Abstract

A new formulation of the Fourier modal method for crossed gratings with symmetry considerations is established by using the group-theoretic approach that we have developed recently. Considering crossed gratings with the C2 symmetry (invariance after rotation about the normal of the mean grating plane through angle π), we present in detail the construction of the new algorithm, illustrate the improved computation efficiency, and discuss its application. It is shown theoretically and numerically that when the grating is Littrow mounted and the truncated reciprocal lattice of the diffracted field also has the C2 symmetry, the maximum effective truncation number of the algorithm is doubled and the computation time is reduced by a factor of 4. The time saving factor is increased to 8 for the special case of normal incidence.

© 2005 Optical Society of America

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References

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  1. R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  2. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  3. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  4. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A, Pure Appl. Opt. 5, 345–355 (2003).
    [CrossRef]
  5. G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A, Pure Appl. Opt. (Suppl.) 4, S145–S149 (2002).
    [CrossRef]
  6. Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]
  7. Ph. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [CrossRef]
  8. C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43–50 (2004).
    [CrossRef]
  9. B. Bai, L. Li, “Reduction of computation time for crossed-grating problems: a group-theoretic approach,” J. Opt. Soc. Am. A 21, 1886–1894 (2004).
    [CrossRef]
  10. W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).
  11. J. V. Smith, Geometrical and Structural Crystallography (Wiley, New York, 1982).
  12. See, for example, J. F. Cornwell, ed., Group Theory in Physics: an Introduction (Academic, San Diego, Calif., 1997), App. C, pp. 299–318.
  13. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  14. See, for example, R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1.
  15. The derivation of matrix Gin Ref. 3, in the case of ζ≠0,depended on the validity of an unproven hypothesis. In Ref. 4a different form of matrix Gwas rigorously derived. However, the numerical examples reported in Ref. 4showed that the two forms of matrix Ghave more or less the same convergence rate. Here for simplicity we use the old form.
  16. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  17. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003).
    [CrossRef]
  18. A. Sentenac, Ph. Lalanne, D. Maystre, “Symmetry properties of the field transmitted by inductive grids,” J. Mod. Opt. 47, 2323–2333 (2000).
    [CrossRef]

2004 (2)

C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43–50 (2004).
[CrossRef]

B. Bai, L. Li, “Reduction of computation time for crossed-grating problems: a group-theoretic approach,” J. Opt. Soc. Am. A 21, 1886–1894 (2004).
[CrossRef]

2003 (2)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A, Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

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

2002 (1)

G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A, Pure Appl. Opt. (Suppl.) 4, S145–S149 (2002).
[CrossRef]

2000 (1)

A. Sentenac, Ph. Lalanne, D. Maystre, “Symmetry properties of the field transmitted by inductive grids,” J. Mod. Opt. 47, 2323–2333 (2000).
[CrossRef]

1997 (2)

1996 (3)

1994 (1)

1993 (1)

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bai, B.

Bräuer, R.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Falter, C.

W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).

Granet, G.

G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A, Pure Appl. Opt. (Suppl.) 4, S145–S149 (2002).
[CrossRef]

Lalanne, Ph.

A. Sentenac, Ph. Lalanne, D. Maystre, “Symmetry properties of the field transmitted by inductive grids,” J. Mod. Opt. 47, 2323–2333 (2000).
[CrossRef]

Ph. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
[CrossRef]

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Lemercier-Lalanne, D.

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Li, L.

Ludwig, W.

W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).

Maystre, D.

A. Sentenac, Ph. Lalanne, D. Maystre, “Symmetry properties of the field transmitted by inductive grids,” J. Mod. Opt. 47, 2323–2333 (2000).
[CrossRef]

Noponen, E.

Plumey, J.

G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A, Pure Appl. Opt. (Suppl.) 4, S145–S149 (2002).
[CrossRef]

Sentenac, A.

A. Sentenac, Ph. Lalanne, D. Maystre, “Symmetry properties of the field transmitted by inductive grids,” J. Mod. Opt. 47, 2323–2333 (2000).
[CrossRef]

Smith, J. V.

J. V. Smith, Geometrical and Structural Crystallography (Wiley, New York, 1982).

Turunen, J.

Wrede, R. C.

See, for example, R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1.

Zhou, C.

C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43–50 (2004).
[CrossRef]

J. Mod. Opt. (2)

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

A. Sentenac, Ph. Lalanne, D. Maystre, “Symmetry properties of the field transmitted by inductive grids,” J. Mod. Opt. 47, 2323–2333 (2000).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (2)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A, Pure Appl. Opt. 5, 345–355 (2003).
[CrossRef]

C. Zhou, L. Li, “Formulation of Fourier modal method of symmetric crossed gratings in symmetric mountings,” J. Opt. A, Pure Appl. Opt. 6, 43–50 (2004).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (Suppl.) (1)

G. Granet, J. Plumey, “Parametric formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. A, Pure Appl. Opt. (Suppl.) 4, S145–S149 (2002).
[CrossRef]

J. Opt. Soc. Am. A (7)

Opt. Commun. (1)

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Other (5)

See, for example, R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1.

The derivation of matrix Gin Ref. 3, in the case of ζ≠0,depended on the validity of an unproven hypothesis. In Ref. 4a different form of matrix Gwas rigorously derived. However, the numerical examples reported in Ref. 4showed that the two forms of matrix Ghave more or less the same convergence rate. Here for simplicity we use the old form.

W. Ludwig, C. Falter, Symmetries in Physics: Group Theory Applied to Physical Problems (Springer, Berlin, 1988).

J. V. Smith, Geometrical and Structural Crystallography (Wiley, New York, 1982).

See, for example, J. F. Cornwell, ed., Group Theory in Physics: an Introduction (Academic, San Diego, Calif., 1997), App. C, pp. 299–318.

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Figures (6)

Fig. 1
Fig. 1

Crossed grating with the C 2 symmetry is illuminated by a linearly polarized plane wave. An oblique Cartesian coordinate system Ox 1 x 2 x 3 is established so that its x 1 and x 2 axes are along the two periodic directions and the x 3 axis is parallel to the normal of the grating plane. A rectangular coordinate system Oxyz is also given, whose x and z axes coincide with the x 1 and x 3 axes, respectively.

Fig. 2
Fig. 2

Top view of a crossed grating with the C 2 symmetry. The covariant basis vectors b ρ and the contravariant basis vectors b ρ of the oblique coordinate system are also given.

Fig. 3
Fig. 3

Schematic illustration of the two symmetry modes of a crossed grating with the C 2 symmetry. In each mode, the symmetrical distribution of the electric vectors projected onto the x 1 x 2 plane is shown by the arrows. The solid arrows indicate the vectors with the same x 3 components, whose sign is opposite to that indicated by the open arrow.

Fig. 4
Fig. 4

Four cases of truncated reciprocal lattices with the C 2 symmetry when a parallelogramic truncation scheme is adopted. The two sides of the truncated zone are parallel to the contravariant basis vectors of the oblique coordinate system. The points within the dashed boundary in (d) represent schematically the Fourier items retained in numerical computation.

Fig. 5
Fig. 5

Convergence of the (-1, -2)th, (-2, -1)th, and (+1, 0)th reflected orders of the metallic crossed grating with the C 2 symmetry, which are computed by both NA and OA.

Fig. 6
Fig. 6

Comparison of the computation times of NA and OA.

Tables (2)

Tables Icon

Table 1 Character Table of Point Group C 2 a

Tables Icon

Table 2 Four Cases of Truncated Reciprocal Lattices with C 2 Symmetry a

Equations (59)

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a [ 1 ] = 2 2 ( 1 ,   1 ) , a [ 2 ] = 2 2 ( 1 ,   - 1 ) .
T ( e ) = 1 1 , T ( c 2 ) = 1 - 1 .
M co ( e ) = M ctr ( e ) = 1 1 1 ,
M co ( c 2 ) = M ctr ( c 2 ) = - 1 - 1 1 ,
M co ( g ) E [ i ] ( M ctr ( g ) - 1 r ) = T ii ( g ) E [ i ] ( r )
( i = 1 ,   2 ;   g = e ,   c 2 ) ,
E 1 [ 1 ] ( x 1 ,   x 2 ,   x 3 ) = - E 1 [ 1 ] ( - x 1 ,   - x 2 ,   x 3 ) ,
E 2 [ 1 ] ( x 1 ,   x 2 ,   x 3 ) = - E 2 [ 1 ] ( - x 1 ,   - x 2 ,   x 3 ) ,
E 3 [ 1 ] ( x 1 ,   x 2 ,   x 3 ) = E 3 [ 1 ] ( - x 1 ,   - x 2 ,   x 3 ) ,
E 1 [ 2 ] ( x 1 ,   x 2 ,   x 3 ) = E 1 [ 2 ] ( - x 1 ,   - x 2 ,   x 3 ) ,
E 2 [ 2 ] ( x 1 ,   x 2 ,   x 3 ) = E 2 [ 2 ] ( - x 1 ,   - x 2 ,   x 3 ) ,
E 3 [ 2 ] ( x 1 ,   x 2 ,   x 3 ) = - E 3 [ 2 ] ( - x 1 ,   - x 2 ,   x 3 ) .
E * ( r ) = t 1 E [ 1 ] ( r ) + t 2 E [ 2 ] ( r ) ,
t i = ( a * ,   a [ i ] ) = 2 2   a ( i = 1 ,   2 ) .
E ρ [ 1 ] ( r ) = E ρ [ 1 , 1 ] ( r ) + E ρ [ 1 , 2 ] ( r ) ( ρ = 1 ,   2 ) ,
E ρ [ 1 , j ] ( r ) = m , n E ρ mn [ 1 , j ] ( x 3 ) exp [ i ( α m [ , j ] x 1 + β n [ , j ] x 2 ) ]
( j = 1 ,   2 ) ,
α m [ , j ] = α 0 [ , j ] + mK 1 , β n [ , j ] = β 0 [ , j ] + nK 2 ;
K 1 = 2 π / d 1 , K 2 = 2 π / d 2 ;
α 0 [ , 1 ] = - α 0 [ , 2 ] = k ( + 1 ) sin   θ   cos   φ α 0 ,
β 0 [ , 1 ] = - β 0 [ , 2 ] = k ( + 1 ) sin   θ   sin ( φ + ζ ) β 0 ;
E ρ [ 1 ] ( r ) = m , n { E ρ mn [ 1 , 1 ] ( x 3 ) exp [ i ( α m x 1 + β n x 2 ) ] + E ρ ( - m ) ( - n ) [ 1 , 2 ] ( x 3 ) exp [ i ( - α m x 1 - β n x 2 ) ] } ,
α 0 = pK 1 / 2 , β 0 = qK 2 / 2 ,
L 1 - = - L 1 - s ,
L 1 + = L 1 - s if α 0 = sK 1 L 1 - s - 1 if α 0 = s + 1 2 K 1
- α m = α - 2 s - m , - β n = β - 2 t - n .
E ρ [ 1 ] ( r ) = m = L 1 - L 1 + n = L 2 - L 2 + E ρ mn [ 1 ] ( x 3 ) exp [ i ( α m x 1 + β n x 2 ) ] ,
E ρ [ 1 ] ( - x 1 ,   - x 2 ,   x 3 )
= m = L 1 - L 1 + n = L 2 - L 2 + E ρ mn [ 1 ] ( x 3 ) exp [ i ( - α m x 1 - β n x 2 ) ]
= m = L 1 - L 1 + n = L 2 - L 2 + E ρ ( - 2 s - m ) ( - 2 t - n ) [ 1 ] ( x 3 ) × exp [ i ( α m x 1 + β n x 2 ) ] .
E ρ mn [ 1 ] ( x 3 ) = - E ρ ( - 2 s - m ) ( - 2 t - n ) [ 1 ] ( x 3 ) ( ρ = 1 ,   2 ) .
I [ 1 , 2 ] = M co ( c 2 ) I [ 1 , 1 ] ,
E ρ [ 1 ] ( r ) = I ρ [ 1 , 1 ] exp [ i ( α 0 x 1 + β 0 x 2 - γ 00 ( + 1 ) x 3 ) ] - I ρ [ 1 , 1 ] exp [ i ( α - 2 s x 1 + β - 2 t x 2 - γ ( - 2 s ) ( - 2 t ) ( + 1 ) x 3 ) ] + m = L 1 - L 1 + n = L 2 - L 2 + R ρ mn [ 1 ] exp [ i ( α m x 1 + β n x 2 + γ mn ( + 1 ) x 3 ) ] ( ρ = 1 ,   2 ;   x 3 > h ) ,
E ρ [ 1 ] ( r ) = m = L 1 - L 1 + n = L 2 - L 2 + T ρ mn [ 1 ] exp [ i ( α m x 1 + β n x 2 - γ mn ( - 1 ) x 3 ) ] ( ρ = 1 ,   2 ;   x 3 < 0 ) ,
I 1 [ 1 , 1 ] = 2 2   ( cos   ψ   cos   θ   cos   φ + sin   ψ   sin   φ ) ,
I 2 [ 1 , 1 ] = 2 2   [ cos   ψ   cos   θ   sin ( ζ + φ ) - sin   ψ   cos ( ζ + φ ) ] ;
( α m 2 + β n 2 - 2 α m β n sin   ζ ) sec 2   ζ + γ mn ( ± 1 ) 2 = k ( ± 1 ) 2 .
Re [ γ mn ( ± 1 ) ] + Im [ γ mn ( ± 1 ) ] > 0 .
( FG - μ k 0 2 cos 2   ζ   γ 2 ) E 1 j [ 1 ] E 2 j [ 1 ] = 0 .
F
= - μ k 0 2 sin   ζ δ ij + U ˜ 11 ( i ,   j ) μ k 0 2 δ ij + U ˜ 12 ( i ,   j ) - μ k 0 2 δ ij + U ˜ 21 ( i ,   j ) μ k 0 2 sin   ζ δ ij + U ˜ 22 ( i ,   j ) ,
G
= - α i β i δ ij + V ˜ 11 ( i ,   j ) α i 2 δ ij + V ˜ 12 ( i ,   j ) - β i 2 δ ij + V ˜ 21 ( i ,   j ) α i β i δ ij + V ˜ 22 ( i ,   j ) ,
W ˜ pq ( i ,   j ) = W pq ( i ,   j ) - W pq ( i ,   j ¯ )
( W = U ,   V ;   p ,   q = 1 ,   2 ) ,
U 11 ( i ,   j ) = α i ε i , j - 1 β j ,
U 12 ( i ,   j ) = - α i ε i , j - 1 α j ,
U 21 ( i ,   j ) = β i ε i , j - 1 β j ,
U 22 ( i ,   j ) = - β i ε i , j - 1 α j ,
V 11 ( i ,   j ) = μ k 0 2 sin   ζ 1 ε i , j - 1 ,
V 12 ( i ,   j ) = - μ k 0 2 cos 2   ζ ε i , j + sin 2   ζ 1 ε i , j - 1 ,
V 21 ( i ,   j ) = μ k 0 2 cos 2   ζ ε i , j + sin 2   ζ 1 ε i , j - 1 ,
V 22 ( i ,   j ) = - μ k 0 2 sin   ζ 1 ε i , j - 1 .
E ρ mn [ 2 ] ( x 3 ) = E ρ ( - 2 s - m ) ( - 2 t - n ) [ 2 ] ( x 3 ) ( ρ = 1 ,   2 ) ,
E ρ [ 2 ] ( r ) = I ρ [ 2 , 1 ] exp [ i ( α 0 x 1 + β 0 x 2 - γ 00 ( + 1 ) x 3 ) ] + I ρ [ 2 , 1 ] exp [ i ( α - 2 s x 1 + β - 2 t x 2 - γ ( - 2 s ) ( - 2 t ) ( + 1 ) x 3 ) ] + m = L 1 - L 1 + n = L 2 - L 2 + R ρ mn [ 2 ] exp [ i ( α m x 1 + β n x 2 + γ mn ( + 1 ) x 3 ) ] ( ρ = 1 ,   2 ;   x 3 > h ) ,
W ˜ pq ( i ,   j ) = W pq ( i ,   j ) + W pq ( i ,   j ¯ )
( W = U , V ;   p ,   q = 1 , 2 ) .
E ρ mn * = 2 2   a ( E ρ mn [ 1 ] + E ρ mn [ 2 ] ) ( ρ = 1 ,   2 ) ,
lim L 1 , L 2   2 ( 2 L 1 L 2 + L 1 + L 2 + 1 ) 3 [ ( 2 L 1 + 1 ) ( 2 L 2 + 1 ) ] 3 = 1 4 ,

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