Abstract

The defocused weak-object transfer function of a partially coherent bright-field microscope is calculated. For weak defocus, this can be expressed analytically. Use of this transfer function for phase restoration (quantitative phase retrieval) from images of weak mixed phase-amplitude objects is discussed.

© 2004 Optical Society of America

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References

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  1. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A. 231, 91–103 (1955).
    [CrossRef]
  2. H. H. Hopkins, “On the diffraction theory of optical images,” Proc. R. Soc. London Ser. A 217, 408–432 (1953).
    [CrossRef]
  3. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent object,” Physica 9, 686–693 (1942).
    [CrossRef]
  4. C. J. R. Sheppard, T. Wilson, “Fourier imaging of phase information in conventional and scanning microscopes,” Philos. Trans. R. Soc. London 295, 513–536 (1980).
    [CrossRef]
  5. B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  6. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  7. C. J. R. Sheppard, “Three-dimensional phase imaging with the intensity transport equation,” Appl. Opt. 41, 5951–5955 (2002).
    [CrossRef] [PubMed]
  8. E. D. Barone-Nugent, A. Barty, K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
    [CrossRef] [PubMed]
  9. M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  10. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  11. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [CrossRef]
  12. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  13. C. J. R. Sheppard, M. D. Sharma, “Integrated intensity, and imaging through scattering media,” J. Mod. Opt. 48, 1517–1525 (2001).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1989).
  15. C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
    [CrossRef]
  16. C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscopy with extended depth of field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
    [CrossRef]

2002 (2)

E. D. Barone-Nugent, A. Barty, K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

C. J. R. Sheppard, “Three-dimensional phase imaging with the intensity transport equation,” Appl. Opt. 41, 5951–5955 (2002).
[CrossRef] [PubMed]

2001 (1)

C. J. R. Sheppard, M. D. Sharma, “Integrated intensity, and imaging through scattering media,” J. Mod. Opt. 48, 1517–1525 (2001).
[CrossRef]

1998 (1)

1987 (1)

1985 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

1983 (2)

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscopy with extended depth of field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

M. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
[CrossRef]

1980 (1)

C. J. R. Sheppard, T. Wilson, “Fourier imaging of phase information in conventional and scanning microscopes,” Philos. Trans. R. Soc. London 295, 513–536 (1980).
[CrossRef]

1967 (1)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A. 231, 91–103 (1955).
[CrossRef]

1953 (1)

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

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent object,” Physica 9, 686–693 (1942).
[CrossRef]

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Barty, K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

Barty, A.

E. D. Barone-Nugent, A. Barty, K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1989).

Cox, I. J.

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscopy with extended depth of field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

Frieden, B. R.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Hamilton, D. K.

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscopy with extended depth of field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

Hopkins, H. H.

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A. 231, 91–103 (1955).
[CrossRef]

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

Matthews, H. J.

Nugent, K. A.

E. D. Barone-Nugent, A. Barty, K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
[CrossRef]

Paganin, D.

Roberts, A.

Sharma, M. D.

C. J. R. Sheppard, M. D. Sharma, “Integrated intensity, and imaging through scattering media,” J. Mod. Opt. 48, 1517–1525 (2001).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard, “Three-dimensional phase imaging with the intensity transport equation,” Appl. Opt. 41, 5951–5955 (2002).
[CrossRef] [PubMed]

C. J. R. Sheppard, M. D. Sharma, “Integrated intensity, and imaging through scattering media,” J. Mod. Opt. 48, 1517–1525 (2001).
[CrossRef]

C. J. R. Sheppard, H. J. Matthews, “Imaging in high-aperture optical systems,” J. Opt. Soc. Am. A 4, 1354–1360 (1987).
[CrossRef]

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscopy with extended depth of field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “Fourier imaging of phase information in conventional and scanning microscopes,” Philos. Trans. R. Soc. London 295, 513–536 (1980).
[CrossRef]

Streibl, N.

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Teague, M.

Wilson, T.

C. J. R. Sheppard, T. Wilson, “Fourier imaging of phase information in conventional and scanning microscopes,” Philos. Trans. R. Soc. London 295, 513–536 (1980).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1989).

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent object,” Physica 9, 686–693 (1942).
[CrossRef]

Appl. Opt. (1)

J. Microsc. (1)

E. D. Barone-Nugent, A. Barty, K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

C. J. R. Sheppard, M. D. Sharma, “Integrated intensity, and imaging through scattering media,” J. Mod. Opt. 48, 1517–1525 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
[CrossRef]

Opt. Lett. (1)

Philos. Trans. R. Soc. London (1)

C. J. R. Sheppard, T. Wilson, “Fourier imaging of phase information in conventional and scanning microscopes,” Philos. Trans. R. Soc. London 295, 513–536 (1980).
[CrossRef]

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent object,” Physica 9, 686–693 (1942).
[CrossRef]

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

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

C. J. R. Sheppard, D. K. Hamilton, I. J. Cox, “Optical microscopy with extended depth of field,” Proc. R. Soc. London Ser. A 387, 171–186 (1983).
[CrossRef]

Proc. R. Soc. London Ser. A. (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. R. Soc. London Ser. A. 231, 91–103 (1955).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, UK, 1989).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

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Figures (4)

Fig. 1
Fig. 1

In-focus WOTF for a partially coherent bright-field microscope with coherence parameter S.

Fig. 2
Fig. 2

Defocused WOTF for a partially coherent bright-field microscope with coherence parameter S.

Fig. 3
Fig. 3

(Negative of the) imaginary part of the WOTF for a weakly defocused, partially coherent bright-field microscope with coherence parameter S.

Fig. 4
Fig. 4

(Negative of the) imaginary part of the WOTF for a weakly defocused, partially coherent bright-field microscope with coherence parameter S, after restoration with an inverse Laplacian filter.

Equations (18)

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S = n c sin   α c n o sin   α o ,
C W ( l ) = π S 2 , 0 l 1 - S ,
C W ( l ) = S 2 arccos l 2 + S 2 - 1 2 lS - l 2 + S 2 - 1 2 l×  S 2 - l 2 + S 2 - 1 2 l 2 1 / 2 + arccos l 2 - S 2 + 1 2 l - l 2 - S 2 + 1 2 l×  1 - l 2 - S 2 + 1 2 l 2 1 / 2 ,
1 - S l 1 + S .
P ( ρ ) = exp ( - 1 2 iu ρ 2 ) , ρ < 1 ,
u = 2 W 20 = 4 kz   sin 2 ( α o / 2 ) ,
C W ( l ;   0 ) = - ( s + l / 2 ) s - l / 2 2 S 2 - l 2 + x 2 1 / 2 d x = π S 2
C W ( l ;   u ) = - ( S + l / 2 ) S - l / 2 2 S 2 - l 2 + x 2 1 / 2×  exp ( iulx ) d x ,
0 l 1 - S ,
C W ( l ;   u ) = - ( 1 - S 2 ) / 2 l S - l / 2 2 S 2 - l 2 + x 2 1 / 2×  exp ( iulx ) d x + l / 2 - 1 - ( 1 - S 2 ) / 2 l 2 1 - l 2 - x 2 1 / 2×  exp ( iulx ) d x ,
1 - S l 1 + S .
C Wi = - 1 2 π ul 2 S 2 , 0 l 1 - S ,
C Wi = - ul 2 2 S 2   arccos l 2 + S 2 - 1 2 lS - arccos l 2 - S 2 + 1 2 l - u 6 l S 2 - l 2 + S 2 - 1 2 l 2 1 / 2×  ( 1 - S 2 ) 2 - l 2 2   ( 1 + l 2 + 7 S 2 ) - 1 - l 2 - S 2 + 1 2 l 2 1 / 2×  ( 1 - S 2 ) 2 - l 2 2   ( 7 + l 2 + S 2 ) ,
1 - S l 1 + S .
F ( l ) = - 1 2 π ul 2 S 2 ,
C Wir = 1 , 0 l 1 - S ,
C Wir = 1 2 π S 2 S 2   arccos l 2 + S 2 - 1 2 lS - arccos l 2 - S 2 + 1 2 l + 1 6 l 3 π S 2 S 2 - l 2 + S 2 - 1 2 l 2 1 / 2×  ( 1 - S 2 ) 2 - l 2 2   ( 1 + l 2 + 7 S 2 ) - 1 - l 2 - S 2 + 1 2 l 2 1 / 2×  ( 1 - S 2 ) 2 - l 2 2   ( 7 + l 2 + S 2 ) ,
1 - S l 1 + S .

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