Abstract

For the visualization of phase objects by use of a differentiation filter, the phase variation is changed into the intensity variation by differentiation, and then the differentiated image is integrated. In the method used in practice, the differentiated image has been recorded on a film, and then integrated by use of a filter. In this paper, however, the image differentiated by means of a filter is entered into a computer using a CCD camera and then integrated. As a result, the method provides rapid on-line processing. We have performed computer simulations and a detailed analysis of the differentiation filter and also provide the experimental results of three-dimensional image visualization.

© 2003 Optical Society of America

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References

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  1. Joseph W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, Calif., 1996).
  2. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [CrossRef] [PubMed]
  3. H. H. Hopkins, “A note on the theory of phase-contrast images,” Proc. Phys. Soc. London, Sect. B. 66, 331–333 (1953).
    [CrossRef]
  4. M. Françon, Progress in Microscopy (Pergamon, New York, 1961).
  5. L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).
  6. L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theor. IT-6, 386–400 (1960).
    [CrossRef]
  7. J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).
  8. R. A. Sprague, Brian J. Thompson, “Quantitative visualization of large variation phase objects,” Appl. Opt. 11, 1469–1479 (1972).
    [CrossRef] [PubMed]
  9. J. Lancis, T. Szoplik, E. Tajahuerce, V. Climent, M. Fernandez-Alonso, “Fractional derivative Fourier plane filter for phase-change visualization,” Appl. Opt. 36, 7461–7462 (1997).
    [CrossRef]
  10. T. Szoplik, V. Climent, E. Tajahuerce, J. Lancis, M. Fernandez-Alonso, “Phase-change visualization in two-dimensional phase objects with a semiderivative real filter,” Appl. Opt. 37, 5472–5478 (1998).
    [CrossRef]

1998 (1)

1997 (1)

1972 (1)

1960 (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theor. IT-6, 386–400 (1960).
[CrossRef]

1955 (1)

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

1953 (1)

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

Climent, V.

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theor. IT-6, 386–400 (1960).
[CrossRef]

DeVelis, J. B.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

Fernandez-Alonso, M.

Françon, M.

M. Françon, Progress in Microscopy (Pergamon, New York, 1961).

Goodman, Joseph W.

Joseph W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, Calif., 1996).

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]

Lancis, J.

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theor. IT-6, 386–400 (1960).
[CrossRef]

Martin, L. C.

L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theor. IT-6, 386–400 (1960).
[CrossRef]

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theor. IT-6, 386–400 (1960).
[CrossRef]

Reynolds, G. O.

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

Sprague, R. A.

Szoplik, T.

Tajahuerce, E.

Thompson, Brian J.

Zernike, F.

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

Appl. Opt. (3)

IRE Trans. Inf. Theor. (1)

L. J. Cutrona, E. N. Leith, C. J. Palermo, L. J. Porcello, “Optical data processing and filtering systems,” IRE Trans. Inf. Theor. IT-6, 386–400 (1960).
[CrossRef]

Proc. Phys. Soc. London, Sect. B. (1)

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

Science (1)

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

Other (4)

Joseph W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, San Francisco, Calif., 1996).

M. Françon, Progress in Microscopy (Pergamon, New York, 1961).

L. C. Martin, The Theory of the Microscope (American Elsevier, New York, 1966).

J. B. DeVelis, G. O. Reynolds, Theory and Applications of Holography (Addison-Wesley, Reading, Mass., 1967).

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Figures (15)

Fig. 1
Fig. 1

Arrangement of optical differentiation.

Fig. 2
Fig. 2

Shape of the wavefront used for computer simulation: (a) 3D shape of the phase, (b) derivative of the phase at y = 0.

Fig. 3
Fig. 3

Amplitude distribution of the wave in the rear focal plane of the lens.

Fig. 4
Fig. 4

Amplitude transmittance of three types of differentiation filter with a 1 mm (filter 1), 400 μm (filter 2) and 100 μm (filter 3).

Fig. 5
Fig. 5

Calculated image intensity distribution for (a) filter 1, (b) filter 2, and (c) filter 3 x0=W2.

Fig. 6
Fig. 6

Relationship between x and image intensity at y = 0 for the three types of filters.

Fig. 7
Fig. 7

Calculated image intensity distribution for the bias point x 0 = W.

Fig. 8
Fig. 8

Experimental setup.

Fig. 9
Fig. 9

Differentiation filter.

Fig. 10
Fig. 10

Transmittance of the filter: (a) intensity transmittance, (b) amplitude transmittance.

Fig. 11
Fig. 11

Image of the sample on the image plane: (a) with no filter, (b) with the differentiation filter.

Fig. 12
Fig. 12

Images of bubbles: (a) with no filter, (b) with the differentiation filter, differentiation with respect to x, (c) with the differentiation filter, differentiation with respect to -x.

Fig. 13
Fig. 13

Calculated 3D structure of the bubbles when integrating the derivative of the phase.

Fig. 14
Fig. 14

Calculated 3D structure of the bubbles when integrating the derivative of the phase and ignoring the small phase derivative.

Fig. 15
Fig. 15

Calculated 3D structure of the bubble in the center of Fig. 12.

Equations (12)

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i1x1, y1=A expiϕx1, y1,
i2-x2, y2=1iλfexpi πλf1-afx22+y22×Fx2λf, y2λf,
i2+x2, y2=tx2, y2i2-x2, y2=αx0+x21iλfexpi πλf1-af×x22+y22Fx2λf, y2λf,
i3x3, y3=i2+x2, y21iλbexpi πλbx22+y22 =1iλc  αx0+x21iλf×expi πλf1-afx22+y22×Fx2λf, y2λfexpi πλbx2-x32+y2-y32dx2dy2,=1mexpi πλbx32+y32α×x0i11m x3, 1m y3+λfi2π×i11m x3, 1m y31m x3,
i3x3, y3=1mexpi πλbx32+y32α×Ax0+λf2π ϕ1m x3, 1m y3×expiϕ1m x3, 1m y3,
|i3x3, y3|=αAmλf2πϕ0+ϕ1m x3, 1m y3,
i2-x2, y2=1iλfexpi πλf1-afx22+y22×δx2λf-γ2π, y2λf.
x2=γλf2π.
Δγ2πλf W.
Δγ2π Wλf=2π 100 μm160 mm×633 nm=2π rad/mm.
ϕ0-Δγ2=2πλfx0-W2.
Δγ2π Wλf20π rad/mm.

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