Abstract

The spectroscopic ellipsometry of lamellar gratings made of lossless dielectric materials is studied numerically by using the rigorous coupled-wave method with the use of Li’s Fourier factorization rules [ J. Opt. Soc. Am. A 13, 1870 ( 1996)], which are known to improve the convergence on the analyses of metallic gratings. Numerical results show that the calculation method also provides fast convergence on lossless gratings, and accurate values of the ellipsometric angles are obtained in very short computation times. Moreover, estimation of grating parameters is investigated by using a cost function defined by the average distance on the Poincaré sphere, and it is shown that the computation required for accurate estimation is possible in reasonable computation time.

© 2005 Optical Society of America

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References

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  1. P. Drude, “Bestimmung der optischen Konstanten der Metalle,” Ann. Phys. (Leipzig) 39, 481–554 (1890).
    [CrossRef]
  2. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
  3. J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).
  4. 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]
  5. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  6. L. Li, “Reformulation of the Fourier model method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
    [CrossRef]
  7. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  8. K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
    [CrossRef]
  9. E. Popov, M. Nevière, “Maxwell equation in Fourier space: fast converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  10. K. Watanabe, “Fast converging formulation of differential theory for non-smooth gratings made of anisotropic materials,” Radio Sci. 38, doi: (2003).
    [CrossRef]
  11. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  12. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  13. K. Postava, T. Yamaguchi, “Optical functions of low-k materials for interlayer dielectrics,” J. Appl. Phys. 89, 2189–2193 (2001).
    [CrossRef]

2003 (2)

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

K. Watanabe, “Fast converging formulation of differential theory for non-smooth gratings made of anisotropic materials,” Radio Sci. 38, doi: (2003).
[CrossRef]

2002 (1)

2001 (2)

2000 (1)

1998 (1)

L. Li, “Reformulation of the Fourier model method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

1996 (1)

1982 (1)

1978 (1)

1890 (1)

P. Drude, “Bestimmung der optischen Konstanten der Metalle,” Ann. Phys. (Leipzig) 39, 481–554 (1890).
[CrossRef]

Aoyama, M.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

Drude, P.

P. Drude, “Bestimmung der optischen Konstanten der Metalle,” Ann. Phys. (Leipzig) 39, 481–554 (1890).
[CrossRef]

Gaylord, T. K.

Horie, M.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

Knop, K.

Kovácová, E.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

Li, L.

L. Li, “Reformulation of the Fourier model method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

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

Mistri´k, J.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

Moharam, M. G.

Nevière, M.

Petit, R.

Pištora, J.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

Popov, E.

Postava, K.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

K. Postava, T. Yamaguchi, “Optical functions of low-k materials for interlayer dielectrics,” J. Appl. Phys. 89, 2189–2193 (2001).
[CrossRef]

Šmatko, V.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

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]

Vlcek, J.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

Watanabe, K.

K. Watanabe, “Fast converging formulation of differential theory for non-smooth gratings made of anisotropic materials,” Radio Sci. 38, doi: (2003).
[CrossRef]

K. Watanabe, R. Petit, M. Nevière, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

Yamaguchi, T.

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

K. Postava, T. Yamaguchi, “Optical functions of low-k materials for interlayer dielectrics,” J. Appl. Phys. 89, 2189–2193 (2001).
[CrossRef]

Ann. Phys. (Leipzig) (1)

P. Drude, “Bestimmung der optischen Konstanten der Metalle,” Ann. Phys. (Leipzig) 39, 481–554 (1890).
[CrossRef]

J. Appl. Phys. (1)

K. Postava, T. Yamaguchi, “Optical functions of low-k materials for interlayer dielectrics,” J. Appl. Phys. 89, 2189–2193 (2001).
[CrossRef]

J. Mod. Opt. (1)

L. Li, “Reformulation of the Fourier model method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Appl. (1)

J. Pištora, T. Yamaguchi, J. Vlček, J. Mistrı́k, M. Horie, V. Šmatko, E. Kováčová, K. Postava, M. Aoyama, “Spectral ellipsometry of binary optic gratings,” Opt. Appl. 33, 251–262 (2003).

Radio Sci. (1)

K. Watanabe, “Fast converging formulation of differential theory for non-smooth gratings made of anisotropic materials,” Radio Sci. 38, doi: (2003).
[CrossRef]

Other (2)

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]

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the lamellar grating under consideration.

Fig. 2
Fig. 2

Convergence of the ellipsometric angles computed by the present formulation for a dielectric lamellar grating.

Fig. 3
Fig. 3

Spectral curves of the ellipsometric angles with different groove breadths. (a) Spectral dependences of Ψ 0 , (b) spectral dependences of Δ 0 .

Fig. 4
Fig. 4

Calculation time to compute one set of spectral curves of the ellipsometric angles.

Fig. 5
Fig. 5

Values of the cost function at the reference point that is used to compose the artificial measured data.

Fig. 6
Fig. 6

Contour maps of the cost function in terms of the grating depth and the groove breadth. (a) Wide view, (b) close view near the reference point.

Equations (36)

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E x ( x ,   y ) = n = - N N E x , n ( y ) exp ( ik 0 α n x ) ,
α n = sin   θ + n   λ d ,
e x ( y ) = ( E x , - N ( y ) E x , N ( y ) ) T ,
d d y e z = ik 0 h x ,
d d y h x = ik 0 ( - X 2 ) e z ,
d d y h z = - ik 0 1 - 1 e x ,
d d y e x = ik 0 ( X - 1 X - I ) h z ,
d 2 d y 2 e z ( y ) = - k 0 2 C g ( s ) e z ( y ) ,
d 2 d y 2 h z ( y ) = - k 0 2 C g ( p ) h z ( y ) ,
C g ( s ) = - X 2 ,
C g ( p ) = 1 - 1 ( I - X - 1 X ) .
f ( f ) ( y ) = Q g ( f ) a g - ( f ) ( y ) a g + ( f ) ( y ) ,
f ( s ) ( y ) = e z ( y ) h x ( y ) ,
f ( p ) ( y ) = h z ( y ) e x ( y ) ,
Q g ( s ) = P g ( s ) P g ( s ) - P g ( s ) Y g ( s ) P g ( s ) Y g ( s ) ,
Q g ( p ) = P g ( p ) P g ( p ) 1 / P g ( p ) Y g ( p ) - 1 / P g ( p ) Y g ( p ) ,
P g ( f ) = ( p g , 1 ( f ) p g , 2 N + 1 ( f ) ) ,
( Y g ( f ) ) ν , μ = δ ν , μ β g , ν ( f ) ,
a g ± ( f ) ( y ) = U g ( f ) ( ± y ) a g ± ( f ) ( 0 ) ,
( U g ( f ) ( y ) ) ν , μ = δ ν , μ exp ( ik 0 β g , ν ( f ) y ) .
f ( f ) ( y ) = Q c ( f ) a c - ( f ) ( y ) a c + ( f ) ( y ) ,
f ( f ) ( y ) = Q s ( f ) a s - ( f ) ( y ) ,
a g - ( f ) ( h ) U g ( f ) ( h ) a g + ( f ) ( 0 ) = G c ( f ) a c - ( f ) ( h ) a c + ( f ) ( h ) ,
U g ( f ) ( h ) a g - ( f ) ( h ) a g + ( f ) ( 0 ) = G s ( f ) a s - ( f ) ( 0 ) ,
G r ( f ) = Q g ( f ) - 1 Q r ( f ) .
a c + ( f ) ( h ) a s - ( f ) ( 0 ) = S 11 ( f ) S 21 ( f ) a c - ( f ) ( h ) ,
S 11 ( f ) S 21 ( f ) = - G c , 22 ( f ) U g ( f ) ( h ) G s , 21 ( f ) U g ( f ) ( h ) G c , 12 ( f ) - G s , 11 ( f ) - 1 × G c , 21 ( f ) - U g ( f ) ( h ) G c , 11 ( f ) ,
G c ( f ) = G c , 11 ( f ) G c , 12 ( f ) G c , 21 ( f ) G c , 22 ( f ) ,
G s ( f ) = G s , 11 ( f ) G s , 21 ( f ) .
Ψ n = arctan ( S 11 ( p ) ) n + N + 1 , N + 1 ( S 11 ( s ) ) n + N + 1 , N + 1 ,
Δ n = - I ln ( S 11 ( p ) ) n + N + 1 , N + 1 ( S 11 ( s ) ) n + N + 1 , N + 1 ,
sub ( λ ) = 1 + A λ 2 λ 2 - B 2 ,
d d y h z = - ik 0 e x ,
d d y e x = ik 0 X 1 X - I h z ,
l ( λ m ) = 2   arcsin ( { 1 - cos [ 2 Ψ 0 , e ( λ m ) ] cos [ 2 Ψ 0 , m ( λ m ) ] - sin [ 2 Ψ 0 , e ( λ m ) ] sin [ 2 Ψ 0 , m ( λ m ) ] cos [ Δ 0 , e ( λ m ) - Δ 0 , m ( λ m ) ] } / 2 ) 1 / 2 .
c = 1 M m = 1 M l ( λ m ) ,

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