Abstract

We derive an adjoint shape optimization algorithm with a compound figure of merit and demonstrate its use with both gradient descent and Levenberg-Marquart updates for the case of SiO2-buried SOI coplanar waveguide crossings. We show that a smoothing parameter, basis function width, can be used to eliminate small feature sizes with a small cost to device performance. The Levenberg-Marquardt update produces devices with larger bandwidth. A waveguide crossing with simulated performance values of > 60 dB cross power extinction ratio and > −0.08 dB through power over the 1500-1600 nm band is presented. A fabricated device is measured to have a maximum of −0.06 dB through power and a 50 dB cross power extinction ratio.

© 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 (2)

N. Sapra, D. Vercruysse, L. Su, K. Yang, J. Skarda, A. Piggott, and J. Vuckovic, “Inverse design and demonstration of broadband grating couplers,” IEEE J. Sel. Top. Quantum Electron. 25(3), 1–7 (2019).
[Crossref]

Z. Yu, A. Feng, X. Xi, and X. Sun, “Inverse-designed low-loss and wideband polarization-insensitive silicon waveguide crossing,” Opt. Lett. 44(1), 77–79 (2019).
[Crossref]

2018 (4)

2017 (1)

A. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
[Crossref]

2016 (1)

2015 (2)

A. Piggott, J. Lu, T. Babinec, K. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4(1), 7210 (2015).
[Crossref]

A. Piggott, J. Lu, K. Lagoudakis, J. Petykiewicz, T. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
[Crossref]

2014 (2)

2013 (5)

2012 (1)

2011 (1)

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5(2), 308–321 (2011).
[Crossref]

2010 (3)

2007 (1)

2006 (1)

2004 (5)

J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag. 52(2), 397–407 (2004).
[Crossref]

E. Kerrinckx, L. Bigot, M. Douay, and Y. Quiquempois, “Photonic crystal fiber design by means of a genetic algorithm,” Opt. Express 12(9), 1990–1995 (2004).
[Crossref]

L. Sanchis, A. Håkansson, D. López-Zanón, J. Bravo-Abad, and J. Sánchez-Dehesa, “Integrated optical devices design by genetic algorithm,” Appl. Phys. Lett. 84(22), 4460–4462 (2004).
[Crossref]

B. West and S. Honkanen, “MMI devices with weak guiding designed in three dimensions using a genetic algorithm,” Opt. Express 12(12), 2716–2722 (2004).
[Crossref]

H. Liu, H. Tam, P. Wai, and E. Pun, “Low-loss waveguide crossing using a multimode interference structure,” Opt. Commun. 241(1-3), 99–104 (2004).
[Crossref]

2003 (1)

2002 (1)

S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

2001 (1)

2000 (1)

M. Giles and N. Pierce, “An introduction to the adjoint approach to design,” Flow, Turbul. Combust. 65(3/4), 393–415 (2000).
[Crossref]

1998 (1)

1997 (1)

K. Warnick, R. Selfridge, and D. Arnold, “Teaching electromagnetic field theory using differential forms,” IEEE Trans. Educ. 40(1), 53–68 (1997).
[Crossref]

Arnold, D.

K. Warnick, R. Selfridge, and D. Arnold, “Teaching electromagnetic field theory using differential forms,” IEEE Trans. Educ. 40(1), 53–68 (1997).
[Crossref]

Babinec, T.

A. Piggott, J. Lu, T. Babinec, K. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4(1), 7210 (2015).
[Crossref]

A. Piggott, J. Lu, K. Lagoudakis, J. Petykiewicz, T. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
[Crossref]

Baehr-Jones, T.

Y. Ma, Y. Zhang, S. Yang, A. Novack, R. Ding, A. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for SOI optical interconnect,” Opt. Express 21(24), 29374–29382 (2013).
[Crossref]

Y. Ma, Y. Liu, R. Ding, T. Baehr-Jones, P. Magill, H. Guan, A. Gazman, Q. Li, K. Bergman, and M. Hochberg, “Optimized silicon photonics components for high-performance interconnect systems,” in IEEE Photonics Conference (IPC), Reston, VA, pp. 353–354 (2015).

Baets, R.

Bauer, T.

A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
[Crossref]

Beausoleil, R.

Bergman, K.

Y. Ma, Y. Liu, R. Ding, T. Baehr-Jones, P. Magill, H. Guan, A. Gazman, Q. Li, K. Bergman, and M. Hochberg, “Optimized silicon photonics components for high-performance interconnect systems,” in IEEE Photonics Conference (IPC), Reston, VA, pp. 353–354 (2015).

Bhargava, S.

C. Lalau-Keraly, S. Bhargava, O. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693–21701 (2013).
[Crossref]

S. Bhargava and E. Yablonovitch, “Multi-objective inverse design of sub-wavelength optical focusing structures for heat assisted magnetic recording,” Proc. SPIE 9201, Optical Data Storage 2014, 92010M (5 September 2014).

Bigot, L.

Bock, P.

Bogaerts, W.

Bravo-Abad, J.

L. Sanchis, A. Håkansson, D. López-Zanón, J. Bravo-Abad, and J. Sánchez-Dehesa, “Integrated optical devices design by genetic algorithm,” Appl. Phys. Lett. 84(22), 4460–4462 (2004).
[Crossref]

Burger, S.

P.-I. Schneider, X. Santiago, V. Soltwisch, M. Hammerschmidt, S. Burger, and C. Rockstuhl, “Benchmarking five global optimization approaches for nano-optical shape optimization and parameter reconstruction,” https://arxiv.org/abs/1809.06674 .

Cai, J.

Cerrina, F.

Cheben, P.

Chen, C.-H.

C.-H. Chen and C.-H. Chiu, “Taper-integrated multimode-interference based waveguide crossing design,” IEEE J. Quantum Electron. 46(11), 1656–1661 (2010).
[Crossref]

Chen, R.

Chen, Y.-F.

Chen, Z.

Chiu, C.-H.

C.-H. Chen and C.-H. Chiu, “Taper-integrated multimode-interference based waveguide crossing design,” IEEE J. Quantum Electron. 46(11), 1656–1661 (2010).
[Crossref]

Chu, P.

A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
[Crossref]

Cox, J.

A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
[Crossref]

Davids, P.

A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
[Crossref]

Delage, A.

Densmore, A.

DeRose, C.

A. Jones, C. DeRose, A. Lentine, D. Trotter, A. Starbuck, and R. Norwood, “Ultra-low crosstalk, CMOS compatible waveguide crossings for densely integrated photonic interconnection networks,” Opt. Express 21(10), 12002–12013 (2013).
[Crossref]

A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
[Crossref]

Ding, R.

Y. Ma, Y. Zhang, S. Yang, A. Novack, R. Ding, A. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for SOI optical interconnect,” Opt. Express 21(24), 29374–29382 (2013).
[Crossref]

Y. Ma, Y. Liu, R. Ding, T. Baehr-Jones, P. Magill, H. Guan, A. Gazman, Q. Li, K. Bergman, and M. Hochberg, “Optimized silicon photonics components for high-performance interconnect systems,” in IEEE Photonics Conference (IPC), Reston, VA, pp. 353–354 (2015).

Ding, Y.

Douay, M.

Dumon, P.

Fan, S.

Fattal, D.

Feng, A.

Fink, Y.

S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Frandsen, L.

Frellsen, L.

Gauger, N.

Gazman, A.

Y. Ma, Y. Liu, R. Ding, T. Baehr-Jones, P. Magill, H. Guan, A. Gazman, Q. Li, K. Bergman, and M. Hochberg, “Optimized silicon photonics components for high-performance interconnect systems,” in IEEE Photonics Conference (IPC), Reston, VA, pp. 353–354 (2015).

Giles, M.

M. Giles and N. Pierce, “An introduction to the adjoint approach to design,” Flow, Turbul. Combust. 65(3/4), 393–415 (2000).
[Crossref]

Guan, H.

Y. Ma, Y. Liu, R. Ding, T. Baehr-Jones, P. Magill, H. Guan, A. Gazman, Q. Li, K. Bergman, and M. Hochberg, “Optimized silicon photonics components for high-performance interconnect systems,” in IEEE Photonics Conference (IPC), Reston, VA, pp. 353–354 (2015).

Håkansson, A.

L. Sanchis, A. Håkansson, D. López-Zanón, J. Bravo-Abad, and J. Sánchez-Dehesa, “Integrated optical devices design by genetic algorithm,” Appl. Phys. Lett. 84(22), 4460–4462 (2004).
[Crossref]

Hall, T.

Hammerschmidt, M.

P.-I. Schneider, X. Santiago, V. Soltwisch, M. Hammerschmidt, S. Burger, and C. Rockstuhl, “Benchmarking five global optimization approaches for nano-optical shape optimization and parameter reconstruction,” https://arxiv.org/abs/1809.06674 .

Han, H.-L.

Haus, H.

Hochberg, M.

Y. Ma, Y. Zhang, S. Yang, A. Novack, R. Ding, A. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for SOI optical interconnect,” Opt. Express 21(24), 29374–29382 (2013).
[Crossref]

Y. Ma, Y. Liu, R. Ding, T. Baehr-Jones, P. Magill, H. Guan, A. Gazman, Q. Li, K. Bergman, and M. Hochberg, “Optimized silicon photonics components for high-performance interconnect systems,” in IEEE Photonics Conference (IPC), Reston, VA, pp. 353–354 (2015).

Honkanen, S.

Hosseini, A.

Ibanescu, M.

S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Ippen, E.

M. Popovic, E. Ippen, and F. Kärtner, “Low-loss Bloch waves in open structures and highly compact, efficient Si waveguide-crossing arrays,” in IEEE LEOS Ann. Mtg., Lake Buena Vista, FL, pp. 56–57 (2007).

Janz, S.

Jensen, J.

J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5(2), 308–321 (2011).
[Crossref]

Jiang, J.

Jin, W.

S. Molesky, Z. Lin, A. Piggott, W. Jin, J. Vuckovic, and A. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

Joannopoulos, J.

S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

S. Johnson, C. Manolatou, S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23(23), 1855–1857 (1998).
[Crossref]

Johnson, S.

S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

S. Johnson, C. Manolatou, S. Fan, P. Villeneuve, J. Joannopoulos, and H. Haus, “Elimination of cross talk in waveguide intersections,” Opt. Lett. 23(23), 1855–1857 (1998).
[Crossref]

Jones, A.

A. Jones, C. DeRose, A. Lentine, D. Trotter, A. Starbuck, and R. Norwood, “Ultra-low crosstalk, CMOS compatible waveguide crossings for densely integrated photonic interconnection networks,” Opt. Express 21(10), 12002–12013 (2013).
[Crossref]

A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
[Crossref]

Kärtner, F.

M. Popovic, E. Ippen, and F. Kärtner, “Low-loss Bloch waves in open structures and highly compact, efficient Si waveguide-crossing arrays,” in IEEE LEOS Ann. Mtg., Lake Buena Vista, FL, pp. 56–57 (2007).

Kerrinckx, E.

Kimerling, L.

Kwong, D.

Lagoudakis, K.

A. Piggott, J. Lu, T. Babinec, K. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4(1), 7210 (2015).
[Crossref]

A. Piggott, J. Lu, K. Lagoudakis, J. Petykiewicz, T. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
[Crossref]

Lalau-Keraly, C.

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A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
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Liu, Y.

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J. Lu and J. Vuckovic, “Nanophotonic computational design,” Opt. Express 21(11), 13351–13367 (2013).
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J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express 18(4), 3793–3804 (2010).
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Ma, Y.

Y. Ma, Y. Zhang, S. Yang, A. Novack, R. Ding, A. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for SOI optical interconnect,” Opt. Express 21(24), 29374–29382 (2013).
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A. Piggott, J. Lu, K. Lagoudakis, J. Petykiewicz, T. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
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A. Piggott, J. Lu, T. Babinec, K. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4(1), 7210 (2015).
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Y. Liu, J. Shainline, X. Zeng, and M. Popovic, “Ultra-low-loss CMOS-compatible waveguide crossing arrays based on multimode Bloch waves and imaginary coupling,” Opt. Lett. 39(2), 335–338 (2014).
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N. Sapra, D. Vercruysse, L. Su, K. Yang, J. Skarda, A. Piggott, and J. Vuckovic, “Inverse design and demonstration of broadband grating couplers,” IEEE J. Sel. Top. Quantum Electron. 25(3), 1–7 (2019).
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A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
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Schneider, P.-I.

P.-I. Schneider, X. Santiago, V. Soltwisch, M. Hammerschmidt, S. Burger, and C. Rockstuhl, “Benchmarking five global optimization approaches for nano-optical shape optimization and parameter reconstruction,” https://arxiv.org/abs/1809.06674 .

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S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
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P.-I. Schneider, X. Santiago, V. Soltwisch, M. Hammerschmidt, S. Burger, and C. Rockstuhl, “Benchmarking five global optimization approaches for nano-optical shape optimization and parameter reconstruction,” https://arxiv.org/abs/1809.06674 .

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A. Jones, C. DeRose, A. Lentine, D. Trotter, A. Starbuck, and R. Norwood, “Ultra-low crosstalk, CMOS compatible waveguide crossings for densely integrated photonic interconnection networks,” Opt. Express 21(10), 12002–12013 (2013).
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N. Sapra, D. Vercruysse, L. Su, K. Yang, J. Skarda, A. Piggott, and J. Vuckovic, “Inverse design and demonstration of broadband grating couplers,” IEEE J. Sel. Top. Quantum Electron. 25(3), 1–7 (2019).
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A. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
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H. Liu, H. Tam, P. Wai, and E. Pun, “Low-loss waveguide crossing using a multimode interference structure,” Opt. Commun. 241(1-3), 99–104 (2004).
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A. Jones, C. DeRose, A. Lentine, D. Trotter, A. Starbuck, and R. Norwood, “Ultra-low crosstalk, CMOS compatible waveguide crossings for densely integrated photonic interconnection networks,” Opt. Express 21(10), 12002–12013 (2013).
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A. Piggott, J. Lu, K. Lagoudakis, J. Petykiewicz, T. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
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J. Lu and J. Vuckovic, “Nanophotonic computational design,” Opt. Express 21(11), 13351–13367 (2013).
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J. Lu and J. Vuckovic, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20(7), 7221–7236 (2012).
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J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express 18(4), 3793–3804 (2010).
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H. Liu, H. Tam, P. Wai, and E. Pun, “Low-loss waveguide crossing using a multimode interference structure,” Opt. Commun. 241(1-3), 99–104 (2004).
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K. Warnick, R. Selfridge, and D. Arnold, “Teaching electromagnetic field theory using differential forms,” IEEE Trans. Educ. 40(1), 53–68 (1997).
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S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
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Appl. Phys. Lett. (1)

L. Sanchis, A. Håkansson, D. López-Zanón, J. Bravo-Abad, and J. Sánchez-Dehesa, “Integrated optical devices design by genetic algorithm,” Appl. Phys. Lett. 84(22), 4460–4462 (2004).
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Flow, Turbul. Combust. (1)

M. Giles and N. Pierce, “An introduction to the adjoint approach to design,” Flow, Turbul. Combust. 65(3/4), 393–415 (2000).
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IEEE J. Quantum Electron. (1)

C.-H. Chen and C.-H. Chiu, “Taper-integrated multimode-interference based waveguide crossing design,” IEEE J. Quantum Electron. 46(11), 1656–1661 (2010).
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IEEE J. Sel. Top. Quantum Electron. (1)

N. Sapra, D. Vercruysse, L. Su, K. Yang, J. Skarda, A. Piggott, and J. Vuckovic, “Inverse design and demonstration of broadband grating couplers,” IEEE J. Sel. Top. Quantum Electron. 25(3), 1–7 (2019).
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IEEE Trans. Antennas Propag. (1)

J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Trans. Antennas Propag. 52(2), 397–407 (2004).
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IEEE Trans. Educ. (1)

K. Warnick, R. Selfridge, and D. Arnold, “Teaching electromagnetic field theory using differential forms,” IEEE Trans. Educ. 40(1), 53–68 (1997).
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J. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Laser Photonics Rev. 5(2), 308–321 (2011).
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Nat. Photonics (2)

A. Piggott, J. Lu, K. Lagoudakis, J. Petykiewicz, T. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015).
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S. Molesky, Z. Lin, A. Piggott, W. Jin, J. Vuckovic, and A. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018).
[Crossref]

Opt. Commun. (1)

H. Liu, H. Tam, P. Wai, and E. Pun, “Low-loss waveguide crossing using a multimode interference structure,” Opt. Commun. 241(1-3), 99–104 (2004).
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Opt. Express (15)

A. Jones, C. DeRose, A. Lentine, D. Trotter, A. Starbuck, and R. Norwood, “Ultra-low crosstalk, CMOS compatible waveguide crossings for densely integrated photonic interconnection networks,” Opt. Express 21(10), 12002–12013 (2013).
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H.-L. Han, H. Li, X.-P. Zhang, A. Liu, T.-Y. Lin, Z. Chen, H.-B. Lu, M.-H. Lu, X.-P. Liu, and Y.-F. Chen, “High performance ultra-compact SOI waveguide crossing,” Opt. Express 26(20), 25602–25610 (2018).
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L. Frellsen, Y. Ding, O. Sigmund, and L. Frandsen, “Topology optimized mode multiplexing in silicon-on-insulator photonic wire waveguides,” Opt. Express 24(15), 16866–16873 (2016).
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A. Michaels and E. Yablonovitch, “Inverse design of near unity efficiency perfectly vertical grating couplers,” Opt. Express 26(4), 4766–4779 (2018).
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A. Michaels and E. Yablonovitch, “Leveraging continuous material averaging for inverse electromagnetic design,” Opt. Express 26(24), 31717–31737 (2018).
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B. West and S. Honkanen, “MMI devices with weak guiding designed in three dimensions using a genetic algorithm,” Opt. Express 12(12), 2716–2722 (2004).
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G. Roelkens, D. Van Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-Silicon overlay,” Opt. Express 14(24), 11622–11630 (2006).
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J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express 18(4), 3793–3804 (2010).
[Crossref]

J. Lu and J. Vuckovic, “Objective-first design of high-efficiency, small-footprint couplers between arbitrary nanophotonic waveguide modes,” Opt. Express 20(7), 7221–7236 (2012).
[Crossref]

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

Y. Ma, Y. Zhang, S. Yang, A. Novack, R. Ding, A. Lim, G.-Q. Lo, T. Baehr-Jones, and M. Hochberg, “Ultralow loss single layer submicron silicon waveguide crossing for SOI optical interconnect,” Opt. Express 21(24), 29374–29382 (2013).
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C. Lalau-Keraly, S. Bhargava, O. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693–21701 (2013).
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A. Niederberger, D. Fattal, N. Gauger, S. Fan, and R. Beausoleil, “Sensitivity analysis and optimization of sub-wavelength optical gratings using adjoints,” Opt. Express 22(11), 12971–12981 (2014).
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Phys. Rev. E (1)

S. Johnson, M. Ibanescu, M. Skorobogaitiy, O. Weisberg, J. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E 65(6), 066611 (2002).
[Crossref]

Sci. Rep. (2)

A. Piggott, J. Lu, T. Babinec, K. Lagoudakis, J. Petykiewicz, and J. Vuckovic, “Inverse design and implementation of a wavelength demultiplexing grating coupler,” Sci. Rep. 4(1), 7210 (2015).
[Crossref]

A. Piggott, J. Petykiewicz, L. Su, and J. Vuckovic, “Fabrication-constrained nanophotonic inverse design,” Sci. Rep. 7(1), 1786 (2017).
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Other (10)

S. Bhargava and E. Yablonovitch, “Multi-objective inverse design of sub-wavelength optical focusing structures for heat assisted magnetic recording,” Proc. SPIE 9201, Optical Data Storage 2014, 92010M (5 September 2014).

Y. Ma, Y. Liu, R. Ding, T. Baehr-Jones, P. Magill, H. Guan, A. Gazman, Q. Li, K. Bergman, and M. Hochberg, “Optimized silicon photonics components for high-performance interconnect systems,” in IEEE Photonics Conference (IPC), Reston, VA, pp. 353–354 (2015).

P.-I. Schneider, X. Santiago, V. Soltwisch, M. Hammerschmidt, S. Burger, and C. Rockstuhl, “Benchmarking five global optimization approaches for nano-optical shape optimization and parameter reconstruction,” https://arxiv.org/abs/1809.06674 .

O. Miller, PhD thesis (2012), University of California at Berkeley, http://optoelectronics.eecs.berkeley.edu/ThesisOwenMiller.pdf

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A. Lentine, C. DeRose, P. Davids, N. Martinez, W. Zortman, J. Cox, A. Jones, D. Trotter, A. Pomerene, A. Starbuck, D. Savignon, T. Bauer, M. Wiwi, and P. Chu, “Silicon photonics platform for national security applications,” in IEEE Aerospace Conference, Big Sky, MT, (2015)..
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Figures (9)

Fig. 1.
Fig. 1. Depiction of the general adjoint shape optimization problem. Here Ω is the domain with background permittivity ɛ1. An electromagnetic forward source illuminates a structure Ψ with boundary ∂Ψ, permittivity ɛ2 and outward surface normal $\mathop{r_n}\limits^{\rightharpoonup} $. The figure of merit is defined over a subdomain of Ω, Φ. The goal is to optimize the figure of merit by applying successive small deformations to ∂Ψ using a shape gradient calculated from fields at the shape boundary which are determined by illumination from the forward source followed by illumination from the adjoint source. Adapted from [17].
Fig. 2.
Fig. 2. (a) Depiction of triangular basis functions used to represent the piecewise linear polygon shape with the basis function for index 5 shown in green bold, (b) the basis function weights for the x-coordinates of an ellipse, (c) the basis function weights for the y-coordinates of an ellipse and (d) the ellipse shown with black solid circles at vertex points with two weight functions shown along the edge of the ellipse with different basis function widths (BFW).
Fig. 3.
Fig. 3. Schematic of the shape optimization problem for the 90-degree SOI coplanar waveguide crossing where the goal is to maximize the through power while minimizing the cross power. The initial crossing shape is shown here and consists of single-mode waveguide feeds connecting to a short linear taper into a multimode section with smoothing at the interior corners.
Fig. 4.
Fig. 4. First quadrant waveguide crossing polygon shape profiles for different basis function widths (BFW). The shapes for 200 nm and 300 nm BFW are offset by 200 and 400 nm in each direction, respectively, for clarity. A large improvement in manufacturability (i.e. no narrow gaps) occurs by increasing the BFW from 100 to 200 nm.
Fig. 5.
Fig. 5. Simulated waveguide crossing performance parameters, worst-case aggregated over the design wavelengths 1520 nm, 1550 nm and 1580 nm, versus iteration number: (a) through power, (b) cross power extinction ratio (ER), (c) radiated power and (d) reflected power. Note the large dip in device performance near iteration 620 for both update methods. This is due to the transition from 2D to 3D simulation and optimization. The Levenberg-Marquardt (LM) update yields waveguide crossings superior to gradient descent (GD) in terms of through power and radiated power however the GD crossing has an extinction ratio about 3 dB larger and a reflected power about 5 dB smaller.
Fig. 6.
Fig. 6. Simulated (3D domain) performance parameter spectra for different figure-of-merit (FOM) weight combinations and for gradient descent (GD) and Levenberg-Marquardt (LM) optimization methods: (a) through power, (b) cross power extinction ratio, (c) reflected power and (d) radiated power. For the cases tested here LM produces devices whose performance parameters are more broadband than GD except for the extinction ratio in the FOM = (1, -1.5) case.
Fig. 7.
Fig. 7. Top-down SEM micrograph of the resist profile of an optimized waveguide crossing with total width of 10 µm.
Fig. 8.
Fig. 8. Layout for the edge-coupled (a) through and (b) cross power tests. The through power test consists of waveguide crossing arrays with 10 µm tapers on the unused ports to reduce back reflections. The cross power test consists of a single waveguide crossing with the output waveguides jogged upwards (not shown) by 2 mm to ensure that the scattered input light does not interfere with the measurement.
Fig. 9.
Fig. 9. Measured and simulated performance parameter spectra for a device designed with 250 nm Si thickness and given + 35 nm shape bias on all sides but fabricated with roughly 230 nm thickness: (a) through power and (b) cross power extinction ratio. The simulated parameters are calculated for shapes that have been biased by growing or shrinking the shapes in 2.5 nm increments along the vertex normal vectors from -50 nm up to 50 nm. The through power matches up well with the 0 nm bias curve. The measured cross power extinction ratio peaks at 50 dB at 1600 nm, is lower than the simulated 0 bias peak and is shifted towards longer wavelengths.

Tables (1)

Tables Icon

Table 1. Normalized Simulated Performance Parameters for Optimized SOI Waveguide Crossings for Different basis function widths (BFW) and references for comparison. Aggregation is performed over the design wavelengths 1520, 1550 and 1580 nm, except for Ref. [41] where the worst case over 1100-1300 nm free space wavelength was taken.

Equations (32)

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F t o t p i = R e k = 1 N l = 1 M α k l ( j φ F k l Ψ r n p i [ ( ε 2 k ε 1 k ) E A | | k l m T E F | | k l + ( 1 ε 1 k 1 ε 2 k ) D A k l m T D F k l ] d A )
F t o t p i = R e k = 1 N l = 1 M α k l ( j φ F k l Ψ W i n p i [ ( ε 2 k ε 1 k ) E A | | k l m T E F | | k l + ( 1 ε 1 k 1 ε 2 k ) D A k l m T D F k l ] d A )
Δ p p n e w p o l d = l p F
F ( p o l d + Δ p ) = F ( p o l d ) + p F ( p o l d ) T Δ p + 1 2 Δ p T H Δ p + O ( n 3 )
H k l = 2 F p k p l
0 = Δ p ( F ( p o l d ) + p F ( p o l d ) T Δ p + 1 2 Δ p T H Δ p ) = p F ( p o l d ) T + Δ p T H
Δ p = H 1 p F ( p o l d )
H a = l ( p F p F T + λ m a x ( d i a g ( p F p F T ) ) I )
F ( E F ) = Ω f ( E F ) d v
F p i = Ω f E F T E F p i d v
d E F = j ω B F
d H F = j ω D F + J F
d D F = 0
d B F = 0
B F = μ H F
D F = ε E F
A E F = 1 j ω d μ 1 d j ω ε
A E F E F = J F
E F A E F p i + A E F E F p i = J F p i
E F p i = A E F 1 ( J F p i A E F p i E F )
F p i = 2 R e ( Ω f E F T A E F 1 ( A E F p i E F ) d v )
A E F T E A = f E F
F p i = 2 R e ( Ω E A T A E F p i E F d v )
E A T = φ F E A m T
F p i = 2 R e ( φ F Ω E A m T A E F p i E F d v )
A E F p i = 1 j ω d μ 1 p i d j ω ε p i
A E F p i = j ω ε p i
F p i = 2 ω Re ( j φ F Ψ E A m T ε r n r n p i E F d v )
F p i = 2 ω Re ( j φ F Ψ E A m T ε r n r n p i E F d r n d A )
F p i = 2 ω Re ( j φ F Ψ r n p i [ ( ε 2 ε 1 ) E A | | m T E F | | + ( 1 ε 1 1 ε 2 ) D A m T D F ] d A )
F t o t = k = 1 N l = 1 M α k l F k l
F t o t p i = R e k = 1 N l = 1 M α k l ( j φ F k l Ψ r n p i [ ( ε 2 k ε 1 k ) E A | | k l m T E F | | k l + ( 1 ε 1 k 1 ε 2 k ) D A k l m T D F k l ] d A )

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