Abstract

A method for the numerical simulation of partially coherent imaging systems is introduced. The two-dimensional source and pupil functions are decomposed into outer-product sums by the singular value decomposition algorithm, thus reducing the computation of the corresponding nonlinear transform function. The method is computationally efficient when the source and pupil matrices are of sufficiently low rank. Numerical examples are studied and compared with theoretical results.

© 1994 Optical Society of America

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References

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  1. D. N. Nyssonen, “Linewidth measurement with an optical microscope: the effect of operating conditions on the image profile,” Appl. Opt. 16, 2223–2230 (1977).
    [Crossref]
  2. R. Hoffman, L. Gross, “Modulation contrast microscope,” Appl. Opt. 14, 1169–1176 (1975).
    [Crossref] [PubMed]
  3. D. S. Goodman, A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” in Optical Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 108–134 (1988).
  4. R. E. Swing, “Conditions for microdensitometer linearity,” J. Opt. Soc. Am. 62, 199–207 (1972).
    [Crossref]
  5. J. van der Gracht, “Partially coherent image enhancement by source modification,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).
  6. J. van der Gracht, W. T. Rhodes, “partially coherent periodic stop filtering for inspection of periodic objects,” Appl. Opt. 32, 3708–3714 (1993).
    [Crossref] [PubMed]
  7. Y. Ichioka, K. Yamamoto, T. Suzuki, “Defocused image of a periodic complex object in an optical system under partially coherent illumination,” J. Opt. Soc. Am. 66, 932–938 (1976).
    [Crossref]
  8. E. C. Kintner, “Method for the calculation of partially coherent imagery,” Appl. Opt. 17, 2747–2753 (1978).
    [Crossref] [PubMed]
  9. S. Subramanian, “Rapid calculation of defocused partially coherent images,” Appl. Opt. 20, 1854–1857 (1981).
    [Crossref] [PubMed]
  10. H. H. Hopkins, “On the diffraction of optical images,” Proc. R. Soc. London 17, 408–432 (1953).
  11. B. E. A. Saleh, M. Rabbani, “Simulation of partially coherent imagery in the space and frequency domains by modal expansions,” Appl. Opt. 21, 2770–2777 (1982).
    [Crossref] [PubMed]
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 10, pp. 526–532.
  13. B. E. A. Saleh, “Optical bilinear transforms,” Opt. Act. 26, 777–799 (1979).
    [Crossref]
  14. D. S. Goodman, “Stationary optical projectors,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1979).
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, p. 105.
  16. J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985), Chap. 5, p. 180.
  17. A. W. Lohmann, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
    [Crossref]
  18. G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), App. A, pp. 442–452.
  19. D. N. Grimes, B. J. Thompson, “Two-point resolution with partially coherent light,” J. Opt. Soc. Am. 57, 1330–1334 (1967).
    [Crossref] [PubMed]
  20. B. J. Thompson, “Image formation with partially coherent light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 191–202.
    [Crossref]

1993 (1)

1984 (1)

A. W. Lohmann, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

1982 (1)

1981 (1)

1979 (1)

B. E. A. Saleh, “Optical bilinear transforms,” Opt. Act. 26, 777–799 (1979).
[Crossref]

1978 (1)

1977 (1)

1976 (1)

1975 (1)

1972 (1)

1967 (1)

1953 (1)

H. H. Hopkins, “On the diffraction of optical images,” Proc. R. Soc. London 17, 408–432 (1953).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 10, pp. 526–532.

Goodman, D. S.

D. S. Goodman, A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” in Optical Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 108–134 (1988).

D. S. Goodman, “Stationary optical projectors,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1979).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, p. 105.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985), Chap. 5, p. 180.

Grimes, D. N.

Gross, L.

Hoffman, R.

Hopkins, H. H.

H. H. Hopkins, “On the diffraction of optical images,” Proc. R. Soc. London 17, 408–432 (1953).

Ichioka, Y.

Kintner, E. C.

Lohmann, A. W.

A. W. Lohmann, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

Nyssonen, D. N.

Rabbani, M.

Rhodes, W. T.

Rosenbluth, A. E.

D. S. Goodman, A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” in Optical Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 108–134 (1988).

Saleh, B. E. A.

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), App. A, pp. 442–452.

Subramanian, S.

Suzuki, T.

Swing, R. E.

Thompson, B. J.

D. N. Grimes, B. J. Thompson, “Two-point resolution with partially coherent light,” J. Opt. Soc. Am. 57, 1330–1334 (1967).
[Crossref] [PubMed]

B. J. Thompson, “Image formation with partially coherent light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 191–202.
[Crossref]

van der Gracht, J.

J. van der Gracht, W. T. Rhodes, “partially coherent periodic stop filtering for inspection of periodic objects,” Appl. Opt. 32, 3708–3714 (1993).
[Crossref] [PubMed]

J. van der Gracht, “Partially coherent image enhancement by source modification,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 10, pp. 526–532.

Yamamoto, K.

Appl. Opt. (6)

J. Opt. Soc. Am. (3)

Opt. Act. (1)

B. E. A. Saleh, “Optical bilinear transforms,” Opt. Act. 26, 777–799 (1979).
[Crossref]

Proc. IEEE (1)

A. W. Lohmann, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[Crossref]

Proc. R. Soc. London (1)

H. H. Hopkins, “On the diffraction of optical images,” Proc. R. Soc. London 17, 408–432 (1953).

Other (8)

J. van der Gracht, “Partially coherent image enhancement by source modification,” Ph.D. dissertation (Georgia Institute of Technology, Atlanta, Ga., 1991).

D. S. Goodman, A. E. Rosenbluth, “Condenser aberrations in Kohler illumination,” in Optical Laser Microlithography, B. J. Lin, ed., Proc. Soc. Photo-Opt. Instrum. Eng.922, 108–134 (1988).

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988), App. A, pp. 442–452.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1989), Chap. 10, pp. 526–532.

D. S. Goodman, “Stationary optical projectors,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1979).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 6, p. 105.

J. W. Goodman, Statistical Optics (Wiley-Interscience, New York, 1985), Chap. 5, p. 180.

B. J. Thompson, “Image formation with partially coherent light,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1969), Vol. 7, pp. 191–202.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

Graphical interpretation of the calculation of the BLT. The shaded circle represents the source distribution; the two clear circles represent displaced pupil distributions. The area of the overlap must be evaluated for all possible pairs of pupil position.

Fig. 2
Fig. 2

Partially coherent imaging of a two-pinhole object through a clear circular pupil.

Fig. 3
Fig. 3

Spatially discretized version of the circular pupil. Ten outer-product terms are necessary for exact representation.

Fig. 4
Fig. 4

Four central slices of two-point imaging simulation runs. The pupil is circular, and the four curves correspond to four different degrees of illumination coherence. Solid curve, μ = 1.0; dashed-dotted curve, μ = 0.55; dashed curve, μ = 0.20; dotted curve; μ = 0.0.

Fig. 5
Fig. 5

Central slice of two-point image intensity with object illumination degree of coherence of 0.2. The theoretical image intensity (solid curve) is plotted with the simulation result (dashed curve).

Fig. 6
Fig. 6

Two-point image intensity slices computed with truncated pupil outer-product expansion. Solid curve, P = 10 terms; dashed curve, P = 2 terms.

Fig. 7
Fig. 7

Two-point-image intensity slices computed with truncated pupil outer-product expansion. The curve has been expanded to show detail in the region of a lobe to the left of the central portion of the image slice. Solid curve, P = 10 terms; dashed curve, P = 4 terms; dotted curve, P = 2 terms.

Fig. 8
Fig. 8

Plot of the incremental energy Ej associated with the partial image obtained for each successive pupil expansion term. Note that the y axis is actually ten times the log of Ej.

Equations (19)

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I ( x , y ) = O ( x 1 , y 1 ) O * ( x 2 , y 2 ) h ( x - x 1 , y - y 1 ) × h * ( x - x 2 , y - y 2 ) × J ( x 1 - x 2 , y 1 - y 2 ) d x 1 d x 2 d y 1 d y 2 ,
I ˜ ( f , g ) = δ ( f + f 2 - f 1 , g + g 2 - g 1 ) × O ˜ ( f 1 , g 1 ) O ˜ * ( f 2 , g 2 ) × C ( f 1 , g 1 ; f 2 , g 2 ) d f 1 d g 1 d f 2 d g 2 ,
C ( f 1 , g 1 ; f 2 , g 2 ) = J ˜ ( f , g ) H ( f + f 1 , g + g 1 ) × H * ( f + f 2 , g + g 2 ) d f d g ,
C ( f 1 , g 1 ; f 2 , g 2 ) = C X ( f 1 ; f 2 ) C Y ( g 1 ; g 2 ) ,
C X ( f 1 ; f 2 ) = J ˜ X ( f ) H X ( f + f 1 ) H X * ( f + f 2 ) d f ,
C Y ( g 1 ; g 2 ) = J ˜ Y ( g ) H Y ( g + g 1 ) H Y * ( g + g 2 ) d g .
J ˜ ( f , g ) = k = 1 S J ˜ X k ( f ) J ˜ Y k ( g ) .
H ( f , g ) = k = 1 P H X k ( f ) H Y k ( g ) .
C ( f 1 , g 1 ; f 2 , g 2 ) = k = 1 S l = 1 P m = 1 P C X k l m ( f 1 ; f 2 ) C Y k l m ( g 1 ; g 2 ) ,
C X k l m ( f 1 ; f 2 ) = J ˜ X k ( f ) H X l ( f + f 1 ) H X m * ( f + f 2 ) d f ,
C Y k l m ( g 1 ; g 2 ) = J ˜ Y k ( g ) H Y l ( g + g 1 ) H Y m * ( g + g 2 ) d g .
C X ( f 1 ; f 2 ) = [ δ ( f 1 - f 2 ) J ˜ X ( f 1 + f 2 2 ) ] * * [ H X ( f 1 ) H X * ( f 2 ) ] ,
M BSVD = S P 2 [ N 4 2 + 2 N 2 ( log 2 N + 1 ) ] + o ( N 3 ) ,
M BSVD S P 2 N 4 2 .
I 2 pt ( x ) = I 0 J 1 [ π D ( x - x 1 ) / λ F ] 2 + I 0 J 1 [ π D ( x - x 2 ) / λ F ] 2 + μ ( x 1 - x 2 ) × J 1 [ π D ( x - x 1 ) / λ F ] J 1 [ π D ( x - x 2 ) / λ F ] ,
I ˜ ( f , g ) = k = 1 P m = 1 P I ˜ k m ( f , g ) ,
I ˜ k m ( f , g ) = δ ( f + f 2 - f 1 , g + g 2 - g 1 ) O ˜ ( f 1 , g 1 ) × O ˜ * ( f 2 , g 2 ) C X k 1 m ( f 1 ; f 2 ) C Y k 1 m ( g 1 ; g 2 ) × d f 1 d f 2 d g 1 d g 2 .
I ˜ ( f , g ) = p = 1 P I ˜ p ( f , g ) ,
I ˜ p ( f , g ) = m = 1 P I ˜ p m ( f , g ) + I ˜ m p ( f , g ) .

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