Abstract

Existing formulations of the three-dimensional (3-D) diffraction pattern of spherical waves that is produced by a circular aperture are reviewed in the context of 3-D serial-sectioning microscopy. A new formulation for off-axis focal points is introduced that has the desirable properties of increased accuracy for larger field angles, invariance to shifts of the focal point about spheres of constant radius when the detection point is on the sphere for both intensity and amplitude fields, and invariance to shifts in three transformed coordinates for intensity fields. Finally, calculated intensity fields for both on-axis and off-axis focal points are included to illustrate the proposal that the classical 3-D diffraction patterns that have been used as analytical models in 3-D serial-sectioning fluorescence microscopy may not be accurate enough for this application.

© 1989 Optical Society of America

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  1. T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).
  2. G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronson, N. Nanninga, “3-dimensional imaging of biological structures by high-resolution confocal scanning laser microscopy,” Scanning Microsc. 2, 33–40 (1988).
    [PubMed]
  3. D. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
    [Crossref] [PubMed]
  4. J. Bille, B. Schmidt, E. Beck, “Reconstruction of three-dimensional light-microscopic images comparing frequency and spatial domain methods,” in Applications of Digital Image Processing IX, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.697, 349–356 (1986).
    [Crossref]
  5. W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the 13th Northeast Bioengineering Conference, K. Foster ed. (Institute of Electrical and Electronics Engineers, New York, 1987).
  6. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
    [Crossref] [PubMed]
  7. S. Kawata, Y. Ichioka, Iterative image restoration for linearly degraded images. I. Basis,” J. Opt. Soc. Am. 70, 762–768 (1980).
    [Crossref]
  8. S. Kawata, Y. Ichioka, “Iterative image restoration for linearly degraded images. II. Reblurring procedure,”J. Opt. Soc. Am. 70, 768–772 (1980).
    [Crossref]
  9. M. Shantz, “Description and classification of neuronal structure in the frog retina,” doctoral dissertation (California Institute of Technology, Pasadena, Calif., 1976).
  10. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).
  11. M. Born, E. Wolf, Principles of Optics, 8th ed. (Pergamon, New York, 1959).
  12. H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. A (London) 231, 91–103 (1955).
    [Crossref]
  13. A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
    [Crossref]
  14. Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).
  15. Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
    [Crossref]
  16. C. J. R. Sheppard, “Imaging in optical systems of finite Fresnel number,” J. Opt. Soc. Am. A 3, 1428–1432 (1986).
    [Crossref]
  17. H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
    [Crossref]
  18. H. Pedersen, J. Stamnes, “Reciprocity principles of focused wave fields and the modified Debye integral,” Opt. Acta 30, 1437–1454 (1983).
    [Crossref]
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  20. Note that this amplitude is often written as Af/r, which is the amplitude of a spherical wave with relative amplitude Afat a distance rfrom its source. In a microscope, however, the spherical wave incident upon the aperture is formed by the lens system from light emitted by a point in object space. Thus the amplitude at the aperture actually varies as the distance of the object point from the lens system rather than as the distance of the image point from the aperture. For simplicity in this paper we disregard the origin of the amplitude on the aperture and simply refer to it as |u(A)|.
  21. Note that, if the integral is evaluated over the planar aperture rather than over Sf, neither this approximation nor the assumption that all rays passing through the aperture A also pass through Sfneed be made. However, rmust then be approximated by using a binomial expansion. In either case, as long as terms of the same order are dropped, the resultant expressions for the 3-D diffraction field are the same.
  22. This should be accurate for systems with small image space field angles and small image numerical aperture. In fact, as noted in the discussion of the Born–Wolf analysis, integrating over the surface Sdwhile making the approximation of the surface differential dSfrom Eq. (4) yields the same expression for the 3-D diffraction pattern as is obtained by integrating over the planar aperture where the surface integral is exact, as long as terms of the same order are dropped in the expansions.
  23. Actually, Hopkins made more-severe approximations [see Ref. 12, Eq. (13)]. He dropped terms that are independent of ξand η, including the scaling term in the integral, 1/s, and approximated tan a′ = a/zdby sin a′.
  24. P. A. Stokseth, “Properties of a defocused optical system,”J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [Crossref]
  25. As usual, we avoid transforming coordinates to their defined u′ and v′. Otherwise, this is equivalent to Eq. (3.4) of Ref. 15.
  26. B. R. Frieden, “Optical transfer of the three-dimensional object,”J. Opt. Soc. Am. 57, 56–66 (1967).
    [Crossref]
  27. In a typical system in which zd= 160 mm and the size of the detector is 10 mm × 10 mm, this angle would be of the order of 2 deg. What is actually required is that kζ2(zd− zf)/zfzdbe small so that it does not cause large fluctuations over the integral in the complex exponential function. For system parameters a= 1 mm, zf= 160 mm, and λ = 520 nm, by using Eq. (13) we find that (kζ2/zf)max< 0.20. Thus, since (zd− zf)/zd< 2, this requirement is satisfied. When the maximum values of ξand ηon the aperture are used, however, [k(ζ2+ η2)/zf]max≃ 75; thus this term could cause significant fluctuations in the complex exponential function over the integral and must be included in the expansion.
  28. Note that r˜2(r˜·r′)/r′3 is less than or equal to r˜3/r′2, which can be large for large amounts of defocus. In microscopes in which the image space field angles are small, r˜ · r′ is approximately zero when rmis perpendicular to the optic axis. Thus this approximation is quite accurate in (x, y) planes close to the in-focus plane.
  29. The parameters used here were zf= 160 mm, a= 1 mm, and λ= 520 nm.
  30. L. Tella, “The determination of a microscope’s three-dimensional transfer function for use in image reconstruction,” master’s thesis (Worcester Polytechnic Institute, Worcester, Mass., 1985).
  31. Y. Hiraoka, J. W. Sedat, D. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
    [Crossref] [PubMed]

1988 (1)

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronson, N. Nanninga, “3-dimensional imaging of biological structures by high-resolution confocal scanning laser microscopy,” Scanning Microsc. 2, 33–40 (1988).
[PubMed]

1987 (1)

Y. Hiraoka, J. W. Sedat, D. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[Crossref] [PubMed]

1986 (1)

1985 (1)

1984 (2)

D. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[Crossref] [PubMed]

Y. Li, E. Wolf, “Three-dimensional intensity distribution near the focus in systems of different Fresnel numbers,” J. Opt. Soc. Am. A 1, 801–808 (1984).
[Crossref]

1983 (2)

H. Pedersen, J. Stamnes, “Reciprocity principles of focused wave fields and the modified Debye integral,” Opt. Acta 30, 1437–1454 (1983).
[Crossref]

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

1980 (2)

1976 (1)

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

1970 (1)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

1969 (1)

1967 (1)

1955 (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. A (London) 231, 91–103 (1955).
[Crossref]

Agard, D.

Y. Hiraoka, J. W. Sedat, D. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[Crossref] [PubMed]

D. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[Crossref] [PubMed]

Arimoto, A.

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

Beck, E.

J. Bille, B. Schmidt, E. Beck, “Reconstruction of three-dimensional light-microscopic images comparing frequency and spatial domain methods,” in Applications of Digital Image Processing IX, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.697, 349–356 (1986).
[Crossref]

Bille, J.

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
[Crossref] [PubMed]

J. Bille, B. Schmidt, E. Beck, “Reconstruction of three-dimensional light-microscopic images comparing frequency and spatial domain methods,” in Applications of Digital Image Processing IX, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.697, 349–356 (1986).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 8th ed. (Pergamon, New York, 1959).

Brakenhoff, G. J.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronson, N. Nanninga, “3-dimensional imaging of biological structures by high-resolution confocal scanning laser microscopy,” Scanning Microsc. 2, 33–40 (1988).
[PubMed]

Carrington, W.

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the 13th Northeast Bioengineering Conference, K. Foster ed. (Institute of Electrical and Electronics Engineers, New York, 1987).

Erhardt, A.

Fogarty, K.

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the 13th Northeast Bioengineering Conference, K. Foster ed. (Institute of Electrical and Electronics Engineers, New York, 1987).

Frieden, B. R.

Goodman, J. W.

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

Hiraoka, Y.

Y. Hiraoka, J. W. Sedat, D. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[Crossref] [PubMed]

Hopkins, H. H.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. A (London) 231, 91–103 (1955).
[Crossref]

Ichioka, Y.

Kawata, S.

Komitowski, D.

Li, Y.

Nanninga, N.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronson, N. Nanninga, “3-dimensional imaging of biological structures by high-resolution confocal scanning laser microscopy,” Scanning Microsc. 2, 33–40 (1988).
[PubMed]

Pedersen, H.

H. Pedersen, J. Stamnes, “Reciprocity principles of focused wave fields and the modified Debye integral,” Opt. Acta 30, 1437–1454 (1983).
[Crossref]

Schmidt, B.

J. Bille, B. Schmidt, E. Beck, “Reconstruction of three-dimensional light-microscopic images comparing frequency and spatial domain methods,” in Applications of Digital Image Processing IX, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.697, 349–356 (1986).
[Crossref]

Sedat, J. W.

Y. Hiraoka, J. W. Sedat, D. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[Crossref] [PubMed]

Shantz, M.

M. Shantz, “Description and classification of neuronal structure in the frog retina,” doctoral dissertation (California Institute of Technology, Pasadena, Calif., 1976).

Sheppard, C.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Sheppard, C. J. R.

Stamnes, J.

H. Pedersen, J. Stamnes, “Reciprocity principles of focused wave fields and the modified Debye integral,” Opt. Acta 30, 1437–1454 (1983).
[Crossref]

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

Stokseth, P. A.

Tella, L.

L. Tella, “The determination of a microscope’s three-dimensional transfer function for use in image reconstruction,” master’s thesis (Worcester Polytechnic Institute, Worcester, Mass., 1985).

van der Voort, H. T. M.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronson, N. Nanninga, “3-dimensional imaging of biological structures by high-resolution confocal scanning laser microscopy,” Scanning Microsc. 2, 33–40 (1988).
[PubMed]

van Spronson, E. A.

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronson, N. Nanninga, “3-dimensional imaging of biological structures by high-resolution confocal scanning laser microscopy,” Scanning Microsc. 2, 33–40 (1988).
[PubMed]

Wilson, T.

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Wolf, E.

Yzuel, M. J.

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

Zinser, G.

Annu. Rev. Biophys. Bioeng. (1)

D. Agard, “Optical sectioning microscopy: cellular architecture in three dimensions,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[Crossref] [PubMed]

Appl. Opt. (1)

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Acta (3)

H. H. Hopkins, M. J. Yzuel, “The computation of diffraction patterns in the presence of aberrations,” Opt. Acta 17, 157–182 (1970).
[Crossref]

H. Pedersen, J. Stamnes, “Reciprocity principles of focused wave fields and the modified Debye integral,” Opt. Acta 30, 1437–1454 (1983).
[Crossref]

A. Arimoto, “Intensity distribution of aberration-free diffraction patterns due to circular apertures in large F-number optical systems,” Opt. Acta 23, 245–250 (1976).
[Crossref]

Optik (1)

Y. Li, “Encircled energy for systems of different Fresnel numbers,” Optik 64, 207–218 (1983).

Proc. Phys. Soc. A (London) (1)

H. H. Hopkins, “The frequency response of a defocused optical system,” Proc. Phys. Soc. A (London) 231, 91–103 (1955).
[Crossref]

Scanning Microsc. (1)

G. J. Brakenhoff, H. T. M. van der Voort, E. A. van Spronson, N. Nanninga, “3-dimensional imaging of biological structures by high-resolution confocal scanning laser microscopy,” Scanning Microsc. 2, 33–40 (1988).
[PubMed]

Science (1)

Y. Hiraoka, J. W. Sedat, D. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[Crossref] [PubMed]

Other (16)

As usual, we avoid transforming coordinates to their defined u′ and v′. Otherwise, this is equivalent to Eq. (3.4) of Ref. 15.

In a typical system in which zd= 160 mm and the size of the detector is 10 mm × 10 mm, this angle would be of the order of 2 deg. What is actually required is that kζ2(zd− zf)/zfzdbe small so that it does not cause large fluctuations over the integral in the complex exponential function. For system parameters a= 1 mm, zf= 160 mm, and λ = 520 nm, by using Eq. (13) we find that (kζ2/zf)max< 0.20. Thus, since (zd− zf)/zd< 2, this requirement is satisfied. When the maximum values of ξand ηon the aperture are used, however, [k(ζ2+ η2)/zf]max≃ 75; thus this term could cause significant fluctuations in the complex exponential function over the integral and must be included in the expansion.

Note that r˜2(r˜·r′)/r′3 is less than or equal to r˜3/r′2, which can be large for large amounts of defocus. In microscopes in which the image space field angles are small, r˜ · r′ is approximately zero when rmis perpendicular to the optic axis. Thus this approximation is quite accurate in (x, y) planes close to the in-focus plane.

The parameters used here were zf= 160 mm, a= 1 mm, and λ= 520 nm.

L. Tella, “The determination of a microscope’s three-dimensional transfer function for use in image reconstruction,” master’s thesis (Worcester Polytechnic Institute, Worcester, Mass., 1985).

T. Wilson, C. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

J. Bille, B. Schmidt, E. Beck, “Reconstruction of three-dimensional light-microscopic images comparing frequency and spatial domain methods,” in Applications of Digital Image Processing IX, A. G. Tescher, ed., Proc. Soc. Photo-Opt. Instrum. Eng.697, 349–356 (1986).
[Crossref]

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the 13th Northeast Bioengineering Conference, K. Foster ed. (Institute of Electrical and Electronics Engineers, New York, 1987).

M. Shantz, “Description and classification of neuronal structure in the frog retina,” doctoral dissertation (California Institute of Technology, Pasadena, Calif., 1976).

J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK, 1986).

M. Born, E. Wolf, Principles of Optics, 8th ed. (Pergamon, New York, 1959).

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

Note that this amplitude is often written as Af/r, which is the amplitude of a spherical wave with relative amplitude Afat a distance rfrom its source. In a microscope, however, the spherical wave incident upon the aperture is formed by the lens system from light emitted by a point in object space. Thus the amplitude at the aperture actually varies as the distance of the object point from the lens system rather than as the distance of the image point from the aperture. For simplicity in this paper we disregard the origin of the amplitude on the aperture and simply refer to it as |u(A)|.

Note that, if the integral is evaluated over the planar aperture rather than over Sf, neither this approximation nor the assumption that all rays passing through the aperture A also pass through Sfneed be made. However, rmust then be approximated by using a binomial expansion. In either case, as long as terms of the same order are dropped, the resultant expressions for the 3-D diffraction field are the same.

This should be accurate for systems with small image space field angles and small image numerical aperture. In fact, as noted in the discussion of the Born–Wolf analysis, integrating over the surface Sdwhile making the approximation of the surface differential dSfrom Eq. (4) yields the same expression for the 3-D diffraction pattern as is obtained by integrating over the planar aperture where the surface integral is exact, as long as terms of the same order are dropped in the expansions.

Actually, Hopkins made more-severe approximations [see Ref. 12, Eq. (13)]. He dropped terms that are independent of ξand η, including the scaling term in the integral, 1/s, and approximated tan a′ = a/zdby sin a′.

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Figures (7)

Fig. 1
Fig. 1

Diffraction of a spherical wave by an aperture in a plane screen.

Fig. 2
Fig. 2

Diffraction formulation of Born and Wolf.

Fig. 3
Fig. 3

Diffraction formulation of Hopkins.

Fig. 4
Fig. 4

Diffraction formulation of Hopkins and Yzuel.

Fig. 5
Fig. 5

Diffraction formulation of Pedersen and St

Fig. 6
Fig. 6

Diffraction formulation of Sheppard.

Fig. 7
Fig. 7

Log intensity cross section of the 3-D diffraction pattern predicted (a), (c), (e) by an off-axis extension of the Born–Wolf expression and (b), (d), (f) by the spherical coordinate formulation for a typical microscope (see the text).

Equations (67)

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U ( P d / P f ) = - i 2 λ A u ( A ) exp [ - i k ( r - s ) ] s × [ cos ( n ^ , r ) - cos ( n ^ , s ) ] d S ,
U ( P d / P f ) - i A 0 λ A exp [ - i k ( r - s ) ] s d S .
s = [ ( x d - ξ ) 2 + ( y d - η ) 2 + ( z d - ζ ) 2 ] 1 / 2 = { ( x d - ξ ) 2 + ( y d - η ) 2 + [ z f + ( z d - z f - ζ ) ] 2 } 1 / 2 = [ ( x d - ξ ) 2 + ( y d - η ) 2 + z f 2 + ( z d - z f ) 2 + ζ 2 - 2 ζ ( z d - z f ) + 2 z f ( z d - z f ) - 2 z f ζ ] 1 / 2 .
z f - ζ = [ z f 2 - ( ξ 2 + η 2 ) ] 1 / 2 = z f - ξ 2 + η 2 2 z f - ( ξ 2 + η 2 ) 2 8 z f 3 + .
ζ = ξ 2 + η 2 2 z f + ( ξ 2 + η 2 ) 2 8 z f 3 - .
s z f + ( z d - z f ) - x d ξ + y d η z f - ( ξ 2 + η 2 ) ( z d - z f ) 2 z f 2
r - s - ( z d - z f ) + x d ξ + y d η z f + ( ξ 2 + η 2 ) ( z d - z f ) 2 z f 2 .
U ( P d / P f ) - i A 0 λ z f exp [ i k ( z d - z f ) ] S f exp [ - i k ( ξ x d + η y d ) z f ] × exp [ - i k ( ξ 2 + η 2 ) ( z d - z f ) 2 z f 2 ] d S .
ξ = a ρ cos θ ,             0 ρ 1 ,             a = aperture radius , η = a ρ sin θ , x d = R d cos ψ , y d = R d sin ψ , d S = a 2 ρ d ρ d θ / [ 1 - ( a ρ / z f ) 2 ] 1 / 2 .
U - i A 0 λ z f exp [ i k ( z d - z f ) ] 0 1 d ρ ρ exp [ - i k a 2 ρ 2 ( z d - z f ) 2 z f 2 ] × 0 2 π exp [ - i k a ρ R d ( cos θ cos ψ + sin θ sin ψ ) z f ] d θ .
cos θ cos ψ + sin θ sin ψ = cos ( θ - ψ )
0 2 π exp [ - i x cos ( θ - ϕ ) ] d θ = 2 π J 0 ( x )
U BW ( P d / P f ) = - i k a 2 A 0 z f exp [ i k ( z d - z f ) ] 0 1 J 0 ( k a ρ R d z f ) × exp [ - i k a 2 ρ 2 ( z d - z f ) 2 z f 2 ] ρ d ρ .
u = k a 2 ( z d - z f ) z f 2 ,             v = k a R d z f .
s = [ ( x d - ξ ) 2 + ( y d - η ) 2 + ( z d - ζ ) 2 ] 1 / 2 = [ ( x d - ξ ) 2 + ( y d - η ) 2 + z d 2 - 2 z d ζ + ζ 2 ] 1 / 2 .
z d - ζ = [ z d 2 - ( ξ 2 + η 2 ) ] 1 / 2 .
ζ = ξ 2 + η 2 2 z d + ( ξ 2 + η 2 ) 2 8 z d 3 - .
s = [ z d 2 + x d 2 + y d 2 - 2 ( ξ x d + η y d ) + ( ξ 2 + η ) 2 4 z d 2 + ] 1 / 2 .
s z d + x d 2 + y d 2 2 z d - ξ x d + η y d z d .
exp [ - i k ( ξ x d + η y d ) z d ] = exp [ - i k ( ξ u + η v ) ] ,
r = [ z d 2 + ( z d - z f ) 2 - 2 z d ( z d - z f ) + ( z d - z f ) ( ξ 2 + η 2 ) z d + ( z d - z f ) ( ξ 2 + η 2 ) 2 4 z d 3 + ] 1 / 2 .
r z d - ( z d - z f ) + ( ξ 2 + η 2 ) ( z d - z f ) 2 z d 2
r - s - ( z d - z f ) - x d 2 + y d 2 2 z d + ξ x d + η y d z d + ( ξ 2 + η 2 ) ( z d - z f ) 2 z d 2 .
U H ( P d / P f ) = - i k a 2 A 0 z d exp [ i k ( z d - z f + R d 2 2 z d ) ] × 0 1 J 0 ( k a ρ R d z d ) exp [ - i k a 2 ρ 2 ( z d - z f ) 2 z d 2 ] ρ d ρ .
s = [ ( x d - ξ ) 2 + ( y d - η ) 2 + ( z d - ζ ) 2 ] 1 / 2 .
r - s - ( z d - z f + x d 2 + y d 2 2 z d ) + ξ x d + η y d z d + ( ξ 2 + η 2 ) ( z d - z f ) 2 z d z f .
U LW ( P d / P f ) = - i k a 2 A 0 z d exp [ i k ( z d - z f + R d 2 2 z d ) ] × 0 1 J 0 ( k a ρ R d z d ) exp [ - i k a 2 ρ 2 ( z d - z f ) 2 z f z d ] ρ d ρ .
r = [ ξ 2 + η 2 + ( z f - ζ ) 2 ] 1 / 2 .
r = [ ξ 2 + η 2 + z f 2 - z f ( ξ 2 + η 2 ) z d + ( z d - z f ) ( ξ 2 + η 2 ) 2 4 z d 3 + ] 1 / 2 .
r z f + ( ξ 2 + η 2 ) ( z d - z f ) 2 z d z f .
r - s - ( z d - z f + x d 2 + y d 2 2 z d ) + ξ x d + η y d z d + ( ξ 2 + η 2 ) ( z d - z f ) 2 z d z f .
U F ( P d / P f ) = - i A 0 λ r f 0 1 J 0 { k a ρ [ ( x d - x f ) 2 + ( y d - y f ) 2 ] 1 / 2 r f } × exp [ - i k a 2 ρ 2 ( z d - z f ) 2 r f 2 ] ρ d ρ .
x 0 r 0 = x f r f ,             y 0 r 0 = y f r f ,             z d r 0 = z f r f ,
r f = ( x f 2 + y f 2 + z f 2 ) 1 / 2 ,             r 0 = ( x 0 2 + y 0 2 + z d 2 ) 1 / 2 .
r = [ ( x f - ξ ) 2 + ( y f - η ) 2 + ( z f - ζ ) 2 ] 1 / 2 = [ r f 2 - 2 r f ( x 0 ξ + y 0 η + z d ζ ) r 0 + ( ξ 2 + η 2 + ζ 2 ) ] 1 / 2 .
r 0 2 + ( ξ 2 + η 2 + ζ 2 ) - 2 ( x 0 ξ + y 0 η + z d ζ ) = r 0 2
2 ( x 0 ξ + y 0 η + z d ζ ) = ξ 2 + η 2 + ζ 2 .
r = [ r f 2 + ( ξ 2 + η 2 + ζ 2 ) ( r 0 - r f ) r 0 ] 1 / 2 .
r r f + ( ξ 2 + η 2 + ζ 2 ) ( r 0 - r f ) 2 r 0 r f .
s = { [ x 0 - ξ - ( x 0 - x d ) ] 2 + [ y 0 - η - ( y 0 - y d ) ] 2 + ( z d - ζ ) 2 } 1 / 2 .
s = { r 0 2 + ( x d 2 - x 0 2 ) + ( y d 2 - y 0 2 ) + 2 [ ξ ( x 0 - x d ) + η ( y 0 - y d ) ] } 1 / 2 .
s r 0 - x 0 2 - x d 2 2 r 0 - y 0 2 - y d 2 2 r 0 + ξ ( x 0 - x d ) + η ( y 0 - y d ) r 0 .
r - s - ( r 0 - r f ) - x d 2 - x 0 2 2 r 0 - y d 2 - y 0 2 2 r 0 + ξ ( x d - x 0 ) + η ( y d - y 0 ) r 0 + ( ξ 2 + η + ζ 2 ) ( r 0 - r f ) 2 r 0 r f .
- U HY = - i A 0 λ r 0 exp ( i k ) S 0 exp { - i k [ ξ ( x d - x 0 ) + η ( y d - y 0 ) ] r 0 } × exp [ - i k ( ξ 2 + η 2 + ζ 2 ) ( r 0 - r f ) 2 r 0 r f ] d S ,
= r 0 - r f + ( x d 2 - x 0 2 ) + ( y d 2 - y 0 2 ) r 0 .
U HY ( P d / P f ) - i k a 2 A 0 r 0 exp ( i k ) × 0 1 J 0 { k a ρ [ ( x d - x 0 ) 2 + ( y d - y 0 ) 2 ] 1 / 2 r 0 } × exp [ - i k a 2 ρ 2 ( r 0 - r f ) 2 r 0 r f ] ρ d ρ ,
= r 0 - r f + ( x d 2 - x 0 2 ) + ( y d 2 - y 0 2 ) r 0 .
r 0 - r f r 0 r f = 1 r f - 1 r 0 = z d - z f r 0 z f .
r m = ½ ( r d + r f ) = ½ ( x d + x f , y d + y f , z d + z f ) , r ˜ = r d - r f , r = r m - ( ξ , η , 0 ) .
r = [ r 2 + ( r ˜ 2 ) 2 - r ˜ · r ] 1 / 2 ,             s = [ r 2 + ( r ˜ 2 ) 2 + r ˜ · r ] 1 / 2 .
r - s = r [ r ˜ · r r 2 - r ˜ 2 ( r · r ) r 4 + ] .
r - s r ˜ · r r .
U PS ( P d / P f ) - i k A 0 A 1 r exp [ - i k ( r ˜ · r ) r ] d S .
k x = k z ( x m - ξ ) z m ,             k y = k z ( y m - η ) z m ,
U MDI ( P d / P f ) = - i 2 π Ω 1 k z exp [ i k · ( r d - r f ) ] d k x d k y ,
r ˜ · r = [ ½ ( x d + x f ) - ξ , ½ ( y d + y f ) - η , ½ ( z d + z f ) ] × ( x d - x f , y d - y f , z d + z f ) = ½ ( x d 2 - x f 2 ) + ½ ( y d 2 - y f 2 ) + ½ ( z d 2 - z f 2 ) - ξ ( x d - x f ) - η ( y d - y f ) .
U ( P d / P f ) = - i A f λ S r exp [ i k ( r - s ) ] s d S ,
s = [ ( x d - ξ ) 2 + ( y d - η ) 2 + z d 2 ] 1 / 2
r = [ ( x f - ξ ) 2 + ( y f - η ) 2 + z f 2 ] 1 / 2 .
r - s - ( z d - z f ) + ( x f - ξ ) 2 + ( y f - η ) 2 2 z f - ( x d - ξ ) 2 + ( y d - η ) 2 2 z d .
U S ( P d / P f ) = - i k a 2 A 0 z d exp ( i k ) × 0 1 J 0 { k a ρ [ ( x d z d - x f z f ) 2 + ( y d z d - y f z f ) 2 ] 1 / 2 } × exp [ - i k a 2 ρ 2 ( z d - z f ) 2 z f z d ] ρ d ρ ,
= z d - z f + R d 2 2 z d - R f 2 2 z f .
r - s = [ ( x f - ξ ) 2 + ( y f - η ) 2 + ( z f - ζ ) 2 ] 1 / 2 - [ ( x d - ξ ) 2 + ( y d - η ) 2 + ( z d - ζ ) 2 ] 1 / 2 = [ r f - 2 ( x f ξ + y f η ) + ξ 2 + η 2 ] 1 / 2 - [ r d - 2 ( x d ξ + y d η ) + ξ 2 + η 2 ] 1 / 2 .
r - s - ( r d - r f ) + ( x f ξ + y f η ) r f - ( x d ξ + y d η ) r d + ( ξ 2 + η 2 ) ( r d - r f ) 2 r f r d .
U sph ( P d / P f ) = - i k a 2 A 0 r d exp ( i k ) × 0 1 J 0 { k a ρ [ ( x d r d - x f r f ) 2 + ( y d r d - y f r f ) 2 ] 1 / 2 } × exp [ - i k a 2 ρ 2 ( r d - r f ) 2 r f r d ] ρ d ρ ,
= r d - r f .
I sph ( P d / P f ) = U sph U sph * = ( k a 2 A 0 ) 2 r d 2 | 0 1 J 0 { k a ρ [ ( ξ d - ξ f ) 2 + ( η d - η f ) 2 ] 1 / 2 } × exp [ i k a 2 ρ 2 ( ζ d - ζ f ) ρ d ρ ] | 2 .

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