Abstract

The purpose of the method described in this paper is to identify with precision sharp phenomena and to produce reliable solutions for various kinds of one-dimensional grating problems in the resonance domain, without limitation in the geometry or the electromagnetic parameters. The theory lies in the use of finite elements and Rumsey’s variational principle for matching the Rayleigh expansions on both sides of the inhomogeneous region. The numerical implementation is checked by classical tests and by comparisons with existing numerical data. In particular, solutions are given for difficult cases where there is a lack of numerical results. The basic principle of the generalization to two-dimensional gratings and an example of a numerical result are given.

© 1993 Optical Society of America

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  1. D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
    [Crossref]
  2. M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [Crossref]
  3. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,”J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  4. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,”J. Opt. Soc. Am. 3, 1780–1787 (1986).
    [Crossref]
  5. S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
    [Crossref]
  6. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [Crossref]
  7. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Studentlitteratur, Stockholm, 1987).
  8. Dautray-Lions, Analyse Mathématique et Calculs Numériques, Tome VI (Masson, Paris, 1988).
  9. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–402 (1902).
  10. D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 661–724.
  11. M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
    [Crossref]
  12. F. Chataux, N. Chateau, J. P. Hugonin, J. C. Saget, J. Taboury, “High diffraction efficiency gratings in dichromated gelatines for near infrared wavelengths,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 150–157 (1989).
    [Crossref]
  13. G. Bouchitté, R. Petit, “On the concepts of a perfectly conducting material and of a perfectly conducting and infinitely thin screen,” Radio Sci. 24, 13–26 (1989).
    [Crossref]
  14. R. Petit, G. Tayeb, “Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips,” J. Opt. Soc. Am. A 7, 1686–1692 (1990).
    [Crossref]
  15. K. Knop, “Diffraction gratings for color filtering in the zero diffraction order,” Appl. Opt. 17, 3598–3603 (1978).
    [Crossref] [PubMed]
  16. J. 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]
  17. R. C. McPhedran, D. Maystre, “Solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
    [Crossref]
  18. A. Taflove, K. R. Umashankar, “Finite-difference time-domain modeling of electromagnetic wave scattering and interaction problems,”IEEE Antennas Propag. Soc. Newsletter (April1988), pp. 5–20.
  19. A. Bossavit, “Simplicial finite elements for scattering patterns in electromagnetism,” Comput. Method Appl. Mech. Eng. 76, 299–316 (1989).
    [Crossref]
  20. J. C. Nedelec, “Mixed elements in R3,” Numerische Mathematik 35, 315–341 (1980).
    [Crossref]

1992 (1)

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[Crossref]

1990 (1)

1989 (2)

G. Bouchitté, R. Petit, “On the concepts of a perfectly conducting material and of a perfectly conducting and infinitely thin screen,” Radio Sci. 24, 13–26 (1989).
[Crossref]

A. Bossavit, “Simplicial finite elements for scattering patterns in electromagnetism,” Comput. Method Appl. Mech. Eng. 76, 299–316 (1989).
[Crossref]

1988 (1)

A. Taflove, K. R. Umashankar, “Finite-difference time-domain modeling of electromagnetic wave scattering and interaction problems,”IEEE Antennas Propag. Soc. Newsletter (April1988), pp. 5–20.

1986 (1)

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

1981 (2)

J. 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]

1980 (1)

J. C. Nedelec, “Mixed elements in R3,” Numerische Mathematik 35, 315–341 (1980).
[Crossref]

1978 (2)

1977 (1)

R. C. McPhedran, D. Maystre, “Solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[Crossref]

1976 (1)

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[Crossref]

1973 (1)

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–402 (1902).

Adams, J. L.

J. 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]

Andrewartha, J. R.

J. 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]

Bossavit, A.

A. Bossavit, “Simplicial finite elements for scattering patterns in electromagnetism,” Comput. Method Appl. Mech. Eng. 76, 299–316 (1989).
[Crossref]

Botten, J. C.

J. 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]

Bouchitté, G.

G. Bouchitté, R. Petit, “On the concepts of a perfectly conducting material and of a perfectly conducting and infinitely thin screen,” Radio Sci. 24, 13–26 (1989).
[Crossref]

Cadilhac, M.

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

Chataux, F.

F. Chataux, N. Chateau, J. P. Hugonin, J. C. Saget, J. Taboury, “High diffraction efficiency gratings in dichromated gelatines for near infrared wavelengths,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 150–157 (1989).
[Crossref]

Chateau, N.

F. Chataux, N. Chateau, J. P. Hugonin, J. C. Saget, J. Taboury, “High diffraction efficiency gratings in dichromated gelatines for near infrared wavelengths,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 150–157 (1989).
[Crossref]

Craig, M. S.

J. 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]

Dautray-Lions,

Dautray-Lions, Analyse Mathématique et Calculs Numériques, Tome VI (Masson, Paris, 1988).

Gaylord, T. K.

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

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

Gedney, S. D.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[Crossref]

Hugonin, J. P.

F. Chataux, N. Chateau, J. P. Hugonin, J. C. Saget, J. Taboury, “High diffraction efficiency gratings in dichromated gelatines for near infrared wavelengths,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 150–157 (1989).
[Crossref]

Hutley, M. C.

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[Crossref]

Johnson, C.

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Studentlitteratur, Stockholm, 1987).

Knop, K.

Lee, J. F.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[Crossref]

Maystre, D.

D. Maystre, “A new general integral theory for dielectric coated gratings,”J. Opt. Soc. Am. 68, 490–495 (1978).
[Crossref]

R. C. McPhedran, D. Maystre, “Solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[Crossref]

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[Crossref]

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 661–724.

McPhedran, R. C.

J. 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, “Solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[Crossref]

Mittra, R.

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[Crossref]

Moharam, M. G.

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

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

Nedelec, J. C.

J. C. Nedelec, “Mixed elements in R3,” Numerische Mathematik 35, 315–341 (1980).
[Crossref]

Nevière, M.

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

Petit, R.

R. Petit, G. Tayeb, “Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips,” J. Opt. Soc. Am. A 7, 1686–1692 (1990).
[Crossref]

G. Bouchitté, R. Petit, “On the concepts of a perfectly conducting material and of a perfectly conducting and infinitely thin screen,” Radio Sci. 24, 13–26 (1989).
[Crossref]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

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

Saget, J. C.

F. Chataux, N. Chateau, J. P. Hugonin, J. C. Saget, J. Taboury, “High diffraction efficiency gratings in dichromated gelatines for near infrared wavelengths,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 150–157 (1989).
[Crossref]

Taboury, J.

F. Chataux, N. Chateau, J. P. Hugonin, J. C. Saget, J. Taboury, “High diffraction efficiency gratings in dichromated gelatines for near infrared wavelengths,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 150–157 (1989).
[Crossref]

Taflove, A.

A. Taflove, K. R. Umashankar, “Finite-difference time-domain modeling of electromagnetic wave scattering and interaction problems,”IEEE Antennas Propag. Soc. Newsletter (April1988), pp. 5–20.

Tayeb, G.

Umashankar, K. R.

A. Taflove, K. R. Umashankar, “Finite-difference time-domain modeling of electromagnetic wave scattering and interaction problems,”IEEE Antennas Propag. Soc. Newsletter (April1988), pp. 5–20.

Vincent, P.

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–402 (1902).

Appl. Opt. (1)

Appl. Phys. (1)

R. C. McPhedran, D. Maystre, “Solar applications of inductive grids,” Appl. Phys. 14, 1–20 (1977).
[Crossref]

Comput. Method Appl. Mech. Eng. (1)

A. Bossavit, “Simplicial finite elements for scattering patterns in electromagnetism,” Comput. Method Appl. Mech. Eng. 76, 299–316 (1989).
[Crossref]

IEEE Antennas Propag. Soc. Newsletter (1)

A. Taflove, K. R. Umashankar, “Finite-difference time-domain modeling of electromagnetic wave scattering and interaction problems,”IEEE Antennas Propag. Soc. Newsletter (April1988), pp. 5–20.

IEEE Trans. Microwave Theory Tech. (1)

S. D. Gedney, J. F. Lee, R. Mittra, “A combined FEM/MoM to analyze the plane wave diffraction by arbitrary gratings,” IEEE Trans. Microwave Theory Tech. 40, 363–370 (1992).
[Crossref]

J. Opt. Soc. Am. (3)

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

Numerische Mathematik (1)

J. C. Nedelec, “Mixed elements in R3,” Numerische Mathematik 35, 315–341 (1980).
[Crossref]

Opt. Acta (1)

J. 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]

Opt. Commun. (2)

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[Crossref]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[Crossref]

Phil. Mag. (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–402 (1902).

Radio Sci. (1)

G. Bouchitté, R. Petit, “On the concepts of a perfectly conducting material and of a perfectly conducting and infinitely thin screen,” Radio Sci. 24, 13–26 (1989).
[Crossref]

Other (5)

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 661–724.

F. Chataux, N. Chateau, J. P. Hugonin, J. C. Saget, J. Taboury, “High diffraction efficiency gratings in dichromated gelatines for near infrared wavelengths,” in Holographic Optics II: Principles and Applications, G. M. Morris, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1136, 150–157 (1989).
[Crossref]

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

C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Studentlitteratur, Stockholm, 1987).

Dautray-Lions, Analyse Mathématique et Calculs Numériques, Tome VI (Masson, Paris, 1988).

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Figures (13)

Fig. 1
Fig. 1

Notations. ɛ (x, y) and μ(x, y) have a period d with respect to the x axis.

Fig. 2
Fig. 2

Mesh for s polarization.

Fig. 3
Fig. 3

Energy reflected in the zeroth order by a gold grating versus wavelength, with a phenomenon of total absorption at λ = 0.65 μm. Continuous curve, efficiency calculated with the FEM; crosses, results of the integral method.

Fig. 4
Fig. 4

Mesh for a sinusoidal gold grating.

Fig. 5
Fig. 5

Efficiencies of a dielectric phase grating versus angle of incidence. The solid and dotted curves show the zeroth and –first transmitted orders, respectively, obtained from the FEM. The circles and crosses give the same efficiencies calculated with a coupled-wave method.

Fig. 6
Fig. 6

Efficiency in the zeroth order reflected by a grating made of very thin and conductive strips versus compression factor F. Solid curve, results obtained from a right-triangle mesh; dotted curve, results of the isosceles-triangle mesh. The horizontal dashed line stands for asymptotic theory. Top, p polarization; bottom, s polarization.

Fig. 7
Fig. 7

Variation of the shape of the triangular mesh during compression: a, right triangle; b, isosceles triangle.

Fig. 8
Fig. 8

Efficiency in the –first-order Littrow mount of a deep lamellar grating. Solid curve, results obtained for a perfectly conducting grating; dotted curve, results for the aluminum grating; dashed curve, total energy scattered by the aluminum grating.

Fig. 9
Fig. 9

Mesh for an aluminum grating. The bold line shows the grating profile.

Fig. 10
Fig. 10

Energy transmitted in the zeroth order by an inductive grid versus wavelength. Continuous curve, efficiency calculated with the finite-difference method; crosses, results obtained with the FEM.

Fig. 11
Fig. 11

Finite element for s polarization. ai is the edge in front of the point Mi, Mi, is the arbitrary point of ai.

Fig. 12
Fig. 12

Finite element for p polarization. z ^ is the unit vector normal to the plane of the figure, t ^ i is the unit vector tangent to the edge ai, Mi is the middle point of ai, Mi is the node in front of ai.

Fig. 13
Fig. 13

Finite element for two-dimensional gratings. t ^ i is the unit vector tangent to the edge ai, Mi is the middle point of aj, Mi is the arbitrary point of the edge ai, ai, is the edge opposite ai.

Equations (63)

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k ± = k ɛ ± μ ± .
E z = exp { i [ α x - β ( y - h ) ] } + - + B n exp { i [ α n x + β n + ( y - h ) ] } ,
H x = 1 Z + ( - β k + exp { i [ α x - β ( y - h ) ] } + - + β n + k + B n exp { i [ α n x + β n + ( y - h ) ] } ) ;
E z = - + C n exp [ i ( α n x - β n - y ) ] ,
H x = - 1 Z - - + β n - k - C n exp [ i ( α n x - β n - y ) ] ,
α = k + sin ( θ ) ,
β = k + cos ( θ ) ,
α n = α + n 2 π d ,
β n ± = [ ( k ± ) 2 - α n 2 ] 1 / 2 ,
Re { β n ± } + Im { β n ± } > 0 ,
θ n ± = tan - 1 ( α n β n ± ) ,
ρ n + = B n B n ¯ β n + β ,
ρ n - = μ + μ - ( C n C n ¯ β n - β ) .
E z = - + A n exp { i [ α n x - β n + ( y - h ) ] } + - + B n exp { i [ α n x + β n + ( y - h ) ] } ,
H x = 1 Z + ( - - + β n + k + A n exp { i [ α n x - β n + ( y - h ) ] } + - + β n + k + B n exp { i [ α n x + β n + ( y - h ) ] } ) ,
E z = - + D n exp [ i ( α n x + β n - y ) ] + - + C n exp [ i ( α n x - β n - y ) ] ,
H x = 1 Z - { - + β n - k - D n exp [ i ( α n x - β n - y ) ] - - + β n - k - C n exp [ i ( α n x - β n - y ) ] } .
E z = i I e i ϕ i ( x , y ) ,
× E = i ω μ 0 μ ( x , y ) H ,
× H = - i ω ɛ 0 ɛ ( x , y ) E .
Ω ϕ t ¯ z ^ ( × H + i k Z 0 ɛ E ) d x d y = 0.
Ω [ H · × ( ϕ t ¯ z ^ ) + i k Z 0 ϕ t ¯ ɛ E z ] d x d y = - Γ + Γ + ϕ t ¯ H · ( z ^ × n ^ ) d s ,
Ω [ k 2 ɛ ϕ t ¯ E z - 1 μ × ( ϕ t ¯ z ^ ) · × ( E z z ^ ) ] d x d y = i k Z 0 Γ + Γ - ϕ t ¯ H x d s .
E z + = N + ( A n + B n ) exp ( i α n x ) ,
H x + = 1 Z + [ N + β n + k + ( - A n + B n ) exp ( i α n x ) ] ,
E z - = N - ( C n + D n ) exp ( i α n x ) ,
H x - = 1 Z - [ N - β n - k - ( D n - C n ) exp ( i α n x ) ] .
e i = E z ( M i ) exp ( - i α x i ) .
e i L = e i R .
E z ( M ) = q = 1 Q τ q ψ q ( M ) ,
E z ± ( x ) = q = 1 Q τ q ψ q ± ( x ) ,
H x ± ( x ) = q = 1 Q ν q ψ q ± ( x ) ,
ν = 1.53 + 0.048 cos ( 2 π x / d ) ,
E z ( M ) = i I int I + I - I L I R e i ϕ i ( M ) .
e i = E z + ( M i ) exp ( - i α x i ) = N + ( A n + B n ) exp [ i ( α n - α ) x i ] .
e i = E z - ( M i ) exp ( - i α x i ) = N - ( C n + D n ) exp [ i ( α n - α ) x i ] ,
e i = e j ,
E z ( M ) = q = 1 Q τ q ψ q ( M ) ,
E z ± ( x ) = q = 1 Q τ q ψ q ± ( x ) ,
H x ± ( x ) = q = 1 Q ν q ψ q ± ( x ) ,
τ q = { e i if i I int I L A n + B n if n N + C n + D n if n N - ,
ν q = { 0 if i I int I L β n + Z + k + if n N + β n - Z - k - if n N - ,
ψ q ( M ) = { ϕ i if i I int ϕ i L + ϕ i R if i I L w n + if n N + w n - if n N - ,
ψ q ± ( x ) = { 0 if i I int I L exp ( i α n x ) if n N + exp ( i α n x ) if n N - ,
w n ± ( M ) = I + ϕ i ( M ) exp [ i ( α n - α ) x i ] .
ϕ i ( M ) = M M i × a i 2 S exp ( i α x i ) ,
Γ ± - ψ p ¯ ψ q d s = ± d δ p q ,
Ω [ k 2 ɛ ψ p ¯ ψ q - 1 μ × ( ψ p ¯ z ^ ) · × ( ψ q z ^ ) ] d x d y = ψ p , ψ q .
p , q = 1 Q τ q ψ p , ψ q ± ikdZ 0 ν q δ p q = 0 ,
[ G 00 G 0 + G 0 - G + 0 G + + G + - G - 0 G - + G - - ] [ e j B m C m ] = [ S 0 + S 0 - S + + S + - S - + S - - ] [ A m D m ] ,
Q = I int + I L + N + + N - , G 00 i j = ϕ i , ϕ j , G + 0 n j = w + n , ϕ j , G - 0 n j = w - n , ϕ j , G 0 + i m = ϕ i , w + m ,             S 0 + i m = - G 0 + i m , G 0 - i m = ϕ i , w - m ,             S 0 - i m = - G 0 - i m , G + + n m = w + n , w + m + i χ + n δ n m ,             S + + n m = - G + + n m + 2 i χ + n δ n m , G + - n m = w + n , w - m ,             S + - n m = - G + - n m , G - + n m = w - n , w + m ,             S - + n m = - G - + n m , G - - n m = w - n , w - m + i χ - n δ n m ,             S - n m = - G - - n m + 2 i χ - n δ n m , χ ± n = β n ± Z ± k ± k d Z 0 .
E x = - + A n exp { i [ α n x - β n + ( y - h ) ] } + - + B n exp { i [ α n x + β n + ( y - h ) ] } ,
H z = 1 Z + ( - + k + β n + A n exp { i [ α n x - β n + ( y - h ) ] } - - + k + β n + B n exp { i [ α n x + β n + ( y - h ) ] } ) ;
E x = - + D n exp [ i ( α n x + β n y ) ] + - + C n exp [ i ( α n x - β n - y ) ] ,
H z = 1 Z - { - - + k - β n - D n exp [ i ( α n x + β n - y ) ] + - + k - β n - C n exp [ i ( α n x - β n - y ) ] } ,
Ω ( k 2 ɛ ϕ t ¯ E - 1 μ × ϕ t ¯ · × E ) d x d y = i k z 0 Γ + Γ - ϕ t x ¯ H z d s ,
E ( M ) = i I e i ϕ i ( M ) ,
ϕ i ( M ) = M M i × z ^ 2 S exp ( i α x i ) ,
e i = exp ( - i α x i ) l i a i E · t ^ i d s ,
w n ± ( M ) = I ± ϕ i ( M ) sin ( α n - α ) l i ( α n - α ) l i exp [ i ( α n - α ) x i ] t ^ i · x ^ ,
χ ± n = k ± Z ± β n ± k d Z 0 ,
ϕ i ( M ) = M M i × a i 6 V exp [ i ( α x i + γ z i ) ] ,
e i = exp [ - i ( α x i + γ z i ) ] l i a i E · t ^ i d s ,

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