Abstract

A new procedure for fast quantitative extraction of cell parameters from diffraction patterns was deduced from model calculations and applied to cervical gynecological material. Based on radial scans of the diffraction pattern, the technique permits simultaneous determination of the nuclear and cytoplasmic diameters by Fourier analysis of the radial scanning signals after compensation for the intensity falloff by an amplification proportional to the third power of the radial position in the diffraction plane. Illustrative examples of measurements on exfoliated cells of different types are presented.

© 1978 Optical Society of America

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References

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  1. H. Borchers, “Zytologische Besonderheiten von Erkrankungen der leukozytären Systeme und des RHS” in Lehrbuch der klinischen Zytodiagnostik, H. J. Soost, Ed. (Thieme, Stuttgart, 1974), pp. 48–53.
  2. S. F. Patten, Diagnostic Cytology of the Uterine Cervix, Monographs in Clinical Cytology (Karger, Basel, 1969).
  3. B. Dew, T. King, D. Mighdoll, J. Histochem. Cytochem. 22, 685 (1974).
    [Crossref] [PubMed]
  4. W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).
  5. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  6. P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. W. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
    [Crossref] [PubMed]
  7. G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
    [PubMed]
  8. M. R. Loken, R. G. Sweet, L. A. Herzenberg, J. Histochem. Cytochem. 24, 284 (1976).
    [Crossref] [PubMed]
  9. T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
    [Crossref] [PubMed]
  10. G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
    [PubMed]
  11. A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
    [Crossref]
  12. A. Brunsting, P. F. Mullaney, Appl. Opt. 11, 675 (1972).
    [Crossref] [PubMed]
  13. R. A. Meyer, A. Brunsting, Biophys. J. 15, 191 (1975).
    [Crossref] [PubMed]
  14. A. F. Fercher, Optik 43, 129 (1975).
  15. E. Evans, Opt. Commun. 2, 317 (1972).
    [Crossref]
  16. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.
  17. G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).
  18. M. Bartholdi, G. C. Salzman, R. D. Hiebert, G. Seger, Opt. Lett. 1, 223 (1977).
    [Crossref] [PubMed]

1977 (1)

1976 (2)

M. R. Loken, R. G. Sweet, L. A. Herzenberg, J. Histochem. Cytochem. 24, 284 (1976).
[Crossref] [PubMed]

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

1975 (4)

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

R. A. Meyer, A. Brunsting, Biophys. J. 15, 191 (1975).
[Crossref] [PubMed]

A. F. Fercher, Optik 43, 129 (1975).

1974 (1)

B. Dew, T. King, D. Mighdoll, J. Histochem. Cytochem. 22, 685 (1974).
[Crossref] [PubMed]

1972 (2)

1969 (1)

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. W. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

1951 (1)

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Achatz, M.

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

Aden, A. L.

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

Arndt-Jovin, D. J.

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Bartholdi, M.

Betz, J.

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

Borchers, H.

H. Borchers, “Zytologische Besonderheiten von Erkrankungen der leukozytären Systeme und des RHS” in Lehrbuch der klinischen Zytodiagnostik, H. J. Soost, Ed. (Thieme, Stuttgart, 1974), pp. 48–53.

Brunsting, A.

Coulter, J. R.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. W. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Crowell, J. M.

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

Dean, P. W.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. W. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Dew, B.

B. Dew, T. King, D. Mighdoll, J. Histochem. Cytochem. 22, 685 (1974).
[Crossref] [PubMed]

Digweed, M.

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Evans, E.

E. Evans, Opt. Commun. 2, 317 (1972).
[Crossref]

Fercher, A. F.

A. F. Fercher, Optik 43, 129 (1975).

Goad, C. A.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.

Haertel, W.

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

Hansen, K. M.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

Herzenberg, L. A.

M. R. Loken, R. G. Sweet, L. A. Herzenberg, J. Histochem. Cytochem. 24, 284 (1976).
[Crossref] [PubMed]

Hiebert, R. D.

M. Bartholdi, G. C. Salzman, R. D. Hiebert, G. Seger, Opt. Lett. 1, 223 (1977).
[Crossref] [PubMed]

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

Ikeda, H.

W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).

Imasoto, D.

W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).

Ingram, M. L.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

Jovin, T. M.

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Kashida, R.

W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).

Kerker, M.

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

King, T.

B. Dew, T. King, D. Mighdoll, J. Histochem. Cytochem. 22, 685 (1974).
[Crossref] [PubMed]

LaBauve, P. M.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

LaBauve, P. W.

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

Loken, M. R.

M. R. Loken, R. G. Sweet, L. A. Herzenberg, J. Histochem. Cytochem. 24, 284 (1976).
[Crossref] [PubMed]

Martin, J. C.

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

Meyer, R. A.

R. A. Meyer, A. Brunsting, Biophys. J. 15, 191 (1975).
[Crossref] [PubMed]

Mighdoll, D.

B. Dew, T. King, D. Mighdoll, J. Histochem. Cytochem. 22, 685 (1974).
[Crossref] [PubMed]

Morris, S. J.

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Mullaney, P. F.

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

A. Brunsting, P. F. Mullaney, Appl. Opt. 11, 675 (1972).
[Crossref] [PubMed]

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. W. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Naujoks, H.

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

Patten, S. F.

S. F. Patten, Diagnostic Cytology of the Uterine Cervix, Monographs in Clinical Cytology (Karger, Basel, 1969).

Romero, A.

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

Salzman, G. C.

M. Bartholdi, G. C. Salzman, R. D. Hiebert, G. Seger, Opt. Lett. 1, 223 (1977).
[Crossref] [PubMed]

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

Schlüter, G.

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

Schultens, A.

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Seger, G.

M. Bartholdi, G. C. Salzman, R. D. Hiebert, G. Seger, Opt. Lett. 1, 223 (1977).
[Crossref] [PubMed]

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

Sengbusch, G. v.

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

Striker, G.

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

Sweet, R. G.

M. R. Loken, R. G. Sweet, L. A. Herzenberg, J. Histochem. Cytochem. 24, 284 (1976).
[Crossref] [PubMed]

Tanaka, W.

W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).

Trujillo, T. T.

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

Ueno, T.

W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).

Van Dilla, M. A.

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. W. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Watanabe, S.

W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).

Acta Cytol. (1)

G. C. Salzman, J. M. Crowell, J. C. Martin, T. T. Trujillo, A. Romero, P. F. Mullaney, P. W. LaBauve, Acta Cytol. 19, 374 (1975).
[PubMed]

Appl. Opt. (1)

Biophys. J. (1)

R. A. Meyer, A. Brunsting, Biophys. J. 15, 191 (1975).
[Crossref] [PubMed]

Clin. Chem. (1)

G. C. Salzman, J. M. Crowell, C. A. Goad, K. M. Hansen, R. D. Hiebert, P. M. LaBauve, J. C. Martin, M. L. Ingram, P. F. Mullaney, Clin. Chem. 21, 1297 (1975).
[PubMed]

J. Appl. Phys. (1)

A. L. Aden, M. Kerker, J. Appl. Phys. 22, 1242 (1951).
[Crossref]

J. Histochem. Cytochem. (3)

M. R. Loken, R. G. Sweet, L. A. Herzenberg, J. Histochem. Cytochem. 24, 284 (1976).
[Crossref] [PubMed]

T. M. Jovin, S. J. Morris, G. Striker, A. Schultens, M. Digweed, D. J. Arndt-Jovin, J. Histochem. Cytochem. 24, 269 (1976).
[Crossref] [PubMed]

B. Dew, T. King, D. Mighdoll, J. Histochem. Cytochem. 22, 685 (1974).
[Crossref] [PubMed]

Opt. Commun. (1)

E. Evans, Opt. Commun. 2, 317 (1972).
[Crossref]

Opt. Lett. (1)

Optik (1)

A. F. Fercher, Optik 43, 129 (1975).

Rev. Sci. Instrum. (1)

P. F. Mullaney, M. A. Van Dilla, J. R. Coulter, P. W. Dean, Rev. Sci. Instrum. 40, 1029 (1969).
[Crossref] [PubMed]

Other (6)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 64.

G. v. Sengbusch, G. Seger, M. Achatz, J. Betz, W. Haertel, H. Naujoks, G. Schlüter, Acta Cytol. (in press).

W. Tanaka, H. Ikeda, T. Ueno, S. Watanabe, D. Imasoto, R. Kashida, Proceedings International Conference on Automation of Uterine Cancer Cytology, G. L. Wied, Ed. (U. Chicago, Illinois, 1975).

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

H. Borchers, “Zytologische Besonderheiten von Erkrankungen der leukozytären Systeme und des RHS” in Lehrbuch der klinischen Zytodiagnostik, H. J. Soost, Ed. (Thieme, Stuttgart, 1974), pp. 48–53.

S. F. Patten, Diagnostic Cytology of the Uterine Cervix, Monographs in Clinical Cytology (Karger, Basel, 1969).

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Figures (11)

Fig. 1
Fig. 1

Shape (a) and optical density profile (b) of the nucleated model cell assumed.

Fig. 2
Fig. 2

Approximation of the Bessel function J1(X) by J1(X) ≈ (2/π)1/2[{(cos[X − (3π/4)]}/(X)1/2].

Fig. 3
Fig. 3

Illustration of the cellular dimensions reflected in the diffraction pattern.

Fig. 4
Fig. 4

Fourier transform of the approximated radial intensity distribution in the diffraction pattern of a nucleated cell calculated for a diameter ratio of 0.25, a density ratio of 5, and an unlimited radial scanning interval.

Fig. 5
Fig. 5

Fourier transform of the approximated radial intensity distribution in the diffraction pattern of a nucleated cell calculated for a diameter ratio of 0.25, a density ratio of 5, and two different radial scanning intervals comprising (a) twenty diffraction rings of the nucleus and (b) five rings only.

Fig. 6
Fig. 6

Illustration of the cellular dimensions reflected in the diffraction pattern of a cell with an eccentric nucleus.

Fig. 7
Fig. 7

Optical setup for analyzing the Fraunhofer diffraction patterns of biological cells.

Fig. 8
Fig. 8

Result of spectrum analysis of the radial intensity function in the diffraction pattern of a spherical particle of 10.5-μm diam after compensation for the intensity falloff.

Fig. 9
Fig. 9

Result of spectrum analysis of the radial intensity functions in the diffraction patterns of spheres of different diameters between 8 μm and 40 μm.

Fig. 10
Fig. 10

Result of spectrum analysis of the radial intensity function in the diffraction pattern of a symmetric parabasal cell.

Fig. 11
Fig. 11

Result of spectrum analysis of the radial intensity function in the diffraction pattern of an intermediate cell with an eccentric position of the nucleus.

Tables (1)

Tables Icon

Table I Results of Cytoplasmic and Nuclear Diameter Extraction from the Diffraction Patterns of 378 Cervical Cells

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

FT [ D ( r ) ] = ( j a ) / π exp [ j a ( 2 f 2 + R 2 ) ] × 0 d r r D ( r ) 0 2 π d φ exp [ j 2 a R r cos ( ϕ φ ) ] ,
1 / ( 2 π ) 0 2 π exp [ j u cos ( ϕ φ ) ] d φ = J 0 ( u ) ,
A ( R ) = FT [ D ( r ) ] = j 2 a exp [ j a ( 2 f 2 + R 2 ) ] 0 r D ( r ) J 0 ( 2 a R r ) d r .
D ( r ) = D C + D N for 0 r ( d N ) / 2 , D ( r ) = D C for ( d N ) / 2 < r ( d C ) / 2 , D ( r ) = 0 for ( d C ) / 2 < r ,
A ( R ) = j 2 a exp [ j a ( 2 f 2 + R 2 ) ] [ D N 0 ( d N ) / 2 r J 0 ( 2 a R r ) d r + D C 0 ( d C ) / 2 r J 0 ( 2 a R r ) d r ] .
0 X u J 0 ( u ) d u = X J 1 ( X )
A ( R ) = exp [ j a ( 2 f 2 + R 2 ) ] ( j a ) / 2 [ D N d N 2 J 1 ( a d N R ) a d N R ] + D C d C 2 J 1 ( a d C R ) a d C R ] ,
I ( R ) = A ( R ) A * ( R ) = ( a / 2 ) 2 [ D N 2 d N 4 J 1 2 ( a d N R ) ( a d N R ) 2 + D C 2 d C 4 J 1 2 ( a d C R ) ( a d C R ) 2 ] + 2 D N d N 2 D C d C 2 J 1 ( a d N R ) J 1 ( a d C R ) ] .
J 1 ( X ) ( 2 / π ) 1 / 2 cos [ X ( 3 π ) / 4 ] ( X ) 1 / 2 for X > 1 ,
I ( R ) = 1 / ( 2 π a R 3 ) { D N 2 d N cos 2 [ a d N R ( 3 π ) / 4 ] + D C 2 d C cos 2 [ a d C R ( 3 π ) / 4 ] + 2 D N D C ( d C ) 1 / 2 ( d N ) 1 / 2 cos [ a d N R ( 3 π ) / 4 ] cos [ a d C R ( 3 π ) / 4 ] } .
I ( R ) = 1 / ( 4 π a ) 1 / ( R 3 ) { D N 2 d N + D C 2 d C + D N 2 d N cos [ 2 a R d N ( 3 π ) / 2 ] + D C 2 d C cos [ 2 a R d C ( 3 π ) / 2 ] + 2 D N D C ( d N ) 1 / 2 cos [ 2 a R ( d C d N ) 2 ] + 2 D N D C ( d N ) 1 / 2 ( d C ) 1 / 2 cos [ 2 a R ( d C d N ) 2 3 π 2 ] } .
FT [ I ( R ) R 3 ] = 1 2 a ( D N 2 d N + D C 2 d C ) δ ( 0 ) + 1 4 a D N 2 d C δ ( 2 a d C ) + 1 4 a D C 2 d N δ ( 2 a d N ) + 1 4 a D N D C ( d C ) 1 / 2 ( d N ) 1 / 2 δ [ 2 a ( d C + d N ) 2 ] + 1 4 a D N D C ( d C ) 1 / 2 ( d N ) 1 / 2 δ [ 2 a ( d C + d N ) 2 ] ,
A ( R ) = exp [ j a ( 2 f 2 + R 2 ) ] j a 2 [ D N d N 2 J 1 ( a d N R ) a d N R exp ( j 2 a e R ) + D C d C 2 J 1 ( a d C R ) a d C R ] ,
I ( R ) = ( a / 2 ) 2 [ D N 2 d N 4 J 1 2 ( a d N R ) ( a d N R ) 2 + D C 2 d C 4 J 1 2 ( a d C R ) ( a d C R ) 2 + 2 D N d N 2 D C d C 2 J 1 ( a d N R ) a d N R J 1 ( a d C R ) a d C R cos ( 2 a e R ) ] .
I ( R ) = 1 4 π a 1 R 3 ( D N 2 d N + D C 2 d C + D N 2 d N cos ( 2 a R d N 3 π 2 ) + D C 2 d C cos ( 2 a R d C 3 π 2 ) + D N D C ( d N ) 1 / 2 ( d C ) 1 / 2 { cos [ 2 a R ( d C d N 2 e ) ] + cos [ 2 a R ( d C d N 2 + e ) 3 π 2 ] + cos [ 2 a R ( d C + d N 2 e ) 3 π 2 ] + cos [ 2 a R ( d C + d N 2 + e ) 3 π 2 ] } ) .
I tot = 0 I ( R ) d R = 2 π 0 r | D ( r ) | 2 d r ,
I tot = 2 π 0 ( d N ) / 2 r ( D N + D C ) 2 d r + 2 π ( d N ) / 2 ( d C ) / 2 r D C 2 d r , = ( D N + D C ) 2 π d N 2 4 + D C 2 ( π d C 2 4 π d N 2 4 ) , = ( D N + D C ) 2 A N + D C 2 ( A C A N ) ,
I ( 0 ) = 2 λ 2 f 2 ( D C A C + D N A N ) .
FT [ F ( R ) ] = m 1 B 1 exp ( B 1 2 4 ω 2 ) m 2 B 2 exp ( B 2 2 4 ω 2 ) ,
+ F ( R ) d R = 0 .
I ( R ) = R + R F ( R ) I ( R R ) R 3 d R .

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