Abstract

Numerical optimization of photonic devices is often limited by a large design space the finite-differences gradient method requires as many electric field computations as there are design parameters. Adjoint-based optimization can deliver the same gradients with only two electric field computations. Here, we derive the relevant adjoint formalism and illustrate its application for a waveguide slab, and for the design of optical sub-wavelength gratings.

© 2014 Optical Society of America

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  1. J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, 1971).
    [CrossRef]
  2. R. Becker, R. Rannacher, “An optimal control approach to error control and mesh adaption in finite element methods,” Acta Numerica 10, 1–102 (2001).
    [CrossRef]
  3. K. Eriksson, D. Estep, P. Hansbo, C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica 4, 105–158 (1995).
    [CrossRef]
  4. D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” Parallel Data Processing 1, 318–362 (1986).
  5. J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999).
    [CrossRef]
  6. J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft 36(1), 61–74 (1999).
    [CrossRef]
  7. Y. seek Chung, Changyul-Cheon, I.-H. Park, S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on 48, 2289–2296 (2000).
    [CrossRef]
  8. N. Georgieva, S. Glavic, M. Bakr, J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on 50, 2751–2758 (2002).
    [CrossRef]
  9. N. K. Nikolova, H. W. Tam, M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on 52, 1207–1216 (2004).
    [CrossRef]
  10. N. K. Nikolova, Y. Li, Y. Li, M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on 54, 1598–1610 (2006).
    [CrossRef]
  11. G. Veronis, R. W. Dutton, S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters 29, 2288–2290 (2004).
    [CrossRef] [PubMed]
  12. Y. Jiao, S. Fan, D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters 30, 302–304 (2005).
    [CrossRef]
  13. P. Seliger, M. Mahvash, C. Wang, A. F. J. Levi, “Optimization of aperiodic dielectric structures,” Journal of Applied Physics 100, 034310 (2006).
    [CrossRef]
  14. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express 21, 21693–21701 (2013).
    [CrossRef] [PubMed]
  15. O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
    [CrossRef] [PubMed]
  16. D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010).
    [CrossRef]
  17. V. Liu, D. Miller, S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE 101, 484–493 (2013).
    [CrossRef]
  18. J. Lu, Vučković, “Inverse design of nanophotonic structures using complementary convex optimization,” Optics Express 18, 3793–3804 (2010).
    [CrossRef] [PubMed]
  19. R. Courant, K. Friedrichs, H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen 100, 32–74 (1928).
    [CrossRef]
  20. K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” Antennas and Propagation, IEEE Transactions on 14, 302–307 (1966).
    [CrossRef]
  21. D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
    [CrossRef]
  22. G. Roelkens, D. V. Thourhout, R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express 14, 11622–11630 (2006).
    [CrossRef] [PubMed]
  23. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
    [CrossRef]

2014 (1)

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

2013 (2)

V. Liu, D. Miller, S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE 101, 484–493 (2013).
[CrossRef]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express 21, 21693–21701 (2013).
[CrossRef] [PubMed]

2010 (3)

J. Lu, Vučković, “Inverse design of nanophotonic structures using complementary convex optimization,” Optics Express 18, 3793–3804 (2010).
[CrossRef] [PubMed]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010).
[CrossRef]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[CrossRef]

2006 (3)

G. Roelkens, D. V. Thourhout, R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express 14, 11622–11630 (2006).
[CrossRef] [PubMed]

P. Seliger, M. Mahvash, C. Wang, A. F. J. Levi, “Optimization of aperiodic dielectric structures,” Journal of Applied Physics 100, 034310 (2006).
[CrossRef]

N. K. Nikolova, Y. Li, Y. Li, M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on 54, 1598–1610 (2006).
[CrossRef]

2005 (1)

Y. Jiao, S. Fan, D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters 30, 302–304 (2005).
[CrossRef]

2004 (2)

N. K. Nikolova, H. W. Tam, M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on 52, 1207–1216 (2004).
[CrossRef]

G. Veronis, R. W. Dutton, S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters 29, 2288–2290 (2004).
[CrossRef] [PubMed]

2002 (2)

N. Georgieva, S. Glavic, M. Bakr, J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on 50, 2751–2758 (2002).
[CrossRef]

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[CrossRef]

2001 (1)

R. Becker, R. Rannacher, “An optimal control approach to error control and mesh adaption in finite element methods,” Acta Numerica 10, 1–102 (2001).
[CrossRef]

2000 (1)

Y. seek Chung, Changyul-Cheon, I.-H. Park, S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on 48, 2289–2296 (2000).
[CrossRef]

1999 (2)

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999).
[CrossRef]

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft 36(1), 61–74 (1999).
[CrossRef]

1995 (1)

K. Eriksson, D. Estep, P. Hansbo, C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica 4, 105–158 (1995).
[CrossRef]

1986 (1)

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” Parallel Data Processing 1, 318–362 (1986).

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” Antennas and Propagation, IEEE Transactions on 14, 302–307 (1966).
[CrossRef]

1928 (1)

R. Courant, K. Friedrichs, H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen 100, 32–74 (1928).
[CrossRef]

Alonso, J. J.

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999).
[CrossRef]

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft 36(1), 61–74 (1999).
[CrossRef]

Baets, R.

G. Roelkens, D. V. Thourhout, R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express 14, 11622–11630 (2006).
[CrossRef] [PubMed]

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[CrossRef]

Bakr, M.

N. Georgieva, S. Glavic, M. Bakr, J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on 50, 2751–2758 (2002).
[CrossRef]

Bakr, M. H.

N. K. Nikolova, Y. Li, Y. Li, M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on 54, 1598–1610 (2006).
[CrossRef]

N. K. Nikolova, H. W. Tam, M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on 52, 1207–1216 (2004).
[CrossRef]

Bandler, J.

N. Georgieva, S. Glavic, M. Bakr, J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on 50, 2751–2758 (2002).
[CrossRef]

Beausoleil, R. G.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010).
[CrossRef]

Becker, R.

R. Becker, R. Rannacher, “An optimal control approach to error control and mesh adaption in finite element methods,” Acta Numerica 10, 1–102 (2001).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[CrossRef]

Bhargava, S.

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express 21, 21693–21701 (2013).
[CrossRef] [PubMed]

Bienstman, P.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[CrossRef]

Bogaerts, W.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[CrossRef]

Changyul-Cheon,

Y. seek Chung, Changyul-Cheon, I.-H. Park, S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on 48, 2289–2296 (2000).
[CrossRef]

Courant, R.

R. Courant, K. Friedrichs, H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen 100, 32–74 (1928).
[CrossRef]

De Mesel, K.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[CrossRef]

DeLacy, B. G.

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

Dutton, R. W.

G. Veronis, R. W. Dutton, S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters 29, 2288–2290 (2004).
[CrossRef] [PubMed]

Eriksson, K.

K. Eriksson, D. Estep, P. Hansbo, C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica 4, 105–158 (1995).
[CrossRef]

Estep, D.

K. Eriksson, D. Estep, P. Hansbo, C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica 4, 105–158 (1995).
[CrossRef]

Fan, S.

V. Liu, D. Miller, S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE 101, 484–493 (2013).
[CrossRef]

Y. Jiao, S. Fan, D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters 30, 302–304 (2005).
[CrossRef]

G. Veronis, R. W. Dutton, S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters 29, 2288–2290 (2004).
[CrossRef] [PubMed]

Fattal, D.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010).
[CrossRef]

Fiorentino, M.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010).
[CrossRef]

Friedrichs, K.

R. Courant, K. Friedrichs, H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen 100, 32–74 (1928).
[CrossRef]

Georgieva, N.

N. Georgieva, S. Glavic, M. Bakr, J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on 50, 2751–2758 (2002).
[CrossRef]

Glavic, S.

N. Georgieva, S. Glavic, M. Bakr, J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on 50, 2751–2758 (2002).
[CrossRef]

Hahn, S.-Y.

Y. seek Chung, Changyul-Cheon, I.-H. Park, S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on 48, 2289–2296 (2000).
[CrossRef]

Hansbo, P.

K. Eriksson, D. Estep, P. Hansbo, C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica 4, 105–158 (1995).
[CrossRef]

Hinton, G. E.

D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” Parallel Data Processing 1, 318–362 (1986).

Hsu, C. W.

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[CrossRef]

Jameson, A.

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft 36(1), 61–74 (1999).
[CrossRef]

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999).
[CrossRef]

Jiao, Y.

Y. Jiao, S. Fan, D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters 30, 302–304 (2005).
[CrossRef]

Joannopoulos, J. D.

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[CrossRef]

Johnson, C.

K. Eriksson, D. Estep, P. Hansbo, C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica 4, 105–158 (1995).
[CrossRef]

Johnson, S. G.

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[CrossRef]

Krauss, T.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[CrossRef]

Lalau-Keraly, C. M.

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express 21, 21693–21701 (2013).
[CrossRef] [PubMed]

Levi, A. F. J.

P. Seliger, M. Mahvash, C. Wang, A. F. J. Levi, “Optimization of aperiodic dielectric structures,” Journal of Applied Physics 100, 034310 (2006).
[CrossRef]

Lewy, H.

R. Courant, K. Friedrichs, H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen 100, 32–74 (1928).
[CrossRef]

Li, J.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010).
[CrossRef]

Li, Y.

N. K. Nikolova, Y. Li, Y. Li, M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on 54, 1598–1610 (2006).
[CrossRef]

N. K. Nikolova, Y. Li, Y. Li, M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on 54, 1598–1610 (2006).
[CrossRef]

Lions, J. L.

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, 1971).
[CrossRef]

Liu, V.

V. Liu, D. Miller, S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE 101, 484–493 (2013).
[CrossRef]

Lu, J.

J. Lu, Vučković, “Inverse design of nanophotonic structures using complementary convex optimization,” Optics Express 18, 3793–3804 (2010).
[CrossRef] [PubMed]

Mahvash, M.

P. Seliger, M. Mahvash, C. Wang, A. F. J. Levi, “Optimization of aperiodic dielectric structures,” Journal of Applied Physics 100, 034310 (2006).
[CrossRef]

Miller, D.

V. Liu, D. Miller, S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE 101, 484–493 (2013).
[CrossRef]

Miller, D. A. B.

Y. Jiao, S. Fan, D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters 30, 302–304 (2005).
[CrossRef]

Miller, O. D.

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express 21, 21693–21701 (2013).
[CrossRef] [PubMed]

Moerman, I.

D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
[CrossRef]

Nikolova, N. K.

N. K. Nikolova, Y. Li, Y. Li, M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on 54, 1598–1610 (2006).
[CrossRef]

N. K. Nikolova, H. W. Tam, M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on 52, 1207–1216 (2004).
[CrossRef]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010).
[CrossRef]

Park, I.-H.

Y. seek Chung, Changyul-Cheon, I.-H. Park, S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on 48, 2289–2296 (2000).
[CrossRef]

Peng, Z.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010).
[CrossRef]

Qiu, W.

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

Rannacher, R.

R. Becker, R. Rannacher, “An optimal control approach to error control and mesh adaption in finite element methods,” Acta Numerica 10, 1–102 (2001).
[CrossRef]

Reid, M. T. H.

O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
[CrossRef] [PubMed]

Remlinger, M. J.

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999).
[CrossRef]

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft 36(1), 61–74 (1999).
[CrossRef]

Reuther, J.

J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999).
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J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft 36(1), 61–74 (1999).
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J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999).
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Y. seek Chung, Changyul-Cheon, I.-H. Park, S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on 48, 2289–2296 (2000).
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N. K. Nikolova, H. W. Tam, M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on 52, 1207–1216 (2004).
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G. Roelkens, D. V. Thourhout, R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express 14, 11622–11630 (2006).
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D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
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[CrossRef]

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[CrossRef]

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Optics Express (3)

J. Lu, Vučković, “Inverse design of nanophotonic structures using complementary convex optimization,” Optics Express 18, 3793–3804 (2010).
[CrossRef] [PubMed]

C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express 21, 21693–21701 (2013).
[CrossRef] [PubMed]

G. Roelkens, D. V. Thourhout, R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express 14, 11622–11630 (2006).
[CrossRef] [PubMed]

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G. Veronis, R. W. Dutton, S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters 29, 2288–2290 (2004).
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Y. Jiao, S. Fan, D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters 30, 302–304 (2005).
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D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” Parallel Data Processing 1, 318–362 (1986).

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O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014).
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D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002).
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Figures (4)

Fig. 1
Fig. 1

The derivative of the longitudinal wave vector β with respect to the thickness of a waveguide slab can faithfully be reproduced using adjoints: the comparison between the direct calculation (blue line) and adjoint calculation (red dots) of d d h β shows that the two methods deliver exactly the same result. However, the adjoint method requires no further expensive solutions of the eigenvalue system in order to determine the derivatives of β with respect to other design variables. Thus, the adjoint method delivers the relevant gradients of a system depending on multiple design parameters faster than the finite-differences approach. The inset shows a schematic of the waveguide slab structure comprised of the cover material nc, the guiding film nf, and the substrate ns.

Fig. 2
Fig. 2

Our SiO2 grating coupler redirects free-space plane-waves (vertical arrows) to a guided mode inside the waveguide (horizontal arrows). We show the device at (a) a pitch of 0.58 and a duty cycle of 0.2, and (b) at a pitch of 0.68 and a duty cycle of 0.8. Both the pitch and the duty cycle are optimized to maximize the coupling between the incident light and the light inside the waveguide.

Fig. 3
Fig. 3

Adjoint-based optimization convergence: from three initial parameters (stars) in parameter regions of poor optical coupling (blue, green, yellow), adjoint-based optimization evolves the design parameters to the parameter regions of optimal coupling (dark red). The computation of the relevant gradients requires only two Maxwell solutions, independently of the number of design parameters. The finite differences approach involves roughly one Maxwell solution per design variable. Thus, this figure shows that adjoints help optimize the system faster than the finite-differences approach.

Fig. 4
Fig. 4

Electric field in and around the waveguide (outlined in light blue) before and after adjoint-based optimization: (a) the grating with the initial design parameters couples the incident light (top) poorly to the waveguide (inside black box), with a complex amplitude of 0.39 (b) After adjoint-based optimization, the grating couples the light very well to the waveguide (inside black box), with a complex amplitude of 0.88.

Equations (31)

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

d T d p k = i T x i d x i d p k + T p k = T x d x d p k + T p k ,
d x d p k 1 Δ p k ( x ( p Δ p k x ( p ) ) ,
R ( x , p ) = 0 .
d R d p k = R x d x d p k + R p k = 0 .
d T d p k = T x d x d p k + T p k v ( R x d x d p k + R p k ) .
d T d p k = T p k v R p k + ( T x v R x ) d x d p k .
T x = v R x ,
d T d p k = T p k v R p k .
× H = D t + J σ + J × E = B t ,
B = μ H D = ε E J σ = σ E ,
× μ 1 × E + ε 2 E t 2 + σ E t = J t .
R ( E ¯ , p ) = E ¯ i ω J ¯ = 0 ,
E ¯ = × μ 1 × E ¯ ω 2 ε E ¯ .
T ( E ¯ , p ) = 1 N Ω d Ω | E ¯ | 2 = E ¯ 0 E ¯ ,
v ¯ i ω ( 1 i ω E ¯ 0 ) = 0 .
d T d p k = v ε p k E ¯ .
tan ( h κ f ) = γ c + γ s κ f ( 1 γ c γ s κ f 2 ) ,
R = tan ( h κ f ) γ c + γ s κ f ( 1 γ c γ s κ f 2 ) ,
T = β ,
T β = 1 = v R β .
d T d p k = d β d p k = v R p k .
R β = β ( tan ( h κ f ) ) β ( γ c + γ s κ f ( 1 γ c γ s κ f 2 ) ) ,
β ( tan ( h κ f ) ) = h β κ f cos 2 ( h κ f ) ,
β ( γ c + γ s q ) = β γ c + β γ s q γ c + γ s q 2 q β ,
q β = β κ f [ 1 + γ s γ c + γ c γ s + γ c γ s κ f 2 ] .
1 v = β [ h κ f cos 2 ( h κ f ) + ( γ c + γ s ) κ f κ f 2 γ c γ s ( 1 γ s γ c + 1 + γ s γ c + γ c γ s + γ c γ s κ f 2 κ f 2 γ c γ s ) ] ,
v = 1 R β
d β d p k = v R p k .
d R d h = κ f cos 2 ( h κ f )
d β d h = v κ f cos 2 ( h κ f ) .
p = ( p d )

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