Abstract

A high-efficiency inverse design of “digital” subwavelength nanophotonic devices using the adjoint method is proposed. We design a single-mode 3 dB power divider and a dual-mode demultiplexer to demonstrate the efficiency of the proposed inverse design approach, called the digitized adjoint method, for single- and dual-object optimization, respectively. The optimization comprises three stages: 1) continuous variation for an “analog” pattern; 2) forced permittivity biasing for a “quasi-digital” pattern; and 3) a multilevel digital pattern. Compared with the conventional brute-force method, the proposed method can improve design efficiency by about five times, and the performance optimization can reach approximately the same level. The method takes advantages of adjoint sensitivity analysis and digital subwavelength structure and creates a new way for the efficient and high-performance design of compact digital subwavelength nanophotonic devices, which could overcome the efficiency bottleneck of the brute-force method, which is restricted by the number of pixels of a digital pattern, and improve the device performance by extending a conventional binary pattern to a multilevel one.

© 2020 Chinese Laser Press

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References

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    [Crossref]
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    [Crossref]
  3. L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2017).
    [Crossref]
  4. D. Vercruysse, N. V. Sapra, L. Su, R. Trivedi, and J. Vučković, “Analytical level set fabrication constraints for inverse design,” Sci. Rep. 9, 8999 (2019).
    [Crossref]
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    [Crossref]
  8. A. Y. Piggott, J. Petykiewicz, L. Su, and J. Vučković, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7, 1786 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
  11. Z. Yu, H. Cui, and X. Sun, “Genetic-algorithm-optimized wideband on-chip polarization rotator with an ultrasmall footprint,” Opt. Lett. 42, 3093–3096 (2017).
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (7)

D. Vercruysse, N. V. Sapra, L. Su, R. Trivedi, and J. Vučković, “Analytical level set fabrication constraints for inverse design,” Sci. Rep. 9, 8999 (2019).
[Crossref]

R. E. Christiansena, J. Vester-Petersenb, S. P. Madsenb, and O. Sigmund, “A non-linear material interpolation for design of metallic nano-particles using topology optimization,” Comput. Methods Appl. Mech. Engrg. 343, 23–29 (2019).
[Crossref]

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simul. 2, 1900088 (2019).
[Crossref]

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge discovery in nanophotonics using geometric deep learning,” Adv. Intell. Syst. 1, 1900132 (2019).
[Crossref]

A. M. Hammond and R. M. Camacho, “Designing integrated photonic devices using artificial neural networks,” Opt. Express 27, 29620–29628 (2019).
[Crossref]

S. Chugh, S. Ghosh, A. Gulistan, and B. M. A. Rahman, “Machine learning regression approach to the nanophotonic waveguide analyses,” J. Lightwave Technol. 37, 6080–6089 (2019).
[Crossref]

2018 (3)

Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
[Crossref]

W. Chang, L. Lu, X. Ren, D. Li, Z. Pan, M. Cheng, D. Liu, and M. Zhang, “Ultra-compact mode (de)multiplexer based on subwavelength asymmetric Y-junction,” Opt. Express 26, 8162–8170 (2018).
[Crossref]

H. Jia, T. Zhou, X. Fu, J. Ding, and L. Yang, “Inverse-design and demonstration of ultracompact silicon meta-structure mode exchange device,” ACS Photon. 5, 1833–1838 (2018).
[Crossref]

2017 (4)

L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2017).
[Crossref]

Z. Yu, H. Cui, and X. Sun, “Genetic-algorithm-optimized wideband on-chip polarization rotator with an ultrasmall footprint,” Opt. Lett. 42, 3093–3096 (2017).
[Crossref]

K. Xu, L. Liu, X. Wen, W. Sun, N. Zhang, N. Yi, S. Sun, S. Xiao, and Q. Song, “Integrated photonic power divider with arbitrary power ratios,” Opt. Lett. 42, 855–858 (2017).
[Crossref]

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

2016 (3)

2015 (3)

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius <λ0/2,” Opt. Lett. 40, 5750–5753 (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, 378–382 (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]

2013 (2)

2011 (1)

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

Abdollahramezani, S.

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge discovery in nanophotonics using geometric deep learning,” Adv. Intell. Syst. 1, 1900132 (2019).
[Crossref]

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simul. 2, 1900088 (2019).
[Crossref]

Adibi, A.

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simul. 2, 1900088 (2019).
[Crossref]

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge discovery in nanophotonics using geometric deep learning,” Adv. Intell. Syst. 1, 1900132 (2019).
[Crossref]

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]

Bhargava, S.

Camacho, R. M.

Chang, W.

Cheng, M.

Christiansena, R. E.

R. E. Christiansena, J. Vester-Petersenb, S. P. Madsenb, and O. Sigmund, “A non-linear material interpolation for design of metallic nano-particles using topology optimization,” Comput. Methods Appl. Mech. Engrg. 343, 23–29 (2019).
[Crossref]

Chugh, S.

Cui, H.

Deng, L.

Deng, Y.

Y. Deng and J. G. Korvink, “Topology optimization for three-dimensional electromagnetic waves using an edge element-based finite-element method,” Proc. R. Soc. A 472, 20150835 (2016).
[Crossref]

Ding, J.

H. Jia, T. Zhou, X. Fu, J. Ding, and L. Yang, “Inverse-design and demonstration of ultracompact silicon meta-structure mode exchange device,” ACS Photon. 5, 1833–1838 (2018).
[Crossref]

Ding, Y.

Du, L.

Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
[Crossref]

Frandsen, L. H.

Frellsen, L. F.

Fu, S.

Fu, X.

H. Jia, T. Zhou, X. Fu, J. Ding, and L. Yang, “Inverse-design and demonstration of ultracompact silicon meta-structure mode exchange device,” ACS Photon. 5, 1833–1838 (2018).
[Crossref]

Ghosh, S.

Gulistan, A.

Hammond, A. M.

Hemmatyar, O.

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge discovery in nanophotonics using geometric deep learning,” Adv. Intell. Syst. 1, 1900132 (2019).
[Crossref]

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simul. 2, 1900088 (2019).
[Crossref]

Jensen, J. S.

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

Jha, D.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Jia, H.

H. Jia, T. Zhou, X. Fu, J. Ding, and L. Yang, “Inverse-design and demonstration of ultracompact silicon meta-structure mode exchange device,” ACS Photon. 5, 1833–1838 (2018).
[Crossref]

Kiarashinejad, Y.

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge discovery in nanophotonics using geometric deep learning,” Adv. Intell. Syst. 1, 1900132 (2019).
[Crossref]

Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simul. 2, 1900088 (2019).
[Crossref]

Koike-Akino, T.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Kojima, K.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Korvink, J. G.

Y. Deng and J. G. Korvink, “Topology optimization for three-dimensional electromagnetic waves using an edge element-based finite-element method,” Proc. R. Soc. A 472, 20150835 (2016).
[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.

Lei, T.

Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
[Crossref]

Li, D.

Li, F.

Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
[Crossref]

Li, Z.

Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
[Crossref]

Lin, C.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Liu, D.

Liu, L.

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, “Nanophotonic computational design,” Opt. Express 21, 13351–13367 (2013).
[Crossref]

Lu, L.

Madsenb, S. P.

R. E. Christiansena, J. Vester-Petersenb, S. P. Madsenb, and O. Sigmund, “A non-linear material interpolation for design of metallic nano-particles using topology optimization,” Comput. Methods Appl. Mech. Engrg. 343, 23–29 (2019).
[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, 378–382 (2015).
[Crossref]

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius <λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]

Miller, O. D.

Min, C.

Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
[Crossref]

Pan, Z.

Parsons, K.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Petykiewicz, J.

L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2017).
[Crossref]

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.

L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2017).
[Crossref]

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]

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]

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius <λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]

Pourabolghasem, R.

Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge discovery in nanophotonics using geometric deep learning,” Adv. Intell. Syst. 1, 1900132 (2019).
[Crossref]

Qiu, H.

Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
[Crossref]

Rahman, B. M. A.

Ren, X.

Sapra, N. V.

D. Vercruysse, N. V. Sapra, L. Su, R. Trivedi, and J. Vučković, “Analytical level set fabrication constraints for inverse design,” Sci. Rep. 9, 8999 (2019).
[Crossref]

L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2017).
[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]

B. Shen, R. Polson, and R. Menon, “Metamaterial-waveguide bends with effective bend radius <λ0/2,” Opt. Lett. 40, 5750–5753 (2015).
[Crossref]

Sigmund, O.

R. E. Christiansena, J. Vester-Petersenb, S. P. Madsenb, and O. Sigmund, “A non-linear material interpolation for design of metallic nano-particles using topology optimization,” Comput. Methods Appl. Mech. Engrg. 343, 23–29 (2019).
[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]

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

Song, Q.

Su, L.

D. Vercruysse, N. V. Sapra, L. Su, R. Trivedi, and J. Vučković, “Analytical level set fabrication constraints for inverse design,” Sci. Rep. 9, 8999 (2019).
[Crossref]

L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2017).
[Crossref]

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

Sun, S.

Sun, W.

Sun, X.

Tahersima, M. H.

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

Trivedi, R.

D. Vercruysse, N. V. Sapra, L. Su, R. Trivedi, and J. Vučković, “Analytical level set fabrication constraints for inverse design,” Sci. Rep. 9, 8999 (2019).
[Crossref]

Vercruysse, D.

D. Vercruysse, N. V. Sapra, L. Su, R. Trivedi, and J. Vučković, “Analytical level set fabrication constraints for inverse design,” Sci. Rep. 9, 8999 (2019).
[Crossref]

Vester-Petersenb, J.

R. E. Christiansena, J. Vester-Petersenb, S. P. Madsenb, and O. Sigmund, “A non-linear material interpolation for design of metallic nano-particles using topology optimization,” Comput. Methods Appl. Mech. Engrg. 343, 23–29 (2019).
[Crossref]

Vuckovic, J.

D. Vercruysse, N. V. Sapra, L. Su, R. Trivedi, and J. Vučković, “Analytical level set fabrication constraints for inverse design,” Sci. Rep. 9, 8999 (2019).
[Crossref]

L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vučković, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photon. 5, 301–305 (2017).
[Crossref]

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).
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[Crossref]

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Z. Xie, T. Lei, F. Li, H. Qiu, Z. Zhang, H. Wang, C. Min, L. Du, Z. Li, and X. Yuan, “Ultra-broadband on-chip twisted light emitter for optical communications,” Light Sci. Appl. 7, 18001–18006 (2018).
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Y. Kiarashinejad, M. Zandehshahvar, S. Abdollahramezani, O. Hemmatyar, R. Pourabolghasem, and A. Adibi, “Knowledge discovery in nanophotonics using geometric deep learning,” Adv. Intell. Syst. 1, 1900132 (2019).
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Y. Kiarashinejad, S. Abdollahramezani, M. Zandehshahvar, O. Hemmatyar, and A. Adibi, “Deep learning reveals underlying physics of light–matter interactions in nanophotonic devices,” Adv. Theory Simul. 2, 1900088 (2019).
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[Crossref]

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).
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[Crossref]

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

M. H. Tahersima, K. Kojima, T. Koike-Akino, D. Jha, B. Wang, C. Lin, and K. Parsons, “Deep neural network inverse design of integrated photonic power splitters,” Sci. Rep. 9, 1368 (2019).
[Crossref]

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Figures (4)

Fig. 1.
Fig. 1. Schematic diagram of the single-mode 3 dB power divider (before optimization).
Fig. 2.
Fig. 2. 3 dB power divider. The optimized (a) analog and (b) quasi-digital patterns in the first and second stages, respectively. (c) The optimized ternary pattern in which the smaller air cylinders with a radius of 35 nm are highlighted in orange. (d) Simulated excess loss profiles for the three patterns. (e) Measured excess loss profiles and (f) the SEM image of the fabricated device based on the ternary pattern. Inset in (e) shows the simulated steady-state intensity distribution.
Fig. 3.
Fig. 3. Schematic diagram of the dual-mode demultiplexer (before optimization).
Fig. 4.
Fig. 4. Dual-mode demultiplexer. The optimized (a) analog and (b) quasi-digital patterns in the first and second stages, respectively. (c) The optimized ternary pattern in which the smaller air cylinders with a radius of 36 nm are highlighted in orange. (d) Simulated insertion loss and crosstalk profiles for the ternary pattern. Insets show the simulated steady-state intensity distributions of TE0 and TE1 modes, respectively. (e) and (f) Respectively, simulated and measured performance of a mode-division multiplexing system composed of a dual-mode multiplexer and a demultiplexer based on the ternary pattern. (g) and (h) Respectively, SEM images of the fabricated device based on the ternary pattern and the mode-division multiplexing system.

Equations (6)

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FOM=14|S[E(p)×H0(p)¯+E0(p)¯×H(p)]·dS|2SRe[E0(p)×H0(p)¯]·dS,
δFOM=2ε0Vχδεr(p)Re[EA(p)·Eold(p)]d3p,
δεr(p)=Re[EA(p)·Eold(p)],
εrbiased(p)=(1+m)·[εrnew(p)6.5]+6.5.
σ=1Mn=1Mρn,ρn={|εr(n)1|2,1εr(n)<6.5|εr(n)12|2,6.5εr(n)12,
εrnew(p)=εrold(p)+Δ·12[δεr1(p)+δεr2(p)].