Abstract

We show quantitatively through a computer simulation the effect of vector diffraction on the image in optical lithography. The simulation was made with a new program, and the variation of diffraction as the magnification was varied was calculated for various numerical apertures and degrees of defocus. The diffraction at the mask improves the images of lines across the polarizing direction, and the diffraction at the lens improves the images of the lines along the polarizing direction. We investigated the difference between images of lines along and across the polarizing direction as the magnification increased. Our result shows that the effect of the lens is so dominant that the images of lines along the polarizing direction are always better.

© 1998 Optical Society of America

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References

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  1. C. M. Yuan, “Calculation of one-dimensional lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604–1613 (1993).
    [CrossRef]
  2. B. W. Smith, D. G. Flagello, J. R. Summa, L. F. Fuller, “Comparison of scalar and vector diffraction modeling for deep-UV lithography,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 847–957 (1993).
    [CrossRef]
  3. M. S. Yeung, D. Lee, R. Lee, A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 452–463 (1993).
    [CrossRef]
  4. Y. Unno, “Polarization effect of illumination light,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 879–891 (1993).
    [CrossRef]
  5. J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 441–447.
  6. M. Mansuripur, “Distribution of light at near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 3, 2086–2093 (1986).
    [CrossRef]
  7. M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
    [CrossRef]
  8. M. Mansuripur, “Distribution of light at near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 10, 382–388 (1993).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 526–532.
  10. K. K. H. Toh “Two-dimensional Images with effects of lens aberrations,” in SAMPLE Report (University of California, Berkeley, Calif., 1988), pp. 3–6.
  11. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 394.
  12. G. R. Bird, M. Parrish, “The wire grid as a near-infrared polarizer,” J. Opt. Soc. Am. 50, 886–891 (1960).
    [CrossRef]

1993 (2)

C. M. Yuan, “Calculation of one-dimensional lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604–1613 (1993).
[CrossRef]

M. Mansuripur, “Distribution of light at near the focus of high-numerical-aperture objectives,” J. Opt. Soc. Am. A 10, 382–388 (1993).
[CrossRef]

1989 (1)

1986 (1)

1960 (1)

Bird, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 526–532.

Flagello, D. G.

B. W. Smith, D. G. Flagello, J. R. Summa, L. F. Fuller, “Comparison of scalar and vector diffraction modeling for deep-UV lithography,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 847–957 (1993).
[CrossRef]

Fuller, L. F.

B. W. Smith, D. G. Flagello, J. R. Summa, L. F. Fuller, “Comparison of scalar and vector diffraction modeling for deep-UV lithography,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 847–957 (1993).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 394.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 441–447.

Lee, D.

M. S. Yeung, D. Lee, R. Lee, A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 452–463 (1993).
[CrossRef]

Lee, R.

M. S. Yeung, D. Lee, R. Lee, A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 452–463 (1993).
[CrossRef]

Mansuripur, M.

Neureuther, A. R.

M. S. Yeung, D. Lee, R. Lee, A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 452–463 (1993).
[CrossRef]

Parrish, M.

Smith, B. W.

B. W. Smith, D. G. Flagello, J. R. Summa, L. F. Fuller, “Comparison of scalar and vector diffraction modeling for deep-UV lithography,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 847–957 (1993).
[CrossRef]

Summa, J. R.

B. W. Smith, D. G. Flagello, J. R. Summa, L. F. Fuller, “Comparison of scalar and vector diffraction modeling for deep-UV lithography,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 847–957 (1993).
[CrossRef]

Toh, K. K. H.

K. K. H. Toh “Two-dimensional Images with effects of lens aberrations,” in SAMPLE Report (University of California, Berkeley, Calif., 1988), pp. 3–6.

Unno, Y.

Y. Unno, “Polarization effect of illumination light,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 879–891 (1993).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 526–532.

Yeung, M. S.

M. S. Yeung, D. Lee, R. Lee, A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 452–463 (1993).
[CrossRef]

Yuan, C. M.

C. M. Yuan, “Calculation of one-dimensional lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604–1613 (1993).
[CrossRef]

IEEE Trans. Electron Devices (1)

C. M. Yuan, “Calculation of one-dimensional lithographic aerial images using the vector theory,” IEEE Trans. Electron Devices 40, 1604–1613 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Other (7)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 526–532.

K. K. H. Toh “Two-dimensional Images with effects of lens aberrations,” in SAMPLE Report (University of California, Berkeley, Calif., 1988), pp. 3–6.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Mass., 1987), p. 394.

B. W. Smith, D. G. Flagello, J. R. Summa, L. F. Fuller, “Comparison of scalar and vector diffraction modeling for deep-UV lithography,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 847–957 (1993).
[CrossRef]

M. S. Yeung, D. Lee, R. Lee, A. R. Neureuther, “Extension of the Hopkins theory of partially coherent imaging to include thin-film interference effects,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 452–463 (1993).
[CrossRef]

Y. Unno, “Polarization effect of illumination light,” in Optical/Laser Microlithography, J. D. Cuthbert, ed., Proc. SPIE1927, 879–891 (1993).
[CrossRef]

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), pp. 441–447.

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Figures (8)

Fig. 1
Fig. 1

Diffraction pattern of a single slit. (a) q (the slit width over a wavelength) is 5, (b) q is 1, (c) q is 0.5.

Fig. 2
Fig. 2

Maximum difference/maximum intensity of the diffraction pattern compared with the q factor.

Fig. 3
Fig. 3

One-dimensional intensity distribution of a 0.2-μm line-and-space pattern with a N.A. of 0.6, a degree of coherence of 0.5, a wavelength of 0.248 μm, and no defocus. (a) 10:1 Reduction system (magnification, 0.1). The linewidth is eight times as large as the wavelength. (b) 2:1 Reduction system (magnification, 0.5). The linewidth is 1.6 times as large as the wavelength. (c) 1:1 system (magnification = 1) linewidth is 0.8 times as large as the wavelength.

Fig. 4
Fig. 4

Image width of a line across the polarizing direction with magnification: The cutoff intensity is as large as the value at the edge of the image of the line along the polarizing direction: (a) with no defocus, (b) defocus of 0.1 μm, (c) defocus of 0.2 μm.

Fig. 5
Fig. 5

Corner of an elbow pattern’s image: line width, 0.2 μm; illumination polarized along the x axis; N.A., 0.6; degree of coherence, 0.5: (a) 10:1 reduction system (magnification, 0.1), (b) 2:1 reduction system (magnification, 0.5).

Fig. 6
Fig. 6

One-dimensional intensity distribution of lines along and across the polarizing direction. The cutoff lines are x = 0.6 and y = 0.6: (a) 10:1 reduction system (magnification, 0.1), (b) 10:3 reduction system (magnification, 0.3), (c) 2:1 reduction system (magnification, 0.5).

Fig. 7
Fig. 7

Image width of the line across the polarizing direction (y axis) with magnification: the cutoff intensity is as large as the value at the edge of the image of the line along the polarizing direction (x axis). Because the images across the polarizing direction are broadened, the line width is wider than that along the x axis at small magnification: (a) with no defocus, (b) defocus of 0.1 μm, (c) defocus of 0.2 μm.

Fig. 8
Fig. 8

Comparison of the real experimental image with the simulated result: (a) image from the experiment, (b) image from simulation.

Equations (21)

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E x ,   y ,   z = i   exp ikr / 2 π r k ×   n ˆ × E x ,   y exp - i k · r d x d y ,
k = 2 π σ x e 1 ˆ + σ y e 2 ˆ + σ z e 3 ˆ / λ ,     n = e 3 ˆ ,
E x ,   y = t x ,   y exp i Φ x ,   y P x e 1 ˆ + P y e 2 ˆ ,
E x f ,   g = P x 1 - λ 2 f 2 - λ 2 g 2 1 / 2 × i   exp ikr / 2 π r     t x ,   y exp i Φ x ,   y × exp - 2 π i f x + g y d x d y , E y f ,   g = P y 1 - λ 2 f 2 - λ 2 g 2 1 / 2 × i   exp ikr / 2 π r     t x ,   y exp i Φ x ,   y × exp - 2 π i f x + g y d x d y , E z f ,   g = - P z 1 - λ 2 f 2 - λ 2 g 2 1 / 2 × i   exp ikr / 2 π r     t x ,   y exp i Φ x ,   y × exp - 2 π i f x + g y d x d y .
E x f ,   g = P x 1 - m 2 λ 2 f 2 - m 2 λ 2 g 2 1 / 2 F f ,   g , E y f ,   g = P y 1 - m 2 λ 2 f 2 - m 2 λ 2 g 2 1 / 2 F f ,   g , E z f ,   g = - m P x λ f + P y λ g F f ,   g ,
F f ,   g = i   exp ikr / 2 π r     t x ,   y exp i Φ x ,   y × exp - 2 π i fx + gy d x d y = C     t x ,   y exp i Φ x ,   y × exp - 2 π i fx + gy d x d y .
σ x = - m σ x ,     σ y = - m σ y .
Φ xx = 1 / σ 0 2 σ z / σ z σ z σ z σ x 2 + σ y 2 , Φ yx = σ x σ y / σ 0 2 σ z / σ z σ z σ z - 1 , Φ zx = m σ x σ z σ z , Φ xy = σ x σ y / σ 0 2 σ z / σ z σ z σ z - 1 , Φ yy = 1 / σ 0 2 σ z / σ z σ z σ z σ y 2 + σ x 2 , Φ zy = m σ y σ z σ z , Φ xz = - σ x σ z σ z / σ z , Φ yz = - σ y σ z σ z / σ z , Φ zz = - m σ 0 2 σ z / σ z .
Φ xx = 1 - σ x 2 / 1 + σ z σ z , Φ yx = - σ x σ y / 1 + σ z σ z , Φ xy = - σ x σ y / 1 + σ z σ z , Φ yy = 1 - σ y 2 / 1 + σ z σ z , Φ xz = - σ x / σ z , Φ yz = - σ y / σ z .
I x ,   y =   O f ,   g exp - 2 π i fx + gy d f d g .
O f ,   g =   F f ,   g F * f + f ,   g + g × TCC f ,   g ;   f ,   g d f d g ,
TCC f ,   g ;   f ,   g =   J f ,   g K f + f ,   g + g × K * f + f + f ,   g + g + g × d f d g ,
J f ,   g = const . σ 2 / π 0   f 2 + g 2 σ N . A . / λ 2 otherwise .
K f ,   g = exp - i 2 π Φ f ,   g / λ 0   f 2 + g 2 N . A . / λ 2 otherwise .
K i f ,   g = a   P a f ,   g exp - i 2 π Φ f ,   g / λ × Φ ai - λ f ,   - λ g .
P x f ,   g = P x 1 - m 2 λ 2 f 2 - m 2 λ 2 g 2 1 / 2 , P y f ,   g = P y 1 - m 2 λ 2 f 2 - m 2 λ 2 g 2 1 / 2 , P z f ,   g = - m P x λ f + P y λ g .
O f ,   g = O x f ,   g + O y f ,   g + O z f ,   g ,
O i f ,   g =   F f + f ,   g + g F * f ,   g TCC i × f + f ,   g + g ;   f ,   g d f d g ,
TCC i f ,   g ;   f ,   g =   J f ,   g K i f + f ,   g + g K i * × f + f + f ,   g + g + g d f d g .
I σ x = sin π L σ x / λ / π L σ x / λ 2 ;
I σ x = 1 - σ x 2 sin π L σ x / λ / π L σ x / λ 2 .

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