Abstract

Iterative-series solutions have been recently developed for the scattering of p-polarized waves by a grating and for three-dimensional surfaces. However, their numerical behavior has not been investigated. We study the scattering by both dielectric and metallic sinusoidal gratings. We show that this kind of solution yields an accurate and efficient solution in the dielectric case. This solution may also be applied in conical-diffraction problems. Although some resonances have been successfully studied, the algorithm diverges for low values of the amplitude–period ratio in the case of metallic gratings.

© 1990 Optical Society of America

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References

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  1. A. A. Maradudin, “Iterative solutions for electromagnetic scattering by gratings,” J. Opt. Soc. Am. 73, 759–764 (1983).
    [Crossref]
  2. C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
    [Crossref]
  3. J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6437–6441 (1988).
    [Crossref]
  4. F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
    [Crossref]
  5. E. P. Da Silva, G. A. Farias, A. A. Maradudin, “Analysis of three theories of scattering of electromagnetic radiation by gratings,” J. Opt. Soc. Am. A 4, 2022–2024 (1987).
    [Crossref]
  6. R. A. Depine, J. M. Simon, “Comparison between the differential and the integral methods used to solve the grating problem in the H case,” J. Opt. Soc. Am. A 4, 834–838 (1987).
    [Crossref]
  7. T. Inagaki, J. P. Goudonnet, E. T. Arakawa, “Plasma resonance absorption in conical diffraction: effects of groove depth,” J. Opt. Soc. Am. B 3, 992–995 (1986).
    [Crossref]
  8. P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [Crossref]
  9. D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies des réseaux de diffraction: application aux anomalies de plasmons,” J. Opt. (Paris) 8, 165–174 (1977).
    [Crossref]
  10. A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
    [Crossref]
  11. S. Dutta Gupta, G. V. Varada, G. S. Agarwal, “Surface plasmons in two-sided corrugated thin films,” Phys. Rev. B 36, 6331–6335 (1987).
    [Crossref]
  12. C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction Entre Photons et Atomes (InterEditions, Paris, 1988).

1988 (1)

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6437–6441 (1988).
[Crossref]

1987 (3)

1986 (1)

1983 (1)

1981 (1)

A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[Crossref]

1978 (1)

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

1977 (2)

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies des réseaux de diffraction: application aux anomalies de plasmons,” J. Opt. (Paris) 8, 165–174 (1977).
[Crossref]

1972 (1)

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Agarwal, G. S.

S. Dutta Gupta, G. V. Varada, G. S. Agarwal, “Surface plasmons in two-sided corrugated thin films,” Phys. Rev. B 36, 6331–6335 (1987).
[Crossref]

Arakawa, E. T.

Celli, V.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Christy, R. W.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction Entre Photons et Atomes (InterEditions, Paris, 1988).

Da Silva, E. P.

Depine, R. A.

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction Entre Photons et Atomes (InterEditions, Paris, 1988).

Dutta Gupta, S.

S. Dutta Gupta, G. V. Varada, G. S. Agarwal, “Surface plasmons in two-sided corrugated thin films,” Phys. Rev. B 36, 6331–6335 (1987).
[Crossref]

Farias, G. A.

Garcia, N.

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

Goudonnet, J. P.

Greffet, J. J.

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6437–6441 (1988).
[Crossref]

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction Entre Photons et Atomes (InterEditions, Paris, 1988).

Hill, N. R.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Inagaki, T.

Johnson, P. B.

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Lopez, C.

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

Maradudin, A. A.

Marvin, A.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Maystre, D.

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies des réseaux de diffraction: application aux anomalies de plasmons,” J. Opt. (Paris) 8, 165–174 (1977).
[Crossref]

Nevière, M.

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies des réseaux de diffraction: application aux anomalies de plasmons,” J. Opt. (Paris) 8, 165–174 (1977).
[Crossref]

Simon, J. M.

Toigo, F.

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Varada, G. V.

S. Dutta Gupta, G. V. Varada, G. S. Agarwal, “Surface plasmons in two-sided corrugated thin films,” Phys. Rev. B 36, 6331–6335 (1987).
[Crossref]

Wirgin, A.

A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[Crossref]

Yndurain, F. J.

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

J. Opt. (Paris) (1)

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies des réseaux de diffraction: application aux anomalies de plasmons,” J. Opt. (Paris) 8, 165–174 (1977).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

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

Opt. Acta (1)

A. Wirgin, “On Rayleigh’s theory of partially reflecting gratings,” Opt. Acta 28, 1377–1404 (1981).
[Crossref]

Phys. Rev. B (5)

S. Dutta Gupta, G. V. Varada, G. S. Agarwal, “Surface plasmons in two-sided corrugated thin films,” Phys. Rev. B 36, 6331–6335 (1987).
[Crossref]

P. B. Johnson, R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

C. Lopez, F. J. Yndurain, N. Garcia, “Iterative series for calculating the scattering of waves from a hard corrugated surface,” Phys. Rev. B 18, 970–972 (1978).
[Crossref]

J. J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6437–6441 (1988).
[Crossref]

F. Toigo, A. Marvin, V. Celli, N. R. Hill, “Optical properties of rough surfaces: general theory and the small roughness limit,” Phys. Rev. B 15, 5618–5626 (1977).
[Crossref]

Other (1)

C. Cohen-Tannoudji, J. Dupont-Roc, G. Grynberg, Processus d’Interaction Entre Photons et Atomes (InterEditions, Paris, 1988).

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Figures (10)

Fig. 1
Fig. 1

Scattering geometry.

Fig. 2
Fig. 2

Convergence of the total energy E and efficiencies ρ0, α0, and α1 as a function of the number of iterations N using the ISS. The parameters of the calculations are h/λ = 0.1, d/λ = 1.6, = 2.25, θi = 0°, φi = 0°, polarization p.

Fig. 3
Fig. 3

Convergence of the total energy E and efficiencies ρ0, α0, and α1 as a function of the number of iterations N using the ISS. The parameters of the calculations are the same as in Fig. 2 except that h/λ = 0.24.

Fig. 4
Fig. 4

Convergence of the total energy E and efficiencies ρ0, α0, and α1 as a function of the number of iterations N using the ISS. The parameters of the calculations are the same as in Fig. 2 except that h/λ = 0.30.

Fig. 5
Fig. 5

Reflectivity ρ, efficiency of order zero ρ0, absorptivity α, and total energy E computed with the RISS and the ISS for a sinusoidal gold grating. The parameters are h/d = 0.025, = −5.28 + i1.48, λ/d = 0.5 (—, ISS; ×, RISS).

Fig. 6
Fig. 6

Resonant absorption by a sinusoidal silver grating in conical diffraction. The parameters are h = 9 nm, = −18.06 + i0.51, d = 2186 nm, λ = 632.8 nm, Φ = 40°.

Fig. 7
Fig. 7

Resonant absorption by a sinusoidal grating. The total absorption occurs for h/d = 0.0105. The parameters used in this calculation are λ = 0.5 μm, h/d = 0.006, d = 0.41667 μm, = −8.2344 + i0.287.

Fig. 8
Fig. 8

Convergence of the total energy E, with efficiencies ρ0 and α0 for the ISS and the RISS as a function of the number of iterations N. The parameters of the calculations are the same as in Fig. 7.

Fig. 9
Fig. 9

Convergence of the total energy E, with efficiencies ρ0 and α0 for the ISS and the RISS as a function of the number of iterations N. The parameters of the calculations are the same as in Fig. 7, except that h/d = 0.04.

Fig. 10
Fig. 10

Comparison of the domain of convergence of the ISS and the RFS: (a) the s-polarized case, λ = 1 μm, = 2.1, d = 2 μm, θ = 10°; (b) the p-polarized case, λ = 1 μm, = 2.1, d = 2 μm, θ = 10°.

Tables (8)

Tables Icon

Table 1 Comparison between the Iterative-Series Solution, the Rearranged Iterative-Series Solution, and the Rayleigh Fourier Solutiona

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Table 2 Convergence Domain of the Iterative-Series Solution for θi = 0°, ϕi = 0°, = 2.25, Polarization pa

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Table 3 Convergence Domain of the Iterative-Series Solution for θi = 0°, θi = 0°, = 2.25, Polarization sa

Tables Icon

Table 4 Convergence Domain of the Iterative-Series Solution for θi = 80°, ϕi = 0, = 2.25, Polarization pa

Tables Icon

Table 5 Convergence Domain of the Iterative-Series Solution for θi = 0°, ϕi = 0°, = −18.05 + i0.51, Polarization pa

Tables Icon

Table 6 Convergence Domain of the Rearranged Iterative-Series Solution for θi = 0°, ϕi = 0°, = −18.05 + i0.51, Polarization pa

Tables Icon

Table 7 Convergence Domain of the Iterative-Series Solution for θi = 0°, ϕi = 0°, = −5.28 + i1.48, Polarization pa

Tables Icon

Table 8 Convergence Domain of the Rearranged Iterative-Series Solution for θi = 0°, ϕi = 0°, = −5.28 + i1.48, Polarization pa

Equations (36)

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S ( x ) = h cos ( 2 π x / d ) .
E i ( r , t ) = Re { E i exp [ i ( k i · r ω t ) ] } with k i = ( κ i , γ i ) .
γ 2 = ω 2 c 2 κ 2 ,
E t ( r ) = e t ( κ ) exp ( i κ · r i γ t z ) d κ ,
γ t 2 + κ 2 = ω 2 c 2 , Im ( γ t ) > 0.
e t ( κ x , κ y ) = m = e t ( m ) δ ( κ x κ x i m κ g ) δ ( κ y κ y i ) ,
E r ( r ) = e r ( κ ) exp ( i κ · r + i γ r z ) d κ ,
e r ( κ x , κ y ) = m = e r ( m ) δ ( κ x κ x i m κ g ) δ ( κ y κ y i )
γ r 2 + κ 2 = ω 2 c 2 , Im ( γ r ) > 0.
e t ( m ) = n = 0 e t ( n ) ( m ) n ! ,
e r ( m ) = n = 0 e r ( n ) ( m ) n ! .
[ e t , s ( 0 ) e t , p ( 0 ) ] = [ 2 γ r γ r + γ t 0 0 2 n γ r γ r + γ t ] [ e i , s ( 0 ) e i , p ( 0 ) ] .
[ e ( m ) t , s ( n ) e ( m ) t , p ( n ) ] = ( γ t γ r ) q = 1 n ( n q ) i q h q 2 q k = 0 q ( q k ) × ( γ r γ t ) q 1 M ( m , m + q 2 k ) [ e ( m + q 2 k ) t , s ( n q ) e ( m + q 2 k ) t , p ( n q ) ] ,
κ i = ( κ i x , κ i y ) , κ = ( κ i x + m κ g , κ i y ) , κ = [ κ i x + ( m + q 2 k ) κ g , κ i y ] , k 0 = ω / c , ( n q ) = n ! q ! ( n q ) !
M ( m , m + q 2 k ) = [ κ ˆ · κ ˆ γ t n k 0 κ ˆ · ( z ˆ × κ ˆ ) γ r n k 0 k 2 + γ r γ t κ ˆ · ( z ˆ × κ ˆ ) κ κ + ( κ ˆ · κ ˆ ) γ r γ t κ 2 + γ r γ t ] , N ( m , m + q 2 k ) = [ κ ˆ · κ ˆ γ t n k 0 κ ˆ · ( z ˆ × κ ˆ ) γ r k 0 κ ˆ · ( z ˆ × κ ˆ ) κ κ ( κ ˆ · κ ˆ ) γ r γ t n k 0 2 ] ,
n 2 = , Im ( n ) > 0.
[ e ( m ) r , s ( n ) e ( m ) r , p ( n ) ] = ( 1 ) k 0 2 2 γ r q = 0 n ( n q ) ( i ) q h q 2 q k = 0 q ( q k ) × ( γ r + γ t ) q 1 N ( m , m + q 2 k ) [ e ( m + q 2 k ) t , s ( n q ) e ( m + q 2 k ) t , p ( n q ) ] .
ρ m = | e r , s ( m ) | 2 + | e r , p ( m ) | 2 | e i | 2 Re ( γ r m ) Re ( γ i ) ,
α m = | e t , s ( m ) | 2 + | e t , p ( m ) | 2 | e i | 2 Re ( γ t m ) R e ( γ i ) .
m ( α m + ρ m ) = 1
α = 1 Re { ρ m } .
u t ( x , z ) = m = u t ( m ) exp [ i ( κ i + m κ g ) x ] exp [ i γ t ( m ) z ] ,
u r ( x , z ) = m = u r ( m ) exp [ i ( κ i + m κ g ) x ] exp [ i γ r ( m ) z ] .
u r [ x , S ( x ) ] + u i [ x , S ( x ) ] = u t [ x , S ( x ) ] ,
n ( u r + u i ) = 1 n n ( u t ) Γ ,
n u ( x , z ) = ( S x u x u z ) z = S ( x ) [ 1 + ( S x ) 2 ] 1 / 2 ,
α = 1 d cos θ i Im { Γ k 2 0 d u * t [ x , S ( x ) ] ( S x z ) u t [ x , S ( x ) ] } .
u t = n u n exp { i [ α n x γ t n S ( x ) ] } ,
u * t = m u * m exp { i [ α m x γ * tm S ( x ) ] } .
0 d u * t ( S x z ) u t = n , m u n u * m 0 d ( i α n S x + i γ t n ) × exp [ i ( α n α m ) x i ( γ t n γ * tm ) S ( x ) ] d x .
x { exp [ i ( α n α m ) x i ( γ t n γ * tm ) S ( x ) ] } = [ i ( α n α m ) i ( γ t n γ * tm ) S x ] × exp [ i ( α n α m ) x i ( γ t n γ * tm ) S ( x ) ] .
0 d u * t ( S x z ) u t = n , m u n u * m i d [ γ t n + α n ( α n α m ) γ t n γ * tm ] I n m ,
I n m = 1 d 0 d exp [ i ( α n α m ) x i ( γ t n γ * tm ) S ( x ) ] d x ,
α = 1 cos θ i Re { Γ k 2 n , m u n u * m [ γ t n + α n ( α n α m ) γ t n γ * tm ] I n m } .
I n , m = p = 0 N ( i ) p p ! ( γ t n γ * tm ) p S ( p ) ( n m ) ,
S ( p ) ( n ) = 1 d 0 d S p ( x ) exp [ i n 2 π d x ] d x .

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