Abstract

We describe, using a high-numerical-aperture vectorial model, the image formation of phase-contrast microscopes. In particular, imaging of a weak phase object is considered. We show that, partly owing to the fact that phase-contrast microscopes are interference microscopes, their image formation is fundamentally different from that of conventional transmission optical microscopes. Our detailed analysis reveals a number of yet undocumented properties of these microscopes, including that depending on the given configuration, they can exhibit an improved lateral resolution when larger detectors are used in comparison with that obtained for a small detector size. We present numerical examples to explain this phenomenon and discuss our analysis in detail.

© 2004 Optical Society of America

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References

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  1. A. H. Bennet, H. Jupnik, H. Osterberg, O. W. Richards, Phase Contrast Microscopy, Principles and Applications, 1st ed. (Wiley, New York, 1951).
  2. 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]
  3. 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]
  4. P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
    [CrossRef]
  5. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [CrossRef] [PubMed]
  6. R. Liang, J. K. Erwin, M. Mansuripur, “Variation on Zernike’s phase-contrast microscope,” Appl. Opt. 39, 2152–2158 (2000).
    [CrossRef]
  7. C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
    [CrossRef]
  8. P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
    [CrossRef]
  9. P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
    [CrossRef]
  10. N. B. E. Sawyer, S. P. Morgan, M. G. Somekh, C. W. See, “Amplitude and phase microscopy for sizing of spherical particles,” Appl. Opt. 42, 4488–4498 (2003).
    [CrossRef] [PubMed]
  11. V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1919).
  12. R. K. Luneburg, Mathematical Theory of Optics, 1st ed. (U. California Press, Berkeley, Calif., 1964), pp. 321–324.
  13. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplantic system,” Proc. R. Soc. London 253, 358–379 (1959).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 424–428.
  15. O. Haeberlé, H. Furukawa, K. Tenjimbayashi, P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express 11, 2964–2969 (2003).
    [CrossRef] [PubMed]
  16. J.-M. Jin, The Finite Element Method in Electromagnetics, 1st ed. (McGraw-Hill, New York, 1995).
  17. C. J. R. Sheppard, M. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Opt. Commun. 84, 7–13 (1991).
    [CrossRef]

2003 (2)

2000 (2)

R. Liang, J. K. Erwin, M. Mansuripur, “Variation on Zernike’s phase-contrast microscope,” Appl. Opt. 39, 2152–2158 (2000).
[CrossRef]

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

1998 (2)

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

1991 (1)

C. J. R. Sheppard, M. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Opt. Commun. 84, 7–13 (1991).
[CrossRef]

1982 (2)

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]

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

1959 (1)

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

1955 (1)

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

1952 (1)

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]

1919 (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1919).

Barer, R.

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]

Bennet, A. H.

A. H. Bennet, H. Jupnik, H. Osterberg, O. W. Richards, Phase Contrast Microscopy, Principles and Applications, 1st ed. (Wiley, New York, 1951).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 424–428.

Erwin, J. K.

Furukawa, H.

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]

Gu, M.

C. J. R. Sheppard, M. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Opt. Commun. 84, 7–13 (1991).
[CrossRef]

Haeberlé, O.

Higdon, P. D.

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

Ignatowsky, V. S.

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1919).

Jin, J.-M.

J.-M. Jin, The Finite Element Method in Electromagnetics, 1st ed. (McGraw-Hill, New York, 1995).

Jupnik, H.

A. H. Bennet, H. Jupnik, H. Osterberg, O. W. Richards, Phase Contrast Microscopy, Principles and Applications, 1st ed. (Wiley, New York, 1951).

Liang, R.

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, 1st ed. (U. California Press, Berkeley, Calif., 1964), pp. 321–324.

Mansuripur, M.

Morgan, S. P.

Osterberg, H.

A. H. Bennet, H. Jupnik, H. Osterberg, O. W. Richards, Phase Contrast Microscopy, Principles and Applications, 1st ed. (Wiley, New York, 1951).

Richards, B.

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

Richards, O. W.

A. H. Bennet, H. Jupnik, H. Osterberg, O. W. Richards, Phase Contrast Microscopy, Principles and Applications, 1st ed. (Wiley, New York, 1951).

Sawyer, N. B. E.

See, C. W.

Sheppard, C. J. R.

C. J. R. Sheppard, M. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Opt. Commun. 84, 7–13 (1991).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

Somekh, M. G.

Tenjimbayashi, K.

Török, P.

O. Haeberlé, H. Furukawa, K. Tenjimbayashi, P. Török, “The point spread function of optical microscopes imaging through stratified media,” Opt. Express 11, 2964–2969 (2003).
[CrossRef] [PubMed]

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

Wilson, T.

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

Wolf, E.

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 424–428.

Zernike, F.

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

Appl. Opt. (2)

J. Microsc. (1)

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. Mod. Opt. (1)

P. Török, P. D. Higdon, T. Wilson, “Theory for confocal and conventional microscopes imaging small dielectric scatterers,” J. Mod. Opt. 45, 1681–1698 (1998).
[CrossRef]

J. R. Microsc. Soc. (1)

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]

Opt. Commun. (3)

P. Török, P. D. Higdon, T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Commun. 148, 300–315 (1998).
[CrossRef]

P. Török, “Imaging of small birefringent objects by polarised light conventional and confocal microscopes,” Opt. Commun. 181, 7–18 (2000).
[CrossRef]

C. J. R. Sheppard, M. Gu, “Improvement of axial resolution in confocal microscopy using an annular pupil,” Opt. Commun. 84, 7–13 (1991).
[CrossRef]

Opt. Express (1)

Proc. R. Soc. London (1)

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

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

C. J. R. Sheppard, T. Wilson, “The image of a single point in microscopes of large numerical aperture,” Proc. R. Soc. London, Ser. A 379, 145–158 (1982).
[CrossRef]

Science (1)

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

Trans. Opt. Inst. Petrograd (1)

V. S. Ignatowsky, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. Petrograd 1(4), 1–36 (1919).

Other (4)

R. K. Luneburg, Mathematical Theory of Optics, 1st ed. (U. California Press, Berkeley, Calif., 1964), pp. 321–324.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), pp. 424–428.

J.-M. Jin, The Finite Element Method in Electromagnetics, 1st ed. (McGraw-Hill, New York, 1995).

A. H. Bennet, H. Jupnik, H. Osterberg, O. W. Richards, Phase Contrast Microscopy, Principles and Applications, 1st ed. (Wiley, New York, 1951).

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Figures (14)

Fig. 1
Fig. 1

Two common configurations used in phase-contrast microscopy.

Fig. 2
Fig. 2

Using the Debye–Wolf integral to find the field due to a point source focused by an afocal arrangement of lenses.

Fig. 3
Fig. 3

General model used to determine ℰ used in the modified Debye–Wolf integral.

Fig. 4
Fig. 4

Calculating the phase term for the field of a defocused dipole.

Fig. 5
Fig. 5

Images of field intensity of various field components at the detector. Left-hand column, configuration 1; right-hand column, configuration 2. (a) and (b) Directly transmitted field, (c) and (d) scattered light, (e) and (f) total field.

Fig. 6
Fig. 6

Image of a small weak phase object scanned in the xy lateral plane in a phase-contrast microscope by use of a point detector.

Fig. 7
Fig. 7

Definition of FWHP.

Fig. 8
Fig. 8

Plots of FWHP values, taken from lateral scans in the x direction, for a variety of objective lenses. The FWHM values of the lateral point-spread function (along the x direction) of a bright-field scanning microscope are shown for comparison.

Fig. 9
Fig. 9

Detector signal components showing why a point detector yields a wider lateral point-spread function than a large detector does.

Fig. 10
Fig. 10

FWHM of the scattered light component for both Zernike configurations.

Fig. 11
Fig. 11

FWHP of the interaction component of the detector signal for both configurations.

Fig. 12
Fig. 12

Plot of the interaction-field component along the x axis in the detector plane for three scan positions. The field component is normalized by the peak absolute value of the interaction-field component of the upper plot. The upper plot is for an on-axis, in-focus phase object. The middle plot has the phase object at the total signal half-width-at-half-peak position for a point detector, and the lower plot is at the total signal peak for a point detector.

Fig. 13
Fig. 13

Variation of lateral contrast for both configurations with detector radius.

Fig. 14
Fig. 14

Axial FWHP for a 0.95NA, 100× objective as a function of detector radius.

Equations (39)

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T(x, y)=exp[ιϕ(x, y)]1+ιϕ(x, y),
E(rp)=-ιf2λsx2+sy21P(sx, sy) E(sx, sy)sz×exp(ιksrp)dsxdsy,
s=(cosϕsinθ,sinϕsinθ, cosθ),
rp=(rpcosϕp, rpsin ϕp, zp).
R=cosϕsin ϕ0-sin ϕcosϕ0001.
L=cosθ0sin θ010-sin θ0cosθ,
E=R-1(ϕ)L2(θ2)L1-1(θ1)R(ϕ)Edp,
Edp=-rs×(rs×ps).
rs=(cosϕsinθ1,sinϕsinθ1, cosθ1).
Ex=px2 [(1+cosθ1cosθ2)-(1-cosθ1cosθ2)cos2ϕ]-py2 (1-cosθ1cosθ2)sin 2ϕ-pzsin θ1cosθ2cosϕ,
Ey=py2 [(1+cosθ1cosθ2)+(1-cosθ1cosθ2)cos2ϕ]-px2 (1-cosθ1cosθ2)sin 2ϕ-pzsin θ1cosθ2sin ϕ,
Ez=-cosθ1sin θ2(pxcosϕ+pysin ϕ)+pzsin θ1sin θ2.
l2=(-zs)2+f2-2(-zs)fcos(π-θ)=f21-2zsfcosθ+zsf2,
l=f1-2zsfcosθ+zsf21/2f1-2zsfcosθ1/2
lf1-zsfcosθ=f-zscosθ.
sin θ2sin θ1=f1f2=β.
02πsin nϕcosnϕexp[ιρcos(ϕ-γ)]dϕ=2πιnJn(ρ)sin(nγ)cos(nγ),
Ex=A[px(K0A+K2Acos2ϕp)+pyK2Asin 2ϕp-2ιpzK1Acosϕp],
Ey=A[py(K0A-K2Acos2ϕp)+pxK2Asin 2ϕp-2ιpzK1Asin ϕp],
Ez=A[-2ιK1B(pxcosϕp+pysin ϕp)+2pzK0B],
K0A=0αP(θ2)cosθ2cosθ11/2(1+cosθ1cosθ2)×sin θ2J0(krpsin θ2)exp(ιkΘ)dθ2,
K0B=0αP(θ2)cosθ2cosθ11/2sin θ1sin2 θ2J0(krpsin θ2)×exp(ιkΘ)dθ2,
K1A=0αP(θ2)×cosθ2cosθ11/2sin θ1sin θ2cosθ2J1(krpsin θ2)×exp(ιkΘ)dθ2,
K1B=0αP(θ2)cosθ2cosθ11/2cosθ1sin2 θ2J1(krpsin θ2)×exp(ιkΘ)dθ2,
K2A=0αP(θ2)cosθ2cosθ11/2×(1-cosθ1cosθ2)sin θ2J2(krpsin θ2)×exp(ιkΘ)dθ2,
Θ=zpcosθ2-zscosθ1,
K0AI0=0αP(θ2)cosθ2(1+cosθ2)×sin θ2J0(krisin θ2)exp(ιkzicosθ2)dθ2,
K0B0,
K1A0,
K1BI1=0αP(θ2)cosθ2sin2 θ2J1(krisin θ2)×exp(ιkzicosθ2)dθ2,
K2AI2=0αP(θ2)cosθ2(1-cosθ2)×sin θ2J2(krisin θ2)×exp(ιkzicosθ2)dθ2.
Pphase(θ)=γ exp(-ιπ/2)θlθθu1otherwise,
Pring(θ)=1θlθθu0otherwise.
Ei,x=A(I0+I2cos2ϕi),
Ei,y=AI2sin 2ϕi,
Ei,z=-2ιAI1cosϕi.
Et=Es+Ed.
Id=S|Et(rd)|2DdS,
Id=S|Et(rd)|2DdS=S|Es+Ed|2DdS=S(|Es|2+|Ed|2+2R{Es*Ed})DdS=Is+Id+Ii,

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