Abstract

The rigorous coupled-wave analysis technique for describing the diffraction of electromagnetic waves by periodic grating structures is reviewed. Formulations for a stable and efficient numerical implementation of the analysis technique are presented for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction. It is shown that by exploitation of the symmetry of the diffraction problem a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, is produced. The rigorous coupled-wave analysis is shown to be inherently stable. The sources of potential numerical problems associated with underflow and overflow, inherent in digital calculations, are presented. A formulation that anticipates and preempts these instability problems is presented. The calculated diffraction efficiencies for dielectric gratings are shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings. The effect of the number of harmonics on the convergence of the diffraction efficiencies is investigated. More field harmonics are shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction.

© 1995 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  3. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  4. 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]
  5. W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1796 (1986).
    [CrossRef]
  7. M. G. Moharam, “Diffraction analysis of multiplexed holographic gratings,” in Digest of Topical Meeting on Holography (Optical Society of America, Washington, D.C., 1986), pp. 100–103.
  8. 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]
  9. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A. 4, 2061–2080 (1987).
    [CrossRef]
  10. M. G. Moharam, D. A. Pommet, E. B. Grann, T. K. Gaylord, “Stable implementation of the rigorous coupled wave analysis for surface-relief dielectric gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  11. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1187 (1993).
    [CrossRef]

1995 (1)

1993 (1)

1987 (1)

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A. 4, 2061–2080 (1987).
[CrossRef]

1986 (1)

1983 (3)

1982 (1)

1981 (1)

Baird, W. E.

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A. 4, 2061–2080 (1987).
[CrossRef]

Grann, E. B.

Haggans, C. W.

Li, L.

Moharam, M. G.

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

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1796 (1986).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction—E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
[CrossRef]

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]

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

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

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

M. G. Moharam, “Diffraction analysis of multiplexed holographic gratings,” in Digest of Topical Meeting on Holography (Optical Society of America, Washington, D.C., 1986), pp. 100–103.

Pommet, D. A.

Appl. Phys. B (1)

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

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

E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A. 4, 2061–2080 (1987).
[CrossRef]

Other (2)

M. G. Moharam, “Diffraction analysis of multiplexed holographic gratings,” in Digest of Topical Meeting on Holography (Optical Society of America, Washington, D.C., 1986), pp. 100–103.

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 (3)

Fig. 1
Fig. 1

Geometry for the binary rectangular-groove grating diffraction problem analyzed herein.

Fig. 2
Fig. 2

Diffraction-efficiency dependence on the normalized grating depth of a binary dielectric grating (nII = nrd = 2.04, nI = 1) for TE polarization, TM polarization, and conical mount (ϕ = 30° and ψ = 45°) at θ = 10°.

Fig. 3
Fig. 3

Diffraction-efficiency dependence on the number of space harmonics for a binary dielectric grating (nII = nrd = 2.04) for TE polarization, TM polarization, and conical mount (ϕ = 30° and ψ = 45°) at θ = 10°.

Equations (80)

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ɛ ( x ) = h ɛ h exp ( j 2 π h Λ ) ,
ɛ 0 = n rd 2 f + n gr 2 ( 1 f ) , ɛ h = ( n rd 2 f n gr 2 ) sin ( π h f ) π h ,
E inc , y = exp [ j k 0 n I ( sin θ x + cos θ z ) ] ,
E I , y = E inc , y + i R i exp [ j ( k x i x k I , z i z ) ] ,
E II , y = i T i exp { j [ k x i x k II , z i ( z d ) ] } ,
k x i = k 0 [ n I sin θ i ( λ 0 / Λ ) ]
k , z i = { + k 0 [ n 2 ( k x i / k 0 ) 2 ] 1 / 2 k 0 n > k x i j k 0 [ ( k x i / k 0 ) n 2 ] 1 / 2 k x i > k 0 n = I , II .
H = ( j ω μ ) × E ,
E g y = i S y i ( z ) exp ( j k x i x ) ,
H g x = j ( 0 μ 0 ) 1 / 2 i U x i ( z ) exp ( j k x i x ) ,
E g y z = j ω μ 0 H g x ,
H g y z = j ω 0 ɛ ( x ) E g y + H g z x .
S y i z = k 0 U x i , U x i z = ( k x i 2 k 0 ) S y i k 0 p ɛ ( i p ) S y p ,
[ S y / ( z ) U x / ( z ) ] = [ 0 I A 0 ] [ S y U x ] ,
[ 2 S y / ( z ) 2 ] = [ A ] [ S y ] ,
A = K x 2 E ,
S y i ( z ) = m = 1 n w i , m { c m + exp ( k 0 q m z ) + c m exp [ k 0 q m ( z d ) ] } ,
U x i ( z ) = m = 1 n υ i , m { c m + exp ( k 0 q m z ) + c m exp [ k 0 q m z ( z d ) ] } ,
δ i 0 + R i = m = 1 n w i , m [ c m + + c m exp ( k 0 q m d ) ] ,
j [ n I cos θ δ i 0 ( k I , z i / k 0 ) R i ] = m = 1 n υ i , m [ c m + c m exp ( k 0 q m d ) ] ,
[ δ i 0 j n I cos θ δ i 0 ] + [ I j Y I ] [ R ] = [ W WX V VX ] [ c + c ] ,
m = 1 n w i , m [ c m + exp ( k 0 q m d ) + c m ] = T i ,
m = 1 n υ i , m [ c m + exp ( k 0 q m d ) c m ] = j ( k II , z i / k 0 ) T i ,
[ WX W VX V ] [ c + c ] = [ I j Y II ] [ T ] ,
DE r i = R i R i * Re ( k I , z i k 0 n I cos θ ) , DE t i = T i T i * Re ( k II , z i k 0 n I cos θ ) .
H inc , y = exp [ j k 0 n I ( sin θ x + cos θ z ) ] .
H I , y = H inc , y + i R i exp [ j ( k x i x k I , z i z ) ] ,
H II , y = i T i exp { j [ k x i x + k II , z i ( z d ) ] } ,
E = ( j ω 0 n 2 ) × H .
H g y = i U y i ( z ) exp ( j k x i x ) ,
E g x = j ( μ 0 0 ) 1 / 2 i S x i ( z ) exp ( j k x i x ) ,
H g y z = j ω 0 ɛ ( x ) E g x ,
E g x z = j ω μ 0 H g y + E g x x .
[ U y / ( z ) S x / ( z ) ] = [ 0 E B 0 ] [ U y S x ] ,
[ 2 U y / ( z ) 2 ] = [ EB ] [ U y ] ,
B = K x E 1 K x I ,
U y i ( z ) = m = 1 n w i , m { c m + exp ( k 0 q m z ) + c m exp [ k 0 q m ( z d ) ] } ,
S x i ( z ) = m = 1 n υ i , m { c m + exp ( k 0 q m z ) + c m exp [ k 0 q m ( z d ) ] } ,
δ i 0 + R i = m = 1 n w i , m [ c m + + c m exp ( k 0 q m d ) ] ,
j [ ( cos θ n I ) δ i 0 ( k I , z i k 0 n I 2 ) R i ] = m = 1 n υ i , m [ c m + c m exp ( k 0 q m d ) ] ,
[ δ i 0 j δ i 0 cos θ / n I ] + [ I Z I ] [ R ] = [ W WX V VX ] [ c + c ] ,
m = 1 n w i , m [ c m + exp ( k 0 q m d ) + c m ] = T i ,
m = 1 n υ i , m [ c m + exp ( k 0 q m d ) + c m ] = j ( k I , z i k 0 n II 2 ) T i ,
[ WX W VX V ] [ c + c ] = [ I j Z II ] [ T ] ,
DE r i = R i R i * Re ( k I , z i / k 0 n I cos θ ) , DE t i = T i T i * Re ( k II , z i n II 2 ) / ( k 0 cos θ n I ) .
E inc = u exp [ j k 0 n I ( sin θ cos ϕ x + sin θ sin ϕ y + cos θ z ) ] ,
u = ( cos ψ cos θ cos ϕ sin ψ sin ϕ ) x ̂ + ( cos ψ cos θ cos ϕ sin ψ cos ϕ ) y ̂ cos ψ sin θ z ̂ ,
E I = E inc + i R i exp [ j ( k x i x + k y y k I , z i z ) ] ,
E II = i T i exp { j [ k x i x + k y y k II , z i ( z d ) ] }
k x i = k 0 [ n I sin θ cos ϕ i ( λ 0 / Λ ) ] ,
k y = k 0 n I sin θ sin ϕ ,
k , z i = { + [ ( k 0 n ) 2 k x i 2 k y 2 ] 1 / 2 ( k x i 2 + k y 2 ) < k 0 n j [ k x i 2 + k y 2 ( k 0 n ) 2 ] 1 / 2 ( k x i 2 + k y 2 ) 1 / 2 > k 0 n , = I , II .
φ i = tan 1 ( k y / k x i ) .
E g = i [ S x i ( z ) x + S y i ( z ) y + S z i ( z ) z ] exp [ j ( k x i x + k y y ) ] ,
H g = j ( 0 μ 0 ) 1 / 2 i [ U x i ( z ) x + U y i ( z ) y + U z i ( z ) z ] × exp [ j ( k x i x + k y y ) ] .
× E g = j ω μ 0 H g , × H g = j ω 0 ɛ ( x ) E g .
[ S y / ( z ) S x / ( z ) U y / ( z ) U x / ( z ) ] = [ 0 0 K y E 1 K x I K y E 1 K y 0 0 K x E 1 K x I K x E 1 K y K x K y E K y 2 0 0 K x 2 E K x K y 0 0 ] × [ S y S x U y U x ] ,
[ 2 S y / ( z ) 2 2 S x / ( z ) 2 ] = [ K x 2 + DE K y [ E 1 K x E K x ] K x [ E 1 K y E K y ] K y 2 + BE ] [ S y S x ] ,
[ 2 U y / ( z ) 2 2 U x / ( z ) 2 ] = [ K y 2 + EB [ K x E K x E 1 ] K y K y [ K y E K y E 1 ] K x K x 2 + ED ] [ U y U x ] .
[ 2 U x / ( z ) 2 ] = [ k y 2 I + A ] [ U x ] , [ 2 S x / ( z ) 2 ] = [ k y 2 I + BE ] [ S x ] .
U x i ( z ) = m = 1 n w 1 , i , m { c 1 , m + exp ( k 0 w q 1 , m z ) + c 1 , m exp [ k 0 q 1 , m ( z d ) ] } ,
S x i ( z ) = m = 1 n w 2 , i , m { c 2 , m + exp ( k 0 q 2 , m z ) + c 2 , m exp [ k 0 q 2 , m ( z d ) ] } ,
S y i ( z ) = m = 1 n υ 11 , i , m { c 1 , m + exp ( k 0 q 1 , m z ) + c 1 , m exp [ k 0 q 1 , m ( z d ) ] } + m = 1 n υ 12 , i , m { c 2 , m + exp ( k 0 q 2 , m z ) + c 2 , m exp [ k 0 q 2 , m ( z d ) ] } ,
U y i ( z ) = m = 1 n υ 11 , i , m { c 1 , m + exp ( k 0 q 1 , m z ) + c 1 , m exp [ k 0 q 1 , m ( z d ) ] } + m = 1 n υ 22 , i , m { c 2 , m + exp ( k 0 q 2 , m z ) + c 2 , m exp [ k 0 q 2 , m ( z d ) ] } ,
V 11 = A 1 W 1 Q 1 , V 12 = ( k y / k 0 ) A 1 K x W 2 , V 21 = ( k y / k 0 ) B 1 K x E 1 W 1 , V 22 = B 1 W 2 Q 2 ,
sin ψ δ i 0 + R s , i = cos φ i S y i ( 0 ) sin φ i S x i ( 0 ) ,
j [ sin ψ n I cos θ ( k I , z i / k 0 ) R s , i ] = [ cos φ i U x i ( 0 ) + sin φ i U y i ( 0 ) ] ,
cos ψ cos θ j [ k I , z i / ( k 0 n I 2 ) ] R p , i = [ cos φ i S x i ( 0 ) + sin φ i S y i ( 0 ) ] ,
j n I cos ψ + R p , i = [ cos φ i U y i ( 0 ) sin φ i U x i ( 0 ) ] ,
R s , i = cos φ i R y i sin φ i R x i , R p , i = ( j / k 0 ) [ cos φ i ( i k I , z i R x i k x i R z i ) sin φ i ( k y R z i + k I , z i R y i i ) ] .
[ sin ψ δ i 0 j sin ψ n I cos θ δ i 0 j cos ψ n I δ i 0 cos ψ cos θ δ i 0 ] + [ I 0 j Y I 0 0 I 0 j Z I ] [ R s R p ] = [ V s s V s p V s s X 1 V s p X 2 W s s W s p W s s X 1 W s p X 2 W p s W p p W p s X 1 W p p X 2 V p s V p p V p s X 1 V p s X 2 ] [ c 1 + c 1 c 2 + c 2 ] ,
V s s = F c V 11 , W p p = F c V 22 , W s s = F c W 1 + F s V 21 , V p p = F c W 2 + F s V 12 , V s p = F c V 12 F c W 2 , W p s = F c V 21 F s W 1 , W s p = F s V 22 , V p s = F s V 11 ,
cos φ i S y i ( d ) sin φ i S x i ( d ) = T s , i ,
[ cos φ i U x i ( d ) + sin φ i U y i ( d ) ] = j ( k I , z i / k 0 ) T s , i ,
[ cos φ i U y i ( d ) + sin φ i U x i ( d ) ] = T p , i ,
cos φ i S x i ( d ) + sin φ i S y i ( d ) = j ( k I , z i / k 0 n I 2 ) T p , i ,
T s , i = cos φ i T y i sin φ i T x i , T p , i = ( j / k 0 ) [ cos φ i ( k II , z i T x i k x i T z i ) sin φ i ( k II , z i T y i + k y T z i ) ] .
[ V s s X 1 V s p X 2 V s s V s p W s s X 1 W s p X 2 W s s W s p W p s X 1 W p p X 2 W p s W p p V p s X 1 V p p X 2 V p s V p p ] [ c 1 + c 1 c 2 + c 2 ] = [ I 0 j Y II 0 0 I 0 j Z II ] [ T s T p ] .
DE r i = | R s , i | 2 Re ( k I , z i k 0 n I cos θ ) + | R p , i | 2 Re ( k I , z i / n I 2 k 0 n I cos θ ) , DE t i = | T s , i | 2 Re ( k II , z i k 0 n I cos θ ) + | T p , i | 2 Re ( k II , z i / n II 2 k 0 n I cos θ ) .
i ( DE r i + DE t i ) = 1 .

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