Abstract

A computationally efficient implementation of rigorous coupled-wave analysis is presented. The eigenvalue problem for a one-dimensional grating in a conical mounting is reduced to two eigenvalue problems in the corresponding nonconical mounting. This reduction yields two n × n matrices to solve for eigenvalues and eigenvectors, where n is the number of orders retained in the computation. For a two-dimensional grating, the size of the matrix in the eigenvalue problem is reduced to 2n × 2n. These simplifications reduce the computation time for the eigenvalue problem by 8–32 times compared with the original computation time. In addition, we show that with rigorous coupled-wave analysis one analytically satisfies reciprocity by retaining the appropriate choice of spatial harmonics in the analysis. Numerical examples are given for metallic lamellar gratings, pulse-width-modulated gratings, deep continuous surface-relief gratings, and two-dimensional gratings.

© 1995 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  2. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  7. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  8. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  9. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
    [CrossRef]
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  16. N. Chateau, J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
    [CrossRef]
  21. N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propag. AP-17, 747–751 (1969).
    [CrossRef]
  22. R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
    [CrossRef]
  23. R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
    [CrossRef]
  24. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
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  25. S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
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    [CrossRef] [PubMed]

1994 (1)

1993 (3)

1992 (4)

S. T. Han, Y-L Tsao, R. M. Walser, M. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2342–2352 (1992).
[CrossRef]

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. 31, 4453–4458, (1992).
[CrossRef] [PubMed]

E. Noponen, A. Vasara, J. Turunen, J. M. Miller, M. R. Taghizadeh, “Synthetic diffractive optics in the resonance domain,” J. Opt. Soc. Am. A 9, 1206–1213 (1992).
[CrossRef]

1991 (1)

1989 (1)

1986 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1984 (1)

1983 (2)

1982 (2)

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

1981 (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

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

1978 (1)

1977 (1)

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1969 (1)

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propag. AP-17, 747–751 (1969).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Amitay, N.

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propag. AP-17, 747–751 (1969).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Awada, K. A.

Becker, M.

S. T. Han, Y-L Tsao, R. M. Walser, M. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2342–2352 (1992).
[CrossRef]

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Chateau, N.

Chuang, S. L.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Farn, M. W.

Galindo, V.

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propag. AP-17, 747–751 (1969).
[CrossRef]

Gaylord, T. K.

Haggans, C. W.

Haidner, H.

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

Han, S. T.

S. T. Han, Y-L Tsao, R. M. Walser, M. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2342–2352 (1992).
[CrossRef]

Hugonin, J.-P.

Kipfer, P.

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

Knop, K.

Kong, J. A.

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Li, L.

Maystre, D.

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

Miller, J. M.

Moharam, M. G.

Morris, G. M.

Noponen, E.

Pai, D. M.

Peng, S. T.

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Petit, R.

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
[CrossRef]

Raguin, D. H.

Russell, P. St. J.

Stork, W.

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

Streibl, N.

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

Taghizadeh, M. R.

Tamir, T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tayeb, G.

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
[CrossRef]

Tsao, Y-L

S. T. Han, Y-L Tsao, R. M. Walser, M. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2342–2352 (1992).
[CrossRef]

Turunen, J.

Vasara, A.

Walser, R. M.

S. T. Han, Y-L Tsao, R. M. Walser, M. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2342–2352 (1992).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

R. C. McPhedran, D. Maystre, “On the theory and solar application of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Amitay, V. Galindo, “On energy conservation and the method of moments in scattering problems,” IEEE Trans. Antennas Propag. AP-17, 747–751 (1969).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (8)

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
[CrossRef]

Optik (1)

H. Haidner, P. Kipfer, W. Stork, N. Streibl, “Zero-order gratings used as an artificial distributed index medium,” Optik 89, 107–112 (1992).

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Radio Sci. (1)

S. L. Chuang, J. A. Kong, “Wave scattering from a periodic dielectric surface for a general angle of incidence,” Radio Sci. 17, 545–557 (1982).
[CrossRef]

Other (3)

R. Petit, G. Tayeb, “On the use of the energy balance criterion as a check of validity of computations in grating theory,” in Application and Theory of Periodic Structures, Diffraction Gratings, and Moire Phenomena III, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.815, 2–10 (1987).
[CrossRef]

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

M. G. Moharam, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of a diffraction grating in a conical mounting.

Fig. 2
Fig. 2

Diffraction efficiencies of a metallic lamellar grating with TM-polarized light: (a) convergence of the zeroth-order transmission wave at wavelength λ = 0.55, (b) wavelength dependence of the diffraction efficiencies. The values of other grating parameters are Λ = 0.25, d = 0.2, ɛI = ɛII′ = 1.0, ɛIII = 2.25, ɛII = (3.18 − 4.41i)2, duty cycle = 30%, ψ = θ = δ = 0 deg.

Fig. 3
Fig. 3

Structure of a pulse-width-modulated grating. One period that contains I subperiods is shown.

Fig. 4
Fig. 4

Diffraction efficiencies of pulse-width-modulated binary gratings in conical and nonconical mountings: (a) convergence of the +1st diffraction order, with θ = 0 deg, δ = 60 deg, ψ = 90 deg for TE polarization, ψ = 0 deg for TM polarization, d = 1.9; (b) convergence of the +2nd diffraction order, with θ = −20 deg, δ = 60 deg, ψ = 90 deg, d = 3.8; (c) angular dependence of the diffraction efficiencies. The +1st and the +2nd transmitted orders are plotted for d = 1.9 and d = 3.8, respectively. The values of other parameters are λ = 0.5461, Λ = 3, ɛI = ɛII′ = 1.0, ɛIII = ɛII = 2.1316, number of subgrooves I = 16.

Fig. 5
Fig. 5

Scheme of approximating a symmetric grating profile by a stack of lamellar gratings. The vertical lines are equally spaced.

Fig. 6
Fig. 6

Convergence of diffraction efficiencies for sinusoidal dielectric surface-relief gratings: (a) transmitted −1st-order diffraction efficiency, d/Λ = 0.3; (b) transmitted zeroth-order and +3-order diffraction efficiencies, d/Λ = 10; (c) diffraction efficiencies versus orders retained in the computation. The values of other parameters are λ = 1.0, Λ = 2, ɛI = ɛII′ = 1.0, ɛIII = ɛII = 4, ψ = 81.50 deg, θ = 61.12 deg, δ = 17.19 deg.

Fig. 7
Fig. 7

Diffraction efficiencies for a 2D dielectric surface-relief grating with a square-wave cross section along the x and the y axes. The values of other parameters are λ = 1.0, Λx = Λy = 1.2, ɛI = ɛII′ = 1.0, ɛIII = ɛII = 2.25, d = 1.0, ψ = 90 deg, θ = 0 deg, δ = 0 deg, and 50% duty cycle along the x and the y axes.

Tables (1)

Tables Icon

Table 1 Normalized Computation Time of the Eigenvalue Problems in RCWA

Equations (54)

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k 1 = k x x ̂ + k y ŷ + k 1 z = k 1 ( sin α cos δ x ̂ + sin α sin δ ŷ + cos α ) ,
û = u x x ̂ + u y ŷ + u z = ( cos ψ cos α cos δ sin ψ sin δ ) x ̂ + ( cos ψ cos α sin δ + sin ψ cos δ ) ŷ ( cos ψ sin α ) ,
E 1 = û exp ( j k 1 r ) + i R i exp [ j k 1 i ( r + 1 2 d ) ] ,
E 3 = i T i exp [ j k 3 i ( r 1 2 d ) ] ,
k l i = k x i x ̂ + k y ŷ + k z l i , l = 1 , 3 ,
k x i = k x i K ,
k z l i 2 = k l 2 k x i 2 , k y 2 , l = 1 , 3 ,
E 2 x x ̂ + E 2 y ŷ = i [ S x i ( z ) x ̂ + S y i ( z ) ŷ ] exp ( j σ i r ) ,
H 2 x x ̂ + H 2 y ŷ = ɛ 0 μ 0 i [ U x i ( z ) x ̂ + U y i ( z ) ŷ ] exp ( j σ i r ) ,
σ i = k x i x ̂ + k y ŷ
× E 2 = j ω μ 0 H 2 ,
× H 2 = j ω ɛ 0 ɛ ( x ) E 2 ,
d S x i ( z ) d z = j ( k x i / k ) p a i p [ k y U x p ( z ) k x p U y p ( z ) ] j k U y i ( z ) ,
d S y i ( z ) d z = j k U x i ( z ) j ( k y / k ) × p a i p [ k y U x p ( z ) k x p U y p ( z ) ] ,
d U x i ( z ) d z = j ( k x i / k ) [ k y S x i ( z ) k x i S y i ( z ) ] + j k p ɛ ̂ i p S y p ( z ) ,
d U y i ( z ) d z = j k p ɛ ̂ i p S x p ( z ) + j ( k y / k ) [ k y S x i ( z ) k x i S y i ( z ) ] ,
ɛ ( x ) = h ɛ ̂ h exp ( jhKx ) ,
ɛ 1 ( x ) = h a h exp ( jhKx ) .
V ˙ = AV ,
S x i ( z ) = m C m ω 1 , i m exp ( λ m z ) ,
S y i ( z ) = m C m ω 2 , i m exp ( λ m z ) ,
U x i ( z ) = m C m ω 3 , i m exp ( λ m z ) ,
U y i ( z ) = m C m ω 4 , i m exp ( λ m z ) ,
2 z 2 E 2 x = ω 2 μ 0 ɛ 0 ɛ ( x ) E 2 x 2 y 2 E 2 x x { 1 ɛ ( x ) x [ ɛ ( x ) E 2 x ] } ,
2 z 2 E 2 y = 2 x y E 2 x y { 1 ɛ ( x ) x [ ɛ ( x ) E 2 x ] } ( 2 x 2 + 2 y 2 ) E 2 y k 2 ɛ ( x ) E 2 y ,
2 z 2 H 2 x = ( 2 x 2 + 2 y 2 ) H 2 x k 2 ɛ ( x ) H 2 x ,
2 z 2 H 2 y = ɛ ( x ) x [ 1 ɛ ( x ) y H 2 x ] 2 x y H 2 x k 2 ɛ ( x ) H 2 y ɛ ( x ) x × [ 1 ɛ ( x ) x H 2 y ] 2 y 2 H 2 y .
d 2 d z 2 S x i ( z ) = k 2 p ɛ ̂ i p S x p ( z ) + k y 2 S x i ( z ) + k x i p p a i p k x p ɛ ̂ p p S x p ( z ) ,
d 2 d z 2 S y i ( z ) = k x i k y S x i ( z ) + k y p p a i p k x p ɛ ̂ p p S x p ( z ) + ( k x i 2 + k y 2 ) S y i ( z ) k 2 p ɛ ̂ i p S y p ( z ) ,
d 2 d z 2 U x i ( z ) = ( k x i 2 + k y 2 ) U x i ( z ) k 2 p ɛ ̂ i p U x p ( z ) ,
d 2 d z 2 U y i ( z ) = k y p p ɛ ̂ i p k x p a p p U x p ( z ) + k x i k y U x i ( z ) k 2 p ɛ ̂ i p U y p ( z ) + p p ɛ ̂ i p k x p a p p k x p U y p ( z ) + k y 2 U y i ( z ) .
A Ω = Ω λ ,
A Ω = Ω λ 2 ,
[ A 11 0 0 0 A 21 A 22 0 0 0 0 A 33 0 0 0 A 43 A 44 ] [ Ω 11 Ω 12 Ω 13 Ω 14 Ω 21 Ω 22 Ω 23 Ω 24 Ω 31 Ω 32 Ω 33 Ω 34 Ω 41 Ω 42 Ω 43 Ω 44 ] = [ Ω 11 Ω 12 Ω 13 Ω 14 Ω 21 Ω 22 Ω 23 Ω 24 Ω 31 Ω 32 Ω 33 Ω 34 Ω 41 Ω 42 Ω 43 Ω 44 ] [ λ 11 2 0 0 0 0 λ 22 2 0 0 0 0 λ 33 2 0 0 0 0 λ 44 2 ] .
A 11 Ω 11 = Ω 11 λ 11 2 ,
A 33 Ω 32 = Ω 32 λ 22 2 ,
Ω 12 = Ω 14 = Ω 31 = Ω 33 = 0 .
λ 33 = λ 11 , Ω 13 = Ω 11 ,
λ 44 = λ 22 , Ω 34 = Ω 32 .
Ω 21 = Ω 23 = k y [ k 2 T ( ɛ ) F 2 ] 1 F Ω 11 ,
Ω 22 = Ω 24 = Ω 32 L ,
Ω 22 = Ω 43 = j [ k T ( ɛ ) Ω 11 ( k y 2 / k ) Ω 11 + ( k y / k ) F Ω 21 ] L ( E ) ,
Ω 42 = Ω 44 = j ( k y / k ) F Ω 22 L ( H ) ,
Ω = [ Ω ( s ) Ω ( s ) Ω ( u ) Ω ( u ) ] ,
λ 11 2 = λ TM 2 + K y 2 ,
λ 22 2 = λ TE 2 + K y 2 ,
B Ω B = Ω B λ B ,
[ 0 B 12 B 21 0 ] [ Ω B 11 Ω B 12 Ω B 21 Ω B 22 ] = [ Ω B 11 Ω B 12 Ω B 21 Ω B 22 ] [ λ B 11 0 0 λ B 22 ] .
B 12 B 21 Ω B 11 = Ω B 11 λ B 11 λ B 11 ,
B 12 B 21 Ω B 12 = Ω B 12 λ B 22 λ B 22 .
Ω B 12 = Ω B 11 ,
λ B 22 = λ B 11 ,
Ω B 21 = B 21 Ω B 11 λ B 11 1 ,
Ω B 22 = Ω B 21 .

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