Abstract

The diffraction computation of crossed gratings is very slow compared with that of line-space gratings of the same size when using a modal method such as rigorous coupled wave analysis (RCWA) or the Chandezon coordinate transformation method. It is well known that the main bottleneck in terms of computation speed is the solution of an eigenproblem for each RCWA slice or interface in the case of the C-method. Even if the crossed grating contains layers that are periodic only in one direction, usually the full 2D problem has to be solved for this layer in order to connect it to the full system solution. In this paper, a computation schema is presented that takes advantage of the 1D periodicity of layers inside a 2D multilayer grating. This results in a considerable acceleration of the formulation and solution of the eigenproblem for these layers. With this new computation schema the total time required for 1D layers in a 2D layer stack can be greatly reduced.

© 2009 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  10. http://www.netlib.org/lapack/

2003 (2)

B. Gralak, M. de Doof, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655-660 (2003).
[Crossref]

1998 (1)

1997 (1)

1996 (1)

1995 (1)

1981 (1)

I. C. Botten, M. S. Craig, R. C. McPhedran, L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 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).

Adams, L.

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

Andrewartha, J. R.

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

Bischoff, J.

J. Bischoff, “Beiträge zur theoretischen und experimentellen Untersuchung der Lichtbeugung an mikrostrukturierten Mehrschichtsystemen,” Habilitation thesis (Technical University Ilmenau, Germany, 2000).

Botten, I. C.

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

Chandezon, J.

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).

Craig, M. S.

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

de Doof, M.

B. Gralak, M. de Doof, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

Enoch, S.

B. Gralak, M. de Doof, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

Gralak, B.

B. Gralak, M. de Doof, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

Granet, G.

Grann, E. B.

Li, L.

Maystre, D.

B. Gralak, M. de Doof, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

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, L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).

Moharam, M. G.

Pommet, D. A.

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).

Tayeb, G.

B. Gralak, M. de Doof, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

J. Opt. (Paris) (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).

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

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

Phys. Rev. E (1)

B. Gralak, M. de Doof, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

Other (2)

J. Bischoff, “Beiträge zur theoretischen und experimentellen Untersuchung der Lichtbeugung an mikrostrukturierten Mehrschichtsystemen,” Habilitation thesis (Technical University Ilmenau, Germany, 2000).

http://www.netlib.org/lapack/

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Figures (3)

Fig. 1
Fig. 1

General multilayer crossed grating composed of layers with 1D and 2D periodicity.

Fig. 2
Fig. 2

Ratio (standard over new method) of the total computation time (time gain).

Fig. 3
Fig. 3

Ratio (standard over new method) of the computation time for the eigensolution of the 1D layers; a “theoretical” curve with 0.7 * M 2 is added for reference.

Tables (5)

Tables Icon

Table 1 Formation of the Full-System W Matrix in the Case of Lines Parallel to x 1

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Table 2 Formation of the Full-System W Matrix in the Case of Lines Parallel to x 2

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Table 3 Simulated Amplitudes for TE and TM Polarization in Zeroth-Order Reflection for All Gratings

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Table 4 Averaged Total Computation Time in Seconds for All Gratings: Standard versus New Approach

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Table 5 Averaged Computation Time in Seconds for the Eigensolution of the 1D Layers: Standard versus New Approach

Equations (14)

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cos ζ k 0 3 ( E 1 E 2 ) = i F ( H 1 H 2 ) ,
cos ζ k 0 3 ( H 1 H 2 ) = i G ( E 1 E 2 ) ,
F = [ α [ | ϵ | ] 1 β μ k 0 2 sin ζ μ k 0 2 α [ | ϵ | ] 1 α β [ | ϵ | ] 1 β μ k 0 2 μ k 0 2 sin ζ β [ | ϵ | ] 1 α ] ,
G = [ [ | 1 ϵ | ] 1 μ k 0 2 sin ζ α β α 2 μ k 0 2 ( ϵ cos 2 ζ + [ | 1 ϵ | ] 1 sin 2 ζ ) μ k 0 2 ( ϵ cos 2 ζ + [ | 1 ϵ | ] 1 sin 2 ζ ) β 2 α β [ | 1 ϵ | ] 1 μ k 0 2 sin ζ ] .
1 ϵ m n , m n = 1 p 1 p 1 2 + p 1 2 d x 1 1 ϵ ( x 1 ) 1 exp [ i x 1 ( m m ) 2 π p 1 ] ,
1 ϵ = 1 p 2 p 2 2 + p 2 2 d x 2 1 ϵ ( x 1 , x 2 ) exp [ i x 2 ( n n ) 2 π p 2 ] ,
1 ϵ m n , m n = 1 p 2 p 2 2 + p 2 2 d x 2 1 ϵ ( x 2 ) 1 exp [ i x 2 ( n n ) 2 π p 2 ] ,
1 ϵ m , m = 1 p 1 p 1 2 + p 1 2 d x 1 1 ϵ ( x 1 , x 2 ) exp [ i x 1 ( m m ) 2 π p 1 ] .
ϵ [ ϵ ] and ϵ [ 1 ϵ ] 1 .
F m = [ α m ϵ 1 β μ k 0 2 sin ζ μ k 0 2 α m ϵ 1 α m β ϵ 1 β μ k 0 2 μ k 0 2 sin ζ β ϵ 1 α m ] ,
G m = [ [ 1 ϵ ] 1 μ k 0 2 sin ζ α m β α m 2 μ k 0 2 ( ϵ cos 2 ζ + [ 1 ϵ ] 1 sin 2 ζ ) [ 1 ϵ ] 1 μ k 0 2 β 2 α m β [ 1 ϵ ] 1 μ k 0 2 sin ζ ] .
ϵ [ 1 ϵ ] 1 and ϵ [ ϵ ] .
F n = [ α ϵ 1 β n μ k 0 2 sin ζ μ k 0 2 α ϵ 1 α β n ϵ 1 β n μ k 0 2 μ k 0 2 sin ζ β n ϵ 1 α ] ,
G n = [ [ 1 ϵ ] 1 μ k 0 2 sin ζ α β n α 2 [ 1 ϵ ] 1 μ k 0 2 μ k 0 2 ( ϵ cos 2 ζ + [ 1 ϵ ] 1 sin 2 ζ ) β n 2 α β n [ 1 ϵ ] 1 μ k 0 2 sin ζ ] .

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