Abstract

We examine theoretically and experimentally the characteristics of in-focus and out-of-focus images of simple, well-defined phase objects. Theoretical calculations are based on the theory of partial coherence, and a simple calculation for imaging with coherent light demonstrates distinctive aspects of bright-field images. Experiments are performed with a well-corrected microscope, equipped for the precise control of illumination conditions and focus position. Theoretical and experimental results agree, although the contrast in the experimental images is often lower than expected. Also verified by experiment is a (to our knowledge) previously uninvestigated linear response in the intensity modulation of defocused, coherent images of thin, phase objects. The near-focus behavior of phase object images differs in symmetry from the more-familiar behavior of opaque object images.

© 1988 Optical Society of America

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References

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  1. H. H. Hopkins, “Phase objects as seen in the ordinary microscope,” in, Contraste de Phase et Contraste par Interférences, M. Françon, ed. (Editions de la Revue d’Optique Théorique et Instrumentale, Paris, 1952), pp. 142–152.
  2. F. Zernike, “Das Phasenkontrastverfahren bei der Mikroskopischen Beobachtung,” Z. Tech. Phys. 16, 454–457 (1935).
  3. G. Nomarski, “Microinterférometric différentielle à ondes polarisées,” J. Phys. Radium 16, 9S–13S (1955).
  4. D. Brewster, “Report on the recent progress of optics,” in Report of the British Association for the Advancement of Science (Murray, London, 1833), Vol. 2, pp. 308–322.
  5. R. Hooke, “Microscopium,” in Lectures and Collections Made by Robert Hooke (Martyn, London, 1678), pp. 81–112.
  6. 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]
  7. M. Françon, Contraste de Phase en Optique et en Microscopie (Editions de la Revue d’Optique Théorique et Instrumentale, Paris, 1950), pp. 35–39.
  8. H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
    [CrossRef]
  9. N. Streibl, “Three-dimensional imaging in the microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  10. W. N. Charman, “Some experimental measurements of diffraction images in low-resolution microscopy,” J. Opt. Soc. Am. 53, 410–414 (1963).
    [CrossRef]
  11. B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta 12, 391–400 (1965).
    [CrossRef]
  12. M. Born, E. Wolf, Principles of Optics, 6th. ed. (Pergamon, Oxford, 1980).
  13. M. M. O’Toole, “Simulation of optically formed profiles in positive photoresist,” Memo. No. UCB/ERL M79/42, Electronics Research Laboratory (University of California, Berkeley, Calif., 1979), pp. 5–23.
  14. M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
    [CrossRef]
  15. A. E. Rosenbluth, D. Goodman, B. J. Lin, “A critical examination of submicron optical lithography using simulated projection images,” J. Vac. Sci. Technol. B 1, 1190–1195 (1983).
    [CrossRef]
  16. K. J. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), pp. 354–360.
  17. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
    [CrossRef] [PubMed]
  18. E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
    [CrossRef]
  19. J. Goodman, Statistical Optics (Wiley, New York, 1985).
  20. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
    [CrossRef]
  21. See Ref. 11, p. 482.
  22. P. M. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [CrossRef]
  23. See Ref. 11, p. 529.
  24. H. H. Hopkins, “Applications of coherence theory in microscopy and interferometry,” J. Opt. Soc. Am. 47, 508–526 (1957).
    [CrossRef]
  25. K. Yamamoto, Y. Ichioka, T. Suzuki, “Influence of light coherence at the exit pupil of the condenser on the image formation,” Opt. Acta 23, 987–996 (1976).
    [CrossRef]
  26. R. E. Swing, “Conditions for microdensitometer linearity,” J. Opt. Soc. Am. 62, 199–207 (1972).
    [CrossRef]
  27. P. N. T. Unwin, R. Henderson, “Molecular structure determination by electron microscopy of unstained crystalline specimens,” J. Mol. Biol. 94, 425–440 (1975).
    [CrossRef] [PubMed]

1985 (2)

1984 (1)

M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
[CrossRef]

1983 (1)

A. E. Rosenbluth, D. Goodman, B. J. Lin, “A critical examination of submicron optical lithography using simulated projection images,” J. Vac. Sci. Technol. B 1, 1190–1195 (1983).
[CrossRef]

1976 (2)

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]

K. Yamamoto, Y. Ichioka, T. Suzuki, “Influence of light coherence at the exit pupil of the condenser on the image formation,” Opt. Acta 23, 987–996 (1976).
[CrossRef]

1975 (1)

P. N. T. Unwin, R. Henderson, “Molecular structure determination by electron microscopy of unstained crystalline specimens,” J. Mol. Biol. 94, 425–440 (1975).
[CrossRef] [PubMed]

1972 (1)

1970 (1)

E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

1969 (1)

1965 (1)

B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta 12, 391–400 (1965).
[CrossRef]

1963 (1)

1959 (1)

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

1957 (1)

1955 (1)

G. Nomarski, “Microinterférometric différentielle à ondes polarisées,” J. Phys. Radium 16, 9S–13S (1955).

1951 (1)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

1935 (1)

F. Zernike, “Das Phasenkontrastverfahren bei der Mikroskopischen Beobachtung,” Z. Tech. Phys. 16, 454–457 (1935).

Bayer, P. W.

M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
[CrossRef]

Bille, J.

Born, M.

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

Brewster, D.

D. Brewster, “Report on the recent progress of optics,” in Report of the British Association for the Advancement of Science (Murray, London, 1833), Vol. 2, pp. 308–322.

Castleman, K. J.

K. J. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), pp. 354–360.

Charman, W. N.

Erhardt, A.

Evans, E.

E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

Françon, M.

M. Françon, Contraste de Phase en Optique et en Microscopie (Editions de la Revue d’Optique Théorique et Instrumentale, Paris, 1950), pp. 35–39.

Goodman, D.

A. E. Rosenbluth, D. Goodman, B. J. Lin, “A critical examination of submicron optical lithography using simulated projection images,” J. Vac. Sci. Technol. B 1, 1190–1195 (1983).
[CrossRef]

Goodman, D. S.

M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
[CrossRef]

Goodman, J.

J. Goodman, Statistical Optics (Wiley, New York, 1985).

Henderson, R.

P. N. T. Unwin, R. Henderson, “Molecular structure determination by electron microscopy of unstained crystalline specimens,” J. Mol. Biol. 94, 425–440 (1975).
[CrossRef] [PubMed]

Hooke, R.

R. Hooke, “Microscopium,” in Lectures and Collections Made by Robert Hooke (Martyn, London, 1678), pp. 81–112.

Hopkins, H. H.

H. H. Hopkins, “Applications of coherence theory in microscopy and interferometry,” J. Opt. Soc. Am. 47, 508–526 (1957).
[CrossRef]

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

H. H. Hopkins, “Phase objects as seen in the ordinary microscope,” in, Contraste de Phase et Contraste par Interférences, M. Françon, ed. (Editions de la Revue d’Optique Théorique et Instrumentale, Paris, 1952), pp. 142–152.

Ichioka, Y.

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]

K. Yamamoto, Y. Ichioka, T. Suzuki, “Influence of light coherence at the exit pupil of the condenser on the image formation,” Opt. Acta 23, 987–996 (1976).
[CrossRef]

Komitowski, D.

Levenson, M. D.

M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
[CrossRef]

Lin, B. J.

A. E. Rosenbluth, D. Goodman, B. J. Lin, “A critical examination of submicron optical lithography using simulated projection images,” J. Vac. Sci. Technol. B 1, 1190–1195 (1983).
[CrossRef]

Lindsey, S.

M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
[CrossRef]

Nomarski, G.

G. Nomarski, “Microinterférometric différentielle à ondes polarisées,” J. Phys. Radium 16, 9S–13S (1955).

O’Toole, M. M.

M. M. O’Toole, “Simulation of optically formed profiles in positive photoresist,” Memo. No. UCB/ERL M79/42, Electronics Research Laboratory (University of California, Berkeley, Calif., 1979), pp. 5–23.

Richards, B.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Rosenbluth, A. E.

A. E. Rosenbluth, D. Goodman, B. J. Lin, “A critical examination of submicron optical lithography using simulated projection images,” J. Vac. Sci. Technol. B 1, 1190–1195 (1983).
[CrossRef]

Santini, H. A. E.

M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
[CrossRef]

Stokseth, P. M.

Streibl, N.

Suzuki, T.

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]

K. Yamamoto, Y. Ichioka, T. Suzuki, “Influence of light coherence at the exit pupil of the condenser on the image formation,” Opt. Acta 23, 987–996 (1976).
[CrossRef]

Swing, R. E.

Unwin, P. N. T.

P. N. T. Unwin, R. Henderson, “Molecular structure determination by electron microscopy of unstained crystalline specimens,” J. Mol. Biol. 94, 425–440 (1975).
[CrossRef] [PubMed]

Watrasiewicz, B. M.

B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta 12, 391–400 (1965).
[CrossRef]

Wolf, E.

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

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

Yamamoto, K.

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]

K. Yamamoto, Y. Ichioka, T. Suzuki, “Influence of light coherence at the exit pupil of the condenser on the image formation,” Opt. Acta 23, 987–996 (1976).
[CrossRef]

Zernike, F.

F. Zernike, “Das Phasenkontrastverfahren bei der Mikroskopischen Beobachtung,” Z. Tech. Phys. 16, 454–457 (1935).

Zinser, G.

Appl. Opt. (1)

IEEE Trans. Electron Devices (1)

M. D. Levenson, D. S. Goodman, S. Lindsey, P. W. Bayer, H. A. E. Santini, “The phase shifting mask II: imaging simulations and submicrometer resist exposures,” IEEE Trans. Electron Devices ED-31, 753–763 (1984).
[CrossRef]

J. Mol. Biol. (1)

P. N. T. Unwin, R. Henderson, “Molecular structure determination by electron microscopy of unstained crystalline specimens,” J. Mol. Biol. 94, 425–440 (1975).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (5)

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

J. Phys. Radium (1)

G. Nomarski, “Microinterférometric différentielle à ondes polarisées,” J. Phys. Radium 16, 9S–13S (1955).

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

A. E. Rosenbluth, D. Goodman, B. J. Lin, “A critical examination of submicron optical lithography using simulated projection images,” J. Vac. Sci. Technol. B 1, 1190–1195 (1983).
[CrossRef]

Opt. Acta (2)

K. Yamamoto, Y. Ichioka, T. Suzuki, “Influence of light coherence at the exit pupil of the condenser on the image formation,” Opt. Acta 23, 987–996 (1976).
[CrossRef]

B. M. Watrasiewicz, “Theoretical calculations of images of straight edges in partially coherent illumination,” Opt. Acta 12, 391–400 (1965).
[CrossRef]

Opt. Commun. (1)

E. Evans, “Comparison of the diffraction theory of image formation with the three-dimensional, first Born scattering approximation in lens systems,” Opt. Commun. 2, 317–320 (1970).
[CrossRef]

Proc. R. Soc. London Ser. A (2)

H. H. Hopkins, “The concept of partial coherence in optics,” Proc. R. Soc. London Ser. A 208, 263–277 (1951).
[CrossRef]

B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London Ser. A 253, 358–379 (1959).
[CrossRef]

Z. Tech. Phys. (1)

F. Zernike, “Das Phasenkontrastverfahren bei der Mikroskopischen Beobachtung,” Z. Tech. Phys. 16, 454–457 (1935).

Other (10)

H. H. Hopkins, “Phase objects as seen in the ordinary microscope,” in, Contraste de Phase et Contraste par Interférences, M. Françon, ed. (Editions de la Revue d’Optique Théorique et Instrumentale, Paris, 1952), pp. 142–152.

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

M. M. O’Toole, “Simulation of optically formed profiles in positive photoresist,” Memo. No. UCB/ERL M79/42, Electronics Research Laboratory (University of California, Berkeley, Calif., 1979), pp. 5–23.

D. Brewster, “Report on the recent progress of optics,” in Report of the British Association for the Advancement of Science (Murray, London, 1833), Vol. 2, pp. 308–322.

R. Hooke, “Microscopium,” in Lectures and Collections Made by Robert Hooke (Martyn, London, 1678), pp. 81–112.

M. Françon, Contraste de Phase en Optique et en Microscopie (Editions de la Revue d’Optique Théorique et Instrumentale, Paris, 1950), pp. 35–39.

See Ref. 11, p. 482.

See Ref. 11, p. 529.

J. Goodman, Statistical Optics (Wiley, New York, 1985).

K. J. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979), pp. 354–360.

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Figures (12)

Fig. 1
Fig. 1

Microscope coordinates: feyepiece is the focal length of the eyepiece, fobj is the focal length of the objective lens, and fc is the condenser focal length.

Fig. 2
Fig. 2

Microscope schematic. Köhler illumination is obtained with the aux/field lens, located between the field stop and the source. This lens images the source onto the front focal plane of the condenser lens. The condenser stop defines, then, the approximate cone angle of the illumination.

Fig. 3
Fig. 3

Image-analysis system.

Fig. 4
Fig. 4

Alignment of the target for imaging. Averaging of several scans (in ξ) at several different η values improves the measurement precision. Spacing, thickness, and width combine with optical properties to define the object for comparison with theory.

Fig. 5
Fig. 5

Photograph of an image of a test reticle pattern in photoresist. The two squares in the field of view are about 100 μm across. The pattern is in a 0.35-μm-thick layer of resist with an index change of approximately 0.13, for a phase shift of about λ/12 at λ = 0.546 μm. The precise focus position varies across the field of view, since the object is tilted slightly. This photo was recorded, with some contrast enhancement, at approximately −10.0-μm defocus. σ = 0.30.

Fig. 6
Fig. 6

(a)–(d) Theoretical and (e)–(h) experimental results for imaging of a 9.9-deg reticle. The bars in these experiments were separated by 4.0 μm, center to center, and were approximately 1.7 μm in width. The thickness of the resist layer varied for different phase shifts from 0.35 to 1.00 μm. Imaging was with the four values of coherence parameter noted in the figure. σ = (N.A.)condenser/(N.A.)objective.

Fig. 7
Fig. 7

(a), (b) Theoretical and (c), (d) experimental results for imaging of a 60-deg target.

Fig. 8
Fig. 8

(a), (b) Theoretical and (c), (d) experimental results for imaging of a 127-deg target.

Fig. 9
Fig. 9

Contrast of the center location of an in-focus three-bar image, [I(0, 0) − I0]/I0, versus the phase shift Φ for four levels of coherence σ: (a) 0.30, (b) 0.50, (c) 0.70, and (d) 1.00.

Fig. 10
Fig. 10

Contrast of the center location of a defocused three-bar image, [I(0, 0) − I0]/I0, versus the phase shift Φ for four levels of coherence σ: (a) 0.30, (b) 0.50, (c) 0.70, and (d) 1.00 with +10-μm defocus; (e) 1.00, (f) 0.70, (g) 0.50, and (h) 0.30, with −10-μm defocus. Plots at σ = 0.30, 0.50, 0.70 show linear behavior up to almost 30 deg. The symmetry about focus degrades rapidly with increasing σ.

Fig. 11
Fig. 11

Images (a), (c) at σ = 0.30 and (b), (d) at σ = 0.70 of 0.8-μm bars, separated by 2.0 μm, center to center. Φ = 21 deg.

Fig. 12
Fig. 12

Images (a), (c) at σ = 0.30 and (b), (d) at σ = 0.70 for a 5.0-μm bar. Φ = 21 deg.

Equations (28)

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U ( ξ , η ) = B t ( ξ , η ) .
U image ( x , y ) = K U ( ξ , η ) air ( x ξ , y n ) d ξ d η .
x = x image / M
y = y image / M .
air ( x , y ) = P ( μ , ν ) exp [ i W ( μ , ν ) ] exp [ i 2 π ( μ x + ν y ) ] d μ d ν ,
μ = x l / λ f obj
ν = y l / λ f obj
P ( μ , ν ) = 1 within the aperture = 0 elsewhere .
t ( ξ , η ) = exp [ i Φ ( ξ , η ) ] .
t ( ξ , η ) = 1 [ 1 exp ( i Φ ) ] A ( ξ , η ) .
A ( ξ , η ) = 1 within the object = 0 otherwies
a ( μ , ν ) = A ( ξ , η ) exp [ i 2 π ( μ ξ + ν η ) ] d ξ d η .
W ( μ , ν ) = w ( μ 2 + ν 2 ) .
w = 2 π λ δ z ( 1 cos θ ) f obj 2 / r 2 .
I ( x , y ) = K 2 B 2 { 1 2 sin Φ Q 1 ( x , y ) ( 2 2 cos Φ ) Q 2 ( x , y ) + ( 2 2 cos Φ ) [ Q 1 2 ( x , y ) + Q 2 2 ( x , y ) ] } ,
Q 1 ( x , y ) = a ( μ , ν ) sin [ w ( μ 2 + ν 2 ) ] cos [ 2 π ( μ x + ν y ) d μ d ν
Q 2 ( x , y ) = a ( μ , ν ) cos [ w ( μ 2 + ν 2 ) ] cos [ 2 π ( μ x + ν y ) ] d μ d ν .
Q 1 ( x , y ) = 0
Q 2 ( x , y ) = Q 2 2 ( x , y ) .
2 sin Φ Q 1 ( x , y ) ,
I ( x , y ) = K 2 J ( ξ 1 ξ 2 , η 1 η 2 ) t ( ξ 1 , η 1 ) t * ( ξ 2 , η 2 ) × air λ m ( x ξ 1 , y η 1 ) air λ m * ( x ξ 2 , y η 2 ) d ξ 1 d η 1 d ξ 2 d η 2 .
J ( ξ 1 ξ 2 , η 1 η 2 ) = I ( α , β ) exp { i 2 π [ α ( ξ 1 ξ 2 ) + β ( η 1 η 2 ) ] } d α d β ,
α = x s / λ m f c
β = y s / λ m f c .
σ = ( N . A . ) condenser ( N . A . ) objective = n sin α n sin θ .
t ( ξ , η ) = exp [ i ( m / 2 ) sin ( 2 π f ξ ) ] .
u ( μ , ν ) = B [ n = J n ( m / 2 ) δ ( μ n f ) ] δ ( ν ) .
I ( x , y ) = K 2 B 2 J 0 2 ( m / 2 ) { 1 4 [ J 1 ( m / 2 ) / J 0 ( m / 2 ) ] × sin ( w f 2 ) sin ( 2 π f x ) } .

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