Abstract

Most modal diffraction methods are formulated for incident plane waves. In practical applications, the probing beam is focused. Usually, this is simulated by means of numerical integration where Gaussian quadrature formulas are most effective. These formulas require smooth integrands, which is not fulfilled for gratings due to Rayleigh singularities and physical resonances. The violation of this condition entails inaccurate integration results, such as kinks and other artifacts. In this paper, a methodology for the efficient treatment of the numerical integration with improved accuracy is presented. It is based on the subdivision of the aperture along the lines of Rayleigh singularities, mapping of these subapertures into unit squares, and separate application of the Gaussian cubature formulas for each subarea.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  5. X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
    [CrossRef]
  6. J. Bischoff and K. Hehl, “Single feature metrology by means of light scatter analysis,” Proc. SPIE 3050, 574–585 (1997).
    [CrossRef]
  7. J. Bischoff and K. Hehl, “Modeling of optical scatterometry with finite-number-of-periods gratings,” Proc. SPIE 3743, 41–48 (1999).
    [CrossRef]
  8. K. Hirayama, E. N. Glytsis, and T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
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    [CrossRef]
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  15. T. Jankewitz, “Zweidimensionale Kubaturformeln,” Master’s thesis (Universität Erlangen, 1998).
  16. W. Vogt, “Adaptive Verfahren zur numerischen Quadratur und Kubatur,” Report TU Ilmenau, Germany, 2006.
  17. K. Knothe and H. Wessels, Finite Elemente (Springer Verlag, 1991).

2006 (1)

W. Vogt, “Adaptive Verfahren zur numerischen Quadratur und Kubatur,” Report TU Ilmenau, Germany, 2006.

2005 (1)

F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E 72, 016608 (2005).
[CrossRef]

2003 (1)

R. Cools, “An encyclopaedia of cubature formulas,” J. Complex. 19, 445–453 (2003).
[CrossRef]

2002 (1)

W. Neundorf, Numerische Mathematik (Shaker Verlag Aachen2002).

2001 (1)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

1999 (1)

J. Bischoff and K. Hehl, “Modeling of optical scatterometry with finite-number-of-periods gratings,” Proc. SPIE 3743, 41–48 (1999).
[CrossRef]

1998 (1)

T. Jankewitz, “Zweidimensionale Kubaturformeln,” Master’s thesis (Universität Erlangen, 1998).

1997 (3)

1993 (1)

1991 (1)

K. Knothe and H. Wessels, Finite Elemente (Springer Verlag, 1991).

1984 (1)

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Dover, 1984).

1983 (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).
[CrossRef]

1980 (1)

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

1971 (1)

A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, 1971).

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).
[CrossRef]

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).
[CrossRef]

Bao, J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Bischoff, J.

J. Bischoff and K. Hehl, “Modeling of optical scatterometry with finite-number-of-periods gratings,” Proc. SPIE 3743, 41–48 (1999).
[CrossRef]

J. Bischoff and K. Hehl, “Single feature metrology by means of light scatter analysis,” Proc. SPIE 3050, 574–585 (1997).
[CrossRef]

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).
[CrossRef]

Chandezon, J.

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

Cools, R.

R. Cools, “An encyclopaedia of cubature formulas,” J. Complex. 19, 445–453 (2003).
[CrossRef]

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).
[CrossRef]

Davis, P. J.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Dover, 1984).

García de Abajo, F. J.

F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E 72, 016608 (2005).
[CrossRef]

Gaylord, T. K.

Glytsis, E. N.

Gómez-Medina, R.

F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E 72, 016608 (2005).
[CrossRef]

Hehl, K.

J. Bischoff and K. Hehl, “Modeling of optical scatterometry with finite-number-of-periods gratings,” Proc. SPIE 3743, 41–48 (1999).
[CrossRef]

J. Bischoff and K. Hehl, “Single feature metrology by means of light scatter analysis,” Proc. SPIE 3050, 574–585 (1997).
[CrossRef]

Hirayama, K.

Jakatdar, N.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Jankewitz, T.

T. Jankewitz, “Zweidimensionale Kubaturformeln,” Master’s thesis (Universität Erlangen, 1998).

Knothe, K.

K. Knothe and H. Wessels, Finite Elemente (Springer Verlag, 1991).

Li, L.

Maystre, D.

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

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).
[CrossRef]

Moharam, M. G.

Neundorf, W.

W. Neundorf, Numerische Mathematik (Shaker Verlag Aachen2002).

Niu, X.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Rabinowitz, P.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Dover, 1984).

Raoult, G.

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

Sáenz, J. J.

F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E 72, 016608 (2005).
[CrossRef]

Spanos, C. J.

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

Stroud, A. H.

A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, 1971).

Vogt, W.

W. Vogt, “Adaptive Verfahren zur numerischen Quadratur und Kubatur,” Report TU Ilmenau, Germany, 2006.

Wessels, H.

K. Knothe and H. Wessels, Finite Elemente (Springer Verlag, 1991).

IEEE Trans. Semicond. Manuf. (1)

X. Niu, N. Jakatdar, J. Bao, and C. J. Spanos, “Specular spectroscopic scatterometry,” IEEE Trans. Semicond. Manuf. 14, 97–111 (2001).
[CrossRef]

J. Complex. (1)

R. Cools, “An encyclopaedia of cubature formulas,” J. Complex. 19, 445–453 (2003).
[CrossRef]

J. Opt. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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).
[CrossRef]

Phys. Rev. E (1)

F. J. García de Abajo, R. Gómez-Medina, and J. J. Sáenz, “Full transmission through perfect-conductor subwavelength hole arrays,” Phys. Rev. E 72, 016608 (2005).
[CrossRef]

Proc. SPIE (2)

J. Bischoff and K. Hehl, “Single feature metrology by means of light scatter analysis,” Proc. SPIE 3050, 574–585 (1997).
[CrossRef]

J. Bischoff and K. Hehl, “Modeling of optical scatterometry with finite-number-of-periods gratings,” Proc. SPIE 3743, 41–48 (1999).
[CrossRef]

Other (6)

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration (Dover, 1984).

W. Neundorf, Numerische Mathematik (Shaker Verlag Aachen2002).

A. H. Stroud, Approximate Calculation of Multiple Integrals (Prentice-Hall, 1971).

T. Jankewitz, “Zweidimensionale Kubaturformeln,” Master’s thesis (Universität Erlangen, 1998).

W. Vogt, “Adaptive Verfahren zur numerischen Quadratur und Kubatur,” Report TU Ilmenau, Germany, 2006.

K. Knothe and H. Wessels, Finite Elemente (Springer Verlag, 1991).

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Figures (7)

Fig. 1
Fig. 1

Simulated diffraction amplitude (TE polarization) inside a rectangular aperture with NA = 0.07 and 70 ° incident angle for the specular reflection from a line–space grating (left) and a thin film (right). The discontinuity caused by a Rayleigh singularity line can clearly be seen running across the aperture box in the vertical direction close to the center and exhibiting a convex shape.

Fig. 2
Fig. 2

Schematic nonorthogonal crossed grating and incident ray direction.

Fig. 3
Fig. 3

Schematic representation of a Cartesian normal area.

Fig. 4
Fig. 4

Rayleigh singularity lines for a line–space grating illuminated with a circular aperture beam at normal incidence. Three cases can be discerned (see text for further explanation). The numbers denote the subareas generated by splitting the pupil along the lines.

Fig. 5
Fig. 5

Location of a Rayleigh singularity for a line–space grating illuminated with a rectangular aperture beam at oblique incidence ( 70 ° ). The practically more relevant case of a negative order line ( m = 2 ) is shown.

Fig. 6
Fig. 6

Photoresist line–space grating for the numerical test example.

Fig. 7
Fig. 7

Modeled spectra for the grating pattern of Fig. 6. The “xx_sub” curves have been modeled with the NISM approach, whereas the other curves result from standard integration schemas.

Tables (1)

Tables Icon

Table 1 Gaussian (G) and Chebyshev (T) Quadrature Formulas of Degrees 2 and 3 for the Interval ( 1 , 1 ) ( ξ ( 1 , 1 ) )

Equations (52)

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a b f ( x ) d x i = 1 n w i · f ( x i ) .
V f ( P ) d V i = 1 n w i · f ( P i )
W = ( 1 1 1 1 ) ,
P = 1 3 ( ( 1 , 1 ) ( + 1 , 1 ) ( 1 , + 1 ) ( + 1 , + 1 ) ) .
W = 1 81 ( 25 40 25 40 64 40 25 40 25 ) ,
P = 1 5 ( ( 3 , 3 ) ( 0 , 3 ) ( + 3 , 3 ) ( 3 , 0 ) ( 0 , 0 ) ( + 3 , 0 ) ( 3 , + 3 ) ( 0 , + 3 ) ( + 3 , + 3 ) ) .
P = { ( 1 6 , 1 2 ) , ( 1 6 , + 1 2 ) , ( 2 3 , 0 ) } , W = ( 4 3 , 4 3 , 4 3 ) ,
P = { ( 0 , 0 ) , ( ± x 1 , 0 ) , ( ± x 2 , ± y 2 ) } , W = ( W 0 , W 1 , W 1 , W 2 , W 2 , W 2 , W 2 ) ,
sec 2 ξ ( α m 2 + β n 2 2 α m β n · sin ξ ) + γ m n 2 = ε ,
α m = sin θ · cos φ + m · λ d x , β n = sin θ · sin φ + n · λ d y .
γ m n = ( ε sec 2 ξ ( α m 2 + β n 2 2 α m β n · sin ξ ) ) 1 / 2 = 0.
α m 2 + β n 2 2 α m β n · sin ξ = cos 2 ξ .
α m 2 + β n 2 = 1 .
sin 2 θ + 2 sin θ ( m λ d x cos φ + m λ d y sin φ ) + ( m λ d x ) 2 + ( n λ d y ) 2 1 = 0 .
x = sin θ · cos φ = r · cos φ , y = sin θ · sin φ = r · sin φ .
α m = m · λ d x , β n = n · λ d y .
( α m + x ) 2 + ( β n + y ) 2 = 1 .
x s 2 + y s 2 + 2 α m x s + α m 2 1 = 0 .
x s = α m ± 1 y s 2 ,
y c = ± 1 α m 2 .
x 2 + y 2 = NA 2 .
x i = 1 NA 2 α m 2 α m ,
y i = ( 1 NA α m ) · ( 1 + NA α m ) · ( 1 + NA + α m ) · ( NA α m 1 ) 2 α m .
x x = x s ( y = 0 ) = α m ± 1.
y 1 = y i , y 2 = + y i , x 1 ( y ) = α ¯ 1 y 2 , x 2 ( y ) = NA 2 y 2 ,
y ( s , t ) = y 1 + ( y 2 y 1 ) · t = y i ( 2 t 1 ) ,
x ( s , t ) = x 1 ( y ) + s · ( x 2 ( y ) x 1 ( y ) ) = α ¯ 1 y 2 + s · ( NA 2 y 2 α ¯ + 1 y 2 ) = ( α ¯ 1 y i 2 · ( 2 t 1 ) 2 ) · ( 1 s ) + s · NA 2 y i 2 · ( 2 t 1 ) 2 .
I = A f ( x , y ) d x d y = 0 1 0 1 f ( x ( s , t ) , y ( s , t ) ) · | J ( s , t ) | d s d t = 0 1 0 1 g ( s , t ) · | J ( s , t ) | d s d t ,
J ( s , t ) = ( x s y s x t y t ) .
x s = NA 2 y i 2 ( 2 t 1 ) 2 α ¯ + 1 y i 2 ( 2 t 1 ) 2 , y s = 0 , y t = 2 y i .
| J | = x s · y t = 2 y i · ( NA 2 y i 2 ( 2 t 1 ) 2 α ¯ + 1 NA 2 ( 2 t 1 ) 2 ) .
I 1 = 2 y i · 0 1 0 1 ( 1 NA 2 ( 2 t 1 ) 2 α ¯ + NA 2 y i 2 ( 2 t 1 ) 2 ) · g ( s , t ) d s d t .
x ( s , t ) = x 0 + x 1 s + x 2 t + x 3 s t + x 4 · s 2 + x 5 t 2 + x 6 s 2 t + x 7 · s t 2 , y ( s , t ) = y 0 + y 1 s + y 2 t + y 3 s t + y 4 · s 2 + y 5 t 2 + y 6 s 2 t + y 7 · s t 2 .
x ( s , t ) = ( 2 s 1 ) x i + 4 t ( 1 t ) ( 1 2 s ) ( x i α ¯ + 1 ) , y ( s , t ) = ( 2 t 1 ) y i + 4 s ( 1 s ) ( 1 2 t ) ( y i NA ) .
x s = 2 x i 8 t ( 1 t ) ( x i α ¯ + 1 ) , y s = 4 ( y i NA ) ( 1 2 s ) ( 1 2 t ) , x t = 4 ( x i α ¯ + 1 ) ( 1 2 s ) ( 1 2 t ) , y t = 2 y i 8 s ( 1 s ) ( y i NA ) .
x ( s , t ) = x 1 ( y ) + s · ( x 2 ( y ) x 1 ( y ) ) = α ¯ + 1 y 2 + s · ( NA 2 y 2 + α ¯ 1 y 2 ) = ( α ¯ + 1 y i 2 · ( 2 t 1 ) 2 ) · ( 1 s ) + s · NA 2 y i 2 · ( 2 t 1 ) 2 ,
| J | = 2 y i · ( NA 2 y i 2 · ( 2 t 1 ) 2 + α ¯ 1 y i 2 ( 2 t 1 ) 2 ) ,
I 1 = 2 y 0 0 1 0 1 ( NA 2 y i 2 · ( 2 t 1 ) 2 + α ¯ 1 y i 2 ( 2 t 1 ) 2 ) · g ( s , t ) d s d t .
x ( s , t ) = ( 1 2 s ) x i + 4 t ( 1 t ) ( 2 s 1 ) ( x i α ¯ + 1 ) .
A 1 = A 3 = NA 2 ( π 2 sin 1 ( x 0 NA ) ) α ¯ y i + sin 1 ( x i α ¯ ) ± π 2 ,
A 2 = 2 NA 2 · sin 1 ( x i NA ) ± 2 α α i 2 sin 1 ( x i α ¯ ) π .
NA min , max = sin ( θ 0 sin 1 ( NA x ) ) .
λ min , max = 1 NA min , max | m | · d x .
y ( s , t ) = y 1 + ( y 2 y 1 ) · t = NA y ( 2 t 1 ) ,
x ( s , t ) = x 1 ( y ) + s · ( x 2 x 1 ( y ) ) = α ¯ 1 y 2 + s · ( NA max α ¯ + 1 y 2 ) = ( α ¯ 1 NA y 2 · ( 2 t 1 ) 2 ) · ( 1 s ) + s · NA max .
| J | = 2 NA y · ( NA max α ¯ + 1 NA y 2 ( 2 t 1 ) 2 ) ,
I 1 = 2 NA y · 0 1 0 1 ( NA max α ¯ + 1 NA y 2 ( 2 t 1 ) 2 ) · g ( s , t ) d s d t .
x ( s , t ) = x 1 + s · ( x 2 ( y ) x 1 ) = NA min + s · ( α ¯ 1 y 2 NA min ) = ( α ¯ 1 NA y 2 · ( 2 t 1 ) 2 ) · s + ( 1 s ) · NA min ,
| J | = 2 NA y · ( α ¯ 1 NA y 2 ( 2 t 1 ) 2 NA min ) ,
I 1 = 2 NA y · 0 1 0 1 ( α ¯ NA min 1 NA y 2 ( 2 t 1 ) 2 ) · g ( s , t ) d s d t .
A 1 = NA y · ( 2 NA max + 1 NA y 2 2 α ¯ ) + sin 1 ( NA y ) , A 2 = NA y · ( 2 α ¯ 1 NA y 2 2 NA min ) sin 1 ( NA y ) .
λ min = NA min + 1 NA y 2 | m | · d x ,

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