Abstract

We propose a novel electromagnetic analysis scheme for crossed nanophotonic structures. The developed scheme is based on the mathematical modeling with the local Fourier modal analysis and the generalized scattering-matrix method. The mathematical Bloch eigenmodes of two-port block and four-port intersection block structures are analyzed by the local Fourier modal analysis. The interconnections of two-port blocks and four-port intersection block are described by the generalized scattering-matrix method. This scheme provides the linear system theory of general crossed nanophotonic structures.

© 2008 Optical Society of America

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References

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  1. N. Moll, R. Harbers, R. F. Mahrt, and G.-L. Bona, “Integrated all-optical switch in a cross-waveguide geometry,” Appl. Phys. Lett. 88, 171104 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. M. F. Yanik and S. Fan, “Stopping light all-optically,” Phys. Rev. Lett. 92, 083901 (2004).
    [CrossRef] [PubMed]
  4. M. F. Yanik, H. A. Altug, J. Vuckovic, and S. Fan, “Sub-micron all optical digital memory and integration of nano-scale photonic devices without isolators,” J. Lightwave Technol. 22, 2316-2322 (2004).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811-818 (1981).
    [CrossRef]
  8. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [CrossRef]
  9. 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]
  10. W. Jian 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]
  11. H. Kim and B. Lee, “Pseudo-Fourier modal analysis of two-dimensional arbitrarily shaped grating structures,” J. Opt. Soc. Am. A 25, 40-54 (2008).
    [CrossRef]
  12. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  13. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  14. 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]
  15. L. Li, “Mathematical reflections on the Fourier modal method in grating theory,” in Mathematical Modeling in Optical Science, G.Bao, ed. (SIAM, Philadelphia, 2001), Chap. 4.
    [CrossRef]
  16. E. Popov and M. Neviére, “Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence,” Opt. Lett. 25, 598-600 (2000).
    [CrossRef]
  17. E. Popov and M. Neviére, “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]
  18. 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]
  19. E. Popov, M. Neviére, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  20. 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]
  21. 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]
  22. E. L. Tan, “Note on formulation of the enhanced scattering-(transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157-1161 (2002).
    [CrossRef]
  23. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655-660 (2003).
    [CrossRef]
  24. M. G. Moharam and A. B. Greenwell, “Efficient rigorous calculations of power flow in grating coupled surface-emitting devices,” Proc. SPIE 5456, 57-67 (2004).
    [CrossRef]
  25. Q. Cao, P. Lalanne, and J. P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335-338 (2002).
    [CrossRef]
  26. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).
  27. J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gerard, A. Tchelnokov, and P. Fornel, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 1999).
  28. S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Springer, 2002).
  29. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2004).
  30. S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. 90, 051113 (2007).
    [CrossRef]

2008 (1)

2007 (2)

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]

S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. 90, 051113 (2007).
[CrossRef]

2006 (3)

2005 (1)

2004 (3)

M. F. Yanik, H. A. Altug, J. Vuckovic, and S. Fan, “Sub-micron all optical digital memory and integration of nano-scale photonic devices without isolators,” J. Lightwave Technol. 22, 2316-2322 (2004).
[CrossRef]

M. F. Yanik and S. Fan, “Stopping light all-optically,” Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

M. G. Moharam and A. B. Greenwell, “Efficient rigorous calculations of power flow in grating coupled surface-emitting devices,” Proc. SPIE 5456, 57-67 (2004).
[CrossRef]

2003 (3)

2002 (3)

2001 (1)

2000 (2)

1997 (1)

1996 (3)

1995 (1)

1981 (1)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811-818 (1981).
[CrossRef]

Appl. Phys. Lett. (2)

N. Moll, R. Harbers, R. F. Mahrt, and G.-L. Bona, “Integrated all-optical switch in a cross-waveguide geometry,” Appl. Phys. Lett. 88, 171104 (2006).
[CrossRef]

S. Kim, H. Kim, Y. Lim, and B. Lee, “Off-axis directional beaming of optical field diffracted by a single subwavelength metal slit with asymmetric dielectric surface gratings,” Appl. Phys. Lett. 90, 051113 (2007).
[CrossRef]

J. Lightwave Technol. (1)

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. A (15)

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

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

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

E. Popov and M. Neviére, “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]

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

Q. Cao, P. Lalanne, and J. P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335-338 (2002).
[CrossRef]

E. L. Tan, “Note on formulation of the enhanced scattering-(transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157-1161 (2002).
[CrossRef]

L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655-660 (2003).
[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]

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]

W. Jian 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, 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]

H. Kim and B. Lee, “Pseudo-Fourier modal analysis of two-dimensional arbitrarily shaped grating structures,” J. Opt. Soc. Am. A 25, 40-54 (2008).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811-818 (1981).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. Lett. (2)

M. F. Yanik and S. Fan, “Stopping light all-optically,” Phys. Rev. Lett. 92, 083901 (2004).
[CrossRef] [PubMed]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Proc. SPIE (1)

M. G. Moharam and A. B. Greenwell, “Efficient rigorous calculations of power flow in grating coupled surface-emitting devices,” Proc. SPIE 5456, 57-67 (2004).
[CrossRef]

Other (5)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gerard, A. Tchelnokov, and P. Fornel, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 1999).

S. G. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Springer, 2002).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2004).

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

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Figures (25)

Fig. 1
Fig. 1

(a) Schematic of nanophotonic network, (b) basic elements and interconnections.

Fig. 2
Fig. 2

(a) Photonic crystal cross waveguide structure, (b) S-matrix diagram of the photonic crystal cross waveguide structure.

Fig. 3
Fig. 3

(a) Multilayer structure and (b) S-matrix and coupling coefficient matrix operator.

Fig. 4
Fig. 4

Schematics of eigenvalue equations of (a) positive and (b) negative Bloch eigenmodes.

Fig. 5
Fig. 5

(a) Two-dimensional photonic crystal waveguide and the guided Bloch eigenmode: (b) y-polarization electric field distribution, (c) x-polarization magnetic field distribution, (d) z-polarization magnetic field distribution.

Fig. 6
Fig. 6

(a) Two-port α block with finite size, (b) two-port half-infinite α block with right boundary, (c) two-port half-infinite α block with left boundary.

Fig. 7
Fig. 7

(a) Two-port β block with finite size, (b) two-port half-infinite β block with upper boundary, (c) two-port half-infinite β block with lower boundary.

Fig. 8
Fig. 8

(a) Left-to-right and (b) right-to-left directional characterizations. The left and right boundaries are set at z = z and z + , respectively.

Fig. 9
Fig. 9

(a) Intersection block model with PML placed within four waveguide branches, (b) schematic of 4 × 4 S matrix.

Fig. 10
Fig. 10

Permittivity profiles of the intersection blocks of the two-dimensional photonic crystal (a) cross waveguide, (b) T-branch, and (c) 90 ° -bend structures.

Fig. 11
Fig. 11

Dominant eigenmode profiles of the intersection block of the cross waveguide structure: (a) E α , ( 1 ) , y + , (b) E β , ( 1 ) , y + , (c) H α , ( 1 ) , x + , (d) H β , ( 1 ) , x + , (e) H α , ( 1 ) , z + , (f) H β , ( 1 ) , z + .

Fig. 12
Fig. 12

Dominant eigenmode profiles of the intersection block of the photonic crystal T-branch structure: (a) E α , ( 1 ) , y + , (b) E β , ( 1 ) , y + , (c) H α , ( 1 ) , x + , (d) H β , ( 1 ) , x + , (e) H α , ( 1 ) , z + , (f) H β , ( 1 ) , z + .

Fig. 13
Fig. 13

Dominant eigenmode profiles of the intersection block of the photonic crystal 90 ° -bend structure: (a) E α , ( 1 ) , y + , (b) E β , ( 1 ) , y + , (c) H α , ( 1 ) , x + , (d) H β , ( 1 ) , x + , (e) H α , ( 1 ) , z + , (f) H β , ( 1 ) , z + .

Fig. 14
Fig. 14

Excitation of ports (a) 1, (b) 2, (c) 3, and (d) 4.

Fig. 15
Fig. 15

LFMA results of the S-matrix characterization of (a) the intersection block of the photonic crystal cross waveguide structure, (b) the photonic crystal T-branch structure, and (c) the photonic crystal 90 ° -bend structure.

Fig. 16
Fig. 16

Two-port α block interconnection: (a) left and right finite size blocks, (b) left half-infinite and right finite size blocks, (c) left finite size and right half-infinite blocks, (d) left and right half-infinite blocks.

Fig. 17
Fig. 17

Two-port β block interconnection: (a) upper and lower finite size blocks, (b) upper half-infinite and lower finite size blocks, (c) upper finite size and lower half-infinite blocks, and (d) upper and lower half-infinite blocks.

Fig. 18
Fig. 18

Diffraction of the fundamental guided Bloch eigenmode at the right endface of the two-dimensional half-infinite photonic crystal structure: (a) simulation schematic, (b) y-polarization electric field distribution, (c) x-polarization magnetic field distribution, (d) z-polarization magnetic field distribution. Excitation of the fundamental guided Bloch eigenmode at the left endface of the two-dimensional half-infinite photonic crystal structure: (e) simulation schematic, (f) y-polarization electric field distribution, (g) x-polarization magnetic field distribution, (h) z-polarization magnetic field distribution.

Fig. 19
Fig. 19

Transmission and reflection of two-dimensional finite sized photonic crystal waveguide by a normally incident plane wave: (a) simulation schematic, (b) y-polarization electric field distribution, (c) x-polarization magnetic field distribution, (d) z-polarization magnetic field distribution.

Fig. 20
Fig. 20

(a) Interconnection of four two-port blocks and a four-port cross block, (b) extended four-port cross block composed of four two-port blocks and a four-port intersection block.

Fig. 21
Fig. 21

Interconnection through ports (a) 1, (b) 2, (c) 3, and (d) 4.

Fig. 22
Fig. 22

Building the extended four-port cross block: (a) step 1: interconnection of a two-port block to the four-port cross block through port 1, (b) step 2: interconnection of a two-port block to the combined four-port cross block through port 2, (c) step 3: interconnection of a two-port block to the combined four-port cross block through port 3, (d) step 4: building the extended four-port cross block by the interconnection of a two-port block to the combined four-port through port 4.

Fig. 23
Fig. 23

(a) Cross waveguide structure and y-polarization electric field distributions at each step of building the extended four-port cross block by the step-by-step interconnection procedure: steps (b) 1, (c) 2, (d) 3, and (e) 4.

Fig. 24
Fig. 24

(a) T-branch waveguide structure and y-polarization electric field distributions at each step of building the extended four-port cross block by the step-by-step interconnection procedure: steps (b) 1, (c) 2, (d) 3, and (e) 4.

Fig. 25
Fig. 25

(a) 90 ° -bend waveguide structure and y-polarization electric field distributions at each step of building the extended four-port cross block by the step-by-step interconnection procedure: steps (b) 1, (c) 2, (d) 3, and (e) 4.

Equations (146)

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S = ( T R R T ) ,
S = ( T R R T ) .
S = ( S 11 S 21 S 31 S 41 S 12 S 22 S 32 S 42 S 13 S 23 S 33 S 43 S 14 S 24 S 34 S 44 ) .
E ̱ ̃ k ̱ = exp ( j ( k x , 0 x + k y , 0 y + k z , 0 z ) ) E ̱ k ̱ ( x , y , z ) ,
H ̱ ̃ k ̱ = exp ( j ( k x , 0 x + k y , 0 y + k z , 0 z ) ) H ̱ k ̱ ( x , y , z ) ,
[ β w ̱ w ̱ ] = ( T ( 1 , Q ) R ( 1 , Q ) R ( 1 , Q ) T ( 1 , Q ) ) [ w ̱ β w ̱ ] ,
( T ( 1 , Q ) 0 R ( 1 , Q ) I ) [ w ̱ w ̱ ] = β ( I R ( 1 , Q ) 0 T ( 1 , Q ) ) [ w ̱ w ̱ ] .
C g = C ̃ a w ̱ g + β g C ̃ b w ̱ g .
( T ( 1 , Q ) 0 R ( 1 , Q ) I ) [ w ̱ w ̱ ] = β + ( I R ( 1 , Q ) 0 T ( 1 , Q ) ) [ w ̱ w ̱ ] ,
1 β ( T ( 1 , Q ) 0 R ( 1 , Q ) I ) [ w ̱ w ̱ ] = ( I R ( 1 , Q ) 0 T ( 1 , Q ) ) [ w ̱ w ̱ ] .
E ̱ ( q ) , g ( 1 , Q ) = m = M M n = N N [ x ̱ E ( q ) , x , m n g ( 1 , Q ) ( z ) + y ̱ E ( q ) , y , m n g ( 1 , Q ) ( z ) + z ̱ E ( q ) , z , m n g ( 1 , Q ) ( z ) ] exp [ j ( k x , m x + k y , n y ) ] ,
H ̱ ( q ) , g ( 1 , Q ) = m = M M n = N N [ x ̱ H ( q ) , x , m n g ( 1 , Q ) ( z ) + y ̱ H ( q ) , y , m n g ( 1 , Q ) ( z ) + z ̱ H ( q ) , z , m n g ( 1 , Q ) ( z ) ] exp [ j ( k x , m x + k y , n y ) ] ,
exp ( j k z , 0 ( g ) ) E ( 1 ) , x , m n g ( 1 , Q ) ( z ) for 0 z l 1 , 1 exp ( j k z , 0 ( g ) ) E ( 2 ) , x , m n g ( 1 , Q ) ( z ) for l 1 , 1 z l 1 , 2 exp ( j k z , 0 ( g ) ) E ( Q ) , x , m n g ( 1 , Q ) ( z ) for l 1 , Q 1 z l 1 , Q p = H H E x , m , n , p ( g ) exp ( j G z , p z ) ,
E ̱ ( g ) ( x , y , z ) = m = M M n = N N p = H H ( E x , m , n , p ( g ) x ̱ + E y , m , n , p ( g ) y ̱ + E z , m , n , p ( g ) z ̱ ) exp ( j ( k x , m x + k y , n y + k z , p ( g ) z ) ) ,
H ̱ ( g ) ( x , y , z ) = m = M M n = N N p = H H ( H x , m , n , p ( g ) x ̱ + H y , m , n , p ( g ) y ̱ + H z , m , n , p ( g ) z ̱ ) exp ( j ( k x , m x + k y , n y + k z , p ( g ) z ) ) ,
E ̱ ( g ) ± ( x , y , z ) = m = M M n = N N p = M M ( E x , m , n , p ( g ) ± x ̱ + E y , m , n , p ( g ) ± y ̱ + E z , m , n , p ( g ) ± z ̱ ) exp ( j ( k x , m x + k y , n y + k z , p ( g ) ± z ) ) ,
H ̱ ( g ) ± ( x , y , z ) = m = M M n = N N p = M M ( H x , m , n , p ( g ) ± x ̱ + H y , m , n , p ( g ) ± y ̱ + H z , m , n , p ( g ) ± z ̱ ) exp ( j ( k x , m x + k y , n y + k z , p ( g ) ± z ) ) .
( k α , x , m , k α , y , n , k α , z , p ( g ) ) = ( k x , 0 + 2 π T x m , k y , 0 + 2 π T y n , k α , z ( g ) + 2 π T z p ) .
E ̱ α , ( g ) ± ( x , y , z ) = m = M M n = N N p = M M ( E α , x , m , n , p ( g ) ± x ̱ + E α , y , m , n , p ( g ) ± y ̱ + E α , z , m , n , p ( g ) ± z ̱ ) exp ( j ( k α , x , m x + k α , y , n y + k α , z , p ( g ) ± ( z z ) ) ) ,
H ̱ α , ( g ) ± ( x , y , z ) = m = M M n = N N p = M M ( H α , x , m , n , p ( g ) ± x ̱ + H α , y , m , n , p ( g ) ± y ̱ + H α , z , m , n , p ( g ) ± z ̱ ) exp ( j ( k α , x , m x + k α , y , n y + k α , z , p ( g ) ± ( z z ) ) )
( k β , x , m ( g ) , k β , y , n , k β , z , p ) = ( k β , x , 0 ( g ) + 2 π T x m , k y , 0 + 2 π T y n , k z , 0 + 2 π T z p ) .
E ̱ β , ( g ) ± ( x , y , z ) = m = M M n = N N p = M M ( E β , x , m , n , p ( g ) ± x ̱ + E β , y , m , n , p ( g ) ± y ̱ + E β , z , m , n , p ( g ) ± z ̱ ) exp ( j ( k β , x , m ( g ) ± ( x x ) + k β , y , n y + k β , z , p z ) ) ,
H ̱ β , ( g ) ± ( x , y , z ) = m = M M n = N N p = M M ( H β , x , m , n , p ( g ) ± x ̱ + H β , y , m , n , p ( g ) ± y ̱ + H β , z , m , n , p ( g ) ± z ̱ ) exp ( j ( k β , x , m ( g ) ± ( x x ) + k β , y , n y + k β , z , p z ) ) .
ε ( x , y , z ) = ε ( z , y , x ) ,
μ ( x , y , z ) = μ ( z , y , x ) ,
( E x , m , n , p , E y , m , n , p , E z , m , n , p ) = ( E z , p , n , m , E y , p , n , m , E x , p , n , m ) ,
( H x , m , n , p , H y , m , n , p , H z , m , n , p ) = ( H z , p , n , m , H y , p , n , m , H x , p , n , m ) .
U ̱ = m = M M n = N N ( u x , m , n x ̱ + u y , m , n y ̱ + u z , m , n z ̱ ) exp ( j ( k x , m x + k y , n y + k z , m , n ( z z ) ) ) ,
R ̱ = m = M M n = N N ( r x , m , n x ̱ + r y , m , n y ̱ + r z , m , n z ̱ ) exp ( j ( k x , m x + k y , n y k z , m , n ( z z ) ) ) ,
T ̱ = m = M M n = N N ( t x , m , n x ̱ + t y , m , n y ̱ + t z , m , n z ̱ ) exp ( j ( k x , m x + k y , n y + k z , m , n ( z z + ) ) ) .
U ̱ = m = M M n = N N ( u x , m , n x ̱ + u y , m , n y ̱ + u z , m , n z ̱ ) exp ( j ( k x , m x + k y , n y k z , m , n ( z z + ) ) ) ,
R ̱ = m = M M n = N N ( r x , m , n x ̱ + r y , m , n y ̱ + r z , m , n z ̱ ) exp ( j ( k x , m x + k y , n y + k z , m , n ( z z + ) ) ) ,
T ̱ = m = M M n = N N ( t x , m , n x ̱ + t y , m , n y ̱ + t z , m , n z ̱ ) exp ( j ( k x , m x + k y , n y k z , m , n ( z z + ) ) ) .
( W α , h W α , h V α , h V α , h ) ( U R ) = ( W α , + ( 0 ) W α , ( z z + ) V α , + ( 0 ) V α , ( z z + ) ) ( C α , a + C α , a ) at z = z ,
( W α , h W α , h V α , h V α , h ) ( T 0 ) = ( W α , + ( z + z ) W α , ( 0 ) V α , + ( z + z ) V α , ( 0 ) ) ( C α , a + C α , a ) at z = z + ,
( W α , h W α , h V α , h V α , h ) ( 0 T ) = ( W α , + ( 0 ) W α , ( z z + ) V α , + ( 0 ) V α , ( z z + ) ) ( C α , b + C α , b ) at z = z ,
( W α , h W α , h V α , h V α , h ) ( R U ) = ( W α , + ( z + z ) W α , ( 0 ) V α , + ( z + z ) V α , ( 0 ) ) ( C α , b + C α , b ) at z = z + ,
W α , h = ( I 0 0 I ) ,
V α , h = ( 1 ω μ 0 k α , x , m k α , y , n k α , z , m , n 1 ω μ 0 ( k α , z , m , n 2 + k α , x , m 2 ) k α , z , m , n 1 ω μ 0 ( k α , y , n 2 + k α , z , m , n 2 ) k α , z , m , n 1 ω μ 0 k α , y , n k α , x , m k α , z , m , n ) .
W α , + ( z ) = ( p = H H E ̃ α , y , m , n , p ( 1 ) + e j k α , z , p ( 1 ) + z p = H H E ̃ α , y , m , n , p ( M + ) + e j k α , z , p ( M + ) + z p = H H E ̃ α , x , m , n , p ( 1 ) + e j k α , z , p ( 1 ) + z p = H H E ̃ α , x , m , n , p ( M + ) + e j k α , z , p ( M + ) + z ) ,
V α , + ( z ) = ( p = H H H ̃ α , y , m , n , p ( 1 ) + e j k α , z , p ( 1 ) + z p = H H H ̃ α , y , m , n , p ( M + ) + e j k α , z , p ( M + ) + z p = H H H ̃ α , x , m , n , p ( 1 ) + e j k α , z , p ( 1 ) + z p = H H H ̃ α , x , m , n , p ( M + ) + e j k α , z , p ( M + ) + z ) .
W α , ( z ) = ( p = H H E ̃ α , y , m , n , p ( 1 ) e j k α , z , p ( 1 ) z p = H H E ̃ α , y , m , n , p ( M ) e j k α , z , p ( M ) z p = H H E ̃ α , x , m , n , p ( 1 ) e j k α , z , p ( 1 ) z p = H H E ̃ α , x , m , n , p ( M ) e j k α , z , p ( M ) z ) ,
V α , ( z ) = ( p = H H H ̃ α , y , m , n , p ( 1 ) e j k α , z , p ( 1 ) z p = H H H ̃ α , y , m , n , p ( M ) e j k α , z , p ( M ) z p = H H H ̃ α , x , m , n , p ( 1 ) e j k α , z , p ( 1 ) z p = H H H ̃ α , x , m , n , p ( M ) e j k α , z , p ( M ) z ) .
[ C α , a + C α , a ] = ( W α , h 1 W α , + ( 0 ) + V α , h 1 V α , + ( 0 ) W α , h 1 W α , ( z z + ) + V α , h 1 V α , ( z z + ) W α , h 1 W α , + ( z + z ) V α , h 1 V α , + ( z + z ) W α , h 1 W α , ( 0 ) V α , h 1 V α , ( 0 ) ) 1 [ 2 U 0 ] ,
[ C α , b + C α , b ] = ( W α , h 1 W α , + ( 0 ) + V α , h 1 V α , + ( 0 ) W α , h 1 W α , ( z z + ) + V α , h 1 V α , ( z z + ) W α , h 1 W α , + ( z + z ) V α , h 1 V α , + ( z + z ) W α , h 1 W α , ( 0 ) V α , h 1 V α , ( 0 ) ) 1 [ 0 2 U ] .
R = W α , h 1 [ W α , + ( 0 ) C α , a + + W α , ( z z + ) C α , a W α , h ] ,
T = W α , h 1 [ W α , + ( z + z ) C α , a + + W α , ( 0 ) C α , a ] ,
R = W α , h 1 [ W α , + ( z + z ) C α , b + + W α , ( 0 ) C α , b W α , h ] ,
T = W α , h 1 [ W α , + ( 0 ) C α , b + + W α , ( z z + ) C α , b ] .
R = [ ( W α , h ) 1 W α , ( z c ) ( V α , h ) 1 V α , ( z c ) ] 1 [ ( W α , h ) 1 W α , + ( z c ) ( V α , h ) 1 V α , + ( z c ) ] ,
T = [ ( W α , ( z c ) ) 1 W α , h ( V α , ( z c ) ) 1 V α , h ] 1 [ ( W α , ( z c ) ) 1 W α , + ( z c ) ( V α , ( z c ) ) 1 V α , + ( z c ) ] ,
R = [ ( W α , ( z c ) ) 1 W α , h ( V α , ( z c ) ) 1 V α , h ] 1 [ ( W α , ( z c ) ) 1 W α , h + ( V α , ( z c ) ) 1 V α , h ] ,
T = 2 [ ( W α , h ) 1 W α , ( z c ) ( V α , h ) 1 V α , ( z c ) ] 1 .
R = [ ( W α , + ( z c ) ) 1 W α , h + ( V α , + ( z c ) ) 1 V α , h ] 1 [ ( W α , + ( z c ) ) 1 W α , h ( V α , + ( z c ) ) 1 V α , h ] ,
T = 2 [ ( W α , h ) 1 W α , + ( z c ) + ( V α , h ) 1 V α , + ( z c ) ] 1 ,
R = [ ( W α , h ) 1 W α , + ( z c ) + ( V α , h ) 1 V α , + ( z c ) ] 1 [ ( W α , h ) 1 W α , ( z c ) + ( V α , h ) 1 V α , ( z c ) ] ,
T = [ ( W α , + ( z c ) ) 1 W α , h + ( V α , + ( z c ) ) 1 V α , h ] 1 [ ( W α , + ( z c ) ) 1 W α , ( z c ) ( V α , + ( z c ) ) 1 V α , ( z c ) ] .
( Y β , h Y β , h Z β , h Z β , h ) ( U R ) = ( Y β , + ( 0 ) Y β , ( x x + ) Z β , + ( 0 ) Z β , ( x x + ) ) ( C β , a + C β , a ) at x = x ,
( Y β , h Y β , h Z β , h Z β , h ) ( T 0 ) = ( Y β , + ( x + x ) Y β , ( 0 ) Z β , + ( x + x ) Z β , ( 0 ) ) ( C β , a + C β , a ) at x = x + ,
( Y β , h Y β , h Z β , h Z β , h ) ( 0 T ) = ( Y β , + ( 0 ) Y β , ( x x + ) Z β , + ( 0 ) Z β , ( x x + ) ) ( C β , b + C β , b ) at x = x ,
( Y β , h Y β , h Z β , h Z β , h ) ( R U ) = ( Y β , + ( x + x ) Y β , ( 0 ) Z β , + ( x + x ) Z β , ( 0 ) ) ( C β , b + C β , b ) at x = x + ,
Y β , h = ( I 0 0 I ) ,
Z β , h = ( 1 ω μ 0 k β , z , m k β , y , n k β , x , m , n 1 ω μ 0 ( k β , x , m , n 2 + k β , z , m 2 ) k β , x , m , n 1 ω μ 0 ( k β , y , n 2 + k β , x , m , n 2 ) k β , x , m , n 1 ω μ 0 k β , y , n k β , z , m k β , x , m , n ) .
Y β , + ( x ) = ( m = M M E ̃ β , y , m , n , s ( 1 ) + e j k x , m ( 1 ) + x m = M M E ̃ β , y , m , n , s ( M + ) + e j k x , m ( M + ) + x m = M M E ̃ β , z , m , n , s ( 1 ) + e j k x , m ( 1 ) + x m = M M E ̃ β , z , m , n , s ( M + ) + e j k x , m ( M + ) + x ) ,
Z β , + ( x ) = ( m = M M H ̃ β , y , m , n , s ( 1 ) + e j k x , m ( 1 ) + x m = M M H ̃ β , y , m , n , s ( M + ) + e j k x , m ( M + ) + x m = M M H ̃ β , z , m , n , s ( 1 ) + e j k x , m ( 1 ) + x m = M M H ̃ β , z , m , n , s ( M + ) + e j k x , m ( M + ) + x ) .
Y β , ( x ) = ( m = M M E ̃ β , y , m , n , s ( 1 ) e j k x , m ( 1 ) x m = M M E ̃ β , y , m , n , s ( M ) e j k x , m ( M ) x m = M M E ̃ β , z , m , n , s ( 1 ) e j k x , m ( 1 ) x m = M M E ̃ β , z , m , n , s ( M ) e j k x , m ( M ) x ) ,
Z β , ( x ) = ( m = M M H ̃ β , y , m , n , s ( 1 ) e j k x , m ( 1 ) x m = M M H ̃ β , y , m , n , s ( M ) e j k x , m ( M ) x m = M M H ̃ β , z , m , n , s ( 1 ) e j k x , m ( 1 ) x m = M M H ̃ β , z , m , n , s ( M ) e j k x , m ( M ) x ) .
[ C β , a + C β , a ] = ( ( Y β , h 1 Y β , + ( 0 ) + Z β , h 1 Z β , + ( 0 ) ) ( Y β , h 1 Y β , ( x x + ) + Z β , h 1 Z β , ( x x + ) ) ( Y β , h 1 Y β , + ( x + x ) Z β , h 1 Z β , + ( x + x ) ) ( Y β , h 1 Y β , ( 0 ) Z β , h 1 Z β , ( 0 ) ) ) 1 [ 2 U 0 ] ,
[ C β , b + C β , b ] = ( ( Y β , h 1 Y β , + ( 0 ) + Z β , h 1 Z β , + ( 0 ) ) ( Y β , h 1 Y β , ( x x + ) + Z β , h 1 Z β , ( x x + ) ) ( Y β , h 1 Y β , + ( x + x ) Z β , h 1 Z β , + ( x + x ) ) ( Y β , h 1 Y β , ( 0 ) Z β , h 1 Z β , ( 0 ) ) ) 1 [ 0 2 U ] .
R = Y β , h 1 [ Y β , + ( 0 ) C β , a + + Y β , ( x x + ) C β , a Y β , h ] ,
T = Y β , h 1 [ Y β , + ( x + x ) C β , a + + Y β , ( 0 ) C β , a ] ,
R = Y β , h 1 [ Y β , + ( x + x ) C β , b + + Y β , ( 0 ) C β , b Y β , h ] ,
T = Y β , h 1 [ Y β , + ( 0 ) C β , b + + Y β , ( x x + ) C β , b ] .
R = [ ( Y β , h ) 1 Y β , ( x c ) ( Z β , h ) 1 Z β , ( x c ) ] 1 [ ( Y β , h ) 1 Y β , + ( x c ) ( Z β , h ) 1 Z β , + ( x c ) ] ,
T = [ ( Y β , ( x c ) ) 1 Y β , h ( Z β , ( x c ) ) 1 Z β , h ] 1 [ ( Y β , ( x c ) ) 1 Y β , + ( Z β , ( x c ) ) 1 Z β , + ] ,
T = 2 [ ( Y β , h ) 1 Y β , ( x c ) ( Z β , h ) 1 Z β , ( x c ) ] 1 ,
R = [ ( Y β , ( x c ) ) 1 Y β , h ( Z β , ( x c ) ) 1 Z β , h ] 1 [ ( Y β , ( x c ) ) 1 Y β , h + ( Z β , ( x c ) ) 1 Z β , h ] .
R = [ ( Y β , + ( x c ) ) 1 Y β , h + ( Z β , + ( x c ) ) 1 Z β , h ] 1 [ ( Y β , + ( x c ) ) 1 Y β , h ( Z β , + ( x c ) ) 1 Z β , h ] ,
T = 2 [ ( Y β , h ) 1 Y β , + ( x c ) + ( Z β , h ) 1 Z β , + ( x c ) ] 1 ,
R = [ ( Y β , h ) 1 Y β , + ( x c ) + ( Z β , h ) 1 Z β , + ( x c ) ] 1 [ ( Y β , h ) 1 Y β , ( x c ) + ( Z β , h ) 1 Z β , ( x c ) ] ,
T = [ ( Y β , + ( x c ) ) 1 Y β , h + ( Z β , + ( x c ) ) 1 Z β , h ] 1 [ ( Y β , + ( x c ) ) 1 Y β , ( Z β , + ( x c ) ) 1 Z β , ] .
E ̱ ( x , y , z ) = g = 1 M + C α , g + E ̱ α , ( g ) + ( x , y , z ) + g = 1 M C α , g E ̱ α , ( g ) ( x , y , z ) + g = 1 M + C β , g + E ̱ β , ( g ) + ( x , y , z ) + g = 1 M C β , g E ̱ β , ( g ) ( x , y , z ) ,
H ̱ ( x , y , z ) = g = 1 M + C α , g + H ̱ α , ( g ) + ( x , y , z ) + g = 1 M C α , g H ̱ α , ( g ) ( x , y , z ) + g = 1 M + C β , g + H ̱ β , ( g ) + ( x , y , z ) + g = 1 M C β , g H ̱ β , ( g ) ( x , y , z ) ,
exp ( j k α , z , 0 ( g ) ± z ) q = H H ζ α , q ( g ) ± exp [ j 2 π q T z z ] ,
exp ( j k β , x , 0 ( g ) ± x ) q = H H ζ β , q ( g ) ± exp [ j 2 π q T x x ] ,
ζ α , q ( g ) ± = sinc ( k z , 0 ( g ) ± T z 2 π q ) ,
ζ β , q ( g ) ± = sinc ( k x , 0 ( g ) ± T x 2 π q ) .
E ̱ α , ( g ) ± ( x , y , z ) = exp ( j ( k x , 0 x + k y , 0 y ) ) m = M M n = N N s = H H ( E ̃ α , x , m , n , s ( g ) ± x ̱ + E ̃ α , y , m , n , s ( g ) ± y ̱ + E ̃ α , z , m , n , s ( g ) ± z ̱ ) exp ( j ( G x , m x + G y , n y + G z , s z ) ) ,
H ̱ α , ( g ) ± ( x , y , z ) = exp ( j ( k x , 0 x + k y , 0 y ) ) m = M M n = N N s = H H ( H ̃ α , x , m , n , s ( g ) ± x ̱ + H ̃ α , y , m , n , s ( g ) ± y ̱ + H ̃ α , z , m , n , s ( g ) ± z ̱ ) exp ( j ( G x , m x + G y , n y + G z , s z ) ) ,
E ̃ α , x , m , n , s ( g ) ± x ̱ + E ̃ α , y , m , n , s ( g ) ± y ̱ + E ̃ α , z , m , n , s ( g ) ± z ̱ = p = H H ( ζ α , s p ( g ) ± E α , x , m , n , p ( g ) ± x ̱ + ζ α , s p ( g ) ± E α , y , m , n , p ( g ) ± y ̱ + ζ α , s p ( g ) ± E α , z , m , n , p ( g ) ± z ̱ ) exp ( j k z , p ( g ) ± z ) ,
H ̃ α , x , m , n , s ( g ) ± x ̱ + H ̃ α , y , m , n , s ( g ) ± y ̱ + H ̃ α , z , m , n , s ( g ) ± z ̱ = p = H H ( ζ α , s p ( g ) ± H α , x , m , n , p ( g ) ± x ̱ + ζ α , s p ( g ) ± H α , y , m , n , p ( g ) ± y ̱ + ζ α , s p ( g ) ± H α , z , m , n , p ( g ) ± z ̱ ) exp ( j k z , p ( g ) ± z ) ,
E ̱ β , ( g ) ± ( x , y , z ) = exp ( j ( k z , 0 z + k y , 0 y ) ) s = M M n = N N p = H H ( E ̃ β , s , m , n , p ( g ) ± x ̱ + E ̃ β , y , s , n , p ( g ) ± y ̱ + E ̃ β , z , s , n , p ( g ) ± z ̱ ) exp ( j ( G x , s x + G y , n y + G z , p z ) ) ,
H ̱ β , ( g ) ± ( x , y , z ) = exp ( j ( k z , 0 z + k y , 0 y ) ) s = M M n = N N s = H H ( H ̃ β , x , s , n , p ( g ) ± x ̱ + H ̃ β , y , s , n , p ( g ) ± y ̱ + H ̃ β , z , s , n , p ( g ) ± z ̱ ) exp ( j ( G x , s x + G y , n y + G z , p z ) ) ,
E ̃ β , x , s , n , p ( g ) ± x ̱ + E ̃ β , y , s , n , p ( g ) ± y ̱ + E ̃ β , z , s , n , p ( g ) ± z ̱ = m = M M ( ζ β , s m ( g ) ± E β , x , m , n , p ( g ) ± x ̱ + ζ β , s m ( g ) ± E β , y , m , n , p ( g ) ± y ̱ + ζ β , s m ( g ) ± E β , z , m , n , p ( g ) ± z ̱ ) exp ( j k x , m ( g ) ± x ) ,
H ̃ β , x , s , n , p ( g ) + x ̱ + H ̃ β , y , s , n , p ( g ) + y ̱ + H ̃ β , z , s , n , p ( g ) + z ̱ = m = M M ( ζ β , s m ( g ) + H β , x , m , n , p ( g ) + x ̱ + ζ β , s m ( g ) + H β , y , m , n , p ( g ) + y ̱ + ζ β , s m ( g ) + H β , z , m , n , p ( g ) + z ̱ ) exp ( j k x , m ( g ) ± x ) .
U ̱ 1 = m = M M n = N N ( u 1 , x , m , n x ̱ + u 1 , y , m , n y ̱ + u 1 , z , m , n z ̱ ) exp ( j ( k α , x , m x + k α , y , n y + k α , z , m , n ( z z ) ) ) ,
U ̱ 2 = m = M M n = N N ( u 2 , x , m , n x ̱ + u 2 , y , m , n y ̱ + u 2 , z , m , n z ̱ ) exp ( j ( k α , x , m x + k α , y , n y k α , z , m , n ( z + z + ) ) ) ,
U ̱ 3 = m = M M n = N N ( u 3 , x , m , n x ̱ + u 3 , y , m , n y ̱ + u 3 , z , m , n z ̱ ) exp ( j ( k β , x , m , n ( x x ) + k β , y , n y + k β , z , m z ) ) ,
U ̱ 4 = m = M M n = N N ( u 4 , x , m , n x ̱ + u 4 , y , m , n y ̱ + u 4 , z , m , n z ̱ ) exp ( j ( k β , x , m , n ( x x + ) + k β , y , n y + k β , z , m z ) ) .
T ̱ i 1 = m = M M n = N N ( t i 1 , x , m , n x ̱ + t i 1 , y , m , n y ̱ + t i 1 , z , m , n z ̱ ) exp ( j ( k α , x , m x + k α , y , n y k α , z , m , n ( z z ) ) ) ,
T ̱ i 2 = m = M M n = N N ( t i 2 , x , m , n x ̱ + t i 2 , y , m , n y ̱ + t i 2 , z , m , n z ̱ ) exp ( j ( k α , x , m x + k α , y , n y + k α , z , m , n ( z z + ) ) ) ,
T ̱ i 3 = m = M M n = N N ( t i 3 , x , m , n x ̱ + t i 3 , y , m , n y ̱ + t i 3 , z , m , n z ̱ ) exp ( j ( k β , x , m , n ( x x ) + k β , y , n y + k β , z , m z ) ) ,
T ̱ i 4 = m = M M n = N N ( t i 4 , x , m , n x ̱ + t i 4 , y , m , n y ̱ + t i 4 , z , m , n z ̱ ) exp ( j ( k β , x , m , n ( x x + ) + k β , y , n y + k β , z , m z ) ) .
( W α , h W α , h V α , h V α , h ) ( U 1 0 0 0 S 11 S 21 S 31 S 41 ) = ( W ̃ α , + ( z ) W ̃ α , ( z ) V ̃ α , + ( z ) V ̃ α , ( z ) ) ( C α , 1 + C α , 2 + C α , 3 + C α , 4 + C α , 1 C α , 2 C α , 3 C α , 4 ) + ( W ̃ β , + ( z ) W ̃ β , ( z ) V ̃ β , + ( z ) V ̃ β , ( z ) ) ( C β , 1 + C β , 2 + C β , 3 + C β , 4 + C β , 1 C β , 2 C β , 3 C β , 4 ) , at z = z ,
( W α , h W α , h V α , h V α , h ) ( S 12 S 22 S 32 S 42 0 U 2 0 0 ) = ( W ̃ α , + ( z + ) W ̃ α , ( z + ) V ̃ α , + ( z + ) V ̃ α , ( z + ) ) ( C α , 1 + C α , 2 + C α , 3 + C α , 4 + C α , 1 C α , 2 C α , 3 C α , 4 ) + ( W ̃ β , + ( z + ) W ̃ β , ( z + ) V ̃ β , + ( z + ) V ̃ β , ( z + ) ) ( C β , 1 +