Abstract

A spectral element method together with a surface integral equation as the radiation boundary condition is used to simulate the scattering properties of periodic subwavelength slits. The surface integral equation utilizes the periodic Green’s function in the wave number space and is solved by the method of moments, while the interior inhomogeneous medium is modeled by the spectral element method. The solution convergence is found to be exponential; i.e., the error decreases exponentially with the order of basis functions. To our knowledge, such a fast solver with spectral accuracy is new in the scattering problem of periodic structures. Scattering properties of a gold slit grid within the whole wavelength-incidence angle parameter space are investigated, with the confirmation that strong transmission of light through subwavelength slits is achievable.

© 2010 Optical Society of America

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  1. J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
    [CrossRef]
  2. G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).
  3. J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
    [CrossRef]
  4. M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
    [CrossRef]
  5. J. Liu and Q. H. Liu, “A novel radiation boundary condition for finite-element method,” Microwave Opt. Technol. Lett. 49, 1995-2002 (2007).
    [CrossRef]
  6. F. Q. Hu, “A spectral boundary integral equation method for the 2D Helmholtz equation,” J. Comput. Phys. 120, 340-347 (1995).
    [CrossRef]
  7. J. Liu and Q. H. Liu, “A spectral integral method (SIM) for periodic and nonperiodic structures,” IEEE Microw. Wirel. Compon. Lett. 14, 97-99 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2009 (2)

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

H.-S. Leong, J. Guo, R. G. Lindquist, and Q. H. Liu, “Surface plasmon resonance in nanostructured metal films under the Kretschmann configuration” J. Appl. Phys. 106, 124314 (2009).
[CrossRef]

2008 (1)

2007 (3)

C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N.-A. P. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express 15, 8163-8169 (2007).
[CrossRef] [PubMed]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold,” J. Chem. Phys. 127, 189901(E) (2007).
[CrossRef]

J. Liu and Q. H. Liu, “A novel radiation boundary condition for finite-element method,” Microwave Opt. Technol. Lett. 49, 1995-2002 (2007).
[CrossRef]

2006 (2)

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

2005 (2)

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

K. M. Byun, S. J. Kim, and D. Kim, “Design study of highly sensitive nanowire-enhanced surface plasmon resonance biosensors using rigorous coupled wave analysis,” Opt. Express 13, 3737-3742 (2005).
[CrossRef] [PubMed]

2004 (1)

J. Liu and Q. H. Liu, “A spectral integral method (SIM) for periodic and nonperiodic structures,” IEEE Microw. Wirel. Compon. Lett. 14, 97-99 (2004).
[CrossRef]

1998 (1)

1995 (2)

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,“Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 12, 1068-1076 (1995).
[CrossRef]

F. Q. Hu, “A spectral boundary integral equation method for the 2D Helmholtz equation,” J. Comput. Phys. 120, 340-347 (1995).
[CrossRef]

1986 (1)

1968 (3)

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).
[CrossRef]

E. Kretschmann and H. Raether, “Radiative decay of nonradiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135-2136 (1968).

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398-410 (1968).
[CrossRef]

1967 (1)

Y.-Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511-514 (1967).
[CrossRef]

1966 (1)

1964 (1)

T. Tamir, H. C. Wang, and A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. 12, 323-335 (1964).
[CrossRef]

Alleyne, C. J.

Arakawa, E. T.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).
[CrossRef]

Burckhardt, C. B.

Byun, K. M.

Cohen, G. C.

G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).

Cowan, J. J.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).
[CrossRef]

Djurisic, A. B.

Elazar, J. M.

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold,” J. Chem. Phys. 127, 189901(E) (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Gaylord, T. K.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,“Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 12, 1068-1076 (1995).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780-1787 (1986).
[CrossRef]

Grann, E. B.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,“Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 12, 1068-1076 (1995).
[CrossRef]

Guo, J.

H.-S. Leong, J. Guo, R. G. Lindquist, and Q. H. Liu, “Surface plasmon resonance in nanostructured metal films under the Kretschmann configuration” J. Appl. Phys. 106, 124314 (2009).
[CrossRef]

Hamm, R. N.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).
[CrossRef]

Hu, F. Q.

F. Q. Hu, “A spectral boundary integral equation method for the 2D Helmholtz equation,” J. Comput. Phys. 120, 340-347 (1995).
[CrossRef]

Kim, D.

Kim, S. J.

Kirk, A. G.

Kretschmann, E.

E. Kretschmann and H. Raether, “Radiative decay of nonradiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135-2136 (1968).

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold,” J. Chem. Phys. 127, 189901(E) (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Lee, J. -H.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

Leong, H. -S.

H.-S. Leong, J. Guo, R. G. Lindquist, and Q. H. Liu, “Surface plasmon resonance in nanostructured metal films under the Kretschmann configuration” J. Appl. Phys. 106, 124314 (2009).
[CrossRef]

Li, Z.

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

Lindquist, R. G.

H.-S. Leong, J. Guo, R. G. Lindquist, and Q. H. Liu, “Surface plasmon resonance in nanostructured metal films under the Kretschmann configuration” J. Appl. Phys. 106, 124314 (2009).
[CrossRef]

Liu, J.

J. Liu and Q. H. Liu, “A novel radiation boundary condition for finite-element method,” Microwave Opt. Technol. Lett. 49, 1995-2002 (2007).
[CrossRef]

J. Liu and Q. H. Liu, “A spectral integral method (SIM) for periodic and nonperiodic structures,” IEEE Microw. Wirel. Compon. Lett. 14, 97-99 (2004).
[CrossRef]

Liu, Q. H.

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

H.-S. Leong, J. Guo, R. G. Lindquist, and Q. H. Liu, “Surface plasmon resonance in nanostructured metal films under the Kretschmann configuration” J. Appl. Phys. 106, 124314 (2009).
[CrossRef]

J. Liu and Q. H. Liu, “A novel radiation boundary condition for finite-element method,” Microwave Opt. Technol. Lett. 49, 1995-2002 (2007).
[CrossRef]

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

J. Liu and Q. H. Liu, “A spectral integral method (SIM) for periodic and nonperiodic structures,” IEEE Microw. Wirel. Compon. Lett. 14, 97-99 (2004).
[CrossRef]

Luo, M.

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

Majewski, M. L.

Maystre, D.

McPhedran, R. C.

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold,” J. Chem. Phys. 127, 189901(E) (2007).
[CrossRef]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

Moharam, M. G.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,“Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 12, 1068-1076 (1995).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780-1787 (1986).
[CrossRef]

Nicorovici, N. -A. P.

Oliner, A. A.

T. Tamir, H. C. Wang, and A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. 12, 323-335 (1964).
[CrossRef]

Otto, A.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398-410 (1968).
[CrossRef]

Pommet, D. A.

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,“Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 12, 1068-1076 (1995).
[CrossRef]

Raether, H.

E. Kretschmann and H. Raether, “Radiative decay of nonradiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135-2136 (1968).

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

Rakic, A. D.

Ritchie, R. H.

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).
[CrossRef]

Shuler, M. L.

Stern, E. A.

Y.-Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511-514 (1967).
[CrossRef]

Tamir, T.

T. Tamir, H. C. Wang, and A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. 12, 323-335 (1964).
[CrossRef]

Teng, Y. -Y.

Y.-Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511-514 (1967).
[CrossRef]

Wang, H. C.

T. Tamir, H. C. Wang, and A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. 12, 323-335 (1964).
[CrossRef]

Xiao, T.

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

Yoon, S. J.

Appl. Opt. (1)

IEEE Microw. Wirel. Compon. Lett. (1)

J. Liu and Q. H. Liu, “A spectral integral method (SIM) for periodic and nonperiodic structures,” IEEE Microw. Wirel. Compon. Lett. 14, 97-99 (2004).
[CrossRef]

IEEE Trans. Comput.-Aided Des. (1)

J.-H. Lee and Q. H. Liu, “An efficient 3-D spectral element method for Schrodinger equation in nanodevice simulation,” IEEE Trans. Comput.-Aided Des. 24, 1848-1858 (2005).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

J.-H. Lee, T. Xiao, and Q. H. Liu, “A 3-D spectral element method using mixed-order curl conforming vector basis functions for electromagnetic fields,” IEEE Trans. Microwave Theory Tech. 54, 437-444 (2006).
[CrossRef]

T. Tamir, H. C. Wang, and A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. 12, 323-335 (1964).
[CrossRef]

J. Appl. Phys. (1)

H.-S. Leong, J. Guo, R. G. Lindquist, and Q. H. Liu, “Surface plasmon resonance in nanostructured metal films under the Kretschmann configuration” J. Appl. Phys. 106, 124314 (2009).
[CrossRef]

J. Chem. Phys. (2)

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125, 164705 (2006).
[CrossRef] [PubMed]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “Erratum: An analytic model for the optical properties of gold,” J. Chem. Phys. 127, 189901(E) (2007).
[CrossRef]

J. Comput. Phys. (1)

F. Q. Hu, “A spectral boundary integral equation method for the 2D Helmholtz equation,” J. Comput. Phys. 120, 340-347 (1995).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

J. Opt. Soc. Am. A Opt. Image Sci. Vis (1)

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord,“Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A Opt. Image Sci. Vis 12, 1068-1076 (1995).
[CrossRef]

Microwave Opt. Technol. Lett. (1)

J. Liu and Q. H. Liu, “A novel radiation boundary condition for finite-element method,” Microwave Opt. Technol. Lett. 49, 1995-2002 (2007).
[CrossRef]

Opt. Express (2)

Phys. Rev. E (1)

M. Luo, Q. H. Liu, and Z. Li, “Spectral element method for band structures of two-dimensional anisotropic photonic crystals,” Phys. Rev. E 79, 026705 (2009).
[CrossRef]

Phys. Rev. Lett. (2)

Y.-Y. Teng and E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511-514 (1967).
[CrossRef]

R. H. Ritchie, E. T. Arakawa, J. J. Cowan, and R. N. Hamm, “Surface-plasmon resonance effect in grating diffraction,” Phys. Rev. Lett. 21, 1530-1533 (1968).
[CrossRef]

Z. Naturforsch. A (1)

E. Kretschmann and H. Raether, “Radiative decay of nonradiative surface plasmons excited by light,” Z. Naturforsch. A 23, 2135-2136 (1968).

Z. Phys. (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398-410 (1968).
[CrossRef]

Other (2)

G. C. Cohen, Higher-Order Numerical Methods for Transient Wave Equations (Springer, 2001).

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

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Figures (8)

Fig. 1
Fig. 1

Geometry of the 2D scattering problem with 1D periodicity.

Fig. 2
Fig. 2

(a) Geometry of a half space interface. (b) Relative errors of the field pattern, reflection rate, and transmission rate compared to the analytical solution of a half space interface versus the sampling density in terms of the number of points per wavelength.

Fig. 3
Fig. 3

Two configurations of a unit cell of a gold slit grid. The upper and lower half spaces are filled with dielectric materials with reflective indices n 1 and n 2 , respectively. The third material, gold in this case, is deposited under the interface of the two dielectric materials. In system (a), the gold grid is rectangular; the period along the x ̂ direction is L x , the thickness of the gold film is d, and the width of the slit is s. The fill factor is defined as f = s / L x and θ is the angle of incidence of the plane wave. In the system (b), the gold grid is triangular; the period and width of the slit are the same as in system (a), but the height of the gold triangle is 2 d , so that the area of gold is the same as in (a).

Fig. 4
Fig. 4

(a) Transmission rate and (b) absorption rate of a square gold grid with L x = 200   nm , d = 200   nm , and f = 0.2 .

Fig. 5
Fig. 5

(a) Transmission rate and (b) absorption rate of a square gold grid with L x = 200   nm , d = 50   nm , and f = 0.2 .

Fig. 6
Fig. 6

Accuracy analysis of the SEM. Comparison of the reflection rate (a) and absorption rate (b) obtained by the fifth order SEM (blue dotted line), the ninth order SEM (red dashed line), and the FMM (black line) for the system in Fig. 5. (c) Comparison of the reflection rate obtained by the third order SEM (blue dotted line), the fifth order SEM (red dashed line), and the analytical solution (black line) with fill factor f = 0 and 200 nm thick gold film. (d) The relative error of reflection rate versus the order of the SEM. The wavelength of these calculations is 850 nm.

Fig. 7
Fig. 7

(a) Transmission rate and (b) absorption rate of a triangular gold grid with L x = 200   nm , 2 d = 100   nm , and f = 0.2 .

Fig. 8
Fig. 8

Transmission rate of a square gold grid with L x = 600   nm , d = 600   nm , and f = 0.2 .

Equations (24)

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

× [ μ r s 1 ( × E z z ̂ ) ] k 0 2 ε r z E z z ̂ = 0 ,
× [ ε r s 1 ( × H z z ̂ ) ] k 0 2 μ r z H z z ̂ = 0 ,
μ r = [ μ r s 0 0 μ r z ] ,     ε r = [ ε r s 0 0 ε r z ] ,
μ r s = [ μ x x μ x y μ y x μ y y ] μ r z = μ z z , ε r s = [ ε x x ε x y ε y x ε y y ]
Γ d r { ( × W j z ̂ ) [ ε r s 1 ( × H z z ̂ ) ] k 0 2 μ r z W j H z } j k 0 b = 1 , 2 Γ b d x W j M z = 0 ,
S [ H i H b 1 H b 2 ] + T ( 1 ) M b 1 + T ( 2 ) M b 1 = 0 ,
S j , k = Γ d r { ( × W j z ̂ ) [ ε r s 1 ( × W k z ̂ ) ] k 0 2 μ r z W j W k } ,
T j , k ( 1 , 2 ) = j k 0 Γ 1 , 2 d x W j W k .
H z inc ( x , y ) = 1 2 H z ( x , y ) + j k 0 ε r b G p ( b ) [ k b ( x x ) , k b ( y y ) ] M z ( x ) d x ,
( G p ( b ) [ k b ( x x ) , k b ( y y ) ] / y ) y = y H z ( x ) d x = 0
G p ( b ) [ k b ( x x ) , k b ( y y ) ] / y = 0
U ( 1 ) H b 1 + V ( 1 ) M b 1 = F ( 1 ) ,
U ( 2 ) H b 2 + V ( 2 ) M b 2 = F ( 2 ) ,
U j , k ( 1 , 2 ) = 1 2 Γ 1 , 2 d x W j W k ,
V j , k ( 1 , 2 ) = j k 0 ε r b Γ 1 , 2 Γ 1 , 2 W j ( x ) G p ( b ) [ k b ( x x ) , 0 ] W k ( x ) d x d x ,
F j ( 1 , 2 ) = Γ 1 , 2 d x W j H inc .
2 G b ( r r ) + k b 2 G b ( r r ) = δ ( r r ) ,
G b ( r r ) = e j q ( r r ) ( 2 π ) 2 ( q 2 k b 2 ) d q ,
G p b ( r r ) = m e j k x m L x e j q ( r r m L x x ̂ ) ( 2 π ) 2 ( q 2 k b 2 ) d q = m e j k x m L x + j q x m L x e j q ( r r ) ( 2 π ) 2 ( q 2 k b 2 ) d q .
G p b ( r r ) = m e j q x m ( x x ) ( 2 π L x ) I m ( k b , k x ) ,
V j , k ( 1 , 2 ) = j k 0 ε r b m = N N I m ( k b , k x ) [ Γ 1 , 2 e j q x m x W j ( x ) d x ] [ Γ 1 , 2 e j q x m x W k ( x ) d x ] ,
a = 1 r t ,
P d = 1 2 Γ ω ε | E | 2 d r
e = Γ | H z H z exact | 2 d r Γ | H z exact | 2 d r ,

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