Abstract

Phase can be retrieved from intensity measurements with the intensity transport equation. Three-dimensional image formation of weak phase objects based on this method is investigated. It is shown that, although the refractive index of a thin object can be measured, the three-dimensional variation of refractive index of an arbitrary object cannot, in general, be reconstructed, as spatial frequencies with a zero-axial component are not detected. However, this may not be a problem if regions with known refractive index are present in the sample.

© 2002 Optical Society of America

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References

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  1. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quantitative optical phase microscopy,” Opt. Lett. 23, 817–819 (1998).
    [CrossRef]
  2. A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
    [CrossRef]
  3. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. 73, 1434–1441 (1983).
    [CrossRef]
  4. N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49, 6–10 (1984).
    [CrossRef]
  5. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  6. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  7. C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
    [CrossRef]
  8. C. J. R. Sheppard, M. Gu, “The three-dimensional (3-D) transmission cross coefficient for transmission imaging,” Optik 100, 155–158 (1994).
  9. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).
  10. C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994).
    [CrossRef]
  11. M. Gu, C. J. R. Sheppard, “Three-dimensional partially-coherent image formation in confocal microscopes with a finite-sized detector,” J. Mod. Opt. 41, 1701–1715 (1994).
    [CrossRef]
  12. C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope images,” in Image Reconstruction and Restoration, T. J. Schulz, L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
    [CrossRef]
  13. C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 377–390 (1991).
  14. C. J. R. Sheppard, C. J. Cogswell, “Image formation in video-enhanced and confocal DIC microscopy,” in Phase Contrast and Differential Phase Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 64–71 (1992).
    [CrossRef]

2000

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

1998

1994

C. J. R. Sheppard, M. Gu, “The three-dimensional (3-D) transmission cross coefficient for transmission imaging,” Optik 100, 155–158 (1994).

M. Gu, C. J. R. Sheppard, “Three-dimensional partially-coherent image formation in confocal microscopes with a finite-sized detector,” J. Mod. Opt. 41, 1701–1715 (1994).
[CrossRef]

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

1991

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 377–390 (1991).

1989

1986

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

1985

1984

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

1983

1969

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Barty, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

Cogswell, C. J.

C. J. R. Sheppard, C. J. Cogswell, “Image formation in video-enhanced and confocal DIC microscopy,” in Phase Contrast and Differential Phase Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 64–71 (1992).
[CrossRef]

Gu, M.

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The three-dimensional (3-D) transmission cross coefficient for transmission imaging,” Optik 100, 155–158 (1994).

M. Gu, C. J. R. Sheppard, “Three-dimensional partially-coherent image formation in confocal microscopes with a finite-sized detector,” J. Mod. Opt. 41, 1701–1715 (1994).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 377–390 (1991).

C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope images,” in Image Reconstruction and Restoration, T. J. Schulz, L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
[CrossRef]

Kawata, S.

Kawata, Y.

Mao, X. Q.

Nugent, K. A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

Paganin, D.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

Roberts, A.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

Sheppard, C. J. R.

C. J. R. Sheppard, M. Gu, “The three-dimensional (3-D) transmission cross coefficient for transmission imaging,” Optik 100, 155–158 (1994).

C. J. R. Sheppard, M. Gu, Y. Kawata, S. Kawata, “Three-dimensional transfer functions for high-aperture systems,” J. Opt. Soc. Am. A 11, 593–598 (1994).
[CrossRef]

M. Gu, C. J. R. Sheppard, “Three-dimensional partially-coherent image formation in confocal microscopes with a finite-sized detector,” J. Mod. Opt. 41, 1701–1715 (1994).
[CrossRef]

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 377–390 (1991).

C. J. R. Sheppard, X. Q. Mao, “Three-dimensional imaging in a microscope,” J. Opt. Soc. Am. A 6, 1260–1269 (1989).
[CrossRef]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope images,” in Image Reconstruction and Restoration, T. J. Schulz, L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, “Image formation in video-enhanced and confocal DIC microscopy,” in Phase Contrast and Differential Phase Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 64–71 (1992).
[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. R.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

J. Microsc.

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 164, 377–390 (1991).

J. Mod. Opt.

M. Gu, C. J. R. Sheppard, “Three-dimensional partially-coherent image formation in confocal microscopes with a finite-sized detector,” J. Mod. Opt. 41, 1701–1715 (1994).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

A. Barty, K. A. Nugent, A. Roberts, D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175, 329–336 (2000).
[CrossRef]

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

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Lett.

Optik

C. J. R. Sheppard, M. Gu, “The three-dimensional (3-D) transmission cross coefficient for transmission imaging,” Optik 100, 155–158 (1994).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik 72, 131–133 (1986).

Other

C. J. R. Sheppard, M. Gu, “Modeling of 3-D brightfield microscope images,” in Image Reconstruction and Restoration, T. J. Schulz, L. Snyder, eds., Proc. SPIE2302, 352–358 (1994).
[CrossRef]

C. J. R. Sheppard, C. J. Cogswell, “Image formation in video-enhanced and confocal DIC microscopy,” in Phase Contrast and Differential Phase Contrast Imaging Techniques and Applications, M. Pluta, M. Szyjer, eds., Proc. SPIE1846, 64–71 (1992).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Three-dimensional WOTF C Dw (ℓ, s) for phase imaging, determined after axial differentiation.

Fig. 2
Fig. 2

Two-dimensional WOTF C Dw (ℓ) for phase imaging of a thin object, determined after axial differentiation.

Fig. 3
Fig. 3

Three-dimensional WOTF C Pw (ℓ, s) for phase imaging, determined after axial differentiation and resulting from a transverse inverse Laplacian operation.

Fig. 4
Fig. 4

Two-dimensional WOTF C Pw (ℓ) for phase imaging of a thin object, determined after axial differentiation and resulting from a transverse inverse Laplacian operation.

Equations (10)

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Vr=ik2n2r-n02,
Tm= Vrexpikm · rdr,
m=mi+nj+sk.
km=k2-k1,
Ir= Cm; mtmt*mexp-ikm-m · rdmdm,
Ir= Cm; mδmδn+Tm×δmδn+T*mexp-ikm-m · rdmdm = C0, 0, s; 0, 0, sexp-iks-szdsds+2 Cm; 0Tmexp-ikm · rdm+ Cm; mTmT*mexp-ikm-m · rdmdm.
Ir= C0s; sexp-iks-szdsds+2 CwmTmexp-ikm · rdm,
Cw, s=i2π 121+a2-24-s2-s-121-a21/2-121+a2-24-s2-s+121-a21/2,
CDw, s=sλ 121+a2-24-s2-s-121-a21/2-121+a2-24-s2-s+121-a21/2,
CPw, s=sλ4π3 121+a2-24-s2-s-121-a21/2-121+a2-24-s2-s+121-a21/2.

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