Abstract

Rigorous coupled-wave analysis is shown to be valid for a pure reflection grating The analysis is extended to take into consideration an arbitrary phase of the sinusoidal modulation of the pure reflection grating’s dielectric constant The correct method of computing the diffraction efficiencies is presented, and results obtained with the extended and corrected analysis are given

© 1983 Optical Society of America

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References

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  1. M G Moharam and T K Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J Opt Soc Am 71, 811–818 (1981)
    [Crossref]
  2. M G Moharam and T K Gaylord, “Coupled-wave analysis of reflection gratings,” Appl Opt 20, 240–244 (1981)
    [Crossref] [PubMed]

1981 (2)

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

M G Moharam and T K Gaylord, “Coupled-wave analysis of reflection gratings,” Appl Opt 20, 240–244 (1981)
[Crossref] [PubMed]

Gaylord, T K

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

M G Moharam and T K Gaylord, “Coupled-wave analysis of reflection gratings,” Appl Opt 20, 240–244 (1981)
[Crossref] [PubMed]

Moharam, M G

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

M G Moharam and T K Gaylord, “Coupled-wave analysis of reflection gratings,” Appl Opt 20, 240–244 (1981)
[Crossref] [PubMed]

Appl Opt (1)

M G Moharam and T K Gaylord, “Coupled-wave analysis of reflection gratings,” Appl Opt 20, 240–244 (1981)
[Crossref] [PubMed]

J Opt Soc Am (1)

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

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Figures (9)

Fig 1
Fig 1

Geometric configuration for RCW analysis

Fig 2
Fig 2

Reflection efficiency of a PRG at first-Bragg incidence (solid line) Dashed line is the erroneous reflection efficiency calculated with the same data according to Refs 1 and 2

Fig 3
Fig 3

Reflection efficiency of a PRG at second-Bragg incidence (solid line) Dashed line is the erroneous reflection efficiency calculated with the same data according to Refs 1 and 2

Fig 4
Fig 4

Deviations from conservation of power (|∑ιRι|2 + |∑ιTι|2 − 1) obtained with the data of Fig 2 when (top to bottom) 11, 17, and 23 diffraction orders were used in the numerical calculations

Fig 5
Fig 5

Deviations from conservation of power (|∑ιRι|2 + |∑ιTι|2 − 1) obtained with the data of Fig 3 when (top to bottom) 11, 17, and 23 diffraction orders were used in the numerical calculations

Fig 6
Fig 6

Reflection efficiency of PRG with modulation phase α = −90° at first-Bragg incidence (solid line) Dashed line is the reflection efficiency when α = 0° (identical to solid curve in Fig 2)

Fig 7
Fig 7

Reflection efficiency of a PRG with modulation phase α = 180° at first-Bragg incidence (solid line) Dashed line is the reflection efficiency when α = 0°(same as solid curve in Fig 2)

Fig 8
Fig 8

Reflection efficiency of a PRG with modulation phase α = −90° at second-Bragg incidence (solid line) Dashed line is the reflection efficiency when α = 0° (identical to solid curve in Fig 3)

Fig 9
Fig 9

Reflection efficiency of a PRG with modulation phase α = 180° at second-Bragg incidence (solid line) Dashed line is the reflection efficiency when α = 0° (same as solid curve in Fig 3)

Equations (59)

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( x , z ) = 2 + Δ cos [ K ( x sin ϕ + z cos ϕ ) + α ] ,
E 1 = exp [ J ( β 0 x + ξ 10 z ) ] + ι R ι exp [ J ( β ι x ξ 1 ι z ) ]
E 2 = ι S ι ( z ) exp [ J ( β ι x + ξ 2 ι z ) ]
E 3 = ι T ι exp { J [ β ι x + ξ 3 ι ( z d ) ] }
J = 1 , β ι = k 1 sin θ ι K sin ϕ , ξ m ι 2 = k m 2 = β ι 2 , m = 1 , 3 , ξ 2 ι = k 2 cos θ ι K cos ϕ , θ = sin 1 [ ( 1 / 2 ) 1 / 2 sin θ ] , k m = 2 π ( m ) 1 / 2 / λ , m = 1 , 2 , 3 , ι = 0 , ± 1 , ± 2 , ,
2 E 2 + ( 2 π / λ ) 2 ( x , z ) E 2 = 0 ,
( Δ / 8 2 ) d 2 S ¯ ι ( u ) d u 2 = ( cos θ ι μ cos ϕ ) d S ¯ ι ( u ) d u ρ ι ( ι B ) S ¯ ι ( u ) + S ¯ ι + 1 ( u ) exp ( J α ) + S ¯ ι 1 ( u ) exp ( + J α ) ,
S ¯ ι ( u ) = S ι ( z ) , u = J κ z = J π Δ z / [ 2 λ ( 2 ) 1 / 2 ] , μ = λ / [ Λ ( 2 ) 1 / 2 ] , ρ = 2 λ 2 / ( Λ 2 Δ ) = 2 μ 2 2 / Δ , and B = 2 Λ ( 2 ) 1 / 2 cos ( ϕ θ ) / λ = 2 cos ( ϕ θ ) / μ
R ι + δ ι 0 = S ι ( 0 ) ,
ξ 1 ι ( R ι δ ι 0 ) = J d S ι d z ( 0 ) ξ 2 ι S ι ( 0 ) ,
T ι = S ι ( d ) exp ( J ξ 2 ι d ) ,
ξ 3 ι T ι = [ J d S ι d z ( d ) ξ 2 ι S ι ( d ) ] exp ( J ξ 2 ι d ) ,
S ¯ ι ( u ) = S ι ( u ) exp ( J ι α )
( Δ / 8 2 ) d 2 S ι ( u ) d u 2 = ( cos θ ι μ cos ϕ ) d S ι ( u ) d u ρ ι ( ι B ) S ι ( u ) + S ι + 1 ( u ) + S ι 1 ( u )
S ι ( z ) = S ι ( z ) exp ( J ι α ) ,
Ŝ ι ( z ) = S ι ( u ) ,
R ι = R ι exp ( J ι α ) ,
T ι = T ι exp ( J ι α ) ,
R ι + δ ι 0 = Ŝ ι ( 0 ) ,
ξ 1 ι ( R ι δ ι 0 ) = J d Ŝ ι d z ( 0 ) ξ 2 ι Ŝ ι ( 0 ) ,
T ι = Ŝ ι ( d ) exp ( J ξ 2 ι d ) ,
ξ 3 ι T ι = [ J d Ŝ ι d z ( d ) ξ 2 ι Ŝ ι ( d ) ] exp ( J ξ 2 ι d )
DE R ι = Re ( ξ 1 ι / ξ 10 ) | R ι | 2 ,
DE T ι = Re ( ξ 3 ι / ξ 10 ) | T ι | 2
DE R ι = Re ( ξ 1 ι / ξ 10 ) | R ι | 2 ,
DE T ι = Re ( ξ 3 ι / ξ 10 ) | T ι | 2
ι ( DE R ι + DE T ι ) = 1
( z ) = 2 + Δ cos ( K z + α )
R = ι R ι
T = ι T ι
DE R = | R | 2 = | ι R ι | 2
DE T = Re ( ξ 30 / ξ 10 ) | T | 2 = Re ( ξ 30 / ξ 10 ) | 1 T ι | 2
DE R = | ι R ι exp ( J ι α ) | 2 ,
DE T = Re ( ξ 30 / ξ 10 ) | ι T ι exp ( J ι α ) | 2
| ι R ι | 2 + Re ( ξ 30 / ξ 10 ) | ι T ι | 2 = 1
1 = 2 = 3 = 2 25 , Δ = 0 2 , λ = Λ = 0 5 μ m , ( z ) = 2 + Δ cos ( 2 π z / Λ + α )
E 1 = exp [ J ( β 0 x + ξ 10 z ) ] + R exp [ J ( β 0 x ξ 10 z ) ]
E 2 = U ( z ) exp ( J β 0 x )
E 3 = T exp { J [ β 0 x + ξ 30 ( z d ) ] }
R + 1 = U ( 0 ) ,
J ξ 10 ( R 1 ) = d U d z ( 0 ) ,
T = U ( d ) ,
J ξ 30 T = d U d z ( d )
R = ι R ι exp ( J ι α ) ,
U ( z ) = ι U ι ( z ) exp ( J ι α ) ,
T = ι T ι exp ( J ι α )
U ι ( z ) = S ι ( z ) exp ( J ξ 2 ι z ) ,
ξ 2 ι = k 2 cos θ ι K
ι [ d 2 Ŝ ι d z 2 2 J ξ 2 ι d Ŝ ι d z ( ξ 2 ι 2 + β 0 2 k 2 2 ) Ŝ ι + ( 2 π / λ ) 2 ( Δ / 2 ) ( Ŝ ι 1 + Ŝ ι + 1 ) ] × exp ( J ξ 2 ι z ) exp ( J ι α ) = 0
d 2 Ŝ ι d z 2 2 J ξ 2 ι d Ŝ ι d z ( ξ 2 ι 2 + β 0 2 k 2 2 ) Ŝ ι + ( 2 π / λ ) 2 ( Δ / 2 ) ( Ŝ ι 1 + Ŝ ι + 1 ) = 0
ι R ι exp ( J ι α ) + 1 = ι Ŝ ι ( 0 ) exp ( J ι α ) ,
R ι + δ ι 0 = Ŝ ι ( 0 )
S ι ( z ) = Ŝ ι ( z ) exp ( J ι α ) ,
R ι = R ι exp ( J ι α ) ,
T ι = T ι exp ( J ι α ) ,
DE R = ι | R ι | 2 + ι n n ι R ι R n * exp [ J ( ι n ) α ]
1 2 π 0 2 π DE R ( α ) d α = ι | R ι | 2
1 2 π 0 2 π DE Γ ( α ) d α = Re ( ξ 30 / ξ 10 ) ι | T ι | 2
ι | R ι | 2 + Re ( ξ 30 / ξ 10 ) ι | T ι | 2 = 1