Abstract

A new implementation of the coupled-wave method for TM polarization is proposed. We use a second-order differential operator established by Nevière together with a scattering-matrix approach. Thus we obtain for metallic gratings a convergence rate as quick as that in TE polarization.

© 1996 Optical Society of America

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References

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  1. M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. 12, 1068–1076 (1995).
    [CrossRef]
  2. M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
    [CrossRef]
  3. Lifeng Li, Charles W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. 10, 1184–1189 (1993).
    [CrossRef]
  4. M. Nevière, “Sur un formalisme différentiel pour less problèmes de diffraction dans le domain de résonance: application à l’etude des réseaux optiques et de diverses structures périodiques,” thèse de doctorat (Université d’Aix–Marseille, Aix–Marseille, France, 1975).
  5. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer–Verlag, Berlin, 1980), Chap. 4.
    [CrossRef]
  6. Nicolas Chateau, Jean-Paul Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am 11, 1321–1331 (1994).
    [CrossRef]
  7. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. 12, 1097–1103 (1995).
    [CrossRef]
  8. 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]

1995

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

M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. 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]

1994

Nicolas Chateau, Jean-Paul Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am 11, 1321–1331 (1994).
[CrossRef]

1993

Lifeng Li, Charles W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. 10, 1184–1189 (1993).
[CrossRef]

Chandezon, J.

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]

Chateau, Nicolas

Nicolas Chateau, Jean-Paul Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am 11, 1321–1331 (1994).
[CrossRef]

Cotter, N. P. K.

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

Gaylord, T. K.

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

M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[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]

Grann, Eric B.

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

M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

Haggans, Charles W.

Lifeng Li, Charles W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. 10, 1184–1189 (1993).
[CrossRef]

Hugonin, Jean-Paul

Nicolas Chateau, Jean-Paul Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am 11, 1321–1331 (1994).
[CrossRef]

Li, Lifeng

Lifeng Li, Charles W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. 10, 1184–1189 (1993).
[CrossRef]

Moharam, M. G.

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

M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

Nevière, M.

M. Nevière, “Sur un formalisme différentiel pour less problèmes de diffraction dans le domain de résonance: application à l’etude des réseaux optiques et de diverses structures périodiques,” thèse de doctorat (Université d’Aix–Marseille, Aix–Marseille, France, 1975).

Plumey, J. P.

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]

Pommet, Drew A.

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

M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

Preist, T. W.

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

Sambles, J. R.

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

Vincent, P.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer–Verlag, Berlin, 1980), Chap. 4.
[CrossRef]

J. Opt. Soc. Am

Nicolas Chateau, Jean-Paul Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am 11, 1321–1331 (1994).
[CrossRef]

J. Opt. Soc. Am.

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

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

M. G. Moharam, Eric B. Grann, Drew A. Pommet, T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

Lifeng Li, Charles W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. 10, 1184–1189 (1993).
[CrossRef]

Pure Appl. Opt.

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]

Other

M. Nevière, “Sur un formalisme différentiel pour less problèmes de diffraction dans le domain de résonance: application à l’etude des réseaux optiques et de diverses structures périodiques,” thèse de doctorat (Université d’Aix–Marseille, Aix–Marseille, France, 1975).

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer–Verlag, Berlin, 1980), Chap. 4.
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Geometry for the lamellar metallic grating analyzed herein.

Fig. 2
Fig. 2

Scattering by a lamellar grating and S-matrix formulation.

Fig. 3
Fig. 3

Convergence of (a) the negative first-order and (b) the zeroth-order diffraction efficiencies for TM polarization computed with our implementation (circles) and with the implementation of Moharam et al. (crosses).

Tables (1)

Tables Icon

Table 1 Numerical Values of TM Diffraction Efficiencies Computed at Three Different Truncation Orders And for Three Different Groove Depths

Equations (31)

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2 ( x ) = { 3 if 0 < x < f d 1 if f d < x < d ,
2 ( x ) = p 2 p exp ( - i 2 π p d x ) ,
1 2 ( x ) = p ˜ 2 p exp ( - i 2 π p d x ) .
H 1 , z ( y , x ) = m { q [ A 1 q + 1 δ m q exp ( - i k β 1 q + y ) + q A 1 q - 1 δ m q exp ( - i k β 1 q - y ) ] } × exp ( - i k α m x ) ,
E 1 x ( y , x ) = m { i k β 1 q + q [ A 1 q + 1 δ m q exp ( - i k β 1 q + y ) + i k β 1 q - q A 1 q - 1 δ m q exp ( - i k β 1 q - y ) ] } × exp ( - i k α m x ) .
H 3 z ( y , x ) = m ( q { A 3 q + 2 δ m q exp [ - i k β 3 q + ( y + h ) ] + q A 3 q - 2 δ m q exp [ - i k β 3 q - ( y + h ) ] } ) × exp ( - i k α m x ) ,
E 3 x ( y , x ) = m ( q { i k β 3 q + A 3 q + δ m q exp [ - i k β 3 q + ( y + h ) ] + q i k β 3 q - A 3 q - 2 δ m q exp [ - i k β 3 q - ( y + h ) ] } ) × exp ( - i k α m x ) ,
δ m q = { 1 if m = q 0 if m q , β j q + = - β j q - = j - ( sin θ + q λ d ) 2 and Re ( β j q + ) - Im ( β j q + ) > 0
α q = sin θ + q λ d .
x [ 1 k 2 2 ( x ) H 2 z x ] + y [ 1 k 2 2 ( x ) H 2 z y ] + H 2 z = 0.
α m p α p ˜ 2 m - p H 2 z p ( y ) - H 2 z m ( y ) = 1 k 2 p ˜ 2 m - p 2 y 2 [ H 2 z p ( y ) ] .
T = [ ˜ ] - 1 ( I - [ α ] [ ˜ ] [ α ] ) ,
T = [ ˜ ] - 1 ( I - [ α ] [ ] - 1 [ α ] ) .
[ ] [ ˜ ] = I ,
T = [ ] ( I - [ α ] [ ˜ ] [ α ] ) .
H 2 z = m { q [ A 2 q + H 2 z m q +   exp ( - i k r q + y ) + q A 2 q - H 2 z m q - exp ( - i k r q - y ) ] } exp ( - i k α m x ) ,
r q + = - r q - = { r q 2 if r q 2 is real r q 2 with negative imaginary part if r q 2 is complex ,
E 2 x = 1 i ω 2 H 2 z y .
E 2 x = m [ q - i k r q + A 2 q + l ˜ l H 2 z m - l q + exp ( - i k r q + y ) + q i k r q - A 2 q - l ˜ l H 2 z m - l q - exp ( - i k r q - y ) ] × exp ( - i k α m x ) .
E 2 x + = [ ˜ ] [ H 2 z + ] [ r + ]
E 2 x - = [ ˜ ] [ H 2 z - ] [ r - ] ,
H 2 z ( y = 0 ) = m [ q ( A 2 q + 1 H 2 z m q + + q A 2 q - 1 H 2 z m q - ) ] × exp ( - i k α m x ) ,
H 2 z ( y = - h ) = m { q [ A 2 q + 2 H 2 z m q + exp ( i r q + h ) + q A 2 q - 2 H 2 z m q - exp ( i k r q - h ) ] } × exp ( - i k α m x ) .
[ A 1 + 1 A 2 - 1 ] = S 1 [ A 1 - 1 A 2 + 1 ] ,
S 1 = [ I - I E 1 x + - E 1 x - ] - 1 [ - H 2 z - H 2 z + - E 2 x - E 2 x + ] ,
[ A 2 + 2 A 3 - 2 ] = S 2 [ A 2 - 2 A 3 + 2 ] ,
S 2 = [ H 2 z + - H 2 z - E 2 x + - E 2 x - ] - 1 [ I I E 3 x + - E 3 x - ] .
[ A 2 + 1 A 2 - 2 ] = S 12 [ A 2 - 1 A 2 + 2 ]
S 12 = [ 0 ϕ 2 ϕ 2 0 ] ,
[ A 1 + 1 A 3 - 2 ] = S [ A 1 - 1 A 3 + 2 ] .
e q = A 1 q + 1 2 β 1 q + cos θ .

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