Abstract

We present a novel mathematical approach to describe phase contrast imaging. The approach is applied to analyze the interferometric phase contrast technique and phase contrast methods, which are based on spatial filtering of the undiffracted or diffracted light. Our new approach makes it possible to analyze phase contrast imaging achieved by phase perturbation of the diffracted light. We show that phase retarding of the diffracted light is an important advantage in the fabrication of the Fourier filter for spatial filtering.

© 2009 Optical Society of America

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References

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  1. D. Malacara, Optical Shop Testing (Wiley-VCH, 1992).
  2. H. Beyer, Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Port, 1974).
  3. D. Malacara, M. Servn, and Z. Malacara, Interferogram Analysis for Optical Shop Testing (Taylor & Francis, 2005).
    [CrossRef]
  4. F. Zernike, “Diffraction theory of the knife-edge test, and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94, 377-384 (1934).
  5. F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part I,” Physica (Utrecht) 9686-698 (1942).
    [CrossRef]
  6. F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part II.,” Physica (Utrecht) 9, 974-986 (1942).
    [CrossRef]
  7. F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
    [CrossRef] [PubMed]
  8. H. Wolter, “Experimentelle und theoretische Untersuchungen zur Abbildung nichtabsorbierender Objekte,” Ann. Phys. 7, 33-53 (1950).
    [CrossRef]
  9. A. G. Oettle, “Experiments with a variable amplitude and phase microscope,” J. R. Microsc. Soc. 70, 232-254 (1950).
    [CrossRef]
  10. W. Krug, J. Rienitz, and G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, 1964).
  11. C. S. Anderson, “Fringe visibility, irradiance, and accuracy in common path interferometers for visualization of phase disturbances,” Appl. Opt. 34, 7474-7485 (1995).
    [CrossRef] [PubMed]
  12. J. Glückstad and P. C. Mogensen, “Optimal phase contrast imaging in common path interferometry,” Appl. Opt. 40, 268-282 (2001).
    [CrossRef]
  13. M. Teschke and S. Sinzinger, “Modified phase contrast for recording of holographic optical elements,” Opt. Lett. 32, 2067-2069 (2007).
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  14. M. Teschke, R. Heyer, M. Fritzsche, S. Stoebenau, and S. Sinzinger, “Application of an interferometric phase contrast to fabricate arbitrary diffractive optical elements,” Appl. Opt. 47, 2550-2556 (2008).
    [CrossRef] [PubMed]
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
  16. E. Hecht, Optics (Addison Wesley, 2001).
  17. W. Lauterborn and T. Kurz, Coherent Optics: Fundamentals and Applications (Springer, 2003).
  18. H. Beyer, Theorie und Praxis des Phasenkontrastverfahrens (Akademische Verlagsgesellschaft Geest & Port, 1965).
  19. R. Barer, “A vector theory of phase contrast and interference contrast. I. Positive phase contrast,” J. R. Microsc. Soc. 72, 10-30 (1952).
    [CrossRef] [PubMed]
  20. R. Barer, “A vector theory of phase contrast and interference contrast. II. Positive phase contrast,” J. R. Microsc. Soc. 72, 81-98 (1952).
    [CrossRef] [PubMed]
  21. R. Barer, “A vector theory of phase contrast and interference contrast. III. Negative phase contrast,” J. R. Microsc. Soc. 73, 30-39 (1953).
    [CrossRef] [PubMed]
  22. R. Barer, “A vector theory of phase contrast and interference contrast. IV. B-type phase contrast,” J. R. Microsc. Soc. 73, 206-215 (1953).
    [CrossRef]
  23. H. Wolter, “Zur Deutung von Beobachtungen mit dem Phasenkontrastverfahren,” Naturwiss. 37, 272-276 (1950).
    [CrossRef]
  24. A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy-Principles and Applications (Wiley, 1951).
  25. L. J. Golden, “Zernike test. 1: Analytical aspects,” Appl. Opt. 16, 205-213 (1977).
    [CrossRef] [PubMed]
  26. D. J. Goldstein, “A simple quantitative analysis of phase contrast microscopy, not restricted to objects of very low retardation,” J. Microsc. 128, 33-47 (1982).
    [CrossRef] [PubMed]
  27. D. J. Goldstein, Understanding the Light Microscope (Academic, 1999).
  28. S. Osten, S. Krüger, and A. Steinhoff, “Spatial light modulators based on reflective microdisplays,” Tech. Mess. 73, 149-156 (2006).
    [CrossRef]
  29. H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London, Sect. B 66, 331-333 (1953).
    [CrossRef]

2008

2007

2006

S. Osten, S. Krüger, and A. Steinhoff, “Spatial light modulators based on reflective microdisplays,” Tech. Mess. 73, 149-156 (2006).
[CrossRef]

2001

1995

1982

D. J. Goldstein, “A simple quantitative analysis of phase contrast microscopy, not restricted to objects of very low retardation,” J. Microsc. 128, 33-47 (1982).
[CrossRef] [PubMed]

1977

1955

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

1953

R. Barer, “A vector theory of phase contrast and interference contrast. III. Negative phase contrast,” J. R. Microsc. Soc. 73, 30-39 (1953).
[CrossRef] [PubMed]

R. Barer, “A vector theory of phase contrast and interference contrast. IV. B-type phase contrast,” J. R. Microsc. Soc. 73, 206-215 (1953).
[CrossRef]

H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London, Sect. B 66, 331-333 (1953).
[CrossRef]

1952

R. Barer, “A vector theory of phase contrast and interference contrast. I. Positive phase contrast,” J. R. Microsc. Soc. 72, 10-30 (1952).
[CrossRef] [PubMed]

R. Barer, “A vector theory of phase contrast and interference contrast. II. Positive phase contrast,” J. R. Microsc. Soc. 72, 81-98 (1952).
[CrossRef] [PubMed]

1950

H. Wolter, “Zur Deutung von Beobachtungen mit dem Phasenkontrastverfahren,” Naturwiss. 37, 272-276 (1950).
[CrossRef]

H. Wolter, “Experimentelle und theoretische Untersuchungen zur Abbildung nichtabsorbierender Objekte,” Ann. Phys. 7, 33-53 (1950).
[CrossRef]

A. G. Oettle, “Experiments with a variable amplitude and phase microscope,” J. R. Microsc. Soc. 70, 232-254 (1950).
[CrossRef]

1942

F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part I,” Physica (Utrecht) 9686-698 (1942).
[CrossRef]

F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part II.,” Physica (Utrecht) 9, 974-986 (1942).
[CrossRef]

1934

F. Zernike, “Diffraction theory of the knife-edge test, and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Anderson, C. S.

Barer, R.

R. Barer, “A vector theory of phase contrast and interference contrast. III. Negative phase contrast,” J. R. Microsc. Soc. 73, 30-39 (1953).
[CrossRef] [PubMed]

R. Barer, “A vector theory of phase contrast and interference contrast. IV. B-type phase contrast,” J. R. Microsc. Soc. 73, 206-215 (1953).
[CrossRef]

R. Barer, “A vector theory of phase contrast and interference contrast. II. Positive phase contrast,” J. R. Microsc. Soc. 72, 81-98 (1952).
[CrossRef] [PubMed]

R. Barer, “A vector theory of phase contrast and interference contrast. I. Positive phase contrast,” J. R. Microsc. Soc. 72, 10-30 (1952).
[CrossRef] [PubMed]

Bennett, A. H.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy-Principles and Applications (Wiley, 1951).

Beyer, H.

H. Beyer, Theorie und Praxis des Phasenkontrastverfahrens (Akademische Verlagsgesellschaft Geest & Port, 1965).

H. Beyer, Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Port, 1974).

Fritzsche, M.

Glückstad, J.

Golden, L. J.

Goldstein, D. J.

D. J. Goldstein, “A simple quantitative analysis of phase contrast microscopy, not restricted to objects of very low retardation,” J. Microsc. 128, 33-47 (1982).
[CrossRef] [PubMed]

D. J. Goldstein, Understanding the Light Microscope (Academic, 1999).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Hecht, E.

E. Hecht, Optics (Addison Wesley, 2001).

Heyer, R.

Hopkins, H. H.

H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London, Sect. B 66, 331-333 (1953).
[CrossRef]

Jupnik, H.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy-Principles and Applications (Wiley, 1951).

Krug, W.

W. Krug, J. Rienitz, and G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, 1964).

Krüger, S.

S. Osten, S. Krüger, and A. Steinhoff, “Spatial light modulators based on reflective microdisplays,” Tech. Mess. 73, 149-156 (2006).
[CrossRef]

Kurz, T.

W. Lauterborn and T. Kurz, Coherent Optics: Fundamentals and Applications (Springer, 2003).

Lauterborn, W.

W. Lauterborn and T. Kurz, Coherent Optics: Fundamentals and Applications (Springer, 2003).

Malacara, D.

D. Malacara, M. Servn, and Z. Malacara, Interferogram Analysis for Optical Shop Testing (Taylor & Francis, 2005).
[CrossRef]

D. Malacara, Optical Shop Testing (Wiley-VCH, 1992).

Malacara, Z.

D. Malacara, M. Servn, and Z. Malacara, Interferogram Analysis for Optical Shop Testing (Taylor & Francis, 2005).
[CrossRef]

Mogensen, P. C.

Oettle, A. G.

A. G. Oettle, “Experiments with a variable amplitude and phase microscope,” J. R. Microsc. Soc. 70, 232-254 (1950).
[CrossRef]

Osten, S.

S. Osten, S. Krüger, and A. Steinhoff, “Spatial light modulators based on reflective microdisplays,” Tech. Mess. 73, 149-156 (2006).
[CrossRef]

Osterberg, H.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy-Principles and Applications (Wiley, 1951).

Richards, O. W.

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy-Principles and Applications (Wiley, 1951).

Rienitz, J.

W. Krug, J. Rienitz, and G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, 1964).

Schulz, G.

W. Krug, J. Rienitz, and G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, 1964).

Servn, M.

D. Malacara, M. Servn, and Z. Malacara, Interferogram Analysis for Optical Shop Testing (Taylor & Francis, 2005).
[CrossRef]

Sinzinger, S.

Steinhoff, A.

S. Osten, S. Krüger, and A. Steinhoff, “Spatial light modulators based on reflective microdisplays,” Tech. Mess. 73, 149-156 (2006).
[CrossRef]

Stoebenau, S.

Teschke, M.

Wolter, H.

H. Wolter, “Experimentelle und theoretische Untersuchungen zur Abbildung nichtabsorbierender Objekte,” Ann. Phys. 7, 33-53 (1950).
[CrossRef]

H. Wolter, “Zur Deutung von Beobachtungen mit dem Phasenkontrastverfahren,” Naturwiss. 37, 272-276 (1950).
[CrossRef]

Zernike, F.

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part I,” Physica (Utrecht) 9686-698 (1942).
[CrossRef]

F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part II.,” Physica (Utrecht) 9, 974-986 (1942).
[CrossRef]

F. Zernike, “Diffraction theory of the knife-edge test, and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Ann. Phys.

H. Wolter, “Experimentelle und theoretische Untersuchungen zur Abbildung nichtabsorbierender Objekte,” Ann. Phys. 7, 33-53 (1950).
[CrossRef]

Appl. Opt.

J. Microsc.

D. J. Goldstein, “A simple quantitative analysis of phase contrast microscopy, not restricted to objects of very low retardation,” J. Microsc. 128, 33-47 (1982).
[CrossRef] [PubMed]

J. R. Microsc. Soc.

R. Barer, “A vector theory of phase contrast and interference contrast. I. Positive phase contrast,” J. R. Microsc. Soc. 72, 10-30 (1952).
[CrossRef] [PubMed]

R. Barer, “A vector theory of phase contrast and interference contrast. II. Positive phase contrast,” J. R. Microsc. Soc. 72, 81-98 (1952).
[CrossRef] [PubMed]

R. Barer, “A vector theory of phase contrast and interference contrast. III. Negative phase contrast,” J. R. Microsc. Soc. 73, 30-39 (1953).
[CrossRef] [PubMed]

R. Barer, “A vector theory of phase contrast and interference contrast. IV. B-type phase contrast,” J. R. Microsc. Soc. 73, 206-215 (1953).
[CrossRef]

A. G. Oettle, “Experiments with a variable amplitude and phase microscope,” J. R. Microsc. Soc. 70, 232-254 (1950).
[CrossRef]

Mon. Not. R. Astron. Soc.

F. Zernike, “Diffraction theory of the knife-edge test, and its improved form, the phase-contrast method,” Mon. Not. R. Astron. Soc. 94, 377-384 (1934).

Naturwiss.

H. Wolter, “Zur Deutung von Beobachtungen mit dem Phasenkontrastverfahren,” Naturwiss. 37, 272-276 (1950).
[CrossRef]

Opt. Lett.

Physica (Utrecht)

F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part I,” Physica (Utrecht) 9686-698 (1942).
[CrossRef]

F. Zernike, “Phasecontrast, a new method for the microscopic observation of transparent objects. Part II.,” Physica (Utrecht) 9, 974-986 (1942).
[CrossRef]

Proc. Phys. Soc. London, Sect. B

H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London, Sect. B 66, 331-333 (1953).
[CrossRef]

Science

F. Zernike, “How I discovered phase contrast,” Science 121, 345-349 (1955).
[CrossRef] [PubMed]

Tech. Mess.

S. Osten, S. Krüger, and A. Steinhoff, “Spatial light modulators based on reflective microdisplays,” Tech. Mess. 73, 149-156 (2006).
[CrossRef]

Other

W. Krug, J. Rienitz, and G. Schulz, Contributions to Interference Microscopy (Hilger & Watts, 1964).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

E. Hecht, Optics (Addison Wesley, 2001).

W. Lauterborn and T. Kurz, Coherent Optics: Fundamentals and Applications (Springer, 2003).

H. Beyer, Theorie und Praxis des Phasenkontrastverfahrens (Akademische Verlagsgesellschaft Geest & Port, 1965).

A. H. Bennett, H. Osterberg, H. Jupnik, and O. W. Richards, Phase Microscopy-Principles and Applications (Wiley, 1951).

D. J. Goldstein, Understanding the Light Microscope (Academic, 1999).

D. Malacara, Optical Shop Testing (Wiley-VCH, 1992).

H. Beyer, Theorie und Praxis der Interferenzmikroskopie (Akademische Verlagsgesellschaft Geest & Port, 1974).

D. Malacara, M. Servn, and Z. Malacara, Interferogram Analysis for Optical Shop Testing (Taylor & Francis, 2005).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Interferometric phase contrast setup: (a) phase pattern visualized as a gray-level image, (b) intensity pattern of the object wave (OW) in the observation plane [without the superposition with the reference wave (RW)], (c) dc signals of OW and RW before alignment, (d) resulting dc signal after correct alignment, (e) extinction of the dc signal, (f) intensity pattern in the observation plane obtained by interference of the undiffracted part of OW with RW ( δ = π ) .

Fig. 2
Fig. 2

Interference of the undiffracted part of the OW with the RW: (a) destructive interference between the two waves, (b) both waves represented in the vector diagram, (c) vectorial addition of both waves.

Fig. 3
Fig. 3

Phase-contrast based on spatial filtering: (a) generic 4 - f configuration for spatial filtering, (b) complex vector diagram.

Fig. 4
Fig. 4

Phase-contrast based on spatial filtering of the diffracted light.

Fig. 5
Fig. 5

Phase-to-intensity mapping: (a) phase retarding of the diffracted light, (b) phase-retarding of the undiffracted light.

Fig. 6
Fig. 6

Setup for variable phase contrast imaging: (a) no phase retardation, (b) phase-retarding of the undiffracted light, (c) phase-retarding of the diffracted light.

Fig. 7
Fig. 7

Fourier filter: (a) spatial filtering of the undiffracted light, (b) spatial filtering of the diffracted light.

Equations (21)

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OW = exp [ i ϕ ] = 1 + i ϕ 1 2 ϕ 2 1 6 i ϕ 3 + 1 24 ϕ 4 + ,
RW = exp [ i π ] = 1 .
OW + RW = i ϕ 1 2 ϕ 2 1 6 i ϕ 3 + 1 24 ϕ 4 + .
I = i ϕ 1 2 ϕ 2 1 6 i ϕ 3 + 1 24 ϕ 4 + 1 120 i ϕ 5 2 = ( i ϕ 1 2 ϕ 2 1 6 i ϕ 3 + 1 24 ϕ 4 + 1 120 i ϕ 5 ) × ( i ϕ 1 2 ϕ 2 + 1 6 i ϕ 3 + 1 24 ϕ 4 1 120 i ϕ 5 ) = ( i 2 ϕ 2 1 2 i ϕ 3 + 1 6 i 2 ϕ 4 + 1 24 i ϕ 5 1 120 i 2 ϕ 6 ) + ( + 1 2 i ϕ 3 + 1 4 ϕ 4 1 12 i ϕ 5 1 48 ϕ 6 + ) + ( + 1 6 i 2 ϕ 4 + 1 12 i ϕ 5 1 36 i 2 ϕ 6 ) + ( 1 24 i ϕ 5 1 48 ϕ 6 + ) + ( 1 120 i 2 ϕ 6 ) + ( ) = ϕ 2 1 12 ϕ 4 + 1 360 ϕ 6 = 2 ( ϕ 2 2 + ϕ 4 24 ϕ 6 720 ) .
cos x = 1 x 2 2 ! + x 4 4 ! x 6 6 ! + ,
cos x 1 = x 2 2 + x 4 24 x 6 720 + .
I = 2 ( cos x 1 ) = 2 2 cos ϕ .
I = exp [ i ϕ ] + exp [ i π ] 2 = 2 2 cos ϕ .
b exp ( i θ ) = b ( cos θ + i sin θ ) ,
OW = b ( cos θ + i sin θ ) + i ϕ 1 2 ϕ 2 1 6 i ϕ 3 + 1 24 ϕ 4 + .
I = b cos θ + b i sin θ + i ϕ 1 2 ϕ 2 1 6 i ϕ 3 + 1 24 ϕ 4 + 2 = ( b 2 cos 2 θ + b 2 sin 2 θ + 2 b sin θ ϕ b cos θ ϕ 2 + ϕ 2 1 3 b sin θ ϕ 3 + 1 12 b cos θ ϕ 4 1 12 ϕ 4 ) = b 2 + 2 b sin θ ( ϕ 1 6 ϕ 3 ) b cos θ ( ϕ 2 1 12 ϕ 4 ) + ( ϕ 2 1 12 ϕ 4 ) .
sin x = x x 3 3 ! + x 5 5 ! x 7 7 ! + .
I = b 2 + 2 b sin θ ( sin ϕ ) b cos θ ( 2 2 cos ϕ ) + ( 2 2 cos ϕ ) = 2 + b 2 2 cos ϕ + 2 b [ cos ( ϕ θ ) cos θ ] .
I = 2 + 0 2 2 cos ϕ + 2 [ cos ( ϕ θ ) cos θ ] = 2 2 cos ϕ .
I = 2 + 1 2 2 cos ϕ + 2 [ cos ( ϕ π 2 ) cos π 2 ] = 3 2 ( cos ϕ + sin ϕ ) .
a ( cos ψ + i sin ψ ) .
OW = 1 + a i ϕ ( cos ψ + i sin ψ ) 1 2 a ϕ 2 ( cos ψ + i sin ψ ) 1 6 a i ϕ 3 ( cos ψ + i sin ψ ) + .
I = 1 + a i ϕ cos ψ a ϕ sin ψ 1 2 a ϕ 2 cos ψ 1 2 a i ϕ 2 sin ψ 1 6 a i ϕ 3 cos ψ + 1 6 a ϕ 3 sin ψ + 2 = 1 + ( a 2 ϕ 2 cos 2 ψ a ϕ 2 cos ψ 1 12 a 2 ϕ 4 cos 2 ψ ) + ( 2 a ϕ sin ψ + 1 3 a ϕ 3 sin ψ ) + ( a 2 ϕ 2 sin 2 ψ 1 12 a 2 ϕ 4 sin 2 ψ ) = 1 + ( a 2 cos 2 ψ a cos ψ ) ( ϕ 2 1 12 ϕ 4 ) ( 2 a sin ψ ) ( ϕ 1 6 ϕ 3 ) + ( a 2 sin 2 ψ ) ( ϕ 2 1 12 ϕ 4 ) .
I = 1 + ( a 2 cos 2 ψ a cos ψ ) ( 2 2 cos ϕ ) 2 a sin ψ sin ϕ + a 2 sin 2 ψ ( 2 2 cos ϕ ) .
OW = b ( cos θ + i sin θ ) + a i ϕ ( cos ψ + i sin ψ ) 1 2 a ϕ 2 ( cos ψ + i sin ψ ) 1 6 a i ϕ 3 ( cos ψ + i sin ψ ) + .
I = b 2 + b a cos θ cos ψ ( 2 cos ϕ 2 ) + b a sin θ sin ψ ( 2 cos ϕ 2 ) + 2 b a sin θ cos ψ sin ϕ 2 b a cos θ sin ψ sin ϕ + a 2 cos 2 ψ ( 2 2 cos ϕ ) + a 2 sin 2 ψ ( 2 2 cos ϕ ) .

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