Abstract

We present a method for obtaining accurate numerical design sensitivities for metal-optical nanostructures. Adjoint design sensitivity analysis, long used in fluid mechanics and mechanical engineering for both optimization and structural analysis, is beginning to be used for nano-optics design, but it fails for sharp-cornered metal structures because the numerical error in electromagnetic simulations of metal structures is highest at sharp corners. These locations feature strong field enhancement and contribute strongly to design sensitivities. By using high-accuracy FEM calculations and rounding sharp features to a finite radius of curvature we obtain highly-accurate design sensitivities for 3D metal devices. To provide a bridge to the existing literature on adjoint methods in other fields, we derive the sensitivity equations for Maxwell’s equations in the PDE framework widely used in fluid mechanics.

© 2015 Optical Society of America

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References

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2014 (4)

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

A. C. Niederberger, D. A. Fattal, N. R. Gauger, S. Fan, and R. G. Beausoleil, “Sensitivity analysis and optimization of sub-wavelength optical gratings using adjoints,” Opt. Express 22, 12971–12981 (2014).
[Crossref] [PubMed]

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Y. Zhang, O. S. Ahmed, and M. H. Bakr, “Adjoint sensitivity analysis of plasmonic structures using the fdtd method,” Optics letters 39, 3002–3005 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2009 (1)

2008 (1)

2007 (2)

P. Hansen, L. Hesselink, and B. Leen, “Design of a subwavelength bent c-aperture waveguide,” Opt. Lett. 32, 1737–1739 (2007).
[Crossref] [PubMed]

R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization,” Swarm Intell. 1, 33–57 (2007).
[Crossref]

2006 (3)

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in the finite-difference time domain,” Opt. Lett. 31, 2972–2974 (2006).
[Crossref] [PubMed]

E. Abenius and B. Strand, “Solving inverse electromagnetic problems using fdtd and gradient-based minimization,” International journal for numerical methods in engineering 68, 650–673 (2006).
[Crossref]

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

2005 (2)

E. X. Jin and X. Xu, “Obtaining super resolution light spot using surface plasmon assisted sharp ridge nanoaperture,” Appl. Phys. Lett. 86, 111106 (2005).
[Crossref]

J. Xu, T. Xu, J. Wang, and Q. Tian, “Design tips of nanoapertures with strong field enhancement and proposal of novel l-shaped aperture,” Opt. Eng. 44, 018001 (2005).
[Crossref]

2004 (7)

N. A. Pierce and M. B. Giles, “Adjoint and defect error bounding and correction for functional estimates,” J. Comput. Phys. 200, 769–794 (2004).
[Crossref]

N. K. Nikolova, J. W. Bandler, and M. H. Bakr, “Adjoint techniques for sensitivity analysis in high-frequency structure cad,” IEEE Trans. Microwave Theory Tech. 52, 403–419 (2004).
[Crossref]

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

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
[Crossref]

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[Crossref] [PubMed]

L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[Crossref] [PubMed]

X. Shi and L. Hesselink, “Design of a C aperture to achieve λ/10 resolution and resonant transmission,” J. Opt. Soc. Am. B 21, 1305–1317 (2004).
[Crossref]

2003 (2)

X. Shi, L. Hesselink, and R. L. Thornton, “Ultrahigh light transmission through a C-shaped nanoaperture,” Opt. Lett. 28, 1320–1322 (2003).
[Crossref] [PubMed]

O. Sigmund and J. S. Jensen, “Systematic design of phononic band–gap materials and structures by topology optimization,” Philos. Trans. R. Soc. London, Ser. A 361, 1001–1019 (2003).
[Crossref]

2002 (1)

M. B. Giles and E. Süli, “Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality,” Acta Numer. 11, 145–236 (2002).
[Crossref]

2001 (2)

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. ii. fdtd case,” IEEE Trans. Magn. 37, 3255–3259 (2001).
[Crossref]

Y.-S. Chung, J. Ryu, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. i. fetd case,” IEEE Trans. Magn. 37, 3289–3293 (2001).
[Crossref]

2000 (2)

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[Crossref]

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

1999 (1)

A. Jameson, “Re-engineering the design process through computation,” J. Aircraft 36, 36–50 (1999).
[Crossref]

1998 (2)

A. Jameson, L. Martinelli, and N. Pierce, “Optimum aerodynamic design using the navier–stokes equations,” Theor. Comp. Fluid Dyn. 10, 213–237 (1998).
[Crossref]

J. D. Jackson, “Classical electrodynamics,” Classical Electrodynamics 1, 832 (1998).

1997 (1)

M. B. Giles and N. A. Pierce, “Adjoint equations in cfd: duality, boundary conditions and solution behaviour,” AIAA paper 97, 1850 (1997).

1996 (1)

L. Ingber and et al., “Adaptive simulated annealing (asa): Lessons learned,” Control and cybernetics 25, 33–54 (1996).

1995 (1)

A. Jameson, “Optimum aerodynamic design using cfd and control theory,” AIAA paper 1729, 124–131 (1995).

1988 (1)

A. Jameson, “Aerodynamic design via control theory,” J. Sci. Comput. 3, 233–260 (1988).
[Crossref]

1974 (1)

O. Pironneau, “On optimum design in fluid mechanics,” J. Fluid Mech. 64, 97–110 (1974).
[Crossref]

Abenius, E.

E. Abenius and B. Strand, “Solving inverse electromagnetic problems using fdtd and gradient-based minimization,” International journal for numerical methods in engineering 68, 650–673 (2006).
[Crossref]

Ahmed, O.

Ahmed, O. S.

Y. Zhang, O. S. Ahmed, and M. H. Bakr, “Adjoint sensitivity analysis of plasmonic structures using the fdtd method,” Optics letters 39, 3002–3005 (2014).
[Crossref] [PubMed]

Bakr, M.

Bakr, M. H.

Y. Zhang, O. S. Ahmed, and M. H. Bakr, “Adjoint sensitivity analysis of plasmonic structures using the fdtd method,” Optics letters 39, 3002–3005 (2014).
[Crossref] [PubMed]

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

N. K. Nikolova, J. W. Bandler, and M. H. Bakr, “Adjoint techniques for sensitivity analysis in high-frequency structure cad,” IEEE Trans. Microwave Theory Tech. 52, 403–419 (2004).
[Crossref]

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

Bandler, J. W.

N. K. Nikolova, J. W. Bandler, and M. H. Bakr, “Adjoint techniques for sensitivity analysis in high-frequency structure cad,” IEEE Trans. Microwave Theory Tech. 52, 403–419 (2004).
[Crossref]

Beausoleil, R. G.

Bermel, P.

Bhargava, S.

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] [PubMed]

S. Bhargava, O. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of optical antennas for sub-wavelength energy delivery,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), pp. CM2F–2.

S. Bhargava and E. Yablonovitch, “Multi-objective inverse design of sub-wavelength optical focusing structures for heat assisted magnetic recording,” in “SPIE Optical Engineering+ Applications,” (International Society for Optics and Photonics, 2014), 92010M.

Blackwell, T.

R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization,” Swarm Intell. 1, 33–57 (2007).
[Crossref]

Borel, P.

Boyd, S. P.

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

Burr, G.

Cheng, Y.-T.

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

J. B. Leen, P. Hansen, Y.-T. Cheng, and L. Hesselink, “Improved focused ion beam fabrication of near-field apertures using a silicon nitride membrane,” Opt. Lett. 33, 2827–2829 (2008).
[Crossref] [PubMed]

Cheon, C.

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. ii. fdtd case,” IEEE Trans. Magn. 37, 3255–3259 (2001).
[Crossref]

Y.-S. Chung, J. Ryu, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. i. fetd case,” IEEE Trans. Magn. 37, 3289–3293 (2001).
[Crossref]

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[Crossref]

Cho, E.

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

Choa, S.-H.

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

Choi, K. K.

V. Komkov, K. K. Choi, and E. J. Haug, Design Sensitivity Analysis of Structural Systems, vol. 177 (Academic, 1986).

Chung, Y.-S.

Y.-S. Chung, B.-J. Lee, and S.-C. Kim, “Optimal shape design of dielectric micro lens using fdtd and topology optimization,” J. Opt. Soc. Korea 13, 286–293 (2009).
[Crossref]

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. ii. fdtd case,” IEEE Trans. Magn. 37, 3255–3259 (2001).
[Crossref]

Y.-S. Chung, J. Ryu, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. i. fetd case,” IEEE Trans. Magn. 37, 3289–3293 (2001).
[Crossref]

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[Crossref]

Dadash, M. S.

M. S. Dadash, “Simulator independent exact adjoint sensitivity analysis of self-adjoint microwave structures,” Master’s thesis, McMaster University (2011).

Deng, Y.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Fan, S.

Farjadpour, A.

Fattal, D. A.

Frandsen, L.

Ganapati, V.

O. D. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of a nano-scale surface texture for light trapping,” in “CLEO: Science and Innovations,” (Optical Society of America, 2012), pp. CF2J–2.

S. Bhargava, O. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of optical antennas for sub-wavelength energy delivery,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), pp. CM2F–2.

Gauger, N. R.

Giles, M. B.

N. A. Pierce and M. B. Giles, “Adjoint and defect error bounding and correction for functional estimates,” J. Comput. Phys. 200, 769–794 (2004).
[Crossref]

M. B. Giles and E. Süli, “Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality,” Acta Numer. 11, 145–236 (2002).
[Crossref]

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

M. B. Giles and N. A. Pierce, “Adjoint equations in cfd: duality, boundary conditions and solution behaviour,” AIAA paper 97, 1850 (1997).

Hahn, S.-Y.

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. ii. fdtd case,” IEEE Trans. Magn. 37, 3255–3259 (2001).
[Crossref]

Y.-S. Chung, J. Ryu, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. i. fetd case,” IEEE Trans. Magn. 37, 3289–3293 (2001).
[Crossref]

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[Crossref]

Hansen, P.

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

J. B. Leen, P. Hansen, Y.-T. Cheng, and L. Hesselink, “Improved focused ion beam fabrication of near-field apertures using a silicon nitride membrane,” Opt. Lett. 33, 2827–2829 (2008).
[Crossref] [PubMed]

P. Hansen, L. Hesselink, and B. Leen, “Design of a subwavelength bent c-aperture waveguide,” Opt. Lett. 32, 1737–1739 (2007).
[Crossref] [PubMed]

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Adjoint fdtd for nanophotonic device optimization,” in “Joint International Symposium on Optical Memory and Optical Data Storage,” (Optical Society of America, 2011), p. OTuE2.

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Nanophotonic device optimization with adjoint fdtd,” in “CLEO: Applications and Technology,” (Optical Society of America, 2011), p. JTuI61.
[Crossref]

Hansen, P. C.

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

Harpøth, A.

Haug, E. J.

V. Komkov, K. K. Choi, and E. J. Haug, Design Sensitivity Analysis of Structural Systems, vol. 177 (Academic, 1986).

Hesselink, L.

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

J. B. Leen, P. Hansen, Y.-T. Cheng, and L. Hesselink, “Improved focused ion beam fabrication of near-field apertures using a silicon nitride membrane,” Opt. Lett. 33, 2827–2829 (2008).
[Crossref] [PubMed]

P. Hansen, L. Hesselink, and B. Leen, “Design of a subwavelength bent c-aperture waveguide,” Opt. Lett. 32, 1737–1739 (2007).
[Crossref] [PubMed]

X. Shi and L. Hesselink, “Design of a C aperture to achieve λ/10 resolution and resonant transmission,” J. Opt. Soc. Am. B 21, 1305–1317 (2004).
[Crossref]

X. Shi, L. Hesselink, and R. L. Thornton, “Ultrahigh light transmission through a C-shaped nanoaperture,” Opt. Lett. 28, 1320–1322 (2003).
[Crossref] [PubMed]

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Nanophotonic device optimization with adjoint fdtd,” in “CLEO: Applications and Technology,” (Optical Society of America, 2011), p. JTuI61.
[Crossref]

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Adjoint fdtd for nanophotonic device optimization,” in “Joint International Symposium on Optical Memory and Optical Data Storage,” (Optical Society of America, 2011), p. OTuE2.

Ibanescu, M.

Ingber, L.

L. Ingber and et al., “Adaptive simulated annealing (asa): Lessons learned,” Control and cybernetics 25, 33–54 (1996).

Jackson, J. D.

J. D. Jackson, “Classical electrodynamics,” Classical Electrodynamics 1, 832 (1998).

Jameson, A.

A. Jameson, “Re-engineering the design process through computation,” J. Aircraft 36, 36–50 (1999).
[Crossref]

A. Jameson, L. Martinelli, and N. Pierce, “Optimum aerodynamic design using the navier–stokes equations,” Theor. Comp. Fluid Dyn. 10, 213–237 (1998).
[Crossref]

A. Jameson, “Optimum aerodynamic design using cfd and control theory,” AIAA paper 1729, 124–131 (1995).

A. Jameson, “Aerodynamic design via control theory,” J. Sci. Comput. 3, 233–260 (1988).
[Crossref]

Jensen, J.

Jensen, J. S.

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
[Crossref]

O. Sigmund and J. S. Jensen, “Systematic design of phononic band–gap materials and structures by topology optimization,” Philos. Trans. R. Soc. London, Ser. A 361, 1001–1019 (2003).
[Crossref]

Jin, E. X.

E. X. Jin and X. Xu, “Obtaining super resolution light spot using surface plasmon assisted sharp ridge nanoaperture,” Appl. Phys. Lett. 86, 111106 (2005).
[Crossref]

X. Xu, E. X. Jin, and S. M. Uppuluri, “Enhancement of optical transmission through planar nanoapertures in a metal film,” in “Optical Science and Technology, the SPIE 49th Annual Meeting,” (International Society for Optics and Photonics, 2004), pp. 230–243.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (John Wiley & Sons, 2014).

Joannopoulos, J.

Johnson, S. G.

Kennedy, J.

R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization,” Swarm Intell. 1, 33–57 (2007).
[Crossref]

Kim, S.-C.

Komkov, V.

V. Komkov, K. K. Choi, and E. J. Haug, Design Sensitivity Analysis of Structural Systems, vol. 177 (Academic, 1986).

Kristensen, M.

Lalau-Keraly, C. M.

Lee, B.-J.

Leen, B.

Leen, J. B.

J. B. Leen, P. Hansen, Y.-T. Cheng, and L. Hesselink, “Improved focused ion beam fabrication of near-field apertures using a silicon nitride membrane,” Opt. Lett. 33, 2827–2829 (2008).
[Crossref] [PubMed]

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

Li, X.

Li, Y.

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

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

Lions, J. L.

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, vol. 170 (Springer Verlag, 1971).
[Crossref]

Liu, Y.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Liu, Z.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Loop, C.

C. Loop, “Smooth subdivision surfaces based on triangles,” Master’s thesis, University of Utah (1987).

Lu, T.-J.

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

Martinelli, L.

A. Jameson, L. Martinelli, and N. Pierce, “Optimum aerodynamic design using the navier–stokes equations,” Theor. Comp. Fluid Dyn. 10, 213–237 (1998).
[Crossref]

Miller, O.

S. Bhargava, O. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of optical antennas for sub-wavelength energy delivery,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), pp. CM2F–2.

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] [PubMed]

O. D. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of a nano-scale surface texture for light trapping,” in “CLEO: Science and Innovations,” (Optical Society of America, 2012), pp. CF2J–2.

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

Niederberger, A. C.

Nikolova, N. K.

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

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

N. K. Nikolova, J. W. Bandler, and M. H. Bakr, “Adjoint techniques for sensitivity analysis in high-frequency structure cad,” IEEE Trans. Microwave Theory Tech. 52, 403–419 (2004).
[Crossref]

Nomura, T.

Park, I.-H.

Y.-S. Chung, J. Ryu, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. i. fetd case,” IEEE Trans. Magn. 37, 3289–3293 (2001).
[Crossref]

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. ii. fdtd case,” IEEE Trans. Magn. 37, 3255–3259 (2001).
[Crossref]

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[Crossref]

Perederey, E.

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Adjoint fdtd for nanophotonic device optimization,” in “Joint International Symposium on Optical Memory and Optical Data Storage,” (Optical Society of America, 2011), p. OTuE2.

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Nanophotonic device optimization with adjoint fdtd,” in “CLEO: Applications and Technology,” (Optical Society of America, 2011), p. JTuI61.
[Crossref]

Pierce, N.

A. Jameson, L. Martinelli, and N. Pierce, “Optimum aerodynamic design using the navier–stokes equations,” Theor. Comp. Fluid Dyn. 10, 213–237 (1998).
[Crossref]

Pierce, N. A.

N. A. Pierce and M. B. Giles, “Adjoint and defect error bounding and correction for functional estimates,” J. Comput. Phys. 200, 769–794 (2004).
[Crossref]

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

M. B. Giles and N. A. Pierce, “Adjoint equations in cfd: duality, boundary conditions and solution behaviour,” AIAA paper 97, 1850 (1997).

Pironneau, O.

O. Pironneau, “On optimum design in fluid mechanics,” J. Fluid Mech. 64, 97–110 (1974).
[Crossref]

Poli, R.

R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization,” Swarm Intell. 1, 33–57 (2007).
[Crossref]

Rodriguez, A.

Roundy, D.

Ryan, J.

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

Ryu, J.

Y.-S. Chung, J. Ryu, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. i. fetd case,” IEEE Trans. Magn. 37, 3289–3293 (2001).
[Crossref]

Shi, P.

Shi, X.

Sigmund, O.

P. Borel, A. Harpøth, L. Frandsen, M. Kristensen, P. Shi, J. Jensen, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
[Crossref] [PubMed]

L. Frandsen, A. Harpøth, P. Borel, M. Kristensen, J. Jensen, and O. Sigmund, “Broadband photonic crystal waveguide 60 bend obtained utilizing topology optimization,” Opt. Express 12, 5916–5921 (2004).
[Crossref] [PubMed]

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
[Crossref]

O. Sigmund and J. S. Jensen, “Systematic design of phononic band–gap materials and structures by topology optimization,” Philos. Trans. R. Soc. London, Ser. A 361, 1001–1019 (2003).
[Crossref]

Sohn, J.-S.

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

Song, C.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Strand, B.

E. Abenius and B. Strand, “Solving inverse electromagnetic problems using fdtd and gradient-based minimization,” International journal for numerical methods in engineering 68, 650–673 (2006).
[Crossref]

Suh, S.-D.

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

Süli, E.

M. B. Giles and E. Süli, “Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality,” Acta Numer. 11, 145–236 (2002).
[Crossref]

Tam, H. W.

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

Thornton, R. L.

Tian, Q.

J. Xu, T. Xu, J. Wang, and Q. Tian, “Design tips of nanoapertures with strong field enhancement and proposal of novel l-shaped aperture,” Opt. Eng. 44, 018001 (2005).
[Crossref]

Uppuluri, S. M.

X. Xu, E. X. Jin, and S. M. Uppuluri, “Enhancement of optical transmission through planar nanoapertures in a metal film,” in “Optical Science and Technology, the SPIE 49th Annual Meeting,” (International Society for Optics and Photonics, 2004), pp. 230–243.

Vandenberghe, L.

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

Wang, J.

J. Xu, T. Xu, J. Wang, and Q. Tian, “Design tips of nanoapertures with strong field enhancement and proposal of novel l-shaped aperture,” Opt. Eng. 44, 018001 (2005).
[Crossref]

Wu, J.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Wu, Y.

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Xu, J.

J. Xu, T. Xu, J. Wang, and Q. Tian, “Design tips of nanoapertures with strong field enhancement and proposal of novel l-shaped aperture,” Opt. Eng. 44, 018001 (2005).
[Crossref]

Xu, T.

J. Xu, T. Xu, J. Wang, and Q. Tian, “Design tips of nanoapertures with strong field enhancement and proposal of novel l-shaped aperture,” Opt. Eng. 44, 018001 (2005).
[Crossref]

Xu, X.

E. X. Jin and X. Xu, “Obtaining super resolution light spot using surface plasmon assisted sharp ridge nanoaperture,” Appl. Phys. Lett. 86, 111106 (2005).
[Crossref]

X. Xu, E. X. Jin, and S. M. Uppuluri, “Enhancement of optical transmission through planar nanoapertures in a metal film,” in “Optical Science and Technology, the SPIE 49th Annual Meeting,” (International Society for Optics and Photonics, 2004), pp. 230–243.

Yablonovitch, E.

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] [PubMed]

S. Bhargava, O. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of optical antennas for sub-wavelength energy delivery,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), pp. CM2F–2.

S. Bhargava and E. Yablonovitch, “Multi-objective inverse design of sub-wavelength optical focusing structures for heat assisted magnetic recording,” in “SPIE Optical Engineering+ Applications,” (International Society for Optics and Photonics, 2014), 92010M.

O. D. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of a nano-scale surface texture for light trapping,” in “CLEO: Science and Innovations,” (Optical Society of America, 2012), pp. CF2J–2.

Zhang, Y.

Y. Zhang, O. S. Ahmed, and M. H. Bakr, “Adjoint sensitivity analysis of plasmonic structures using the fdtd method,” Optics letters 39, 3002–3005 (2014).
[Crossref] [PubMed]

Zheng, Y.

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Nanophotonic device optimization with adjoint fdtd,” in “CLEO: Applications and Technology,” (Optical Society of America, 2011), p. JTuI61.
[Crossref]

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Adjoint fdtd for nanophotonic device optimization,” in “Joint International Symposium on Optical Memory and Optical Data Storage,” (Optical Society of America, 2011), p. OTuE2.

Acta Numer. (1)

M. B. Giles and E. Süli, “Adjoint methods for pdes: a posteriori error analysis and postprocessing by duality,” Acta Numer. 11, 145–236 (2002).
[Crossref]

AIAA paper (2)

A. Jameson, “Optimum aerodynamic design using cfd and control theory,” AIAA paper 1729, 124–131 (1995).

M. B. Giles and N. A. Pierce, “Adjoint equations in cfd: duality, boundary conditions and solution behaviour,” AIAA paper 97, 1850 (1997).

Appl. Phys. Lett. (2)

J. S. Jensen and O. Sigmund, “Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,” Appl. Phys. Lett. 84, 2022–2024 (2004).
[Crossref]

E. X. Jin and X. Xu, “Obtaining super resolution light spot using surface plasmon assisted sharp ridge nanoaperture,” Appl. Phys. Lett. 86, 111106 (2005).
[Crossref]

Classical Electrodynamics (1)

J. D. Jackson, “Classical electrodynamics,” Classical Electrodynamics 1, 832 (1998).

Control and cybernetics (1)

L. Ingber and et al., “Adaptive simulated annealing (asa): Lessons learned,” Control and cybernetics 25, 33–54 (1996).

Flow Turbul. Combust. (1)

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

IEEE Trans. Magn. (2)

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. ii. fdtd case,” IEEE Trans. Magn. 37, 3255–3259 (2001).
[Crossref]

Y.-S. Chung, J. Ryu, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal design method for microwave device using time domain method and design sensitivity analysis. i. fetd case,” IEEE Trans. Magn. 37, 3289–3293 (2001).
[Crossref]

IEEE Trans. Microwave Theory Tech. (4)

N. K. Nikolova, J. W. Bandler, and M. H. Bakr, “Adjoint techniques for sensitivity analysis in high-frequency structure cad,” IEEE Trans. Microwave Theory Tech. 52, 403–419 (2004).
[Crossref]

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

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

Y.-S. Chung, C. Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” IEEE Trans. Microwave Theory Tech. 48, 2289–2296 (2000).
[Crossref]

International journal for numerical methods in engineering (1)

E. Abenius and B. Strand, “Solving inverse electromagnetic problems using fdtd and gradient-based minimization,” International journal for numerical methods in engineering 68, 650–673 (2006).
[Crossref]

J. Aircraft (1)

A. Jameson, “Re-engineering the design process through computation,” J. Aircraft 36, 36–50 (1999).
[Crossref]

J. Comput. Phys. (1)

N. A. Pierce and M. B. Giles, “Adjoint and defect error bounding and correction for functional estimates,” J. Comput. Phys. 200, 769–794 (2004).
[Crossref]

J. Fluid Mech. (1)

O. Pironneau, “On optimum design in fluid mechanics,” J. Fluid Mech. 64, 97–110 (1974).
[Crossref]

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

J. Opt. Soc. Korea (1)

J. Sci. Comput. (1)

A. Jameson, “Aerodynamic design via control theory,” J. Sci. Comput. 3, 233–260 (1988).
[Crossref]

Nano letters (1)

Y. Zheng, J. Ryan, P. Hansen, Y.-T. Cheng, T.-J. Lu, and L. Hesselink, “Nano-optical conveyor belt, part ii: Demonstration of handoff between near-field optical traps,” Nano letters 14, 2971–2976 (2014).
[Crossref] [PubMed]

Opt. Eng. (1)

J. Xu, T. Xu, J. Wang, and Q. Tian, “Design tips of nanoapertures with strong field enhancement and proposal of novel l-shaped aperture,” Opt. Eng. 44, 018001 (2005).
[Crossref]

Opt. Express (4)

Opt. Lett. (5)

Optics letters (1)

Y. Zhang, O. S. Ahmed, and M. H. Bakr, “Adjoint sensitivity analysis of plasmonic structures using the fdtd method,” Optics letters 39, 3002–3005 (2014).
[Crossref] [PubMed]

Philos. Trans. R. Soc. London, Ser. A (1)

O. Sigmund and J. S. Jensen, “Systematic design of phononic band–gap materials and structures by topology optimization,” Philos. Trans. R. Soc. London, Ser. A 361, 1001–1019 (2003).
[Crossref]

Plasmonics (1)

Y. Deng, Z. Liu, C. Song, J. Wu, Y. Liu, and Y. Wu, “Topology optimization-based computational design methodology for surface plasmon polaritons,” Plasmonics 10, 1–15 (2014).

Swarm Intell. (1)

R. Poli, J. Kennedy, and T. Blackwell, “Particle swarm optimization,” Swarm Intell. 1, 33–57 (2007).
[Crossref]

Theor. Comp. Fluid Dyn. (1)

A. Jameson, L. Martinelli, and N. Pierce, “Optimum aerodynamic design using the navier–stokes equations,” Theor. Comp. Fluid Dyn. 10, 213–237 (1998).
[Crossref]

Other (14)

V. Komkov, K. K. Choi, and E. J. Haug, Design Sensitivity Analysis of Structural Systems, vol. 177 (Academic, 1986).

J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, vol. 170 (Springer Verlag, 1971).
[Crossref]

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Nanophotonic device optimization with adjoint fdtd,” in “CLEO: Applications and Technology,” (Optical Society of America, 2011), p. JTuI61.
[Crossref]

P. Hansen, Y. Zheng, E. Perederey, and L. Hesselink, “Adjoint fdtd for nanophotonic device optimization,” in “Joint International Symposium on Optical Memory and Optical Data Storage,” (Optical Society of America, 2011), p. OTuE2.

O. D. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of a nano-scale surface texture for light trapping,” in “CLEO: Science and Innovations,” (Optical Society of America, 2012), pp. CF2J–2.

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

S. Bhargava and E. Yablonovitch, “Multi-objective inverse design of sub-wavelength optical focusing structures for heat assisted magnetic recording,” in “SPIE Optical Engineering+ Applications,” (International Society for Optics and Photonics, 2014), 92010M.

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

S. Bhargava, O. Miller, V. Ganapati, and E. Yablonovitch, “Inverse design of optical antennas for sub-wavelength energy delivery,” in “CLEO: Science and Innovations,” (Optical Society of America, 2013), pp. CM2F–2.

M. S. Dadash, “Simulator independent exact adjoint sensitivity analysis of self-adjoint microwave structures,” Master’s thesis, McMaster University (2011).

J. Jin, The Finite Element Method in Electromagnetics (John Wiley & Sons, 2014).

C. Loop, “Smooth subdivision surfaces based on triangles,” Master’s thesis, University of Utah (1987).

J. B. Leen, E. Cho, S.-D. Suh, P. C. Hansen, J.-S. Sohn, S.-H. Choa, and L. Hesselink, “90 bent metallic waveguide with a tapered c-shaped aperture for use in hamr,” in “Optical Data Storage 2007,” (International Society for Optics and Photonics, 2007), pp. 66200R.
[Crossref]

X. Xu, E. X. Jin, and S. M. Uppuluri, “Enhancement of optical transmission through planar nanoapertures in a metal film,” in “Optical Science and Technology, the SPIE 49th Annual Meeting,” (International Society for Optics and Photonics, 2004), pp. 230–243.

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Figures (10)

Fig. 1
Fig. 1 Device design schematic. The entire domain Ω is partitioned into subdomains Ω1 and Ω2, with a material discontinuity on the interior interface. During the design process, the material parameters in Ω and the shape of the interface Ω12 and outer boundary Ω may change. The normal vector n points out of outer Ω and from Ω1 to Ω2 along Ω12.
Fig. 2
Fig. 2 Boundary condition on perturbed boundary. Each xΩ becomes x′Ω′, and U must satisfy a new boundary condition on Ω′ (because, for instance, the matrix to extract tangential components of U depends on the slope of Ω).
Fig. 3
Fig. 3 Coordinates on Ω and Ω12. Fields and spatial derivatives will be represented in these coordinates.
Fig. 4
Fig. 4 Electric field enhancement for translating metal box test. The left particle of the dimer will be translated to the left starting from this initial position. The gold permittivity is ε = −44.7234 − 3.3160i.
Fig. 5
Fig. 5 Left: F (a) and ∂F/∂x (b) as the leftmost particle of the dimer is swept along the x axis. The derivative ∂F/∂x is calculated numerically from the samples of F. By coincidence the “Chamfer 1” curves are closely overlapped by the “Fillet” curves. Right: Coarsest meshes (c) and finest meshes (d) used to test convergence of F and DF.
Fig. 6
Fig. 6 Relative error in ∂F/∂p as the mesh is refined (relative error = ‖∂F/∂p − ΔF/Δp2/‖ΔF/Δp2). Slopes are 0.32 (Sharp), 0.59 (Chamfer 1), 0.75 (Chamfer 2), and 2.1 (Fillet).
Fig. 7
Fig. 7 Electric field intensity enhancement in the C-shaped engraving (CSE) under 980 nm x-polarized illumination from above (+z). Its initial dimensions are a = 60 nm and d = 150 nm. The CSE has eight designable parameters, corresponding to displacements of its interior faces. The eight transverse displacement parameters F1 through F8 are illustrated.
Fig. 8
Fig. 8 Objective function, sensitivities and adjoint sensitivities of the CSE. Top: objective functions. Bottom: adjoint (dashed line) and measured (solid line) sensitivities. Left-to-right: sharp, singly-rounded and doubly-rounded structures.
Fig. 9
Fig. 9 Top: Initial (left) and final optimized (right) unconstrained CSE designs. The optimized design is over 3× better than the initial design and 70% better than the original CSE-60 design. Bottom: progress of optimization over 42 iterations. Arrows mark the designs shown above. The horizontal dashed line at F = 79 marks the performance of the hand-tuned CSE-60, surpassed by the sixth iteration.
Fig. 10
Fig. 10 Top: field enhancement in standard CSE-60. Bottom: field enhancement in optimized undercut CSE. The xy cross-section (left) is taken at z = 20 nm. The yz (center) and xz (right) cross-sections are taken through x = 0 and y = 0, respectively. Dimensions are in nm.

Tables (2)

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Table 1 Relative error of ∂F/∂pi for three CSE variants. *Absolute sensitivity to p2 is very low, as F varies about 1% from the smallest to largest value of p2; the relative error in these near-zero values is accordingly very high.

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Table 2 Performance of resonant CSE-60 compared to initial and optimized unconstrained CSE designs.)

Equations (57)

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g u = v f ,
{ A u = f ( Primal system ) A v = g ( Dual system ) .
A ( p ) x = b ( Forward system ) .
F p i = F x x p i for i ( 1 , 2 , ) .
A x p i = b p i A p i x for i ( 1 , 2 , ) .
u i x p i f i b p i A p i x g i ( F x )
A ( p ) v = ( F x )
F p i = v ( b p i A p i x ) for i ( 1 , 2 , ) .
( p ) U = S in Ω B ( p ) U = S Ω on outer Ω B 12 ( p ) U 21 = S Ω 12 on interior Ω 12 .
F = Ω F Ω ( U ) + Ω F Ω ( U ) + Ω 12 F Ω 12 ( U ¯ ) .
d F d p i = ( g , u ) Ω + ( C h , u ) Ω + ( C 12 h 12 , u ¯ ) Ω 12 .
g = ( d F Ω d U ) C h = ( d F Ω d U ) C 12 h 12 = ( d F Ω 12 d U ¯ ) .
( p ) u = f in Ω B ( p ) u = e on outer Ω B 12 ( p ) u 21 = e 12 on interior Ω 12 .
f = p i U + S p i e = B p i U + S Ω p i e 12 = B 12 p i U 21 + S Ω 12 p i .
( p ) v = g in Ω β ( p ) v = h on Ω β 12 ( p ) v 21 = h 12 on interior Ω 12
d F d p = ( v , f ) Ω + ( v , γ e ) Ω + ( v ¯ , γ 12 e 12 ) Ω 12 .
( v , u ) Ω = ( v , u ) Ω + ( v , ( n G ) u ) Ω + ( v , ( n G ) u ) Ω 1 + ( v , ( n G ) u ) Ω 2 .
n G = β C γ B on Ω n G = β 12 C 12 = γ 12 B 12 on Ω 12 .
B U ( x ) = S Ω ( x ) x Ω ( original ) B U ( x ) = S Ω ( x ) x Ω ( perturbed ) .
B U ( x + η ( x ) n ) = S Ω ( x + η ( x ) n ) x Ω .
B p i U ( x ) + B ( η p i n U + U p i ) = S Ω p i ( x )
e = B p i U B η p i n U + S Ω p i on Ω e 12 = B 12 p i U 21 B 12 η p i n U 21 + S Ω p i on Ω 12 .
i ω [ ε 0 0 μ ] [ E H ] + [ 0 × × 0 ] [ E H ] = [ J Ω M Ω ] in Ω [ t 2 T 0 t 1 T 0 ] [ E H ] = M Ω on Ω [ 0 t 2 T 0 t 1 T t 2 T 0 t 1 T 0 ] [ E 21 H 21 ] = [ J Ω 12 M Ω 12 ] on Ω 12 .
B = [ t 2 T 0 t 1 T 0 ] , B 12 = [ 0 t 2 T 0 t 1 T t 2 T 0 t 1 T 0 ] .
F = Ω F Ω ( [ E H ] ) + Ω F Ω ( C [ E H ] ) + Ω 12 F Ω 12 ( C 12 [ E ¯ H ¯ ] ) C = [ 0 t 1 T 0 t 2 T ] C 12 = [ t 1 T 0 t 2 T 0 0 t 1 T 0 t 2 T ] .
F p i = ( g , [ E H ] ) Ω + ( C h , [ E H ] ) Ω + ( C 12 h 12 , [ E ¯ H ¯ ] ) Ω 12
g = [ F Ω E F Ω H ] , C h = [ F Ω E F Ω H ] , C 12 h 12 = [ F Ω 12 E ¯ F Ω 12 H ¯ ] .
i ω [ ε 0 0 μ ] [ E H ] + [ 0 × × 0 ] [ E H ] = f in Ω [ t 2 T 0 t 1 T 0 ] [ E H ] = e on Ω [ 0 t 2 T 0 t 1 T t 2 T 0 t 1 T 0 ] [ E 21 H 21 ] = e 12 on Ω 12 .
B p i = [ n T t 2 η p i 0 n T t 1 η p i 0 ] , B 12 p i = [ 0 n T t 2 η p i 0 n T t 1 η p i n T t 2 η p i 0 n T t 1 η p i 0 ] .
f = i ω [ ε p i 0 0 μ p i ] [ E H ] + [ J Ω p i M Ω p i ] m e = [ ( t 2 η p i ) E n + η p i n E t 2 + M t 1 Ω p i ( t 2 η p i ) E n η p i n E t 1 + M t 2 Ω p i ] , e 12 = [ ( t 2 η p i ) H n 21 η p i n H t 2 21 J t 1 Ω 12 p i ( t 1 η p i ) H n 21 + η p i n H t 1 21 t t 2 Ω 12 p i ( t 2 η p i ) E n 21 η p i n E t 2 21 + M t 1 Ω 12 p i ( t 1 η p i ) E n 21 + η p i n E t 1 21 + M t 2 Ω 12 p i ] .
= i ω [ ε 0 0 μ ] + [ 0 × × 0 ] .
Ω [ ] ( i ω [ ε 0 0 μ ] [ E H ] + [ 0 × × 0 ] [ E H ] ) = Ω ( i ω [ ε 0 0 μ ] [ ] + [ 0 × × 0 ] [ ] ) [ E H ] + Ω [ ] [ 0 n × n × 0 ] [ E H ]
= i ω [ ε 0 0 μ ] + [ 0 × × 0 ]
n G = [ 0 n × n × 0 ] = [ 0 t 1 t 2 T t 2 t 1 T t 1 t 2 T t 2 t 1 T 0 ] .
i ω [ ε 0 0 μ ] [ ] + [ 0 × × 0 ] [ ] = g in Ω [ t 2 T 0 t 1 T 0 ] [ ] = h on Ω [ 0 t 2 T 0 t 1 T t 2 T 0 t 1 T 0 ] [ 21 21 ] = h 12 on Ω 12
F p i = ( [ ] , f ) Ω + ( [ ] , γ e ) Ω + ( [ ] , γ 12 e 12 ) Ω 12 = Ω [ ] ( i ω p i [ ε 0 0 μ ] [ E H ] + p i [ J Ω M Ω ] ) + Ω [ t 1 t 2 ] ( E n [ t 2 t 1 ] η p i + η p i n [ E t 2 E t 1 ] + p i [ M t 1 Ω M t 2 Ω ] ) + Ω 12 [ ¯ t 1 ¯ t 2 ] ( H n 21 [ t 2 t 1 ] η p i + η p i n [ H t 2 21 H t 1 21 ] p i [ J t 1 Ω 12 J t 2 Ω 12 ] ) + Ω 12 [ ¯ t 1 ¯ t 2 ] ( E n 21 [ t 2 t 1 ] η p i + η p i n [ E t 2 21 E t 1 21 ] + p i [ M t 1 Ω 12 M t 2 Ω 12 ] ) .
Ω [ t 1 t 2 ] ( E n [ t 2 t 1 ] η p i + η p i n [ E t 2 E t 1 ] + p i [ M t 1 Ω M t 2 Ω ] ) = Ω t 2 ( t 1 E n η p i ) t 1 ( t 2 E n η p i ) + Ω η p i [ ( t 2 t 1 ) E n t 1 t 2 E n + ( t 1 t 2 ) E n + t 2 t 1 E n + t 1 n E t 2 t 2 n E t 1 ] + Ω t 1 M t 1 Ω p i + t 2 M t 2 Ω p i .
i ω B t 1 = t 2 E n + n E t 2 i ω B t 2 = n E t 1 + t 1 E n i ω 𝒟 n = t 1 t 2 + t 2 t 1
Ω i ω η p i ( t 1 B t 1 + t 2 B t 2 + 𝒟 n E n ) + t 1 M t 1 Ω p i + t 2 M t 2 Ω p i .
F p i = Ω [ ] ( i ω p i [ ε 0 0 μ ] [ E H ] + p i [ J Ω M Ω ] ) + Ω i ω η p i ( 𝒟 n E n + t 1 B t 1 + t 2 B t 2 ) + Ω t 1 M t 1 Ω p i + t 2 M t 2 Ω p i + Ω 12 i ω η p i ( 𝒟 ¯ n E n 21 + ¯ t 1 D t 1 21 + ¯ t 2 D t 2 21 + ¯ n H n 21 ¯ t 1 B t 1 21 ¯ t 2 B t 2 21 ) + Ω 12 ¯ t 1 J t 1 Ω 12 p i ¯ t 2 J t 2 Ω 12 p i + ¯ t 1 M t 1 Ω 12 p i + ¯ t 2 M t 2 Ω 12 p i .
U = [ i ω ε × × i ω μ ] U , v = [ i ω ε × × i ω μ ] v .
u ρ ( π / β ) 1 .
F = 1 A box | E x | 2 + | E y | 2 E 0 2 .
relative error = F / p Δ F / Δ p 2 Δ F / Δ p 2
F = 1 V V E 2 / E 0 2 ,
u = f in Ω B u = e on outer Ω B 12 ( u 2 u 1 ) = e 12 on interior Ω 12
v = g in Ω β v = h on Ω β 12 ( v 2 v 1 ) = h 12 on interior Ω 12 .
I primal = ( g , u ) Ω + ( h , C u ) Ω + ( h 12 , C 12 u ¯ ) Ω 12
I dual = ( v , f ) Ω + ( γ v , e ) Ω + ( γ 12 v ¯ , e 12 ) Ω 12
( v , u ) Ω = ( v , u ) Ω + ( v , ( n G ) u ) Ω + ( v , ( n G ) u ) Ω 1 + ( v , ( n G ) u ) Ω 2 .
( v , u ) Ω = ( v , u ) Ω + ( v , ( n G ) u ) Ω + ( v 1 , ( n G ) u 1 ) Ω 12 ( v 2 , ( n G ) u 2 ) Ω 12 .
0 = I primal I dual = ( g , u ) Ω ( v , f ) Ω + ( h , C u ) Ω + ( h 12 , C 12 u ¯ ) Ω 12 ( γ v , e ) Ω ( γ 12 v ¯ , e 12 ) Ω 12 .
( v , u ) Ω = ( v , u ) Ω ( γ v , B u ) Ω + ( β v , C u ) Ω ( γ 12 v ¯ , B 12 ( u 2 u 1 ) ) Ω 12 + ( β 12 ( v 2 v 1 ) , C 12 u ¯ ) Ω 12 .
n G = β C γ B on Ω 12 ,
( γ 12 v 1 + v 2 2 , B 12 ( u 2 u 1 ) ) Ω 12 ( β 12 ( v 2 + v 1 ) , C 12 u 1 u 2 2 ) Ω 12 = ( v 1 , ( n G ) u 1 ) Ω 12 + ( v 2 , ( n G ) u 2 ) Ω 12 .
Ω 12 [ v 1 v 2 ] T [ n G n G ] [ u 1 u 2 ] = Ω 12 [ v 1 v 2 ] T 1 2 [ β 12 T C 12 γ 12 T B 12 β 12 T C 12 + γ 12 T B 12 β 12 T C 12 γ 12 T B 12 γ 12 T B 12 β 12 T C 12 ] [ u 1 u 2 ] .
n G = β 12 T C 12 = γ 12 T B 12 .

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