Abstract

This work presents a novel finite-element solution to the problem of scattering from multiple two-dimensional holes with side grating in infinite metallic walls. The formulation is based on using the surface integral equation with free-space Green’s function as the boundary constraint. The solution region is divided into interior regions containing each hole or cavity as a side grating and exterior region. The finite-element formulation is applied inside the interior regions to derive a linear system of equations associated with nodal field values. The surface integral equation is then applied at the opening of the holes as a boundary constraint to connect nodes on the boundaries to interior nodes. The technique presented here is highly efficient in terms of computing resources, versatile and accurate in comparison with previously published methods. The near and far fields are generated for different single and multiple hole examples.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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]
  2. H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
    [CrossRef] [PubMed]
  3. H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer, 1988).
  4. D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
    [CrossRef]
  5. S. H. Kang, H. J. Eom, and T. J. Park, “TM-scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Microwave Theory Tech. 41, 895-899 (1993).
    [CrossRef]
  6. T. J. Park, S. H. Kang, and H. J. Eom, “TE scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Antennas Propag. 42, 112-114 (1994).
    [CrossRef]
  7. H. J. Eom, “Slit array antenna,” in Electromagnetic Wave Theory for Boundary-Value Problems (Springer, 2004), pp. 251-257.
  8. J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” in IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
    [CrossRef]
  9. J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
    [CrossRef]
  10. O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
    [CrossRef]
  11. B. Alavikia and O. M. Ramahi, “Finite-element solution of the problem of scattering from cavities in metallic screens using the surface integral equation as a boundary constraint,” J. Opt. Soc. Am. A 26, 1915-1925 (2009).
    [CrossRef]
  12. Ansoft HFSS Version 10.1., Ansoft Corporation, http://www.ansoft.com.

2009 (1)

2002 (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

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]

1994 (1)

T. J. Park, S. H. Kang, and H. J. Eom, “TE scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Antennas Propag. 42, 112-114 (1994).
[CrossRef]

1993 (1)

S. H. Kang, H. J. Eom, and T. J. Park, “TM-scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Microwave Theory Tech. 41, 895-899 (1993).
[CrossRef]

1991 (1)

O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
[CrossRef]

1990 (2)

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” in IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

1978 (1)

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

Alavikia, B.

Auckland, D. T.

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Ebbesen, T. W.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

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]

Eom, H. J.

T. J. Park, S. H. Kang, and H. J. Eom, “TE scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Antennas Propag. 42, 112-114 (1994).
[CrossRef]

S. H. Kang, H. J. Eom, and T. J. Park, “TM-scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Microwave Theory Tech. 41, 895-899 (1993).
[CrossRef]

H. J. Eom, “Slit array antenna,” in Electromagnetic Wave Theory for Boundary-Value Problems (Springer, 2004), pp. 251-257.

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

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]

Harrington, R. F.

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

Jin, J. M.

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” in IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

Kang, S. H.

T. J. Park, S. H. Kang, and H. J. Eom, “TE scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Antennas Propag. 42, 112-114 (1994).
[CrossRef]

S. H. Kang, H. J. Eom, and T. J. Park, “TM-scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Microwave Theory Tech. 41, 895-899 (1993).
[CrossRef]

Lezec, H. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

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]

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Mittra, R.

O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
[CrossRef]

Park, T. J.

T. J. Park, S. H. Kang, and H. J. Eom, “TE scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Antennas Propag. 42, 112-114 (1994).
[CrossRef]

S. H. Kang, H. J. Eom, and T. J. Park, “TM-scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Microwave Theory Tech. 41, 895-899 (1993).
[CrossRef]

Raether, H.

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

Ramahi, O. M.

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]

Volakis, J. L.

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” in IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

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]

IEE Proc., Part H: Microwaves, Antennas Propag. (1)

J. M. Jin and J. L. Volakis, “TM scattering by an inhomogeneously filled aperture in a thick conducting plane,” in IEE Proc., Part H: Microwaves, Antennas Propag. 137, 153-159 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

J. M. Jin and J. L. Volakis, “TE scattering by an inhomogeneously filled thick conducting plane,” IEEE Trans. Antennas Propag. 38, 1280-1286 (1990).
[CrossRef]

O. M. Ramahi and R. Mittra, “Finite element solution for a class of unbounded geometries,” IEEE Trans. Antennas Propag. 39, 244-250 (1991).
[CrossRef]

T. J. Park, S. H. Kang, and H. J. Eom, “TE scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Antennas Propag. 42, 112-114 (1994).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

D. T. Auckland and R. F. Harrington, “Electromagnetic transmission through a filled slit in a conducting plane of finite thickness, TE case,” IEEE Trans. Microwave Theory Tech. 26, 499-505 (1978).
[CrossRef]

S. H. Kang, H. J. Eom, and T. J. Park, “TM-scattering from a slit in a thick conducting screen: Revisited,” IEEE Trans. Microwave Theory Tech. 41, 895-899 (1993).
[CrossRef]

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

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]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297, 820-823 (2002).
[CrossRef] [PubMed]

Other (3)

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

Ansoft HFSS Version 10.1., Ansoft Corporation, http://www.ansoft.com.

H. J. Eom, “Slit array antenna,” in Electromagnetic Wave Theory for Boundary-Value Problems (Springer, 2004), pp. 251-257.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (15)

Fig. 1
Fig. 1

Schematic of the scattering problem from a hole with arbitrary shape in an infinite PEC surface. An ABC or PML is used to truncate the computational domain.

Fig. 2
Fig. 2

Schematic of the scattering problem from a hole in a PEC surface. The dashed line represents the bounded region that contains all inhomogeneities and anisotropies.

Fig. 3
Fig. 3

Schematic of the surface integral contour in the upper half-space and lower half-space of the hole.

Fig. 4
Fig. 4

Schematic showing the extension of the surface integral method to multiple holes with side cavities.

Fig. 5
Fig. 5

Amplitude of total E field at the openings of the hole for a 0.8 λ × 0.5 λ air-filled rectangular hole, TM case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS).

Fig. 6
Fig. 6

Amplitude of total H field at the openings of the hole for a 0.8 λ × 0.5 λ air-filled rectangular hole, TE case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS).

Fig. 7
Fig. 7

Amplitude of total E-Field at the openings of the hole for a 0.7 λ × 0.35 λ Silicon ( ε r = 11.9 ) filled rectangular hole, TM case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS). The 0.42 λ × 0.07 λ PEC strip is positioned at the geometric center of the hole.

Fig. 8
Fig. 8

Amplitude of the total E field at the openings of the holes for five identical 0.4 λ × 0.2 λ air-filled rectangular holes, TM case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS). The holes are separated by 0.4 λ .

Fig. 9
Fig. 9

Amplitude of the total H field at the openings of the holes for five identical 0.4 λ × 0.2 λ air-filled rectangular holes, TE case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS). The holes are separated by 0.4 λ .

Fig. 10
Fig. 10

Amplitude of the far field for five identical 0.4 λ × 0.2 λ air-filled rectangular holes, TM case, θ = 0 , calculated using the method introduced in this work (FEM-TFSIE). The holes are separated by 0.4 λ .

Fig. 11
Fig. 11

Amplitude of the far-field for five identical 0.4 λ × 0.2 λ air-filled rectangular holes, TE case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE). The holes are separated by 0.4 λ .

Fig. 12
Fig. 12

Amplitude of the total E field at the openings of a single 0.5 λ × 0.8 λ hole with three identical 0.5 λ × 0.3 λ air-filled rectangular side cavities, TM case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS). The hole and cavities are separated by 0.5 λ .

Fig. 13
Fig. 13

Amplitude of the total H field at the openings of a single 0.5 λ × 0.8 λ hole with three identical 0.5 λ × 0.3 λ air-filled rectangular side cavities, TE case, θ = 0 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS). The hole and cavities are separated by 0.5 λ .

Fig. 14
Fig. 14

Amplitude of the total H field at the openings of a single 0.5 λ × 0.8 λ hole with three identical 0.5 λ × 0.3 λ air-filled rectangular side cavities, TE case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS). The hole and cavities are separated by 0.5 λ . The distance between ABC and aperture of the hole and cavities is h = 1 λ .

Fig. 15
Fig. 15

Amplitude of the total H field at the openings of a single 0.5 λ × 0.8 λ hole with three identical 0.5 λ × 0.3 λ air-filled rectangular side cavities, TE case, θ = 30 , calculated using the method introduced in this work (FEM-TFSIE), and HFSS (FEM-HFSS). The hole and cavities are separated by 0.5 λ . The distance between ABC and aperture of the hole and cavities, is h = 3 λ .

Equations (78)

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

( 1 p ( x , y ) u t ) + k 0 2 q ( x , y ) u t = g ,
R i = Ω in w i ( ( 1 p ( x , y ) u t ) + k 0 2 q ( x , y ) u t g ) d Ω = 0 ,
R i = Ω in ( 1 p ( x , y ) w i u t k 0 2 q ( x , y ) w i u t + g w i ) d Ω + w i p ( x , y ) u t d Γ = 0 .
R i = Ω in ( 1 p ( x , y ) w i u t k 0 2 q ( x , y ) w i u t + g w i ) d Ω = 0 .
u t = i = 1 m u i t α i ( x , y ) , i = 1 , 2 , 3 , , m ,
R = [ M ] [ U ] [ F ] = 0 ,
M i j = Element ( 1 p ( x , y ) α i ( x , y ) α j ( x , y ) k 0 2 q ( x , y ) α i ( x , y ) α j ( x , y ) ) d Ω ,
F i = Element g α i ( x , y ) d Ω .
[ M i i M i b I M i b II 0 0 M b I i M b I b I 0 M b I o I 0 M b II i 0 M b II b II 0 M b II o II 0 M o I b I 0 M o I o I 0 0 0 M o II b II 0 M o II o II ] [ u i u b I u b II u o I u o II ] = [ F i F b I F b II F o I F o II ] ,
2 E z ( ρ ) + k 0 2 E z ( ρ ) = j ω μ J z ( ρ ) , ρ Ω I ,
2 G e I ( ρ , ρ ) + k 0 2 G e I ( ρ , ρ ) = δ ( ρ ρ ) ,           ρ , ρ Ω I ,
G e I ( ρ , ρ ) = j 4 H 0 2 ( k 0 | ρ ρ source | ) + j 4 H 0 2 ( k 0 | ρ ρ image source | ) .
Ω I G e I ( ρ , ρ ) ( 2 E z ( ρ ) + k 0 2 E z ( ρ ) ) d Ω I = j ω μ Ω I J z ( ρ ) G e I ( ρ , ρ ) d Ω I
Ω I ( E z 2 G e I G e I 2 E z ) d Ω I = Γ I + Γ I ( E z G e I n G e I E z n ) d Γ I ,
Ω I E z ( ρ ) ( 2 G e I + k 0 2 G e I ) d Ω I = j ω μ Ω I J z ( ρ ) G e I d Ω I + Γ I + Γ I ( E z G e I n G e I E z n ) d Γ I .
E z ( ρ ) = j ω μ Ω I J z ( ρ ) G e I ( ρ , ρ ) d Ω I Γ I + Γ I ( E z ( ρ ) G e I ( ρ , ρ ) n G e I ( ρ , ρ ) E z ( ρ ) n ) d Γ I .
E z ( ρ ) = j ω μ Ω I J z ( ρ ) G e I ( ρ , ρ ) d Ω I Aperture I E z ( ρ ) G e I ( ρ , ρ ) n d Γ I .
E z ( ρ ) = E z inc ( ρ ) + E z ref ( ρ ) Aperture I E z ( ρ ) G e I ( ρ , ρ ) n d Γ I .
E z inc = exp ( j k 0 ( x sin θ y cos θ ) ) ,
E z ref = exp ( j k 0 ( x sin θ + y cos θ ) ) ,
E z ( ρ ) = j = 1 n E z j k = 1 2 ψ j k ( x j ) ,
ψ j k ( x ) = { x j Δ x , k = 1 ; 1 x j Δ x , k = 2 . }
[ u o I ] = [ T I ] + [ S I ] [ u b I ] ,
S i j I = x j Δ x x j ψ j 1 ( x j ) | G e I ( x i , y , x j , y ) y | y = 0 d x + x j x j + Δ x ψ j 2 ( x j ) | G e I ( x i , y , x j , y ) y | y = 0 d x ,
| G e I ( x i , y , x j , y ) y | y = 0 = j k 0 y 2 ( x i x j ) 2 + y 2 H 1 2 ( k 0 ( x i x j ) 2 + y 2 ) .
2 E z ( ρ ) + k 0 2 E z ( ρ ) = 0 , ρ Ω II .
2 G e II ( ρ , ρ ) + k 0 2 G e II ( ρ , ρ ) = δ ( ρ ρ ) ρ , ρ Ω II .
E z ( ρ ) = Γ II + Γ II ( E z ( ρ ) G e II ( ρ , ρ ) n G e II ( ρ , ρ ) E z ( ρ ) n ) d Γ II ,
E z ( ρ ) = Aperture II E z ( ρ ) G e II ( ρ , ρ ) n d Γ II .
E z ( ρ ) = Aperture II E z ( ρ ) G e II ( ρ , ρ ) y d x .
[ u o II ] = [ S II ] [ u b II ] ,
S i j II = x j Δ x x j ψ j 1 ( x j ) | G e II ( x i , y , x j , y ) y | y = t d x + x j x j + Δ x ψ j 2 ( x j ) | G e II ( x i , y , x j , y ) y | y = t d x ,
| G e II ( x i , y , x j , y ) y | y = t = j k 0 ( y + t ) 2 ( x i x j ) 2 + ( y + t ) 2 H 1 2 ( k 0 ( x i x j ) 2 + ( y + t ) 2 ) .
[ M i i M i b I M i b II M b I i M b I b I + M b I o I S I 0 M b II i 0 M b II b II M b II o II S II ] [ u i u b I u b II ] = [ F i F b I M b I o I T I F b II ] .
2 H z ( ρ ) + k 0 2 H z ( ρ ) = j ω ε M z ( ρ ) , ρ Ω I ,
2 G h I ( ρ , ρ ) + k 0 2 G h I ( ρ , ρ ) = δ ( ρ ρ ) , ρ , ρ Ω I .
| G h I ( ρ , ρ ) y | y = 0 = 0 .
G h I ( ρ , ρ ) = j 4 H 0 2 ( k 0 | ρ ρ source | ) j 4 H 0 2 ( k 0 | ρ ρ image source | ) .
H z ( ρ ) = j ω ε Ω I M z ( ρ ) G h I ( ρ , ρ ) d Ω I Γ I + Γ I ( H z ( ρ ) G h I ( ρ , ρ ) n G h I ( ρ , ρ ) H z ( ρ ) n ) d Γ I .
H z ( ρ ) = j ω ε Ω I M z ( ρ ) G h I ( ρ , ρ ) d Ω I + Aperture I G h I ( ρ , ρ ) H z ( ρ ) n d Γ I .
H z ( ρ ) = H z inc ( ρ ) + H z ref ( ρ ) + Aperture I G h I ( ρ , ρ ) H z ( ρ ) n d Γ I .
H z inc = exp ( j k 0 ( x sin θ y cos θ ) ) ,
H z ref = exp ( j k 0 ( x sin θ + y cos θ ) ) .
H z ( ρ ) n = H z ( x = x , y ) H z ( x , y ) y y
H z = j = 1 n H z j ψ j ( x j ) ,
ψ j ( x j ) = { 1 , x j Δ x j 2 < x j < x j + Δ x j 2 , 0 , elsewhere . }
S i j I = x j Δ x j 2 x j + Δ x j 2 G h I ( x i , y , x j , y = 0 ) ψ j ( x j ) y y d x ,
G h I ( x i , y , x j , y = 0 ) = j 2 H 0 2 ( k 0 ( x i x j ) 2 + y 2 ) ,
[ u o I ] = [ T I ] [ S I ] { [ u o I ] [ u b I ] } ,
[ u o I ] = { [ 1 ] + [ S I ] } 1 [ T I ] + { [ 1 ] + [ S I ] } 1 [ S I ] [ u b I ] .
2 H z ( ρ ) + k 0 2 H z ( ρ ) = 0 , ρ Ω II
2 G h II ( ρ , ρ ) + k 0 2 G h II ( ρ , ρ ) = δ ( ρ ρ ) ρ , ρ Ω II .
| G h II ( ρ , ρ ) y | y = t = 0 ,
H z ( ρ ) = Γ II + Γ II ( H z ( ρ ) G h II ( ρ , ρ ) n G h II ( ρ , ρ ) H z ( ρ ) n ) d Γ II ,
H z ( ρ ) = Aperture II G h II ( ρ , ρ ) H z ( ρ ) n d Γ II .
H z ( ρ ) = Aperture II G h II ( ρ , ρ ) ( H z ( x = x , y ) H z ( x , y ) y y ) d x
S i j II = x j Δ x j 2 x j + Δ x j 2 G h II ( x i , y , x j , y = t ) ψ j ( x j ) y y d x ,
G h II ( x i , y , x j , y = t ) = j 2 H 0 2 ( k 0 ( x i x j ) 2 + ( y + t ) 2 ) ,
[ u o II ] = [ S II ] { [ u o II ] [ u b II ] } ,
[ u o II ] = { [ 1 ] [ S II ] } 1 [ S II ] [ u b II ] .
[ M i i M i b I M i b II M b I i M b I b I + M b I o I ( 1 + S I ) 1 S I 0 M b II i 0 M b II o II ( 1 S II ) 1 S II ] [ u i u b I u b II ] = [ F i F b I M b I o I ( 1 + S I ) 1 T I F b II ] .
[ M ] ( 1 ) [ u ] ( 1 ) = [ F ] ( 1 ) ,
[ M ] ( 2 ) [ u ] ( 2 ) = [ F ] ( 2 ) ,
[ [ M ] ( 1 ) 0 0 [ M ] ( 2 ) ] [ [ u ] ( 1 ) [ u ] ( 2 ) ] = [ [ F ] ( 1 ) [ F ] ( 2 ) ] .
[ [ u o I ] ( 1 ) [ u o I ] ( 2 ) ] = [ [ T I ] ( 1 ) [ T I ] ( 2 ) ] + [ [ S I ] ( 11 ) [ S I ] ( 12 ) [ S I ] ( 21 ) [ S I ] ( 22 ) ] [ [ u b I ] ( 1 ) [ u b I ] ( 2 ) ]
[ [ u o II ] ( 1 ) [ u o II ] ( 2 ) ] = [ [ S II ] ( 11 ) [ S II ] ( 12 ) [ S II ] ( 21 ) [ S II ] ( 22 ) ] [ [ u b II ] ( 1 ) [ u b II ] ( 2 ) ] ,
[ [ M ] ( 1 ) [ C ] ( 12 ) [ C ] ( 21 ) [ M ] ( 2 ) ] [ [ u ] ( 1 ) [ u ] ( 2 ) ] = [ [ F ] ( 1 ) [ F ] ( 2 ) ] ,
[ C ] ( 12 ) = [ 0 0 0 0 [ M b I o I ] ( 1 ) [ S I ] ( 12 ) 0 0 0 [ M b II o II ] ( 1 ) [ S II ] ( 12 ) ] ,
[ C ] ( 21 ) = [ 0 0 0 0 [ M b I o I ] ( 2 ) [ S I ] ( 21 ) 0 0 0 [ M b II o II ] ( 2 ) [ S II ] ( 21 ) ] ,
[ M ] ( 1 ) = [ [ M i i ] ( 1 ) [ M i b I ] ( 1 ) [ M i b II ] ( 1 ) [ M b I i ] ( 1 ) [ M b I b I ] ( 1 ) + [ M b I o I ] ( 1 ) [ S I ] ( 11 ) 0 [ M b II i ] ( 1 ) 0 [ M b II b II ] ( 1 ) [ M b II o II ] ( 1 ) [ S II ] ( 11 ) ] ,
[ M ] ( 2 ) = [ [ M i i ] ( 2 ) [ M i b I ] ( 2 ) [ M i b II ] ( 2 ) [ M b I i ] ( 2 ) [ M b I b I ] ( 2 ) + [ M b I o I ] ( 2 ) [ S I ] ( 22 ) 0 [ M b II i ] ( 2 ) 0 [ M b II b II ] ( 2 ) [ M b II o II ] ( 2 ) [ S II ] ( 22 ) ] ,
[ F ] ( 1 ) = [ [ F i ] ( 1 ) [ F b I ] ( 1 ) [ M b I o I ] ( 1 ) [ T I ] ( 1 ) [ F b II ] ( 1 ) ] ,
[ F ] ( 2 ) = [ [ F i ] ( 2 ) [ F b I ] ( 2 ) [ M b I o I ] ( 2 ) [ T I ] ( 2 ) [ F b II ] ( 2 ) ] ,
[ u ] ( k ) = [ [ u i ] ( k ) [ u b I ] ( k ) [ u b II ] ( k ) ] ( k = 1 & 2 ) .
[ [ M ] ( 1 ) [ C ] ( 12 ) [ C ] ( 1 N ) [ C ] ( 21 ) [ M ] ( 2 ) [ C ] ( 2 N ) [ C ] ( N 1 ) [ C ] ( N 2 ) [ M ] ( N ) ] [ [ u ] ( 1 ) [ u ] ( 2 ) [ u ] ( N ) ] = [ [ F ] ( 1 ) [ F ] ( 2 ) [ F ] ( N ) ] ,
[ C ] ( p q ) = [ 0 0 0 0 [ M b I o I ] ( p ) [ S I ] ( p q ) 0 0 0 [ M b II o II ] ( p ) [ S II ] ( p q ) ] ,
[ M ] ( p ) = [ [ M i i ] ( p ) [ M i b I ] ( p ) [ M i b II ] ( p ) [ M b I i ] ( p ) [ M b I b I ] ( p ) + [ M b I o I ] ( p ) [ S I ] ( p p ) 0 [ M b II i ] ( p ) 0 [ M b II b II ] ( p ) [ M b II o II ] ( p ) [ S II ] ( p p ) ] ,
[ F ] ( p ) = [ [ F i ] ( p ) [ F b I ] ( p ) [ M b I o I ] ( p ) [ T I ] ( p ) [ F b II ] ( p ) ] ( p , q = 1 N ) .

Metrics