Abstract

The perturbation method is combined with the Rigorous Coupled Wave Analysis (RCWA) to enhance its computational speed. In the original RCWA, a grating is approximated by a stack of lamellar gratings and the number of eigenvalue systems to be solved is equal to the number of subgratings. The perturbation method allows to derive the eigensolutions in many layers from the computed eigensolutions of a reference layer provided that the optical and geometrical parameters of these layers differ only slightly. A trapezoidal grating is considered to evaluate the performance of the method.

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References

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  1. S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  2. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
    [CrossRef]
  3. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).
  4. I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).
  5. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  6. Ph. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  7. G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  8. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  9. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  10. J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
  11. D. M. Pai and K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  12. N. Chateau and J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  13. N. P. van der Aa and R. M. M. Mattheij, “Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA,” J. Opt. Soc. Am. A 24, 2692–2700 (2007).
    [CrossRef]
  14. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
    [CrossRef]
  15. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
  16. L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304–313 (1999)
    [CrossRef]
  17. T. Vallius, “Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization,” J. Opt. Soc. Am. A 19, 1555–1562 (2002).
    [CrossRef]
  18. G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102–2108 (2001).
    [CrossRef]
  19. L.D. Landau and E.M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Ed. Pergamon, New York, (1977).
  20. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, T2, Ed. Hermann, Paris, (1994).
    [PubMed]

2007 (1)

2002 (2)

2001 (1)

1999 (1)

1996 (3)

1994 (1)

1993 (1)

1991 (1)

1983 (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

1982 (1)

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

1981 (2)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).

1980 (1)

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).

1978 (1)

1975 (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Adams, J. L.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).

Andrewartha, J. R.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).

Awada, K. A.

Bertoni, H. L.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Botten, I. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

Chandezon, J.

Chateau, N.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, T2, Ed. Hermann, Paris, (1994).
[PubMed]

Craig, M. S.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, T2, Ed. Hermann, Paris, (1994).
[PubMed]

Gaylord, T. K.

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

Gralak, B.

Granet, G.

Guizal, B.

Hugonin, J.-P.

Knop, K.

Lalanne, Ph.

Laloë, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, T2, Ed. Hermann, Paris, (1994).
[PubMed]

Landau, L.D.

L.D. Landau and E.M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Ed. Pergamon, New York, (1977).

Li, L.

Lifschitz, E.M.

L.D. Landau and E.M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Ed. Pergamon, New York, (1977).

Mattheij, R. M. M.

Maystre, D.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).

McPhedran, R. C.

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).

Moharam, M. G.

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

Morris, G. M.

Nevière, M.

Pai, D. M.

Peng, S. T.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Petit, R.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

Plumey, J.-P.

Popov, E.

Raniriharinosy, K.

Raoult, G.

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

Tamir, T.

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tayeb, G.

Vallius, T.

van der Aa, N. P.

Appl. Opt. (1)

IEEE Trans. Microw. Theory Tech. (1)

S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microw. Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Opt. (Paris) (2)

J. Y. Suratteau, M. Cadilhac, and R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (11)

Ph. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

G. Granet and B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

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

D. M. Pai and K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
[CrossRef]

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

N. P. van der Aa and R. M. M. Mattheij, “Computing shape parameter sensitivity of the field of one-dimensional surface-relief gratings by using an analytical approach based on RCWA,” J. Opt. Soc. Am. A 24, 2692–2700 (2007).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
[CrossRef]

T. Vallius, “Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization,” J. Opt. Soc. Am. A 19, 1555–1562 (2002).
[CrossRef]

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, “Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings,” J. Opt. Soc. Am. A 18, 2102–2108 (2001).
[CrossRef]

Opt. Acta (Lond.) (2)

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta (Lond.) 28, 413–428 (1981).

I. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta (Lond.) 28, 1087–1102 (1981).

Other (2)

L.D. Landau and E.M. Lifschitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed. Ed. Pergamon, New York, (1977).

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Mécanique quantique, T2, Ed. Hermann, Paris, (1994).
[PubMed]

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Figures (3)

Fig. 1
Fig. 1

The grating configuration: two periods are depicted.

Fig. 2
Fig. 2

The staircase approximation: the grating is replaced by a stack of lamellar gratings.

Fig. 3
Fig. 3

Surface grating made of a stack of two dielectric trapezoidal elements. Numerical parameters: λ = 240 nm, d = 480 nm, b1 = 240 nm, b2 = 120 nm, b3 = 80 nm, h1 = 80 nm, h2 = 200 nm, θ = 75°, n1 = n3 = 1.51309, n2 = 1.

Tables (8)

Tables Icon

Table 1 Convergence of the RCWA with respect of the truncation order M Results for the grating described by Figure 3 with (Nc = 220)

Tables Icon

Table 2 Convergence of the RCWA with respect of the number of layers Nc Results for the grating described by Figure 3 with 30 Fourier harmonics (M = 30)

Tables Icon

Table 3 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 4 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 5 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 6 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 7 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Tables Icon

Table 8 Results for the grating described by Figure 3 with 220 layers (Nc = 220) for each trapezoidal element and 30 Fourier harmonics (M = 30)

Equations (38)

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TE : ( x x + y y + k 2 ν 2 ( x , y ) ) E z = 0 ,
TM : ( x 1 ν 2 ( x , y ) x + y 1 ν 2 ( x , y ) y + k 2 ) H z = 0 .
ν j ( x ) = { n 1 , x [ d l j 2 , d + l j 2 ] , n 2 , x [ 0 , d l j 2 ] [ d + l j 2 , d ] .
F ( x , y ) = 1 k 2 y 2 F ( x , y ) ,
TE : = 1 k 2 x 2 + ν 2 ( x ) ,
TM : = 1 k 2 ν 2 ( x ) x 1 ν 2 ( x ) x + ν 2 ( x ) .
F ( x , y ) = ψ ( x ) e ± i k r y ,
ψ ( x ) = r 2 ψ ( x ) ,
ψ ( x + d ) = e i k α d ψ ( x ) .
Im ( r ) < 0 or r > 0 if r is real .
TE : < f , g > E = 1 d x 0 x 0 + d f ¯ ( x ) g ( x ) d x ,
TM : < f , g > H = 1 d x 0 x 0 + d 1 ν 2 ( x ) f ¯ ( x ) g ( x ) d x .
TE : = 1 k 2 d 2 d x 2 + ν 2 ¯ ( x ) ,
TM : = 1 k 2 ν 2 ¯ ( x ) d d x 1 ν 2 ¯ ( x ) d d x + ν 2 ¯ ( x ) ,
f ( x ) = m = 1 a m ψ m ( x ) , where a m = < ψ m , f > .
F ( x , y ) = m = 1 ( A m e i k r m y + B m e i k r m y ) ψ m ( x ) .
ψ m ( x ) = n = n = ψ m n e n ( x ) ,
e n ( x ) = e i k α n x , α n = sin θ + n λ d .
TE : L E = α α ,
TM : L H = ˜ 1 [ I α 1 α ] .
F ( x , y ) = m = 1 m = 2 N + 1 ( A m e i k r m y + B m e i k r m y ) ψ m ( x ) ,
ψ m ( x ) = n = N n = N ψ n m e n ( x ) .
( 0 ) ψ p ( 0 ) ( x ) = λ p ( 0 ) ψ p ( 0 ) ( x )
( 0 ) = 𝒫 = ζ 𝒫 ˜ , ζ 1
λ p = n ζ n λ p ( n ) , ψ p = n ζ n ψ p ( n ) .
( ( 0 ) + ζ 𝒫 ˜ ) n ζ n ψ p ( n ) = m , n ζ m + n λ p ( m ) ψ p ( n ) .
( 0 ) ψ p ( 0 ) = λ p ( 0 ) ψ p ( 0 ) ,
( 0 ) ψ p ( 1 ) + 𝒫 ˜ ψ p ( 0 ) = λ p ( 0 ) ψ p ( 1 ) + λ p ( 1 ) ψ p ( 0 ) ,
( 0 ) ψ p ( 2 ) + 𝒫 ˜ ψ p ( 1 ) = λ p ( 0 ) ψ p ( 2 ) + λ p ( 1 ) ψ p ( 1 ) + λ p ( 2 ) ψ p ( 0 ) .
< ψ p , ψ p > = 1 < ψ p ( 0 ) , ψ p > real .
< ψ p ( 0 ) , ψ p ( 0 ) > = 1 ,
< ψ p ( 0 ) , ψ p ( 1 ) > = < ψ p ( 1 ) , ψ p ( 0 ) > = 0 ,
< ψ p ( 0 ) , ψ p ( 2 ) > = < ψ p ( 2 ) , ψ p ( 0 ) > = 1 2 < ψ p ( 1 ) , ψ p ( 1 ) > .
λ p = λ p ( 0 ) + < ψ p ( 0 ) , 𝒫 ψ p ( 0 ) > ,
ψ p = ψ p ( 0 ) + Σ q p < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > λ p ( 0 ) λ q ( 0 ) .
λ p = λ p ( 0 ) + < ψ p ( 0 ) , 𝒫 ψ p ( 0 ) > + Σ q p < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > < ψ p ( 0 ) , 𝒫 ψ q ( 0 ) > λ p ( 0 ) λ q ( 0 )
ψ p = ψ p ( 0 ) + Σ q p < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > λ p ( 0 ) λ q ( 0 ) ψ q ( 0 ) + Σ q p Σ l p < ψ l ( 0 ) , 𝒫 ψ p ( 0 ) > < ψ q ( 0 ) , 𝒫 ψ l ( 0 ) > ( λ p ( 0 ) λ l ( 0 ) ) ( λ p 0 λ q ( 0 ) ) ψ q ( 0 ) Σ q p < ψ p ( 0 ) , 𝒫 ψ p ( 0 ) > < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > ( λ p ( 0 ) λ q ( 0 ) ) 2 ψ q ( 0 ) 1 2 Σ q p < ψ p ( 0 ) , 𝒫 ψ q ( 0 ) > < ψ q ( 0 ) , 𝒫 ψ p ( 0 ) > ( λ p ( 0 ) λ q ( 0 ) ) 2 ψ q ( 0 ) .
ξ ( n p ) = Int ( log 10 | eff RCWA eff PM ( n p ) | ) ,

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