossref]

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency–domain Maxwell’s equations solvers,” J. Comput. Phys. 231, 3406–3431 (2012).
[Crossref]

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

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

Nat. Photonics (2)

B. Shen, P. Wang, R. Polson, and R. Menon, “An integrated-nanophotonics polarization beamsplitter with 2.4 × 2.4  μm2 footprint,” Nat. Photonics 9, 378–382 (2015).
[Crossref]

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

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. E (1)

O. J. Martin and N. B. Piller, “Electromagnetic scattering in polarizable backgrounds,” Phys. Rev. E 58, 3909 (1998).
[Crossref]

Sci. Rep. (1)

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

Other (4)

O. D. Miller, “Photonic design: from fundamental solar cell physics to computational inverse design,” arXiv:1308.0212 (2013).

D. Pavesi, Silicon Fundamentals for Photonics Applications. Silicon photonics (2004).

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

E. N. Economou, Green’s Functions in Quantum Physics, 2nd ed. (Springer-Verlag, 1990).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1. Concentration of light intensity at a target point. The optimization aims at maximizing light intensity at a target point (the black dot), separated from the source (the red dot) by a distance of one wavelength. The wavelength is λ = 1.55 μm . We consider a scalar electric field perpendicular to the plane. The permittivities allowed for the binary system are respectively 2.25 and 6.25. The space for optimization has a square size with a dimension of 2 λ × 2 λ , divided into 50 nm × 50 nm square pixels.
Fig. 2.
Fig. 2. Performance of various continuous optimization techniques for the light concentration setup in Fig. 1. (a) FOM in a log10 scale as a function of the iteration number; black curve: pixel-by-pixel binary-adjoint optimization guided by gradient information, starting from silica background; green curve: level-set optimization of a binary structure from a random binary configuration; blue and red curves: continuous optimization followed by level-set optimization starting either from silica background or a random starting point, respectively. The constant maximal permittivity change in the domain in each iteration is constrained to be 0.25. The crosses on the curve show the end of continuous optimization. (b) Corresponding to each case in (a), the left column shows the initial structure. The middle column shows the final structure with the corresponding FOM in a log10 scale. The right column shows the electric field intensity distribution for the final structure, in a log10 scale. (c) FOM in a log10 scale as a function of the iteration number. Both curves correspond to a continuous adaptive optimization process, starting from the same initial conditions as continuous optimization of (a), but with a maximal permittivity change between iterations continuously reduced to guarantee a monotonic ascent of the FOM during the optimization process. (d) Corresponding to the two cases in (c), the left column shows the structure at the end of the continuous optimization process. The middle column shows the structure after a thresholding process to create a binary structure. In both columns, the corresponding FOM in a log10 scale for each structure is also shown. The right column shows the electric field intensity distributions in a log10 scale for the final binary structures. Each iteration takes approximately 0.1 s on an Intel core i5-7200u CPU at 2.50 Ghz, RAM 8 GB.
Fig. 3.
Fig. 3. Performance of our Green-function-based optimization techniques for the light concentration setup in Fig. 1. (a) FOM in a log10 scale as a function of the iteration number. The solid lines are from our optimization techniques, starting from a uniform silica design region. Each iteration takes 0.15 s on an Intel(r) core(tm) i5-7200u CPU at 2.50 ghz, RAM 8 GB. Solid black line: the optimization process starts from the first iteration. Other solid color lines: the first 250, 500, 750, or 1000 iterations involve randomly adding high index pixels. The optimization process starts afterwards. The dashed dotted lines are from adaptive continuous optimization included for comparison purposes. (b) Evolution of structure and the corresponding electric field intensity profiles during the optimization process. The structures are taken at the iterations numbers 336, 500, 1250, 2308 along the black line in (a). The FOM in a log10 scale of the final structure is also indicated. (c) The electric field intensity distribution of the final structure shown in (b), as calculated using the FDFD method. (d) The top and bottom rows show the initial and final structures, and the corresponding electric field intensity distribution shown in a log10 scale for the four cases as presented in Fig. 1(a), where the initial structures consist of randomly added pixels. The FOM in a log10 scale of the final structure is also shown.
Fig. 4.
Fig. 4. Mode and frequency splitter. Light coming from the fundamental mode of a 1-μm-wide multimode waveguide is sent into either of the 0.3-μm-wide single-mode waveguides depending on its wavelength. Light with the wavelength λ 1 = 1.31 μm is sent to the left waveguide, and light with the wavelength λ 2 = 1.55 μm is sent to the right waveguide.
Fig. 5.
Fig. 5. Performance of our optimization techniques for the splitter setup in Fig. 4. (a), FOM as a function of iteration number. The red curves are from our approach. The blue curves are from the use of a continuous optimization followed by a level set optimization (FDFD). The design region in the starting structure is empty. (b) The top row shows the initial structure, the final structure, as well as the electric field intensity distribution at the two wavelengths of λ 1 = 1.31 μm and λ 2 = 1.55 μm , respectively, with the corresponding power-coupling efficiencies at each wavelength. The bottom row is the same as the top row, except with our method. (c) Same as (a), for a starting structure with a filled design region. (d) Same as b, for starting structure with a filled design region. Green’s function initialization (FDFD Matrix inversion) requires 130 s per λ , then each iteration time is 0.1 s per λ with our method, as for FDFD (the total domain is similar as previous example, but the design space where Green’s function is updated in our method is reduced), on an Intel core i5-7200u CPU @2.50 Ghz, RAM 8 GB.
Fig. 6.
Fig. 6. Optimization under the constraint of minimum feature sizes, using the example shown in Fig. 1. (a) Optimized structure and the electric field intensity distribution, for a minimum feature size of 50 nm. (b) Same as (a), for a minimum feature size of 100 nm. (c) Same as (a), for a minimum feature size of 150 nm. (d) FOM as a function of iteration number, using our method, for the three cases shown in (a)–(c). The times taken per iteration are 0.15 s, 0.75 s, 1.25 s for the minimum feature sizes of 50 nm, 100 nm, and 150 nm, respectively.

Equations (15)

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

FOM = F ( E ( r 0 ) ) ,
Δ FOM = F ( E ( new ) ( r 0 ) ) F ( E ( old ) ( r 0 ) ) ,
E ( new ) ( r ) = E ( old ) ( r ) + V k 0 2 G ( old ) ( r , r ) Δ ε ( r ) E ( new ) ( r ) d r ,
Δ FOM = F ( E ( old ) ( r 0 ) + k 0 2 V G ( old ) ( r 0 , r ) Δ ε ( r ) E ( new ) ( r ) ) F ( E ( old ) ( r 0 ) ) ,
E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( old ) ( r , r ) Δ ε ( r ) E ( new ) ( r ) .
E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( old ) ( r , r ) Δ ε ( r ) E ( new ) ( r ) .
G ( new ) ( r , r ) = G ( old ) ( r , r ) + k 0 2 V G ( old ) ( r , r ) Δ ε ( r ) G ( new ) ( r , r ) .
G ( new ) ( r , r ) = G ( old ) ( r , r ) + k 0 2 V G ( old ) ( r , r ) Δ ε ( r ) G ( new ) ( r , r ) .
FOM = | S ( E × H m * + E m * × H ) · d S | 2 ,
FOM = F ( E | Ω , E m ) = | Ω ( E · O m ^ E m + E m * · O ^ E ) · d Ω | 2 | η ( E | Ω , E m ) | 2 ,
Δ FOM = F ( E ( old ) + k 0 2 V G ( old ) ( r , r 0 ) T Δ ε ( r ) E ( new ) ( r ) | Ω , E m ) F ( E ( old ) | Ω , E m ) ,
G ( old ) ( r 0 , r ) = G ( old ) ( r , r 0 ) T .
Δ FOM = 2 k 0 2 V Re ( E adj ( old ) ( r ) · Δ ε ( r ) E ( new ) ( r ) ) ,
E adj ( old ) ( r ) · E ( new ) ( r ) = η ( E ( old ) | Ω , E m ) * η ( G ( old ) ( r , r 0 ) T E ( new ) ( r ) | Ω , E m ) .
FOM = 1 2 i = 1 2 | S out i ( E × H out i * + E out i * × H ) · d S | 2 4 [ S in Re ( E in i × H in i * ) · d S ] [ S out i Re ( E out i × H out i * ) · d S ] .

Metrics