Abstract

The pseudo-Fourier modal analysis of two-dimensional arbitrarily shaped grating structures is described. It is shown that the pseudo-Fourier modal analysis has an advantage of improved structure modeling over the conventional rigorous coupled-wave analysis. In the conventional rigorous coupled-wave analysis, grating structures are modeled by the staircase approximation, which is well known to have inherent significant errors under TM polarization. However, in the pseudo-Fourier modal analysis, such a limitation of the staircase approximation can be overcome through the smooth-structure modeling based on two-dimensional Fourier representation. The validity of the claim is proved with some comparative numerical results from the proposed pseudo-Fourier modal analysis and the conventional rigorous coupled-wave analysis.

© 2008 Optical Society of America

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    [CrossRef]
  2. P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [CrossRef]
  3. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  4. L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A, Pure Appl. Opt. 5, 345-355 (2003).
    [CrossRef]
  5. L. Li, "Mathematical reflections on the Fourier modal method in grating theory," in Mathematical Modeling in Optical Science, G.Bao, ed. (SIAM, 2001), Chap. 4.
    [CrossRef]
  6. E. Popov and M. Nevière, "Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence," Opt. Lett. 25, 598-600 (2000).
    [CrossRef]
  7. E. Popov and M. Nevière, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
    [CrossRef]
  8. J. P. Hugonin and P. Lalanne, "Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A 22, 1844-1849 (2005).
    [CrossRef]
  9. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  10. H. Kim, I.-M. Lee, and B. Lee, "Extended scattering matrix method for efficient full parallel implementation of rigorous coupled wave analysis," J. Opt. Soc. Am. A 24, 2313-2327 (2007).
    [CrossRef]
  11. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  12. W. Jiang and R. T. Chen, "Rigorous analysis of diffraction gratings of arbitrary profiles using virtual photonic crystals," J. Opt. Soc. Am. A 23, 2192-2197 (2006).
    [CrossRef]
  13. H. Kim, S. Kim, I.-M. Lee, and B. Lee, "Pseudo-Fourier modal analysis on dielectric slabs with arbitrary longitudinal permittivity and permeability profiles," J. Opt. Soc. Am. A 23, 2177-2191 (2006).
    [CrossRef]
  14. K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
    [CrossRef]
  15. M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-977 (2006).
    [CrossRef]
  16. M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
    [CrossRef]

2007 (1)

2006 (4)

2005 (1)

2003 (2)

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A, Pure Appl. Opt. 5, 345-355 (2003).
[CrossRef]

K. Mehrany and B. Rashidian, "Polynomial expansion of electromagnetic eigenmodes in layered structures," J. Opt. Soc. Am. B 20, 2434-2441 (2003).
[CrossRef]

2002 (1)

2001 (1)

L. Li, "Mathematical reflections on the Fourier modal method in grating theory," in Mathematical Modeling in Optical Science, G.Bao, ed. (SIAM, 2001), Chap. 4.
[CrossRef]

2000 (2)

1997 (1)

1996 (2)

1981 (1)

Chamanzar, M.

M. Chamanzar, K. Mehrany, and B. Rashidian, "Legendre polynomial expansion for analysis of linear one-dimensional inhomogeneous optical structures and photonic crystals," J. Opt. Soc. Am. B 23, 969-977 (2006).
[CrossRef]

M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

Chen, R. T.

Gaylord, T. K.

Gralak, B.

Hugonin, J. P.

Jiang, W.

Kim, H.

Kim, S.

Lalanne, P.

Lee, B.

Lee, I.-M.

Li, L.

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A, Pure Appl. Opt. 5, 345-355 (2003).
[CrossRef]

L. Li, "Mathematical reflections on the Fourier modal method in grating theory," in Mathematical Modeling in Optical Science, G.Bao, ed. (SIAM, 2001), Chap. 4.
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
[CrossRef]

Mehrany, K.

Moharam, M. G.

Nevière, M.

Popov, E.

Rashidian, B.

Tayeb, G.

IEEE Trans. Antennas Propag. (1)

M. Chamanzar, K. Mehrany, and B. Rashidian, "Planar diffraction analysis of homogeneous and longitudinally inhomogeneous gratings based on Legendre expansion of electromagnetic fields," IEEE Trans. Antennas Propag. 54, 3686-3694 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

L. Li, "Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors," J. Opt. A, Pure Appl. Opt. 5, 345-355 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
[CrossRef]

L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
[CrossRef]

E. Popov and M. Nevière, "Grating theory: new equations in Fourier space leading to fast converging results for TM polarization," J. Opt. Soc. Am. A 17, 1773-1784 (2000).
[CrossRef]

J. P. Hugonin and P. Lalanne, "Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization," J. Opt. Soc. Am. A 22, 1844-1849 (2005).
[CrossRef]

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
[CrossRef]

H. Kim, I.-M. Lee, and B. Lee, "Extended scattering matrix method for efficient full parallel implementation of rigorous coupled wave analysis," J. Opt. Soc. Am. A 24, 2313-2327 (2007).
[CrossRef]

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

W. Jiang and R. T. Chen, "Rigorous analysis of diffraction gratings of arbitrary profiles using virtual photonic crystals," J. Opt. Soc. Am. A 23, 2192-2197 (2006).
[CrossRef]

H. Kim, S. Kim, I.-M. Lee, and B. Lee, "Pseudo-Fourier modal analysis on dielectric slabs with arbitrary longitudinal permittivity and permeability profiles," J. Opt. Soc. Am. A 23, 2177-2191 (2006).
[CrossRef]

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

Opt. Lett. (1)

Other (1)

L. Li, "Mathematical reflections on the Fourier modal method in grating theory," in Mathematical Modeling in Optical Science, G.Bao, ed. (SIAM, 2001), Chap. 4.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Multilayer modeling of the grating to be analyzed.

Fig. 2
Fig. 2

(a) Target grating structure, (b) structure modeling in the RCWA, (c) structure modeling in the PFMA.

Fig. 3
Fig. 3

Analysis of a single layer.

Fig. 4
Fig. 4

Bidirectional characterization of the nth layer: (a) left-to-right characterization, (b) right-to-left characterization.

Fig. 5
Fig. 5

Bidirectional characterization of a multilayer composed of two neighboring layers: (a) left-to-right characterization, (b) right-to-left characterization.

Fig. 6
Fig. 6

Subwavelength metallic triangle grating structure.

Fig. 7
Fig. 7

Comparison of the PFMA and the RCWA simulation results: (a) E y obtained by the PFMA, (b) E y obtained by the RCWA, (c) E x obtained by the PFMA, (d) E x obtained by the RCWA, (e) E z obtained by the PFMA, (f) E z obtained by the RCWA, (g) E z obtained by the PFMA, (h) E z obtained by the RCWA.

Fig. 8
Fig. 8

Subwavelength metallic triangle grating structure.

Fig. 9
Fig. 9

z-direction electric field distribution, E z , obtained by the RCWA with the staircase approximation of (a) 4 levels, (b) 8 levels, (c) 16 levels, (d) 32 levels, (e) 64 levels, and (f) 128 levels.

Fig. 10
Fig. 10

Oblique incidence: (a) E y obtained by the PFMA, (b) E y obtained by the RCWA, (c) E x obtained by the PFMA, (d) E x obtained by the RCWA, (e) E z obtained by the PFMA, (f) E z obtained by the RCWA.

Fig. 11
Fig. 11

Normal incidence: (a) E y obtained by the PFMA, (b) E y obtained by the RCWA, (c) E x obtained by the PFMA, (d) E x obtained by the RCWA, (e) E z obtained by the PFMA, (f) E z obtained by the RCWA.

Fig. 12
Fig. 12

Trapezoid.

Equations (85)

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ε ( n ) ( x ) = g ε ̃ g ( n ) exp ( j G x , g x ) ,
μ ( n ) ( x ) = g μ ̃ g ( n ) exp ( j G x , g x ) ,
ε ̂ ( n ) ( x , z ) = g , h ε ̃ g h ( n ) exp [ j ( G x , g x + G z , h z ) ] ,
μ ̂ ( n ) ( x , z ) = g , h μ ̃ g h ( n ) exp [ j ( G x , g x + G z , h z ) ] ,
ε ̂ ( n ) ( x , z ) = ε ( n ) ( x , z ) q = δ ( z q Δ d ) ,
μ ̂ ( n ) ( x , z ) = μ ( n ) ( x , z ) q = δ ( z q Δ d ) .
α ̂ ( n ) ( x , z ) = 1 ε ̂ ( n ) ( x , z ) = g , h α ̃ g h ( n ) exp [ j ( G x , g x + G z , h z ) ] ,
β ̂ ( n ) ( x , z ) = 1 μ ̂ ( n ) ( x , z ) = g , h β ̃ g h ( n ) exp [ j ( G x , g x + G z , h z ) ] .
E ̱ I = E ̱ i n c + E ̱ R = E ̱ i n c + h = H H ( r x , h x ̱ + r y , h y ̱ + r z , h z ̱ ) × exp [ j ( k x , h x + k y y k I , z , h z ) ] ,
E ̱ I I = E ̱ T = h = H H ( t x , h x ̂ + t y , h y ̂ + t z , h z ̂ ) × exp [ j ( k x , h x + k y y + k I I , z , h ( z Δ d ) ) ] ,
E ̱ i n c = E ̱ 0 ( u x , s x ̱ + u y , s y ̱ + u z , s z ̱ ) exp [ j ( k x , s x + k y y + k I , z , s z ) ] .
k x , s = k x , 0 + G x , s , for H s H ,
k I , z , s = ( k 0 n I ) 2 ( k x , s ) 2 ( k y ) 2 ,
k I I , z , s = ( k 0 n I I ) 2 ( k x , h ) 2 ( k y ) 2 ,
k x , 0 = k 0 n I sin θ cos ϕ ,
k y = k 0 n I sin θ sin ϕ ,
E ̱ k ̱ = exp [ j ( k x , 0 x + k y y + k z , 0 z ) ] E ̱ ̂ k ̱ ( x , y , z ) ,
H ̱ k ̱ = exp [ j ( k x , 0 x + k y y + k z , 0 z ) ] H ̱ ̂ k ̱ ( x , y , z ) ,
E ̱ k ̱ = E x x ̱ + E y y ̱ + E z z ̱ = exp ( j ( k x , 0 x + k y y + k z , 0 z ) ) E ̱ k ̱ = exp ( j ( k x , 0 x + k y y + k z , 0 z ) ) h = H H q = Q Q ( E x , h , q x ̱ + E y , h , q y ̱ + E z , h , q z ̱ ) exp ( j ( G x , h x + G z , q z ) ) ,
H ̱ k ̱ = H x x ̱ + H y y ̱ + H z z ̱ = exp ( j ( k x , 0 x + k y y + k z , 0 z ) ) H ̱ k ̱ ( x , y , z ) = exp ( j ( k x , 0 x + k y y + k z , 0 z ) ) j ε 0 μ 0 h = H H q = Q Q ( H x , h , q x ̱ + H y , h , q y ̱ + H z , h , q z ̱ ) exp ( j ( G x , h x + G z , q z ) ) .
× E ̱ = ( y E z z E y ) x ̱ + ( z E x x E z ) y ̱ + ( x E y y E x ) z ̱ = j ω μ 0 μ ( x , z ) ( H x x ̱ + H y y ̱ + H z z ̱ ) ,
× H ̱ = ( y H z z H y ) x ̱ + ( z H x x H z ) y ̱ + ( x H y y H x ) z ̱ = j ω ε 0 ε ( x , z ) ( E x x ̱ + E y y ̱ + E z z ̱ ) .
j k z , q E y , h , q = k 0 g p μ ̃ h g , q p H x , g , p + j k y E z , h , q ,
j k z , q E x , h , q = k 0 g p μ ̃ h g , q p H y , h , p + j k x , h E z , h , q ,
g p ε ̃ h g , q p E z , g , p = j k x , h k 0 H y , h , q + j k y k 0 H x , h , q ,
j k z , q k 0 H y , h , q = g p ε ̃ h g , q p E x , g , p + j k y k 0 H z , h , q ,
j k z , q k 0 H x , h , q = g p ε ̃ h g , q p E y , g , p + j k x , h k 0 H z , h , q ,
g p μ ̃ h g , q p H z , g , p = j k x , h k 0 E y , h , q + j k y k 0 E x , h , q ,
k z , q = k z , 0 + G z , q .
ε ͇ = ( ε ͇ 0 ε ͇ 1 ε ͇ 2 H ε ͇ 1 ε ͇ 0 ε ͇ 2 H + 1 ε ͇ 2 H ε ͇ 2 H 1 ε ͇ 0 ) ,
ε ͇ k = ( ε ̃ k , 0 ε ̃ k , 1 ε ̃ k , 2 Q ε ̃ k , 1 ε ̃ k , 0 ε ̃ k , 2 Q + 1 ε ̃ k , 2 Q ε ̃ k , 2 Q 1 ε ̃ k , 0 ) .
K ͇ x = ( k x , H k 0 I ͇ ( 2 Q + 1 ) 0 0 0 k x , H + 1 k 0 I ͇ ( 2 Q + 1 ) 0 0 0 0 0 k x , H k 0 I ͇ ( 2 Q + 1 ) ) ,
K ͇ y = ( k y k 0 ) I ͇ ( 2 Q + 1 ) ( 2 H + 1 ) ,
K ͇ z = ( [ k z , q k 0 ] 0 0 0 [ k z , q k 0 ] 0 0 [ k z , q k 0 ] 0 0 0 [ k z , q k 0 ] ) ,
[ k z , q k 0 ] = ( k z , Q k 0 0 0 0 k z , ( Q 1 ) k 0 0 0 0 0 0 k z , Q k 0 ) .
K ͇ z = k z , 0 k 0 I ͇ ( 2 Q + 1 ) ( 2 H + 1 ) + G ͇ z ,
G ͇ z = ( [ G z , q k 0 ] 0 0 0 [ G z , q k 0 ] 0 0 [ G z , q k 0 ] 0 0 0 [ G z , q k 0 ] ) ,
[ G z , q k 0 ] = ( G z , Q k 0 0 0 0 G z , ( Q 1 ) k 0 0 0 0 0 0 G z , Q k 0 ) .
E ̱ x = [ [ E x , H , Q E x , H , Q ] [ E x , H + 1 , Q E x , H + 1 , Q ] [ E x , H , Q E x , H , Q ] ] t .
[ ε ͇ 0 0 0 0 0 0 ε ͇ 0 0 0 0 0 0 ε ͇ 0 0 0 0 0 0 μ ͇ 0 0 0 0 0 0 μ ͇ 0 0 0 0 0 0 μ ͇ ] [ E ̱ x E ̱ y E ̱ z H ̱ x H ̱ y H ̱ z ] = [ 0 0 0 0 j K ͇ z j K ͇ y 0 0 0 j K ͇ z 0 j K ͇ x 0 0 0 j K ͇ y j K ͇ x 0 0 j K ͇ z j K ͇ y 0 0 0 j K ͇ z 0 j K ͇ x 0 0 0 j K ͇ y j K ͇ x 0 0 0 0 ] [ E ̱ x E ̱ y E ̱ z H ̱ x H ̱ y H ̱ z ] .
[ ε ͇ ( x ) 0 0 0 0 0 0 ε ͇ ( y ) 0 0 0 0 0 0 ε ͇ ( z ) 0 0 0 0 0 0 μ ͇ ( x ) 0 0 0 0 0 0 μ ͇ ( y ) 0 0 0 0 0 0 μ ͇ ( z ) ] [ E ̱ x E ̱ y E ̱ z H ̱ x H ̱ y H ̱ z ] = [ 0 0 0 0 j K ͇ z j K ͇ y 0 0 0 j K ͇ z 0 j K ͇ x 0 0 0 j K ͇ y j K ͇ x 0 0 j K ͇ z j K ͇ y 0 0 0 j K ͇ z 0 j K ͇ x 0 0 0 j K ͇ y j K ͇ x 0 0 0 0 ] [ E ̱ x E ̱ y E ̱ z H ̱ x H ̱ y H ̱ z ] .
E ̱ z = ε ͇ ( z ) 1 ( j K ͇ x H ̱ y + j K ͇ y H ̱ x ) ,
H ̱ z = μ ͇ ( z ) 1 ( j K ͇ x E ̱ y + j K ͇ y E ̱ x ) ,
[ 0 0 K ͇ y ε ͇ ( z ) 1 K ͇ x μ ͇ ( x ) K ͇ y ε ͇ ( z ) 1 K ͇ y 0 0 μ ͇ ( y ) + K ͇ x ε ͇ ( z ) 1 K ͇ x K ͇ x ε ͇ ( z ) 1 K ͇ y K ͇ y μ ͇ ( z ) 1 K ͇ x ε ͇ ( x ) K ͇ y μ ͇ ( z ) 1 K ͇ y 0 0 ε ͇ ( y ) + K ͇ x μ ͇ ( z ) 1 K ͇ x K ͇ x μ ͇ ( z ) 1 K ͇ y 0 0 ] [ E ̱ y E ̱ x H ̱ y H ̱ x ] = j k z , 0 k 0 [ E ̱ y E ̱ x H ̱ y H ̱ x ] + [ j G ͇ z 0 0 0 0 j G ͇ z 0 0 0 0 j G ͇ z 0 0 0 0 j G ͇ z ] [ E ̱ y E ̱ x H ̱ y H ̱ x ] .
[ j G ͇ z 0 K ͇ y ε ( z ) 1 K ͇ x μ ͇ ( x ) K ͇ y ε ͇ ( z ) 1 K ͇ y 0 j G ͇ z μ ͇ ( y ) + K ͇ x ε ͇ ( z ) 1 K ͇ x K ͇ x ε ͇ ( z ) 1 K ͇ y K ͇ y μ ͇ ( z ) 1 K ͇ x ε ͇ ( x ) K ͇ y μ ͇ ( z ) 1 K ͇ y j G ͇ z 0 ε ͇ ( y ) + K ͇ x μ ͇ ( z ) 1 K ͇ x K ͇ x μ ͇ ( z ) 1 K ͇ y 0 j G ͇ z ] [ E ̱ y E ̱ x H ̱ y H ̱ x ] = j k z , 0 k 0 [ E ̱ y E ̱ x H ̱ y H ̱ x ] .
ε ͇ ( x ) = t x ε ͇ + ( 1 t x ) α ͇ 1 ,
ε ͇ ( y ) = t y ε ͇ + ( 1 t y ) α ͇ 1 ,
ε ͇ ( z ) = t z ε ͇ + ( 1 t z ) α ͇ 1 ,
μ ͇ ( x ) = s x μ ͇ + ( 1 s x ) β ͇ 1 ,
μ ͇ ( y ) = s y μ ͇ + ( 1 s y ) β ͇ 1 ,
μ ͇ ( z ) = s z μ ͇ + ( 1 s z ) β ͇ 1 .
G z 2 Re ( k z , 0 ) < G z 2 .
E ̱ ( g ) + ( x , y , z ) = h = H H q = Q Q ( E x , h , q ( g ) + x ̱ + E y , h , q ( g ) + y ̂ + E z , h , q ( g ) + z ̂ ) × exp [ j ( k x , h x + k y y + k z , q ( g ) + z ) ] ,
E ̱ ( g ) ( x , y , z ) = h = H H q = Q Q ( E x , h , q ( g ) x ̱ + E y , h , q ( g ) y ̂ + E z , h , q ( g ) z ̂ ) × exp { j [ k x , h x + k y y + k z , q ( g ) ( z Δ d ) ] } ,
H ̱ ( g ) + ( x , y , z ) = j ε 0 μ 0 h = H H q = Q Q ( H x , h , q ( g ) + x ̱ + H y , h , q ( g ) + y ̂ + H z , h , q ( g ) + z ̂ ) × exp [ j ( k x , h x + k y y + k z , q ( g ) + z ) ] ,
H ̱ ( g ) ( x , y , z ) = j ε 0 μ 0 h = H H q = Q Q ( H x , h , q ( g ) x ̱ + H y , h , q ( g ) y ̂ + H z , h , q ( g ) z ̂ ) × exp { j [ k x , h x + k y y + k z , q ( g ) ( z Δ d ) ] } .
E ̱ ( x , y , z ) = g = 1 M + C g + E ̱ ( g ) + ( x , y , z ) + g = 1 M C g E ̱ ( g ) ( x , y , z ) ,
H ̱ ( x , y , z ) = g = 1 M + C g + H ̱ ( g ) + ( x , y , z ) + g = 1 M C g H ̱ ( g ) ( x , y , z ) ,
[ I 0 I 0 0 I 0 I k x , h k y k 0 k I , z , h ( k I , z , h 2 + k x , h 2 ) k 0 k I , z , h k x , h k y k 0 k I , z , h ( k I , z , h 2 + k x , h 2 ) k 0 k I , z , h ( k y 2 + k I , z , h 2 ) k 0 k I , z , h k y k x , h k 0 k I , z , h ( k y 2 + k I , z , h 2 ) k 0 k I , z , h k y k x , h k 0 k I , z , h ] [ u y δ h s u x δ h s R y , h R x , h ]
= ( q = Q Q E y , h , q ( 1 ) + q = Q Q E y , h , q ( M + ) + q = Q Q E y , h , q ( 1 ) e j ( q G z + k z ( 1 ) ) Δ d q = Q Q E y , h , q ( M ) e j ( q G z + k z ( M ) ) Δ d q = Q Q E x , h , q ( 1 ) + q = Q Q E x , h , q ( M + ) + q = Q Q E x , h , q ( 1 ) e j ( q G z + k z ( 1 ) ) Δ d p = Q Q E x , h , q ( M ) e j ( q G z + k z ( M ) ) Δ d j q = Q Q H y , h , q ( 1 ) + j q = Q Q H y , h , q ( M + ) + j q = Q Q H y , h , q ( 1 ) e j ( q G z + k z ( 1 ) ) Δ d j p = Q Q H y , h , q ( M ) e j ( q G z + k z ( M ) ) Δ d j p = Q Q H x , h , q ( 1 ) + j q = Q Q H x , h , q ( M + ) + j q = Q Q H x , h , q ( 1 ) e j ( q G z + k z ( 1 ) ) Δ d j p = Q Q H x , h , q ( M ) e j ( q G z + k z ( M ) ) Δ d ) [ C 1 + C M + + C 1 C M ] ,
[ I 0 0 I k x , h k y k 0 k I I , z , h ( k I I , z , h 2 + k x , h 2 ) k 0 k I I , z , h ( k y 2 + k I I , z , h 2 ) k 0 k I I , z , h k y k x , h k 0 k I I , z , h ] [ T y , h T x , h ]
= ( q = Q Q E y , h , q ( 1 ) + e j ( q G z + k z ( 1 ) + ) Δ d q = Q Q E y , h , q ( M + ) + e j ( q G z + k z ( M + ) + ) Δ d q = Q Q E y , h , q ( 1 ) q = Q Q E y , h , q ( M ) q = Q Q E x , h , q ( 1 ) + e j ( q G z + k z ( 1 ) + ) Δ d q = Q Q E x , h , q ( M + ) + e j ( q G z + k z ( M + ) + ) Δ d q = Q Q E x , h , q ( 1 ) p = Q Q E x , h , q ( M ) j q = Q Q H y , h , q ( 1 ) + e j ( q G z + k z ( 1 ) + ) Δ d j q = Q Q H y , h , q ( M + ) + e j ( q G z + k z ( M + ) + ) Δ d j q = Q Q H y , h , q ( 1 ) j p = Q Q H y , h , q ( M ) j p = Q Q H x , h , q ( 1 ) + e j ( q G z + k z ( 1 ) + ) Δ d j q = Q Q H x , h , q ( M + ) + e j ( q G z + k z ( M + ) + ) Δ d j q = Q Q H x , h , q ( 1 ) j p = Q Q H x , h , q ( M ) ) [ C 1 + C M + + C 1 C M ] .
( W h W h V h V h ) ( U ( n , n ) R ( n , n ) ) = ( W + ( n ) W ( n ) X ( n ) V + ( n ) V ( n ) X ( n ) ) ( C a ( n , n ) + C a ( n , n ) ) ,
( W + ( n ) X + ( n ) W ( n ) V + ( n ) X + ( n ) V ( n ) ) ( C a ( n , n ) + C a ( n , n ) ) = ( W h W h V h V h ) ( T ( n , n ) 0 ) ,
W h = ( I 0 0 I ) ,
V h = ( [ k x , h k y k 0 k I , z , h ] [ ( k I , z , h 2 + k x , h 2 ) k 0 k I , z , h ] [ ( k y 2 + k I , z , h 2 ) k 0 k I , z , h ] [ k y k x , h k 0 k I , z , h ] ) ,
( a H 0 0 a H ) .
( C a ( n , n ) + C a ( n , n ) ) = ( W h 1 W + ( n ) + V h 1 V + ( n ) ( W h 1 W ( n ) + V h 1 V ( n ) ) X ( n ) ( W h 1 W + ( n ) V h 1 V + ( n ) ) X + ( n ) W h 1 W ( n ) V h 1 V ( n ) ) 1 ( 2 U ( n , n ) 0 ) ,
R ( n , n ) = W h 1 [ W + ( n ) C a ( n , n ) + + W ( n ) X ( n ) C a ( n , n ) W h U ( n , n ) ] ,
T ( n , n ) = W h 1 [ W + ( n ) X + ( n ) C a ( n , n ) + + W ( n ) C a ( n , n ) ] .
C a ( n , n ) = ( C a ( n , n ) + C a ( n , n ) ) t .
S ( n , n ) = ( T ( n , n ) R ( n , n ) R ( n , n ) T ( n , n ) ) .
R ( n , n + 1 ) = R ( n , n ) + T ( n , n ) [ ( I R ( n + 1 , n + 1 ) R ( n , n ) ) 1 ] R ( n + 1 , n + 1 ) T ( n , n ) ,
T ( n , n + 1 ) = T ( n + 1 , n + 1 ) [ ( I R ( n , n ) R ( n + 1 , n + 1 ) ) 1 ] T ( n , n ) ,
R ( n , n + 1 ) = R ( n + 1 , n + 1 ) + T ( n + 1 , n + 1 ) [ ( I R ( n , n ) R ( n + 1 , n + 1 ) ) 1 ] R ( n , n ) T ( n + 1 , n + 1 ) ,
T ( n , n + 1 ) = T ( n , n ) [ ( I R ( n + 1 , n + 1 ) R ( n , n ) ) 1 ] T ( n + 1 , n + 1 ) .
C a ( n , n + 1 ) = ( C a , 1 ( n , n + 1 ) C a , 2 ( n , n + 1 ) ) ,
C b ( n , n + 1 ) = ( C b , 1 ( n , n + 1 ) C b , 2 ( n , n + 1 ) ) ,
( C a , 1 ( n , n + 1 ) C b , 1 ( n , n + 1 ) ) = ( C a ( n , n ) + C b ( n , n ) ( I R ( n + 1 , n + 1 ) R ( n , n ) ) 1 R ( n + 1 , n + 1 ) T ( n , n ) C b ( n , n ) ( I R ( n + 1 , n + 1 ) R ( n , n ) ) 1 T ( n + 1 , n + 1 ) ) ,
( C a , 2 ( n , n + 1 ) C b , 2 ( n , n + 1 ) ) = ( C a ( n + 1 , n + 1 ) ( I R ( n , n ) R ( n + 1 , n + 1 ) ) 1 T ( n , n ) C b ( n + 1 , n + 1 ) + C a ( n + 1 , n + 1 ) ( I R ( n , n ) R ( n + 1 , n + 1 ) ) 1 R ( n , n ) T ( n + 1 , n + 1 ) ) .
( C a ( n , n + 1 ) C b ( n , n + 1 ) ) = ( C a ( n , n ) C b ( n , n ) ) * ( C a ( n + 1 , n + 1 ) C b ( n + 1 , n + 1 ) ) .
Γ ( x , y ) = { 1 for ( x , y ) trapezoid 0 for ( x , y ) trapezoid .
F ( f x , f y ) = ( j ( y 2 y 1 ) 2 π f x ) { e 2 π f x b e j 2 π ( f x b c f y ) ( y 1 + y 2 2 ) sinc [ ( f x b c f y ) ( y 2 y 1 ) ] + e 2 π f x a e j 2 π ( f x a c + f y ) ( y 1 + y 2 2 ) sinc [ ( f x a c + f y ) ( y 2 y 1 ) ] } ;
F ( f x , f y ) = ( a + b c ) e i 2 π f y c { [ j 2 π ( c y 2 ) f y 1 ] e j 2 π f y ( c y 2 ) [ j 2 π ( c y 1 ) f y 1 ] e j 2 π f y ( c y 1 ) ( 2 π f y ) 2 } ;
F f x , f y = { ( a + b ) ( y 2 y 1 ) ( 2 c y 1 y 2 ) 2 c } .

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