Abstract

In part I of this series of papers, we introduced an efficient pixel-by-pixel optimization technique, utilizing the Dyson’s equation in a Green’s function formalism. This technique directly incorporates material and minimum feature size constraints in practical photonic devices. The implementation in part I, however, requires full storage of the Green’s function, which is costly. In this paper, we further develop this optimization technique by showing that only the diagonal part of the Green’s function needs to be stored, and by showing that the optimization process can be implemented using standard commercially available electromagnetic solvers. The development here should enable a wider use of this optimization technique.

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

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References

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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)

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]

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]

2013 (1)

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)

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]

1994 (1)

1988 (1)

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

1987 (1)

1980 (2)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[Crossref]

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Electromagn. Compat. EMC-22, 191–202 (1980).
[Crossref]

1966 (1)

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

Bhargava, S.

Boutami, S.

Dereux, 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]

Dutton, R. W.

Fan, S.

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]

Jiao, Y.

Jin, J. M.

J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 2015).

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.

Lockwood, D. J.

D. J. Lockwood and L. Pavesi, “Silicon fundamentals for photonics applications,” in Silicon Photonics (Springer, 2004), pp. 1–50.

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]

Martin, O. J.

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).

Pavesi, L.

D. J. Lockwood and L. Pavesi, “Silicon fundamentals for photonics applications,” in Silicon Photonics (Springer, 2004), pp. 1–50.

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]

Seldowitz, M. A.

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]

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.

Taflove, A.

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Electromagn. Compat. EMC-22, 191–202 (1980).
[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]

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]

Yablonovitch, E.

Yaghjian, A. D.

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[Crossref]

Yee, K.

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

Appl. Opt. (1)

Astrophys. J. (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[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, 302–307 (1966).
[Crossref]

IEEE Trans. Electromagn. Compat. (1)

A. Taflove, “Application of the finite-difference time-domain method to sinusoidal steady-state electromagnetic-penetration problems,” IEEE Trans. Electromagn. Compat. EMC-22, 191–202 (1980).
[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]

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, 3406–3431 (2012).
[Crossref]

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

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

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 (2)

Opt. Lett. (2)

Phys. Rev. E (1)

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

Proc. IEEE (1)

A. D. Yaghjian, “Electric dyadic Green’s functions in the source region,” Proc. IEEE 68, 248–263 (1980).
[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)

Fullwave® and Synopsis®, “Rsoft Design,”.

J. M. Jin, The Finite Element Method in Electromagnetics (Wiley, 2015).

D. J. Lockwood and L. Pavesi, “Silicon fundamentals for photonics applications,” in Silicon Photonics (Springer, 2004), pp. 1–50.

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

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Figures (5)

Fig. 1.
Fig. 1. 2D TE example in homogeneous background. (a) Schematic of the setup of the optimization, which aims to maximize light intensity at a target point, separated from an emitting source by a distance of one wavelength ( λ = 1.55 μm ). We consider a binary system with two permittivities of 2.25 and 6.25, respectively. The design space is a 2 λ -sized square, divided in 50-nm-sized square pixels, with the source at the center. (b) Optimized structure obtained in part I [12] with the DDA approach. (c) FOM as a function of iteration number. (d) Optimized structure. (c)–(d) are obtained using the approach described in this paper. (e) Electric field of the optimized structure, as determined using FDTD simulation.
Fig. 2.
Fig. 2. 2D TE example in complex background. (a) Schematic of the setup of the optimization: light incident as the fundamental mode of a 1-μm-wide multimode input waveguide is routed to two 0.3-μm-wide single-mode output waveguides in a wavelength-dependent fashion. Light at wavelength λ 1 = 1.31 μm is routed to the left, and light at the wavelength λ 2 = 1.55 μm is routed to the right. (b) Optimized structure obtained in part I [12] with the DDA approach. (c) FOM as a function of iteration number, starting from a uniform silica background in the design region. (d) Optimized structure. (c)–(d) are obtained using the approach described in this paper. (e) Corresponding electric fields at wavelengths λ 1 = 1.31 μm (left panel) and λ 2 = 1.55 μm (right panel), as obtained using FDTD simulations of the optimized structure.
Fig. 3.
Fig. 3. 2D TM example in complex background. Same as in Fig. 2, but for TM polarization. (a) FOM as a function of iteration number, starting from a uniform silica background in the design region. (b) Optimized structure. (c) Corresponding magnetic fields at wavelengths λ 1 = 1.31 μm (left panel) and λ 2 = 1.55 μm (right panel), as obtained using FDTD simulations of the optimized structure.
Fig. 4.
Fig. 4. 3D example for optimization of a slab structure. (a) Schematic of the setup of the optimization, which aims to maximize light intensity at a target point, separated from a y -polarized emitting source dipole by a distance of a half-wavelength ( λ = 1.55 μm ), with the source at the center. Source and target point are located in the middle of the slab along the y direction. We consider a binary system with two permittivities of 2.25 (silica) and 12.25 (silicon), respectively. The design space is a 1.55 μm × 0.3 μm × 1.55 μm slab, divided in 50-nm-sized cubic pixels. (b) FOM as a function of iteration number, starting from a uniform silica background. (c) Optimized structure, using the approach described in this paper. (d) Ey electric field of the optimized structure in the middle of the slab, as determined using FDTD simulation.
Fig. 5.
Fig. 5. 3D example of a slab structure in complex background with minimum feature size technological constraints. (a) Schematic of the setup of the optimization, which aims to maximize conversion of TE 00 mode into TE 01 mode, for 1-μm-wide 0.3-μm-thick input and output waveguides ( λ = 1.55 μm ). We consider a binary system with two permittivities of 2.25 (silica) and 12.25 (silicon), respectively. The design space is a 1 μm × 0.3 μm × 1 μm slab, divided in 50-nm-sized cubic pixels. (b) Optimized structures, using the approach described in this paper, for minimum feature sizes of (i) 50 nm, (ii) 100 nm, and (iii) 200 nm. (c) Ex electric field of the optimized structures in the middle of the slab, as determined using FDTD simulations, with indicated conversion efficiencies.

Equations (22)

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

FOM = | E ( r 0 ) | 2 .
Δ FOM = | E ( new ) ( r 0 ) | 2 | E ( old ) ( r 0 ) | 2 ,
E ( new ) ( r ) = E ( old ) ( r ) + V k 0 2 G ( old ) ( r , r ) Δ ε ( r ) E ( new ) ( r ) d r ,
Δ FOM = | E ( old ) ( r 0 ) + k 0 2 V G ( old ) ( r 0 , r ) Δ ε ( r ) E ( new ) ( r ) | 2 | E ( old ) ( r 0 ) | 2 .
G ( old ) ( r 0 , r ) = G ( old ) ( r , r 0 ) T ,
E adj ( old ) ( r ) = G ( old ) ( r , r 0 ) E ( old ) ( r 0 ) * ,
Δ FOM = 2 k 0 2 V Re ( E adj ( old ) ( 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 ) .
E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) E ( new ) ( r b ) .
G ( new ) ( r , r ) = G ( old ) ( r , r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) G ( new ) ( r b , r ) .
G ( new ) ( r b , r ) = G ( old ) ( r b , r ) + k 0 2 V G ( old ) ( r b , r b ) Δ ε ( r ) G ( new ) ( r b , r ) .
G ( new ) ( r , r ) = G ( old ) ( r , r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) G ( new ) ( r b , r ) .
G ( new ) ( r b , r ) = G ( new ) ( r , r b ) T .
E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) E ( new ) ( r b ) .
{ E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) E ( new ) ( r b ) , E adj ( new ) ( r ) = E adj ( old ) ( r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) E adj ( new ) ( r b ) .
{ E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) E ( new ) ( r b ) , E adj ( new ) ( r ) = E adj ( old ) ( r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) E adj ( new ) ( r b ) , E Green ( new ) ( r ) = E Green ( old ) ( r ) + k 0 2 V G ( old ) ( r , r b ) Δ ε ( r b ) E Green ( new ) ( r b ) ,
E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( new ) ( r , r b ) Δ ε ( r b ) E ( old ) ( r b ) .
{ E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( new ) ( r , r b ) Δ ε ( r b ) E ( old ) ( r b ) , E adj ( new ) ( r ) = E adj ( old ) ( r ) + k 0 2 V G ( new ) ( r , r b ) Δ ε ( r b ) E adj ( old ) ( r b ) ,
{ E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 V G ( new ) ( r , r b ) Δ ε ( r b ) E ( old ) ( r b ) , E adj ( new ) ( r ) = E adj ( old ) ( r ) + k 0 2 V G ( new ) ( r , r b ) Δ ε ( r b ) E adj ( old ) ( r b ) , E Green ( new ) ( r ) = E Green ( old ) ( r ) + k 0 2 V G ( new ) ( r , r b ) Δ ε ( r b ) E Green ( old ) ( r b ) .
Δ FOM = 2 k 0 2 block d r Re ( E adj ( old ) ( r ) · Δ ε ( r ) E ( new ) ( r ) ) ,
E ( new ) ( r ) = E ( old ) ( r ) + k 0 2 block d r 1 G ( old ) ( r , r 1 ) Δ ε ( r 1 ) E ( new ) ( r 1 ) .
G ( new ) ( r 2 , r 1 ) = G ( old ) ( r 2 , r 1 ) + k 0 2 V G ( old ) ( r 2 , r b ) Δ ε ( r b ) G ( new ) ( r b , r 1 ) .

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