Abstract

We propose a methodology for a systematic design of grating couplers for efficient excitation of surface plasmons at metal–dielectric interfaces. The methodology is based on a two-dimensional topology optimization formulation based on the H-polarized scalar Helmholtz equation and finite-element method simulations. The efficiency of the method is demonstrated by optimized designs for input and output grating couplers for an Ag-SiO2 interface. The results indicate that slanted grove gratings may raise the coupling efficiency above 68% where the highest previously reported value was 50%.

© 2010 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).
  2. W. L. Barnes, “Turning the tables on surface plasmons,” Nature Mater. 3, 588–589 (2004).
    [CrossRef]
  3. P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The role of surface plasmons in organic light-emitting diodes,” IEEE J. Quantum Electron. 8, 378–386 (2002).
    [CrossRef]
  4. V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
    [CrossRef]
  5. H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63 (2007).
    [CrossRef] [PubMed]
  6. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [CrossRef] [PubMed]
  7. T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
    [CrossRef]
  8. C. Peng and W. A. Challener, “Input-grating couplers for narrow Gaussian beam: influence of groove depth,” Opt. Express 12, 6481–6490 (2004).
    [CrossRef] [PubMed]
  9. G. Leveque and O. J. F. Martin, “Numerical study and optimization of a diffraction grating for surface plasmon excitation,” Proc. SPIE 5927, 246–254 (2005).
  10. Y. Huang, “Optimal design for the grating coupler of surface plasmons,” in Proceedings of the COMSOL Conference (2008), 4980http://www.comsol.com/papers/4980.
  11. J. Lu, C. Petre, E. Yablonovitch, and J. Conway, “Numerical optimization of a grating coupler for the efficient excitation of surface plasmons at an Ag-SiO2 interface,” J. Opt. Soc. Am. B 24, 2268–2272 (2007).
    [CrossRef]
  12. M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer-Verlag, 2003).
  13. M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
    [CrossRef]
  14. T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids 41, 77–107 (2003).
    [CrossRef]
  15. J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Actuators, A 76, 463–469 (1999).
    [CrossRef]
  16. P. I. Borel, A. Harpøth, L. H. Frandsen, M. Kristensen, J. S. Jensen, P. Shi, and O. Sigmund, “Topology optimization and fabrication of photonic crystal structures,” Opt. Express 12, 1996–2001 (2004).
    [CrossRef] [PubMed]
  17. O. Sigmund, “Design of multiphysics actuators using topology optimization—Part I: One-material Structures,” Comput. Methods Appl. Mech. Eng. 190, 6577–6604 (2001).
    [CrossRef]
  18. U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromech. Syst. 6, 99–106 (1997).
    [CrossRef]
  19. O. Sigmund and S. Torquato, “Composites with extremal thermal expansion coefficients,” Appl. Phys. Lett. 69, 3203–3205 (1996).
    [CrossRef]
  20. M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wireless Compon. Lett. 11, 152–154 (2001).
    [CrossRef]
  21. J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
    [CrossRef]
  22. O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
    [CrossRef]
  23. J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
    [CrossRef]
  24. K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
    [CrossRef]
  25. EURYI, http://www.esf.org/euryi.
  26. TopAnt, http://www.topant.dtu.dk.
  27. http://www.topopt.dtu.dk.

2009 (1)

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

2008 (2)

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Y. Huang, “Optimal design for the grating coupler of surface plasmons,” in Proceedings of the COMSOL Conference (2008), 4980http://www.comsol.com/papers/4980.

2007 (3)

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[CrossRef]

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63 (2007).
[CrossRef] [PubMed]

J. Lu, C. Petre, E. Yablonovitch, and J. Conway, “Numerical optimization of a grating coupler for the efficient excitation of surface plasmons at an Ag-SiO2 interface,” J. Opt. Soc. Am. B 24, 2268–2272 (2007).
[CrossRef]

2005 (2)

G. Leveque and O. J. F. Martin, “Numerical study and optimization of a diffraction grating for surface plasmon excitation,” Proc. SPIE 5927, 246–254 (2005).

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

2004 (3)

2003 (3)

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer-Verlag, 2003).

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids 41, 77–107 (2003).
[CrossRef]

2002 (2)

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The role of surface plasmons in organic light-emitting diodes,” IEEE J. Quantum Electron. 8, 378–386 (2002).
[CrossRef]

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

2001 (2)

O. Sigmund, “Design of multiphysics actuators using topology optimization—Part I: One-material Structures,” Comput. Methods Appl. Mech. Eng. 190, 6577–6604 (2001).
[CrossRef]

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wireless Compon. Lett. 11, 152–154 (2001).
[CrossRef]

1999 (1)

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Actuators, A 76, 463–469 (1999).
[CrossRef]

1997 (1)

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromech. Syst. 6, 99–106 (1997).
[CrossRef]

1996 (1)

O. Sigmund and S. Torquato, “Composites with extremal thermal expansion coefficients,” Appl. Phys. Lett. 69, 3203–3205 (1996).
[CrossRef]

1988 (2)

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

1977 (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Atwater, H. A.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63 (2007).
[CrossRef] [PubMed]

Barnes, W. L.

W. L. Barnes, “Turning the tables on surface plasmons,” Nature Mater. 3, 588–589 (2004).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The role of surface plasmons in organic light-emitting diodes,” IEEE J. Quantum Electron. 8, 378–386 (2002).
[CrossRef]

Bendsøe, M. P.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer-Verlag, 2003).

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

Borel, P. I.

Borrvall, T.

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids 41, 77–107 (2003).
[CrossRef]

Bouwstra, S.

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Actuators, A 76, 463–469 (1999).
[CrossRef]

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromech. Syst. 6, 99–106 (1997).
[CrossRef]

Challener, W. A.

Conway, J.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Ferry, V. E.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Frandsen, L. H.

Guest, J. K.

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

Harpøth, A.

Hobson, P. A.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The role of surface plasmons in organic light-emitting diodes,” IEEE J. Quantum Electron. 8, 378–386 (2002).
[CrossRef]

Huang, Y.

Y. Huang, “Optimal design for the grating coupler of surface plasmons,” in Proceedings of the COMSOL Conference (2008), 4980http://www.comsol.com/papers/4980.

Jensen, J. S.

Jonsmann, J.

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Actuators, A 76, 463–469 (1999).
[CrossRef]

Kikuchi, N.

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

Koshiba, M.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wireless Compon. Lett. 11, 152–154 (2001).
[CrossRef]

Kristensen, M.

Larsen, U. D.

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromech. Syst. 6, 99–106 (1997).
[CrossRef]

Leveque, G.

G. Leveque and O. J. F. Martin, “Numerical study and optimization of a diffraction grating for surface plasmon excitation,” Proc. SPIE 5927, 246–254 (2005).

Lu, J.

Martin, O. J. F.

G. Leveque and O. J. F. Martin, “Numerical study and optimization of a diffraction grating for surface plasmon excitation,” Proc. SPIE 5927, 246–254 (2005).

Pacifici, D.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Peng, C.

Peng, S. T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Petersson, J.

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids 41, 77–107 (2003).
[CrossRef]

Petre, C.

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Sage, I.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The role of surface plasmons in organic light-emitting diodes,” IEEE J. Quantum Electron. 8, 378–386 (2002).
[CrossRef]

Sasaki, S.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wireless Compon. Lett. 11, 152–154 (2001).
[CrossRef]

Shi, P.

Sigmund, O.

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[CrossRef]

J. S. Jensen and O. Sigmund, “Topology optimization of photonic crystal structures: a high-bandwidth low-loss T-junction waveguide,” J. Opt. Soc. Am. B 22, 1191–1198 (2005).
[CrossRef]

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

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer-Verlag, 2003).

O. Sigmund, “Design of multiphysics actuators using topology optimization—Part I: One-material Structures,” Comput. Methods Appl. Mech. Eng. 190, 6577–6604 (2001).
[CrossRef]

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Actuators, A 76, 463–469 (1999).
[CrossRef]

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromech. Syst. 6, 99–106 (1997).
[CrossRef]

O. Sigmund and S. Torquato, “Composites with extremal thermal expansion coefficients,” Appl. Phys. Lett. 69, 3203–3205 (1996).
[CrossRef]

Svanberg, K.

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

Sweatlock, L. A.

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Tamir, T.

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Torquato, S.

O. Sigmund and S. Torquato, “Composites with extremal thermal expansion coefficients,” Appl. Phys. Lett. 69, 3203–3205 (1996).
[CrossRef]

Tsuji, Y.

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wireless Compon. Lett. 11, 152–154 (2001).
[CrossRef]

Wasey, J. A. E.

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The role of surface plasmons in organic light-emitting diodes,” IEEE J. Quantum Electron. 8, 378–386 (2002).
[CrossRef]

Yablonovitch, E.

Appl. Phys. (1)

T. Tamir and S. T. Peng, “Analysis and design of grating couplers,” Appl. Phys. 14, 235–254 (1977).
[CrossRef]

Appl. Phys. Lett. (1)

O. Sigmund and S. Torquato, “Composites with extremal thermal expansion coefficients,” Appl. Phys. Lett. 69, 3203–3205 (1996).
[CrossRef]

Comput. Methods Appl. Mech. Eng. (3)

O. Sigmund, “Design of multiphysics actuators using topology optimization—Part I: One-material Structures,” Comput. Methods Appl. Mech. Eng. 190, 6577–6604 (2001).
[CrossRef]

J. K. Guest, “Topology optimization with multiple phase projection,” Comput. Methods Appl. Mech. Eng. 199, 123–135 (2009).
[CrossRef]

M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in optimal design using a homogenization method,” Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988).
[CrossRef]

IEEE J. Quantum Electron. (1)

P. A. Hobson, J. A. E. Wasey, I. Sage, and W. L. Barnes, “The role of surface plasmons in organic light-emitting diodes,” IEEE J. Quantum Electron. 8, 378–386 (2002).
[CrossRef]

IEEE Microw. Wireless Compon. Lett. (1)

M. Koshiba, Y. Tsuji, and S. Sasaki, “High-performance absorbing boundary conditions for photonic crystal waveguide simulations,” IEEE Microw. Wireless Compon. Lett. 11, 152–154 (2001).
[CrossRef]

Int. J. Numer. Methods Fluids (1)

T. Borrvall and J. Petersson, “Topology optimization of fluids in Stokes flow,” Int. J. Numer. Methods Fluids 41, 77–107 (2003).
[CrossRef]

J. Microelectromech. Syst. (1)

U. D. Larsen, O. Sigmund, and S. Bouwstra, “Design and fabrication of compliant mechanisms and material structures with negative Poisson’s ratio,” J. Microelectromech. Syst. 6, 99–106 (1997).
[CrossRef]

J. Opt. Soc. Am. B (2)

Nano Lett. (1)

V. E. Ferry, L. A. Sweatlock, D. Pacifici, and H. A. Atwater, “Plasmonic nanostructure design for efficient light coupling into solar cells,” Nano Lett. 8, 4391–4397 (2008).
[CrossRef]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef] [PubMed]

Nature Mater. (1)

W. L. Barnes, “Turning the tables on surface plasmons,” Nature Mater. 3, 588–589 (2004).
[CrossRef]

Opt. Express (2)

Proc. SPIE (1)

G. Leveque and O. J. F. Martin, “Numerical study and optimization of a diffraction grating for surface plasmon excitation,” Proc. SPIE 5927, 246–254 (2005).

Sci. Am. (1)

H. A. Atwater, “The promise of plasmonics,” Sci. Am. 296, 56–63 (2007).
[CrossRef] [PubMed]

Sens. Actuators, A (1)

J. Jonsmann, O. Sigmund, and S. Bouwstra, “Compliant thermal microactuators,” Sens. Actuators, A 76, 463–469 (1999).
[CrossRef]

SIAM J. Optim. (1)

K. Svanberg, “A class of globally convergent optimization methods based on conservative convex separable approximations,” SIAM J. Optim. 12, 555–573 (2002).
[CrossRef]

Struct. Multidiscip. Optim. (1)

O. Sigmund, “Morphology-based black and white filters for topology optimization,” Struct. Multidiscip. Optim. 33, 401–424 (2007).
[CrossRef]

Other (6)

EURYI, http://www.esf.org/euryi.

TopAnt, http://www.topant.dtu.dk.

http://www.topopt.dtu.dk.

M. P. Bendsøe and O. Sigmund, Topology Optimization: Theory, Methods and Applications (Springer-Verlag, 2003).

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Y. Huang, “Optimal design for the grating coupler of surface plasmons,” in Proceedings of the COMSOL Conference (2008), 4980http://www.comsol.com/papers/4980.

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Figures (4)

Fig. 1
Fig. 1

A surface plasmon induced in the interface between a metal and a dielectric. + and − signs represent regions with low and high surface charge densities, respectively. The electric field E oscillates in the ( x , y ) -plane making the surface plasmons transverse magnetic in character (hence magnetic field H z in the z-direction).

Fig. 2
Fig. 2

Computational models for the input and output couplers containing a dielectric ( Ω SiO 2 ) , a metal ( Ω Ag ) , and a design domain ( Ω des ) . A plane wave is excited at Γ inp . A surface plasmon will be induced in the interface between the two media and the energy flux is measured in the domain Ω out . To avoid reflections from the boundaries, PML domains ( Ω PML ) and absorbing boundary condition Γ abs have been introduced. All interior boundaries have continuous boundary condition.

Fig. 3
Fig. 3

Optimized designs.

Fig. 4
Fig. 4

FE simulations showing the absolute magnetic field for the optimized designs.

Equations (17)

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

( ϵ r 1 H z ( r ) ) + k 0 2 H z ( r ) = 0     in   Ω ,
x ( s y s x ϵ r 1 H z ( r ) x ) + y ( s x s y ϵ r 1 H z ( r ) y ) + s x s y k 0 2 H z ( r ) = 0     in   Ω PML ,
n ( ϵ r 1 H z ( r ) ) = 0     on   Γ con ,
n ( ϵ r 1 H z ( r ) ) + i k 0 ϵ r 1 H z = 0     on   Γ abs ,
n ( ϵ r 1 H z ( r ) ) = i k 0 ϵ r 1 [ 2   exp ( ( x x 0 ) 2 2 ( 1 2 σ ) 2 ) H 0 H z ]     on   Γ inp ,
n ( ϵ r 1 H z ( r ) ) = i k 0 ϵ r 1 [ 2   exp ( | k y , Ag | | y 0 y | ) H 0 H z ]     on   Γ inp ,
ϵ r ( ρ e ) = ϵ Ag + ρ e ( ϵ SiO 2 ϵ Ag ) i [ ϵ Ag + ρ e ( ϵ SiO 2 ϵ Ag ) ] + i 4 d ( ρ e 2 ρ e ) ,
max ρ Φ ( H z ( ρ ̃ ) ) = P out = Ω out 1 2 L x ϵ 0 ω R ( i ϵ r 1 H z H z ) n d r , s. t. Helmholtz   equation
1 V Ω des Ω des ρ ̃ ( r ) d r β 0 ,     0 ρ ( r ) 1 ,
Efficiency = P out exp ( 1 ) [ 1 exp ( 1 ) ] P inp .
Efficiency = P out exp ( 2 k x , SP L SP ) P inp ,
S ( ρ ) u = f ,
d Φ d ρ ̃ e = Φ ρ ̃ e + 2 R [ λ T ( S ρ ̃ e u f ρ ̃ e ) ] ,
S T λ = Φ u .
Φ u = Ω out ( A u + A u ) d r ,
A u = 1 4 L x ϵ 0 ω i ϵ r 1 u n ,
A u = 1 4 L x ϵ 0 ω i ϵ r 1 u n .

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