Abstract

We show that the adjoint variable method can be combined with the multi-frequency finite-difference frequency-domain method for efficient sensitivity calculations, enabling the systematic optimization of active nanophotonic devices. As a proof of principle demonstration, we have optimized a dynamic isolator structure in two-dimensions, resulting in the reduction of the length of the modulated regions by a factor of two, while retaining good performance in the isolation ratio and insertion loss.

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

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References

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    [Crossref]
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    [Crossref]
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2017 (4)

D. A. Miller, “Attojoule optoelectronics for low-energy information processing and communications,” J. Lightwave Technol. 35(3), 346–396 (2017).
[Crossref]

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
[Crossref] [PubMed]

T. Hughes, G. Veronis, K. P. Wootton, R. J. England, and S. Fan, “Method for computationally efficient design of dielectric laser accelerator structures,” Opt. Express 25(13), 15414–15427 (2017).
[Crossref] [PubMed]

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

2016 (3)

2015 (1)

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(6), 374–377 (2015).
[Crossref]

2014 (4)

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

J. Andkjær, V. E. Johansen, K. S. Friis, and O. Sigmund, “Inverse design of nanostructured surfaces for color effects,” J. Opt. Soc. Am. 31(1), 164–174 (2014).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

2013 (1)

2012 (4)

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain maxwell’s equations solvers,” J. Comput. Phys. 231(8), 3406–3431 (2012).
[Crossref]

R. C. Rumpf, “Simple implementation of arbitrarily shaped total-field/scattered-field regions in finite-difference frequency-domain,” Prog. Electromagn. Res. B 36, 221–248 (2012).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref] [PubMed]

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1(1), 17–22 (2012).
[Crossref]

2011 (1)

2010 (1)

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

2009 (3)

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[Crossref] [PubMed]

P. Dong, S. Liao, D. Feng, H. Liang, D. Zheng, R. Shafiiha, C.-C. Kung, W. Qian, G. Li, X. Zheng, A. V. Krishnamoorthy, and M. Asghari, “Low vpp, ultralow-energy, compact, high-speed silicon electro-optic modulator,” Opt. Express 17(25), 22484–22490 (2009).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[Crossref]

2008 (2)

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008).
[Crossref]

2004 (3)

2002 (1)

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microw. Theory Tech. 50(12), 2751–2758 (2002).
[Crossref]

2000 (1)

B. Troyanovsky, Z. Yu, and R. W. Dutton, “Physics-based simulation of nonlinear distortion in semiconductor devices using the harmonic balance method,” Comput. Methods in Appl. Mech. Eng. 181(4), 467–482 (2000).
[Crossref]

1992 (1)

K. Inoue, “Four-wave mixing in an optical fiber in zero-dispersion wavelength region,” J. Lightwave Technol. 10(11), 1553–1561 (1992).
[Crossref]

1986 (1)

K. S. Kundert and A. Sangiovanni-Vincentelli, “Simulation of nonlinear circuits in the frequency domain,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 5(4), 521–535. (1986).
[Crossref]

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas. Propag. 14(3), 302–307 (1966).
[Crossref]

1964 (1)

Alù, A.

M. Tymchenko, D. Sounas, and A. Alù, “Composite floquet scattering matrix for the analysis of time-modulated systems,” in IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (IEEE, 2017), pp. 65–66.

Andkjær, J.

J. Andkjær, V. E. Johansen, K. S. Friis, and O. Sigmund, “Inverse design of nanostructured surfaces for color effects,” J. Opt. Soc. Am. 31(1), 164–174 (2014).
[Crossref]

Aplet, L.

Asghari, M.

Aydin, K.

F. Callewaert, S. Butun, Z. Li, and K. Aydin, “Inverse design of an ultra-compact broadband optical diode based on asymmetric spatial mode conversion,” Sci. Rep. 6, 32577 (2016).
[Crossref] [PubMed]

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(6), 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. Microw. Theory Tech. 50(12), 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. Microw. Theory Tech. 50(12), 2751–2758 (2002).
[Crossref]

Beals, M.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Bernardis, S.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics(Academic, 2003).

Boyd, S.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
[Crossref]

Brongersma, M. L.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[Crossref] [PubMed]

Butun, S.

F. Callewaert, S. Butun, Z. Li, and K. Aydin, “Inverse design of an ultra-compact broadband optical diode based on asymmetric spatial mode conversion,” Sci. Rep. 6, 32577 (2016).
[Crossref] [PubMed]

Cai, W.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[Crossref] [PubMed]

Callewaert, F.

F. Callewaert, S. Butun, Z. Li, and K. Aydin, “Inverse design of an ultra-compact broadband optical diode based on asymmetric spatial mode conversion,” Sci. Rep. 6, 32577 (2016).
[Crossref] [PubMed]

Carson, J.

Cerjan, A.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Chen, S.-W.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Cheng, J.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Cohen, O.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Ding, Y.

Dong, P.

Doshay, S.

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
[Crossref] [PubMed]

Dutton, R. W.

G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. 29(19), 2288–2290 (2004).
[Crossref] [PubMed]

B. Troyanovsky, Z. Yu, and R. W. Dutton, “Physics-based simulation of nonlinear distortion in semiconductor devices using the harmonic balance method,” Comput. Methods in Appl. Mech. Eng. 181(4), 467–482 (2000).
[Crossref]

England, R. J.

Esterhazy, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Fan, J. A.

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
[Crossref] [PubMed]

Fan, S.

T. Hughes, G. Veronis, K. P. Wootton, R. J. England, and S. Fan, “Method for computationally efficient design of dielectric laser accelerator structures,” Opt. Express 25(13), 15414–15427 (2017).
[Crossref] [PubMed]

Y. Shi, W. Shin, and S. Fan, “Multi-frequency finite-difference frequency-domain algorithm for active nanophotonic device simulations,” Optica 3(11), 1256–1259 (2016).
[Crossref]

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref] [PubMed]

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain maxwell’s equations solvers,” J. Comput. Phys. 231(8), 3406–3431 (2012).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[Crossref]

G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. 29(19), 2288–2290 (2004).
[Crossref] [PubMed]

Fang, K.

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref] [PubMed]

Fejer, M. M.

Feng, D.

Frandsen, L. H.

Frellsen, L. F.

Friis, K. S.

J. Andkjær, V. E. Johansen, K. S. Friis, and O. Sigmund, “Inverse design of nanostructured surfaces for color effects,” J. Opt. Soc. Am. 31(1), 164–174 (2014).
[Crossref]

Gardes, F. Y.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Ge, L.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

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. Microw. Theory Tech. 50(12), 2751–2758 (2002).
[Crossref]

Glavic, S.

N. K. Georgieva, S. Glavic, M. H. Bakr, and J. W. Bandler, “Feasible adjoint sensitivity technique for EM design optimization,” IEEE Trans. Microw. Theory Tech. 50(12), 2751–2758 (2002).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method(Artech House, 2005).

Hsieh, I.-W.

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008).
[Crossref]

Hsu, S. S.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Hu, Y.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Hughes, T.

Inoue, K.

K. Inoue, “Four-wave mixing in an optical fiber in zero-dispersion wavelength region,” J. Lightwave Technol. 10(11), 1553–1561 (1992).
[Crossref]

Jensen, J. S.

Johansen, V. E.

J. Andkjær, V. E. Johansen, K. S. Friis, and O. Sigmund, “Inverse design of nanostructured surfaces for color effects,” J. Opt. Soc. Am. 31(1), 164–174 (2014).
[Crossref]

Johnson, S. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Jones, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Kimerling, L. C.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Krishnamoorthy, A. V.

Kundert, K. S.

K. S. Kundert and A. Sangiovanni-Vincentelli, “Simulation of nonlinear circuits in the frequency domain,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 5(4), 521–535. (1986).
[Crossref]

Kung, C.-C.

Kurz, J. R.

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(6), 374–377 (2015).
[Crossref]

Langrock, C.

Lanzillotti-Kimura, N. D.

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1(1), 17–22 (2012).
[Crossref]

Li, G.

Li, K.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Li, Z.

F. Callewaert, S. Butun, Z. Li, and K. Aydin, “Inverse design of an ultra-compact broadband optical diode based on asymmetric spatial mode conversion,” Sci. Rep. 6, 32577 (2016).
[Crossref] [PubMed]

Liang, H.

Liao, L.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Liao, S.

Liertzer, M.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Lipson, M.

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

Liu, A.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Liu, D.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Liu, J.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

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(6), 374–377 (2015).
[Crossref]

J. Lu and J. Vučković, “Nanophotonic computational design,” Opt. Express 21(11), 13351–13367 (2013).
[Crossref] [PubMed]

Ma, R.-M.

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1(1), 17–22 (2012).
[Crossref]

Makris, K. G.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Mashanovich, G.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Mashanovich, G. Z.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Melenk, J. M.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Michel, J.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Miller, D. A.

Mizumoto, T.

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008).
[Crossref]

Nedeljkovic, M.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Nicolaescu, R.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Nussenzveig, P.

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

Osgood, R. M.

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008).
[Crossref]

Paniccia, M.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Petykiewicz, J.

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

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(6), 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] [PubMed]

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(6), 374–377 (2015).
[Crossref]

Pomerene, A.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Qian, W.

Reed, G. T.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Rotter, S.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Roussev, R. V.

Rubin, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Rumpf, R. C.

R. C. Rumpf, “Simple implementation of arbitrarily shaped total-field/scattered-field regions in finite-difference frequency-domain,” Prog. Electromagn. Res. B 36, 221–248 (2012).
[Crossref]

Samara-Rubio, D.

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Sangiovanni-Vincentelli, A.

K. S. Kundert and A. Sangiovanni-Vincentelli, “Simulation of nonlinear circuits in the frequency domain,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 5(4), 521–535. (1986).
[Crossref]

Sell, D.

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
[Crossref] [PubMed]

Shafiiha, R.

Shi, Y.

Shin, W.

Y. Shi, W. Shin, and S. Fan, “Multi-frequency finite-difference frequency-domain algorithm for active nanophotonic device simulations,” Optica 3(11), 1256–1259 (2016).
[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(8), 3406–3431 (2012).
[Crossref]

Shoji, Y.

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008).
[Crossref]

Sigmund, O.

Sorger, V. J.

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1(1), 17–22 (2012).
[Crossref]

Sounas, D.

M. Tymchenko, D. Sounas, and A. Alù, “Composite floquet scattering matrix for the analysis of time-modulated systems,” in IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (IEEE, 2017), pp. 65–66.

Stone, A. D.

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Su, L.

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

Sun, R.

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method(Artech House, 2005).

Thomson, D. J.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

Troyanovsky, B.

B. Troyanovsky, Z. Yu, and R. W. Dutton, “Physics-based simulation of nonlinear distortion in semiconductor devices using the harmonic balance method,” Comput. Methods in Appl. Mech. Eng. 181(4), 467–482 (2000).
[Crossref]

Tymchenko, M.

M. Tymchenko, D. Sounas, and A. Alù, “Composite floquet scattering matrix for the analysis of time-modulated systems,” in IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (IEEE, 2017), pp. 65–66.

Tzuang, L. D.

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

Vandenberghe, L.

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
[Crossref]

Veronis, G.

Vuckovic, J.

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

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(6), 374–377 (2015).
[Crossref]

J. Lu and J. Vučković, “Nanophotonic computational design,” Opt. Express 21(11), 13351–13367 (2013).
[Crossref] [PubMed]

Wang, F.

White, J. S.

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[Crossref] [PubMed]

Wilson, P. R.

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Wootton, K. P.

Yang, J.

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
[Crossref] [PubMed]

Yang, R.

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
[Crossref] [PubMed]

Yee, K.

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas. Propag. 14(3), 302–307 (1966).
[Crossref]

Yokoi, H.

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008).
[Crossref]

Yu, Z.

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref] [PubMed]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[Crossref]

B. Troyanovsky, Z. Yu, and R. W. Dutton, “Physics-based simulation of nonlinear distortion in semiconductor devices using the harmonic balance method,” Comput. Methods in Appl. Mech. Eng. 181(4), 467–482 (2000).
[Crossref]

Zhang, X.

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1(1), 17–22 (2012).
[Crossref]

Zheng, D.

Zheng, X.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008).
[Crossref]

Comput. Methods in Appl. Mech. Eng. (1)

B. Troyanovsky, Z. Yu, and R. W. Dutton, “Physics-based simulation of nonlinear distortion in semiconductor devices using the harmonic balance method,” Comput. Methods in Appl. Mech. Eng. 181(4), 467–482 (2000).
[Crossref]

IEEE Trans. Antennas. Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas. Propag. 14(3), 302–307 (1966).
[Crossref]

IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. (1)

K. S. Kundert and A. Sangiovanni-Vincentelli, “Simulation of nonlinear circuits in the frequency domain,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst. 5(4), 521–535. (1986).
[Crossref]

IEEE Trans. Microw. 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. Microw. Theory Tech. 50(12), 2751–2758 (2002).
[Crossref]

J. Comput. Phys. (1)

W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain maxwell’s equations solvers,” J. Comput. Phys. 231(8), 3406–3431 (2012).
[Crossref]

J. Lightwave Technol. (2)

K. Inoue, “Four-wave mixing in an optical fiber in zero-dispersion wavelength region,” J. Lightwave Technol. 10(11), 1553–1561 (1992).
[Crossref]

D. A. Miller, “Attojoule optoelectronics for low-energy information processing and communications,” J. Lightwave Technol. 35(3), 346–396 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

J. Andkjær, V. E. Johansen, K. S. Friis, and O. Sigmund, “Inverse design of nanostructured surfaces for color effects,” J. Opt. Soc. Am. 31(1), 164–174 (2014).
[Crossref]

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

Nano Lett. (2)

W. Cai, J. S. White, and M. L. Brongersma, “Compact, high-speed and power-efficient electrooptic plasmonic modulators,” Nano Lett. 9(12), 4403–4411 (2009).
[Crossref] [PubMed]

D. Sell, J. Yang, S. Doshay, R. Yang, and J. A. Fan, “Large-angle, multifunctional metagratings based on freeform multimode geometries,” Nano Lett. 17(6), 3752–3757 (2017).
[Crossref] [PubMed]

Nanophotonics (2)

V. J. Sorger, N. D. Lanzillotti-Kimura, R.-M. Ma, and X. Zhang, “Ultra-compact silicon nanophotonic modulator with broadband response,” Nanophotonics 1(1), 17–22 (2012).
[Crossref]

G. T. Reed, G. Z. Mashanovich, F. Y. Gardes, M. Nedeljkovic, Y. Hu, D. J. Thomson, K. Li, P. R. Wilson, S.-W. Chen, and S. S. Hsu, “Recent breakthroughs in carrier depletion based silicon optical modulators,” Nanophotonics 3(4–5), 229–245 (2014).
[Crossref]

Nat. Photonics (5)

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(6), 374–377 (2015).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3(2), 91–94 (2009).
[Crossref]

J. Liu, M. Beals, A. Pomerene, S. Bernardis, R. Sun, J. Cheng, L. C. Kimerling, and J. Michel, “Waveguide-integrated, ultralow-energy GeSi electro-absorption modulators,” Nat. Photonics 2(7), 433–437 (2008).
[Crossref]

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4(8), 518–526 (2010).
[Crossref]

L. D. Tzuang, K. Fang, P. Nussenzveig, S. Fan, and M. Lipson, “Non-reciprocal phase shift induced by an effective magnetic flux for light,” Nat. Photonics 8(9), 701–705 (2014).
[Crossref]

Nature (1)

A. Liu, R. Jones, L. Liao, D. Samara-Rubio, D. Rubin, O. Cohen, R. Nicolaescu, and M. Paniccia, “A high-speed silicon optical modulator based on a metal–oxide–semiconductor capacitor,” Nature 427(6975), 615–618 (2004).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (2)

Optica (1)

Phys. Rev. A (1)

S. Esterhazy, D. Liu, M. Liertzer, A. Cerjan, L. Ge, K. G. Makris, A. D. Stone, J. M. Melenk, S. G. Johnson, and S. Rotter, “Scalable numerical approach for the steady-state ab initio laser theory,” Phys. Rev. A 90(2), 023816 (2014).
[Crossref]

Phys. Rev. Lett. (1)

K. Fang, Z. Yu, and S. Fan, “Photonic Aharonov-Bohm effect based on dynamic modulation,” Phys. Rev. Lett. 108(15), 153901 (2012).
[Crossref] [PubMed]

Prog. Electromagn. Res. B (1)

R. C. Rumpf, “Simple implementation of arbitrarily shaped total-field/scattered-field regions in finite-difference frequency-domain,” Prog. Electromagn. Res. B 36, 221–248 (2012).
[Crossref]

Sci. Rep. (2)

F. Callewaert, S. Butun, Z. Li, and K. Aydin, “Inverse design of an ultra-compact broadband optical diode based on asymmetric spatial mode conversion,” Sci. Rep. 6, 32577 (2016).
[Crossref] [PubMed]

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

Other (4)

R. W. Boyd, Nonlinear Optics(Academic, 2003).

S. Boyd and L. Vandenberghe, Convex Optimization(Cambridge University, 2004).
[Crossref]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-difference Time-domain Method(Artech House, 2005).

M. Tymchenko, D. Sounas, and A. Alù, “Composite floquet scattering matrix for the analysis of time-modulated systems,” in IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting (IEEE, 2017), pp. 65–66.

Supplementary Material (1)

NameDescription
» Visualization 1       The video shows the convergence process of our optimized structure.

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Figures (5)

Fig. 1
Fig. 1 Design of a dynamic optical isolator. (a) Band structure of a silicon slab waveguide. The width of the slab waveguide is 1.1μm. The frequency and wavevector are normalized with respect to a = 1.0μm. The red and blue dots represent the lowest-order even and odd modes, respectively. The shaded region represents the radiative modes. (b) Optical isolator design. The width of the central waveguide is 2.0μm. Modulation is applied to the upper half of the waveguide and is marked as the gray regions. The modulation phase difference between the left and right sides is π/2. (c) Forward propagation. An even mode is injected from the left side of the device. The electric field remains in the even mode after passing through the device. (d) Backward propagation. The even mode is injected from the right side of the device and is converted to the odd mode upon passing through the device.
Fig. 2
Fig. 2 Optimization Setup. The objective of the optimization is to design the dielectric structure in the regions as indicated by the red dashed lines, such that the lengths of the modulated regions are reduced by a factor of 2 as compared with the original structure shown in Fig. 1(b).
Fig. 3
Fig. 3 The flowchart of the optimization process
Fig. 4
Fig. 4 The optimized optical isolator. (a) Optimized structure. The colored regions (including grays and blue regions) represent the silicon device, which is surrounded by free space. The gray regions indicate the modulated regions. Notice that the length of the modulated region (enclosed by the green lines) is half the length in the original structure as shown in Fig. 1(b). (b) Forward propagation. An even mode is injected from the left side of the device. The electric field remains in the even mode after passing through the device. (c) Backward propagation. The even mode is injected from the right side of the device and is converted to the odd mode upon passing through the device. (d) Objective function values as a function of iteration steps. (e) Binary characterization values as a function of iteration steps (see Visualization 1 for successive convergence of the optimized structure).
Fig. 5
Fig. 5 (a) The spectrum of the isolation ratio for each isolator structure. (b) The spectrum of the insertion loss for each isolator structure. The red lines are the spectra for the original structure as shown in Fig. 1. The black lines are for the reduced-length structure before optimization as shown in Fig. 2. And the blue lines are for the optimized structure as shown in Fig. 4.

Equations (21)

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ϵ ( r , t ) = ϵ s ( r ) + δ ( r ) cos ( Ω t + ϕ ( r ) ) ,
E ˜ ( r , t ) = Re { n E ( r , ω n ) e i ω n t } .
× μ 1 × E ( ω n ) ω n 2 ϵ s E ( ω n ) 1 2 ω n 2 δ e i ϕ E ( ω n 1 ) 1 2 ω n 2 δ e i ϕ E ( ω n + 1 ) = i ω 0 J ( ω 0 ) δ n 0 .
( A N C N , N + 1   0     0 C 0 , 1 A 0 C 0 , 1 0     0   C N , N 1 A N ) ( E ( ω N ) E ( ω 0 ) E ( ω N ) ) = ( 0 i ω 0 J ( ω 0 ) 0 ) ,
A E = i ω 0 J .
δ ( E ( γ ) ) γ = Ε E γ .
A γ E + A E γ = 0
E γ = A 1 A γ E .
( E ( γ ) ) γ = ( E A 1 ) A γ E = E ^ T A γ E ,
A T E ^ = ( E ) T .
( E f , E b ) γ = E ^ f T A γ E f E ^ b T A γ E b ,
T 0 n = 1 2 ω n Re d y E ( x p , y , ω n ) × H * ( x p , y , ω n ) x ^ ,
C = 10 log 10 T 0 0 T 0 0 .
L = 10 log 10 T 0 0 .
( E f , E b , ϵ s ) = ˜ ( E f , E b ) + κ ( i , j ) ( ϵ Si ϵ i j ) ( ϵ i j ϵ 0 ) .
˜ ( E f , E b ) = W 1 T 0 0 2 W 2 ( 1 T 0 0 ) 2 + W 3 ( 1 T 1 0 ) 2 + W 4 T 0 1 2 + W 5 ( n R 0 n ) 2 + W 6 ( n R n 0 ) 2
minimize ϵ s ( E f , E b , ϵ s ) subject to  ϵ 0 ϵ i j ϵ Si .
Δ ϵ i j : = α ϵ i j + η Δ ϵ i j , ϵ i j : = ϵ i j Δ ϵ i j ,
ϵ i j = { ϵ 0 ϵ i j ϵ 0 ϵ Si ϵ i j ϵ Si .
ϵ i j = { ϵ 0 ϵ i j ( ϵ 0 + ϵ Si ) / 2 ϵ Si otherwise ,
B = mean ( i , j ) | 2 ϵ i j ϵ 0 ϵ Si ϵ 0 1 | .

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