Abstract

A new formulation of the Chandezon method for crossed gratings is presented. In the nonorthogonal translation coordinate system, an arbitrary field in a homogeneous source-free region can be expressed as the sum of a TE field and a TM field. It is shown that the whole solution can be derived from the eigensolutions of an operator independent of the polarization. In addition, use is made of the S-matrix formalism to include multilayer coated crossed gratings with parallel faces. Numerical results are given for sinusoidal crossed gratings and pyramidal gratings.

© 1998 Optical Society of America

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References

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  1. P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
    [CrossRef]
  2. O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
    [CrossRef]
  3. M. G. Moharam, “Coupled-wave analysis of two dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
    [CrossRef]
  4. R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  5. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  6. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  7. L. Li, “New modal method by Fourier expansion for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  8. G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
    [CrossRef]
  9. G. Granet, “Diffraction par des surfaces bipériodiques: résolution en coordonnées non orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
    [CrossRef]
  10. J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
    [CrossRef]
  11. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7.
  12. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  13. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  14. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–847 (1982).
    [CrossRef]
  15. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  16. G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  17. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1034 (1996).
    [CrossRef]
  18. J. J. Greffet, C. Baylard, P. Versaevel, “Diffraction of electromagnetic waves by crossed gratings: a series solution,” Opt. Lett. 17, 1740–1742 (1992).
    [CrossRef] [PubMed]
  19. S. T. Han, Y. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
  20. E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  21. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
    [CrossRef] [PubMed]
  22. S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  23. J. P. Plumey, B. Guizal, J. Chandezon, “The coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
    [CrossRef]

1997 (2)

1996 (3)

1995 (4)

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

G. Granet, “Diffraction par des surfaces bipériodiques: résolution en coordonnées non orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

1994 (1)

1993 (3)

1992 (2)

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1982 (1)

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–847 (1982).
[CrossRef]

1979 (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

1978 (1)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Baylard, C.

Becker, M. F.

Botten, L. C.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7.

Bräuer, R.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bruno, O. P.

Bryngdahl, O.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Chandezon, J.

J. P. Plumey, B. Guizal, J. Chandezon, “The coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–847 (1982).
[CrossRef]

Cornet, G.

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–847 (1982).
[CrossRef]

Cotter, N. P. K.

Derrick, G. H.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7.

Dupuis, M. T.

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–847 (1982).
[CrossRef]

Granet, G.

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

G. Granet, “Diffraction par des surfaces bipériodiques: résolution en coordonnées non orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Greffet, J. J.

Guizal, B.

Han, S. T.

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Harris, J. B.

Li, L.

Magnusson, R.

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–847 (1982).
[CrossRef]

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

McPhedran, R. C.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7.

Moharam, M. G.

M. G. Moharam, “Coupled-wave analysis of two dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
[CrossRef]

Morris, G. M.

Nevière, M.

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

Noponen, E.

Peng, S.

Plumey, J. P.

J. P. Plumey, B. Guizal, J. Chandezon, “The coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Popov, E.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

Preist, T. W.

Reitich, F.

Sambles, J. R.

Thorpe, R. N.

Tsao, Y.

Turunen, J.

Versaevel, P.

Vincent, P.

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

Walser, R. M.

Wang, S. S.

Watts, R. A.

Appl. Opt. (2)

Appl. Phys. (1)

G. H. Derrick, R. C. McPhedran, D. Maystre, M. Nevière, “Crossed gratings: a theory and its applications,” Appl. Phys. 18, 39–52 (1979).
[CrossRef]

J. Opt. Soc. Am. A (10)

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

L. Li, “New modal method by Fourier expansion for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

O. P. Bruno, F. Reitich, “Numerical solution of diffraction problems: a method of variation of boundaries. III. Doubly periodic gratings,” J. Opt. Soc. Am. A 10, 2551–2562 (1993).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, R. N. Thorpe, R. A. Watts, “Optical response of bigratings,” J. Opt. Soc. Am. A 13, 2041–2049 (1996).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. A 72, 839–847 (1982).
[CrossRef]

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[CrossRef]

L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1034 (1996).
[CrossRef]

S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

J. P. Plumey, B. Guizal, J. Chandezon, “The coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610–617 (1997).
[CrossRef]

Opt. Acta (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of the anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (2)

P. Vincent, “A finite-difference method for dielectric and conducting crossed gratings,” Opt. Commun. 26, 293–296 (1978).
[CrossRef]

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Opt. Lett. (1)

Pure Appl. Opt. (2)

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

G. Granet, “Diffraction par des surfaces bipériodiques: résolution en coordonnées non orthogonales,” Pure Appl. Opt. 4, 777–793 (1995).
[CrossRef]

Other (4)

M. G. Moharam, “Coupled-wave analysis of two dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
[CrossRef]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 7.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

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Figures (10)

Fig. 1
Fig. 1

Multilayer coated crossed grating.

Fig. 2
Fig. 2

Definition of the incident plane wave.

Fig. 3
Fig. 3

S matrix at an interface.

Fig. 4
Fig. 4

S matrix of a layer.

Fig. 5
Fig. 5

Reflectivity versus period for a gold sinusoidal crossed grating with normally incident light of 0.65-µm wavelength. Curve 1, h=0.040 µm; curve 2, h=0.055 µm; curve 3, h=0.070 µm. M=N=2.

Fig. 6
Fig. 6

Reflectivity versus incidence angle for a sinusoidal crossed grating with 3333 lines/mm along the x and z directions. M=N=4. λ=632.8 nm.

Fig. 7
Fig. 7

Same as Fig. 6, but for a 3400-line/mm spacing.

Fig. 8
Fig. 8

Reflectivity versus incidence angle for a modulated layer bounded by vacuum. TE polarization case. M=N=3.

Fig. 9
Fig. 9

Same as Fig. 8, but for the TM polarization case.

Fig. 10
Fig. 10

Reflectivity versus period for a modulated layer above a gold substrate with normally incident light of 0.65-µm wavelength, with e1=1 µm and h=0.070 µm. Curve 1, ν1=0.8-i4; curve 2, ν1=1.5-i3.3; curve 3, ν1=0.158-i3.3986. M=N=2. The points marked by circles correspond to efficiencies calculated without the coating layer.

Tables (6)

Tables Icon

Table 1 Comparison of the Computational Speed of the Codes Based on Two Formulations of the C Methoda

Tables Icon

Table 2 Comparison of the Numerical Results of the Present Study with the Results of Bruno and Reitich (Ref. 2) for the Perfectly Conducting Sinusoidal Crossed Grating of Eq. (71) under Normal Incidence with a Wavelength-to-Period Ratio λ/d=0.83

Tables Icon

Table 3 Convergence Study of the Zero eflected Order for the Perfectly Conducting Sinusoidal Crossed Grating of Eq. (71), with Various Height-to-Period Ratios, under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.83a

Tables Icon

Table 4 Convergence Study of the Efficiencies for a Perfectly Conducting Pyramidal Crossed Grating with h=0.5, dx=dz=1, θ=0, ϕ=0, δ=90, and λ=0.4368

Tables Icon

Table 5 Comparison among Different Methods for a Pyramidal Dielectric Grating with dx=1.5, dz=1, h=0.25, λ=1.533, ν=1.5, θ=30°, ϕ=45°, and δ=90°

Tables Icon

Table 6 Reflected Efficiencies of a 22-Layer Coated Grating

Equations (91)

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yj=-i<jei+a(x, z),
a(x, z)=p,qapq exp-i2πpxdx+qzdz,
apq=1dxdz0dx0dza(u, v) expi2πpudx+qvdzdudv.
kx=k sin θ cos ϕ,ky=-k cos θ,
kz=-k sin θ sin ϕ,
uˆ=(cos δ cos θ cos ϕ-sin δ sin ϕ)xˆ+(cos δ sin θ)yˆ+(cos δ cos θ sin ϕ+sin δ cos ϕ)zˆ,
ψj(x, y, z)=αγAαγ exp[-ik(αx+βjy+γz)]
α2+βj2+γ2=νj2
α=αm=α0+m λdx,γ=γn=γ0+n λdz,
α0=sin θ cos ϕ,γ0=-sin θ sin ϕ.
βj mn2=νj2-αm2-γn2.
Eyj=m,nAj mn+TM exp[-ik(αmx+βj mn+y+γnz)]+m,nAj mn-TM exp[-ik(αmx+βj mn-y+γnz)],
Hyj=m,nAj mn+TE exp[-ik(αmx+βj mn+y+γnz)]+m,nAj mn-TE exp[-ik(αmx+βj mn-y+γnz)],
βj mn+
=(νj2-αm2-γn2)1/2ifνj2-αm2-γn20-i(αm2+γn2-νj2)1/2ifνj2-αm2-γn2<0,
βj mn-
=-βj mn+.
ψjR=m,n(Aj mn+TEψj mn+R TE+Aj mn+TMψj mn+R TM) ×exp[-ik(αmx+βj mn+y+γnz)]+m,n(Aj mn-TEψj mn-R TE+Aj mn-TMψj mn-R TM) ×exp[-ik(αmx+βj mn-y+γnz)],
ψjR=EzjZHxjZHzjExj,
ψj mn+R TE=1βj mn2-νj2αmβj mn+αmβj mn+γn-γn,
ψj mn+R TM=1βj mn2-νj2βj mn+γn-νj2γnνj2αmβj mn+αm,
Z=μ0/0.
x1=x,x2=u=y-a(x, z),x3=z.
gij=1-ax0-ax1+ax2+az2-az0-az1.
ξijkjEk=-ikggijZHj,
ξijkjZHk=ikν2ggijEj,
ξijk=+1if(i, j, k)isanevenpermutationof(1, 2, 3)-1if(i, j, k)isanoddpermutationof(1, 2, 3)0otherwise;
2E3ZH1ZH3E1=L1,3E3ZH1ZH3E1,
S(x1, x2, x3)=S(x1, x3)exp(-ikrx2),
(22+ω2μ)E3=-(iωμg122+iωμ1)H2+(23-ω2μg32)E2,
(22+ω2μ)H1=(21-ω2μg12)H2-(iω3+iωg322)E2,
(22+ω2μ)H3=(23-ω2μg32)H2+(2iωg12+iω1)E2,
(22+ω2μ)E1=(iωμ3+iωμg322)H2+(21-ω2μg12)E2.
g2222ϕ+2[(g211+1g21)ϕ]+2[(g233+3g23)ϕ]
+12ϕ+32ϕ+ω2μϕ=0.
ψ=sTEψTE+sTMψTM,
(22+ω2μ)ψTE
=-(iωμg122+iωμ1)ϕ(21-ω2μg12)ϕ(23-ω2μg32)ϕ(iωμg322+iωμ3)ϕif22+ω2μ0,
ψTE=ϕ±iωϕ00if22+ω2μ=0;
(22+ω2μ)ψTM
=(23-ω2μg32)ϕ-(iωg322+iω3)ϕ(iωg122+iω1)ϕ(21-ω2μg12)ϕif22+ω2μ0,
ψTM=00ϕ±iωμϕif22+ω2μ=0.
LA 1,3 uϕϕ=LB 1,3ϕϕ,
LA 1,3
=-g12 x-xg12-g23 z-zg23-ikg22ik0,
LB 1,3
=-k2ν2-2x2-2z2001.
ϕ(x, u, z)=ϕ(x, z)exp(-ikru).
ϕ(x, z)=m,nϕmn exp[-ik(αmx+γnz)].
ϕ(x, z)=m=-Mm=+Mn=-Nn=+Nϕmn exp[-ik(αmx+γnz)].
g12=m,ngmn12 exp-i2πmxdx+nzdz,
g23=m,ngmn23 exp-i2πmxdx+nzdz,
g22=m,ngmn22 exp-i2πmxdx+nzdz,
gmn12=-i 2πmdxamn,
gmn23=-i 2πndzamn,
gmn22=δmn+u,vgm-u,n-v12guv12+u,vgm-u,n-v23guv23,
p,q(-gm-p,n-q12αp-gm-p,n-q12αm-gm-p,n-q23
×γq-gm-p,n-q23γn)rjϕmn+p,qgm-p,n-q22rjϕmn
=(νj2-αm2-γn2)ϕmn,
rjϕmn=ϕmn.
Arϕϕ=Bϕϕ,
ϕj(x, u, z)=m=-Mm=+Mn=-Nn=+Nl=1l=2Lϕj, mn l exp(-ikrjlu) ×exp(-ikαmx)exp(-ikγnz),
limM,N rj pqMN=βj pq.
m=-Mm=+Mn=-Nn=+Np,qUjϕj mn pq exp(-ikrj pqu)
×exp(-ikαmx)exp(-ikγnz),
Uj={(p, q)[-M, M]×[-N, N],νj2-αp2-γq2>0}
p,q exp{-ikβj pq[u+a(x, z)]}×exp(-ikαpx)exp(-ikγqz).
ϕj(x, u, z)
=m=-Mm=+Mn=-Nn=+Np,qUjϕj mn pq+R exp(-ikβj pq+u) ×exp(-ikαmx)exp(-ikγnz)+m=-Mm=+Mn=-Nn=+Np,qVj+ϕj mn pq+exp(-ikrj pq+u) ×exp(-ikαmx)exp(-ikγnz)+m=-Mm=+Mn=-Nn=+Np,qUjϕj mn pq-Rexp(-ikβj pq-u) ×exp(-ikαmx)exp(-ikγnz)+m=-Mm=+Mn=-Nn=+Np,qVj-ϕj mn pq- exp(-ikrj pq-u) ×exp(-ikαmx)exp(-ikγnz),
ϕj mn pq±R=1dxdz0dx0dz exp[-ik(βj pq±)a(u, v)] ×exp(-ikαp+mu)exp(-ikγq+nv)dudv,
Vj+={(p, q)([-M, M]×[-N, N]-Uj),Im(rj pq)<0},
Vj-={(p, q)([-M, M]×[-N, N]-Uj),Im(rj pq)>0}.
ψjl=m,n p,qAj pq+l TEψj mn pq+TE exp[-ikrj pq+(u-ul)] ×exp(-ikαmx)exp(-ikγnz)+m,n p,qAj pq+l TMψj mn pq+TM exp[-ikrj pq+(u-ul)] ×exp(-ikαmx)exp(-ikγnz)+m,n p,qAj pq-l TEψj mn pq-TE exp[-ikrj pq-(u-ul)] ×exp(-ikαmx)exp(-ikγnz)+m,n p,qAj pq-l TMψj mn pq-TM exp[-ikrj pq-(u-ul)] ×exp(-ikαmx)exp(-ikγnz),
A1 pq+0 TEA1 pq+0 TMAQ+1 pq-Q TEAQ pq-Q TM=SA1 pq-0 TEA1 pq-0 TMAQ+1 pq+Q TEAQ pq+Q TM,
AQ pq+Q TE=AQ pq+Q TM=0pandq,
A1 pq-0 TE=0ifporq0cos δ,ifpandq=0,
A1 pq-0 TM=0ifporq0sin δifpandq=0.
ψjj(uj)=ψj+1j(uj).
Aj-1+j TEAj-1+j TMAj-j TEAj-j TM=Sj-1,jjAj-1-j TEAj-1-j TMAj+j TEAj+j TM,
Sj-1,jj=(ψj-1-j TEψj-1+j TMψj-j TEψj-j TM)-1 ×(ψj-1-j TEψj-1-j TMψj+j TEψj+j TM).
ψj+j TE=(ψj pq+j TE)=Ezj pq+jHxj pq+jHzj pq+jExj pq+j.
Aj+j TEAj+j TMAj-(j+1) TEAj-(j+1) TM=Sjj,j+1Aj-j TEAj-j TMAj+(j+1) TEAj+(j+1) TM,
Sji,j+1=00φj+0000φj+φj-0000φj-00.
Nu=12(EzHx*-Hz*Ex),
Pj pq±=1dx1dz0dx0dzNj pqu± dxdz=12mnEzj mn pq±TEHxj mn pq*±TE-mnHzj mn pq*±TEExj mn pq±TE+12mnEzj mn pq±TMHxj mn pq*±TM-mnHzj mn pq*±TMExj mn pq±TM.
epqr=(|A0 pq+0 TE|2+|A0 pq+0 TM|2) P0 pq+P0 pq-,
epqt=(|AQ pq-Q TE|2+|AQ pq+Q TM|2) PQ pq-P0 00-.
a(x, z)=h4sin 2πxd+sin 2πzd.
=1-p,qU0epqr-p,qUQepqt.
ρj=ejνj/λ.
sin θ=12dλ.

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