Abstract

A fast computation algorithm for fine-pixel aerial images is presented that modifies the transmission cross coefficient approach of Hopkins as a product of two matrices. The spatial frequency of the image is calculated by the sum of diagonal and off-diagonal elements of the matrix. Let N, NF, and M be the number of the point sources, the sampling number for fast Fourier transform, and the sampling number in the spatial frequency domain ranging twice the pupil size, respectively. The calculation time of this method is proportional to BN[(M1)/2]4, while that of a conventional source integration method is 2ANNFlog2NF, where A and B are constants and generally B<A. If NF is sufficiently greater than M or M is small enough, which is the fine-pixel condition, this method runs faster than the source integration method. If the coherence factor is 0.9 and M55, this method runs faster than the source integration even under the Nyquist sampling condition.

© 2010 Optical Society of America

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  1. H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.
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    [CrossRef]
  6. E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (Electrical Engineering and Computer Science, University of California, Berkeley, 1998).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  14. K. Yamazoe, “Fast fine-pixel aerial image calculation by matrix representation of Hopkins equation,” presented at the 19th Lithography Workshop, Coeur d’Alene, Idaho, USA, 28 June–2 July 2009.
  15. T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
    [CrossRef]
  16. S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
    [CrossRef]
  17. Y. Lian and X. Zhou, “Fast and accurate computation of partially coherent imaging by stacked pupil shift operator,” Proc. SPIE 7488, 74883G (2009).
    [CrossRef]

2010 (1)

2009 (1)

Y. Lian and X. Zhou, “Fast and accurate computation of partially coherent imaging by stacked pupil shift operator,” Proc. SPIE 7488, 74883G (2009).
[CrossRef]

2008 (1)

2006 (1)

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

2005 (1)

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

2002 (1)

A. S. Ostrovsky, O. Ramos-Romero, and G. Martínez-Niconoff, “Fast algorithm for bilinear transforms in optics,” Rev. Mex. Fís. 48, 186–191 (2002).

1996 (1)

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

1982 (1)

1978 (1)

1953 (1)

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

Ausschnitt, C. P.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

Brunner, T.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Butt, S.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Chen, C.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Cobb, N. B.

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (Electrical Engineering and Computer Science, University of California, Berkeley, 1998).

Corliss, D.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1964), Vol. 3, Chap. 3.
[CrossRef]

Gau, T.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), Chap. 7.

Ho, B.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

Hsieh, H.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Huang, J.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Ke, C.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Kintner, E.

Ku, Y.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Lian, Y.

Y. Lian and X. Zhou, “Fast and accurate computation of partially coherent imaging by stacked pupil shift operator,” Proc. SPIE 7488, 74883G (2009).
[CrossRef]

Lin, B. J.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Martínez-Niconoff, G.

A. S. Ostrovsky, O. Ramos-Romero, and G. Martínez-Niconoff, “Fast algorithm for bilinear transforms in optics,” Rev. Mex. Fís. 48, 186–191 (2002).

Neureuther, A. R.

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.

Ostrovsky, A. S.

A. S. Ostrovsky, O. Ramos-Romero, and G. Martínez-Niconoff, “Fast algorithm for bilinear transforms in optics,” Rev. Mex. Fís. 48, 186–191 (2002).

Rabbani, M.

Ramos-Romero, O.

A. S. Ostrovsky, O. Ramos-Romero, and G. Martínez-Niconoff, “Fast algorithm for bilinear transforms in optics,” Rev. Mex. Fís. 48, 186–191 (2002).

Saleh, B. E. A.

Smith, M.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Socha, R. J.

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

Wiltshire, T.

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

Yamazoe, K.

Yen, A.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Yeung, M.

M. Yeung, “Modeling aerial images in two and three dimensions,” in Proceedings of Kodak Microelectronics Seminar: Interface ’85, Kodak Publ. G-154 (Eastman Kodak, 1986), pp. 115–126.

Yu, S.

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

Zhou, X.

Y. Lian and X. Zhou, “Fast and accurate computation of partially coherent imaging by stacked pupil shift operator,” Proc. SPIE 7488, 74883G (2009).
[CrossRef]

Appl. Opt. (2)

J. Microlith. Microfab. Microsyst. (2)

T. Brunner, D. Corliss, S. Butt, T. Wiltshire, C. P. Ausschnitt, and M. Smith, “Laser bandwidth and other sources of focus blur in lithography,” J. Microlith. Microfab. Microsyst. 5, 043003 (2006).
[CrossRef]

S. Yu, B. J. Lin, A. Yen, C. Ke, J. Huang, B. Ho, C. Chen, T. Gau, H. Hsieh, and Y. Ku, “Thin-film optimization strategy in high numerical aperture optical lithography, part 1: principles,” J. Microlith. Microfab. Microsyst. 4, 043003(2005).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Vac. Sci. Technol. B (1)

R. J. Socha and A. R. Neureuther, “Propagation effects of partially coherence in optical lithography,” J. Vac. Sci. Technol. B 14, 3724–3729 (1996).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “On the diffraction theory of optical image,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
[CrossRef]

Proc. SPIE (1)

Y. Lian and X. Zhou, “Fast and accurate computation of partially coherent imaging by stacked pupil shift operator,” Proc. SPIE 7488, 74883G (2009).
[CrossRef]

Rev. Mex. Fís. (1)

A. S. Ostrovsky, O. Ramos-Romero, and G. Martínez-Niconoff, “Fast algorithm for bilinear transforms in optics,” Rev. Mex. Fís. 48, 186–191 (2002).

Other (7)

N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph.D. dissertation (Electrical Engineering and Computer Science, University of California, Berkeley, 1998).

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.

J. W. Goodman, Statistical Optics, 1st ed. (Wiley-Interscience, 1985), Chap. 7.

M. Yeung, “Modeling aerial images in two and three dimensions,” in Proceedings of Kodak Microelectronics Seminar: Interface ’85, Kodak Publ. G-154 (Eastman Kodak, 1986), pp. 115–126.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, E.Wolf, ed. (North-Holland, 1964), Vol. 3, Chap. 3.
[CrossRef]

E. L. O’Neill, Introduction to Statistical Optics (Dover, 2003), Chap. 8.

K. Yamazoe, “Fast fine-pixel aerial image calculation by matrix representation of Hopkins equation,” presented at the 19th Lithography Workshop, Coeur d’Alene, Idaho, USA, 28 June–2 July 2009.

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Figures (6)

Fig. 1
Fig. 1

Example of the diagonal sum operator D.

Fig. 2
Fig. 2

Schematic view of how to calculate the matrix Z in accordance with the example in Subsection 2D. Illumination from f 4 is the normal incident, while illumination from f 5 is the oblique incident that shifts the object spectrum. By stacking B 1 1 D and B 2 1 D , we can obtain B ex 1 D in Eq. (15).

Fig. 3
Fig. 3

(a) An object is represented by the combination of transparent rectangles placed on an opaque background. (b) Partially coherent illumination. Coherence factor σ is set to 0.9. Each pixel represents a mutually incoherent point source. The white circle indicates the pupil edge.

Fig. 4
Fig. 4

(a) Aerial images obtained by this method. (b) Difference in the aerial images obtained by this method and source integration.

Fig. 5
Fig. 5

Comparison of the simulation time in the M = 63 case. As the illumination, coherence factors of 0.35 ( N = 97 ) and 1.00 ( N = 749 ) were used. Under the Nyquist sampling condition, the pixel size is 69.9 nm , while the feature width is 100 nm .

Fig. 6
Fig. 6

(a) Light intensity inside the photoresist at the y = 0 cross section. The resist depth is represented by z. (b) Averaged image of all light intensities in the photoresist.

Equations (23)

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I ( x , y ) = ϕ | E | ϕ ,
I ( x , y ) = ϕ | Z | ϕ ,
( f 1 f 2 f 3 f 4 f 5 f 6 f 7 ) = ( 2 4 3 2 3 0 2 3 4 3 2 ) .
| ϕ 1 D = ( exp ( i 2 π f 1 x ) exp ( i 2 π f 2 x ) exp ( i 2 π f 7 x ) ) T ,
I 1 D ( x ) = ϕ 1 D | Z 1 D | ϕ 1 D ,
Z 1 D = ( Z 11 Z 12 Z 17 Z 21 Z 22 Z 71 Z 72 Z 77 ) .
I 1 D ( x ) = k = 3 3 C k exp ( i 2 π k Δ f x ) ,
C 0 = Z 11 + Z 22 + + Z 77 .
C 1 = Z 12 + Z 23 + + Z 67 .
I ^ 1 D ( f ) = D [ Z 1 D ] .
B 1 D = ( S 1 a ^ ( f 1 f 1 ) P ( f 1 ) S 1 a ^ ( f 2 f 1 ) P ( f 2 ) S 1 a ^ ( f M f 1 ) P ( f M ) S 2 a ^ ( f 1 f 2 ) P ( f 1 ) S 2 a ^ ( f 2 f 2 ) P ( f 2 ) S 2 a ^ ( f M f 2 ) P ( f M ) S N a ^ ( f 1 f N ) P ( f 1 ) S N a ^ ( f 2 f N ) P ( f 2 ) S N a ^ ( f M f N ) P ( f M ) ) ,
Y [ I ^ ( f , g ) ] T = D [ Z ] .
I ^ ( f , g ) = Y 1 [ D [ Z ] T ] .
P ( f ) = ( 0 0 1 1 1 0 0 ) .
B ex 1 D = ( 0 0 a ^ 3 a ^ 4 a ^ 5 0 0 0 0 a ^ 2 a ^ 3 a ^ 4 0 0 ) .
Z ex 1 D = ( B ex 1 D ) B ex 1 D = ( 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a ^ 2 * a ^ 2 + a ^ 3 * a ^ 3 a ^ 3 * a ^ 4 + a ^ 2 * a ^ 3 a ^ 3 * a ^ 5 + a ^ 2 * a ^ 4 0 0 0 0 a ^ 4 * a ^ 3 + a ^ 3 * a ^ 2 a ^ 3 * a ^ 3 + a ^ 4 * a ^ 4 a ^ 4 * a ^ 5 + a ^ 3 * a ^ 4 0 0 0 0 a ^ 5 * a ^ 3 + a ^ 4 * a ^ 2 a ^ 5 * a ^ 4 + a ^ 4 * a ^ 3 a ^ 4 * a ^ 4 + a ^ 5 * a ^ 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ) .
I ^ e x 1 D ( f ) = D [ Z ex 1 D ] = ( 0 a ^ 4 * a ^ 2 + a ^ 5 * a ^ 3 a ^ 3 * a ^ 2 + 2 a ^ 4 * a ^ 3 + a ^ 5 * a ^ 4 a ^ 2 * a ^ 2 + 2 a ^ 3 * a ^ 3 + 2 a ^ 4 a ^ 4 * + a ^ 5 a ^ 5 * a ^ 2 * a ^ 3 + 2 a ^ 3 * a ^ 4 + a ^ 4 * a ^ 5 a ^ 3 * a ^ 5 + a ^ 2 * a ^ 4 0 ) T .
B red 1 D = ( a ^ 3 a ^ 4 a ^ 5 a ^ 2 a ^ 3 a ^ 4 ) .
Z = 1 N d i = 1 N d B i B i = 1 N d i = 1 N d Z i .
I ^ ( f , g ) = Y 1 [ D [ Z ] T ] .
P ave = 1 N d ( P 1 P 2 P N d ) .
T ave = 1 N d i = 1 N d P i P i .
{ d k = j = 1 m ( k m ) m j , j + ( k m ) if k m , d k = j = 1 m + ( k m ) m j + ( k m ) , j if k < m .

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