Abstract

The recent reformulation of the coupled-wave method by Lalanne and Morris [ J. Opt. Soc. Am. A 13, 779 ( 1996)] and by Granet and Guizal [ J. Opt. Soc. Am. A 13, 1019 ( 1996)], which dramatically improves the convergence of the method for metallic gratings in TM polarization, is given a firm mathematical foundation in this paper. The new formulation converges faster because it uniformly satisfies the boundary conditions in the grating region, whereas the old formulations do so only nonuniformly. Mathematical theorems that govern the factorization of the Fourier coefficients of products of functions having jump discontinuities are given. The results of this paper are applicable to any numerical work that requires the Fourier analysis of products of discontinuous periodic functions.

© 1996 Optical Society of America

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References

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  1. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  3. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  4. P. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  5. G. Granet, 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]
  6. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  7. L. Li, “Fourier factorization of a product of discontinuous periodic functions,” submitted to SIAM J. Anal. Math.
  8. A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, 1977), Vol. 1, Chap. 4, Sec. 8, p. 159.
  9. G. H. Hardy, Divergent Series (Oxford U. Press, London, 1949), Chap. 10, Secs. 12–15, pp. 239–246.
  10. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
    [CrossRef]
  11. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  12. L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A (to be published).
  13. See, for example, G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 5, Sec. 7, pp. 125–135, or G. Heinig, K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators (Birkhäuser Verlag, Basel, Switzerland, 1984), Chap. 1, pp. 14–33, and the references therein.

1996 (2)

1995 (1)

1993 (1)

1982 (2)

1978 (1)

Chandezon, J.

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A (to be published).

Cornet, G.

Dupuis, M. T.

Gaylord, T. K.

Golub, G. H.

See, for example, G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 5, Sec. 7, pp. 125–135, or G. Heinig, K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators (Birkhäuser Verlag, Basel, Switzerland, 1984), Chap. 1, pp. 14–33, and the references therein.

Granet, G.

Grann, E. B.

Guizal, B.

Haggans, C. W.

Hardy, G. H.

G. H. Hardy, Divergent Series (Oxford U. Press, London, 1949), Chap. 10, Secs. 12–15, pp. 239–246.

Knop, K.

Lalanne, P.

Li, L.

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[CrossRef]

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A (to be published).

L. Li, “Fourier factorization of a product of discontinuous periodic functions,” submitted to SIAM J. Anal. Math.

Maystre, D.

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Van Loan, C. F.

See, for example, G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 5, Sec. 7, pp. 125–135, or G. Heinig, K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators (Birkhäuser Verlag, Basel, Switzerland, 1984), Chap. 1, pp. 14–33, and the references therein.

Vincent, P.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[CrossRef]

Zygmund, A.

A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, 1977), Vol. 1, Chap. 4, Sec. 8, p. 159.

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Other (6)

L. Li, “Fourier factorization of a product of discontinuous periodic functions,” submitted to SIAM J. Anal. Math.

A. Zygmund, Trigonometric Series (Cambridge U. Press, Cambridge, 1977), Vol. 1, Chap. 4, Sec. 8, p. 159.

G. H. Hardy, Divergent Series (Oxford U. Press, London, 1949), Chap. 10, Secs. 12–15, pp. 239–246.

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 101–121.
[CrossRef]

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A (to be published).

See, for example, G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 5, Sec. 7, pp. 125–135, or G. Heinig, K. Rost, Algebraic Methods for Toeplitz-like Matrices and Operators (Birkhäuser Verlag, Basel, Switzerland, 1984), Chap. 1, pp. 14–33, and the references therein.

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Figures (4)

Fig. 1
Fig. 1

Periodic, piecewise-constant medium. The periodicity of the permittivity is d, and its discontinuities are located at x = ±d1/2.

Fig. 2
Fig. 2

Graphs of ΦM(x) in the neighborhood of x = 0 for (a) M = 10, (b) M = 100, and (c) M = 1000. Note the change of scale for the horizontal axes.

Fig. 3
Fig. 3

(a) Graph of h(M)(x) that is Fourier factorized by the finite Laurent rule, with f(x) given by Eq. (24) g(x) = 1/f(x), and M = 200. (b) Enlarged view of Fig. 3(a) in the neighborhood of x = π/2. The straight horizontal line in Fig. 3(b) is obtained by the inverse rule.

Fig. 4
Fig. 4

(a) Schematic representations of functions f(x) and g(x) in Eqs. (24) and (25) and their product h(x) in order of decreasing line thickness. Here a = 6 and b = 2. (b) Function h(M)(x), with M = 200, in the neighborhood of x = π/2. The oscillatory curve is obtained by Laurent’s rule, and the nonoscillatory line is obtained by the inverse rule.

Equations (36)

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1 i d H z n d y = - k 0 m n - m E x m ,
1 i d E x n d y = - k 0 μ 0 H z n + α n k 0 m ( 1 ) n - m α m H z m ,
d 2 H z n d y 2 = m n - m p [ α m ( 1 ) m - p α p - μ 0 k 0 2 δ m p ] H z p .
1 i d H z n d y = - k 0 m 1 n m - 1 E x m ,
1 i d E x n d y = - k 0 μ 0 H z n + α n k 0 m n m - 1 α m H z m ,
d 2 H z n d y 2 = m 1 n m - 1 p ( α m m p - 1 α p - μ 0 k 0 2 δ m p ) H z p ,
d 2 H z n d y 2 = m n - m p ( α m m p - 1 α p - μ 0 k 0 δ m p ) H z p .
h ( x ) = f ( x ) g ( x )
U f = { x j f ( x j + 0 ) f ( x j - 0 ) ,             j = 1 , 2 , }
U f g = U f U g
h ( x p - 0 ) = h ( x p + 0 )             ( x p U f g ) ,
f ^ j = f ( x j + 0 ) - f ( x j - 0 ) ,
h n = m = - + f n - m g m .
h ( x ) = n = - + h n exp ( i n x ) = n = - + m = - + f n - m g m exp ( i n x ) .
h ( x ) = lim N n = - N N ( lim M m = - M M f n - m g m ) exp ( i n x ) .
Laurent ' s rule :             h n ( M ) = m = - M M f n - m g m ,
h ( M ) ( x ) = n = - M M h n ( M ) exp ( i n x ) ,
h M ( x ) = n = - M M h n exp ( i n x ) .
h ( ) ( x ) = h ( x ) .
h ( M ) ( x ) = h M ( x ) - x p U f g f ^ p g ^ p 2 π 2 Φ M ( x - x p ) + o ( 1 ) ,
Φ M ( x ) = n = 1 M cos n x n m > M 1 m - n .
lim M Φ M ( x ) = 0             ( x 0 ) ,
lim M Φ M ( 0 ) = π 2 4 .
Inverse Rule :             h n ( M ) = m = - M M 1 f n m ( M ) - 1 g m
f ^ p g ^ p = - h ( x p ) ( 1 - α ) 2 α .
f ( x ) = { a a 2 , x < π 2 π 2 < x π ,             ( a 0 ) ,
g ( x ) = { b ( 1 - x π ) 2 b ( 1 - x π ) , x < π 2 π 2 < x π ,             ( b 0 ) .
1 i H z y = - k 0 E x ,
1 i E x y = - k 0 μ 0 H z - 1 k 0 x ( 1 H z x ) ,
- 2 H z y 2 = [ x ( 1 H z x ) + μ 0 k 0 2 H z ] .
- 2 E z y 2 = 2 E z x 2 + μ 0 k 0 2 E z .
k z 2 E x - 2 E x y 2 = x [ 1 x ( E x ) ] + μ 0 k 0 2 E x ,
2 E x n y 2 = k z 2 E x n + m ( α n n m - 1 α m ) - μ 0 k 0 2 δ n m ) p 1 m p - 1 E x p .
l = - M M m - l ( 1 ) l - n = δ m n + Δ m n ,
Δ m n = l > M m - l ( 1 ) l - n .
( M ) - 1 1 ( M ) ,             M .

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