Abstract

The influence of birefringence caused by rotationally symmetric stress distribution in a high-resolution projection optical system is investigated. The general form of the pupil function is derived based on the Jones matrix calculation, expressing the wave front as a combination of the two orthogonal polarization components. Assuming a linearly polarized incident beam, it is found that the main polarization portion of the wave front exiting the projection lens has astigmatic aberration in the Seidel region and shows phase singularity at four pupil points at which the amplitude transmittance becomes zero.

© 1998 Optical Society of America

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References

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  1. P. Rai-Choudhury, ed., Handbook of Microlithography, Micromachining, and Microfabrication, Volume 1: Microlithography (SPIE Optical Engineering Press, Bellingham, Wash., 1997), Chap. 1.
  2. H. Shinonaga, M. Arakawa, “Stepper exposure system for the quarter-micrometer age,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 754–766 (1996).
    [CrossRef]
  3. K. Suzuki, S. Wakamoto, K. Nishi, “KrF step-and-scan exposure system using higher-NA projection lens,” Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 767–779 (1996).
    [CrossRef]
  4. M. Rothchild, D. J. Ehrlich, D. C. Shaver, “Effects of excimer laser irradiation on the transmission, index of refraction, and density of ultraviolet grade fused silica,” Appl. Phys. Lett. 55, 1276–1278 (1989).
    [CrossRef]
  5. R. Schenker, W. Oldham, “Effects of compaction on 193 nm lithographic system performance,” J. Vac. Sci. Technol. B 14, 3709–3713 (1996).
    [CrossRef]
  6. D. C. Allan, C. Smith, N. F. Borrelli, T. P. Seward, “193-nm excimer-laser-induced densification of fused silica,” Opt. Lett. 21, 1960–1962 (1996).
    [CrossRef] [PubMed]
  7. N. F. Borrelli, C. Smith, D. C. Allan, T. P. Seward, “Densification of fused silica under 193-nm excitation,” J. Opt. Soc. Am. B 14, 1606–1615 (1997).
    [CrossRef]
  8. S. M. Rekhson, “Thermal stresses, relaxation, and hysteresis in glass,” J. Am. Ceram. Soc. 76, 1113–1123 (1993).
    [CrossRef]
  9. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  10. W. Primak, D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient,” J. Appl. Phys. 30, 779–788 (1959).
    [CrossRef]
  11. E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1992), Chap. 10.
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chaps. 5 and 9.
  13. Ref. 12, Chap. 3.
  14. D. L. Fried, J. L. Vaughn, “Branch cuts in the phase function,” Appl. Opt. 31, 2865–2882 (1992).
    [CrossRef] [PubMed]
  15. F. S. Roux, “Diffractive lens with a null in the center of its focal point,” Appl. Opt. 32, 4191–4192 (1993).
    [CrossRef] [PubMed]
  16. M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
    [CrossRef]
  17. M. Gu, X. S. Gan, “Fresnel diffraction by a circular plane wave with optical phase singularities and its effect on the intensity distribution in the focal plane of a lens,” Optik 105, 51–56 (1997).
  18. H. Aben, J. Josepson, “Strange interference blots in the interferometry of inhomogeneous objects,” Appl. Opt. 36, 7172–7179 (1997).
    [CrossRef]
  19. R. Kingslake, “The interferometer patterns due to the primary aberrations,” Trans. Opt. Soc. London 27, 94–105 (1926).
    [CrossRef]

1997 (3)

1996 (2)

R. Schenker, W. Oldham, “Effects of compaction on 193 nm lithographic system performance,” J. Vac. Sci. Technol. B 14, 3709–3713 (1996).
[CrossRef]

D. C. Allan, C. Smith, N. F. Borrelli, T. P. Seward, “193-nm excimer-laser-induced densification of fused silica,” Opt. Lett. 21, 1960–1962 (1996).
[CrossRef] [PubMed]

1994 (1)

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

1993 (2)

S. M. Rekhson, “Thermal stresses, relaxation, and hysteresis in glass,” J. Am. Ceram. Soc. 76, 1113–1123 (1993).
[CrossRef]

F. S. Roux, “Diffractive lens with a null in the center of its focal point,” Appl. Opt. 32, 4191–4192 (1993).
[CrossRef] [PubMed]

1992 (1)

1989 (1)

M. Rothchild, D. J. Ehrlich, D. C. Shaver, “Effects of excimer laser irradiation on the transmission, index of refraction, and density of ultraviolet grade fused silica,” Appl. Phys. Lett. 55, 1276–1278 (1989).
[CrossRef]

1959 (1)

W. Primak, D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient,” J. Appl. Phys. 30, 779–788 (1959).
[CrossRef]

1941 (1)

1926 (1)

R. Kingslake, “The interferometer patterns due to the primary aberrations,” Trans. Opt. Soc. London 27, 94–105 (1926).
[CrossRef]

Aben, H.

Allan, D. C.

Arakawa, M.

H. Shinonaga, M. Arakawa, “Stepper exposure system for the quarter-micrometer age,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 754–766 (1996).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chaps. 5 and 9.

Borrelli, N. F.

Collett, E.

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1992), Chap. 10.

Ehrlich, D. J.

M. Rothchild, D. J. Ehrlich, D. C. Shaver, “Effects of excimer laser irradiation on the transmission, index of refraction, and density of ultraviolet grade fused silica,” Appl. Phys. Lett. 55, 1276–1278 (1989).
[CrossRef]

Fried, D. L.

Gan, X. S.

M. Gu, X. S. Gan, “Fresnel diffraction by a circular plane wave with optical phase singularities and its effect on the intensity distribution in the focal plane of a lens,” Optik 105, 51–56 (1997).

Gu, M.

M. Gu, X. S. Gan, “Fresnel diffraction by a circular plane wave with optical phase singularities and its effect on the intensity distribution in the focal plane of a lens,” Optik 105, 51–56 (1997).

Harris, M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Hill, C. A.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Jones, R. C.

Josepson, J.

Kingslake, R.

R. Kingslake, “The interferometer patterns due to the primary aberrations,” Trans. Opt. Soc. London 27, 94–105 (1926).
[CrossRef]

Nishi, K.

K. Suzuki, S. Wakamoto, K. Nishi, “KrF step-and-scan exposure system using higher-NA projection lens,” Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 767–779 (1996).
[CrossRef]

Oldham, W.

R. Schenker, W. Oldham, “Effects of compaction on 193 nm lithographic system performance,” J. Vac. Sci. Technol. B 14, 3709–3713 (1996).
[CrossRef]

Post, D.

W. Primak, D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient,” J. Appl. Phys. 30, 779–788 (1959).
[CrossRef]

Primak, W.

W. Primak, D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient,” J. Appl. Phys. 30, 779–788 (1959).
[CrossRef]

Rekhson, S. M.

S. M. Rekhson, “Thermal stresses, relaxation, and hysteresis in glass,” J. Am. Ceram. Soc. 76, 1113–1123 (1993).
[CrossRef]

Rothchild, M.

M. Rothchild, D. J. Ehrlich, D. C. Shaver, “Effects of excimer laser irradiation on the transmission, index of refraction, and density of ultraviolet grade fused silica,” Appl. Phys. Lett. 55, 1276–1278 (1989).
[CrossRef]

Roux, F. S.

Schenker, R.

R. Schenker, W. Oldham, “Effects of compaction on 193 nm lithographic system performance,” J. Vac. Sci. Technol. B 14, 3709–3713 (1996).
[CrossRef]

Seward, T. P.

Shaver, D. C.

M. Rothchild, D. J. Ehrlich, D. C. Shaver, “Effects of excimer laser irradiation on the transmission, index of refraction, and density of ultraviolet grade fused silica,” Appl. Phys. Lett. 55, 1276–1278 (1989).
[CrossRef]

Shinonaga, H.

H. Shinonaga, M. Arakawa, “Stepper exposure system for the quarter-micrometer age,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 754–766 (1996).
[CrossRef]

Smith, C.

Suzuki, K.

K. Suzuki, S. Wakamoto, K. Nishi, “KrF step-and-scan exposure system using higher-NA projection lens,” Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 767–779 (1996).
[CrossRef]

Vaughan, J. M.

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Vaughn, J. L.

Wakamoto, S.

K. Suzuki, S. Wakamoto, K. Nishi, “KrF step-and-scan exposure system using higher-NA projection lens,” Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 767–779 (1996).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chaps. 5 and 9.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

M. Rothchild, D. J. Ehrlich, D. C. Shaver, “Effects of excimer laser irradiation on the transmission, index of refraction, and density of ultraviolet grade fused silica,” Appl. Phys. Lett. 55, 1276–1278 (1989).
[CrossRef]

J. Am. Ceram. Soc. (1)

S. M. Rekhson, “Thermal stresses, relaxation, and hysteresis in glass,” J. Am. Ceram. Soc. 76, 1113–1123 (1993).
[CrossRef]

J. Appl. Phys. (1)

W. Primak, D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient,” J. Appl. Phys. 30, 779–788 (1959).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

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

R. Schenker, W. Oldham, “Effects of compaction on 193 nm lithographic system performance,” J. Vac. Sci. Technol. B 14, 3709–3713 (1996).
[CrossRef]

Opt. Commun. (1)

M. Harris, C. A. Hill, J. M. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Opt. Lett. (1)

Optik (1)

M. Gu, X. S. Gan, “Fresnel diffraction by a circular plane wave with optical phase singularities and its effect on the intensity distribution in the focal plane of a lens,” Optik 105, 51–56 (1997).

Trans. Opt. Soc. London (1)

R. Kingslake, “The interferometer patterns due to the primary aberrations,” Trans. Opt. Soc. London 27, 94–105 (1926).
[CrossRef]

Other (6)

P. Rai-Choudhury, ed., Handbook of Microlithography, Micromachining, and Microfabrication, Volume 1: Microlithography (SPIE Optical Engineering Press, Bellingham, Wash., 1997), Chap. 1.

H. Shinonaga, M. Arakawa, “Stepper exposure system for the quarter-micrometer age,” in Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 754–766 (1996).
[CrossRef]

K. Suzuki, S. Wakamoto, K. Nishi, “KrF step-and-scan exposure system using higher-NA projection lens,” Optical Microlithography IX, G. E. Fuller, ed., Proc. SPIE.2726, 767–779 (1996).
[CrossRef]

E. Collett, Polarized Light: Fundamentals and Applications (Marcel Dekker, New York, 1992), Chap. 10.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chaps. 5 and 9.

Ref. 12, Chap. 3.

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Figures (10)

Fig. 1
Fig. 1

Plate model describing the influence of rotationally symmetric birefringence on the transmitting wave front entering point P(r, θ).

Fig. 2
Fig. 2

Force equilibrium relationship for a small fan-shaped structure defined around point P(r, θ).

Fig. 3
Fig. 3

Optical configuration used for the derivation of the pupil function.

Fig. 4
Fig. 4

Distribution of A x (ρ, α), the amplitude transmittance for the x-polarization component, calculated for the y-polarized illumination beam assuming that the lens has a birefringence characterized by φ(ρ) = 2πρ2.

Fig. 5
Fig. 5

Distribution of A y (ρ, α), the amplitude transmittance for the y-polarization component, calculated with the same conditions as those of A x (ρ, α).

Fig. 6
Fig. 6

Distribution of W y (ρ, α) (m = 0), the wave-front aberration for the y-polarization component having the amplitude transmittance distribution of A y (ρ, α). Phase changes along the cross sections A′–A, B′–B, C′–C, and D′–D are also presented.

Fig. 7
Fig. 7

(a) Modified wave front created by adding π in 2 /2 ≤ ρ ≤ 1 to the distribution of Fig. 6. (b) Wave front created by subtracting π in 2 /2 ≤ ρ ≤ 1 from the distribution of Fig. 6. In (c) and (d) the thick curves represent the locations where the phase jumps of 2π occur.

Fig. 8
Fig. 8

N represents an ordinary point around which the phase value is continuously connected on a closed path, whereas S represents one of the singular points given by (ρ, α) = ( 2 /2, π/4).

Fig. 9
Fig. 9

Phase change on a circle around point S. The value is restricted between -π/2 and π/2 in (a) whereas the curve is continuously connected at γ = 3π/4, 7π/4 in (b).

Fig. 10
Fig. 10

Fringe patterns that are expected to be observed for the y-polarization component after the influence of astigmatic aberration is removed.

Equations (32)

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Ψ r = 2 π λ   Ch σ θ r - σ r r ,
- σ r - d σ r 2 r - d r 2 d θ + σ r + d σ r 2 r + d r 2 d θ - σ θ d θ d r = 0 ,
σ θ r - σ r r = r σ r r
σ θ r - σ r r = σ r r + r σ r r .
Ψ 0 = Ψ 0 = 0
Φ r = 2 π λ   Ch r σ θ r - σ r r ,
Φ 0 = Φ 0 = 0
p x , k p y , k = R - θ k exp i Φ k r k / 2 0 0 exp - i Φ k r k / 2 × R θ k p x , k - 1 p y , k - 1 ,
R θ k = cos   θ k sin   θ k - sin   θ k cos   θ k ,
M k = R - θ k exp i Φ k r k / 2 0 0 exp - i Φ k r k / 2 R θ k ,
p x ρ ,   α p y ρ ,   α = k = N , - 1 1   M k p x , 0 p y , 0 ,
θ 1 = θ 2 = = θ N - 1 = θ N = α
p x , 0 p y , 0 = 0 1 .
p x ρ ,   α p y ρ ,   α = R - α × k = N , - 1 1 exp i Φ k r k / 2 0 0 exp - i Φ k r k / 2 × R α 0 1 .
k = N , - 1 1 exp i Φ k r k / 2 0 0 exp - i Φ k r k / 2 = exp i 2 k = 1 N   Φ k r k 0 0 exp - i 2 k = 1 N   Φ k r k
p x ρ ,   α p y ρ ,   α = i   sin φ ρ / 2 sin 2 α cos φ ρ / 2 - i   sin φ ρ / 2 cos 2 α
φ ρ k = 1 N   Φ k r k .
p x ρ ,   α p y ρ ,   α = A x ρ ,   α exp iW x ρ ,   α A y ρ ,   α exp iW y ρ ,   α ,
A x ρ ,   α = sin φ ρ / 2 sin 2 α , W x ρ ,   α = π / 2 ,
A y ρ ,   α = 1 - sin 2 φ ρ / 2 sin 2 2 α 1 / 2 , W y ρ ,   α = arctan - tan φ ρ / 2 cos 2 α + m π ,
φ ρ = 2   n = 2   a n ρ n
A x ρ ,   α a 2 ρ 2   sin 2 α ,     W x ρ ,   α = π / 2 ,
A y ρ ,   α 1 ,     W y ρ ,   α - a 2 ρ 2   cos 2 α ,
φ ρ = 2 π ρ 2 0 ρ 1 ,
A x ρ ,   α = sin π ρ 2 sin 2 α ,
A y ρ ,   α = 1 - sin 2 π ρ 2 sin 2 2 α 1 / 2 ,
W y ρ ,   α = arctan - tan π ρ 2 cos 2 α + m π .
ρ   cos   α = 1 / 2 + δ   cos   γ ,     ρ   sin   α = 1 / 2 + δ   sin   γ ,
tan π ρ 2 - 1 π δ cos   γ + sin   γ ,
cos 2 α δ cos   γ - sin   γ 1 / 2 + δ cos   γ + sin   γ .
ω s γ arctan 2 cos   γ - sin   γ π cos   γ + sin   γ + m π
W fringe ρ ,   α = W y ρ ,   α + π ρ 2   cos 2 α + 12 π ρ 2 ,

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