Abstract

A method for separating the phase distribution from the image of a semitransparent sample observed with a phase-contrast microscope is proposed. Three images of the same sample are used for calculating the phase distribution, one observed by a normal objective lens and the others observed by a bright-contrast objective and a dark-contrast objective. The method works even for samples that are not weak phase objects. The phase (or refractive-index) distribution is quantitatively reconstructed by numerically solving a set of linear equations for every pixel of the sample image. The method of phase reconstruction is mathematically derived and is explained with graphs in the complex amplitude space. Experimental results are also presented.

© 1992 Optical Society of America

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References

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  1. F. Zernike, “Das Phasenkontrastverfahren bei der mikroskopischen Beobachtung,”Z. Tech. Phys. 16, 454–457 (1935).
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  3. H. Osterberg, “The polanret microscope,”J. Opt. Soc. Am. 37, 726–730 (1947).
    [Crossref] [PubMed]
  4. W. G. Driscoll, W. Vaughan, Handbook of Optics (McGraw-Hill, New York, 1978), Chap. 6.
  5. W. Lang, Nomarski Differential Interference-Contrast Microscopy (reprint: Zeiss, 7082 Oberkochen, Germany, 1975).

1947 (1)

1935 (1)

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

Driscoll, W. G.

W. G. Driscoll, W. Vaughan, Handbook of Optics (McGraw-Hill, New York, 1978), Chap. 6.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Lang, W.

W. Lang, Nomarski Differential Interference-Contrast Microscopy (reprint: Zeiss, 7082 Oberkochen, Germany, 1975).

Osterberg, H.

Vaughan, W.

W. G. Driscoll, W. Vaughan, Handbook of Optics (McGraw-Hill, New York, 1978), Chap. 6.

Zernike, F.

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

J. Opt. Soc. Am. (1)

Z. Tech. Phys. (1)

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

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

W. G. Driscoll, W. Vaughan, Handbook of Optics (McGraw-Hill, New York, 1978), Chap. 6.

W. Lang, Nomarski Differential Interference-Contrast Microscopy (reprint: Zeiss, 7082 Oberkochen, Germany, 1975).

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Figures (16)

Fig. 1
Fig. 1

Schematic optical setup of phase-contrast system.

Fig. 2
Fig. 2

Graphic expression of complex transmittances of a sample and the amplitude distribution of a bright-contrast image. The complex transmittance t(x, y) {=|t(x, y)|exp[(x, y)]} at a position (x, y) is shown by an arrow. The transmittances and the amplitude distribution are shown by the hatched area and the crosshatched area, respectively. The values of phase retardation and attenuation of the phase plate are B and αB, respectively.

Fig. 3
Fig. 3

Graphic expression of complex transmittances of a sample and the amplitude distribution of a dark-contrast imaging. The values of phase retardation and attenuation of the phase plate are D and αD, respectively.

Fig. 4
Fig. 4

Phase estimation from intensities of three images: (a) an absorption image, (b) a bright-contrast image, (c) a dark-contrast image. The estimated transmittance is given by the intersection of three circles. The radii of the three circles correspond to the square roots of the three image intensities.

Fig. 5
Fig. 5

Block diagram of experimental system. Three types of image are observed by exchanging the objectives of the microscope. DMA, direct memory access.

Fig. 6
Fig. 6

Observed images of a Spirogyra isolate that contains spiral chloroplasts in its cell and a nucleus at the center of the cell (the wavelength λ of the observation light is 550 nm): (a) absorption image, (b) bright-contrast image, (c) dark-contrast image.

Fig. 7
Fig. 7

Estimated phase distribution of Spirogyra from three observed images (Fig. 6). The unevenness of the brightness of the phase-contrast images [Figs. 6(b) and 6(c)] that was due to the absorption effect is neglected in the estimated distribution.

Fig. 8
Fig. 8

Observed images of Euglena gracillis (λ = 550 nm): (a) absorption image, (b) bright-contrast image, (c) dark-contrast image.

Fig. 9
Fig. 9

Estimated phase distribution of Euglena gracillis from three observed images (Fig. 8).

Fig. 10
Fig. 10

Histogram image of the estimated complex-transmittance distribution of Euglena gracillis. The histogram is drawn in complex space similar to that of Fig. 2. The distribution is classified into four major portions: Parts A, B, C, and D correspond to the nucleus, the cytoplasm, and a vacuole of Euglena gracillis and the background of the observed image, respectively.

Fig. 11
Fig. 11

Observed images of Giemsa-stained human blood cells (λ = 650 nm). One white corpuscle, which has a nucleus, and some red corpuscles are shown: (a) absorption image, (b) bright-contrast image, (c) dark-contrast image.

Fig. 12
Fig. 12

Estimated phase distribution of human blood cells (λ = 650 nm).

Fig. 13
Fig. 13

Observed images of Giemsa-stained human blood cells (λ = 450 nm): (a) absorption image, (b) bright-contrast image, (c) dark-contrast image.

Fig. 14
Fig. 14

Estimated phase distribution of human blood cells (λ = 450 nm).

Fig. 15
Fig. 15

(a) Phase spectra and (b) absorption spectra of human blood cells. □, white corpuscles; ■, red corpuscles.

Fig. 16
Fig. 16

Correlation functions of observed images: (a) function of raw images in which the peak was located off center and (b) function of corrected images in which the peak was located on center.

Equations (21)

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t ( x , y ) = t ( x , y ) exp [ j ϕ ( x , y ) ]
S ( ξ , η ) = δ ( ξ , η ) ,
U B ( ξ , η ) = { α B T ( 0 , 0 ) exp ( j B ) for ξ = η = 0 T ( ξ , η ) otherwise ,
u B ( x , y ) = U B ( 0 , 0 ) δ ( ξ , η ) exp [ 2 π j ( x ξ + y η ) ] d ξ d η + [ U B ( ξ , η ) - U B ( 0 , 0 ) δ ( ξ , η ) ] exp [ 2 π j ( x ξ + y η ) ] d ξ d η = α B t ¯ exp ( j B ) + [ t ( x , y ) - t ¯ ] ,
t ¯ = T ( 0 , 0 ) = t ( x , y ) d x d y .
i B ( x , y ) = u B ( x , y ) 2 .
t ( x , y ) 1 + j ϕ ( x , y ) .
i B ( x , y ) 1 + 2 ϕ ( x , y ) .
i D ( x , y ) 1 - 2 ϕ ( x , y ) .
i N ( x , y ) = t ( x , y ) 2 ,
i B ( x , y ) = t ( x , y ) - t ¯ + α B t ¯ exp ( j B ) 2 ,
i D ( x , y ) = t ( x , y ) - t ¯ + α D t ¯ exp ( j D ) 2 ,
B = t ¯ - α B t ¯ exp ( j B ) = ( t ¯ - α B t ¯ cos B ) - j α B t ¯ sin B = B exp ( j γ B ) .
D = t ¯ - α D t ¯ exp ( j D ) = ( t ¯ - α D t ¯ cos D ) - j α D t ¯ sin D = D exp ( j γ D ) .
i B ( x , y ) = t ( x , y ) - B 2 = t ( x , y ) 2 + B 2 - 2 t ( x , y ) × B cos [ γ B - ϕ ( x , y ) ] ,
i D ( x , y ) = t ( x , y ) - D 2 = t ( x , y ) 2 + D 2 - 2 t ( x , y ) × D cos [ γ D - ϕ ( x , y ) ] .
B 0 ,
D 0 ,
sin ( γ B - γ D ) 0.
{ t ^ ( x , y ) - [ i N ( x , y ) ] 1 / 2 } 2 + { t ^ ( x , y ) - B - [ i B ( x , y ) ] 1 / 2 } 2 + { t ^ ( x , y ) - D - [ i D ( x , y ) ] 1 / 2 } 2 min .
t ¯ = t ( x , y ) d x d y = [ i N ( x , y ) ] 1 / 2 d x d y .

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