Abstract

We present an adjoint-based optimization for electromagnetic design. It embeds commercial Maxwell solvers within a steepest-descent inverse-design optimization algorithm. The adjoint approach calculates shape derivatives at all points in space, but requires only two “forward” simulations. Geometrical shape parameterization is by the level set method. Our adjoint design optimization is applied to a Silicon photonics Y-junction splitter that had previously been investigated by stochastic methods. Owing to the speed of calculating shape derivatives within the adjoint method, convergence is much faster, within a larger design space. This is an extremely efficient method for the design of complex electromagnetic components.

© 2013 OSA

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References

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  1. P. Sandborn, N. Quack, N. Hoghooghi, J. B. Chou, J. Ferrara, S. Gambini, B. Behroozpour, L. Zhu, B. Boser, C. Chang-Hasnain, and M. C. Wu, “Linear frequency chirp generation employing opto-electronic feedback loop and integrated Silicon photonics,” in CLEO, (2013) pp. 5–6.
  2. A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron.E85, 1033–1038 (2002).
  3. Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express21(1), 1310–1316 (2013).
    [CrossRef] [PubMed]
  4. P. Sanchis, P. Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galán, A. Brimont, and J. Martí, “Highly efficient crossing structure for silicon-on-insulator waveguides,” Opt. Lett.34(18), 2760–2762 (2009).
    [CrossRef] [PubMed]
  5. Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
    [CrossRef]
  6. T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
    [CrossRef] [PubMed]
  7. M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. In Appl. M. 71, 197–224 (1988).
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  9. T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids41(1), 77–107 (2003).
    [CrossRef]
  10. J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev.5(2), 308–321 (2011).
    [CrossRef]
  11. P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006).
    [CrossRef]
  12. W. R. Frei, D. A. Tortorelli, and H. T. Johnson, “Geometry projection method for optimizing photonic nanostructures,” Opt. Lett.32(1), 77–79 (2007).
    [CrossRef] [PubMed]
  13. V. Liu and S. Fan, “Compact bends for multi-mode photonic crystal waveguides with high transmission and suppressed modal crosstalk,” Opt. Express21(7), 8069–8075 (2013).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim.48(5), 3191–3223 (2010).
    [CrossRef]
  16. G. Strang, Computational Science and Engineering, (Wellesley-Cambridge, 2007).
  17. O. D. Miller, PhD thesis (2012), University of California at Berkeley, http://optoelectronics.eecs.berkeley.edu/ThesisOwenMiller.pdf
  18. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
    [CrossRef] [PubMed]
  19. Lumerical FDTD Solutions, www.lumerical.com
  20. S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys.79(1), 12–49 (1988).
    [CrossRef]
  21. http://optoelectronics.eecs.berkeley.edu/PhotonicInverseDesign/

2013 (3)

2011 (2)

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev.5(2), 308–321 (2011).
[CrossRef]

2010 (1)

A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim.48(5), 3191–3223 (2010).
[CrossRef]

2009 (1)

2007 (2)

2006 (1)

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006).
[CrossRef]

2004 (1)

2003 (1)

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids41(1), 77–107 (2003).
[CrossRef]

2002 (2)

A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron.E85, 1033–1038 (2002).

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

1988 (1)

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys.79(1), 12–49 (1988).
[CrossRef]

Baba, T.

A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron.E85, 1033–1038 (2002).

Baehr-Jones, T.

Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express21(1), 1310–1316 (2013).
[CrossRef] [PubMed]

Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
[CrossRef]

Balram, K. C.

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Borrvall, T.

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids41(1), 77–107 (2003).
[CrossRef]

Brimont, A.

Brongersma, M. L.

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Cuesta, F.

Dutton, R. W.

Fan, S.

Fink, Y.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

Frei, W. R.

Fukazawa, T.

A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron.E85, 1033–1038 (2002).

Galán, J. V.

Galland, C.

Griol, A.

Håkansson, A.

Hochberg, M.

Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express21(1), 1310–1316 (2013).
[CrossRef] [PubMed]

Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
[CrossRef]

Ibanescu, M.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

Jensen, J. S.

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev.5(2), 308–321 (2011).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

Johnson, H. T.

Johnson, S. G.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

Levi, A. F. J.

A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim.48(5), 3191–3223 (2010).
[CrossRef]

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006).
[CrossRef]

Lim, A. E. J.

Lim, E. J.

Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
[CrossRef]

Liu, V.

Lo, G.

Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
[CrossRef]

Lo, G. Q.

Ly-Gagnon, D. S.

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Mahvash, M.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006).
[CrossRef]

Martí, J.

Miller, D. A. B.

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Osher, S.

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys.79(1), 12–49 (1988).
[CrossRef]

Petersson, J.

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids41(1), 77–107 (2003).
[CrossRef]

Rosen, I. G.

A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim.48(5), 3191–3223 (2010).
[CrossRef]

Sakai, A.

A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron.E85, 1033–1038 (2002).

Sanchis, P.

Seliger, P.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006).
[CrossRef]

Sethian, J. A.

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys.79(1), 12–49 (1988).
[CrossRef]

Sigmund, O.

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev.5(2), 308–321 (2011).
[CrossRef]

Skorobogatiy, M. A.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

Strang, G.

G. Strang, Computational Science and Engineering, (Wellesley-Cambridge, 2007).

Tanemura, T.

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Tortorelli, D. A.

Veronis, G.

Villalba, P.

Wahl, P.

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Wang, C.

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006).
[CrossRef]

Weisberg, O.

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

White, J. S.

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Yang, S.

Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express21(1), 1310–1316 (2013).
[CrossRef] [PubMed]

Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
[CrossRef]

Zhang, Y.

Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
[CrossRef]

Y. Zhang, S. Yang, A. E. J. Lim, G. Q. Lo, C. Galland, T. Baehr-Jones, and M. Hochberg, “A compact and low loss Y-junction for submicron silicon waveguide,” Opt. Express21(1), 1310–1316 (2013).
[CrossRef] [PubMed]

Computational Science and Engineering (1)

G. Strang, Computational Science and Engineering, (Wellesley-Cambridge, 2007).

IEEE Photon. Technol. Lett. (1)

Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M. Hochberg, “A CMOS-compatible, low-loss and low-crosstalk silicon waveguide crossing,” IEEE Photon. Technol. Lett.25(5), 422–425 (2013).
[CrossRef]

IEICE Trans. Electron. (1)

A. Sakai, T. Fukazawa, and T. Baba, “Low loss ultra-small branches in a silicon photonic wire waveguide,” IEICE Trans. Electron.E85, 1033–1038 (2002).

Int. J. Numer. Methods Fluids (1)

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids41(1), 77–107 (2003).
[CrossRef]

J. Appl. Phys. (1)

P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” J. Appl. Phys.100(3), 034310 (2006).
[CrossRef]

J. Comput. Phys. (1)

S. Osher and J. A. Sethian, “Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys.79(1), 12–49 (1988).
[CrossRef]

Las. Photon. Rev. (1)

J. S. Jensen and O. Sigmund, “Topology optimization for nano-photonics,” Las. Photon. Rev.5(2), 308–321 (2011).
[CrossRef]

Nano Lett. (1)

T. Tanemura, K. C. Balram, D. S. Ly-Gagnon, P. Wahl, J. S. White, M. L. Brongersma, and D. A. B. Miller, “Multiple-wavelength focusing of surface plasmons with a nonperiodic nanoslit coupler,” Nano Lett.11(7), 2693–2698 (2011).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (3)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002).
[CrossRef] [PubMed]

SIAM J. Contr. Optim. (1)

A. F. J. Levi and I. G. Rosen, “A novel formulation of the adjoint method in the optimal design of quantum electronic devices,” SIAM J. Contr. Optim.48(5), 3191–3223 (2010).
[CrossRef]

Other (6)

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

Lumerical FDTD Solutions, www.lumerical.com

http://optoelectronics.eecs.berkeley.edu/PhotonicInverseDesign/

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Meth. In Appl. M. 71, 197–224 (1988).

M. P. Bendsoe and O. Sigmund, Topology Optimization Theory, Methods and Applications. (Springer, 2003)

P. Sandborn, N. Quack, N. Hoghooghi, J. B. Chou, J. Ferrara, S. Gambini, B. Behroozpour, L. Zhu, B. Boser, C. Chang-Hasnain, and M. C. Wu, “Linear frequency chirp generation employing opto-electronic feedback loop and integrated Silicon photonics,” in CLEO, (2013) pp. 5–6.

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Figures (6)

Fig. 1
Fig. 1

Adjoint method schematic: two simulations are needed for every iteration; the direct and the adjoint simulation. Sources for each simulation are drawn in red.

Fig. 2
Fig. 2

Top view of the optimized silicon splitter geometry obtained after 51 iterations of the Steepest Descent algorithm. Only the designable region geometry was allowed to change. The Silicon waveguide is 220nm thick, and the cladding is Silicon dioxide.

Fig. 3
Fig. 3

Coupling efficiency evolution during the optimization. The switch from 2d to 3d FDTD is visible at iteration 41. For comparison, the previous record of ref [3]. was −0.13dB and required 1500 simulations.

Fig. 4
Fig. 4

Geometry evolution during the optimization process and total coupling efficiency to the output waveguides. Iter indicates the iteration number, and the insertion loss is given in dB. The optimization is first carried out using a 2d approximation with an effective waveguide index = 2.8, which mimics the 3d in-plane propagation constant. The final iterative steps are carried out in full 3d FDTD.

Fig. 5
Fig. 5

Simulated field intensity |E|2 for the optimized structure at λ = 1550nm for a slice in the middle of the device.

Fig. 6
Fig. 6

Simulated insertion loss of the optimized device for wavelengths between 1.5 and 1.6 µm. The broad operating spectrum of the device is a good indicator of the robustness of the design.

Equations (9)

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

FoM= | E( x 0 ) | 2
ΔFoM=Re[ E old ( x 0 ) ¯ ΔE( x 0 ) ]
ΔE( x 0 )= G EP ¯ ¯ ( x 0 ,x) p ind = ε 0 Δ ε r ΔV G EP ¯ ¯ ( x 0 ,x) E new (x)
ΔFoM Δ ε r = ε 0 ΔVRe[ E old ( x 0 ) ¯ ( G EP ¯ ¯ ( x 0 ,x) E old (x) ) ]
ΔFoM Δ ε r =Re[ ( ε 0 ΔV G EP ¯ ¯ (x, x 0 ) E old ( x 0 ) ¯ ) E old (x) ]Re[ E adj (x) E old (x) ]
E adj (x)= ε 0 ΔV G EP ¯ ¯ (x, x 0 ) E old ( x 0 ) ¯
FoM= 1 8 | E× H m ¯ dS+ E m ¯ ×HdS | 2 Re( E m × H m ¯ ) dS
E adj (x)=A ( G EP ¯ ¯ (x,x') H m (x') ¯ ×n G EM ¯ ¯ (x,x') n× E m (x') ¯ μ 0 ) dS
A= 1 4 ε 0 ΔV ( E old × H m ¯ dS+ E m ¯ × H old dS ) ¯ Re( E m × H m ¯ ) dS

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