Abstract

This paper presents a method for evaluating the image interaction of two object patterns of arbitrary shape in partially coherent imaging. The interaction is shown to be the interference of the eigenfunctions of the transmission cross coefficient at the image plane convolved with each of the object patterns. For two arbitrary patterns, the ith eigenfunction convolved with one object pattern interferes only with the ith eigenfunction convolved with the second object pattern. This method is useful for understanding how object feature shapes influence image interactions, and examples are simulated to evaluate the method. A method to determine constructive and destructive interference areas is also introduced.

© 2010 Optical Society of America

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  7. A. R. Neureuther, P. Flanner III, and S. Shen, “Coherence of defect interactions with features in optical imaging,” J. Vac. Sci. Technol. B 5, 308–312 (1987).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, 1980), Chap. 10.
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    [CrossRef]
  10. N. B. Cobb, “Fast optical and process proximity correction algorithms for integrated circuit manufacturing,” Ph. D. dissertation (University of California, Berkeley, 1998).
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    [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.
  13. R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
    [CrossRef]
  14. K. Yamazoe, Y. Sekine, and T. Honda, “Fast computation of constructive and destructive interference areas in partially coherent imaging for resolution enhancement in optical microlithography,” Appl. Opt. 48, 1419–1424 (2009).
    [CrossRef] [PubMed]
  15. C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
    [CrossRef]

2009

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

K. Yamazoe, Y. Sekine, and T. Honda, “Fast computation of constructive and destructive interference areas in partially coherent imaging for resolution enhancement in optical microlithography,” Appl. Opt. 48, 1419–1424 (2009).
[CrossRef] [PubMed]

2008

2004

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

1987

A. R. Neureuther, P. Flanner III, and S. Shen, “Coherence of defect interactions with features in optical imaging,” J. Vac. Sci. Technol. B 5, 308–312 (1987).
[CrossRef]

1966

1958

1957

1953

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

Born, M.

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

Chan, T. T.

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

Clifford, C. H.

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

Cobb, N. B.

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

Conley, W.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Corcoran, N.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Flanner, P.

A. R. Neureuther, P. Flanner III, and S. Shen, “Coherence of defect interactions with features in optical imaging,” J. Vac. Sci. Technol. B 5, 308–312 (1987).
[CrossRef]

Fung Chen, J.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Goldberg, K. A.

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.

Goodman, W.

W. Goodman, Statistical Optics, Wiley Classical Library (Wiley, 2000), Chap. 5.

Hollerbach, U.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Honda, T.

Hopkins, H. H.

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

Hsu, S.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Laidig, T.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Levinson, H. J.

H. J. Levinson, Principles of Lithography, 1st ed. (SPIE, 2001), Chaps. 1 and 2.

Liang, T.

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

Mochi, I.

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

Neureuther, A. R.

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

A. R. Neureuther, P. Flanner III, and S. Shen, “Coherence of defect interactions with features in optical imaging,” J. Vac. Sci. Technol. B 5, 308–312 (1987).
[CrossRef]

Sekine, Y.

Shen, S.

A. R. Neureuther, P. Flanner III, and S. Shen, “Coherence of defect interactions with features in optical imaging,” J. Vac. Sci. Technol. B 5, 308–312 (1987).
[CrossRef]

Shi, X.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Socha, R.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Thompson, B. J.

van den Broeke, D.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Wampler, K. E.

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Wiraatmadja, S.

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

Wolf, E.

Wong, A. K.

A. K. Wong, Resolution Enhancement Technology in Optical Microlithography, Vol. TT47 of SPIE Tutorial Texts in Optical Engineering (SPIE, 2001), Chap. 3.
[CrossRef]

Yamazoe, K.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

A. R. Neureuther, P. Flanner III, and S. Shen, “Coherence of defect interactions with features in optical imaging,” J. Vac. Sci. Technol. B 5, 308–312 (1987).
[CrossRef]

C. H. Clifford, S. Wiraatmadja, T. T. Chan, A. R. Neureuther, K. A. Goldberg, I. Mochi, and T. Liang, “Comparison of fast three-dimensional simulation and actinic inspection for extreme ultraviolet masks with buried defects and absorber features,” J. Vac. Sci. Technol. B 27, 2888–2893 (2009).
[CrossRef]

Proc. R. Soc. London, Ser. A

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

Proc. SPIE

R. Socha, D. van den Broeke, S. Hsu, J. Fung Chen, T. Laidig, N. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML™),” Proc. SPIE 5377, 222–240 (2004).
[CrossRef]

Other

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

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Chap. 2.

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

H. J. Levinson, Principles of Lithography, 1st ed. (SPIE, 2001), Chaps. 1 and 2.

A. K. Wong, Resolution Enhancement Technology in Optical Microlithography, Vol. TT47 of SPIE Tutorial Texts in Optical Engineering (SPIE, 2001), Chap. 3.
[CrossRef]

W. Goodman, Statistical Optics, Wiley Classical Library (Wiley, 2000), Chap. 5.

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Figures (12)

Fig. 1
Fig. 1

Object patterns a 1 ( x , y ) and a 2 ( x , y ) , which are clear apertures placed on opaque background. The central spacing 2 d is 240 nm , and the width of patterns w is 120 nm .

Fig. 2
Fig. 2

Two types of illuminations used. Each pixel shows a point source, which is mutually incoherent. The white circle shows the pupil edge. The illumination shown in (b) is obtained by rotating (a) by 90 deg , limited by coarse pixelation.

Fig. 3
Fig. 3

First eigenfunction of the TCC, calculated by the illumination shown in Fg. 2a. The maximum value is normalized to be unity. (a) ϕ 1 ( f , g ) defined at the pupil plane. The white circle indicates the pupil. (b) Φ 1 ( x , y ) defined in the image plane, obtained by the Fourier transform of (a). This figure is again used later in Section 4 as the approximate point-to-point interaction Ω ( x , y ) from the illumination in Fig. 2a.

Fig. 4
Fig. 4

Two-dimensional illustrations of (a) E 1 1 ( x , y ) = Φ 1 ( x , y ) a 1 ( x , y ) and (b) E 2 1 ( x , y ) = Φ 1 ( x , y ) a 2 ( x , y ) , where Φ 1 ( x , y ) is shown in Fig. 3b, and object patterns a 1 ( x , y ) and a 2 ( x , y ) are shown in Fig. 1. The maximum value is normalized to unity in both figures, but the upper limit of the plots is limited to 0.5 to emphasize the sidelobes.

Fig. 5
Fig. 5

The image interaction Γ 12 ( x , y ) calculated by Eq. (10) with the object shown in Fig. 1 and the illumination shown in Fig. 2a.

Fig. 6
Fig. 6

Schematic view to determine the point-to-point interaction. The image plane is divided into small segments. The black and gray dots show the reference point and the test point, respectively. Changing the test point calculating the interaction, we can obtain the interaction with respect to the reference point.

Fig. 7
Fig. 7

Point-to-point interaction Ω ( x , y ) calculated with the illumination in Fig. 2a. The maximum value is normalized to unity, but the upper limit of the plot is limited to 0.3 to emphasize the sidelobes.

Fig. 8
Fig. 8

Comparison of different methods to calculated the point-to-point interaction. (a) The difference between the method in [13] and this method: [ Ω ( x , y ) Ω ( x , y ) ] . (b) The difference between the method in [14] and this method: [ Ψ ( x , y ) Ω ( x , y ) ] .

Fig. 9
Fig. 9

Approximation of Fig. 5 by setting N = 1 in Eq. (14): Γ 12 N = 1 ( x , y ) .

Fig. 10
Fig. 10

Interaction of a defect with a line pattern by different sources. Both figures are normalized by the maximum value of the aerial image calculated by the line with the defect pattern and the illumination shown in Fig. 2a. (a) Illumination shown in Fig. 2a was used. (b) Illumination shown in Fig. 2b was used.

Fig. 11
Fig. 11

Interaction of a defect simulated with the illumination shown in Fig. 2a. The value is normalized by the maximum value of the aerial image. All patterns are placed on opaque background. Two lines possess 100% transmittance with 0 phase. Two defects also have 100% transmittance, but one has π 2 phase and the other has π 2 phase.

Fig. 12
Fig. 12

Through-focus interaction of a defect simulated with the illumination shown in Fig. 2a. The value is normalized by the maximum value of the aerial image at best focus. All patterns are placed on opaque background. Two lines possess 100% transmittance with 0 phase. Five square defects also have 100% transmittance; the center one has ( 2 3 ) π phase and the others have π 3 phase. (a) 100 nm defocus. (b) 100 nm defocus.

Equations (14)

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I ( x , y ) = a ̂ | T | a ̂ ,
I ( x , y ) = i = 1 N μ i F T [ ϕ i ( f , g ) a ̂ ( f , g ) ] F T [ ϕ i ( f , g ) a ̂ ( f , g ) ] * = i = 1 N μ i [ Φ i ( x , y ) a ( x , y ) ] [ Φ i ( x , y ) a ( x , y ) ] * ,
a ( x , y ) = a 1 ( x , y ) + a 2 ( x , y ) .
a ( x , y ) = rect ( x + d w , y w ) + rect ( x d w , y w ) ,
I ( x , y ) = I 1 ( x , y ) + I 2 ( x , y ) + Γ 12 ( x , y ) ,
a ̂ ( f , g ) = F T [ a ( x , y ) ] = F T [ a 1 ( x , y ) ] + F T [ a 2 ( x , y ) ] = a ̂ 1 ( f , g ) + a ̂ 2 ( f , g ) .
I ( x , y ) = a ̂ | T | a ̂ = a ̂ 1 + a ̂ 2 | T | a ̂ 1 + a ̂ 2 = a ̂ 1 | T | a ̂ 1 + a ̂ 2 | T | a ̂ 2 + a ̂ 1 | T | a ̂ 2 + a ̂ 2 | T | a ̂ 1 = a ̂ 1 | T | a ̂ 1 + a ̂ 2 | T | a ̂ 2 + 2 Re [ a ̂ 1 | T | a ̂ 2 ] = I 1 ( x , y ) + I 2 ( x , y ) + Γ 12 ( x , y ) ,
Γ 12 ( x , y ) = 2 Re { i = 1 N μ i [ Φ i ( x , y ) a 1 ( x , y ) ] [ Φ i ( x , y ) a 2 ( x , y ) ] * } .
E j i ( x , y ) = μ i [ Φ i ( x , y ) a j ( x , y ) ] .
Γ 12 ( x , y ) = 2 Re [ i = 1 N E 1 i ( x , y ) E 2 i * ( x , y ) ] .
Ω ( x , y ) = 2 Re [ i = 1 N μ i Φ i ( x , y ) Φ i * ( 0 , 0 ) ] .
Ω ( x , y ) 2 Re [ λ i Φ 1 ( x , y ) Φ 1 * ( 0 , 0 ) ] = C Re [ Φ 1 ( x , y ) ] ,
Ψ ( x , y ) = C F T [ S ( f , g ) ] F T [ P ( f , g ) ] ,
Γ 12 N ( x , y ) = 2 Re { i = 1 N μ i [ Φ i ( x , y ) a 1 ( x , y ) ] [ Φ i ( x , y ) a 2 ( x , y ) ] * } .

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