Abstract

A novel hybrid technique based on the boundary integral-equation method is proposed for studying the surface plasmon polariton behaviors in two-dimensional periodic structures. Considering the periodicity property of the problem, we use the plane-wave expansion concept and the periodic boundary condition instead of using the periodic Green’s function. The diffraction efficiency can then be readily calculated once the equivalent electric and magnetic currents are solved that avoids invoking the numerical calculation of the radiation integral. The numerical validity is verified with the cases of highly conducting materials and practical metals. Numerical convergence can be easily achieved even in the case of a large incident angle as 80°. Based on the numerical scheme, a metal-dielectric wavy structure is designed for enhancing the transmittance of optical signal through the structure. The excitation of the coupled surface plasmon polaritons for the high transmission is demonstrated.

© 2007 Optical Society of America

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References

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  1. H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).
  2. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
    [CrossRef]
  3. K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
    [CrossRef]
  4. S. D. Gedney and R. Mittra, "Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution," IEEE Trans. Antennas Propagat. 39, 1605-1614 (1991).
    [CrossRef]
  5. K. Yashiro and S. Ohkawa, "Boundary element method for electromagnetic scattering from cylinders," IEEE Trans. Antennas Propagat. 33, 383-389 (1985).
    [CrossRef]
  6. L. C. Trintinalia and H. Ling, "Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme," IEEE Trans. Antennas Propagat. 52, 2253-2261 (2004).
    [CrossRef]
  7. T. Sondergaard, S. I. Bozhevolnyi, and A. Boltasseva, "Theoretical analysis of ridge grating for long-range surface plasmon polaritons," Phys. Rev. B 73, 045320 (2006).
    [CrossRef]
  8. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of metallic surface-relief gratings," J. Opt. Soc. Am. A 3, 1780-1796 (1986).
    [CrossRef]
  9. E. Popov, B. Chernov, M. Nevière, and N. Bonod, "Differential theory: Application to highly conducting gratings," J. Opt. Soc. Am. A 21, 199-206 (2004).
    [CrossRef]
  10. K. Watanabe, "Study of the differential theory of lamellar gratings made of highly conducting materials," J. Opt. Soc. Am. A 23, 69-72 (2006).
    [CrossRef]
  11. E. Popov, M. Nevie`re, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  12. K. Yasumoto and K. Yoshitomi, "Efficient calculation of lattice sums for free-space periodic Green’s function," IEEE Trans. Antennas Propagat. 47, 1050-1055 (1999).
    [CrossRef]
  13. H. Rogier and D. De Zutter, "A fast converging series expansion for the 2-d periodic Green’s function based on perfectly matched layers," IEEE Trans. Microwave Theory Tech. 52, 1199-1206 (2004).
    [CrossRef]
  14. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).
  15. K.-M. Chen, "A mathematical formulation of the equivalence principle," IEEE Trans. Microwave Theory Tech. 37, 1576-1581 (1989).
    [CrossRef]
  16. C. A. Balanis, Advanced Engineering Electromagnetics (John Wiley & Sons, New York, 1989).
  17. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  18. D. K. Gifford and D. G. Hall, "Extraordinary transmission of organic photoluminescence through an otherwise opaque metal layer via surface plasmon cross coupling," Appl. Phys. Lett. 80, 3679-3681 (2002).
    [CrossRef]
  19. S. Wedge and W. L. Barnes, "Surface plasmon-polariton mediated light emission through thin metal films," Opt. Express 12, 3673-3685 (2004).
    [CrossRef] [PubMed]
  20. C. Bonnand, J. Bellessa, C. Symonds, and J. C. Plenet, "Polaritonic emission via surface plasmon cross coupling," Appl. Phys. Lett. 89, 231119 (2006).
    [CrossRef]
  21. U. Schroter and D. Heitmann, "Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration," Phys. Rev. B 60, 4992-4999 (1999).
    [CrossRef]
  22. I. R. Hooper and J. R. Sambles, "Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces," Phys. Rev. B 70, 045421 (2004).
    [CrossRef]
  23. D. Crouse and P. Keshavareddy, "Role of optical and surface plasmon modes in enhanced transmission and applications," Opt. Express 13, 7760-7771 (2005).
    [CrossRef] [PubMed]

2006 (3)

T. Sondergaard, S. I. Bozhevolnyi, and A. Boltasseva, "Theoretical analysis of ridge grating for long-range surface plasmon polaritons," Phys. Rev. B 73, 045320 (2006).
[CrossRef]

C. Bonnand, J. Bellessa, C. Symonds, and J. C. Plenet, "Polaritonic emission via surface plasmon cross coupling," Appl. Phys. Lett. 89, 231119 (2006).
[CrossRef]

K. Watanabe, "Study of the differential theory of lamellar gratings made of highly conducting materials," J. Opt. Soc. Am. A 23, 69-72 (2006).
[CrossRef]

2005 (1)

2004 (5)

I. R. Hooper and J. R. Sambles, "Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces," Phys. Rev. B 70, 045421 (2004).
[CrossRef]

L. C. Trintinalia and H. Ling, "Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme," IEEE Trans. Antennas Propagat. 52, 2253-2261 (2004).
[CrossRef]

E. Popov, B. Chernov, M. Nevière, and N. Bonod, "Differential theory: Application to highly conducting gratings," J. Opt. Soc. Am. A 21, 199-206 (2004).
[CrossRef]

S. Wedge and W. L. Barnes, "Surface plasmon-polariton mediated light emission through thin metal films," Opt. Express 12, 3673-3685 (2004).
[CrossRef] [PubMed]

H. Rogier and D. De Zutter, "A fast converging series expansion for the 2-d periodic Green’s function based on perfectly matched layers," IEEE Trans. Microwave Theory Tech. 52, 1199-1206 (2004).
[CrossRef]

2002 (2)

D. K. Gifford and D. G. Hall, "Extraordinary transmission of organic photoluminescence through an otherwise opaque metal layer via surface plasmon cross coupling," Appl. Phys. Lett. 80, 3679-3681 (2002).
[CrossRef]

E. Popov, M. Nevie`re, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

1999 (2)

U. Schroter and D. Heitmann, "Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration," Phys. Rev. B 60, 4992-4999 (1999).
[CrossRef]

K. Yasumoto and K. Yoshitomi, "Efficient calculation of lattice sums for free-space periodic Green’s function," IEEE Trans. Antennas Propagat. 47, 1050-1055 (1999).
[CrossRef]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

1996 (1)

1991 (1)

S. D. Gedney and R. Mittra, "Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution," IEEE Trans. Antennas Propagat. 39, 1605-1614 (1991).
[CrossRef]

1989 (1)

K.-M. Chen, "A mathematical formulation of the equivalence principle," IEEE Trans. Microwave Theory Tech. 37, 1576-1581 (1989).
[CrossRef]

1986 (1)

1985 (1)

K. Yashiro and S. Ohkawa, "Boundary element method for electromagnetic scattering from cylinders," IEEE Trans. Antennas Propagat. 33, 383-389 (1985).
[CrossRef]

1966 (1)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

Barnes, W. L.

Bellessa, J.

C. Bonnand, J. Bellessa, C. Symonds, and J. C. Plenet, "Polaritonic emission via surface plasmon cross coupling," Appl. Phys. Lett. 89, 231119 (2006).
[CrossRef]

Boltasseva, A.

T. Sondergaard, S. I. Bozhevolnyi, and A. Boltasseva, "Theoretical analysis of ridge grating for long-range surface plasmon polaritons," Phys. Rev. B 73, 045320 (2006).
[CrossRef]

Bonnand, C.

C. Bonnand, J. Bellessa, C. Symonds, and J. C. Plenet, "Polaritonic emission via surface plasmon cross coupling," Appl. Phys. Lett. 89, 231119 (2006).
[CrossRef]

Bonod, N.

Bozhevolnyi, S. I.

T. Sondergaard, S. I. Bozhevolnyi, and A. Boltasseva, "Theoretical analysis of ridge grating for long-range surface plasmon polaritons," Phys. Rev. B 73, 045320 (2006).
[CrossRef]

Chen, K.-M.

K.-M. Chen, "A mathematical formulation of the equivalence principle," IEEE Trans. Microwave Theory Tech. 37, 1576-1581 (1989).
[CrossRef]

Chernov, B.

Crouse, D.

De Zutter, D.

H. Rogier and D. De Zutter, "A fast converging series expansion for the 2-d periodic Green’s function based on perfectly matched layers," IEEE Trans. Microwave Theory Tech. 52, 1199-1206 (2004).
[CrossRef]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Gaylord, T. K.

Gedney, S. D.

S. D. Gedney and R. Mittra, "Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution," IEEE Trans. Antennas Propagat. 39, 1605-1614 (1991).
[CrossRef]

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Gifford, D. K.

D. K. Gifford and D. G. Hall, "Extraordinary transmission of organic photoluminescence through an otherwise opaque metal layer via surface plasmon cross coupling," Appl. Phys. Lett. 80, 3679-3681 (2002).
[CrossRef]

Gralak, B.

Hall, D. G.

D. K. Gifford and D. G. Hall, "Extraordinary transmission of organic photoluminescence through an otherwise opaque metal layer via surface plasmon cross coupling," Appl. Phys. Lett. 80, 3679-3681 (2002).
[CrossRef]

Heitmann, D.

U. Schroter and D. Heitmann, "Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration," Phys. Rev. B 60, 4992-4999 (1999).
[CrossRef]

Hooper, I. R.

I. R. Hooper and J. R. Sambles, "Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces," Phys. Rev. B 70, 045421 (2004).
[CrossRef]

Keshavareddy, P.

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Li, L.

Ling, H.

L. C. Trintinalia and H. Ling, "Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme," IEEE Trans. Antennas Propagat. 52, 2253-2261 (2004).
[CrossRef]

Mittra, R.

S. D. Gedney and R. Mittra, "Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution," IEEE Trans. Antennas Propagat. 39, 1605-1614 (1991).
[CrossRef]

Moharam, M. G.

Nevie`re, M.

Nevière, M.

Ohkawa, S.

K. Yashiro and S. Ohkawa, "Boundary element method for electromagnetic scattering from cylinders," IEEE Trans. Antennas Propagat. 33, 383-389 (1985).
[CrossRef]

Plenet, J. C.

C. Bonnand, J. Bellessa, C. Symonds, and J. C. Plenet, "Polaritonic emission via surface plasmon cross coupling," Appl. Phys. Lett. 89, 231119 (2006).
[CrossRef]

Popov, E.

Rogier, H.

H. Rogier and D. De Zutter, "A fast converging series expansion for the 2-d periodic Green’s function based on perfectly matched layers," IEEE Trans. Microwave Theory Tech. 52, 1199-1206 (2004).
[CrossRef]

Sambles, J. R.

I. R. Hooper and J. R. Sambles, "Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces," Phys. Rev. B 70, 045421 (2004).
[CrossRef]

Schroter, U.

U. Schroter and D. Heitmann, "Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration," Phys. Rev. B 60, 4992-4999 (1999).
[CrossRef]

Sondergaard, T.

T. Sondergaard, S. I. Bozhevolnyi, and A. Boltasseva, "Theoretical analysis of ridge grating for long-range surface plasmon polaritons," Phys. Rev. B 73, 045320 (2006).
[CrossRef]

Symonds, C.

C. Bonnand, J. Bellessa, C. Symonds, and J. C. Plenet, "Polaritonic emission via surface plasmon cross coupling," Appl. Phys. Lett. 89, 231119 (2006).
[CrossRef]

Tayeb, G.

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Trintinalia, L. C.

L. C. Trintinalia and H. Ling, "Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme," IEEE Trans. Antennas Propagat. 52, 2253-2261 (2004).
[CrossRef]

Watanabe, K.

Wedge, S.

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Yashiro, K.

K. Yashiro and S. Ohkawa, "Boundary element method for electromagnetic scattering from cylinders," IEEE Trans. Antennas Propagat. 33, 383-389 (1985).
[CrossRef]

Yasumoto, K.

K. Yasumoto and K. Yoshitomi, "Efficient calculation of lattice sums for free-space periodic Green’s function," IEEE Trans. Antennas Propagat. 47, 1050-1055 (1999).
[CrossRef]

Yee, K. S.

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

Yoshitomi, K.

K. Yasumoto and K. Yoshitomi, "Efficient calculation of lattice sums for free-space periodic Green’s function," IEEE Trans. Antennas Propagat. 47, 1050-1055 (1999).
[CrossRef]

Appl. Phys. Lett. (2)

C. Bonnand, J. Bellessa, C. Symonds, and J. C. Plenet, "Polaritonic emission via surface plasmon cross coupling," Appl. Phys. Lett. 89, 231119 (2006).
[CrossRef]

D. K. Gifford and D. G. Hall, "Extraordinary transmission of organic photoluminescence through an otherwise opaque metal layer via surface plasmon cross coupling," Appl. Phys. Lett. 80, 3679-3681 (2002).
[CrossRef]

IEEE Trans. Antennas Propagat. (5)

K. S. Yee, "Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media," IEEE Trans. Antennas Propagat. 14, 302-307 (1966).
[CrossRef]

S. D. Gedney and R. Mittra, "Analysis of the electromagnetic scattering by thick gratings using a combined FEM/MM solution," IEEE Trans. Antennas Propagat. 39, 1605-1614 (1991).
[CrossRef]

K. Yashiro and S. Ohkawa, "Boundary element method for electromagnetic scattering from cylinders," IEEE Trans. Antennas Propagat. 33, 383-389 (1985).
[CrossRef]

L. C. Trintinalia and H. Ling, "Integral equation modeling of multilayered doubly-periodic lossy structures using periodic boundary condition and a connection scheme," IEEE Trans. Antennas Propagat. 52, 2253-2261 (2004).
[CrossRef]

K. Yasumoto and K. Yoshitomi, "Efficient calculation of lattice sums for free-space periodic Green’s function," IEEE Trans. Antennas Propagat. 47, 1050-1055 (1999).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

H. Rogier and D. De Zutter, "A fast converging series expansion for the 2-d periodic Green’s function based on perfectly matched layers," IEEE Trans. Microwave Theory Tech. 52, 1199-1206 (2004).
[CrossRef]

K.-M. Chen, "A mathematical formulation of the equivalence principle," IEEE Trans. Microwave Theory Tech. 37, 1576-1581 (1989).
[CrossRef]

J. Opt. Soc. Am. A (5)

Nature (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, "Extraordinary optical transmission through sub-wavelength hole arrays," Nature 391, 667-669 (1998).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (3)

U. Schroter and D. Heitmann, "Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration," Phys. Rev. B 60, 4992-4999 (1999).
[CrossRef]

I. R. Hooper and J. R. Sambles, "Coupled surface plasmon polaritons on thin metal slabs corrugated on both surfaces," Phys. Rev. B 70, 045421 (2004).
[CrossRef]

T. Sondergaard, S. I. Bozhevolnyi, and A. Boltasseva, "Theoretical analysis of ridge grating for long-range surface plasmon polaritons," Phys. Rev. B 73, 045320 (2006).
[CrossRef]

Other (3)

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941).

C. A. Balanis, Advanced Engineering Electromagnetics (John Wiley & Sons, New York, 1989).

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Figures (10)

Fig. 1.
Fig. 1.

A unit cell of a 2D periodic structure. The unit cell repeats itself in the x direction, with the solid lines for its true boundaries or interfaces. The red dashed lines represent virtual boundaries for applying the periodic boundary conditions and the interior-exterior connection.

Fig. 2.
Fig. 2.

An alternative version of Fig. 1 to focus on the discussions of the interior-exterior interfaces.

Fig. 3.
Fig. 3.

A reflection-type metallic-lamellar grating with a grating period a, a groove depth d, and a groove width g. The incident wave is p-polarized with an incident angle θi .

Fig. 4.
Fig. 4.

Reflectance of the -1 order versus the groove width, calculated with (a) the coupled-wave method (CWM) and with (b) the proposed method.

Fig. 5.
Fig. 5.

Zero-order reflectance versus wavelength. Solid lines denote the results of our method and empty squares represent those calculated with the CWM. (a) θi = 0°. (b) θi = 80°.

Fig. 6.
Fig. 6.

Schematic illustration of a wavy layered structure. The grating period is a, and the amplitude of the sinusoidal variation is A. The cover layer, with its thickness denoted by tc , has a refractive index nc . The thickness of the Ag film is tm .

Fig. 7.
Fig. 7.

(a). Zero-order transmittance and (b) power loss versus wavelength. The results are calculated for different Ag-film thicknesses. (c) Magnitude distributions of the magnetic field at the transmittance peak of tm = 44 nm (labeled by (1) in (a)) and the two peaks of tm = 28 nm (labeled by (2) and (3) in (a)). (d) Distributions of the time-average Poynting vector corresponding to the three plots of (c).

Fig. 8.
Fig. 8.

(a). Zero-order transmittance and (b) power loss versus wavelength. The results are calculated for different amplitudes of the sinusoidal shape.

Fig. 9.
Fig. 9.

(a). Zero-order transmittance and (b) power loss versus wavelength. The results are calculated for different thicknesses of the cover layer. (c) Magnitude distributions of the magnetic field at the peaks of tc = 100 nm, tc = 70 nm, and tc = 50 nm.

Fig. 10.
Fig. 10.

(a). Zero-order transmittance and power loss versus wavelength. The structure parameters used are a = 218 nm, A = 20 nm, tm = 28 nm, tc = 70 nm, and nc = 2.7. (b) Magnitude distribution of the magnetic field at λ 0 = 630 nm. (c) Distribution of the time-average Poynting vector at λ 0 = 630 nm.

Equations (17)

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

{ E ( ρ ) = 2 E inc ( ρ ) + 2 { iωμ J ( ρ ) φ ρ ρ M ( ρ ) × φ ρ ρ [ i ωε · J ( ρ ) ] φ ( ρ , ρ ) } dl . H ( ρ ) = 2 H inc ( ρ ) + 2 [ iωε M ( ρ ) φ ρ ρ + J ( ρ ) × φ ρ ρ ] dl
φ ρ ρ = i 4 H 0 ( 1 ) ( k ρ ρ ) .
{ E = E inc + E ref = E inc x y + = L L t ̂ η h e i ( k x + ℓG ) x e i k y ( y y 0 ) H = H inc + H ref = z ̂ H inc x y + = L L z ̂ h e i ( k x + ℓG ) x e i k y ( y y 0 ) . k y = k 2 ( k x + ℓG ) 2 , Re ( k y ) 0 , Im ( k y ) 0
k ̂ = x ̂ ( k x + ℓG ) k y ̂ k y k .
{ J ( x ) = n = 0 N x ̂ J n f n ( x ) , N = 2 L + 1 J ( x ) = ( y ̂ ) × H x y 0 .
f n , 1 = { x x n 1 x n x n 1 , x n 1 x x n 0 , otherwise , f n , 2 = { x n + 1 x x n + 1 x n , x n x x n + 1 . 0 , otherwise
n = 0 N x ̂ J n f n ( x n ) = y ̂ × [ z ̂ H inc x n y 0 + = L L z ̂ h e i ( k x + ℓG ) x n ] .
{ J n = H inc x n y 0 + = L L P nℓ h h = n = 0 N 1 [ Q ℓn J n Q ℓn H inc x n y 0 ] ,
{ E = E inc x y + = L L n = 0 N 1 t ̂ η [ Q ℓn J n Q ℓn H inc x n y 0 ] e i ( k x + ℓG ) x e i k y ( y y 0 ) H = z ̂ H inc x y + = L L n = 0 N 1 z ̂ [ Q ℓn J n Q ℓn H inc x n y 0 ] e i ( k x + ℓG ) x e i k y ( y y 0 ) .
J j , p = β p j n ( J n j s = 1 2 t ̂ n , s j α n , s j f n , s j ) ,
M j , p = β p j n ( M n j s = 1 2 z ̂ α n , s j f n , s j )
{ T i E { E p } = T i E { 2 E inc p + j ( enclosing region p ) 2 c j [ μ p J j , p φ p M j , p × φ p i ω ε p ( · J j , p ) φ p ] dl } = b i p + j ( enclosing region p ) [ A ij p J j + B ij p M j ] T i H { H p } = T i H { 2 H inc p + j ( enclosing region p ) 2 c j [ ε p M j , p φ p + J j , p × φ p ] dl } = c i p + j ( enclosing region p ) [ C ij p J j + D ij p M j ]
{ A ij , mn p = 2 r = 1 2 s = 1 2 α m , r i α n , s j c i t ̂ m , r i f m , r i · { c j [ μ p β p j t ̂ n , s j f n , s j φ p i ω ε p β p j · ( t ̂ n , s j f n , s j ) φ p ] dl } dl B ij , mn p = 2 r = 1 2 s = 1 2 α m , r i α n , s j c i t ̂ m , r i f m , r i · { c j [ z ̂ β p j f n , s j × φ p ] dl } dl C ij , mn p = 2 r = 1 2 s = 1 2 α m , r i α n , s j c i z ̂ f m , r i · { c j [ t ̂ n , s j β p j f n , s j × φ p ] dl } dl D ij , mn p = 2 r = 1 2 s = 1 2 α m , r i α n , s j c i f m , r i { c j [ ε p β p j f n , s j φ p ] dl } dl b i , m p = 2 r = 1 2 α m , r i c i t ̂ m , r i f m , r i · E inc p dl c i , m p = 2 r = 1 2 α m , r i c i z ̂ f m , r i · H inc p dl
{ T i = 1 E { E p = 0 } = T i = 1 E { E inc p = 0 x y 0 + = L L n = 0 N 1 1 t ̂ l η [ Q ℓn J n j = 1 Q ℓn H inc p = 0 x n y 0 ] e i ( k x + ℓG ) x } = b i = 1 p = 0 + A i = 1 , j = 1 p = 0 J j = 1 T i = 1 H { H p = 0 } = T i = 1 H { z ̂ H inc p = 0 x y 0 + = L L n = 0 N 1 1 z ̂ [ Q ℓn J n j = 1 Q ℓn H inc p = 0 x n y 0 ] e i ( k x + ℓG ) x } = c i = 1 p = 0 + C i = 1 , j = 1 p = 0 J j = 1 ,
{ A i = 1 , j = 1 , mn p = 0 = r = 1 2 α m , r i = 1 t ̂ m , r i = 1 f m , r i = 1 · [ = L L t ̂ η Q ℓn e i ( k x + ℓG ) x ] dx C i = 1 , j = 1 , mn p = 0 = r = 1 2 α m , r i = 1 t ̂ m , r i = 1 f m , r i = 1 · [ = L L z ̂ Q ℓn e i ( k x + ℓG ) x ] dx b i = 1 , m p = 0 = r = 1 2 α m , r i = 1 t ̂ m , r i = 1 f m , r i = 1 · [ E inc p = 0 x y 0 = L L n = 0 N 1 1 t ̂ η Q ℓn H inc p = 0 x n y 0 e i ( k x + ℓG ) x ] dx c i = 1 , m p = 0 = r = 1 2 α m , r i = 1 t ̂ m , r i = 1 f m , r i = 1 · [ z ̂ H inc p = 0 x y 0 = L L n = 0 N 1 1 z ̂ Q ℓn H inc p = 0 x n y 0 e i ( k x + ℓG ) x ] dx
[ A B C D ] [ J M ] = [ b c ] ,
k x = n ( a ) + k SPP .

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