Abstract

The coordinate transformation method for modeling surface-relief gratings is reformulated with use of the recent results [ J. Opt. Soc. Am. A 13, 1870 ( 1996)] on the Fourier factorization of products that contain discontinuous periodic functions. The matrix operator of the eigenvalue problem in the traditional formulation is modified following the correct Fourier factorization procedures. In addition, a new and simpler matrix operator is derived. Both the modified old operator and the new operator greatly improve the convergence of the coordinate transformation method for gratings whose profiles have sharp edges.

© 1996 Optical Society of America

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  1. J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  2. 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]
  3. E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
    [CrossRef]
  4. S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
    [CrossRef]
  5. E. Popov, M. Nevière, “Surface-enhanced second-harmonics generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
    [CrossRef]
  6. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994);errata, J. Opt. Soc. Am. A 13, 543 (1996).
    [CrossRef]
  7. N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
    [CrossRef]
  8. G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  9. T. W. Preist, N. P. K. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
    [CrossRef]
  10. See L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996),and L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (1996).
    [CrossRef]
  11. J. B. Harris, T. W. Preist, J. R. Sambles, “Differential formalism for multilayer diffraction gratings made with uniaxial materials,” J. Opt. Soc. Am. A 12, 1965–1973 (1995).
    [CrossRef]
  12. K. Knop, “Rigorous diffraction theory for transmission gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  13. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  14. 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]
  15. 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]
  16. 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]
  17. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  18. R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1, Sec. 5, pp. 65–85.
  19. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 3, p. 47,and Chap. 6, p. 119.
  20. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2583–2591 (1993).
    [CrossRef]
  21. G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Appl. Computat. Electromagn. Soc. J. 9, 90–100 (1994).

1996 (4)

1995 (4)

1994 (3)

1993 (2)

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, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2583–2591 (1993).
[CrossRef]

1991 (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

1986 (2)

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
[CrossRef]

1982 (1)

1980 (1)

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

1978 (1)

Bryan-Brown, G. P.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Chandezon, J.

See L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996),and L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

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]

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

Cornet, G.

Cotter, N. P. K.

Dupuis, M. T.

Elston, S. J.

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Gaylord, T. K.

Granet, G.

See L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996),and L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (1996).
[CrossRef]

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]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Guizal, B.

Haggans, C. W.

Harris, J. B.

Knop, K.

Lalanne, P.

Li, L.

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

See L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996),and L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (1996).
[CrossRef]

L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994);errata, J. Opt. Soc. Am. A 13, 543 (1996).
[CrossRef]

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, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2583–2591 (1993).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

Maystre, D.

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]

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

Moharam, M. G.

Morris, G. M.

Nevière, M.

Plumey, J. P.

See L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996),and L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Popov, E.

E. Popov, M. Nevière, “Surface-enhanced second-harmonics generation in nonlinear corrugated dielectrics: new theoretical approaches,” J. Opt. Soc. Am. B 11, 1555–1564 (1994).
[CrossRef]

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 3, p. 47,and Chap. 6, p. 119.

Preist, T. W.

Raoult, G.

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

Sambles, J. R.

Tayeb, G.

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Appl. Computat. Electromagn. Soc. J. 9, 90–100 (1994).

Wrede, R. C.

R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1, Sec. 5, pp. 65–85.

Appl. Computat. Electromagn. Soc. J. (1)

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Appl. Computat. Electromagn. Soc. J. 9, 90–100 (1994).

J. Opt. (Paris) (2)

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

E. Popov, L. Mashev, “Conical diffraction mounting generalization of a rigorous differential method,” J. Opt. (Paris) 17, 175–180 (1986).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (10)

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
[CrossRef]

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]

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]

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]

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-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994);errata, J. Opt. Soc. Am. A 13, 543 (1996).
[CrossRef]

N. P. K. Cotter, T. W. Preist, J. R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995).
[CrossRef]

T. W. Preist, N. P. K. Cotter, J. R. Sambles, “Periodic multilayer gratings of arbitrary shape,” J. Opt. Soc. Am. A 12, 1740–1748 (1995).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2583–2591 (1993).
[CrossRef]

J. B. Harris, T. W. Preist, J. R. Sambles, “Differential formalism for multilayer diffraction gratings made with uniaxial materials,” J. Opt. Soc. Am. A 12, 1965–1973 (1995).
[CrossRef]

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

Phys. Rev. B (1)

S. J. Elston, G. P. Bryan-Brown, J. R. Sambles, “Polarization conversion from diffraction gratings,” Phys. Rev. B 44, 6393–6400 (1991).
[CrossRef]

Pure Appl. Opt. (2)

See L. Li, G. Granet, J. P. Plumey, J. Chandezon, “Some topics in extending the C method to multilayer-coated gratings of different profiles,” Pure Appl. Opt. 5, 141–156 (1996),and L. Li, “Periodic multilayer gratings of arbitrary shape: comment,” J. Opt. Soc. Am. A 13, 1475–1476 (1996).
[CrossRef]

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Other (2)

R. C. Wrede, Introduction to Vector and Tensor Analysis (Dover, New York, 1972), Chap. 1, Sec. 5, pp. 65–85.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962), Chap. 3, p. 47,and Chap. 6, p. 119.

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Figures (6)

Fig. 1
Fig. 1

Grating profile function a(x) for a trapezoidal grating (upper graph) and its derivative a ˙ ( x ) (lower graph).

Fig. 2
Fig. 2

Geometrical demonstration of the continuity of Gu and Gυ and the discontinuity of Gυ and Gu at υ = υ0, where a ˙ ( υ ) is discontinuous. The figures are drawn for an a(υ) that has a caret shape in the neighborhood of υ0. The subscripts ± denote the field components and the basis vectors at υ0 ± 0. (a) Contravariant components in covariant basis, (b) covariant components in covariant basis, (c) covariant components in contravariant basis, (d) contravariant components in contravariant basis.

Fig. 3
Fig. 3

Convergence of the sum of diffraction efficiencies by the old formulation (thin curves) and the new formulations (thick curves) of the C method for a grating that has the following parameters: h = d = d3 = 1.7, d1 = d2 = 0.5, n1 = 0 + i5, n2 = 1.0, wavelength λ = 1.0; the angle of incidence is at −1 order Littrow. Solid curves, TM polarization; dashed curves, TE polarization.

Fig. 4
Fig. 4

Same as in Fig. 3, except that h = d = 2.0, d1 = 0.25, d2 = 1.0, d3 = 1.25, n1 = 0.3 + i7.0, and the angle of incidence is at 45°.

Fig. 5
Fig. 5

Same as in Fig. 4, except that n1 = 1.5. The data points, open for TE and filled for TM, were obtained by substituting a ˙ 2 for a ˙ a ˙ in Eq. (30).

Fig. 6
Fig. 6

Same as in Fig. 3, except that d1 = d2 = 0.1, n1 = 1.5, and θ = 0.

Tables (3)

Tables Icon

Table 1 Diffraction Efficiencies of Individual Diffraction Orders Computed by the New Formulations for the Case of Fig. 3 (N = 81)

Tables Icon

Table 2 Diffraction Efficiencies of Individual Diffraction Orders Computed by the New Formulations for the Case of Fig. 4 (N = 81)

Tables Icon

Table 3 Diffraction Efficiencies of Individual Diffraction Orders Computed by the New Formulations for the Case of Fig. 5 (N = 81)

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

υ = x , u = y a ( x ) , w = z .
g i j = [ 1 a ˙ 0 a ˙ 1 + a ˙ 2 0 0 0 1 ] .
e ijk j E k = i k 0 μ g g i j H j , e ijk j H k = i k 0 g g i j E j ,
H υ u H u υ = i k 0 E z ,
E z u = i μ k 0 ( H υ a ˙ H u ) ,
E z υ = i μ k 0 [ a ˙ H υ ( 1 + a ˙ 2 ) H u ] ,
E z u = D ( υ ) E z υ + i μ k 0 C ( υ ) H υ ,
H υ u = i μ k 0 { μ k 0 2 E z + υ [ C ( υ ) E z υ ] } + υ [ D ( υ ) H υ ] ,
C ( υ ) = 1 1 + a ˙ 2 ( υ ) , D ( υ ) = a ˙ ( υ ) 1 + a ˙ 2 ( υ ) .
ϕ ϕ m , υ i α m ,
f g n f m n g n
1 i d d u F = M F ,
M = [ D m n α n μ k 0 C m n 1 μ k 0 ( μ k 0 2 δ m n α m C m n α n ) α m D m n ] ,
f ( x 0 + 0 ) f ( x 0 0 ) 0 , g ( x 0 + 0 ) g ( x 0 0 ) 0.
f ( x 0 0 ) g ( x 0 0 ) = f ( x 0 + 0 ) g ( x 0 + 0 ) .
h n ( P ) = m = P P f n m g m ,
h n ( P ) = m = P P 1 f n m ( P ) 1 g m ,
υ = υ + , u = u + + [ a ˙ ( υ 0 + 0 ) a ˙ ( υ 0 0 ) ] υ + + u 0 , w = w + ,
G υ + = x i υ + G i = G υ + [ a ˙ ( υ 0 + 0 ) a ˙ ( υ 0 0 ) ] G u ,
G u + = x i u + G i = G u ,
G υ ( υ 0 + 0 ) G υ ( υ 0 0 ) ,
G u ( υ 0 + 0 ) = G u ( υ 0 0 ) ,
( G υ a ˙ G u ) υ 0 + 0 = ( G υ a ˙ G u ) υ 0 0 .
υ υ 0 + 0 υ υ 0 0 ,
u υ 0 + 0 = u υ 0 0 ,
( υ a ˙ u ) υ 0 + 0 = ( υ a ˙ u ) υ 0 0 .
b υ ± = x ̂ + a ˙ ( υ 0 ± 0 ) ŷ , b u ± = b u = ŷ ,
b ± υ = b υ = x ̂ , b ± u = ŷ a ˙ ( υ 0 ± 0 ) x ̂ ,
d H υ n d u i α n H u n = i k 0 E z n ,
d E z n d u = i μ k 0 ( H υ n m [ a ˙ ] n m H u m ) ,
E z υ = i μ k 0 [ H u + a ˙ ( H υ a ˙ H u ) ] ,
α n E z n = μ k 0 [ H u n + m [ a ˙ ] n m ( H υ m p [ a ˙ ] m p H u p ) ] .
M = [ a ˙ α μ k 0 1 μ k 0 ( μ k 0 2 α α ) α a ˙ ] ,
= ( 1 + a ˙ a ˙ ) 1 .
( 2 x 2 + 2 y 2 + μ k 0 2 ) E z = 0.
[ ( υ a ˙ u ) 2 + 2 u 2 + μ k 0 2 ] E z = 0 ,
m [ l ( i α n δ n l [ a ˙ ] n l d d u ) ( i α l δ l m [ a ˙ ] l m d d u ) + ( d 2 d u 2 + μ k 0 2 ) δ n m ] E z m = 0 .
Q n = 1 i d E z n d u ,
M 1 i d V d u = V ,
M = [ 1 β 2 ( α a ˙ + a ˙ α ) 1 β 2 ( 1 + a ˙ a ˙ ) 1 0 ]
H υ = i μ k 0 [ a ˙ E z υ ( 1 + a ˙ 2 ) E z u ] = i μ k 0 [ E z u + a ˙ ( E z υ a ˙ E z u ) ] .
H υ n = 1 μ k 0 m ( β n 2 λ δ n m + α n [ a ˙ ] n m ) E z m ,

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